American Institute of Aeronautics and Astronautics 1
Delta Wing Vortex Burst Behavior Under Dynamic Freestream Part 2 - Deceleration and Acceleration at
Moderate and Fast Pitch Rates
Roy Y Myose
Department of Aerospace Engineering Wichita State University Wichita KS 67260-0044
and
Ismael Herondagger
Eclipse Aviation Corporation Albuquerque NM 87106
A series of experiments on a 70-degree delta wing was conducted at Wichita State University to study the effect on the vortex burst position when simultaneous dynamic pitch and unsteady freestream velocity were used The aim was to better understand the relationship between the freestream velocity and the time constants involved in the movement of the vortex burst point Experiments indicated that a change in the freestream velocity changed the forward progression of the vortex burst Under pitch-up conditions deceleration resulted in a momentarily retardation in the forward progression of the burst while acceleration resulted in a faster progression toward the apex
Introduction Recent interest in highly maneuverable military aircraft capable of operating over a large range of angles of attack has refocused attention on delta-shaped wings One of the hallmark features of delta-shaped wings and strakes is the presence of a pair of vortices called leading-edge vortices At non-zero angles of attack there is a pressure difference between the upper suction surface and the lower pressure surface which causes a flow around the leading-edges The flow detaches along the leading-edge into a shear layer that curls up into a spiral The center of the spiral is tight enough that it forms in essence a pair of strong counter-rotating vortices These leading-edge vortices induce velocities on the flow field and additional suction over the delta wing which can account for up to 30 of the total lift at moderate angles of attack1 For example a 70-degree swept delta wing continues to increase its lift until about 40-degrees angle of attack2 In comparison symmetric two dimensional airfoils typically stall out at around 10- to 15-degrees angle of attack Unfortunately there are limits to the benefits produced by the leading-edge vortices As the angle of attack is increased there is a sudden breakdown in vortex structure followed by degeneration into a non-coherent turbulent-like flow This phenomena also known as vortex bursting can be defined as a sudden expansion in radial size and an abrupt decrease in the axial velocity of the vortex3 Once this occurs lift is no longer enhanced aft of the burst point Thus the development and subsequent breakdown of leading-edge vortices is crucial to the performance of delta wing aircraft The effect of vortex bursting is to reduce the lift generated by the delta wing If the delta wing is pitched to a given angle of attack and then maintained at that angle until the transient flow features die down it is said to be tested under ldquostaticrdquo conditions As this process is repeated at increasing values of α the vortex burst will be located at ldquofixedrdquo positions that are closer to the apex of the delta wing as discussed earlier Compare this to the dynamic situation where the delta wing is continuously pitched never allowing the flow features to become steady Under dynamic conditions where the delta wing is pitched upwards at a given rate the Professor Associate Fellow AIAA dagger Aerodynamicist Member AIAA
AIAA Atmospheric Flight Mechanics Conference and Exhibit20 - 23 August 2007 Hilton Head South Carolina
AIAA 2007-6726
Copyright copy 2007 by Myose and Heron Published by the American Institute of Aeronautics and Astronautics Inc with permission
American Institute of Aeronautics and Astronautics 2
location of the vortex burst is farther towards the trailing edge compared to the same angle of attack under static conditions4-7 This produces a phase lag in the burst location allowing transient values of lift to exceed those obtained during static testing Similarly pitching down the delta wing results in a vortex burst location forward of the static case This produces a phase lead and a reduction in lift compared to similar static angles of attack This introduces the notion of a hysteresis effect or time delay where there is a difference in the measured lift coefficient values if the angle of attack is increasing or decreasing8 The magnitude of the phase lag or lead increases as the pitch rate increases7 The faster the pitch up rate the higher the angle of attack before the vortex burst appears over the surface of the delta wing Here the non-dimensional pitch rate κ is defined as κ = (dαdt)c (2Uinfin) (1) Modern combat aircraft use either slender delta wings or highly swept leading-edge extensions (ie strakes) that harness vorticity These aircraft take advantage of the phase lag hysteresis effect in an attempt to increase the performance envelope In many cases the increase in performance has led to aircraft with ldquohyper-agilityrdquo or the ability to maneuver at very fast rates Take for example the case of the Su-27 aircraft undergoing a Cobra maneuver In this case the aircraft enters the pitch up phase at 190 knots indicated airspeed (kias) During a 2- to 5-second time frame the aircraft reaches 90-degrees angle of attack or more while the airspeed drops substantially to about 70 kias The aircraft subsequently points nose down in order to accelerate and exit at a much lower angle of attack910 In order to properly simulate ldquohyper-agilerdquo maneuvers such as the Cobra researchers have been focusing lately on experiments and simulations that are closer to the Reynolds number spectrum of full-scale aircraft11 This involves the use of large expensive and heavily mechanized models and mounts that are necessary to replicate the high-rate maneuvers in pitch (andor yaw and roll)12 Even with this however it is seldom possible to replicate the exact full-scale Reynolds numbers Another problem with this approach is the control of the freestream velocity fluctuation that naturally occurs during many of these ldquohyper-agilerdquo maneuvers Many water and wind tunnels are not capable of decelerating their flow velocity rapidly enough especially those large enough to accommodate the mounts necessary to accomplish the full-scale Reynolds numbers and the motions being simulated At the opposite edge of the Reynolds number spectrum renewed interest in MAVrsquos and UAVrsquos have promoted research into time-dependent methods of achieving high wing-loadings high maneuverability and small physical size With these issues in mind the goal of this investigation is to subject a known delta wing to a simultaneous pitching maneuver under a dynamic freestream condition The vortex burst position is then measured thus quantifying the vortex-burst propagation rate The present investigation is part of an on-going project to investigate the effect of dynamic freestream Past work has described the development of the towing system13 some preliminary work14 and fast pitch-up under deceleration15 The present paper will summarize the results for the full data set of deceleration and acceleration at pitch-down and pitch-up at different rates Experimental Method An existing water tunnel facility was used to create a towing facility where dynamic pitching and dynamic freesteam could be obtained The illustration on the left-hand side of Figure 1 shows a schematic diagram of the water tunnel located in Wichita State Universityrsquos National Institute of Aviation Research16 This facility is a closed-loop tunnel containing 3500 gallons of water and consists of a 2- by 3- by 6-feet test section The facility has excellent optical access providing two side views a bottom view and an end view The water tunnel is capable of producing flow velocities up to 1 fts using an impeller pump driven by a 5-hp variable-speed motor For the purposes of the present investigation however the tunnel was used as a simple water tank with the pump turned off The photograph on the right-hand side of Figure 1 shows the towing system with an inverted delta wing mounted on a carriage (which is not visible) The carriage itself rides on top of a track which is built into an aluminum frame A lightweight ldquofalse ceilingrdquo is built into the frame to suppress bow waves which may be created by towing the model support strut if there were a free surface Towing speeds of 04 fts + 5 is obtained by pulling the carriage with a spring-tensioned nylon wire and a DC motor The delta wing model is mounted upside down such that ldquopitch-uprdquo involves rotating the delta wing apex down towards the floor of the water tunnel This arrangement reduces the likelihood of carriage derailment since the delta wing provides
American Institute of Aeronautics and Astronautics 3
additional down force for the carriage wheels Visible in the photograph of Figure 1 is the tandem strut mounting mechanism for the delta wing The larger diameter strut is attached to a hinge on the delta wingrsquos pressure side The smaller diameter strut located on the downstream side is attached to a cam mechanism on the carriage which is driven by a small DC motor This allows the delta wing to be pivoted about the 50 root chord location at non-dimensional pitch rates up to k=02 Dye flow visualization was used to identify the vortex burst locations The flow visualization image was video-taped and information such as the towing carriage velocity and delta wing angle of attack were also recorded The recorded images and relevant information were subsequently analyzed using a computer-assisted image analysis software tool Additional details about the towing system measurement technique results of the validation tests and some of the initial pitch-up results are presented in references 13 through 15 In the present investigation both deceleration and acceleration is considered In this regard it is appropriate to define the velocity ratio which is a comparison between the minimum and maximum velocities In the case of deceleration the delta wing is initially towed at the maximum velocity of 04 fts and decelerated to a slower speed of either 02 fts (for a velocity ratio of 05) or 03 fts (for a velocity ratio of 075) In the case of acceleration the delta wing is brought to the maximum velocity of 04 fts starting from a slower speed of either 02 fts (for a velocity ratio of 05) or 03 fts (for a velocity ratio of 075) There are a few limitations with this experimental set-up that need to be acknowledged 1- The location of the vortex burst is accomplished by identifying where the core flares out (bubble burst) or the location of the first sharp kink (spiral burst) As such the identified location may or may not coincide with the actual core stagnation point The assumption that the two are close if not coincident has been done in the past 2- Some of the features observed cannot be explored further using this system thus a certain amount of interpretation must be exercised Nevertheless the uncertainty should be within plusmn 005c 3- The synchronization of the start of the slow-down process is accomplished manually A switch is actuated when the angle of attack α corresponding to that particular test case is observed in the AD conversion display Through practice it is possible to attain a repeatability of α 15 degrees Fast Pitch-up Results amp Discussion Figure 2 shows a comparison of the results for four experimental runs under acceleration (κ=02 velocity ratio 05 dvdt=0092 fts2 30 lt α lt 50 degrees) and four experimental runs under deceleration For convenience a positive outcome is one where the propagation of the vortex burst is delayed or arrested while a negative outcome is one where the burst moves forward faster (or jumps) than normal for a steady-velocity pitch-up The figure shows that deceleration has a positive effect of retarding the forward progression of the vortex burst this being in evidence by the level-off in the scatter plot at about sc=05 (4 secs elapsed time) The acceleration on the other hand shows a negative effect with a definite forward jump in the scatter plot Comparing with the constant velocity case and extrapolating linearly shows that at least 1 to 15 seconds can be gained before the burst has progressed to a spatial location (for example s=04c) when subjected to a deceleration Figure 3 (κ=02 velocity ratio 075 dvdt=0046 fts2 30 lt α lt 50 degrees) indicates that slower accelerations still have a negative effect on the burst location as a forward jump in the scatter is visible starting at about 5 seconds The slow deceleration in this case however does not contribute greatly to the delay in the forward propagation of the burst location as the scatter plot lies almost parallel to the linear extrapolation for a constant speed pitch-up Figure 4 (κ=02 velocity ratio 05 dvdt=0092 fts2 45 lt α lt beyond 55 degrees) indicates that acceleration has a negative effect deceleration has a mild positive effect and the angle of attack where the change in the slope of the scatter plot (ie the change in propagation velocity) also has an effect If the reader equates elapsed time with angle of attack α and compares Figure 2 to the current figure the point in time where the burst location experiences a change in slope is the same for acceleration and deceleration in Figure 2 but different in Figure 4 the acceleration precipitates a forward jump sooner than the deceleration arrests the burst progression Figure 5 (κ=02 velocity ratio 075 dvdt=0046 fts2 45 lt α lt beyond 55 degrees) indicates that at the higher angle extremes changes in velocity have little impact on the burst position There is a slight change in
American Institute of Aeronautics and Astronautics 4
the slope at approximately 5 seconds elapsed time This discontinuity appears in both accelerating and decelerating experiments In Figure 6 (κ=02 velocity ratio 075 dvdt=0092 fts2 30 lt α lt 40 degrees) the deceleration appears to have a mild positive effect So mild in fact that it is difficult to say whether the downward inflection is the result of the acceleration or just the normal change in slope that occurs when κ changes value (Recall that κ changes value in proportion to the freestream velocity) The acceleration does not appear to have any changes as the slope remained quasi-linear throughout the maneuver To summarize pitching up at κ=02 at different angle of attack ranges and velocity ratios produces a mild to strong negative effect on the burst location when accelerating This negative effect is almost independent of the actual acceleration or range of α over which it occurs The positive delay in the burst movement on the other hand appears to be strongest when the delta wing experiences a strong deceleration (dvdt=0092 fts2) over a large range of α values (30 to 50 degrees) This effect becomes much smaller at slower decelerations andor limited values in the α range This implies that perhaps instabilities (whether inherent or external to the vortex core) could help precipitate the forward jump (as in Figure 4) making it easier to lose lift than to keep lift Part 115 of this two part paper series presented a possible explanation why a positive delay in the burst movement may be obtained through deceleration (ie reduction in freestream speed) A brief review of that discussion is repeated here Considering the different velocity components shown in Figure 7 the velocity component that is directly responsible for vorticity production at the leading edge is given by VS = Uinfin sin α cos β (2) It can be seen that cos β is constant for a given installation (ie no sideslip component β=0 and cos β=1) Taking the time differential dVS dt = (dUinfindt sin α + Uinfin dαdt cos α) cos β (3) Both dUinfindt and dαdt are for the purposes of these experiments quasi-constant numbers The relative magnitudes of both numbers set a balance The change in the leading edge perpendicular velocity is therefore given by the two terms in the parenthesis Figure 8 shows the effect of various terms for the deceleration case presented in Figure 2 (dUinfindt= -0092 fts2 and dαdt= 916 degs= 016 rads at κ= 02) When the deceleration begins in this case a large change in the character of the curves occurs The increase in velocity perpendicular to the leading edge is initially driven by the Uinfin dαdt cosα term However the velocity perpendicular to the leading edge is reduced when deceleration is introduced (ie with the inclusion of the dUinfindt sin α term) This in turn reduces the formation of vorticity at its source It should be noted that Figure 8 does not include the effect of the time delay it takes to convect the vorticity from the leading edge into the vortex core This time delay will be non-existent at the apex and increases linearly with chord location towards the trailing edge Thus the velocity calculated in Figure 8 is first experienced at the apex then further down the length of the vortex core and at the trailing edge last Nevertheless a reduction in vortex production with deceleration should result in a positive delay in the vortex burst movement Figure 9 shows the effect of various terms for the acceleration case corresponding to the situation of Figure 2 (dUinfindt = -0092 fts2 and dαdt= 916 degs= 016 rads at κ= 02) When acceleration begins in this case a large change in the character of the curve occurs once again This time however the velocity perpendicular to the leading edge is increased when acceleration is introduced (ie with the inclusion of the dUinfindt sin α term) This would lead to an increase in the production of vorticity at its source This in turn is likely to lead to the negative result of early vortex burst (ie movement of the burst closer to the apex) seen in Figure 2 Results amp Discussion for Pitch-up at Moderate Rate Pitching up at half the pitch rate (κ=01) such as is the case in Figure 10 (κ=01 velocity ratio 05 dvdt=0046 fts2 30 lt α lt 50 degrees) produces results where the deceleration has a clear positive effect on the burst location The scatter is also increased Notice that the deceleration produces a slope (at 7 seconds elapsed time) close to that of a constant speed pitch-up at κ=02 which is intuitively correct when one considers that since Uinfin occurs in the denominator reducing it to frac12 of the initial magnitude should double the value of κ
American Institute of Aeronautics and Astronautics 5
In Figure 11 the acceleration does not appear to change the slope of the propagation curves On the other hand the deceleration produces a mild retardation in the burst propagation In one run (Exp 3 black circular symbols) a clearly visible change in slope occurred at 5 seconds The other two runs produced a barely discernible change in slope at approximately 45 seconds Notice that both Figures 10 and 11 present the results of experiments at the higher dvdt value (0046 fts2) but the range of α values tested for Figure 11 starts at a very high value of α thus the deceleration appears to have a smaller effect when started at a higher angle of attack Figures 12 and 13 are similar in that the experiments occurred at the lower value of dvdt tested (0023 fts2) In both cases it is hard to observe any clear effect of the change in velocity as there is no discernible change in slope particularly during the deceleration portion The acceleration of the experiments in Figure 12 did change the slope towards the end (at approximately 5 seconds elapsed time) but the change was to flatten (delay) the forward motion of the burst location This change in slope was probably due to the fact that the pitch-up was close to the end point To summarize pitching up at κ=01 at different ranges and velocity ratios produced mild to non-observable negative and positive effects Both effects appeared to occur when the delta wing experienced a strong deceleration (dvdt=0046 fts2) over a large range of α values (30 to 50 degrees) At slower decelerations andor limited values in the α range the effects became almost unobservable in the scatter plots due in part to the increased scatter Instabilities (whether inherent or external to the vortex core) or other mechanisms (ie laminarity in the viscous regions) could play a more dominant effect at the low Reynolds numbers experienced under these conditions by the delta wing Pitch-down Results amp Discussion Figures 14 through 16 present the results of a pitch-down at κ=01 under different values of dvdt and α ranges The deceleration does appear to move the burst back towards the trailing edge (a positive effect) faster in the case where dvdt is high (0092 fts2) In the other two cases where dvdt has intermediate and low values there is no appreciable difference between accelerating and decelerating Scatter is well controlled in all three figures Thus the benefits of decelerating the delta wing to delay the forward progression of the burst appear to be confined to the pitch-up regime of flight Summary A series of experiments on a 70-degree delta wing was conducted at Wichita State University to study the effect on the vortex burst position when simultaneous dynamic pitch and unsteady freestream velocity were used The aim was to better understand the relationship between the freestream velocity and the time constants involved in the movement of the vortex burst point Decelerating from 04 to 02 fts (ie a velocity ratio of 05) a reduction in the vortex burst propagation velocity is apparent at about 15 seconds after the freestream velocity begins to decelerate One possible means to non-dimensionalize the time axis is to utilize a non-dimensional timeframe trsquo = (t a) (ω c) (4) where t = elapsed time a = acceleration ω = pitch rate and c = chord Using this definition the reduction in forward propagation of the vortex burst location occurs at (t = 15 seconds which corresponds to) trsquo= 0863 when the delta wing is decelerated with a velocity ratio of 05 during pitch-up between 15 to 55 degrees at a rate of κ = 02 Accelerating from 02 to 04 ftsec (ie a minimum to maximum velocity ratio of 05) and pitching up at κ = 02 between 15 to 55 degrees there was a slight downward inflection in the burst location curves at approximately 175 and 2 seconds (trsquo= 101 to 115 respectively) after the beginning of the velocity ramp-up (acceleration from 02 fts) During a small deceleration from 04 to 03 fts (ie a velocity ratio of 075) and pitching up at a rate κ = 02 between 15 to 55 degrees a reduction in the vortex burst propagation velocity was apparent at approximately 1 second after the freestream velocity began to change (trsquo= 0288) This retardation delays the forward propagation of the vortex burst (towards the apex) The forward propagation of the burst location resumes at about 45 seconds elapsed time (trsquo= 1294) and this appeared to be at the same propagation rate (ie the slope of the burst) compared to the beginning of the experiment
American Institute of Aeronautics and Astronautics 6
It appears that at κ = 02 for large ranges of angle of attack α deceleration of the freestream velocity appeared to arrest the motion of the burst location anywhere between 15 and 25 seconds after the deceleration began to take place (ie 0863 lt trsquolt 1438) The positive delay in the burst movement appeared to be strongest when the delta wing experienced a strong deceleration (dvdt=0092 fts2) over a large range of α values (30 to 50 degrees) This effect became much smaller at slower decelerations andor limited values in the α range This implies that perhaps instabilities (whether inherent or external to the vortex core) could help precipitate the forward jump making it easier to lose lift than to keep lift It appears that acceleration had the effect of increasing the forward propagation of the vortex burst along the leading edge vortex core At different α ranges and velocity ratios accelerating produced a mild to strong negative effect on the burst location when accelerating This negative effect was almost independent of the actual acceleration or range of α over which it occurred The results obtained during pitch up at the slower rate of κ = 01 for different combinations of acceleration and deceleration exhibited the same general features as those observed during the faster pitch-ups deceleration in the freestream velocity appeared to have a beneficial effect ie it arrested the forward motion of the burst location (thereby delaying the propagation of the vortex burst towards the apex) Acceleration in the freestream velocity appeared to have the opposite effect that of pushing the burst location forward A second trait observed during the slower pitch-up tests was that the degree of scatter in the data increased In some cases there were large fluctuations along the time axis where the forward propagation accelerated In other cases slightly different curves (one for each experiment performed) were traced each having identical slope but separated from its neighbor by a few fractions of a second This indicated that at the slower pitching rate the experiment was more sensitive to either external noise (such as unavoidable vibrations in the carriage being transmitted to the delta wing) or to something instrinsic within the flow phenomenon The experiment as it was performed could not isolate the cause of this variability References 1 Polhamus E ldquoA Concept of the Vortex Lift of Sharp-Edge Delta Wings Based on a Leading-Edge
Suction Analogyrdquo NASA TN D-3767 Dec 1966 2 Earnshaw PB and Lawford JA ldquoLow-Speed Wind Tunnel Experiments on a Series of Sharp-Edged
Delta Wingsrdquo Aeronautical Research Council R M No 3424 1964 3 Erickson GE ldquoWater-Tunnel Studies of Leading-Edge Vorticesrdquo AIAA Journal of Aircraft Vol 19
No 6 1982 pp 442-448 4 LeMay WP Batill SM and Nelson RC ldquoVortex Dynamics on a Pitching Delta Wingrdquo AIAA
Journal of Aircraft Vol 27 No 2 1990 pp 131-138 5 Miller LS and Gile BE ldquoEffects of Blowing on Delta Wing Vortices During Dynamic Pitchingrdquo
AIAA Journal of Aircraft Vol 30 No 3 1993 pp 334-339 6 Rediniotis O K Klute S M Hoang N T and Telionis D P ldquoDynamic Pitch-Up of a Delta Wingrdquo
AIAA Journal Vol 32 No 4 1994 pp 716-725 7 Myose RY Hayashibara S Yeong PC and Miller LS ldquoEffect of Canards on Delta Wing Vortex
Breakdown during Dynamic Pitchingrdquo AIAA Journal of Aircraft Vol 32 No 2 1997 pp 168-173 8 Al-Garni AZ Ahmed SA Sahin AZ and Al-Garni AM ldquoAn Experimental Study of a 65-Degree
Delta Wing with Different Pitching Ratesrdquo Canadian Aeronautics and Space Journal Vol 47 No 2 2001 pp 85-93
9 Skow A M ldquoAn Analysis of the Su-27 Flight Demonstration at the 1989 Paris Airshowrdquo SAE Paper 90-1001 1990
10 Kolano E ldquoFlying the Flankerrdquo Flight Journal httpflightjournalcomarticlessu27Su27_5asp 11 Ericsson L E ldquoEffect of Fuselage Geometry on Delta-Wing Vortex Breakdownrdquo AIAA Journal of
Aircraft Vol 35 No 6 Nov 1998 12 Schaeffler N W and Telionis D P ldquoApex Flap Control of Dynamic Delta Wing Maneuversrdquo AIAA
Paper 96-0662 1996 13 Heron I and Myose R Y ldquoDevelopment of a Dynamic Pitch and Unsteady Freestream Delta Wing
Mount For A Water Tunnelrdquo AIAA Paper 2003-3527 June 2003
American Institute of Aeronautics and Astronautics 7
14 Heron I and Myose R Y ldquoVortex Burst Behavior Under Dynamic Freestreamrdquo AIAA Paper 2005-0063 January 2005
15 Heron I and Myose R Y ldquoDelta Wing Vortex Burst Behavior Under Dynamic Freestream Part 1 ndash Fast Pitch-up During Decelerationrdquo AIAA paper 2007-6725 August 2007
16 Johnson BL ldquoFacility Description of the Walter H Beech Memorial 7 x 10 foot Low-Speed Wind Tunnelrdquo AR93-1 National Institute for Aviation Research Wichita State University June 1993
Figure 1 Schematic diagram of Wichita State 2- by 3-feet water tunnel (left)16 and photograph of test section showing the towing system with an inverted delta wing mounted on carriage (right)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 30 - 50 deg
Figure 2 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 8
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-36
Figure 3 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 45 - 55+ deg
Figure 4 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 9
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc (s
= 0
at A
pex)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-38
Figure 5 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 30 to 40 degFig 5-39
Figure 6 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (40 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 10
Figure 7 Velocity Components on Leading Edge of Delta Wing
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 04 to 02 fts
dV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 8 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Deceleration (Ideal Representation of Fig 2rsquos Deceleration Case)
Half-apex angle ζ = 90 - Λ Angle of Attack α
Uinfin
Angle of Yaw β
Sweep Angle Λ
VS = Uinfin sin α cos β
VLE = Uinfin (cos α cos ζ cos β - sin ζ sin β)
VLE = Uinfin (cos α cos ζ cos β + sin ζ sin β)
VLE = Uinfin (cos α cos β sin ζ - sin β cos ζ)
VLE = Uinfin (cos α cos β sin ζ + sin β cos ζ)
ζ
American Institute of Aeronautics and Astronautics 11
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
01
012
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 02 to 04 ftsdV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 9 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Acceleration (Ideal Representation of Fig 2rsquos Acceleration Case)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 30 - 50 deg
k=02 slopek=01 slope
Figure 10 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
American Institute of Aeronautics and Astronautics 12
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 45 - 55+ deg k=02 slopek=01 slope
Figure 11 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Vel change from 30-50 degFig 5-42
k=02 slopek=01 slope
Figure 12 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 2
location of the vortex burst is farther towards the trailing edge compared to the same angle of attack under static conditions4-7 This produces a phase lag in the burst location allowing transient values of lift to exceed those obtained during static testing Similarly pitching down the delta wing results in a vortex burst location forward of the static case This produces a phase lead and a reduction in lift compared to similar static angles of attack This introduces the notion of a hysteresis effect or time delay where there is a difference in the measured lift coefficient values if the angle of attack is increasing or decreasing8 The magnitude of the phase lag or lead increases as the pitch rate increases7 The faster the pitch up rate the higher the angle of attack before the vortex burst appears over the surface of the delta wing Here the non-dimensional pitch rate κ is defined as κ = (dαdt)c (2Uinfin) (1) Modern combat aircraft use either slender delta wings or highly swept leading-edge extensions (ie strakes) that harness vorticity These aircraft take advantage of the phase lag hysteresis effect in an attempt to increase the performance envelope In many cases the increase in performance has led to aircraft with ldquohyper-agilityrdquo or the ability to maneuver at very fast rates Take for example the case of the Su-27 aircraft undergoing a Cobra maneuver In this case the aircraft enters the pitch up phase at 190 knots indicated airspeed (kias) During a 2- to 5-second time frame the aircraft reaches 90-degrees angle of attack or more while the airspeed drops substantially to about 70 kias The aircraft subsequently points nose down in order to accelerate and exit at a much lower angle of attack910 In order to properly simulate ldquohyper-agilerdquo maneuvers such as the Cobra researchers have been focusing lately on experiments and simulations that are closer to the Reynolds number spectrum of full-scale aircraft11 This involves the use of large expensive and heavily mechanized models and mounts that are necessary to replicate the high-rate maneuvers in pitch (andor yaw and roll)12 Even with this however it is seldom possible to replicate the exact full-scale Reynolds numbers Another problem with this approach is the control of the freestream velocity fluctuation that naturally occurs during many of these ldquohyper-agilerdquo maneuvers Many water and wind tunnels are not capable of decelerating their flow velocity rapidly enough especially those large enough to accommodate the mounts necessary to accomplish the full-scale Reynolds numbers and the motions being simulated At the opposite edge of the Reynolds number spectrum renewed interest in MAVrsquos and UAVrsquos have promoted research into time-dependent methods of achieving high wing-loadings high maneuverability and small physical size With these issues in mind the goal of this investigation is to subject a known delta wing to a simultaneous pitching maneuver under a dynamic freestream condition The vortex burst position is then measured thus quantifying the vortex-burst propagation rate The present investigation is part of an on-going project to investigate the effect of dynamic freestream Past work has described the development of the towing system13 some preliminary work14 and fast pitch-up under deceleration15 The present paper will summarize the results for the full data set of deceleration and acceleration at pitch-down and pitch-up at different rates Experimental Method An existing water tunnel facility was used to create a towing facility where dynamic pitching and dynamic freesteam could be obtained The illustration on the left-hand side of Figure 1 shows a schematic diagram of the water tunnel located in Wichita State Universityrsquos National Institute of Aviation Research16 This facility is a closed-loop tunnel containing 3500 gallons of water and consists of a 2- by 3- by 6-feet test section The facility has excellent optical access providing two side views a bottom view and an end view The water tunnel is capable of producing flow velocities up to 1 fts using an impeller pump driven by a 5-hp variable-speed motor For the purposes of the present investigation however the tunnel was used as a simple water tank with the pump turned off The photograph on the right-hand side of Figure 1 shows the towing system with an inverted delta wing mounted on a carriage (which is not visible) The carriage itself rides on top of a track which is built into an aluminum frame A lightweight ldquofalse ceilingrdquo is built into the frame to suppress bow waves which may be created by towing the model support strut if there were a free surface Towing speeds of 04 fts + 5 is obtained by pulling the carriage with a spring-tensioned nylon wire and a DC motor The delta wing model is mounted upside down such that ldquopitch-uprdquo involves rotating the delta wing apex down towards the floor of the water tunnel This arrangement reduces the likelihood of carriage derailment since the delta wing provides
American Institute of Aeronautics and Astronautics 3
additional down force for the carriage wheels Visible in the photograph of Figure 1 is the tandem strut mounting mechanism for the delta wing The larger diameter strut is attached to a hinge on the delta wingrsquos pressure side The smaller diameter strut located on the downstream side is attached to a cam mechanism on the carriage which is driven by a small DC motor This allows the delta wing to be pivoted about the 50 root chord location at non-dimensional pitch rates up to k=02 Dye flow visualization was used to identify the vortex burst locations The flow visualization image was video-taped and information such as the towing carriage velocity and delta wing angle of attack were also recorded The recorded images and relevant information were subsequently analyzed using a computer-assisted image analysis software tool Additional details about the towing system measurement technique results of the validation tests and some of the initial pitch-up results are presented in references 13 through 15 In the present investigation both deceleration and acceleration is considered In this regard it is appropriate to define the velocity ratio which is a comparison between the minimum and maximum velocities In the case of deceleration the delta wing is initially towed at the maximum velocity of 04 fts and decelerated to a slower speed of either 02 fts (for a velocity ratio of 05) or 03 fts (for a velocity ratio of 075) In the case of acceleration the delta wing is brought to the maximum velocity of 04 fts starting from a slower speed of either 02 fts (for a velocity ratio of 05) or 03 fts (for a velocity ratio of 075) There are a few limitations with this experimental set-up that need to be acknowledged 1- The location of the vortex burst is accomplished by identifying where the core flares out (bubble burst) or the location of the first sharp kink (spiral burst) As such the identified location may or may not coincide with the actual core stagnation point The assumption that the two are close if not coincident has been done in the past 2- Some of the features observed cannot be explored further using this system thus a certain amount of interpretation must be exercised Nevertheless the uncertainty should be within plusmn 005c 3- The synchronization of the start of the slow-down process is accomplished manually A switch is actuated when the angle of attack α corresponding to that particular test case is observed in the AD conversion display Through practice it is possible to attain a repeatability of α 15 degrees Fast Pitch-up Results amp Discussion Figure 2 shows a comparison of the results for four experimental runs under acceleration (κ=02 velocity ratio 05 dvdt=0092 fts2 30 lt α lt 50 degrees) and four experimental runs under deceleration For convenience a positive outcome is one where the propagation of the vortex burst is delayed or arrested while a negative outcome is one where the burst moves forward faster (or jumps) than normal for a steady-velocity pitch-up The figure shows that deceleration has a positive effect of retarding the forward progression of the vortex burst this being in evidence by the level-off in the scatter plot at about sc=05 (4 secs elapsed time) The acceleration on the other hand shows a negative effect with a definite forward jump in the scatter plot Comparing with the constant velocity case and extrapolating linearly shows that at least 1 to 15 seconds can be gained before the burst has progressed to a spatial location (for example s=04c) when subjected to a deceleration Figure 3 (κ=02 velocity ratio 075 dvdt=0046 fts2 30 lt α lt 50 degrees) indicates that slower accelerations still have a negative effect on the burst location as a forward jump in the scatter is visible starting at about 5 seconds The slow deceleration in this case however does not contribute greatly to the delay in the forward propagation of the burst location as the scatter plot lies almost parallel to the linear extrapolation for a constant speed pitch-up Figure 4 (κ=02 velocity ratio 05 dvdt=0092 fts2 45 lt α lt beyond 55 degrees) indicates that acceleration has a negative effect deceleration has a mild positive effect and the angle of attack where the change in the slope of the scatter plot (ie the change in propagation velocity) also has an effect If the reader equates elapsed time with angle of attack α and compares Figure 2 to the current figure the point in time where the burst location experiences a change in slope is the same for acceleration and deceleration in Figure 2 but different in Figure 4 the acceleration precipitates a forward jump sooner than the deceleration arrests the burst progression Figure 5 (κ=02 velocity ratio 075 dvdt=0046 fts2 45 lt α lt beyond 55 degrees) indicates that at the higher angle extremes changes in velocity have little impact on the burst position There is a slight change in
American Institute of Aeronautics and Astronautics 4
the slope at approximately 5 seconds elapsed time This discontinuity appears in both accelerating and decelerating experiments In Figure 6 (κ=02 velocity ratio 075 dvdt=0092 fts2 30 lt α lt 40 degrees) the deceleration appears to have a mild positive effect So mild in fact that it is difficult to say whether the downward inflection is the result of the acceleration or just the normal change in slope that occurs when κ changes value (Recall that κ changes value in proportion to the freestream velocity) The acceleration does not appear to have any changes as the slope remained quasi-linear throughout the maneuver To summarize pitching up at κ=02 at different angle of attack ranges and velocity ratios produces a mild to strong negative effect on the burst location when accelerating This negative effect is almost independent of the actual acceleration or range of α over which it occurs The positive delay in the burst movement on the other hand appears to be strongest when the delta wing experiences a strong deceleration (dvdt=0092 fts2) over a large range of α values (30 to 50 degrees) This effect becomes much smaller at slower decelerations andor limited values in the α range This implies that perhaps instabilities (whether inherent or external to the vortex core) could help precipitate the forward jump (as in Figure 4) making it easier to lose lift than to keep lift Part 115 of this two part paper series presented a possible explanation why a positive delay in the burst movement may be obtained through deceleration (ie reduction in freestream speed) A brief review of that discussion is repeated here Considering the different velocity components shown in Figure 7 the velocity component that is directly responsible for vorticity production at the leading edge is given by VS = Uinfin sin α cos β (2) It can be seen that cos β is constant for a given installation (ie no sideslip component β=0 and cos β=1) Taking the time differential dVS dt = (dUinfindt sin α + Uinfin dαdt cos α) cos β (3) Both dUinfindt and dαdt are for the purposes of these experiments quasi-constant numbers The relative magnitudes of both numbers set a balance The change in the leading edge perpendicular velocity is therefore given by the two terms in the parenthesis Figure 8 shows the effect of various terms for the deceleration case presented in Figure 2 (dUinfindt= -0092 fts2 and dαdt= 916 degs= 016 rads at κ= 02) When the deceleration begins in this case a large change in the character of the curves occurs The increase in velocity perpendicular to the leading edge is initially driven by the Uinfin dαdt cosα term However the velocity perpendicular to the leading edge is reduced when deceleration is introduced (ie with the inclusion of the dUinfindt sin α term) This in turn reduces the formation of vorticity at its source It should be noted that Figure 8 does not include the effect of the time delay it takes to convect the vorticity from the leading edge into the vortex core This time delay will be non-existent at the apex and increases linearly with chord location towards the trailing edge Thus the velocity calculated in Figure 8 is first experienced at the apex then further down the length of the vortex core and at the trailing edge last Nevertheless a reduction in vortex production with deceleration should result in a positive delay in the vortex burst movement Figure 9 shows the effect of various terms for the acceleration case corresponding to the situation of Figure 2 (dUinfindt = -0092 fts2 and dαdt= 916 degs= 016 rads at κ= 02) When acceleration begins in this case a large change in the character of the curve occurs once again This time however the velocity perpendicular to the leading edge is increased when acceleration is introduced (ie with the inclusion of the dUinfindt sin α term) This would lead to an increase in the production of vorticity at its source This in turn is likely to lead to the negative result of early vortex burst (ie movement of the burst closer to the apex) seen in Figure 2 Results amp Discussion for Pitch-up at Moderate Rate Pitching up at half the pitch rate (κ=01) such as is the case in Figure 10 (κ=01 velocity ratio 05 dvdt=0046 fts2 30 lt α lt 50 degrees) produces results where the deceleration has a clear positive effect on the burst location The scatter is also increased Notice that the deceleration produces a slope (at 7 seconds elapsed time) close to that of a constant speed pitch-up at κ=02 which is intuitively correct when one considers that since Uinfin occurs in the denominator reducing it to frac12 of the initial magnitude should double the value of κ
American Institute of Aeronautics and Astronautics 5
In Figure 11 the acceleration does not appear to change the slope of the propagation curves On the other hand the deceleration produces a mild retardation in the burst propagation In one run (Exp 3 black circular symbols) a clearly visible change in slope occurred at 5 seconds The other two runs produced a barely discernible change in slope at approximately 45 seconds Notice that both Figures 10 and 11 present the results of experiments at the higher dvdt value (0046 fts2) but the range of α values tested for Figure 11 starts at a very high value of α thus the deceleration appears to have a smaller effect when started at a higher angle of attack Figures 12 and 13 are similar in that the experiments occurred at the lower value of dvdt tested (0023 fts2) In both cases it is hard to observe any clear effect of the change in velocity as there is no discernible change in slope particularly during the deceleration portion The acceleration of the experiments in Figure 12 did change the slope towards the end (at approximately 5 seconds elapsed time) but the change was to flatten (delay) the forward motion of the burst location This change in slope was probably due to the fact that the pitch-up was close to the end point To summarize pitching up at κ=01 at different ranges and velocity ratios produced mild to non-observable negative and positive effects Both effects appeared to occur when the delta wing experienced a strong deceleration (dvdt=0046 fts2) over a large range of α values (30 to 50 degrees) At slower decelerations andor limited values in the α range the effects became almost unobservable in the scatter plots due in part to the increased scatter Instabilities (whether inherent or external to the vortex core) or other mechanisms (ie laminarity in the viscous regions) could play a more dominant effect at the low Reynolds numbers experienced under these conditions by the delta wing Pitch-down Results amp Discussion Figures 14 through 16 present the results of a pitch-down at κ=01 under different values of dvdt and α ranges The deceleration does appear to move the burst back towards the trailing edge (a positive effect) faster in the case where dvdt is high (0092 fts2) In the other two cases where dvdt has intermediate and low values there is no appreciable difference between accelerating and decelerating Scatter is well controlled in all three figures Thus the benefits of decelerating the delta wing to delay the forward progression of the burst appear to be confined to the pitch-up regime of flight Summary A series of experiments on a 70-degree delta wing was conducted at Wichita State University to study the effect on the vortex burst position when simultaneous dynamic pitch and unsteady freestream velocity were used The aim was to better understand the relationship between the freestream velocity and the time constants involved in the movement of the vortex burst point Decelerating from 04 to 02 fts (ie a velocity ratio of 05) a reduction in the vortex burst propagation velocity is apparent at about 15 seconds after the freestream velocity begins to decelerate One possible means to non-dimensionalize the time axis is to utilize a non-dimensional timeframe trsquo = (t a) (ω c) (4) where t = elapsed time a = acceleration ω = pitch rate and c = chord Using this definition the reduction in forward propagation of the vortex burst location occurs at (t = 15 seconds which corresponds to) trsquo= 0863 when the delta wing is decelerated with a velocity ratio of 05 during pitch-up between 15 to 55 degrees at a rate of κ = 02 Accelerating from 02 to 04 ftsec (ie a minimum to maximum velocity ratio of 05) and pitching up at κ = 02 between 15 to 55 degrees there was a slight downward inflection in the burst location curves at approximately 175 and 2 seconds (trsquo= 101 to 115 respectively) after the beginning of the velocity ramp-up (acceleration from 02 fts) During a small deceleration from 04 to 03 fts (ie a velocity ratio of 075) and pitching up at a rate κ = 02 between 15 to 55 degrees a reduction in the vortex burst propagation velocity was apparent at approximately 1 second after the freestream velocity began to change (trsquo= 0288) This retardation delays the forward propagation of the vortex burst (towards the apex) The forward propagation of the burst location resumes at about 45 seconds elapsed time (trsquo= 1294) and this appeared to be at the same propagation rate (ie the slope of the burst) compared to the beginning of the experiment
American Institute of Aeronautics and Astronautics 6
It appears that at κ = 02 for large ranges of angle of attack α deceleration of the freestream velocity appeared to arrest the motion of the burst location anywhere between 15 and 25 seconds after the deceleration began to take place (ie 0863 lt trsquolt 1438) The positive delay in the burst movement appeared to be strongest when the delta wing experienced a strong deceleration (dvdt=0092 fts2) over a large range of α values (30 to 50 degrees) This effect became much smaller at slower decelerations andor limited values in the α range This implies that perhaps instabilities (whether inherent or external to the vortex core) could help precipitate the forward jump making it easier to lose lift than to keep lift It appears that acceleration had the effect of increasing the forward propagation of the vortex burst along the leading edge vortex core At different α ranges and velocity ratios accelerating produced a mild to strong negative effect on the burst location when accelerating This negative effect was almost independent of the actual acceleration or range of α over which it occurred The results obtained during pitch up at the slower rate of κ = 01 for different combinations of acceleration and deceleration exhibited the same general features as those observed during the faster pitch-ups deceleration in the freestream velocity appeared to have a beneficial effect ie it arrested the forward motion of the burst location (thereby delaying the propagation of the vortex burst towards the apex) Acceleration in the freestream velocity appeared to have the opposite effect that of pushing the burst location forward A second trait observed during the slower pitch-up tests was that the degree of scatter in the data increased In some cases there were large fluctuations along the time axis where the forward propagation accelerated In other cases slightly different curves (one for each experiment performed) were traced each having identical slope but separated from its neighbor by a few fractions of a second This indicated that at the slower pitching rate the experiment was more sensitive to either external noise (such as unavoidable vibrations in the carriage being transmitted to the delta wing) or to something instrinsic within the flow phenomenon The experiment as it was performed could not isolate the cause of this variability References 1 Polhamus E ldquoA Concept of the Vortex Lift of Sharp-Edge Delta Wings Based on a Leading-Edge
Suction Analogyrdquo NASA TN D-3767 Dec 1966 2 Earnshaw PB and Lawford JA ldquoLow-Speed Wind Tunnel Experiments on a Series of Sharp-Edged
Delta Wingsrdquo Aeronautical Research Council R M No 3424 1964 3 Erickson GE ldquoWater-Tunnel Studies of Leading-Edge Vorticesrdquo AIAA Journal of Aircraft Vol 19
No 6 1982 pp 442-448 4 LeMay WP Batill SM and Nelson RC ldquoVortex Dynamics on a Pitching Delta Wingrdquo AIAA
Journal of Aircraft Vol 27 No 2 1990 pp 131-138 5 Miller LS and Gile BE ldquoEffects of Blowing on Delta Wing Vortices During Dynamic Pitchingrdquo
AIAA Journal of Aircraft Vol 30 No 3 1993 pp 334-339 6 Rediniotis O K Klute S M Hoang N T and Telionis D P ldquoDynamic Pitch-Up of a Delta Wingrdquo
AIAA Journal Vol 32 No 4 1994 pp 716-725 7 Myose RY Hayashibara S Yeong PC and Miller LS ldquoEffect of Canards on Delta Wing Vortex
Breakdown during Dynamic Pitchingrdquo AIAA Journal of Aircraft Vol 32 No 2 1997 pp 168-173 8 Al-Garni AZ Ahmed SA Sahin AZ and Al-Garni AM ldquoAn Experimental Study of a 65-Degree
Delta Wing with Different Pitching Ratesrdquo Canadian Aeronautics and Space Journal Vol 47 No 2 2001 pp 85-93
9 Skow A M ldquoAn Analysis of the Su-27 Flight Demonstration at the 1989 Paris Airshowrdquo SAE Paper 90-1001 1990
10 Kolano E ldquoFlying the Flankerrdquo Flight Journal httpflightjournalcomarticlessu27Su27_5asp 11 Ericsson L E ldquoEffect of Fuselage Geometry on Delta-Wing Vortex Breakdownrdquo AIAA Journal of
Aircraft Vol 35 No 6 Nov 1998 12 Schaeffler N W and Telionis D P ldquoApex Flap Control of Dynamic Delta Wing Maneuversrdquo AIAA
Paper 96-0662 1996 13 Heron I and Myose R Y ldquoDevelopment of a Dynamic Pitch and Unsteady Freestream Delta Wing
Mount For A Water Tunnelrdquo AIAA Paper 2003-3527 June 2003
American Institute of Aeronautics and Astronautics 7
14 Heron I and Myose R Y ldquoVortex Burst Behavior Under Dynamic Freestreamrdquo AIAA Paper 2005-0063 January 2005
15 Heron I and Myose R Y ldquoDelta Wing Vortex Burst Behavior Under Dynamic Freestream Part 1 ndash Fast Pitch-up During Decelerationrdquo AIAA paper 2007-6725 August 2007
16 Johnson BL ldquoFacility Description of the Walter H Beech Memorial 7 x 10 foot Low-Speed Wind Tunnelrdquo AR93-1 National Institute for Aviation Research Wichita State University June 1993
Figure 1 Schematic diagram of Wichita State 2- by 3-feet water tunnel (left)16 and photograph of test section showing the towing system with an inverted delta wing mounted on carriage (right)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 30 - 50 deg
Figure 2 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 8
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-36
Figure 3 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 45 - 55+ deg
Figure 4 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 9
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc (s
= 0
at A
pex)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-38
Figure 5 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 30 to 40 degFig 5-39
Figure 6 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (40 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 10
Figure 7 Velocity Components on Leading Edge of Delta Wing
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 04 to 02 fts
dV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 8 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Deceleration (Ideal Representation of Fig 2rsquos Deceleration Case)
Half-apex angle ζ = 90 - Λ Angle of Attack α
Uinfin
Angle of Yaw β
Sweep Angle Λ
VS = Uinfin sin α cos β
VLE = Uinfin (cos α cos ζ cos β - sin ζ sin β)
VLE = Uinfin (cos α cos ζ cos β + sin ζ sin β)
VLE = Uinfin (cos α cos β sin ζ - sin β cos ζ)
VLE = Uinfin (cos α cos β sin ζ + sin β cos ζ)
ζ
American Institute of Aeronautics and Astronautics 11
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
01
012
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 02 to 04 ftsdV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 9 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Acceleration (Ideal Representation of Fig 2rsquos Acceleration Case)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 30 - 50 deg
k=02 slopek=01 slope
Figure 10 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
American Institute of Aeronautics and Astronautics 12
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 45 - 55+ deg k=02 slopek=01 slope
Figure 11 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Vel change from 30-50 degFig 5-42
k=02 slopek=01 slope
Figure 12 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 3
additional down force for the carriage wheels Visible in the photograph of Figure 1 is the tandem strut mounting mechanism for the delta wing The larger diameter strut is attached to a hinge on the delta wingrsquos pressure side The smaller diameter strut located on the downstream side is attached to a cam mechanism on the carriage which is driven by a small DC motor This allows the delta wing to be pivoted about the 50 root chord location at non-dimensional pitch rates up to k=02 Dye flow visualization was used to identify the vortex burst locations The flow visualization image was video-taped and information such as the towing carriage velocity and delta wing angle of attack were also recorded The recorded images and relevant information were subsequently analyzed using a computer-assisted image analysis software tool Additional details about the towing system measurement technique results of the validation tests and some of the initial pitch-up results are presented in references 13 through 15 In the present investigation both deceleration and acceleration is considered In this regard it is appropriate to define the velocity ratio which is a comparison between the minimum and maximum velocities In the case of deceleration the delta wing is initially towed at the maximum velocity of 04 fts and decelerated to a slower speed of either 02 fts (for a velocity ratio of 05) or 03 fts (for a velocity ratio of 075) In the case of acceleration the delta wing is brought to the maximum velocity of 04 fts starting from a slower speed of either 02 fts (for a velocity ratio of 05) or 03 fts (for a velocity ratio of 075) There are a few limitations with this experimental set-up that need to be acknowledged 1- The location of the vortex burst is accomplished by identifying where the core flares out (bubble burst) or the location of the first sharp kink (spiral burst) As such the identified location may or may not coincide with the actual core stagnation point The assumption that the two are close if not coincident has been done in the past 2- Some of the features observed cannot be explored further using this system thus a certain amount of interpretation must be exercised Nevertheless the uncertainty should be within plusmn 005c 3- The synchronization of the start of the slow-down process is accomplished manually A switch is actuated when the angle of attack α corresponding to that particular test case is observed in the AD conversion display Through practice it is possible to attain a repeatability of α 15 degrees Fast Pitch-up Results amp Discussion Figure 2 shows a comparison of the results for four experimental runs under acceleration (κ=02 velocity ratio 05 dvdt=0092 fts2 30 lt α lt 50 degrees) and four experimental runs under deceleration For convenience a positive outcome is one where the propagation of the vortex burst is delayed or arrested while a negative outcome is one where the burst moves forward faster (or jumps) than normal for a steady-velocity pitch-up The figure shows that deceleration has a positive effect of retarding the forward progression of the vortex burst this being in evidence by the level-off in the scatter plot at about sc=05 (4 secs elapsed time) The acceleration on the other hand shows a negative effect with a definite forward jump in the scatter plot Comparing with the constant velocity case and extrapolating linearly shows that at least 1 to 15 seconds can be gained before the burst has progressed to a spatial location (for example s=04c) when subjected to a deceleration Figure 3 (κ=02 velocity ratio 075 dvdt=0046 fts2 30 lt α lt 50 degrees) indicates that slower accelerations still have a negative effect on the burst location as a forward jump in the scatter is visible starting at about 5 seconds The slow deceleration in this case however does not contribute greatly to the delay in the forward propagation of the burst location as the scatter plot lies almost parallel to the linear extrapolation for a constant speed pitch-up Figure 4 (κ=02 velocity ratio 05 dvdt=0092 fts2 45 lt α lt beyond 55 degrees) indicates that acceleration has a negative effect deceleration has a mild positive effect and the angle of attack where the change in the slope of the scatter plot (ie the change in propagation velocity) also has an effect If the reader equates elapsed time with angle of attack α and compares Figure 2 to the current figure the point in time where the burst location experiences a change in slope is the same for acceleration and deceleration in Figure 2 but different in Figure 4 the acceleration precipitates a forward jump sooner than the deceleration arrests the burst progression Figure 5 (κ=02 velocity ratio 075 dvdt=0046 fts2 45 lt α lt beyond 55 degrees) indicates that at the higher angle extremes changes in velocity have little impact on the burst position There is a slight change in
American Institute of Aeronautics and Astronautics 4
the slope at approximately 5 seconds elapsed time This discontinuity appears in both accelerating and decelerating experiments In Figure 6 (κ=02 velocity ratio 075 dvdt=0092 fts2 30 lt α lt 40 degrees) the deceleration appears to have a mild positive effect So mild in fact that it is difficult to say whether the downward inflection is the result of the acceleration or just the normal change in slope that occurs when κ changes value (Recall that κ changes value in proportion to the freestream velocity) The acceleration does not appear to have any changes as the slope remained quasi-linear throughout the maneuver To summarize pitching up at κ=02 at different angle of attack ranges and velocity ratios produces a mild to strong negative effect on the burst location when accelerating This negative effect is almost independent of the actual acceleration or range of α over which it occurs The positive delay in the burst movement on the other hand appears to be strongest when the delta wing experiences a strong deceleration (dvdt=0092 fts2) over a large range of α values (30 to 50 degrees) This effect becomes much smaller at slower decelerations andor limited values in the α range This implies that perhaps instabilities (whether inherent or external to the vortex core) could help precipitate the forward jump (as in Figure 4) making it easier to lose lift than to keep lift Part 115 of this two part paper series presented a possible explanation why a positive delay in the burst movement may be obtained through deceleration (ie reduction in freestream speed) A brief review of that discussion is repeated here Considering the different velocity components shown in Figure 7 the velocity component that is directly responsible for vorticity production at the leading edge is given by VS = Uinfin sin α cos β (2) It can be seen that cos β is constant for a given installation (ie no sideslip component β=0 and cos β=1) Taking the time differential dVS dt = (dUinfindt sin α + Uinfin dαdt cos α) cos β (3) Both dUinfindt and dαdt are for the purposes of these experiments quasi-constant numbers The relative magnitudes of both numbers set a balance The change in the leading edge perpendicular velocity is therefore given by the two terms in the parenthesis Figure 8 shows the effect of various terms for the deceleration case presented in Figure 2 (dUinfindt= -0092 fts2 and dαdt= 916 degs= 016 rads at κ= 02) When the deceleration begins in this case a large change in the character of the curves occurs The increase in velocity perpendicular to the leading edge is initially driven by the Uinfin dαdt cosα term However the velocity perpendicular to the leading edge is reduced when deceleration is introduced (ie with the inclusion of the dUinfindt sin α term) This in turn reduces the formation of vorticity at its source It should be noted that Figure 8 does not include the effect of the time delay it takes to convect the vorticity from the leading edge into the vortex core This time delay will be non-existent at the apex and increases linearly with chord location towards the trailing edge Thus the velocity calculated in Figure 8 is first experienced at the apex then further down the length of the vortex core and at the trailing edge last Nevertheless a reduction in vortex production with deceleration should result in a positive delay in the vortex burst movement Figure 9 shows the effect of various terms for the acceleration case corresponding to the situation of Figure 2 (dUinfindt = -0092 fts2 and dαdt= 916 degs= 016 rads at κ= 02) When acceleration begins in this case a large change in the character of the curve occurs once again This time however the velocity perpendicular to the leading edge is increased when acceleration is introduced (ie with the inclusion of the dUinfindt sin α term) This would lead to an increase in the production of vorticity at its source This in turn is likely to lead to the negative result of early vortex burst (ie movement of the burst closer to the apex) seen in Figure 2 Results amp Discussion for Pitch-up at Moderate Rate Pitching up at half the pitch rate (κ=01) such as is the case in Figure 10 (κ=01 velocity ratio 05 dvdt=0046 fts2 30 lt α lt 50 degrees) produces results where the deceleration has a clear positive effect on the burst location The scatter is also increased Notice that the deceleration produces a slope (at 7 seconds elapsed time) close to that of a constant speed pitch-up at κ=02 which is intuitively correct when one considers that since Uinfin occurs in the denominator reducing it to frac12 of the initial magnitude should double the value of κ
American Institute of Aeronautics and Astronautics 5
In Figure 11 the acceleration does not appear to change the slope of the propagation curves On the other hand the deceleration produces a mild retardation in the burst propagation In one run (Exp 3 black circular symbols) a clearly visible change in slope occurred at 5 seconds The other two runs produced a barely discernible change in slope at approximately 45 seconds Notice that both Figures 10 and 11 present the results of experiments at the higher dvdt value (0046 fts2) but the range of α values tested for Figure 11 starts at a very high value of α thus the deceleration appears to have a smaller effect when started at a higher angle of attack Figures 12 and 13 are similar in that the experiments occurred at the lower value of dvdt tested (0023 fts2) In both cases it is hard to observe any clear effect of the change in velocity as there is no discernible change in slope particularly during the deceleration portion The acceleration of the experiments in Figure 12 did change the slope towards the end (at approximately 5 seconds elapsed time) but the change was to flatten (delay) the forward motion of the burst location This change in slope was probably due to the fact that the pitch-up was close to the end point To summarize pitching up at κ=01 at different ranges and velocity ratios produced mild to non-observable negative and positive effects Both effects appeared to occur when the delta wing experienced a strong deceleration (dvdt=0046 fts2) over a large range of α values (30 to 50 degrees) At slower decelerations andor limited values in the α range the effects became almost unobservable in the scatter plots due in part to the increased scatter Instabilities (whether inherent or external to the vortex core) or other mechanisms (ie laminarity in the viscous regions) could play a more dominant effect at the low Reynolds numbers experienced under these conditions by the delta wing Pitch-down Results amp Discussion Figures 14 through 16 present the results of a pitch-down at κ=01 under different values of dvdt and α ranges The deceleration does appear to move the burst back towards the trailing edge (a positive effect) faster in the case where dvdt is high (0092 fts2) In the other two cases where dvdt has intermediate and low values there is no appreciable difference between accelerating and decelerating Scatter is well controlled in all three figures Thus the benefits of decelerating the delta wing to delay the forward progression of the burst appear to be confined to the pitch-up regime of flight Summary A series of experiments on a 70-degree delta wing was conducted at Wichita State University to study the effect on the vortex burst position when simultaneous dynamic pitch and unsteady freestream velocity were used The aim was to better understand the relationship between the freestream velocity and the time constants involved in the movement of the vortex burst point Decelerating from 04 to 02 fts (ie a velocity ratio of 05) a reduction in the vortex burst propagation velocity is apparent at about 15 seconds after the freestream velocity begins to decelerate One possible means to non-dimensionalize the time axis is to utilize a non-dimensional timeframe trsquo = (t a) (ω c) (4) where t = elapsed time a = acceleration ω = pitch rate and c = chord Using this definition the reduction in forward propagation of the vortex burst location occurs at (t = 15 seconds which corresponds to) trsquo= 0863 when the delta wing is decelerated with a velocity ratio of 05 during pitch-up between 15 to 55 degrees at a rate of κ = 02 Accelerating from 02 to 04 ftsec (ie a minimum to maximum velocity ratio of 05) and pitching up at κ = 02 between 15 to 55 degrees there was a slight downward inflection in the burst location curves at approximately 175 and 2 seconds (trsquo= 101 to 115 respectively) after the beginning of the velocity ramp-up (acceleration from 02 fts) During a small deceleration from 04 to 03 fts (ie a velocity ratio of 075) and pitching up at a rate κ = 02 between 15 to 55 degrees a reduction in the vortex burst propagation velocity was apparent at approximately 1 second after the freestream velocity began to change (trsquo= 0288) This retardation delays the forward propagation of the vortex burst (towards the apex) The forward propagation of the burst location resumes at about 45 seconds elapsed time (trsquo= 1294) and this appeared to be at the same propagation rate (ie the slope of the burst) compared to the beginning of the experiment
American Institute of Aeronautics and Astronautics 6
It appears that at κ = 02 for large ranges of angle of attack α deceleration of the freestream velocity appeared to arrest the motion of the burst location anywhere between 15 and 25 seconds after the deceleration began to take place (ie 0863 lt trsquolt 1438) The positive delay in the burst movement appeared to be strongest when the delta wing experienced a strong deceleration (dvdt=0092 fts2) over a large range of α values (30 to 50 degrees) This effect became much smaller at slower decelerations andor limited values in the α range This implies that perhaps instabilities (whether inherent or external to the vortex core) could help precipitate the forward jump making it easier to lose lift than to keep lift It appears that acceleration had the effect of increasing the forward propagation of the vortex burst along the leading edge vortex core At different α ranges and velocity ratios accelerating produced a mild to strong negative effect on the burst location when accelerating This negative effect was almost independent of the actual acceleration or range of α over which it occurred The results obtained during pitch up at the slower rate of κ = 01 for different combinations of acceleration and deceleration exhibited the same general features as those observed during the faster pitch-ups deceleration in the freestream velocity appeared to have a beneficial effect ie it arrested the forward motion of the burst location (thereby delaying the propagation of the vortex burst towards the apex) Acceleration in the freestream velocity appeared to have the opposite effect that of pushing the burst location forward A second trait observed during the slower pitch-up tests was that the degree of scatter in the data increased In some cases there were large fluctuations along the time axis where the forward propagation accelerated In other cases slightly different curves (one for each experiment performed) were traced each having identical slope but separated from its neighbor by a few fractions of a second This indicated that at the slower pitching rate the experiment was more sensitive to either external noise (such as unavoidable vibrations in the carriage being transmitted to the delta wing) or to something instrinsic within the flow phenomenon The experiment as it was performed could not isolate the cause of this variability References 1 Polhamus E ldquoA Concept of the Vortex Lift of Sharp-Edge Delta Wings Based on a Leading-Edge
Suction Analogyrdquo NASA TN D-3767 Dec 1966 2 Earnshaw PB and Lawford JA ldquoLow-Speed Wind Tunnel Experiments on a Series of Sharp-Edged
Delta Wingsrdquo Aeronautical Research Council R M No 3424 1964 3 Erickson GE ldquoWater-Tunnel Studies of Leading-Edge Vorticesrdquo AIAA Journal of Aircraft Vol 19
No 6 1982 pp 442-448 4 LeMay WP Batill SM and Nelson RC ldquoVortex Dynamics on a Pitching Delta Wingrdquo AIAA
Journal of Aircraft Vol 27 No 2 1990 pp 131-138 5 Miller LS and Gile BE ldquoEffects of Blowing on Delta Wing Vortices During Dynamic Pitchingrdquo
AIAA Journal of Aircraft Vol 30 No 3 1993 pp 334-339 6 Rediniotis O K Klute S M Hoang N T and Telionis D P ldquoDynamic Pitch-Up of a Delta Wingrdquo
AIAA Journal Vol 32 No 4 1994 pp 716-725 7 Myose RY Hayashibara S Yeong PC and Miller LS ldquoEffect of Canards on Delta Wing Vortex
Breakdown during Dynamic Pitchingrdquo AIAA Journal of Aircraft Vol 32 No 2 1997 pp 168-173 8 Al-Garni AZ Ahmed SA Sahin AZ and Al-Garni AM ldquoAn Experimental Study of a 65-Degree
Delta Wing with Different Pitching Ratesrdquo Canadian Aeronautics and Space Journal Vol 47 No 2 2001 pp 85-93
9 Skow A M ldquoAn Analysis of the Su-27 Flight Demonstration at the 1989 Paris Airshowrdquo SAE Paper 90-1001 1990
10 Kolano E ldquoFlying the Flankerrdquo Flight Journal httpflightjournalcomarticlessu27Su27_5asp 11 Ericsson L E ldquoEffect of Fuselage Geometry on Delta-Wing Vortex Breakdownrdquo AIAA Journal of
Aircraft Vol 35 No 6 Nov 1998 12 Schaeffler N W and Telionis D P ldquoApex Flap Control of Dynamic Delta Wing Maneuversrdquo AIAA
Paper 96-0662 1996 13 Heron I and Myose R Y ldquoDevelopment of a Dynamic Pitch and Unsteady Freestream Delta Wing
Mount For A Water Tunnelrdquo AIAA Paper 2003-3527 June 2003
American Institute of Aeronautics and Astronautics 7
14 Heron I and Myose R Y ldquoVortex Burst Behavior Under Dynamic Freestreamrdquo AIAA Paper 2005-0063 January 2005
15 Heron I and Myose R Y ldquoDelta Wing Vortex Burst Behavior Under Dynamic Freestream Part 1 ndash Fast Pitch-up During Decelerationrdquo AIAA paper 2007-6725 August 2007
16 Johnson BL ldquoFacility Description of the Walter H Beech Memorial 7 x 10 foot Low-Speed Wind Tunnelrdquo AR93-1 National Institute for Aviation Research Wichita State University June 1993
Figure 1 Schematic diagram of Wichita State 2- by 3-feet water tunnel (left)16 and photograph of test section showing the towing system with an inverted delta wing mounted on carriage (right)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 30 - 50 deg
Figure 2 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 8
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-36
Figure 3 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 45 - 55+ deg
Figure 4 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 9
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc (s
= 0
at A
pex)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-38
Figure 5 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 30 to 40 degFig 5-39
Figure 6 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (40 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 10
Figure 7 Velocity Components on Leading Edge of Delta Wing
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 04 to 02 fts
dV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 8 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Deceleration (Ideal Representation of Fig 2rsquos Deceleration Case)
Half-apex angle ζ = 90 - Λ Angle of Attack α
Uinfin
Angle of Yaw β
Sweep Angle Λ
VS = Uinfin sin α cos β
VLE = Uinfin (cos α cos ζ cos β - sin ζ sin β)
VLE = Uinfin (cos α cos ζ cos β + sin ζ sin β)
VLE = Uinfin (cos α cos β sin ζ - sin β cos ζ)
VLE = Uinfin (cos α cos β sin ζ + sin β cos ζ)
ζ
American Institute of Aeronautics and Astronautics 11
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
01
012
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 02 to 04 ftsdV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 9 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Acceleration (Ideal Representation of Fig 2rsquos Acceleration Case)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 30 - 50 deg
k=02 slopek=01 slope
Figure 10 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
American Institute of Aeronautics and Astronautics 12
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 45 - 55+ deg k=02 slopek=01 slope
Figure 11 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Vel change from 30-50 degFig 5-42
k=02 slopek=01 slope
Figure 12 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 4
the slope at approximately 5 seconds elapsed time This discontinuity appears in both accelerating and decelerating experiments In Figure 6 (κ=02 velocity ratio 075 dvdt=0092 fts2 30 lt α lt 40 degrees) the deceleration appears to have a mild positive effect So mild in fact that it is difficult to say whether the downward inflection is the result of the acceleration or just the normal change in slope that occurs when κ changes value (Recall that κ changes value in proportion to the freestream velocity) The acceleration does not appear to have any changes as the slope remained quasi-linear throughout the maneuver To summarize pitching up at κ=02 at different angle of attack ranges and velocity ratios produces a mild to strong negative effect on the burst location when accelerating This negative effect is almost independent of the actual acceleration or range of α over which it occurs The positive delay in the burst movement on the other hand appears to be strongest when the delta wing experiences a strong deceleration (dvdt=0092 fts2) over a large range of α values (30 to 50 degrees) This effect becomes much smaller at slower decelerations andor limited values in the α range This implies that perhaps instabilities (whether inherent or external to the vortex core) could help precipitate the forward jump (as in Figure 4) making it easier to lose lift than to keep lift Part 115 of this two part paper series presented a possible explanation why a positive delay in the burst movement may be obtained through deceleration (ie reduction in freestream speed) A brief review of that discussion is repeated here Considering the different velocity components shown in Figure 7 the velocity component that is directly responsible for vorticity production at the leading edge is given by VS = Uinfin sin α cos β (2) It can be seen that cos β is constant for a given installation (ie no sideslip component β=0 and cos β=1) Taking the time differential dVS dt = (dUinfindt sin α + Uinfin dαdt cos α) cos β (3) Both dUinfindt and dαdt are for the purposes of these experiments quasi-constant numbers The relative magnitudes of both numbers set a balance The change in the leading edge perpendicular velocity is therefore given by the two terms in the parenthesis Figure 8 shows the effect of various terms for the deceleration case presented in Figure 2 (dUinfindt= -0092 fts2 and dαdt= 916 degs= 016 rads at κ= 02) When the deceleration begins in this case a large change in the character of the curves occurs The increase in velocity perpendicular to the leading edge is initially driven by the Uinfin dαdt cosα term However the velocity perpendicular to the leading edge is reduced when deceleration is introduced (ie with the inclusion of the dUinfindt sin α term) This in turn reduces the formation of vorticity at its source It should be noted that Figure 8 does not include the effect of the time delay it takes to convect the vorticity from the leading edge into the vortex core This time delay will be non-existent at the apex and increases linearly with chord location towards the trailing edge Thus the velocity calculated in Figure 8 is first experienced at the apex then further down the length of the vortex core and at the trailing edge last Nevertheless a reduction in vortex production with deceleration should result in a positive delay in the vortex burst movement Figure 9 shows the effect of various terms for the acceleration case corresponding to the situation of Figure 2 (dUinfindt = -0092 fts2 and dαdt= 916 degs= 016 rads at κ= 02) When acceleration begins in this case a large change in the character of the curve occurs once again This time however the velocity perpendicular to the leading edge is increased when acceleration is introduced (ie with the inclusion of the dUinfindt sin α term) This would lead to an increase in the production of vorticity at its source This in turn is likely to lead to the negative result of early vortex burst (ie movement of the burst closer to the apex) seen in Figure 2 Results amp Discussion for Pitch-up at Moderate Rate Pitching up at half the pitch rate (κ=01) such as is the case in Figure 10 (κ=01 velocity ratio 05 dvdt=0046 fts2 30 lt α lt 50 degrees) produces results where the deceleration has a clear positive effect on the burst location The scatter is also increased Notice that the deceleration produces a slope (at 7 seconds elapsed time) close to that of a constant speed pitch-up at κ=02 which is intuitively correct when one considers that since Uinfin occurs in the denominator reducing it to frac12 of the initial magnitude should double the value of κ
American Institute of Aeronautics and Astronautics 5
In Figure 11 the acceleration does not appear to change the slope of the propagation curves On the other hand the deceleration produces a mild retardation in the burst propagation In one run (Exp 3 black circular symbols) a clearly visible change in slope occurred at 5 seconds The other two runs produced a barely discernible change in slope at approximately 45 seconds Notice that both Figures 10 and 11 present the results of experiments at the higher dvdt value (0046 fts2) but the range of α values tested for Figure 11 starts at a very high value of α thus the deceleration appears to have a smaller effect when started at a higher angle of attack Figures 12 and 13 are similar in that the experiments occurred at the lower value of dvdt tested (0023 fts2) In both cases it is hard to observe any clear effect of the change in velocity as there is no discernible change in slope particularly during the deceleration portion The acceleration of the experiments in Figure 12 did change the slope towards the end (at approximately 5 seconds elapsed time) but the change was to flatten (delay) the forward motion of the burst location This change in slope was probably due to the fact that the pitch-up was close to the end point To summarize pitching up at κ=01 at different ranges and velocity ratios produced mild to non-observable negative and positive effects Both effects appeared to occur when the delta wing experienced a strong deceleration (dvdt=0046 fts2) over a large range of α values (30 to 50 degrees) At slower decelerations andor limited values in the α range the effects became almost unobservable in the scatter plots due in part to the increased scatter Instabilities (whether inherent or external to the vortex core) or other mechanisms (ie laminarity in the viscous regions) could play a more dominant effect at the low Reynolds numbers experienced under these conditions by the delta wing Pitch-down Results amp Discussion Figures 14 through 16 present the results of a pitch-down at κ=01 under different values of dvdt and α ranges The deceleration does appear to move the burst back towards the trailing edge (a positive effect) faster in the case where dvdt is high (0092 fts2) In the other two cases where dvdt has intermediate and low values there is no appreciable difference between accelerating and decelerating Scatter is well controlled in all three figures Thus the benefits of decelerating the delta wing to delay the forward progression of the burst appear to be confined to the pitch-up regime of flight Summary A series of experiments on a 70-degree delta wing was conducted at Wichita State University to study the effect on the vortex burst position when simultaneous dynamic pitch and unsteady freestream velocity were used The aim was to better understand the relationship between the freestream velocity and the time constants involved in the movement of the vortex burst point Decelerating from 04 to 02 fts (ie a velocity ratio of 05) a reduction in the vortex burst propagation velocity is apparent at about 15 seconds after the freestream velocity begins to decelerate One possible means to non-dimensionalize the time axis is to utilize a non-dimensional timeframe trsquo = (t a) (ω c) (4) where t = elapsed time a = acceleration ω = pitch rate and c = chord Using this definition the reduction in forward propagation of the vortex burst location occurs at (t = 15 seconds which corresponds to) trsquo= 0863 when the delta wing is decelerated with a velocity ratio of 05 during pitch-up between 15 to 55 degrees at a rate of κ = 02 Accelerating from 02 to 04 ftsec (ie a minimum to maximum velocity ratio of 05) and pitching up at κ = 02 between 15 to 55 degrees there was a slight downward inflection in the burst location curves at approximately 175 and 2 seconds (trsquo= 101 to 115 respectively) after the beginning of the velocity ramp-up (acceleration from 02 fts) During a small deceleration from 04 to 03 fts (ie a velocity ratio of 075) and pitching up at a rate κ = 02 between 15 to 55 degrees a reduction in the vortex burst propagation velocity was apparent at approximately 1 second after the freestream velocity began to change (trsquo= 0288) This retardation delays the forward propagation of the vortex burst (towards the apex) The forward propagation of the burst location resumes at about 45 seconds elapsed time (trsquo= 1294) and this appeared to be at the same propagation rate (ie the slope of the burst) compared to the beginning of the experiment
American Institute of Aeronautics and Astronautics 6
It appears that at κ = 02 for large ranges of angle of attack α deceleration of the freestream velocity appeared to arrest the motion of the burst location anywhere between 15 and 25 seconds after the deceleration began to take place (ie 0863 lt trsquolt 1438) The positive delay in the burst movement appeared to be strongest when the delta wing experienced a strong deceleration (dvdt=0092 fts2) over a large range of α values (30 to 50 degrees) This effect became much smaller at slower decelerations andor limited values in the α range This implies that perhaps instabilities (whether inherent or external to the vortex core) could help precipitate the forward jump making it easier to lose lift than to keep lift It appears that acceleration had the effect of increasing the forward propagation of the vortex burst along the leading edge vortex core At different α ranges and velocity ratios accelerating produced a mild to strong negative effect on the burst location when accelerating This negative effect was almost independent of the actual acceleration or range of α over which it occurred The results obtained during pitch up at the slower rate of κ = 01 for different combinations of acceleration and deceleration exhibited the same general features as those observed during the faster pitch-ups deceleration in the freestream velocity appeared to have a beneficial effect ie it arrested the forward motion of the burst location (thereby delaying the propagation of the vortex burst towards the apex) Acceleration in the freestream velocity appeared to have the opposite effect that of pushing the burst location forward A second trait observed during the slower pitch-up tests was that the degree of scatter in the data increased In some cases there were large fluctuations along the time axis where the forward propagation accelerated In other cases slightly different curves (one for each experiment performed) were traced each having identical slope but separated from its neighbor by a few fractions of a second This indicated that at the slower pitching rate the experiment was more sensitive to either external noise (such as unavoidable vibrations in the carriage being transmitted to the delta wing) or to something instrinsic within the flow phenomenon The experiment as it was performed could not isolate the cause of this variability References 1 Polhamus E ldquoA Concept of the Vortex Lift of Sharp-Edge Delta Wings Based on a Leading-Edge
Suction Analogyrdquo NASA TN D-3767 Dec 1966 2 Earnshaw PB and Lawford JA ldquoLow-Speed Wind Tunnel Experiments on a Series of Sharp-Edged
Delta Wingsrdquo Aeronautical Research Council R M No 3424 1964 3 Erickson GE ldquoWater-Tunnel Studies of Leading-Edge Vorticesrdquo AIAA Journal of Aircraft Vol 19
No 6 1982 pp 442-448 4 LeMay WP Batill SM and Nelson RC ldquoVortex Dynamics on a Pitching Delta Wingrdquo AIAA
Journal of Aircraft Vol 27 No 2 1990 pp 131-138 5 Miller LS and Gile BE ldquoEffects of Blowing on Delta Wing Vortices During Dynamic Pitchingrdquo
AIAA Journal of Aircraft Vol 30 No 3 1993 pp 334-339 6 Rediniotis O K Klute S M Hoang N T and Telionis D P ldquoDynamic Pitch-Up of a Delta Wingrdquo
AIAA Journal Vol 32 No 4 1994 pp 716-725 7 Myose RY Hayashibara S Yeong PC and Miller LS ldquoEffect of Canards on Delta Wing Vortex
Breakdown during Dynamic Pitchingrdquo AIAA Journal of Aircraft Vol 32 No 2 1997 pp 168-173 8 Al-Garni AZ Ahmed SA Sahin AZ and Al-Garni AM ldquoAn Experimental Study of a 65-Degree
Delta Wing with Different Pitching Ratesrdquo Canadian Aeronautics and Space Journal Vol 47 No 2 2001 pp 85-93
9 Skow A M ldquoAn Analysis of the Su-27 Flight Demonstration at the 1989 Paris Airshowrdquo SAE Paper 90-1001 1990
10 Kolano E ldquoFlying the Flankerrdquo Flight Journal httpflightjournalcomarticlessu27Su27_5asp 11 Ericsson L E ldquoEffect of Fuselage Geometry on Delta-Wing Vortex Breakdownrdquo AIAA Journal of
Aircraft Vol 35 No 6 Nov 1998 12 Schaeffler N W and Telionis D P ldquoApex Flap Control of Dynamic Delta Wing Maneuversrdquo AIAA
Paper 96-0662 1996 13 Heron I and Myose R Y ldquoDevelopment of a Dynamic Pitch and Unsteady Freestream Delta Wing
Mount For A Water Tunnelrdquo AIAA Paper 2003-3527 June 2003
American Institute of Aeronautics and Astronautics 7
14 Heron I and Myose R Y ldquoVortex Burst Behavior Under Dynamic Freestreamrdquo AIAA Paper 2005-0063 January 2005
15 Heron I and Myose R Y ldquoDelta Wing Vortex Burst Behavior Under Dynamic Freestream Part 1 ndash Fast Pitch-up During Decelerationrdquo AIAA paper 2007-6725 August 2007
16 Johnson BL ldquoFacility Description of the Walter H Beech Memorial 7 x 10 foot Low-Speed Wind Tunnelrdquo AR93-1 National Institute for Aviation Research Wichita State University June 1993
Figure 1 Schematic diagram of Wichita State 2- by 3-feet water tunnel (left)16 and photograph of test section showing the towing system with an inverted delta wing mounted on carriage (right)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 30 - 50 deg
Figure 2 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 8
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-36
Figure 3 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 45 - 55+ deg
Figure 4 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 9
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc (s
= 0
at A
pex)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-38
Figure 5 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 30 to 40 degFig 5-39
Figure 6 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (40 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 10
Figure 7 Velocity Components on Leading Edge of Delta Wing
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 04 to 02 fts
dV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 8 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Deceleration (Ideal Representation of Fig 2rsquos Deceleration Case)
Half-apex angle ζ = 90 - Λ Angle of Attack α
Uinfin
Angle of Yaw β
Sweep Angle Λ
VS = Uinfin sin α cos β
VLE = Uinfin (cos α cos ζ cos β - sin ζ sin β)
VLE = Uinfin (cos α cos ζ cos β + sin ζ sin β)
VLE = Uinfin (cos α cos β sin ζ - sin β cos ζ)
VLE = Uinfin (cos α cos β sin ζ + sin β cos ζ)
ζ
American Institute of Aeronautics and Astronautics 11
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
01
012
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 02 to 04 ftsdV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 9 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Acceleration (Ideal Representation of Fig 2rsquos Acceleration Case)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 30 - 50 deg
k=02 slopek=01 slope
Figure 10 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
American Institute of Aeronautics and Astronautics 12
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 45 - 55+ deg k=02 slopek=01 slope
Figure 11 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Vel change from 30-50 degFig 5-42
k=02 slopek=01 slope
Figure 12 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 5
In Figure 11 the acceleration does not appear to change the slope of the propagation curves On the other hand the deceleration produces a mild retardation in the burst propagation In one run (Exp 3 black circular symbols) a clearly visible change in slope occurred at 5 seconds The other two runs produced a barely discernible change in slope at approximately 45 seconds Notice that both Figures 10 and 11 present the results of experiments at the higher dvdt value (0046 fts2) but the range of α values tested for Figure 11 starts at a very high value of α thus the deceleration appears to have a smaller effect when started at a higher angle of attack Figures 12 and 13 are similar in that the experiments occurred at the lower value of dvdt tested (0023 fts2) In both cases it is hard to observe any clear effect of the change in velocity as there is no discernible change in slope particularly during the deceleration portion The acceleration of the experiments in Figure 12 did change the slope towards the end (at approximately 5 seconds elapsed time) but the change was to flatten (delay) the forward motion of the burst location This change in slope was probably due to the fact that the pitch-up was close to the end point To summarize pitching up at κ=01 at different ranges and velocity ratios produced mild to non-observable negative and positive effects Both effects appeared to occur when the delta wing experienced a strong deceleration (dvdt=0046 fts2) over a large range of α values (30 to 50 degrees) At slower decelerations andor limited values in the α range the effects became almost unobservable in the scatter plots due in part to the increased scatter Instabilities (whether inherent or external to the vortex core) or other mechanisms (ie laminarity in the viscous regions) could play a more dominant effect at the low Reynolds numbers experienced under these conditions by the delta wing Pitch-down Results amp Discussion Figures 14 through 16 present the results of a pitch-down at κ=01 under different values of dvdt and α ranges The deceleration does appear to move the burst back towards the trailing edge (a positive effect) faster in the case where dvdt is high (0092 fts2) In the other two cases where dvdt has intermediate and low values there is no appreciable difference between accelerating and decelerating Scatter is well controlled in all three figures Thus the benefits of decelerating the delta wing to delay the forward progression of the burst appear to be confined to the pitch-up regime of flight Summary A series of experiments on a 70-degree delta wing was conducted at Wichita State University to study the effect on the vortex burst position when simultaneous dynamic pitch and unsteady freestream velocity were used The aim was to better understand the relationship between the freestream velocity and the time constants involved in the movement of the vortex burst point Decelerating from 04 to 02 fts (ie a velocity ratio of 05) a reduction in the vortex burst propagation velocity is apparent at about 15 seconds after the freestream velocity begins to decelerate One possible means to non-dimensionalize the time axis is to utilize a non-dimensional timeframe trsquo = (t a) (ω c) (4) where t = elapsed time a = acceleration ω = pitch rate and c = chord Using this definition the reduction in forward propagation of the vortex burst location occurs at (t = 15 seconds which corresponds to) trsquo= 0863 when the delta wing is decelerated with a velocity ratio of 05 during pitch-up between 15 to 55 degrees at a rate of κ = 02 Accelerating from 02 to 04 ftsec (ie a minimum to maximum velocity ratio of 05) and pitching up at κ = 02 between 15 to 55 degrees there was a slight downward inflection in the burst location curves at approximately 175 and 2 seconds (trsquo= 101 to 115 respectively) after the beginning of the velocity ramp-up (acceleration from 02 fts) During a small deceleration from 04 to 03 fts (ie a velocity ratio of 075) and pitching up at a rate κ = 02 between 15 to 55 degrees a reduction in the vortex burst propagation velocity was apparent at approximately 1 second after the freestream velocity began to change (trsquo= 0288) This retardation delays the forward propagation of the vortex burst (towards the apex) The forward propagation of the burst location resumes at about 45 seconds elapsed time (trsquo= 1294) and this appeared to be at the same propagation rate (ie the slope of the burst) compared to the beginning of the experiment
American Institute of Aeronautics and Astronautics 6
It appears that at κ = 02 for large ranges of angle of attack α deceleration of the freestream velocity appeared to arrest the motion of the burst location anywhere between 15 and 25 seconds after the deceleration began to take place (ie 0863 lt trsquolt 1438) The positive delay in the burst movement appeared to be strongest when the delta wing experienced a strong deceleration (dvdt=0092 fts2) over a large range of α values (30 to 50 degrees) This effect became much smaller at slower decelerations andor limited values in the α range This implies that perhaps instabilities (whether inherent or external to the vortex core) could help precipitate the forward jump making it easier to lose lift than to keep lift It appears that acceleration had the effect of increasing the forward propagation of the vortex burst along the leading edge vortex core At different α ranges and velocity ratios accelerating produced a mild to strong negative effect on the burst location when accelerating This negative effect was almost independent of the actual acceleration or range of α over which it occurred The results obtained during pitch up at the slower rate of κ = 01 for different combinations of acceleration and deceleration exhibited the same general features as those observed during the faster pitch-ups deceleration in the freestream velocity appeared to have a beneficial effect ie it arrested the forward motion of the burst location (thereby delaying the propagation of the vortex burst towards the apex) Acceleration in the freestream velocity appeared to have the opposite effect that of pushing the burst location forward A second trait observed during the slower pitch-up tests was that the degree of scatter in the data increased In some cases there were large fluctuations along the time axis where the forward propagation accelerated In other cases slightly different curves (one for each experiment performed) were traced each having identical slope but separated from its neighbor by a few fractions of a second This indicated that at the slower pitching rate the experiment was more sensitive to either external noise (such as unavoidable vibrations in the carriage being transmitted to the delta wing) or to something instrinsic within the flow phenomenon The experiment as it was performed could not isolate the cause of this variability References 1 Polhamus E ldquoA Concept of the Vortex Lift of Sharp-Edge Delta Wings Based on a Leading-Edge
Suction Analogyrdquo NASA TN D-3767 Dec 1966 2 Earnshaw PB and Lawford JA ldquoLow-Speed Wind Tunnel Experiments on a Series of Sharp-Edged
Delta Wingsrdquo Aeronautical Research Council R M No 3424 1964 3 Erickson GE ldquoWater-Tunnel Studies of Leading-Edge Vorticesrdquo AIAA Journal of Aircraft Vol 19
No 6 1982 pp 442-448 4 LeMay WP Batill SM and Nelson RC ldquoVortex Dynamics on a Pitching Delta Wingrdquo AIAA
Journal of Aircraft Vol 27 No 2 1990 pp 131-138 5 Miller LS and Gile BE ldquoEffects of Blowing on Delta Wing Vortices During Dynamic Pitchingrdquo
AIAA Journal of Aircraft Vol 30 No 3 1993 pp 334-339 6 Rediniotis O K Klute S M Hoang N T and Telionis D P ldquoDynamic Pitch-Up of a Delta Wingrdquo
AIAA Journal Vol 32 No 4 1994 pp 716-725 7 Myose RY Hayashibara S Yeong PC and Miller LS ldquoEffect of Canards on Delta Wing Vortex
Breakdown during Dynamic Pitchingrdquo AIAA Journal of Aircraft Vol 32 No 2 1997 pp 168-173 8 Al-Garni AZ Ahmed SA Sahin AZ and Al-Garni AM ldquoAn Experimental Study of a 65-Degree
Delta Wing with Different Pitching Ratesrdquo Canadian Aeronautics and Space Journal Vol 47 No 2 2001 pp 85-93
9 Skow A M ldquoAn Analysis of the Su-27 Flight Demonstration at the 1989 Paris Airshowrdquo SAE Paper 90-1001 1990
10 Kolano E ldquoFlying the Flankerrdquo Flight Journal httpflightjournalcomarticlessu27Su27_5asp 11 Ericsson L E ldquoEffect of Fuselage Geometry on Delta-Wing Vortex Breakdownrdquo AIAA Journal of
Aircraft Vol 35 No 6 Nov 1998 12 Schaeffler N W and Telionis D P ldquoApex Flap Control of Dynamic Delta Wing Maneuversrdquo AIAA
Paper 96-0662 1996 13 Heron I and Myose R Y ldquoDevelopment of a Dynamic Pitch and Unsteady Freestream Delta Wing
Mount For A Water Tunnelrdquo AIAA Paper 2003-3527 June 2003
American Institute of Aeronautics and Astronautics 7
14 Heron I and Myose R Y ldquoVortex Burst Behavior Under Dynamic Freestreamrdquo AIAA Paper 2005-0063 January 2005
15 Heron I and Myose R Y ldquoDelta Wing Vortex Burst Behavior Under Dynamic Freestream Part 1 ndash Fast Pitch-up During Decelerationrdquo AIAA paper 2007-6725 August 2007
16 Johnson BL ldquoFacility Description of the Walter H Beech Memorial 7 x 10 foot Low-Speed Wind Tunnelrdquo AR93-1 National Institute for Aviation Research Wichita State University June 1993
Figure 1 Schematic diagram of Wichita State 2- by 3-feet water tunnel (left)16 and photograph of test section showing the towing system with an inverted delta wing mounted on carriage (right)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 30 - 50 deg
Figure 2 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 8
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-36
Figure 3 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 45 - 55+ deg
Figure 4 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 9
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc (s
= 0
at A
pex)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-38
Figure 5 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 30 to 40 degFig 5-39
Figure 6 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (40 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 10
Figure 7 Velocity Components on Leading Edge of Delta Wing
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 04 to 02 fts
dV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 8 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Deceleration (Ideal Representation of Fig 2rsquos Deceleration Case)
Half-apex angle ζ = 90 - Λ Angle of Attack α
Uinfin
Angle of Yaw β
Sweep Angle Λ
VS = Uinfin sin α cos β
VLE = Uinfin (cos α cos ζ cos β - sin ζ sin β)
VLE = Uinfin (cos α cos ζ cos β + sin ζ sin β)
VLE = Uinfin (cos α cos β sin ζ - sin β cos ζ)
VLE = Uinfin (cos α cos β sin ζ + sin β cos ζ)
ζ
American Institute of Aeronautics and Astronautics 11
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
01
012
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 02 to 04 ftsdV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 9 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Acceleration (Ideal Representation of Fig 2rsquos Acceleration Case)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 30 - 50 deg
k=02 slopek=01 slope
Figure 10 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
American Institute of Aeronautics and Astronautics 12
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 45 - 55+ deg k=02 slopek=01 slope
Figure 11 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Vel change from 30-50 degFig 5-42
k=02 slopek=01 slope
Figure 12 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 6
It appears that at κ = 02 for large ranges of angle of attack α deceleration of the freestream velocity appeared to arrest the motion of the burst location anywhere between 15 and 25 seconds after the deceleration began to take place (ie 0863 lt trsquolt 1438) The positive delay in the burst movement appeared to be strongest when the delta wing experienced a strong deceleration (dvdt=0092 fts2) over a large range of α values (30 to 50 degrees) This effect became much smaller at slower decelerations andor limited values in the α range This implies that perhaps instabilities (whether inherent or external to the vortex core) could help precipitate the forward jump making it easier to lose lift than to keep lift It appears that acceleration had the effect of increasing the forward propagation of the vortex burst along the leading edge vortex core At different α ranges and velocity ratios accelerating produced a mild to strong negative effect on the burst location when accelerating This negative effect was almost independent of the actual acceleration or range of α over which it occurred The results obtained during pitch up at the slower rate of κ = 01 for different combinations of acceleration and deceleration exhibited the same general features as those observed during the faster pitch-ups deceleration in the freestream velocity appeared to have a beneficial effect ie it arrested the forward motion of the burst location (thereby delaying the propagation of the vortex burst towards the apex) Acceleration in the freestream velocity appeared to have the opposite effect that of pushing the burst location forward A second trait observed during the slower pitch-up tests was that the degree of scatter in the data increased In some cases there were large fluctuations along the time axis where the forward propagation accelerated In other cases slightly different curves (one for each experiment performed) were traced each having identical slope but separated from its neighbor by a few fractions of a second This indicated that at the slower pitching rate the experiment was more sensitive to either external noise (such as unavoidable vibrations in the carriage being transmitted to the delta wing) or to something instrinsic within the flow phenomenon The experiment as it was performed could not isolate the cause of this variability References 1 Polhamus E ldquoA Concept of the Vortex Lift of Sharp-Edge Delta Wings Based on a Leading-Edge
Suction Analogyrdquo NASA TN D-3767 Dec 1966 2 Earnshaw PB and Lawford JA ldquoLow-Speed Wind Tunnel Experiments on a Series of Sharp-Edged
Delta Wingsrdquo Aeronautical Research Council R M No 3424 1964 3 Erickson GE ldquoWater-Tunnel Studies of Leading-Edge Vorticesrdquo AIAA Journal of Aircraft Vol 19
No 6 1982 pp 442-448 4 LeMay WP Batill SM and Nelson RC ldquoVortex Dynamics on a Pitching Delta Wingrdquo AIAA
Journal of Aircraft Vol 27 No 2 1990 pp 131-138 5 Miller LS and Gile BE ldquoEffects of Blowing on Delta Wing Vortices During Dynamic Pitchingrdquo
AIAA Journal of Aircraft Vol 30 No 3 1993 pp 334-339 6 Rediniotis O K Klute S M Hoang N T and Telionis D P ldquoDynamic Pitch-Up of a Delta Wingrdquo
AIAA Journal Vol 32 No 4 1994 pp 716-725 7 Myose RY Hayashibara S Yeong PC and Miller LS ldquoEffect of Canards on Delta Wing Vortex
Breakdown during Dynamic Pitchingrdquo AIAA Journal of Aircraft Vol 32 No 2 1997 pp 168-173 8 Al-Garni AZ Ahmed SA Sahin AZ and Al-Garni AM ldquoAn Experimental Study of a 65-Degree
Delta Wing with Different Pitching Ratesrdquo Canadian Aeronautics and Space Journal Vol 47 No 2 2001 pp 85-93
9 Skow A M ldquoAn Analysis of the Su-27 Flight Demonstration at the 1989 Paris Airshowrdquo SAE Paper 90-1001 1990
10 Kolano E ldquoFlying the Flankerrdquo Flight Journal httpflightjournalcomarticlessu27Su27_5asp 11 Ericsson L E ldquoEffect of Fuselage Geometry on Delta-Wing Vortex Breakdownrdquo AIAA Journal of
Aircraft Vol 35 No 6 Nov 1998 12 Schaeffler N W and Telionis D P ldquoApex Flap Control of Dynamic Delta Wing Maneuversrdquo AIAA
Paper 96-0662 1996 13 Heron I and Myose R Y ldquoDevelopment of a Dynamic Pitch and Unsteady Freestream Delta Wing
Mount For A Water Tunnelrdquo AIAA Paper 2003-3527 June 2003
American Institute of Aeronautics and Astronautics 7
14 Heron I and Myose R Y ldquoVortex Burst Behavior Under Dynamic Freestreamrdquo AIAA Paper 2005-0063 January 2005
15 Heron I and Myose R Y ldquoDelta Wing Vortex Burst Behavior Under Dynamic Freestream Part 1 ndash Fast Pitch-up During Decelerationrdquo AIAA paper 2007-6725 August 2007
16 Johnson BL ldquoFacility Description of the Walter H Beech Memorial 7 x 10 foot Low-Speed Wind Tunnelrdquo AR93-1 National Institute for Aviation Research Wichita State University June 1993
Figure 1 Schematic diagram of Wichita State 2- by 3-feet water tunnel (left)16 and photograph of test section showing the towing system with an inverted delta wing mounted on carriage (right)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 30 - 50 deg
Figure 2 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 8
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-36
Figure 3 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 45 - 55+ deg
Figure 4 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 9
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc (s
= 0
at A
pex)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-38
Figure 5 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 30 to 40 degFig 5-39
Figure 6 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (40 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 10
Figure 7 Velocity Components on Leading Edge of Delta Wing
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 04 to 02 fts
dV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 8 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Deceleration (Ideal Representation of Fig 2rsquos Deceleration Case)
Half-apex angle ζ = 90 - Λ Angle of Attack α
Uinfin
Angle of Yaw β
Sweep Angle Λ
VS = Uinfin sin α cos β
VLE = Uinfin (cos α cos ζ cos β - sin ζ sin β)
VLE = Uinfin (cos α cos ζ cos β + sin ζ sin β)
VLE = Uinfin (cos α cos β sin ζ - sin β cos ζ)
VLE = Uinfin (cos α cos β sin ζ + sin β cos ζ)
ζ
American Institute of Aeronautics and Astronautics 11
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
01
012
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 02 to 04 ftsdV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 9 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Acceleration (Ideal Representation of Fig 2rsquos Acceleration Case)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 30 - 50 deg
k=02 slopek=01 slope
Figure 10 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
American Institute of Aeronautics and Astronautics 12
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 45 - 55+ deg k=02 slopek=01 slope
Figure 11 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Vel change from 30-50 degFig 5-42
k=02 slopek=01 slope
Figure 12 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 7
14 Heron I and Myose R Y ldquoVortex Burst Behavior Under Dynamic Freestreamrdquo AIAA Paper 2005-0063 January 2005
15 Heron I and Myose R Y ldquoDelta Wing Vortex Burst Behavior Under Dynamic Freestream Part 1 ndash Fast Pitch-up During Decelerationrdquo AIAA paper 2007-6725 August 2007
16 Johnson BL ldquoFacility Description of the Walter H Beech Memorial 7 x 10 foot Low-Speed Wind Tunnelrdquo AR93-1 National Institute for Aviation Research Wichita State University June 1993
Figure 1 Schematic diagram of Wichita State 2- by 3-feet water tunnel (left)16 and photograph of test section showing the towing system with an inverted delta wing mounted on carriage (right)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 30 - 50 deg
Figure 2 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 8
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-36
Figure 3 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 45 - 55+ deg
Figure 4 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 9
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc (s
= 0
at A
pex)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-38
Figure 5 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 30 to 40 degFig 5-39
Figure 6 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (40 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 10
Figure 7 Velocity Components on Leading Edge of Delta Wing
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 04 to 02 fts
dV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 8 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Deceleration (Ideal Representation of Fig 2rsquos Deceleration Case)
Half-apex angle ζ = 90 - Λ Angle of Attack α
Uinfin
Angle of Yaw β
Sweep Angle Λ
VS = Uinfin sin α cos β
VLE = Uinfin (cos α cos ζ cos β - sin ζ sin β)
VLE = Uinfin (cos α cos ζ cos β + sin ζ sin β)
VLE = Uinfin (cos α cos β sin ζ - sin β cos ζ)
VLE = Uinfin (cos α cos β sin ζ + sin β cos ζ)
ζ
American Institute of Aeronautics and Astronautics 11
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
01
012
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 02 to 04 ftsdV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 9 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Acceleration (Ideal Representation of Fig 2rsquos Acceleration Case)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 30 - 50 deg
k=02 slopek=01 slope
Figure 10 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
American Institute of Aeronautics and Astronautics 12
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 45 - 55+ deg k=02 slopek=01 slope
Figure 11 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Vel change from 30-50 degFig 5-42
k=02 slopek=01 slope
Figure 12 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 8
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-36
Figure 3 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Accl 02 to 04 fts - Exp 4Const Speed k=02Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 45 - 55+ deg
Figure 4 Burst Comparison Pitch Up at κ=02 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 9
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc (s
= 0
at A
pex)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-38
Figure 5 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 30 to 40 degFig 5-39
Figure 6 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (40 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 10
Figure 7 Velocity Components on Leading Edge of Delta Wing
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 04 to 02 fts
dV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 8 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Deceleration (Ideal Representation of Fig 2rsquos Deceleration Case)
Half-apex angle ζ = 90 - Λ Angle of Attack α
Uinfin
Angle of Yaw β
Sweep Angle Λ
VS = Uinfin sin α cos β
VLE = Uinfin (cos α cos ζ cos β - sin ζ sin β)
VLE = Uinfin (cos α cos ζ cos β + sin ζ sin β)
VLE = Uinfin (cos α cos β sin ζ - sin β cos ζ)
VLE = Uinfin (cos α cos β sin ζ + sin β cos ζ)
ζ
American Institute of Aeronautics and Astronautics 11
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
01
012
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 02 to 04 ftsdV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 9 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Acceleration (Ideal Representation of Fig 2rsquos Acceleration Case)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 30 - 50 deg
k=02 slopek=01 slope
Figure 10 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
American Institute of Aeronautics and Astronautics 12
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 45 - 55+ deg k=02 slopek=01 slope
Figure 11 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Vel change from 30-50 degFig 5-42
k=02 slopek=01 slope
Figure 12 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 9
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc (s
= 0
at A
pex)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Fig 5-38
Figure 5 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Accl 03 to 04 fts - Exp 4Const Speed k=02Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 30 to 40 degFig 5-39
Figure 6 Burst Comparison Pitch Up at κ=02 velocity ratio of 075 Acceleration Deceleration from initial speed (30 deg) to final speed (40 deg)
dvdt asymp 0046 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 10
Figure 7 Velocity Components on Leading Edge of Delta Wing
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 04 to 02 fts
dV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 8 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Deceleration (Ideal Representation of Fig 2rsquos Deceleration Case)
Half-apex angle ζ = 90 - Λ Angle of Attack α
Uinfin
Angle of Yaw β
Sweep Angle Λ
VS = Uinfin sin α cos β
VLE = Uinfin (cos α cos ζ cos β - sin ζ sin β)
VLE = Uinfin (cos α cos ζ cos β + sin ζ sin β)
VLE = Uinfin (cos α cos β sin ζ - sin β cos ζ)
VLE = Uinfin (cos α cos β sin ζ + sin β cos ζ)
ζ
American Institute of Aeronautics and Astronautics 11
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
01
012
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 02 to 04 ftsdV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 9 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Acceleration (Ideal Representation of Fig 2rsquos Acceleration Case)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 30 - 50 deg
k=02 slopek=01 slope
Figure 10 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
American Institute of Aeronautics and Astronautics 12
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 45 - 55+ deg k=02 slopek=01 slope
Figure 11 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Vel change from 30-50 degFig 5-42
k=02 slopek=01 slope
Figure 12 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 10
Figure 7 Velocity Components on Leading Edge of Delta Wing
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 04 to 02 fts
dV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 8 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Deceleration (Ideal Representation of Fig 2rsquos Deceleration Case)
Half-apex angle ζ = 90 - Λ Angle of Attack α
Uinfin
Angle of Yaw β
Sweep Angle Λ
VS = Uinfin sin α cos β
VLE = Uinfin (cos α cos ζ cos β - sin ζ sin β)
VLE = Uinfin (cos α cos ζ cos β + sin ζ sin β)
VLE = Uinfin (cos α cos β sin ζ - sin β cos ζ)
VLE = Uinfin (cos α cos β sin ζ + sin β cos ζ)
ζ
American Institute of Aeronautics and Astronautics 11
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
01
012
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 02 to 04 ftsdV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 9 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Acceleration (Ideal Representation of Fig 2rsquos Acceleration Case)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 30 - 50 deg
k=02 slopek=01 slope
Figure 10 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
American Institute of Aeronautics and Astronautics 12
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 45 - 55+ deg k=02 slopek=01 slope
Figure 11 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Vel change from 30-50 degFig 5-42
k=02 slopek=01 slope
Figure 12 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 11
42383432622181410602
-008
-006
-004
-002
0
002
004
006
008
01
012
10 15 20 25 30 35 40 45 50 55 60
Angle of Attack α (deg)
C
hang
e in
Vel
ocity
at L
eadi
ng E
dge
(fts
^2)
000
005
010
015
020
025
030
035
040
045
Vel
ocity
(fts
)
Time (sec)at κ= 02
dUinfin dt sin(α)
Uinfindα dt cos(α)
V perpendicular to LE =Uinfin sin(α) cos(β)
Uinfin = 02 to 04 ftsdV dt = dUinfindt sin α + Uinfin dαdt cos α cos β First Term = Second Term =
Figure 9 Example of Perpendicular Velocity at Leading Edge Constant Pitch-Up Rate
and Acceleration (Ideal Representation of Fig 2rsquos Acceleration Case)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Vel change from 30 - 50 deg
k=02 slopek=01 slope
Figure 10 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
American Institute of Aeronautics and Astronautics 12
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 45 - 55+ deg k=02 slopek=01 slope
Figure 11 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Vel change from 30-50 degFig 5-42
k=02 slopek=01 slope
Figure 12 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 12
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3Decl 04 to 02 fts - Exp 4
Vel change from 45 - 55+ deg k=02 slopek=01 slope
Figure 11 Burst Comparison Pitch Up at κ=01 velocity ratio of 05 Acceleration Deceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Const Speed k=02Const Speed k=01Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Vel change from 30-50 degFig 5-42
k=02 slopek=01 slope
Figure 12 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (30 deg) to final speed (50 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 13
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3Decl 04 to 03 fts - Exp 4
Vel change from 45 - 55+ deg
Figure 13 Burst Comparison Pitch Up at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (45 deg) to final speed (beyond 55 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-44
Figure 14 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (40 deg)
dvdt asymp 0023 fts2
dvdt asymp 0092 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
dvdt asymp 0046 fts2
dvdt asymp 0023 fts2
American Institute of Aeronautics and Astronautics 14
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)Accl 02 to 04 fts - Exp 1Accl 02 to 04 fts - Exp 2Accl 02 to 04 fts - Exp 3Decl 04 to 02 fts - Exp 1Decl 04 to 02 fts - Exp 2Decl 04 to 02 fts - Exp 3
Fig 5-45
Figure 15 Burst Comparison Pitch Down at κ=01 velocity ratio of 05 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
00
01
02
03
04
05
06
07
08
09
10
0 1 2 3 4 5 6 7 8 9 10Elapsed Time (s)
Bur
st L
ocat
ion
sc
(s=
0 at
Ape
x)
Accl 03 to 04 fts - Exp 1Accl 03 to 04 fts - Exp 2Accl 03 to 04 fts - Exp 3Decl 04 to 03 fts - Exp 1Decl 04 to 03 fts - Exp 2Decl 04 to 03 fts - Exp 3
Fig 5-46
Figure 16 Burst Comparison Pitch Down at κ=01 velocity ratio of 075 AccelerationDeceleration from initial speed (50 deg) to final speed (30 deg)
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