Experimental and Computational Steady and UnsteadyTransonic Flows about a Thick Airfoil
Lionel L. Levy Jr.*NASA Ames Research Center, Moffett Field, Calif.
An experimental and computational investigation of the steady and unsteady transonic flowfields about athick airfoil is described. An operational computer code for solving the two-dimensional, compressible Navier-Stokes equations for flow over airfoils was modified to include solid-wall, slip-flow boundary conditions toproperly assess the code and help guide the development of improved turbulence models. Steady and unsteadyfiowfieids about an 18% thick circular arc airfoil at Mach numbers of 0.720, 0.754, and 0.783 and a chordReynolds number of 11 x 106 are predicted and compared with experiment. Results from comparisons withexperimental pressure and skin-friction distributions show improved agreement when including test-section wallboundaries in the computations. Steady-flow results were in good quantitative agreement with experimental datafor flow conditions which result in relatively small regions of separated flow. For flows with larger regions ofseparated flow, improvements in turbulence modeling are required before good agreement with experiment willbe obtained. For the first time, computed results for unsteady turbulent flows with separation caused by a shockwave were obtained which qualitatively reproduce the time-dependent aspects of experiments. Features such asthe intensity and reduced frequency of airfoil surface-pressure fluctuations, oscillatory regions of trailing-edgeand shock-induced separation, and the Mach number range for unsteady flows were all qualitatively reproduced.
NomenclatureA = Van Driest damping lengthCf = skin-friction coefficient, Tw/qQOCp = pressure coefficient^-/;)/^c = airfoil chorde — internal energy/ = reduced frequency, 2?r (frequency) (c/2)/u^t - mixing length of turbulenceM — Mach numbern,t = local coordinates normal and tangential to test-
section wallA/? = normal distance between image point and test-
section wallp = pressurePi = total pressureAp = instantaneous p—mean value of pq = total velocity, (u2 + v 2 ) ' / 2
q^ = freestream dynamic pressure, (!/2)p„ u29
R — radius of curvatureReCt0a = chord Reynolds number, P^^^C/JJL^T = temperature/ =timeutv = velocity components in x and y directions,
respectivelyx,y =axes parallel and normal, repectively, to airfoil
chord with origin at airfoil leading edgeyDS = location of separation streamline7 = ratio of specific heats<5 = thickness of viscous region€ = eddy viscosity coefficientp = molecular viscosityT = shear stress\l/ = angle whose tangent is the local slope of the test-
Index categories: Computational Methods; Nonsteady Aero-dynamics; Transonic Flow.
* Research Scientist. Member AIAA.
Subscriptsb = wall boundary point outside of test section/ = image point inside o f test sectionoo = freestream valuen,t = components normal and tangential to test-section
wallw = airfoil surface or test-section walld = value at outer edge of viscous region
I. Introduction
THE NASA Ames Research Center has initiated combinedresearch programs in experimental and computational
fluid dynamics for testing and guiding the development ofturbulence modeling within regions of separated flows.!
Toward this end, the flowfield about an 18% thick circulararc airfoil is being experimentally documented over a range ofMach and Reynolds numbers for which the major features offlow separation important to turbulence modeling arepresent, including weak and strong shock-wave boundary-layer interactions and both trailing-edge and shock-inducedboundary-layer separation. Results to date for pressure andskin-friction distributions have beem compared in Refs. 2 and3, respectively, with computed results from a two-dimensional code for time-dependent solutions of Reynolds-averaged, compressible Navier-Stokes equations.4 Thecomputer code includes additional equations for turbulencemodeling and applies only to free-flight conditions; it doesnot include test-section wall boundary conditions. Con-sequently, comparisons of experimental and computed resultsmade in Refs. 2 and 3 are considered to show qualitativelycorrect trends. Before different 'turbulence models can bequantitatively tested against the pressure and skin-frictiondata from these references, the proper outer wall boundaryconditions must be incorporated into the computer code.
This paper describes the modifications to the computercode of Refs. 4 and 5 to include solid-wall inviscid boundaryconditions. Results are presented which show differences inpressure and skin-friction distributions over an 18% thickcircular arc airfoil, calculated by using free-flight boundaryconditions and tfre solid-wall test-section boundaries of theexperiments reported in Refs. 2 and 3. Experimental resultsare included for comparison.
JUNE 1978 COMPUTATIONAL TRANSONIC FLOWS ABOUT A THICK AIRFOIL 565
As reported in Ref. 2, both steady and unsteady flows wereobserved for specific ranges of freestream Mach numbers andchord Reynolds numbers. For example, for a given chordReynolds number, there was observed a low subsonic Machnumber range (above that for locally sonic flow on the airfoil)and a high subsonic Mach number range for which the localflowfieid was steady. In the intermediate Mach number range,the local flowfieid was unsteady and periodic.
The original intent in applying the modified computer codewith wall boundary conditions was to obtain a solution forcomparison with experiment at freestream conditions withineach steady-flow regime near the boundaries,of the unsteady-flow regime. As noted in Ref. 2, definition of the low Machnumber boundary of the unsteady-flow regime is subject tohysteresis in the flowfieid. The initial selection of freestreamconditions for a solution in the low Mach number steady-flowregime was inadvertently made in the region of hysteresis.Surprisingly, an unsteady flowfieid solution was obtained.The results are included in this paper and demonstrate, for thefirst time, the capabilities of the computer code to reproducequalitatively the time-dependent aspects of experimentallyobserved unsteady turbulent flows with both weak and strongshock-wave boundary-layer interactions. Features importantto airplane buffet such as oscillations in shock-wave locationand strength and alternate regions of trailing-edge and shock-induced boundary-layer separation are reproduced.
II. Analysis
Simulation MethodThe transonic flowfieid about the airfoil was simulated
numerically using a form of the computer program describedin Ref. 5. The program utilizes an explicit finite-differencemethod6"8 to solve the time-dependent, two-dimensional,Reynolds averaged form of the Navier-Stokes equationsapplicable to compressible turublent flows. The turbulence isassumed in equilibrium with the mean flow and is modeledusing an algebraically expressed eddy viscosity model. Thecomputer program also incorporates the efficient numericalsolver for Navier-Stokes equations described in Ref. 9. Theflow and turbulence-modeling equations are repeated here forcompleteness.
The flow equations in time-dependent form are
- <7dvol+Ot
f H-nds =^5
where
u= pu
pv
q == uex + vev
Expressions for the eddy viscosity coefficient in the regionsof the boundary layer and wake indicated in Fig. 1 are listedbelow.
Inner region (I)
e=0.41y[l-exv(-y/A)]
where the Van Driest damping length is
*>* ^W ( Pw=26—— (-—- )^ Ir /
Outer region (II) of boundary layer and wake
f)T1=0.09 (yt-
4- o
In region II over the separated bubble (regions III and IV),£ is frozen at the value evaluated at the first computationalchordwise station upstream of separation.
Separation bubble wall region (III)
/ y-yw \pe=(pe)DS(——— }\yDs~yw '
Separation bubble wake region (IV)
pe=(pe)D S
The control volume, mesh, and boundary conditions aredescribed in detail in Ref. 5. Briefly, the airfoil, initially atrest, is impulsively started at time zero at the desiredfreestream Mach number and pressure. The control volume,- 12 and + 8 chords in the* direction and ± 6 chords in the ydirection, is divided into a 78x35 mesh. The flowfieiddevelopment within this volume is followed in time until itattains a steady state. For the present code, airfoil, andfreestream conditions, this requires a time equivalent to themean flow traveling about 9 chord lengths past the airfoil(w00/7e=9). At the far upstream and transverse boundaries,the flow is assumed uniform and at freestream conditions. Atthe downstream boundary, all gradients in the flow directionare assumed negligible. The airfoil is assumed impermeable(no-slip boundary conditions) and adiabatic, and the pressuregradient normal to the surface is assumed zero.
Modifications to Include Tunnel-Wall Boundary ConditionsUse of the experimental data of Refs. 2 and 3 to assess
existing turbulence models and help develop improved modelsnecessitated the choice of solid-wall boundary conditions.Computer storage and time significantly influenced thedecision that the initial effort be based upon inviscid slip-flowboundary conditions.
The quantities ax and ov are the normal stresses in thex and ydirections, respectively, ?x, "ev are unit vectors in orthogonalxty space, and n is a unit normal vector to the surface elementds about the volume element vol.
The turbulence modeling is incorporated in the shear stressterms rxv and ryx in the form of an eddy viscosity coefficient eas
Fig. 1 Boundary-layer and wake regions for turbulence model.
566 L.L. LEVY JR. AIAA JOURNAL
To minimize the modifications, the wall boundaries aresuperimposed upon the computational mesh used to obtainfree-flight solutions. Computations with the walls are madefor the entire mesh. However, at each of the computationalmesh points just outside the wall boundaries, the dependentvariables (horizontal and vertical components of velocity,internal energy, and pressure) are updated at each time step tosatisfy the inviscid boundary conditions. This procedure isonly slightly inefficient, since approximately 90% of thecomputational time is used in resolving the airfoil boundary-layer flow. Inclusion of the wall boundaries results in an 8%increase in computer time over that required for a free-flightsolution.
In the present approach the coordinates, slope, and localwail curvature of the test section studied are used as inputs tothe modified computer program. The program then finds allcomputational mesh points (mesh centroids) just outside theboundaries. Next, the image points of these boundary cen-troids are constructed normal to and inside the test sectionflowfield. Values of the dependent variables at the meshpoints in the flowfield which most closely surround eachimage point are used for interpolation to obtain values of thedependent variables at the image points. These values of thedependent variables, at each time step, are used in con-junction with boundary conditions to continually update thedependent variables at the boundary centroids. Specifically,the horizontal and vertical components of velocity at theboundary points, uh and vb, respectively, are determined suchthat, locally, the total velocity q is tangent to the walls (i.e.,no flow through the walls). With reference to Fig. 2 theseconditions are written
qnf)=-qn.; q»w = 0 (2)
From Eqs. (1) and (2) and the slope of the wall, it can beshown that
(3)
(4)
The internal energy eb at each boundary point is readilydetermined using the assumption of an adiabatic wall. Thus,
eb=€j =ew (5)
The pressure/?/, at the boundary points is determined from thenormal momentum equation
ub = iij ( cos2 \l/ — sin2 \// ) +2Vj sin\l/
vb = 2uj sin^ cos^ — y/ (• cos2 \l/ — sin2 0 )
( dpdn
=_d n / w R (6)
Equation (6) can be rewritten by discretizing the partialderivative, making use of the equation of state, and ex-pressing*?^ as a function of the known quantities, uh V;, and
(y-l)ewRw(7)
Finally, by assuming that pw = l/i(pb +p,), and with use ofEq. (5), the pressure at each boundary centroid is given by
_ 1-BPb — , , ,-, Pi
where
B =
(8)
(9)
WALL
Fig. 2 Flowfieid and geometrical relationships for wali boundaryconditions.
and
(10)
To verify that the numerical algorithm for the wallboundary conditions was correct, two streamlines from asolution of a lifting airfoil in free flight were selected as upperand lower "walls" of a test channel. A solution was thenobtained at identical freestream conditions for the airfoil inthe presence of these walls. Within the accuracy of thenumerical method, the flowfield about the airfoil should bethe same for both solutions. This is demonstrated to be thecase by the pressure distributions on the airfoil surfaces andalong the free-flight streamlines and channel walls shown inFig. 3.
III. Results and DiscussionThe upper and lower test-section walls used in the ex-
periments reported in Refs. 2 and 3 were contoured to theshape of streamlines computed using an inviscid code.Streamlines used were one-half of the channel height awayfrom the chordline of an 18% thick circular arc airfoil at afreestream Mach number M^ of 0.775. These walls werediverged slightly to compensate for channel-wall boundary-layer growth and were unchanged for tests at all Mach andReynolds numbers. Thus, many of the experimental data ofRefs. 2 and 3 were obtained at off-design conditions. Con-sequently, use of these experimental data to test the validity ofturbulence modeling requires that the computer code containthe boundary conditions of the experiment. The actual wallcoordinates of the experimental apparatus were used. Allsubsequent results are for an 18% thick circular arc airfoil at0 deg angle of attack, and only results for half the flowfieldare presented (with the exception of computed Mach contoursof the entire flowfield). To avoid numerical difficulties, allcomputed results were obtained for an 18% thick circular arcairfoil with 1 % chord nose radius.
Computed Effect of Wa!I BoundariesFigure 4 shows the actual channel walls and streamlines,
through the beginning of the contoured portion of the walls,obtained from free-flight viscous solutions atM^ =0.783 and0.720 and a chord Reynolds number Rec^ of 11 x 106. Notethat an expanded ordinate is used to emphasize the wall andstreamline shapes. The channel walls are a reasonablerepresentation of the streamlines from the A/a, =0.783,
./tec,<x> =11 x 106 solution. Consequently, these freestreamconditions constitute a near-design test condition, and theresults in Fig. 5 for free-flight and wall boundary conditionsdemonstrate that the respective differences between pressureand skin-friction distributions are small. Comparison of
JUNE 1978 COMPUTATIONAL TRANSONIC FLOWS ABOUT A THICK AIRFOIL 567
UPPERSURFACE
.2 .8.4 .6x/c
a) AIRFOIL SURFACE PRESSURES
CP 0
b} STREAMLINE AND WALL-BOUNDARY PRESSURES
Fig. 3 Calculated pressure distributions for a lifting airfoil in freeflight and in a test channel with walls contoured to the shape of free-flight streamlines; Korn-Garabedian airfoil, A/oo = 0.756,Recoo = 2 ix l0 6 , a = -1.54deg.
a) PRESSURE
b) SKIN FRICTION
Fig. 5 Computed pressure and skin-friction distributions on thecircular arc airfoil with free-flight and channel wall boundary con-ditions, M^ -0.783, Recoo = II X 106, a = 0deg.
Fig. 4 Channel wall and streamlines through beginning coordinateof wall for the 18% thick circular arc airfoil, a = 0 deg.
either set of computed results in Fig. 5 with experiment toassess the adequacy of the computer code and turbulencemodel would lead to essentially the same conclusions for thisnear-design case.
For the off-design conditions of Mx= 0.720 andRec>00 = 11 x 106, the results in Fig. 4 show that the actualchannel walls provide more "open area" about the airfoilthan would exist had the walls been contoured to the shape ofstreamlines obtained from the free-flight viscous solution.Consequently, compared to free-flight results, the pressurefield computed about the airfoil would be less severe whenincluding the actual channel-wail boundaries. This isdemonstrated by the computed pressure and skin-frictiondistributions shown in Fig. 6. These results indicate thenecessity for a computer code with proper boundary con-ditions when using experimental results to assess the adequacyof a code and turbulence model. Obviously, comparisons atoff-design conditions between experimental data andsolutions with channel walls would lead to a differentassessment of the code and turbulence model than wouldsimilar computations using free-flight boundary conditions.
CHANNEL WALLSFREE FLIGHT
a) PRESSURE
-.0010 .2 .4 .6 .8 1.0
x/c
b) SKIN FRICTION
Fig. 6 Computed pressure and skin-friction distributions on thecircular arc airfoil with free-flight and channel wall boundary con-ditions, A/^ = 0.0720, Recoo = 11 X 106 a = 0 deg.
Comparisons with ExperimentAs noted in the Introduction, during the experiments
reported in Ref. 2 both steady and unsteady flows were ob-served for specific ranges of freestream Mach numbers andchord Reynolds numbers. Figure 7 has been reproduced from
Fig. 7 Experimental flow domains for the 18% thick circular arcairfoil.
b) Unsteady flow, qscsliaiory separation
r) Steady HJ.JW, shock-induced separation
Fig. 8 Boundary-layer separation on the circular arc airfoil from ;shadowgraph movie.
Ref. 2 to identify the Reynolds number and Mach numberdomains within which three distinctively different types offlow have been documented for the 18% thick circular arcairfoil. The steady flow in the low Mach number domain wascharacterized by a weak shock wave near midchord andtrailing-edge boundary-layer separation. This flow picture isdepicted by the left-most sketch in Fig. 7. A more detailedpicture of the trailing edge is illustrated by a selected framefrom a shadowgraph movie, reproduced from Ref. 10, in Fig.8a. The unsteady flow domain shown in Fig. 7 includes theregion of flow hysteresis in the lower Mach number rangenoted earlier. The flow in this domain was characterized byperiodic oscillations in shock-wave location and intensity andconcomitant oscillations between trailing-edge and shock-induced separation of the boundary layer. This flowphenomenon is depicted by the middle sketches in Fig. 7 and
a) Mx = 0.720
b) MM = 0.754
Cp
O i ! 1 I i I I.2 .4 .6
x/cc) Mx = 0.783
1 1.8 1.
Fig. 9 Computed and experimental pressure distributions on thecircular arc airfoil, ReCQO = 11 x 106.
the movie frames in Fig. 8b. In the high Mach number,steady-flow domain, a shock wave occurs near the 70% chordstation sufficiently strong to induce boundary-layerseparation at the base of the shock wave; see the right-mostsketch in Fig. 7 and movie frame in Fig. 8c. Indicated by thesquare symbols in Fig. 7 are three sets of freestrearn con-ditions, M^ = 0.720, 0.754, 0.783 andReCi„ = 11 x 106, one ineach flow domain, for which computer solutions with wallboundaries have been obtained for comparisons withavailable experimental data.
The flowfields in the two steady flow domains, in actuality,are pseudo-steady. Specifically, the shock-wave location andintensity, the point of boundary-layer separation, and thepressure and skin-friction distributions do experience smallexcursions about respective mean values in both the ex-periment and the computations. The amplitudes of theseexcursions are an order of magnitude or more smaller than inthe unsteady flow domain. This will become apparent indiscussions of subsequent results.
Experimental pressure distributions from Ref. 2 and thepresent computed results are shown in Fig. 9 for the three sets
JUNE 1978 COMPUTATIONAL TRANSONIC FLOWS ABOUT A THICK AIRFOIL 569
of freestream conditions just noted. The symbols denote themean values of the experiment, and the bars on the symbols atx/c = 0.25, 0.50, and 0.775 represent maximum and minimumvalues of fluctuations about the mean; they do not denote theuncertainty of the measured mean values. (The pressurefluctuations at x/c = Q.25 were obtained by McDevitt. J 3 jMean values from the computations are denoted by the solidlines and the range of fluctuations by the shaded areas. Forthe steady-flow cases at M^ =0.720 and 0.783, Figs. 9a and9c, respectively, the computed mean values are the averagesover a time equivalent to several chord lengths of mean flowtravel beyond that for which the solution is normally con-sidered to have converged to a steady state. Computed resultsfor the unsteady-flow case at M^ ^0.754, Fig. 9b, are fromtime histories of four consecutive cycles of flowfieldoscillation (approximately 32 chord lengths).
The steady flowfield at M^ ^0.720 is characterized-by aweak shock wave and trailing-edge separation (see Figs. 7 and8a). The computed results in Fig. 9a are in good agreementwith experiment over most of the airfoil. Failure of theseresults to better predict the pressures in the separated regionnear the trailing edge is attributed to inadequate turbulencemodeling in this region. The amplitudes of the measuredpressure fluctuations at x/c = 0.25 and 0.50 lie within thesymbols and are not shown. The mean results for the com-putations shown in Fig. 9a are the same results shown in Fig.6a for the channel walls. A comparision of the computed free-flight pressure distribution shown in Fig. 6a with the ex-perimental data in Fig. 9a would indicate poor agreement inthe range 0.45 <x/c< 0.70. In this case, one woulderroneously suspect deficiencies in the computer code and/orturbulence model. The results of these contrasting com-parisons emphasize the necessity for computer codes with theproper boundary conditions.
The steady flowfield at M^ =0.783 is characterized by astrong shock wave and shock-induced separation (see Figs. 7and 9c). The computed results in Fig. 9c (and Fig. 5a) are inexcellent agreement with experiment ahead of the shock wave(x/c - 0.675). The large differences between the computed andexperimental mean results in the region of the shock wave andaft in the region of shock-induced separation again are at-tributed to inadequate turbulence modeling. In the separatedregion at x/c = 0.775, it is interesting to note the goodagreement between the computed and measured magnitudesof the pressure fluctuations. Here, also, the fluctuationmagnitudes at x/c = 0.25 and 0.50 are within the symbols.
The unsteady flowfield at M^ =0.754 is characterized byperiodic shock-wave oscillations and oscillations in boundary-layer separation between the trailing-edge and shock-inducedtype (see Figs. 7 and 8b). The calculated and experimentalmean pressures agree well over the forward half of the airfoil.The similarity in the trends of the variation of the magnitudeof pressure fluctuations about the mean value stronglysuggests the possibility that the wave form of the experimentalpressure fluctuations also may be reproduced by thecalculations. This is shown to be the case in Fig. 10. Ex-perimental and computed time histories of the instantaneouspressure oscillations about the mean pressure (normalized bythe channel total pressure) on the upper and lower airfoilsurfaces at two chord stations are reproduced from Ref. 10.The qualitative agreement between the different wave formsat two chordwise stations is surprisingly good considering thatthe computed unsteady results were obtained using simplealgebraic eddy viscosity to model turbulence. The 180-degphase difference between the dynamic pressures on upper andlower airfoil surfaces at identical chord stations demonstratethat the oscillatory unsteadiness is an asymmetricphenomenon, both in the experiment and in the com-putations. The reduced frequency of the surface pressureoscillations determined from the numerical solution, / = 0.40,differs from that reported in Ref. 2 (/ = 0.49) only by about20%.
REF. 10UPPER SURFACE
EXPERIMENT
COMPUTATION 0
-.2LOWER SURFACE
EXPERIMENT
COMPUTATION 0
-.2
0 9 18 27CHORDS TRAVELED
0 9 18 27CHORDS TRAVELED
Fig. 10 Computed and experimental surface-pressure lime historieson the circular arc airfoil, Mx = 0.754, Recoo = 11X106.
c ) STEADY FLOW, SHOCK-INDUCED SEPARATION. N^ - 0.783
CHORDS TRAVELED
Fig. 11 Mach contours in the flow field about the circular arc airfoilfrom a computer movie.
Further evidence that Navier-Stokes type computer codeswith simple turbulence models reproduce shock-wave andboundary-layer spearation features for steady flows and arecapable of qualitatively reproducing the time-dependentaspects of these features for unsteady flows is presented inFig. 11. Selected frames illustrating the three distinctivelydifferent types of flow just discussed are reproduced from acomputer movie of Mach contours. The elapsed timecorresponds to the chord lengths traveled by the mean flowduring one cycle of the oscillatory flow shown in Fig. lib.Coalescence of the near-vertical contours over the latter halfof the airfoil indicates the formation of a shock wave—thecloser the contours, the stronger the shock wave. Thehorizontal contours aft of the shock waves and in the wakeare in the outer region of the boundary layer and wakedenoted as region II in Fig. 1.
A careful comparison of Fig. 11 with the shadowgraphs inFig. 8 reveals marked similarities between the computed andexperimental flowfield features for both steady and unsteadyflows. Figures 8a and l la depict steady flow, wi th 'a weakshock wave and trailing-edge separation. Figures 8c and lieshow a strong shock wave with a large region of flowseparation emanating from the base of the shock wave. Theflow similarities between Figs. 8b and l ib for the unsteady-flow case further demonstrate that the oscillatory un-steadiness is an asymmetric phenomenon. For example, the
570 L. L. LEVY JR. AIAA JOURNAL
shock wave forms near the trailing edge just above a region oftrailing-edge separation (note the upper surface in the secondframe in Fig. l ib and the t = 0.62 ms frame in Fig. 8b). Theshock wave increases in strength as the local airfoil surfacevelocities ahead of the shock increase. The increased strengthgives rise to shock-induced separation, and the shock waveand separated region begin to move forward. The localsurface velocities upstream of the shock continue to increaseand stabilize in a maximum velocity distribution. As the shockcontinues forward into a region of locally lower velocities, itdiminishes in strength and vanishes as the separation pointreverts to the trailing edge to complete the cycle. Meanwhile,the identical process is occurring on the lower surface 180 degout of phase.
b) Moo = 0.754
O EXPERIMENT, REF. 3
——— COMPUTED
.4 -6x/c
Computed and experimental skin-friction distributions areshown in Fig. 12 for the three flow conditions of interest. Thecomputed results were obtained from the same sets offlowfield data described earlier in discussing the pressuredistributions. It is not surprising, therefore,, that the trends inthe computed magnitudes of the skin-friction fluctuations(shaded areas) about the mean values (solid lines) are similarto those noted for the pressure fluctuations. Experimentaldata are available from Ref. 3 only for Mx =0.783. Thedistribution of mean values of skin friction is shown in Fig.12c. As in the case of the pressure distribution (see Fig. 9c),the agreement between the computed and measured values isgood ahead of the shock wave. The poor agreement indefining the shock-wave location and aft in the separated-flow region is again attributed to deficiencies in the turbulencemodel.
As discussed in Ref. 3, the skin-friction gages used respondto a skin-friction parameter which is a combination of theflow quantities ILWPWTW. Because fluctuating pressuremeasurements were not recorded simultaneously with thefluctuating skin-friction data, it was not possible to determinethe magnitudes of the maximum and minimum excursions inskin friction from dynamic records of jnH,pM,7M, (pw ~pw).Therefore, only the calculated and experimentally deducedvalues of the root-mean-square variations in the unsteadyskin-friction parameter are compared here. The rms valuesshown in Fig. 13 have been normalized by a value near themidchord ahead of the shock wave. The sign of the ordinate inFig. 13 corresponds to that of the local mean value of the skinfriction. Just as these results for mean skin-friction show, thecomputed unsteady skin-friction results indicate the shock tobe further downstream than in the experiment. However, thetrends showing a rapid increase in dynamic activity ahead ofthe shock wave and an increased magnitude in the fluc-tuations downstream in the shock-induced separation regionare similar.
The present results comprise a substantial amount ofcircumstantial evidence which indicates, for the first time,that Navier-Stokes type computer codes are capable ofreproducing the time-dependent aspects of experimentallydocumented unsteady turbulent flows with shock-inducedseparation. However, just as the experimenter alwaysquestions whether unsteady flows are caused by someresonant tunnel phenomena, so the numerical analystnaturally inquires if computed unsteady-flow results arecaused by the .wall boundary conditions, numerical inac-curacies, or computer-code asymmetries. The experimenterdemonstrates that the unsteady flow is aerodynamically in-duced by using records of the aerodynamically differentairfoil and wind-tunnel frequencies (see, e.g., Refs. 2 and 11)and by testing airfoils of different chord lengths and ob-serving a constant airfoil reduced frequency (see, e.g., Refs.11 and 12). The numerical analyst can make similar
—O— EXPERIMENT, REF. 3— O—COMPUTED
c) Mw = 0.783
Fig. 12 Computed and experimental skin-friction distributions onthe circular arc airfoil, Rec ^ = 11 x 106.
Fig. 13 Computed and experimental unsteady skin-frictionparameters for the circular arc airfoil, M^ = 0.783, Reccx = 11X 106.
JUNE 1978 COMPUTATIONAL TRANSONIC FLOWS ABOUT A THICK AIRFOIL 571
EXPERIMENT, ReCr00 = 8x106
COMPUTED, ReCoo = 11 x 106
Fig. 14 Computed and experimental pressure distributions on thecircular arc airfoil with a quarter-chord trailing-edge splitter plate,M = 0.754.
demonstrations in addition to checks for numerical accuracyand asymmetry.
The computed unsteady flows were determined not to havebeen induced by including the channel walls as boundaryconditions. This was determined from the results of a solutionfor the present airfoil at M^ =0.754 and Recoo = 11 x 106 infree flight. A periodic unsteady-flow solution was obtained.The reduced frequency and magnitudes of the airfoil surfacepressure oscillations were slightly different from similarresults for the solution including the channel walls and did notagree as well with experiment. Similar to the approach takenin Refs. 11 and 12, a solution with wall boundaries was ob-tained at the present unsteady-flow free-stream conditions forthe same airfoil with one-half the present chord. An unsteady-flow solution was obtained and the frequency of the airfoilsurface pressure oscillations was twice that for the longerchord airfoil; thus the reduced frequency was constant at/=0.40.
A cursory investigation of the unsteady-flow solutions hasrevealed that they are, in fact, a result of viscous phenomenaand not a result of numerical inaccuracies. For example,inviscid computations for the same M^ produced a steady-flow solution. A viscous solution for a half airfoil atM^^0.754 and Rec<x = llxl06 produced a steady-flowsolution. The half-model boundary conditions, however,preclude communication of pressure waves across the airfoilwake and provide a solid boundary upon which the separatedboundary layer can reattach. A similar, more physicallyrealistic test is to experimentally and computationally reducesuch communication across the wake and provide a solidboundary by using a trailing-edge splitter plate of sufficientlength to produce a steady flow. It was determined ex-perimentally that a one-quarter-chord trailing-edge splitterplate stopped the unsteady flow at all Reynolds numbers forwhich unsteady flow occurred without the splitter plate atMx =0.754 (see Fig. 7).!3 Viscous computations for the sameairfoil and splitter plate at M^ =0.754 and ReC}00 = 11 x 106
produced a steady-flow solution. The computed and ex-perimental airfoil pressure distributions are shown in Fig.14.13 The agreement is very good except near the airfoiltrailing edge. These results demonstrate that the present codeand turbulence model can predict the flow field about anairfoil with a large pressure jump (shock wave) and trailing-edge separation. The disagreement in the small separatedregion near the trailing edge is again attributed to inadequateturbulence modeling in this region.
A careful examination of the present computer coderevealed one obvious coding error which introduced a largenumerical asymmetry and two subtle coding errors that
possibly may introduce small numerical asymmetries. Thecomputer code was corrected for the coding error which hadintroduced the large numerical asymmetry, the present un-steady-flow case was re-run, and essentially the same un-steady-flow solution was obtained. Correcting the code forthe two subtle coding errors has proven more time consumingthan originally anticipated. Since eliminating the possiblesources of small numerical asymmetries should not alter thequalitative aspects of the agreement with experiment, thecomputed results obtained to date are being published withoutdelay.
Even though there is strong circumstantial evidence that thepresent computed unsteady-flow results qualitativelyreproduce the physical features of similar real flows, untilsimilar results are obtained with a computer code with nonumerical asymmetries, the present computed unsteady-flowresults should be considered preliminary.
IV. Concluding RemarksA successful initial effort has been made in including wall
boundary conditions in a computer code for solving the two-dimensional, compressible Navier-Stokes equations for flowover airfoils, albeit slip-flow wall boundary conditions wereused. While inclusion of the wall boundary conditions doesprovide improved agreement with experimental results, it doesnot alter the conclusion of earlier investigators that improvedturbulence models are required before existing codes cancorrectly predict the flow features characteristic of strongshock-wave boundary-layer interactions with relatively largeregions of separated flow. Present results indicate that, withthe availability of computer codes with proper boundaryconditions, an improved tool is now in hand for quantitativelytesting different kinds of turbulence models against data fromtwo-dimensional, fully documented experiments.
Evidence is also in hand which shows (for the first time tothe author's knowledge) that Navier-Stokes type computercodes are capable of reproducing the time-dependent aspectsof unsteady turbulent flows involving weak and strong shock-wave boundary-layer interactions. Present computed resultsindicate that the intensity of airfoil surface-pressure and skin-friction fluctuations, the reduced frequency of pressurefluctuations, oscillatory regions of trailing-edge and shock-induced separation, and the Mach number range for unsteadyflows can be qualitatively reproduced. These results inspireconfidence that once turbulence models are developed withwhich the improved computer codes can predict experimentalsteady-flow results, the codes can then be used to study thetime-dependent aspects of unsteady flows and hence provideinsight into unsteady aerodynamic phenomena such asbuffeting, inlet buzz, and rotating helicopter blades.
References1 Marvin, J. G., "Experiments Planned Specifically for Developing
Turbulence Models in Computations of Flow Fields AroundAerodynamic Shapes," AGARD-CP-210, Oct. 1976.
2McDevitt, J. B., Levy, L. L. Jr. and Deiwert, G. S., "TransonicFlow About a Thick Circular-Arc Airfoil," AIAA Journal, Vol. 14,May 1976, pp. 606-613.
3Rubesin, M. W., Okuno, A. F., Levy, L. L. Jr., McDevitt, J. B.,and Seegmiller, H. L., "An Experimental and Computational in-vestigation of the Flow Field About a Transonic Airfoil in Super-critical Flow with Turbulent Boundary-Layer Separation/' 10thCongress of the International Council of the Aeronautical Sciences,Ottawa, Canada, Oct. 4-8, 1976; also NASA TM X-73,157, July 1976.
4 Deiwert, G. S., "Computation of Separated Transonic TurbulentFlows," AIAA paper 75-829, Hartford, Conn., June 1975.
5 Deiwert, G. S., "On the Prediction of Viscous Phenomena inTrankonic Flows/' Project SQUID Workshop on Transonic FlowProblems in Turbomachinery, Monterey, Calif., Feb. 1975.
6Deiwert, G. S., "Numerical Simulation of High ReynoldsNumber Transonic Flows," AIAA Journal, Vol. 13, Oct. 1975, pp.1354-1359.
572 L. L. LEVY JR. AIAA JOURNAL
7Deiwert, G. S., "High Reynolds Number Transonic FlowSimulation," Lecture Notes in Physics, Vol. edited by Springer-Verlag, 1975, p. 132.
8Baldwin, B. S., MacCormack, R. W., and Deiwert, G. S.,"Numerical Techniques for the Solution of the Compressible Navier-Stokes Equations and Implementation of Turbulence Models,"AGARD-LSP-73, March 1975.
9MacCormack, R. W., "An Efficient Numerical Method for Solv-ing the Time-Dependent Compressible Navier Stokes Equations atHigh Reynolds Number," NASATM X-73,129, July 1976.
IOSeegmil!er, H. L., Marvin, J. G., and Levy, L. L. Jr., "Steadyand Unsteady Transonic Flow," AIAA Paper 78-160, Huntsville,Ala., Jan. 1978.
!iPolentz, P. P., Page, W. A., and Levy, L. L,, Jr., "UnsteadyNormal-Force Characteristic of Selected NACA Profiles at HighSubsonic Mach Numbers," NACA RM A55C02, May 1955.
12Finke, K., "Unsteady Shock-Wave Boundary-Layer Interactionon Profiles in Transonic Flow," AGARD-CPP-168, Paper 28, May1975.
13McDevitt, J. B., Private communication on NASA Ames ex-periment, 1977.
From the AIAA Progress in Astronautics and Aeronautics Series..
Edited by Leroy S. Fletcher, University of Virginia
The science and technology of heat transfer const i tute an established and well-formed discipline. Al though one wouldexpect relatively l i t t l e change in the heat transfer field in view of its apparent m a t u r i t y , it so happens that new develop-ments are taking place rapidly in certain branches of heat transfer as a result of the demands of rocket and spacecraftdesign. The established "textbook" theories of radiation, convection, and conduction simply do not encompass the un-derstanding required to deal wi th the advanced problems raised by rocket and spacecraft condit ions. Moreover, researchengineers concerned with such problems have discovered that it is necessary to clarify some fundamental processes in thephysics of matter and radiation before acceptable technological solutions can be produced. As a resu l t , these advancedtopics in heat transfer have been given a new name in order to characterize both the fundamental science involved and thequant i t a t ive nature of the investigation. The name is Thermophysics. Any heat transfer engineer who wishes to be able tocope wi th advanced problems, in heat transfer, in radiat ion, in convection, or in conduction, whether for spacecraft designor for any other technical purpose, must acquire some knowledge of this new field.
Volume 59 and Volume 60 of the Series offer a coordinated series of original papers representing some of the latestdevelopments in the field. In Volume 59, the topics covered arc 1) The Aerothcrmal Envi ronment , par t icu la r lyaerodynamic heating combined with radiation exchange and chemical reaction; 2) Plume Radiation, wi th special referenceto the emissions characteristic of the jet components; and 3) Thermal Protection Systems, especially for intense heat ingconditions. Volume 60 is concerned wi th : 1) Heat Pipes, a widely used but rather in t r ica te means for in te rna l temperaturecontrol; 2) Heat Transfer, especially in complex situations; and 3) Thermal Control Systems, a description of sophisticatedsystems designed to control the flow of heat within a vehicle so as to maintain a specified temperature environment.
Volume 59—432 pp. ,6x9, ill us. $20.00 Mem. $35.00 ListVolume 60—398pp., 6 x 9, ///MS. 520.00 Mem. $35.00 List
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