Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, January 1996A9618735, AIAA Paper 96-0780

The critical allowable height of a backward-facing step

Jamal A. MasadHigh Technology Corp., Hampton, VA

AIAA 34th Aerospace Sciences Meeting and Exhibit, Reno, NV Jan 15-18, 1996

The critical allowable height of a roughness element on a surface is defined as the minimum height at which transitiononset location begins to move upstream. A numerical criterion is presented for defining the critical allowable height ofa backward-facing step on a flat plate. The critical allowable height of the step has a nonzero value only when the stepis located before branch I of the most amplified frequency of flow over a smooth surface. Variations of the criticalallowable height of a backward-facing step with freestream Mach and Reynolds numbers, and the slope of the step arequantified. It is found that the critical allowable height of the step increases by increasing the freestream Mach number,by decreasing the unit Reynolds number, by decreasing the slope of the step, and by moving the step upstream.(Author)

Page 1

AIAA-96-0780

The Critical Allowable Height of a Backward-Facing Step

Jamal A. MasadHigh Technology Corporation

Hampton, VA 23666

Abstract

The critical allowable height of a roughness el-ement on a surface is defined as the minimumheight at which transition onset location begins tomove upstream. A numerical criterion is presentedfor defining the critical allowable height of a back-ward-facing step on a flat plate. The critical allow-able height of the step has a nonzero value onlywhen the step is located before branch I of the mostamplified frequency of flow over a smooth surface.Variations of the critical allowable height of a back-ward-facing step with freestream Mach andReynolds numbers, and slope of the step are quanti-fied. It is found that the critical allowable height ofthe step increases by increasing the freestreamMach number, by decreasing the unit Reynoldsnumber, by decreasing the slope of the step, and bymoving the step upstream.

1. IntroductionThe critical allowable height of a roughness el-

ement is defined as the minimum height at whichtransition onset location begins to move upstream.It follows from this definition that if the height ofthe roughness element is lower than the critical al-lowable height then the transition onset location isthe same as that in flow over a smooth surface. Fornatural laminar flow (NLF) and laminar flowcontrol (LFC) applications, there is a need for accu-rate determination of critical allowable heights ofunavoidable roughness elements such as steps andgaps at junctions.

An experimental criterion for predicting allow-able two-dimensional (2-D) roughness elements in 2-D flow is that of Carmichael.1 Canniehael's crite-rion is valid for a roughness element in the form of asingle wave or multiple waves. For a single wave,Carmichael's criterion is given by

*•* r -|0.5' 59000cA* A* Re™ (1)

where k is the double-amplitude wave height, A isthe wavelength, c is the chord length and Rec is thefreestream chord's Reynolds number. The value ofk*l £ given by equation (1) determines the criticalallowable value of the surface wave. Carmichael1defined this critical value of the wave as the mini-mum k /A which prevents the attainment of lami-nar flow to the trailing edge. The data base pointsof Carmichael's criterion include effects of compress-ibility, suction, and pressure gradient, however,these effects are not represented in the criterion.Furthermore, the wave-type roughness elementwhich is the subject of Carmichael's criterion haslittle importance in real applications.

The work on criteria for the allowable dimen-sions of steps and gaps came from the X-21 experi-ments2 and the T-34C experiments.3 The criteriabased on the X-21 experiments determined thestep/gap critical Reynolds number Refacritical =

k*U^/v* where k* is the height of the step orlength of the gap at which the first turbulent burstsoccurred far downstream from step/gap, Ut, is thefreestream streamwise velocity, and v* is the fluidkinematic viscosity. It is clear that the Rek&itical cr^~teria do not account for the important effect of thelocation of the step on the surface. Furthermore,the Re^critiaai f°r a gap determines the critical gap'slength independent of both the gap's height (or ef-fective height) and its location on the surface.

Flight experiments on a T-34C were conducted3

to develop an RekiCriticai criterion for a rounded for-ward-facing step (FFS) in 2-D flow. The used stepwas located at the 5-percent chord location of thelower surface. The height of the FFS was 0.027in.and the rounding has a 0.02-in. radius. The condi-tions flown resulted in an Re^criticai °f about 2700.However, this value is valid only for the consideredstep location.

Research Scientist, Senior Member AIAA

Co^yrigxtO 1996 by the American Institute of Aeronautics andAstronautics, Inc. All right* reserved.

A correlation of transition for 2-D roughnesselements in 2-D flow is that of Fage.4 Page's corre-lation is for low-speed 2-D flow over a single rough-ness element on a flat plate or an airfoil. Theroughness elements considered in the experimentaldata base of Page's correlation include smoothbulges, smooth hollows, flat ridges, and archedridges. In the experimental data base of Page's cor-relation, the freestream Reynolds number based onthe distance from the leading edge to the center ofthe roughness element varied from 0.5 million to 1million, and in this range, results of the e^ methodagreed5'6-7 well with the results of the correlation ofFage. However, Page's correlation is a tool fortransition prediction in the presence of relativelylarge-height roughness elements rather than a toolfor predicting the critical allowable dimensions ofroughness elements.

A widely used empirical criterion for transitionin flow over a roughness element is the Rek crite-rion. In this criterion Re/, is given by

Ret=k= *'<£ (2)

where k is the roughness element's height, Uk isthe flow streamwise velocity at height k* in absenceof the roughness element, and v*k is the fluid kine-matic viscosity at height k* in absence of the rough-ness element. It follows from equation (2) that in-creasing k increases Rek. Increasing the flow unitReynolds number increases Uk in equation (2)which increases Rek. Increasing the freestreamMach number thickens the boundary layer and,therefore, reduces U*k which results in a reducedRek. Moreover, moving the roughness element up-stream in the region of the leading edge decreasesUk which decreases Rek. However, it is clear fromequation (2) that the Rek criterion doesn't accountfor the length of the roughness element. The Rekcriterion is more used for three-dimensional rough-ness elements and in boundary-layer tripping appli-cations rather than for predicting the critical allow-able height of a 2-D roughness element.Furthermore, even in boundary-layer tripping appli-cations, the experimental values of Rek show a sig-nificant scatter.

Dovgal and Kozlov8 conducted a controlled(forced) experiment on the influence of 2-D stepsand humps on the stability of low-speed flow over aflat surface. A vibrating ribbon was placed up-stream of a roughness element to introduce 2-Dsmall-amplitude disturbances into the developing

boundary layer. In the steps cases, the center of thestep was located 500mm downstream of the leadingedge of the plate, the freestream velocity was 6 m/s,the freestream Reynolds number based on the dis-tance from the leading edge to the center of the stepwas 2 x 105, and FFS and backward-facing steps(BFS) were considered. Two step heights were used: 0.9 and 2.2mm and the vibrating ribbon was ex-cited by three different frequencies f * : 60, 76, and94 Hz. Dovgal and Kozlov8 reported the streamwisevariation of the integrated growth rate (2V-factor) ofthe instability wave. Comparisons9 of the results ofa combination of interacting boundary layer (IBL)theory and linear stability theory (LST) with the ex-perimental data of Dovgal and Kozlov8 demon-strated very good agreement for some of the cases.The heights, of the steps considered in the experi-ments of Dovgal and Kozlov are large compared tothe critical allowable heights. For example, the IBLtheory predicts flow separation in the presence of aFFS or a BFS of a 2.2mm height. Furthermore, theobjective of the experiment of Dovgal and Kozlovwas to study the stability and receptivity character-istics of flow over a roughness element rather thanquantifying the movement of transition onset loca-tion with varying the height of a roughness element

2. Formulation and Methods of SolutionWe consider a two-dimensional compressible

subsonic flow over a BFS on a flat plate. We use theIBL theory which accounts for the viscous-inviscidinteraction to analyze the flow field. (See Davis.10)The accuracy of IBL code that we use in this workhas been validated in an earlier work11 by compar-ing its results with results from a Navier-Stokessolver for subsonic flow over a step. The equationfor the step is given by

y=>

180k

where erf is the error function and

(3)

(4)

U^, is the dimensional freestream velocity, v^ is thedimensional freestream fluid's kinematic viscosity,

L* is the distance from the leading edge of the plateto the center of the step and x = x*lL* where x* isthe distance measured from the leading edge of theplate. The nondimensional height k is equal to&*/L* where k* is the dimensional height of thestep. The slope of the step is s which is the smallangle (in degrees) between the flat surface and thestep's surface, it is positive for a FFS and negativefor a BFS.

In stability analysis, small, unsteady two-di-mensional disturbances are superimposed on thecomputed mean-flow quantities. For subsonic flowwith freestream Mach number up to 0.8 we onlyconsider 2-D disturbances. Mack10 showed that themost amplified waves in 2-D flow with a free-stream Mach number AfM of less than approxi-mately 0.8 are 2-D waves. Next, the total quantitiesare substituted into the Navier-Stokes equations,the equations for the basic state are subtracted out,the quasi-parallel assumption is invoked, and theequations are linearized with respect to the distur-bance quantities. The disturbance quantities areassumed to have the normal-mode form, so that adisturbance quantity q is

Q. = ~a^ + complex conjugate (5)

The streamwise coordinate is x, t is the time, a anda> are generally complex. In the stability analysisand the computations throughout this work, the ref-erence length is 5* =^vl**/LC , the reference veloc-ity is t£, the reference time is S*/U^, the refer-ence temperature is the free-stream temperatureT^, the reference viscosity is the free-stream dy-namic viscosity [£,, and the pressure is madenondimensional with respect to p*JU^ (where pi isthe free-stream density). The viscosity varies withtemperature in accordance with Sutherland's for-mula; the specific heat at constant pressure C*p isconstant, and the Prandtl number is constant andequal to 0.72. For temporal stability, a is real, and<9 = <Br+£(»,• is complex, where the real part car isthe disturbance frequency and its imaginary part o^is the temporal growth rate. For the spatial stabil-ity of the 2-D flow considered in this work, <o is real,and a = ar +10^ is complex, where the real part aris the streamwise wave number and the negative ofthe imaginary part -c^ is the spatial growth rate.The frequency co is related to the dimensional circu-lar frequency <o* through a> = at*5* fU^, which leads(with the definition of 5*) to

F = o r " or

(6)

(7)

f is the dimensional circular frequency in cps (Hz)and

(8)

(9)

Because a? and /* are fixed for a certain physicalwave as it is convected downstream, F is also fixedfor the same wave.

The normal-mode form given in equation (5)separates the streamwise and temporal variations.The resulting equations and boundary conditionsform an eigenvalue problem that can be solved nu-merically. For the results presented in this work,the computations were performed with an adaptivesecond-order-accurate finite-difference scheme withdeferred correction.13

By solving the linear instability eigenvalueproblem, we obtain the disturbance-wave growthrate as a function of location on the surface. Thetransition onset location is then empirically corre-lated with the location at which the integratedgrowth rate (2V factor) of the disturbance wavereaches a certain value. This is the empirical JV-fac-tor transition criterion proposed by Smith andGamberoni14 based on experimental data. (See alsoJaffe et al.15) For flow over a flat plate, transitionwas found to occur when the N factor reached avalue close to 9; we use this value throughout thiswork to correlate the location of transition onset.We denote the value of Rex at which the N factorreaches a value of 9 by (Rex)Nm9-

HI. Results

A. Stability and Transition in Flow Over a StepWe start by presenting the streamwise distri-

bution of the pressure coefficient Cp in flow over aBFS. We consider incompressible flow over a BFSwith Re - 106. The slope of the BFS is s = -5 itsheight is k = 0.005 and it is centered at x = 1. The

streamwise variation of Cp for the separating flowover the BFS is shown in figure 1. The verticaldashed lines are the streamwise boundaries of theseparation bubble. Far upstream and far down-stream of the step, the pressure coefficient ap-proaches that of the flow over a smooth flat plate;hence we expect that the stability characteristics inthese regions approach those of the flow over asmooth flat plate. We have three regions of pres-sure gradient; a short favorable-pressure-gradientregion, followed by a strong adverse-pressure-gradi-ent region that could cause the flow to separate, andanother region of favorable pressure gradient.Consequently, the step will have a stabilizing influ-ence in the favorable-pressure-gradient regions anda destabilizing influence in the adverse-pressure-gradient region.

In Figure 2, we compare the neutral stabilitycurve for the flow over a flat plate without a rough-ness element with that for the flow in the presenceof a BFS. Far away form the step, the neutral curveapproaches that of the Blasius flow, as expected.The first region of favorable pressure gradient di-vides the unstable region into two regions. Thesecond favorable-pressure-gradient region reducesthe instability. Moreover, the strong adverse pres-sure gradient, which is responsible for separation,causes part of the unstable region to extend over awide band of frequencies. The unstable high-fre-quency disturbances in the flow over the step are astrong indication of the invistid nature of the insta-bility in the separation region. This instability issimilar to mixing and shear-layer instabilities. Inour search for branch I of the neutral stability curvein the adverse-pressure-gradient region of figure 2,we encountered an interval of R in which the stabil-ity code converged on negative frequencies, whichmeans that in this interval the flow in unstable re-gardless of how small is the frequency. The wavenumbers (Figure 2b) associated with these unstablelow-frequency disturbances are very small.

Next, we present results which demonstratethe differences between the effects of a BFS and aFFS on transition. We performed IBL and LST cal-culations at AC = 0.8, a freestream Reynoldsnumber of Re = 1.8 x 106 and steps slopes of +5.Variations of the predicted transition onsetReynolds number using the eN method with thesteps height k are compared in Figure 3. The con-sidered location of the steps places them down-stream of branch I of the most-amplified frequencyin flow over a smooth flat plate. Therefore, there is

no critical allowable height at these conditions (thecritical height is zero) and the transition onset loca-tion begins to move upstream as soon as the step'sheight exceeds the zero value. It is clear from figure3 that at relatively low heights of the steps, the BFStriggers transition more than the FFS. At moderateheights of the steps, the FFS triggers transitionmuch more than the BFS. At large heights wherethe transition onset location saturates to a valueclose to the location of the steps (tripping heights),the BFS and FFS result in close transition onsetReynolds numbers. The filled circles in figure 3 in-dicate that the flow separated and then reattachedwhereas the hollow circles indicate that the flowremained attached. The circles indicate the pointsat which actual calculations were made and thecircles are connected by straight lines. Note infigure 3 that at large heights of the steps, the pre-dicted transition onset Reynolds numbers saturateto almost a constant value which is close to theReynolds number value based on L*; the distancefrom the leading edge to the center of the step.

B. The Critical Allowable Height of 3 BFSFor a small height roughness element with Re

(equation 4) less than the Reynolds number at thefirst neutral point of the most amplified frequency,the roughness element has a destabilizing effect inits vicinity, however, any accumulated growth dueto the roughness element gets canceled by the sub-sequent decay in the region between the end of theroughness element and the first neutral point of themost amplified frequency. This is demonstrated inFigures 4 and 5 which show, respectively, thestreamwise variations of growth rate and N factorfor incompressible flow over a BFS with height k =0.0005. It is clear from figure 4 that the first ampli-fied region which is due to the adverse pressuregradient created by the BFS gets completely can-celed by the subsequent decay between the end ofthe step and the first neutral point. This results ina negative value for the N factor which is reset tozero (Figure 5). Therefore, for a small step heightand low values of Re, the transition Reynoldsnumber is the same as that of flow over a smoothsurface. Note that at the same Re, increasing theroughness element's height extends its accumulateddestabilizing effect closer and closer to the firstneutral point. At a certain height of the roughnesselement the accumulated growth due to the rough-ness element will not be canceled completely by thesubsequent decay. The roughness element's height

at which the sum of the accumulated growth andsubsequent decay just becomes positive at the firstneutral point is taken as the critical allowableheight of the roughness element. The streamwisevariations of growth rates and N factors of instabil-ity wave in the presence of such critical height areshown in figures 6 and 7, respectively. In figure 6,the critical height for the incompressible flow over aBFS with slope s = -4 and at Re = 5 x 106 is k =0.00077. Note in figure 7 that the N factor justreaches zero at the end of the damped region be-tween the end of the step and the first neutral pointof the most amplified frequency. For step heightslarger than the critical height, the growth due to thepresence of the step overcomes the subsequent decaybetween the end of the step and the first neutralpoint of the most-amplified frequency (figure 8).This results in a positive net value for the N factorat the first neutral point (figure 9).

Results of the predictions of our numericaldefinition of the critical allowable height are shownin figure 10. The results in figure 10 show variationof the predicted transition onset Reynolds numberwith the height of the BFS. The variation is charac-terized by an almost constant transition onsetReynolds number up to the critical height at whichthe predicted transition onset Reynolds numberdrops at a relatively high rate. The result of our nu-merical criterion for the critical height of the BFS isshown by the filled circle in figure 10. It is clearthat the numerical criterion picks accurately the ac-tual critical allowable height.

For roughness elements such as steps and gaps,the critical allowable height depends on the step's orgap's location, its streamwise extent, and on theflow freestream Mach and unit Reynolds numbers.If the combination of roughness location at V andflow freestream unit Reynolds number result in avalue of Re larger than the Reynolds number at thefirst neutral point of the most amplified frequency,then, there is no allowable height (the allowableheight is zero) and any nonzero height will movetransition onset location upstream. Therefore, thevalue of the parameter Re compared to the value ofthe Reynolds number at the first neutral point ofthe most amplified frequency of flow over a smoothsurface constitutes a very important considerationin regard to the critical allowable roughness ele-ment's height. This consideration was severelyoverlooked in the development of experimental cri-teria for allowable roughness elements heights.

We used our numerical criterion for definingthe critical allowable height of a BFS to quantify thevariation of such height kcru with the step's slope,freestream Reynolds number Re, and freestreamMach number Mn in the subsonic range. Note thatthe effect of Re (equation 4) is a combination of twoeffects; the effect of the freestream unit Reynoldsnumber and the effect of the step's location.Variations of &m-t with Re for four step slopes areshown in figures 11-13 for AC = 0, 0.4, and 0.8. Itis clear from these figures that increasing the slopeof the step decreases its critical allowable height.However, at relatively large values of Re the effectof the step's slope on kerit decreases. At Af» = 0(figure 11) and .Re = 0.5 x 106, the effect of the slopeon kcrit is very small and kcrit almost saturates to aconstant value regardless of the step's slope.Figures 11-13 demonstrate clearly that decreasingRe increases the critical allowable height of a BFS.This means that decreasing the unit Reynoldsnumber and moving the BFS upstream result inlarger values for kcrit. The effect of freestream Machnumber Mm on &cr# is shown clearly in figure 14 fors = -6. Increasing M^ increases kcrjt. It can be seenfrom comparing figures 11-13 with each other thatthis is also the case at other values of s. This resultis consistent with the overall stabilizing effect ofcompressibility in flow over a roughness element.7

References

1. B. H. Carmichael, "Surface Waviness Criteriafor Swept and Unswept Laminar Suction Wings,"Northrop Aircraft Report No. NOR-59-438 (BLC-123), 1957.

2. Anon, "Final Report on LFC Aircraft DesignData Laminar Flow Control DemonstrationProgram," Northrop Corp. NOR-67-136 (availablefrom DTIC as AD 819 317), 1967.

3. B. J. Holmes, C. J. Obara, G. L. Martin, and C.S. Domack, "Manufacturing Tolerances," in LaminarFlow Aircraft Certification, NASA CP-2413, pp. 171-183,1986.

4. A. Fage, " The Smallest Size of SpanwiseSurface Corrugation Which Affects Boundary LayerTransition on an Airfoil," British AeronauticalResearch Council 2120,1943.

5. A. H. Nayfeh, S. A. Ragab, and A. A. Al-Maaitah, "Effect of Bulges on the Stability ofBoundary Layers," Physics of Fluids, Vol. 31, pp.796-806,1988.

6. T. Cebeci, and D. A. Egan, "Prediction ofTransition Due to Isolated Roughness," AIAAJournal, Vol. 27, pp. 870-875,1989.

7. J. A. Masad and V. lyer, "Transition Predictionand Control in Subsonic Flow Over a Hump,"Physics of Fluids, Vol. 6, No. 1, pp. 313-327, 1994.

8. A. V. Dovgal and V. V. Kozlov, "HydrodynamicInstability and Receptivity of Small ScaleSeparation Regions," in Laminar-TurbulentTransition (D. Arnal, and R. Michel, eds.), Springer-Verlag, Berlin, 1990.

9. J. A. Masad and A. H. Nayfeh, "The Influenceof Imperfections on the Stability of SubsonicBoundary Layers," in Instabilities and Turbulencein Engineering Flows (D. E. Ashpis, T. B. Gatski,and R. Hirsh, eds.), Kluwer Academic Publishers,pp. 65-82, 1993.

10. R. T. Davis, "A Procedure for Solving theCompressible Interacting Boundary LayerEquations for Subsonic and Supersonic Flows,"AIAA Paper No. 84-1614,1984.

11. S. A. Ragab, A. H. Nayfeh, and R. C. Krishna,"Stability of Compressible Boundary Layers Over aSmooth Backward- and Forward-Facing Step,"AIAA Paper No. 90-1449,1990.

12. L. M. Mack, "Boundary-Layer StabilityTheory," Jet Propulsion Laboratory, Pasadena, CA,Document No. 900-277 (Rev. A), 1969.

13. V. Pereyra, "PASVA3: An Adaptive Finite-Difference Fortran Program for First-OrderNonlinear Ordinary Boundary-Value Problems,"Lect. Notes Comput. Sd., Vol. 76,1976, p. 67.

14. A. M. O. Smith, and N. Gamberoni, "TransitionPressure Gradient, and Stability Theory," DouglasAircraft Co. Inc. Report No. ES 25388, El Segundo,CA, 1956.

15. N. A. Jaffe, T. T. Okamura, A. M. O. Smith,"Determination of Spatial Amplification Factors andtheir Application to Predicting Transition," AIAA J.,Vol. 8, No. 2,1970, pp. 301-508.

0.025 -

0.000 -

•0.025 -

750 1000 1250

R

10°x 250

200

150

100 -

50

(a)

500 1000 1500 2000

R

Figure 1. Streamwise distribution of pressure co-efficient for incompressible flow over a BFS with k =0.005, s = -5, and Re = 106.

Figure 2. Neutral stability curves for 2-D distur-bances in incompressible flow over a BFS with k =0.005, s = -5. and Re = 106. (a) frequency -Reynoldsnumber domain and

25<

0.50 r

0.40 -

0.30 -

0.20 -

0.10 -

0.00

(b)

500 1000 1500 2000 2500

R

= 1.8x10-5 , +5

_L0.002 0.004

k

0.006

2(b). streamwise wavenumber-Reynolds number Figure 3.Variation of predicted transitiondomain. Reynolds number with the height of a backward-

and a forward-facing step on a flat plate.

0.010

0.005

"S 0.000

s -0.005

-0.010

-0.015500 1000

R1500 2000

0.20

0.10

0.00

k = 0.0005c/te = 5x10£

s = -4M =0

600 800

R1000

Figure 4. Streamwise variation of growth rate for Figure 5. Streamwise variation of N factor for thea step's height less than the critical height. growth rates of figure 2.

0.010 r

0.005

0.000

•0.005

•0.010

-0.015

1.0 r

500 1000

R1500 2000

-0.5 -

-1.0600 1200

Figures. Streamwise variation of growth rate for Figure 7. Streamwise variation of N factor for thea step's height equal to the critical height. growth rates of figure 4.

<§0.015 r

0.010

0.005

0.000

-0.005

-0.010

-0.015500 1000

R1500 2000 600 800 1000

R1200 1400

FigureS. Streamwise variation of growth rate for Figure 9. Streamwise variation of N factor for thea step's height larger than the critical height growth rates of figure 6.

1()6X3.30

3.25

o\<•;

3.15

3.10

3.05

3.00

s= -4

_!_ i

vcrif

0.0050

0.0040

0.0030

0.0020

0.0010

0.0000 0.0002 0.0004 0.0006

k

0.0008 0.00100.0000

0.30 0.35

M =0

0.40 0.45

Re0.50 o

X 1

Figure 10. Variation of predicted transition Figure 11. Variation of the critical height of a BFSReynolds number with the height of a backward-fac- with freestreaffi Reynolds number at M.. = 0.ing step on a flat plate. The filled circle indicatesthe prediction of our criterion for the critical allow-able height.

"crit

0.015

0.010

0.005

.,=-3

JkL = 0.4

0.010 r

0.008 -

0.006 -

0.004 -

0.002 -

AiL. = 0.8

0.30 0.35 0.40 0.45 0.50

Re0.55

x106

0.0000.30

Figure 12. Variation of the critical height of a BFS Figure 13. Variation of the critical height of a BFSwith freestream Reynolds number at Afw = 0.4. with freestream Reynolds number at Af« = 0.8.

0.008

0.006

0.004

0.002

0.000

s = -6

_i—t—i—u.0.30 0.35 0.40 0.45 0.50 0.55

Re x10c

Figure 14. Variation of the critical height of a BFSwith freestream Reynolds number of slope s = -6.

10