new approach for the calculation of transitional flows
TRANSCRIPT
AIAA JOURNALVol. 31, No. 4, April 1993
New Approach for the Calculation of Transitional Flows
T. Wayne Young* and Eric S. Warren*North Carolina State University, Raleigh, North Carolina 27695
Julius E. HarristNASA Langley Research Center, Hampton, Virginia 23681
andH. A. HassanJ
North Carolina State University, Raleigh, North Carolina 27695
In spite of many attempts at modeling natural transition, it has not been possible to predict the streamwiseintensities. A procedure is developed that incorporates some results of linear stability theory into one-equationand stress model formulations. The stresses resulting from fluctuations in the transitional region have turbulent,laminar (nonturbulent), and large eddy components. Comparison with Schubauer and Klebanoff s experimentshave shown that the nonturbulent and large eddy components have a large influence on the streamwiseintensities and little influence on the skin friction. Finally, predictions of the one-equation model were as goodas those obtained by the stress model.
Introduction
T HE region of transition from laminar to turbulent flowis, in some ways, the least understood and the most
important region of fluid flow. In this region, skin frictionand heat transfer coefficients change quite rapidly and attainvalues that exceed those in the fully turbulent region. In gen-eral, there are three modes of transition. The first mode isnatural transition, which is a result of the Tollmien-Schlicht-ing (TS) mode of instability. Bypass transition, followingMorkovin,1 is caused by large disturbances in the mean flow.The third mode is the separated flow transition that occurs inseparated laminar flows. This investigation deals with naturaltransition.
Models designed to predict transitional flows have beenreviewed by Arnal,2 by Narasimha,3 and more recently byScheuerer.4 A number of the models make use of the intermit-tency factor 7, which is defined as the fraction of the time thatthe flow is turbulent at a given point.
Attempts at predicting streamwise intensities were made by,among others, Jiang et al.,5 who used a one-equation model,and by Arnal and Juillen,6 who employed a k - e model. InRef. 5, empirical modifications were made to the eddy viscos-ity, the Karman constant, and the length scale in the outerregion. Comparisons with measured streamwise intensitiespointed to discrepancies in the near wall and outer regions ofthe boundary layer, particularly for stations central to thetransition region. The predictions of Ref. 6 were somewhatpoor.
The aim of this investigation is to develop a new approachbased on the Navier-Stokes equations to model transitionalflows. Two models are employed. The first is a one-equationeddy viscosity model that makes use of the turbulent kineticenergy equation. The second is an abbreviated stress model inwhich additional equations are solved for the normal stresses.
Received April 29, 1992; presented as Paper 92-2669 at the AIAA10th Applied Aerodynamics Conference, Palo Alto, CA, June 22-24,1992; revision received Sept. 4, 1992; accepted for publication Sept. 5,1992. Copyright © 1992 by the American Institute of Aeronautics andAstronautics, Inc. All rights reserved.
*Research Assistant, Mechanical and Aerospace Engineering De-partment. Student Member AIAA.
t Senior Research Scientist, Theoretical Flow Physics Branch, FluidDynamics Division. Associate Fellow AIAA.
JProfessor, Mechanical and Aerospace Engineering Department.Associate Fellow AIAA.
Both of the approaches do not require specification of anyspecial initial conditions or length scales. They do require spec-ification of freestream intensities.
The present investigation addresses the more challengingcase of natural transition. In particular, emphasis is placed onmodeling the Schubauer and Klebanoff7 experiment. The startof transition was specified according to the experiment. Anexpression for intermittency was obtained from Dhawan andNarasimha.8 The experimental freestream intensity was 0.03percent.
Formulation of the ProblemFluctuations in the Transitional Region
If 7 represents the fraction of the time that the flow isturbulent, then the mean streamwise velocity Um is
(1)
where Ut is the mean turbulent velocity and Ug the meannonturbulent or laminar velocity. Measurements by Kuan andWang9 showed that nonturbulent profiles are not Blasius pro-files for flows over flat plates. Moreover, turbulent profilesare not the traditional fully developed profiles. As will bediscussed subsequently, this is confirmed by calculations. Atany given instant, the streamwise fluctuation in the transi-tional region u 'r is given by
u'r = u - Um
If u is the nonturbulent velocity, then
ufr =
and
Similarly, if u is the turbulent velocity, then
7^ = ~^? + (1 - y)2(Ut - Ue)2
(2)
(3)
The expression given in Eq. (3) is obtained 7 of the time andthe expression in Eq. (2) is obtained (1-7) of the time.Therefore,
IT? = - 7(1 - (4)
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630 YOUNG, WARREN, HARRIS, AND HASSAN: NEW APPROACH FOR TRANSITIONAL FLOWS
In general, one can write
and
when
kr = ykt + (1 - y)kt +
AC// = Utti - Uy
(5)
(6)
(7)
Traditionally, terms resulting from nonturbulent fluctua-tions have never been calculated. This does not rule out use ofprocedures used to calculate turbulent flows. The terms A£//Al//are a result of large eddies. Their calculation requires specifi-cation of the turbulent and nonturbulent profiles in the transi-tional region.
Governing EquationsThe equations governing turbulent mean flow are the Rey-
nolds averaged Navier-Stokes equations. When Favre's massaveraging is used, the compressible equations resemble theincompressible equations:
0 (8)
tt + [ pUiUj + Ofjl
(pE))t + [ pUjH + QJ - Uj
where
-(r ,7-pW?W ;)b=0 (9)
-pu'lu'j) + pu]h"}j = 0
-P
(10)
OD
1.00
0.75
0.50
0.25X/C = 0.4375
=0.0
0.75
0.50
0.25
0.050 0.100 0.150 0.200Y(inches)
0.75
0.50
0.25
1.00
0.75
0.50
0.25
X/C = 0.4792=0.16
1.00
0.75
0.50
0.25
0.050 0.100 0.150 0.200Y(inches)
0.75
oiX/C = 0.5208
=0.50
0.50
0.25
X/C = 0.5625=0.82
0.10 0.20 0.30 0.40Y(inches)
X/C = 0.6250=0.98
0.10 0.20 0.30 0.40 0.50 0.60 0.70Y(inches)
X/C = 0.6667=1-0
0.050 0.100 0.150 0.200 0.250 0.300 0.25 0.50Y(inches) Y(inches)
Fig. 1 Typical velocity profiles. .
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YOUNG, WARREN, HARRIS, AND HASSAN: NEW APPROACH FOR TRANSITIONAL FLOWS 631
0.0075
0.0050
0.0025
2.5000E6Rex
Fig. 2 Typical skin friction.
5.0000E6
0.075
0.050»
0.025
0.000
0.100
0.075
0.050
0.025
0.000
0.150
0.125
0.100"i 0.075i
0.050
0.025
0.000
0
— CalculationExperiment
b s~^°
X/C = 0.4792=0.16
0.25Y(inches)
0.50
X/C = 0.50=0.32
0.25Y(inches)
0.50
In the preceding equations, p is the density, iii the mean velocity,E and H the total energy and total enthalpy, P the pressure, fthe temperature, 7 the ratio of specific heats, and p and X thecoefficients of molecular viscosity and thermal conductivity,respectively.
The one-equation model employed follows closely that ofRef. 10. Thus, the turbulent kinetic energy equation is takenas
(pk\t
= puju!uitk-(pe)where
(12)
p/2
-PU'-U" = HttfiJ + Ujj - 2/3dijUm>m] - 2/3dijpk
0.150
0.125
.0.100
\ 0.075f-
0.050
0.025
n nnn
>.o
0
', o
o0 o o o
f- ^"^v ° X/C = 0.541 7
N S O Y = °-70
N. 0: . . . . i . >^_ .° • 9 . o .
0.25 0.50Y(inches)
0.150
0.125
0.100»
0.075
0.050
0.025
0.000
0.075
.0.050
3
X/C = 0.5625Y =0.82
0.25 0.50Y(inches)
0.75
X/C = 0.5208 0.025Y =0.50
0.000
X/C = 0.6250=0-98
0.25 0.50 0.25 0.50Y(inches) Y(inches)
Fig. 3 Streamwise turbulent intensities; contributions of the turbulent normal stresses.
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632 YOUNG, WARREN, HARRIS, AND HASSAN: NEW APPROACH FOR TRANSITIONAL FLOWS
0
CalculationExperiment
0.25Y(inches)
X/C = 0.4792
= 0.16
•MM
0.50
0.25Y(inches)
0.50
0.25Y(inches)
X/C = 0.5208= 0.50
ana
0.50
0.175
0.150
0.125
0.100
0.075
0.050
0.025
0.000
0.075
•0.050
0.025
X/C = 0.5417
=0.70
0.25Y(inches)
0.50
X/C = 0.5625
Y =0.82
0.25Y(inches)
0.50
X/C = 0.6667= 1.0
0.25Y(inches)
0.50
Fig. 4 Streamwise turbulent intensities; influence of large scales [Eq. (18)].
(13)
^ and £e are the viscosity and dissipation length scales10
(C; = 0.09 and Ck = 0.25).This form of the governing equations can be used for low-
and high-speed flows. For the current application, the densityis essentially constant and the terms resulting from compress-ibility corrections can be ignored.
The stress model employed follows closely that of Ref. 11.Because of the complexity of the stress model, these equationswill not be reproduced here. Note that in Ref. 11, no e or o>equations were used because the length scale was specified.
Contribution of the TS WavesExamination of Eq. (5) shows that one needs to model the
turbulent contributions, the nonturbulent contributions, andthe large scales. The turbulent terms are discussed in theprevious section. In general, one can develop a similar formu-
lation for the nonturbulent terms. Traditionally, this has notbeen done. Because the nonturbulent terms are a result of thepresence of the TS waves, an approximate approach thatmakes use of some of the results of linear stability theory willbe adopted here.
The dominant disturbance frequency at breakdown is wellpredicted by the frequency of the TS waves having the maxi-mum amplification rate. Using the work of Obremski et al.,12
Walker13 showed that the locus of maximum amplificationrate can be correlated by
(uv/U2e) = 3.2/te;*3/2 (14)
where Ue is the edge velocity, Re8* the Reynolds number basedon the displacement thickness 6*, arid co the frequency. Itshould be noted that Eq. (14) is based on calculations usinglinear stability theory for the Falkner-Skan profiles and not on
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YOUNG, WARREN, HARRIS, AND HASSAN: NEW APPROACH FOR TRANSITIONAL FLOWS 633
experiment. The quantities co and £ determine a length scale 4,given by
4 = < (15)
where a is a constant to be determined later. Thus, one way toaccount for the presence of the TS waves and the resultingnonturbulent fluctuations is to solve an equation for kg similarto that of Kt but with ̂ replaced by 4. This was not done here.Instead it was assumed that both turbulent and nonturbulentcontributions can be obtained by replacing ^ in Eq. (13) by
^ = (1 - T)4 + ygt^ (is)
where ̂ is the turbulent viscosity length scale. This approachgives the transitional shear stress directly.
Before transition, where 7 = 0, ^ = 4. Therefore, an esti-mate of a can be obtained from a calculation of Streamwise
intensities before transition and comparison with the data ofRef. 7. Using this procedure, a value of 0.07 is obtained.
Contribution of the Large ScalesIn order to determine the contribution of the larger scales,
i.e., terms At//At// in Eq. (5), a number of procedures wereconsidered. The first assumes that the nonturbulent profilesare Blasius profiles and the turbulent profiles are the fullydeveloped turbulent profiles. As will be seen from the resultssection, this assumption is not satisfactory.
The second procedure considered made use of the boundarylayer code developed by Harris.14 This code was exercised inthe following manner. Starting from an assumed transitionpoint xt and an assumed 7 distribution, calculations werecarried out downstream of xt by assuming that the flow iseither laminar or turbulent. The choice is determined bywhether 7 is larger than a random number Rf. If 7 > Rf, theturbulent version of the code is used to calculate the flow
0.150
0.125
0.100
0.075
0.050
0.025
0.000
oCalculationExperiment
X/C = 0.4792=0.16
0.10 0.20 0.30 0.40Y(inches)
0.10 0.20 0.30 0.40Y(inches)
X/C = 0.5208
=0.50
X/C = 0.5417
=0.70
0.25Y(inches)
0.50
X/C = 0.5625
Y =0.82
0.25 0.50Y(inches)
0.75
0.075
.0.050
0.025
0.000
X/C = 0.6667
'-Y--JJL
0.10 0.20 0.30 0.40 0.50 0.25 0.50Y(inches) Y(inches)
Fig. 5 Streamwise turbulent intensities—stress model; influence of large scales [Eq. (18)].
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634 YOUNG, WARREN, HARRIS, AND HASSAN: NEW APPROACH FOR TRANSITIONAL FLOWS
0.075 -
0.050 -
0.025
0.000
o
CalculationExperiment
X/C = 0.4792=0.16
0.25Y(inches)
0.50
0.25Y(inches)
0.50
X/C = 0.5208=0.50
0.25Y(inches)
0.50
0.075
.0.050
0.025
0.000
X/C = 0.5417=0.70
0.25Y(inches)
0.50
X/C = 0.5625=0.82
0.25 0.50Y(inches)
0.75
X/C = 0.6250Y =0.98
0.25 0.50Y(inches)
0.75
Fig. 6 Streamwise turbulent intensities; influence of large scales [Eq. (17)].
there. If, on the other hand, 7 < Rf, then the flow is treatedas laminar. Because of the marching nature of boundary layercodes, calculated laminar profiles quickly become similar tothe turbulent profiles. As a result, AC// calculated from thisprocedure were not satisfactory, and results of this procedureare not included.
The third procedure assumes AC// as
// = C//-(C//)7=0 (17)
The rationale for this choice stems from the fact that (C//)T=0is representative of the nonturbulent profiles and C// are notmuch different from turbulent profiles. In the absence of aconditional sampling procedure, it is difficult to calculateactual turbulent and nonturbulent profiles in the transitionalregion.
Results and Discussion
An upwind procedure similar to that developed in Ref. 11 isused for both one-equation and stress models. Because of thelow Mach numbers considered, grid sequencing was used toaccelerate convergence. All comparisons are made with theexperiments of Ref. 7. Unless otherwise noted, calculationsare based on the one-equation model.
Velocity profiles and skin friction calculations were some-what insensitive to the contribution resulting from TS wavesand the large scales. Moreover, they were insensitive to thetype model employed, i.e., one-equation vs stress model. Fig-ures 1 and 2 show a comparison of a representative calculationwith experiment. Figure 1 compares velocities, whereas Fig. 2compares skin friction. As the figures show, both predictionsare in good agreement with experiments. Thus, if the objective
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YOUNG, WARREN, HARRIS, AND HASSAN: NEW APPROACH FOR TRANSITIONAL FLOWS 635
0.075
0.050§
0.025
0.000
0.100
0.075
0.050
0.025
0.000
0
CalculationExperiment
0.150
0.125
_.0.100
0.075
X/C = 0.4792 0.050
Y =0.16' 0.025
0 * • i • 0.0000.25
Y(inches)0.50
X/C = 0.50=0.32
0.25Y(inches)
0.50
0.150
0.125
0.100
0.075
0.050
0.025
0.000
0.075
.0.050
X/C = 0.5208 Q.025Y = 0.50
0.000
X/C = 0.5417
7 =0.70
0.25 0.50Y(inches)
X/C = 0.5625
Y =0.82
0.25 0.50Y(inches)
0.75
X/C = 0.6250
=0.98
0.25 0.50 0.25 0.50Y(inches) Y(inches)
Fig. 7 Streamwise turbulent intensities; influence of large scales [Eq. (17)] and TS correction.
0.75
is the calculation of skin friction and velocity profiles, one canapproximate the transitional shear stress term by the first terminEq.(5).
Initially, solutions were obtained using only the first term ofEq. (5). Figure 3 compares the resulting Streamwise intensitieswith experiments. As the figure shows, large discrepanciesexist, especially in the near wall region. This shows that ap-proximating transitional normal stresses by their turbulentcomponent is not a good approximation.
The remaining discussion will concentrate on the influenceof the TS waves and large scales on intensity. Assumptionsregarding large scales are considered first. Figures 4 and 5were obtained using the one-equation model and the stressmodel, respectively, for the case where At// is given by
At// (18)
where Ut and Ug assume fully developed turbulent profiles andBlasius profiles, respectively. In this calculation, the contribu-tion of the TS correction was neglected. As the figures show,this particular choice of At// overpredicts Streamwise intensi-ties. Thus, such an assumption is unsatisfactory. In addition,nothing is gained by using a stress model in place of a one-equation model. Because of this, subsequent comparisons willbe restricted to results of the one-equation model.
The next set of calculations assumes that the large scales aremodeled by Eq. (17). Figure 6 shows a calculation withoutallowing for TS waves, whereas Fig. 7 allows for the TSwaves. The two figures show that allowing for the presence ofthe TS waves improves predictions in the early stage of transi-tion. Moreover, they show that the presence of the spikes inthe Streamwise intensities are a direct result of the large scaleeddies. Further, the choice of Eq. (17) to represent the contri-
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636 YOUNG, WARREN, HARRIS, AND HASSAN: NEW APPROACH FOR TRANSITIONAL FLOWS
butions of the large scales appears to be a satisfactory approx-imation.
Concluding RemarksThis investigation shows that large scales and TS waves have
little influence on the transitional skin friction but have aprofound influence on the intensities. The approach used toaccount for both of these effects is somewhat approximate.More elaborate models would be needed to improve the pre-sent predictions. It is expected that both of these mechanismswill not be very important when the bypass mechanism is ineffect, i.e., at high freestream turbulent intensities.
AcknowledgmentsThis work is supported in part by the following grants:
NASA Cooperative Agreement NCCI-22; the HypersonicAerodynamics Program Grant NAGW-1022, funded jointlyby NASA, the Air Force Office of Scientific Research, and theOffice of Naval Research; and the Mars Mission ResearchCenter, funded by NASA Grant NAGW-1331. The authorswould like to acknowledge helpful discussions with T. Zang ofthe NASA Langley Research Center, N. Chokani of NorthCarolina State University, and T. Wang of Clemson University.
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Boundary Layer," Progress in Aerospace Sciences, Vol. 22, Jan.1985, pp. 29-80.
4Scheuerer, G., "Numerical Prediction of Boundary Layers andTransitional Flows," Boundary Layers in Turbomachines, Von Kar-man Inst. for Fluid Dynamics, Lecture Ser. 1991-06, Rhode-SaintGenese, Belgium, Sept. 1991.
5Jiang, H., Moore, J. G., and Moore, J., "Low Reynolds NumberOne-Equation Turbulence Modelling for Prediction of TransitionalFlows Over a Flat Plate," AIAA Paper 90-0242, Jan. 1990.
6Arnal, D., and Juillen, J. C., "Etude Experimentale et Theoriquede la Transition de la Couche Limite," La Recherche Aerospatiale,Vol. 2, March-April 1977, pp. 79-88.
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HGaffney, R. L., Jr., Salas, M. D., and Hassan, H. A., "AnAbbreviated Reynolds Stress Turbulence Model for Airfoil Flows,"AIAA Paper 90-1468, June 1990.
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13Walker, G. J., "Transitional Flow on Axial Turbomachine Blad-ing," AIAA Journal, Vol. 27, No. 5, 1989, pp. 595-602.
14Harris, J. E., "Numerical Solutions of Equations for Compress-ible Laminar, Transitional and Turbulent Boundary Layers and Com-parisons with Experimental Data," NASA TR R-368, 1971.
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