a numerical research on a compressibility-correlated langtry's transition model for double...

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Chinese Journal of Aeronautics

journal homepage: www.elsevier.com/locate/cja

Chinese Journal of Aeronautics 24 (2011) 249-257

A Numerical Research on a Compressibility-correlated Langtry’s Transition Model for Double Wedge Boundary Layer Flows

ZHANG Xiaodong , GAO Zhenghong* School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China

Received 27 May 2010; revised 30 July 2010; accepted 7 December 2010

Abstract

A new compressibility correlation is introduced in the Langtry’s local variable-based transition model to investigate the phe-nomenon on double wedge shock/boundary layer interactions. The computational analysis compared with experimental data has been made to assess the influence of the wall temperature and the leading edge nose radius on a hypersonic double wedge boundary layer. It has been found that the laminar boundary layer separation occurs on the first ramp. Furthermore, the wall temperature and the leading edge nose radius have remarkable influence on the separation characteristics in the kink. Compari-son of the calculated pressure coefficient distribution and the boundary layer profile with the experimental data shows that betterresults can be achieved when using the modified transition model.

Keywords: transition model; boundary layer; hypersonic double wedge flat; compressibility correlation; wall temperature; blun-ness radius

1. Introduction1

Boundary layer transition plays an important role in hypersonic flow. The accuracy of the prediction in the transition has a great influence on the design of the thermal protection system (TPS) of super/hypersonic aircraft, scramjets and re-entry vehicle.

As a rapid development in computational fluid dy-namics (CFD), applying the CFD techniques to aircraft design has become a preliminary way to improve the design level for the aircraft. The traditional method for predicting transition is the en method[1]. As it is hard to couple with the CFD unstructured solver, its applica-tion is restricted. Since the large eddy simulation (LES) and direct numerical simulation (DNS) require too many computing resources, they are still difficult to be applied to engineering practice. Therefore, the research of developing a transition model satisfying

*Corresponding author. Tel.: +86-29-88495971. E-mail address: [email protected]

1000-9361/$ - see front matter © 2011 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(11)60030-7

the demand of engineering design still needs more attention[2-3]. In 2004, Langtry and Menter, et al.[4-6]

developed a correlation-based transition model using local variables which coupled with SST k- turbulent model[7]. This method combines the advantages of low Reynolds number turbulence model and the transition empirical correlation[8], and forms a framework for implementation of transition correlations applying lo-cal variables to general-purpose CFD methods. How-ever the transition correlations in this model are origi-nally from a low-speed, incompressible flat plate tran-sition experiment, without further consideration of the Mach number effect, hence this model cannot directly be applied to solve the super/hypersonic problem. Therefore the authors in this paper introduce a com-pressibility correlation into the original model based on some hypersonic tunnel test data[9-10]. The modified model is applied to a double wedge compression cor-ner configuration at the inlet of scramjet vehicle like Hyper X-43[11]. The transition flow and the shock wave/boundary layer interaction are numerically simulated and analyzed in this paper.

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2. Description of tRe transition model

The tRe model is based on two transport equa-tions: one for intermittency and the other for a transi-tion onset criterion in terms of local momentum thick-ness Reynolds number tRe . The intermittency equation is used to define the time ratio of the turbu-lent or laminar flow and trigger the transition proc-ess[12]. The tRe equation is designed as the criterion of transition onset position, which realizes its localiz-able process.

The tRe transport equation (Eq.(1)) is developed in order to introduce free-stream information and avoid the additional non-local operations.

tt tt t t

( )( ) ( )j j

ReRe RePt x t x

(1) where is the density, Uj the velocity component in x,y and z derections, xj the displacement component in x,y and z derections, P j the source term of Eq.(1), j aconstant, the molecular viscosity coefficient, and tthe eddy viscosity coefficient.

The intermittency transport equation is used to con-trol the onset and development during the transition process, which is expressed as

t( )( )j j j

P Et x x x

(2)

where E is the destruction/relaminarization source term, and f a constant.

The source term in the intermittency equation is de-fined as follows

0.5length a 1 onset e 1

vonset 1

c

( ) (1 )

2.193

P F c S F c

ReFRe

(3)

where Flength is an empirical correlation that controls the length of the transition region, S the absolute value of strain rate, Fonset a function which is used to trigger the intermittency production, Fonset 1 the location where the local vorticity Reynolds number exceeds the local transition onset critia, ca1 and ce1 are the coefficients, Rev

is the vorticity Reynolds number, and Re c is the transi-tion onset momentum thickness Reynolds number.

Finally, the effective intermittency eff which is solved from Eq.(2) will be used to control the source term in K-equation (Eq.(4)) of SST turbulence model, and the eddy viscosity will be modified to simulate the transition process in the end[13].

t

eff sep

eff

max( , )

k k kj j j

k k

kk kP Dt x x x

P P

(4)

where Pk and kD are the production and destruction terms for the turbulence kinetic energy equation , k the kinetic energy, sep is the separation intermittency, and

k is a constant. In Langtry’s transition model, he provides an em-

pirical correlation for transition onset momentum thick-ness Reynolds number as a function of turbulence in-tensity Tu and pressure gradient which is defined as

1.5

t

2

0.671

2

3 1.5

35.0

( , ) ( ) ( )

0.219 61 173.51 589.428 1.3( )

331.50( 0.565 8) 1.3

1 ( 12.986 123.66

( ) 405.689 )e 0

1 0.275(1 e )e

Tu

Re f Tu E Tu F

Tu TuTuE Tu

Tu Tu

F

0.5

2

0

( / ) (d /d )

Tu

U s (5)

where is the momentum thickness, v the dynamic viscosity coefficient, U the local velocity, and s the streamline direction.

As this correlation is originally obtained from in-compressive flat plate transition experiments, it is in-appropriate to use this transition model in hypersonic problems. The transition onset momentum thickness Re t will increase from 800 to 3 500 as the Mach number increases from 1 to 12, which is found in tran-sition experiments of hypersonic sharp cone from Langley wing tunnel. This means using the original correlation transition will be triggered too earlier[14].(see Fig.1, where Re is the momentum thickness Reynolds number, Mae the external Mach number, TTthe reference temperature)

Fig.1 Transition experiment on sharp cone Re vs Mae[14].

This paper introduces a local Mach number into the Re t correlation based on some hypersonic wind tun-

Uj

Uj

Uj

k

f

No.3 ZHANG Xiaodong et al. / Chinese Journal of Aeronautics 24(2011) 249-257 · 251 ·

nel transition experiments[14-15], and proposes a new correlation as follows

t

3 2

( , , ) ( ) ( ) ( )

( ) 0.009 87 0.144 07 0.751 09 1[0,12]

Re C Tu Ma E Tu F G Ma

G Ma Ma Ma MaMa

(6)In order to validate this correlation, the hypersonic

double wedge flat will be numerically investigated in this paper (see Fig.2, where 1, 2, cowl and iso are the angles between corresponding control surface and horizontal plane, hth is the height of the scramjet in-take.).

Fig.2 Sketmatic of engine inlet of hypersonic aircraft model.

3. Experimental Setup in Model and Numerical Method

There are some researches on different patterns of the shock-shock interaction around a re-entry vehicle in recent years, and several typical shock-shock inter-actions around the configuration are shown in Fig.3[16].

Fig.3 Schematic presentation of foreshock-aftershock in-teractions around a re-entry vehicle (Schramm 1999)[17].

One of those typical configurations is spiked-blunt bodies, which is widely used in the applications of hypersonic aeronautics, including axisymmetric jet inlets with conical centre body, ballistic missiles drag reduction by spike, and ablation associated with re- entry problems. Panaras, et al.[16] conducted a very detailed research on these spiked-blunt bodies by using LES in 2009. He studied two modes around the spiked-blunt bodies, pulsation mode and oscillation mode.

In the pulsation mode, the conical separation bubble formed on the concave part of the body periodically inflates and radically expands, taking a hemispherical shape. In the oscillation mode the unsteadiness is

milder. The conical foreshock envelops the separation bubble, and the accompanying shear layer oscillates laterally and its shape changes periodically from con-cave to convex.

In addition, there are some alternative configura-tions on hypersonic vehicle control surfaces, e.g., cone-flare, double cones and hollow cylinder-flare. Neuenhahm, et al.[10] investigated the influence of the leading edge radius and the wall temperature on the hypersonic boundary layer of a double wedge flat in Aachen University TH2 hypersonic shock tunnel in 2006. The inlet conditions for the performed experi-ments are given in Table 1 and Fig.4, in which is farfield density, T farfield temperature, U the free-stream velocity, ttest run time of experiment in the wind tunnel, Rex, farfield stream Reynolds number, Rnthe leading edge nose radius, l1 the length of the first wedge, and l2 the length of the second wedge

Table 1 Inlet conditions on leading edge plate Parameter Value

Ma 8.1 P /Pa 520/(kg·m 3) 0.017 1 T /K 106

U /(m·s 1) 1 635 Rex, /m 1 3.8 106

Tu/% 0.5 ttest/ms 2.0

Fig.4 Schematic of shock/boundary layer interaction on a blunt double wedge configuration.

The first ramp has an angle of 9 and the second one 20.5 . The length of the first ramp is L=180 mm, and the second one is 250 mm. Neuenhahm, et al.[10] se-lected the first ramp leading radius in the experiment as 0 mm (sharp leading edge), 0.5 mm and 1.0 mm respectively. The model surface temperature is evalu-ated by resistance heating elements up to 760 K, giv-ing a homogenous temperature distribution over the model surface, and the wall temperature is chosen as Tw=300, 600, 760 K respectively.

The commercial CFD software FLUENT is used here combining with user-defined function (UDF) in this paper. The density-based implicit solver is selected for compressible hypersonic flow. An implicit Roe- FDS (flux difference splitting) method is employed in this paper. The second upwind scheme provides opti-mum accuracy for the spatial discretization of the Na-


vier-Stokes equation. The H-grid is adopted for the sharp leading edge case whereas the C-grid is adopted for the blunt leading edge case. 45 000 grid points are used. The wall normal expansion ratio is 1.1, the y+ isbetween 0.01-0.50, the computation grid and the inlet condition setting are shown in Figs.5-6. The inlet con-dition for tRe should be changed as the introduction of Ma correlation, and the boundary condition of tRe[18-19] is obtained by substituting Tuinlet, Mainlet, =0 in Eq.(5). The criterion of convergence is by monitoring the surface friction coefficient distribution until it does not change as the number of iterations increase. The iteration number for solving flow field is about 8 000, and the maximum courant-friedrich-levy (CFL) num-ber is close to 6.0 during the iterations.

Fig.5 Grid and boundary conditions for sharp leading edge.

Fig.6 Grid and boundary conditions for blunt leading edge.

4. Computational Analysis in Transition Model and Comparison with Experimental Results

In order to elucidate the effect of the new correla-tion, Fig.7 shows the comparison of surface pressure coefficients Cp using various models respectively in sharp leading edge case at Tw=300 K. The separation region is predicted to be a little smaller by the Lang-try’s original transition model, because the onset value

of transition obtained by the incompressive correla-tions is much lower compared to the value in hyper-sonic condition, which will trigger transition earlier. The early transition will result in the deficiency of simulating separation zone, because the turbulent boundary layer is more stable compared to the laminar case. As the new correlation with compressive amend-ment will increase the onset value of transition in the hypersonic condition, the computed surface pressure distribution matches the experiment much better. Small difference can be identified due to some side-flow that occurs in the experiment.

Fig.7 Comparison of pressure coefficient between experi-ment data[10] and different models for Rn= 0 mm and Tw=300 K.

It is easy to see that the SST turbulent model cannot capture the separation phenomenon in the kink (see Fig.8). This is because that the SST turbulent model is designed for the turbulent boundary layer, the transi-tion occurs very quickly before the flow arrives at the second ramp, and the turbulent boundary layer will undergo more adverse pressure to separation.

Fig.8 Difference of pressure contour computation results between SST and transition model.

Fig.9 shows that the transition model could capture this shock wave/boundary layer interaction induced by separation phenomenon, which brings notable change in the kink.


Fig.10 is a schlieren image of the shock boundary layer and separation in the whole flow field from the experiment.

A study of grid dependence on the transition result is also performed here. Three grids are given as fol-lows: 1) 25 000 grids with the average y+ = 0.10; 2) 45 000 grids with the average y+=0.10; 3) 45 000 grids with the average y+=0.01. The results from those grids are shown in Fig.11.

Fig.9 Separation phenomenon predicted by transition model in kink.

Fig.10 Schlieren image of shock boundary layer and sepa-ration in whole flow field.

Fig.11 Results for different grid models for Rn= 0 mm and Tw=300 K.

From Fig.11, it is easy to observe that the result is improved as the grid is refined, and the separation lo-cation shifts upstream as the grid density increases. In Ref.[13], Langtry mentioned that if the y+ values were between 0.001 and 1, there would be no effect on the

solution, and the numerical results proved his conclu-sion. However, if the y+ values are below 0.001, the transition location appears to move downstream. This is presumably caused by the large surface value of the omega, which is scaled with the first grid height. For more detail, refer to Ref.[13].

The influence of different leading edge radii on pressure coefficient at Tw=300 K is demonstrated in Fig.12. The length of separation bubble increases as the leading edge radius changes from 0 mm (sharp) to 0.5 mm. Further increase of the leading edge radius does not affect the separation bubble length apparently. The pressure coefficient for the sharp leading edge case increases much more rapidly than the other two blunt cases after the boundary layer is separated in the kink. The computational results agree well with the experiment.

Fig.13 shows the influence on the pressure coeffi-cients of various wall temperatures for blunt leading

Fig.12 Numerical simutlation of effect of different leading edge radii at Tw=300 K on pressure distribution comparing with experiment data[10].

Fig.13 Wall temperature effect on pressure distribution for Rn=1.0 mm.


edge Rn=1.0 mm. As the wall temperature gets ele-vated, the separation bubble length increases and the separation location shifts upstream. The reattachment position for both elevated wall temperature cases is shifted downstream but the difference in both cases seems to be negligible. Furthermore, the maximum reattachment pressure is reduced. This is because high temperature reduces the density and thus increases the boundary layer height, which has a destabilizing effect. The pressure rises over the separation shock and then reaches a plateau value. Subsequently, the pressure increases strongly over the reattachment shock and reaches a maximum value. Both experimental and nu-merical results show that the pressure coefficient in-creases more rapidly as the wall temperature de-creases. The modified transition model gives improved pressure distribution results which match well with the experiment.

The Stanton number is used to describe the thermal load distribution, which is defined as

waw w

aw w w

( )( ) pq TSt k U C T T

yU h h

where qw is the heat flux on the wall, hw the enthalpy value on the wall, haw the adiabatic wall enthalpy, and Taw the adiabatic wall temperature.

As the run time of experiment in the wind tunnel is very short, it is hard to get the adiabatic wall enthalpy haw, so Neuenhahm used the total enthalpy h0 in his paper. The Stanton number distribution can be used for identifying a laminar boundary layer and the separa-tion location. The first decrease in the Stanton number curve indicates a laminar separation, and the reattach-ment location can be determined by the first increase in the heat flux.

Figs.14-15 show the Stanton number distributions for the sharp leading edge case at different wall tem-peratures. In Fig.15, “VD Tw” represents the wall temperature predicted by van Driest theory. The mini-

Fig.14 Numerical results of Stanton number distributions for sharp leading edge at different wall tempera-tures.

Fig.15 Experiment results[10] of Stanton number distribu-tions for sharp leading edge at different wall tem-peratures.

mum and maximum of the heat flux coefficients de-crease when the wall temperature increases. The im-proved transition model gives more accurate separa-tion location compared with the data from wind tunnel test. However, the heat flux coefficient seems a little suppressed after the flow reattaches, which is because the source term in the intermittency Eq.(3) is not fully developed after the flow reattachment, but is derived from the original incompressible Blasuis equation (see Appendix A).

In order to investigate the effects of the leading edge bluntness on the shock wave/boundary layer interac-tion, the leading edge region is examined first (see Figs.16-21). It is found that increasing the leading edge bluntness will alter the shock shape and increase

Fig.16 Numerical result of density distribution for Rn= 0 mm and Tw=300 K.

Fig.17 Schlieren image of leading edge region for Rn= 0 mm and Tw=300 K.


Fig.18 Numerical result of density distribution for Rn= 0.5 mm and Tw=300 K.

Fig.19 Schlieren image of leading edge region for Rn= 0.5 mm and Tw=300 K.

Fig.20 Numerical result of density distribution for Rn= 1.0 mm and Tw=300 K.

Fig.21 Schlieren image of leading edge region for Rn= 1.0 mm and Tw=300 K.

the height of shock boundary layer. The schlieren im-age from the experiment shows that the shock shape is a straight line with little curvature at the leading edge for the sharp leading edge case. As the leading edge radius increases, it alters to a highly curved shock for the blunt cases. The increased curvature of the shock also increases the entropy layer’s height. Although the color-schlieren image can be obtained from the com-putational flow field data, it is too complicated, there-fore the numerical results of the density distribution from the transition model are introduced here, which show the same trend as the experiments.

The turbulent flow reattaches to the wall at the be-ginning of the second ramp, which is found in the ex-periment. Because of the intense compressibility effect in the second ramp, the aerodynamic heat rises very quickly (see Fig.22). As the transition process has al-ready been finished in the separation region, turbu-lence enhances the heat exchange in the boundary layer (see Fig.23). Considering this factor, it is not surprising that a great change in the temperature field occurs at the beginning of the second ramp.

Fig.22 Temperature flow field in kink for case of Rn= 0.5 mm leading edge radius.

Fig.23 Shock/boundary layer interaction phenomenon for Rn= 0.5 mm and Tw=300 K.

5. Conclusions

In this paper, the shock/boundary layer interactions and other flow phenomena over a double wedge are


investigated by employing a local variable-based tran-sition model proposed by Langtry. After introducing a new compressibility correlation, the effects of two parameters, the wall temperature and the bluntness of the leading edge on the flow phenomena are examined. Compared with the parameter of the flow field in the experiment, the improved transition model captures subtle changes in the boundary layer. The computa-tional results match well with the experiment.

Although this tRe transition model is essen-tially based on aerodynamic experimental data, pre-vious researches show that the 3D distributed rough-ness, the leading edge radius and the wall temperature also have significant influences on the hypersonic transition. Further researches are still needed to im-prove and validate the presented model. Investigation shows that this model can be coupled with the turbu-lence model conveniently. Although the transition mechanism is not fully understood, the proposed model has much room to improve for engineering application. This transition model is of good value in the future aerodynamic design and CFD transition prediction.

Acknowledgments

I would like to express my profound gratitude to Dr. Bowen Zhong for revising the paper writing.

References

[1] van Ingen J L. A suggested semi-empirical method for the calculation of the boundary layer transition region. Ph.D. thesis, Delft University of Technology, 1956: 8-26.

[2] Suzen Y B, Huang P G, Volino R J. A comprehensive CFD study of transitional flows in low-pressure tur-bines under a wide range of operating conditions. AIAA-2003-3591, 2003.

[3] Walters D K, Cokljat D. A three-equation eddy-viscosity model for Reynolds-averaged Navier-Stokes simula-tions of transitional flow. Journal of Fluids Engineering 2008; 130(12): 121401.1-121401.14 .

[4] Menter F R, Langtry R B, Likki S R, et al. A correla-tion based transition model using local variables Part 1: model formulation. Journal of Turbomachinery 2006; 128(3): 413-422.

[5] Langtry R B, Menter F R, Likki S R, et al. A correla-tion based transition model using local variables Part 2: test cases and industrial applications. Journal of Tur-bomachinery 2006; 128(3): 423-434.

[6] Langtry R B, Menter F R. Correlation-based transition modeling for unstructured parallelized computational fluid dynamics codes. AIAA Journal 2009; 47(12): 2894-2906.

[7] Wilcox D C. Turbulence modeling for CFD. Canada: DCW Industries, 1993.

[8] Mayle R E. The role of laminar-turbulent transition in gas turbine engines. ASME Journal of Turbomachinery 1991; 113(4): 509-537.

[9] Reinartz B U, Ballmann J. Numerical investigation of

wall temperature and entropy layer effects on double wedge shock/boundary layer interactions. AIAA-2006- 8137, 2006.

[10] Neuenhahm T, Olivier H. Influence of the wall tem-perature and entropy layer effects on double wedge shock boundary layer interactions. AIAA-2006-8136, 2006.

[11] Holden M S. Boundary-layer displacement and lead-ing-edge bluntness effects on attached and separated laminar boundary layers in a compression corner. Part 11: experimental study. AIAA Journal 1971; 9(1): 84- 93.

[12] Dhawan D, Narasimha R. Some properties of boundary layer flow during transition from laminar to turbulent motion. Journal of Fluid Mechanics 1958; 3(4): 418-436.

[13] Langtry R B. A correlation-based transition model using local variables for unstructured parallelized CFD codes. Ph.D thesis, University Stuttgart, 2006.

[14] Reshotko E. Is Re e a meaningful transition crite-rion? AIAA -2007-943, 2007.

[15] Zhang X D, Gao Z H. Numerical discussion on com-plete empirical correlation in Langtry’s transition mod-el. Applied Mathematics and Mechanics 2010; 31(5): 575-584.

[16] Panaras A G, Drikakis D. High-speed unsteady flows around spiked-blunt bodies. Journal of Fluid Mechan-ics 2009; 632(1): 69-96.

[17] Schramm J M, Eitelberg G. Shock boundary layer in-teraction in hypersonic high enthalpy flow on a double wedge. 22nd International Symposium on Shock Waves. 1999.

[18] Chen Y. The transition prediction in aerodynamics computation. M.S. thesis, Northwestern Ploytechnical University, 2008. [in Chinese]

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Biographies:

ZHANG Xiaodong Born in 1984, he received B.S. degree from Northwestern Polytechnical University in 2006, and then became a Ph.D. candidate there. His main research in-terest lies in the investigation of turbulence models and tran-sition models. E-mail: [email protected]

GAO Zhenghong Born in 1960, she is a professor in Northwestern Polytechnical University. Her main research interest lies in flight vehicle design and flight control. E-mail: [email protected]

Appendix A:

One of the most important works in Langtry’s pa-per[4-6] is to localize the momentum thickness Reynolds number in his model. The momentum thickness Rey-nolds number is hard to couple with the unstructured solver however there are some relations between the momentum thickness Reynolds number and the vortic-ity or alternatively the strain-rate Reynolds number which is defined here:


2 2 2

vy y yuRe S

y (A1)

where is the absolute value of vorticity. The maximum of the profile in the boundary layer is

proportional to the momentum thickness Reynolds number and can be related to the transition correlations as follows:

vmax( )2.193

ReRe (A2)

This transition criterion will be described in detail. First we start from Blasuis equation which is used to solve incompressible flat at zero angle of attack under zero pressure gradient:

1 02

f ff (A3)

where f is the non-dimensional stream function. For the solid wall, the boundary condition of this

equation is defined as follows:

w w0: 0, 0: 1

f f f ff

(A4)

( )

xu yy Rex x

ufu

(A5)

where is the non-dimensional similarity parameter. The numerical solution for f ( ), f ( ) and f ( )

could be obtained by the Runge-Kutta method. Ac-cording to the definition of Eq.(A5), the absolute value of vorticity in the boundary layer has the following form:

xx

Reu yu f Rey x x

(A6)

The solution for thickness of boundary layer in laminar boundary layer from Eq.(A3) is expressed as

0.55.35.3

x

x xu Re

(A7)

Substitution of Eqs.(A6)-(A7) into the definition of vorticity Reynolds number yields

2 2

v 5.3xu Rey y yRe fx

(A8)

The solution for momentum thickness of boundary layer in laminar boundary layer from Eq.(A3) can be written as

0.50.6640.664

x

x xu Re

(A9)

The momentum thickness Reynolds number is de-fined as

uRe (A10)

Substitution of Eqs.(A8)-(A10) into v 2.193Re /Re yields

2v 2.193 5.3

2.193 0.664xRe Re y yf

Re (A11)

Then the relationship between v 2.193Re Re and y/ could be obtained at any location along stream- wise direction, which is shown in Fig.A1.

Fig.A1 Scaled vorticity Reynolds number Rev profile in a Blasius boundary layer.

Based on the observation from Fig.A1, it is easy to use the vorticity Reynolds number to trigger the onset of transition.

a numerical research on a compressibility-correlated langtry's transition model for double...

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