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Improved γ-Reθt model for heat transfer prediction of hypersonic boundary layer transition
International Journal of Heat and Mass Transfer 107 (2017) 329–338
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Improved c-Reht model for heat transfer prediction of hypersonic boundary layer transition Zihui Hao a, Chao Yan a,⇑, Yupei Qin a, Ling Zhou b a b
School of Aeronautics Science and Engineering, Beihang University, 100191 Beijing, China School of Energy and Power Engineering, Huazhong University of Science & Technology, 430074 Wuhan, China
a r t i c l e
i n f o
Article history: Received 1 July 2016 Received in revised form 13 November 2016 Accepted 14 November 2016
Keywords: Heat transfer rate Boundary layer transition c-Reht model Hypersonic Momentum thickness Reynolds number
a b s t r a c t An improved c-Reht model has been developed to predict the heat transfer of hypersonic boundary layer transition in this paper. A new correlation of momentum thickness Reynolds number for hypersonic boundary layer flow is first presented using local flow variables. In the correlation, the Mach number and Reynolds number are introduced to reflect the compressible effect of hypersonic flow. The function Fonset1 used to control the transition onset as well as several relevant model parameters are also modified to make the c-Reht model suitable for hypersonic flow. Two test cases including the hypersonic flow over a flat plate and the X-51A forebody with different Reynolds numbers and wind tunnel noise levels are employed to assess the performance of the improved c-Reht model. Compared with the original model and various experiments, the improved c-Reht model can successfully predict the changes of Stanton number and heat transfer rate caused by the boundary layer transition. The model can also reasonably reflect the effects of freestream Reynolds number and wind tunnel noise level on hypersonic boundary layer transition. By analyzing the distributions of heat transfer rate on the X-51A forebody, the windward side transition is found to be related to the separation in the compression corner and large separation can cause earlier transition onset in boundary layer. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Boundary layer transition is the process from laminar to turbulence flow. Since the aeroheating of turbulence is much higher than that of laminar especially in hypersonic flow, improper thermal protection design may greatly reduce the performance of aerospace vehicles. Therefore, the accuracy of boundary layer transition prediction is of great importance for thermal protection in the design of hypersonic vehicles. In addition, boundary layer transition is also important for separation and drag prediction. Up to now, various numerical methods for boundary layer transition prediction have been developed, such as semi-empirical eN method, direct numerical simulation (DNS), large eddy simulation (LES), transition models and so on. Among those, eN method can describe the linear stability process accurately. However, it still needs more work to apply the method to complex 3D configurations. As it is hard to couple eN method with modern CFD methods, aeroheating cannot be predicted with the transition prediction at the same time. DNS can help to understand the mechanisms of boundary layer transition, such as the mechanisms of receptivity, ⇑ Corresponding author. E-mail address: [email protected] (C. Yan). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.11.052 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
the evolution process of turbulent spots. However, due to the extraordinarily large computational requirements, DNS is not clearly yet able to be applied in engineering design. LES approach has been applied to predict flow transition on a turbine blade [1], while it still requires a large amount of computing resources relative to transition models. Based on RANS equations, transition models offer a compromise between the accuracy and calculation expense, and they have been widely applied in engineering. In recent years, transition models have been developed rapidly. Low Reynolds number models [2] have a certain ability to predict transition. However, Wilcox [3] proves that the ability of transition prediction is attributed to the mathematical properties of the turbulence model. Walters [4] introduces a ‘‘splat mechanism” to describe the growth of laminar kinetic energy kL in the kT-kL-x method. In order to predict natural transition and bypass transition with Walters’ method, some special modifications based on local mean flow and laminar kinetic energy are made in the kL production term and the transition production term. Considering the intermittency and Mack modes instability, Wang and Fu [5–7] propose a three equations k-x-c method. Considering the effects of Reynolds number and nose bluntness, Zhou [8] improves the performance of the original method by reconstructing the c transport equation. The k-x-c model has a good performance in predicting
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Nomenclature Uj
c ceff q S X k Pk Dk Pc Dc Pht Re1 Reh Rev Rehc e ht Re xj
velocity vector intermittency factor effective intermittency factor fluid density absolute value of strain rate, (2SijSij)1/2 absolute value of vorticity, (2XijXij)1/2 turbulence kinetic energy production term of k transport equation dissipation term of k transport equation production term of c transport equation dissipation term of c transport equation source term of Reht transport equation freestream Reynolds number momentum thickness Reynolds number vorticity Reynolds number critical momentum thickness Reynolds number local transition onset momentum thickness Reynolds number (obtained from a transport equation) Cartesian coordinates
hypersonic boundary layer transition validated by Kong [9], Hao [10], Xiao [11] and others. Langtry and Menter [12–15] propose a correlation-based c-Reht transition model, which is built strictly on local variables and suitable for unstructured grids and massively parallel execution. The method can predict natural transition, bypass transition and the transition induced by separation. Recently, Grabe [16] and Langtry [17] extend this method to predict cross flow transition. Among the methods above, the c-Reht model is widely coupled with some commercial softwares, such as FLUENT, FUN3D, SU2, OVERFLOW, etc., and it may be the most popular method for subsonic transition prediction. In recent years, some researchers have attempted to extend the c-Reht model to predict hypersonic boundary layer transition. For example, You [18] applies the c-Reht model in hypersonic flows. Langtry and Menter’s correlations are replaced by Krause’s correlations and the effect of pressure gradient is taken into consideration in the transitional flow in the research of You [18]. Frauholz [19] couples the c-Reht transition model with the SSG/LRR x turbulence model, and new correlations of Rehc and Flenght are implemented in the c-Reht transition model. The modified model shows good performance in predicting hypersonic boundary layer transition of scramjets. This paper focuses on the c-Reht model for heat transfer prediction in hypersonic transitional flows. The original c-Reht model is built based on low speed flow, and a correlation about the momentum thickness Reynolds number is obtained from a Blasius boundary layer, which limits the application of the original method for hypersonic flow. Therefore, the original model is unsuitable for predicting hypersonic boundary layer transition. A new method is proposed in this paper to extend the original c-Reht model to hypersonic flow. The main idea is different from previous researchers’ work. A new correlation of momentum thickness Reynolds number for hypersonic boundary layer flow is first presented in this paper and the correlation is applied in the c-Reht model. The performance of the improved c-Reht model is assessed by two cases at different freestream Reynolds numbers. The prediction ability of hypersonic boundary layer transition at different wind tunnel noise levels is validated in this paper as well. All the predicted results are compared with the experimental results.
y Ma Tu P0 T0 Tw T h Q St
l lt Fonset Flength d Reht
distance to nearest wall Mach number freestream turbulence intensity stagnation pressure stagnation temperature wall temperature temperature momentum thickness heat transfer rate Stanton number molecular viscosity eddy viscosity transition onset function transition length function boundary layer thickness transition onset momentum thickness Reynolds number (based on freestream conditions)
2. Transition model 2.1. Original c-Reht transition model The original c-Reht model, proposed by Langtry and Menter [12] in 2005, is built with local correlations and easy to couple with modern CFD codes. The model is based on the shear stress transport (SST) turbulence model. Two additional transport equations are supplemented for transition prediction: one is for the intermittency factor c, and the other is for the local transition onset e ht , as follows: momentum thickness Reynolds number Re
@ðqcÞ @ qU j c @ l @c þ Pc Dc þ ¼ lþ t @t @xj rc @xj @xj
" # e ht e ht Þ @ qU j Re e ht @ðq Re @ @ Re þ Pht ¼ rht ðl þ lt Þ þ @t @xj @xj @xj
ð1Þ
ð2Þ
where q, Uj and xj stand for fluid density, velocity vector and Cartesian coordinates respectively. l and lt are molecular viscosity and turbulence eddy viscosity respectively. rc, rht are model constants e ht equations, and rc = 1.0, rht = 2.0. for c and Re Pc is the production term of c equation as shown below:
Pc ¼ F length ca1 qS½cF onset 0:5 ð1 ce1 cÞ
ð3Þ
e ht based on the flat plates T3A, T3B, where Flength is a correlation of Re and the test cases from Schubauer and Klebanof [20]. The function of Flength is used to control the length of transition zone and it is first published by Langtry and Menter [15]. S represents the strain-rate magnitude. ca1, ce1 are model constants, here ca1 = 2.0, ce1 = 1.0. The transition onset is controlled by the following functions: Rem F onset1 ¼ 2:193Re hc F onset2 ¼ min maxðF onset1 ; F 4onset1 Þ; 2:0 R 3 T F onset3 ¼ max 1 2:5 ;0
F onset ¼ max ðF onset2 F onset3 ; 0Þ
ð4Þ
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e ht and it is the critical Reynolds where Rehc is a correlation of Re number where the intermittency initially starts to increase in the boundary layer. Rev represents the vorticity Reynolds number:
Rem ¼ qSy l
2
or
qXy2 l
ð5Þ
Ec represents the dissipation term of c transport equation:
E ¼ ca2 qcF turb ðce2 c 1Þ F turb ¼ e
R 4 T 4
ð6Þ ð7Þ
where X is the vorticity magnitude. The source term Pht of the transition momentum thickness Reynolds number transport equation is defined as follows:
Pht ¼ cht
q t
~ ht ð1:0 F ht Þ Reht Re
ð8Þ
By using a blending function Fht, which equals one in the bound-
e ht to ary layer and zero in the freestream, Pht is designed to force Re match Reht outside the boundary layer. Reht is the transition onset momentum thickness Reynolds number, which is based on freestream conditions.
Reht ¼
qht U 0 l
ð10Þ
When coupling the transition equations with SST turbulence model, the transport equation of turbulence kinetic energy k is modified as follows:
@k @ @ @ ek D ek þP ðqkÞ þ ðquj kÞ ¼ l þ rk lt @t @xj @xj @xj e k ¼ c Pk ; D e k ¼ min maxðc ; 0:1Þ; 1:0 Dk P eff eff
l
ð12Þ
The correlation in Eq. (12), based on the observation of a Blasius boundary layer, is first proposed by Wilcox [21]. The application of the correlation can avoid additional nonlocal boundary layer parameters for transition prediction. This idea has also been used as a reference in other transition methods based on local variables. For example, Wang and Fu introduce the X in the effective length scale in the k-x-c model, which contains the information of boundary layer thickness. Fig. 1 shows the scaled vorticity Reynolds number Rev profile in a laminar boundary layer. The data comes from the computational results of a flat plate at Reynolds number 6 106/m. The freestream Mach numbers equal to 0.3 and 6 respectively. For the subsonic flow, the Rev,max is closed to the 2.193 times the magnitude of Reh, while the correlation is obviously unsuitable for hypersonic flow as shown in Fig. 1. A new correlation of Rev,max and Reh for hypersonic flow is presented as follows:
K ¼ 2:1 0:0895 Ma1
ð13Þ
D ¼ 300 þ 0:0000275 Re1
h i Reht ¼ 1173:51 589:428Tu þ 0:2196 Fðkht Þ; Tu 6 1:3 Tu2
qh2 dU kht ¼ t l ds
qUh Rem;max ¼ 2:193Reh ¼ 2:193
Rem;max ¼ KReh þ D ð9Þ
where ht is the transition onset momentum thickness and it can be obtained from an empirical correlation as follows:
Reht ¼ 331:50½Tu 0:56580:671 Fðkht Þ; Tu 6 1:3
transition onset. Among the three functions, Fonset1 is the most important term, which is based on a correlation as follows:
where Reh is the momentum thickness Reynolds number which is not a local flow variable. A boundary layer parameters identification method is used to obtain Reh accurately in the following test case. A detailed description of the boundary layer parameters identification method is presented in Ref. [22]. The location where Ue = 0.99U1 is defined as the boundary layer edge in this paper. In the new correlation, the Mach number and Reynolds number are introduced to reflect the compressible effect of hypersonic flow. Between the two parameters, the freestream Reynolds number is shown as follows:
Re1 ¼ ð11Þ
The equation of the specific turbulence dissipation rate x is the same with the original SST turbulence model. 2.2. Modification of the c-Reht model for hypersonic flow The original c-Reht model is proposed mainly based on experimental data in low speed flow, and the excellent performance of the method has been validated by Langtry and Menter especially in subsonic flow. So far, the c-Reht model has been widely used for general aviation applications. Langtry and Menter point out that the main missing extensions are crossflow instabilities and high speed flow correlation in the original method [15]. Many researchers have tried to extend the c-Reht model to hypersonic flow in recent years. Among those, You [18] and Frauholz [19] have successfully extended the original c-Reht model in predicting hypersonic boundary layer transition. They put emphasis on the modification with new correlations of Rehc and Flenght, however, the correlation about Rev,max and Reh which is unsuitable for hypersonic flow is ignored in their attempt. In this paper, the ignored correlation is taken into account and the detailed modification of the model for hypersonic flow is presented as follows. In the original c-Reht model, the function Fonset, composed of three functions (Fonset1, Fonset2 and Fonset3), is used to control the
q1 U 1 Lref l
ð14Þ
where Lref is the reference length and equals one in this paper. A flat plate case is used to obtain the relationship between Rev,max and Reh in hypersonic flow. The flow parameters are listed in Table 1. The wall temperature Tw equals 300K. Fig. 2 shows the comparison of the new and the original correlations with computational results in hypersonic flow. The new
Fig. 1. Scaled vorticity Reynolds number Rev profile in a laminar boundary layer.
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Z. Hao et al. / International Journal of Heat and Mass Transfer 107 (2017) 329–338 Table 1 Flow parameters for the flat plate model. Re1(1/m) 6
2 10 4 106 6 106 8 106
Ma1 4,6,8 4,6,8 4,6,8 4,6,8
correlation shows a good agreement with the computational results, however, the original correlation has an obvious deviation in Fig. 2. It can be concluded that the new correlation is more suitable for hypersonic flow than the original one. Both the new and the original correlations are linear. The obvious difference is that the intercept term of the original correlation is zero while it is a function of Re1 in the new correlation. When the new correlation is directly applied in the c-Reht method instead of the original one, the intermittency near the wall in turbulent region is obviously less than the normal level. That will lead to an insufficient viscosity in the region near wall. In order to solve the problem, a function Fr is constructed as follows:
Fr ¼ e
5Rev 2D2
ð15Þ
Fr is a function of Rev, similar to Fturb in the construction. The function Fturb can be found in the dissipation term of c transport equation. Finally, Fonset1 is modified as follows in the improved c-Reht model:
F onset1 ¼
Rem Dð1 F r Þ KRehc
ð16Þ
A flat plate is used to analyze the characteristic of function Fr. The length of the flat plate is 1500 mm and the freestream conditions are: Ma1 = 6.0, Re1 = 2.6 106/m. Fig. 3(a) shows the computational grid and Mach number contour within x = 1300 mm–1500 mm, which is in the turbulent region. Fig. 3 (b) shows the distribution of the function Fr. The value of Fr equals one near the wall and the region outside the boundary layer edge. It is zero in the most region within the boundary layer. Fig. 3 (c) and (d) show the comparison of the results with and without the application of Fr. Adopting Fr, the intermittency backs to the normal value in Fig. 3(d). Several relevant model parameters are also recalibrated to make the c-Reht model suitable for hypersonic flow as follows:
F length ¼ 20;
ce2 ¼ 20
ð17Þ
Other model parameters and a detailed formulation of the c-Reht transition model can be found in Refs. [13–15]. 3. Test cases Two cases have been selected to assess the performance of the improved c-Reht method for hypersonic flows in this section. One is a hypersonic flat plate and the other is the X-51A forebody configuration. The X-51A test case is analyzed with different wind tunnel noise levels and freestream Reynolds numbers respectively. Fig. 2. Comparison of the new and the original correlations with computational results in hypersonic flow.
3.1. A flat plate at Ma1 = 6 In this test case, hypersonic boundary layer flow over a flat plate is analyzed to evaluate the effectiveness of the improved c-Reht transition model. The flat plate model is the same as that shown in Fig. 3(a) and the flow conditions are consistent with the experimental study of Mee [23]. The wall temperature is Tw = 300K,
the freestream turbulence intensity is Tu = 0.4% and other detailed flow parameters are shown in Table 2. Grid convergence analysis is necessary for numerical study. Four grids with successively increasing resolutions are used to analyze the grid convergence of the improved method. The details are
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0.002
St
0.0015 0.001
Coarse Medium Fine Finest
0.0005 0
500
x(mm)
1000
1500
Fig. 4. Grid convergence analysis.
(a) Mach number contour and computational grid
(b) Distribution of the function Fr
(c) Intermittency factor without the application of Fr
(d) Intermittency factor with the application of Fr
Fig. 3. The test case for the function Fr.
3.2. X-51A forebody at different wind tunnel noise levels The test case is a 20% scale X-51A forebody model, which is tested in the Boeing/AFOSR Mach-6 wind tunnel [24]. The results reported here are under the conditions of Ma1 = 6 and a = 4°.
Table 2 Flow parameters for Mee’s flat plate model.
Ma Static pressure kPa Static temperature K Static density kg/m3 Unit Reynolds number m1
grid shows significant deviation from the others, while the results of the fine and finest grids coincide with each other very well. Here the position where the Stanton number departs from the laminar value is defined as the transition onset. The locations of transition onset are almost the same except the coarse grid. By the comparisons above, the grid convergence is established when the grid resolution reaches the fine grid or finest grid levels. In addition, the grids of the following test cases are all based on the finest grid resolution level. Fig. 5 compares the experimental [23] and computational Stanton numbers at three different Reynolds numbers. The locations of transition onset, predicted with the improved method, are almost consistent with the experimental results. Transition onset location moves downstream and the length of transition zone becomes longer with decreasing freestream Reynolds number. As to the original method, there is no transition onset in Fig. 5(a) and (b) and transition onset delays in Fig. 5(c) compared with the experiment. Therefore, the improved method is more excellent than the original one in predicting Stanton number and transition onset. While one point is needed to state that the Stanton numbers, predicted by both the improved and the original methods, are a little larger than the experimental result in the turbulent region. It should be on account of the performance of the turbulence model and some work about those will be conducted in the future study.
Low Re
Middle Re
High Re
6.3 2.8 570 0.017 1.7 106
6.2 5.4 690 0.027 2.6 106
6.1 12.1 800 0.053 4.9 106
Table 3 Grid resolutions for convergence analysis. Grids
Number of cells in axial direction
Number of cells in normal direction
y+
Coarse Medium Fine Finest
170 210 250 290
70 110 150 190
2.0 1.0 0.7 0.4
listed in Table 3 and y+ decreases from 2.0 to 0.4 with increasing resolutions. Fig. 4 shows the comparison of the Stanton number St obtained with four different grids: coarse, medium, fine and finest. The flow parameters are the middle Re condition. The result of the coarse
Fig. 5. Stanton number distributions at different Reynolds numbers.
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Fig. 7. Basic flow and the wall pressure distribution of the X-51A forebody.
Fig. 6. Computational domain of the X-51A forebody model.
Fig. 6 shows the computational domain. With y+ = 0.4 and 2.83 million cells (half model), the grid resolution is sufficient to obtain grid-converged results. The computational parameters are listed in Table 4, which are used to simulate the quiet and noisy wind tunnel conditions respectively. The basic flow and the wall pressure distribution of the X-51A forebody configuration are presented in Fig. 7. The flow separation occurs on the sideward and a shock wave generates near the compression corner on the windward side. Comparing the distribution of surface pressure, the pressure on the second compression ramp increases obviously and the pressure on the leeward side is relatively lower than that on the windward side. Figs. 8 and 9 compare the computational results obtained by the improved and the original methods with the experimental measurements of Borg [24]. Specifically, Fig. 8 shows the comparison under the quiet wind tunnel condition while Fig. 9 shows that under the noisy wind tunnel condition. The experimental measurements are presented using wall temperature, which is hard to be obtained accurately for current computations. Because the properties of the model material and radiation heat transfer should be taken into account when obtaining the wall temperature in computations. Therefore, obtaining the wall temperature would take more efforts than obtaining the heat transfer rate. Considering that, this paper puts emphasis on the prediction of transition onset. The heat transfer rate shows the same trend with the wall temperature when transition occurs, so heat transfer rate is adopted to identify the transition onset in this test case. For the quiet wind tunnel condition, the experimental measurement and the computational results of the two methods are all laminar as shown in Fig. 8. While for the noisy wind tunnel condition, the wall temperature increases obviously at x = 18 cm–24 cm in the experimental measurement and the transition onset is in a ‘‘V” shape as shown in Fig. 9(a). Comparing the results of the improved method in Fig. 9 (b) with the experimental measurement, the trends of the transition onset are almost the same. However, the result of the original method is still in a laminar state. Therefore, the improved model predicts the transition at different wind tunnel noise levels more accurately than the original c-Reht method. Fig. 10 shows the centerline wall temperatures of the experimental measurement under both quiet and noisy wind tunnel con-
(a) Wall temperature of experiment [24]
(b) Heat transfer rate of improved method
(c) Heat transfer rate of original method Fig. 8. Comparison of transition onset under quiet wind tunnel condition (Re1 = 6.6 106/m).
ditions. Fig. 11 shows the centerline heat transfer rate of the computational results. Under the quiet wind tunnel condition, no transition was predicted by the original method or the improved method, which is consistent with the experiment. For the noisy wind tunnel condition, the heat transfer rate predicted by the original method coincides with the laminar result, while the heat transfer rate predicted by the improved method changes at three different positions. The first position where the heat transfer rate departs from the laminar at x = 12.5 cm, the second position where the heat transfer rate has a sudden rise at about x = 18.0 cm, and the third position where there is a peak in the heat transfer rate at x = 26.0 cm. For the experimental measurement, the corresponding three positions are about at x = 13.1 cm, 17.8 cm and 26.8 cm as shown in Fig. 10. The first position of the experimental results is determined according to the measured data at different
Table 4 Computational parameters for X-51A at different wind tunnel noise levels.
Quiet tunnel Noise tunnel
Re1(1/m)
T0(K)
P0(kPa)
Tu
Tw(K)
6.6 106 7.4 106
418 424
586 621
0.05% 1.2%
300 300
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335
(a) Wall temperature of experiment [24]
(b) Heat transfer rate of improved method Fig. 11. Heat transfer rate distributions of computation along windward centerline under different wind tunnel conditions.
(c) Heat transfer rate of original method Fig. 9. Comparison of transition onset under noisy wind tunnel condition (Re1 = 7.4 106/m).
314
6
Re=6.6x10 /m,Quiet 6 Re=7.4x10 /m,Noisy
312
(a) Windward T(K)
310
26.8cm 17.8cm
308
13.1cm 306
(b) Leeward
304
302
12
14
16
18
20
22
24
26
28
30
32
34
Fig. 12. Distribution of surface streamlines and heat transfer rate under noisy wind tunnel condition.
x(cm) Fig. 10. Temperature distributions of experiment along windward centerline under different wind tunnel conditions [24].
Reynolds numbers in Ref. [24]. Compared with the experimental results, the relative errors of the prediction results by the improved method are less than 5%. Fig. 12 presents the distribution of surface streamlines and heat transfer rate under noisy condition. On the windward side, there is an obvious separation in the compression corner and the transition onset presents an ‘‘M” shape. Fig. 13 shows the amplified separation in the compression corner, and it can be seen that the separation is larger near the sideward and smaller near the centerline region. There is an obvious relationship between the size of separation and the transition onset. The size of the separation near the two peaks of the ‘‘M” shape is a little larger than other regions. As transition onset is greatly related to the vorticity Reynolds number in the c-Reht model, the vorticity Reynolds number distribution at x = 145 mm is shown in Fig. 14 for further analysis. The cross section location of x = 145 mm is shown in Fig. 15, which is downstream of the separation and upstream of the transition
Fig. 13. Separation in the compression corner.
Fig. 14. Distribution of vorticity Reynolds number at x = 145 mm.
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Fig. 15. Location of cross section at x = 145 mm.
onset. Compared with the shape of separation in Fig. 13, the vorticity Reynolds number distribution has an obvious relevance with the separation. The vorticity Reynolds number is larger following the large separation. It can be concluded that the large separation causes the transition onset to move forward.
3.3. X-51A forebody at different freestream Reynolds numbers In order to assess the performance of the improved c-Reht method at different Reynolds numbers, Fig. 16 compares the computational results with the experimental measurements on the leeward side of the forebody at Re1 = 3.2 106/m, 7.1 106/m and 11.7 106/m. The experimental results are presented using wall temperature and the computational results are shown with heat transfer rate. No transition is observed at Re1 = 3.2 106/m. As Re1 increases, transition onset moves forward on the leeward side of the forebody. The transition onset is in a ‘‘K” shape near the cen-
terline on the leeward side. The shape and the trend of the computational transition onsets show a good agreement with the experimental measurements. Therefore, the improved c-Reht method can effectively predict the transition onset at different Reynolds numbers. Figs. 17 and 18 present the transition distributions along the leeward centerline. The positions of transition onset and the peaks of heat transfer rate are marked by arrows. The transition distributions predicted by the improved c-Reht method show an agreement with the experiment at Re1 = 3.2 106/m and 11.7 106/m. However, there is a little delay of transition onset at Re1 = 7.1 106/m. The positions of computational transition onset are at x = 14.8 cm and 21.4 cm for Re1 = 11.7 106/m and Re1 = 7.1 106/m respectively, while the experimental results are about x = 15.1 cm and 18.4 cm respectively. As for the peaks positions at Re1 = 11.7 106/m, they are at about x = 24.2 cm and x = 24.0 cm for computation and experiment respectively. Figs. 19 and 20 show the heat transfer rate contours on the windward side and the heat transfer rate curves along the windward centerline at different Reynolds numbers respectively. Transition doesn’t take palce at Re1 = 3.2 106/m, that is the same as the result on leeward side. The transition onset is in a ‘‘V” shape distribution in the range of x = 18 cm 24 cm at Re1 = 7.1 106/ m, while in a ‘‘W” shape in the range of x = 13.5 cm 15 cm at Re1 = 11.7 106/m. Through the above comparisons, it can be concluded that the improved c-Reht method can successfully predict the boundary layer transition of hypersonic flow at different wind tunnel noise levels and freestream Reynolds numbers.
(a) Temperature of experiment, Re∞=3.2×106/m
(b) Heat transfer rate, Re∞=3.2×106/m
(c) Temperature of experiment, Re∞=7.1×106/m
(d) Heat transfer rate, Re∞=7.1×106/m
(e) Temperature of experiment, Re∞=11.7×106/m
(f)
Heat transfer rate, Re∞=11.7×106/m
Fig. 16. Comparison of transition onset on the leeward side at different Reynolds numbers. (left: experimental results [24], right: computational results with improved c-Reht method).
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307
6
Re= 3.2 x106/m Re= 7.1 x106/m Re=11.7x106/m
306.5
8
306
24cm
15.1cm
Q(kW/m2)
T(K)
305.5
Re= 3.2 x10 ,Transition Re= 7.1 x106,Transition Re=11.7x106,Transition Re= 3.2 x1066,Laminar Re= 7.1 x10 6,Laminar Re=11.7x10 ,Laminar
305 304.5
6
4
18.4cm 304
2 303.5 303 14
16
18
20
22
24
26
28
30
32
34
0
x(cm) Fig. 17. Temperature distributions of experiment along leeward centerline at different Reynolds numbers [24].
12
14
16
18
20
22
24
26
28
30
32
34
x(cm) Fig. 20. Heat transfer rate distributions of computation along windward centerline at different Reynolds numbers.
4. Conclusion In this paper, the the heat transfer of Two test cases have the improved c-Reht drawn as follows.
Fig. 18. Heat transfer rate distributions of computation along leeward centerline at different Reynolds numbers.
original c-Reht model is extended to predict hypersonic boundary layer transition flow. been selected to assess the performance of transition model. Some conclusions can be
(1) A new correlation of the momentum thickness Reynolds number based on the hypersonic flow is first presented in this paper. In the correlation, the Mach number and Reynolds number are introduced to reflect the compressible effect of hypersonic flow. The function Fonset1 used to control the transition onset and several relevant model parameters are also modified to make the c-Reht model suitable for hypersonic flow. (2) Compared with the original model, the improved c-Reht model can successfully predict the changes of the Stanton number and the heat transfer rate caused by boundary layer transition. The model can also reasonably reflect the effects of freestream Reynolds number and wind tunnel noise level on hypersonic boundary layer transition. (3) The heat transfer rate distributions of the X-51A forebody are analyzed in this paper. The windward side transition is found to be related to the separation in the compression corner and large separation can cause earlier transition onset in boundary layer. (4) Although the improved c-Reht model shows a good performance in predicting hypersonic boundary layer transition of a flat plate and the X-51A forebody, more test cases still need to be considered in the future work to further validate the performance of the improved c-Reht model.
Acknowledgement
Fig. 19. Distributions of heat transfer rate on the windward side at different Reynolds numbers.
The authors acknowledge the technical support provided by National Laboratory for Computational Fluid Dynamics in this investigation.
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References [1] Y.F. Wang, F. Chen, H.P. Liu, et al., Large eddy simulation of unsteady transitional flow on the low-pressure turbine blade, Sci. China 57 (9) (2014) 1761–1768. [2] V.C. Patel, W. Rodi, G. Scheuerer, Turbulence models for near-wall and low Reynolds number flows: a review, AIAA J. 23 (9) (2012) 1308–1319. [3] D. Wilcox, Turbulence Modelling for CFD, DCW Industries, La Canada, CA, 1992, Chap 3. [4] D.K. Walters, D. Cokljat, A three-equation eddy-viscosity model for Reynoldsaveraged Navier-Stokes simulations of transitional flow, J. Fluids Eng. 130 (2008) 121401. [5] L. Wang, S. Fu, Modelling flow transition in a hypersonic boundary layer with Reynolds-averaged Navier-Stokes approach, Sci. China-Phys. Mech. Astron. 52 (2009) 768–774. [6] S. Fu, L. Wang, RANS modeling of high-speed aerodynamic flow transition with consideration of stability theory, Prog. Aerosp. Sci. 58 (2013) 36–59. [7] L. Wang, S. Fu, A. Carnarius, et al., A modular RANS approach for modelling laminar–turbulent transition in turbomachinery flows, Int. J. Heat Fluid Flow 34 (2012) 62–69. [8] L. Zhou, C. Yan, Z.H. Hao, et al., Improved k-x-c model for hypersonic boundary layer transition prediction, Int. J. Heat Mass Transfer 94 (2016) 380–389. [9] W.X. Kong, Investigation of Hypersonic Boundary Layer Transition Modeling, Beihang University, 2013 (Ph.D. thesis). [10] Z.H. Hao, C. Yan, L. Zhou, Parametric study of a k-x-c model in prediction hypersonic boundary-layer flow transition, Chin. J. Theor. Appl. Mech. 47 (2015) 215–222. [11] Z.X. Xiao, M.H. Zhang, L.H. Xiao, et al., Studies of roughness-induced transition using three-equation k-x-c transition/turbulence model, in: 43rd Fluid Dynamics Conference, San Diego, 2013, June 24–27. [12] R.B. Langtry, F.R. Menter, Transition modeling for general CFD applications in aeronautics, AIAA Paper, 2005, AIAA-2005-522.
[13] R.B. Langtry, F.R. Menter, S.R. Likki, et al., A correlation-based transition model using local variables—Part II: Test cases and industrial applications, J. Turbo 128 (2006) 423–434. [14] F.R. Menter, R.B. Langtry, S. Völker, Transition modelling for general purpose CFD codes, Flow Turbul. Comb. 77 (2006) 277–303. [15] R.B. Langtry, F.R. Menter, Correlation-based transition modeling for unstructured parallelized computational fluid dynamics codes, AIAA J. 47 (2009) 2894–2906. [16] C. Grabe, A. Krumbein, Extension of the c-Reht model for prediction of crossflow transition, AIAA Paper, 2014, AIAA-2014-1269. [17] R.B. Langtry, K. Sengupta, D.T. Yeh, et al., Extending the c-Reht local correlation based transition model for crossflow effects, AIAA Paper, 2015, AIAA-20152474. [18] Y.C. You, H. Luedeke, T. Eggers, Application of the c-Reht transition model in high speed flows, in: 18th AIAA/3AF International Space Planes and Hypersonic Systems and Technologies Conference, Tours, France, 2012, Sep 24–28. [19] S. Frauholz, B. Reinartz, S. Muller, et al., Transition prediction for scramjets using c-Reht model coupled to two turbulence models, J. Propul. Power 31 (2015) 5. [20] G.B. Schubauer, P.S. Klebanoff, Contribution on the mechanics of boundary layer transition, NACATN 3489 (1955). [21] D.C. Wilcox, Turbulence model transition prediction, AIAA J. 13 (2) (1975) 241–243. [22] Z.H. Hao, C. Yan, L. Zhou, et al., Development of boundary layer parameters identification method for transition prediction with complex grids, Proc. IMechE Part G: J. Aerosp. Eng. (2016), http://dx.doi.org/10.1177/ 0954410016660872. [23] D.J. Mee, Boundary-layer transition measurements in hypervelocity flows in a shock tunnel, AIAA J. 40 (2002) 1542–1548. [24] M.P. Borg, Instability and transition on the X-51 (Ph.D. thesis), Purdue University School of Aeronautics & Astronautics, 2009.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Improved c-Reht model for heat transfer prediction of hypersonic boundary layer transition Zihui Hao a, Chao Yan a,⇑, Yupei Qin a, Ling Zhou b a b
School of Aeronautics Science and Engineering, Beihang University, 100191 Beijing, China School of Energy and Power Engineering, Huazhong University of Science & Technology, 430074 Wuhan, China
a r t i c l e
i n f o
Article history: Received 1 July 2016 Received in revised form 13 November 2016 Accepted 14 November 2016
Keywords: Heat transfer rate Boundary layer transition c-Reht model Hypersonic Momentum thickness Reynolds number
a b s t r a c t An improved c-Reht model has been developed to predict the heat transfer of hypersonic boundary layer transition in this paper. A new correlation of momentum thickness Reynolds number for hypersonic boundary layer flow is first presented using local flow variables. In the correlation, the Mach number and Reynolds number are introduced to reflect the compressible effect of hypersonic flow. The function Fonset1 used to control the transition onset as well as several relevant model parameters are also modified to make the c-Reht model suitable for hypersonic flow. Two test cases including the hypersonic flow over a flat plate and the X-51A forebody with different Reynolds numbers and wind tunnel noise levels are employed to assess the performance of the improved c-Reht model. Compared with the original model and various experiments, the improved c-Reht model can successfully predict the changes of Stanton number and heat transfer rate caused by the boundary layer transition. The model can also reasonably reflect the effects of freestream Reynolds number and wind tunnel noise level on hypersonic boundary layer transition. By analyzing the distributions of heat transfer rate on the X-51A forebody, the windward side transition is found to be related to the separation in the compression corner and large separation can cause earlier transition onset in boundary layer. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Boundary layer transition is the process from laminar to turbulence flow. Since the aeroheating of turbulence is much higher than that of laminar especially in hypersonic flow, improper thermal protection design may greatly reduce the performance of aerospace vehicles. Therefore, the accuracy of boundary layer transition prediction is of great importance for thermal protection in the design of hypersonic vehicles. In addition, boundary layer transition is also important for separation and drag prediction. Up to now, various numerical methods for boundary layer transition prediction have been developed, such as semi-empirical eN method, direct numerical simulation (DNS), large eddy simulation (LES), transition models and so on. Among those, eN method can describe the linear stability process accurately. However, it still needs more work to apply the method to complex 3D configurations. As it is hard to couple eN method with modern CFD methods, aeroheating cannot be predicted with the transition prediction at the same time. DNS can help to understand the mechanisms of boundary layer transition, such as the mechanisms of receptivity, ⇑ Corresponding author. E-mail address: [email protected] (C. Yan). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.11.052 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
the evolution process of turbulent spots. However, due to the extraordinarily large computational requirements, DNS is not clearly yet able to be applied in engineering design. LES approach has been applied to predict flow transition on a turbine blade [1], while it still requires a large amount of computing resources relative to transition models. Based on RANS equations, transition models offer a compromise between the accuracy and calculation expense, and they have been widely applied in engineering. In recent years, transition models have been developed rapidly. Low Reynolds number models [2] have a certain ability to predict transition. However, Wilcox [3] proves that the ability of transition prediction is attributed to the mathematical properties of the turbulence model. Walters [4] introduces a ‘‘splat mechanism” to describe the growth of laminar kinetic energy kL in the kT-kL-x method. In order to predict natural transition and bypass transition with Walters’ method, some special modifications based on local mean flow and laminar kinetic energy are made in the kL production term and the transition production term. Considering the intermittency and Mack modes instability, Wang and Fu [5–7] propose a three equations k-x-c method. Considering the effects of Reynolds number and nose bluntness, Zhou [8] improves the performance of the original method by reconstructing the c transport equation. The k-x-c model has a good performance in predicting
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Nomenclature Uj
c ceff q S X k Pk Dk Pc Dc Pht Re1 Reh Rev Rehc e ht Re xj
velocity vector intermittency factor effective intermittency factor fluid density absolute value of strain rate, (2SijSij)1/2 absolute value of vorticity, (2XijXij)1/2 turbulence kinetic energy production term of k transport equation dissipation term of k transport equation production term of c transport equation dissipation term of c transport equation source term of Reht transport equation freestream Reynolds number momentum thickness Reynolds number vorticity Reynolds number critical momentum thickness Reynolds number local transition onset momentum thickness Reynolds number (obtained from a transport equation) Cartesian coordinates
hypersonic boundary layer transition validated by Kong [9], Hao [10], Xiao [11] and others. Langtry and Menter [12–15] propose a correlation-based c-Reht transition model, which is built strictly on local variables and suitable for unstructured grids and massively parallel execution. The method can predict natural transition, bypass transition and the transition induced by separation. Recently, Grabe [16] and Langtry [17] extend this method to predict cross flow transition. Among the methods above, the c-Reht model is widely coupled with some commercial softwares, such as FLUENT, FUN3D, SU2, OVERFLOW, etc., and it may be the most popular method for subsonic transition prediction. In recent years, some researchers have attempted to extend the c-Reht model to predict hypersonic boundary layer transition. For example, You [18] applies the c-Reht model in hypersonic flows. Langtry and Menter’s correlations are replaced by Krause’s correlations and the effect of pressure gradient is taken into consideration in the transitional flow in the research of You [18]. Frauholz [19] couples the c-Reht transition model with the SSG/LRR x turbulence model, and new correlations of Rehc and Flenght are implemented in the c-Reht transition model. The modified model shows good performance in predicting hypersonic boundary layer transition of scramjets. This paper focuses on the c-Reht model for heat transfer prediction in hypersonic transitional flows. The original c-Reht model is built based on low speed flow, and a correlation about the momentum thickness Reynolds number is obtained from a Blasius boundary layer, which limits the application of the original method for hypersonic flow. Therefore, the original model is unsuitable for predicting hypersonic boundary layer transition. A new method is proposed in this paper to extend the original c-Reht model to hypersonic flow. The main idea is different from previous researchers’ work. A new correlation of momentum thickness Reynolds number for hypersonic boundary layer flow is first presented in this paper and the correlation is applied in the c-Reht model. The performance of the improved c-Reht model is assessed by two cases at different freestream Reynolds numbers. The prediction ability of hypersonic boundary layer transition at different wind tunnel noise levels is validated in this paper as well. All the predicted results are compared with the experimental results.
y Ma Tu P0 T0 Tw T h Q St
l lt Fonset Flength d Reht
distance to nearest wall Mach number freestream turbulence intensity stagnation pressure stagnation temperature wall temperature temperature momentum thickness heat transfer rate Stanton number molecular viscosity eddy viscosity transition onset function transition length function boundary layer thickness transition onset momentum thickness Reynolds number (based on freestream conditions)
2. Transition model 2.1. Original c-Reht transition model The original c-Reht model, proposed by Langtry and Menter [12] in 2005, is built with local correlations and easy to couple with modern CFD codes. The model is based on the shear stress transport (SST) turbulence model. Two additional transport equations are supplemented for transition prediction: one is for the intermittency factor c, and the other is for the local transition onset e ht , as follows: momentum thickness Reynolds number Re
@ðqcÞ @ qU j c @ l @c þ Pc Dc þ ¼ lþ t @t @xj rc @xj @xj
" # e ht e ht Þ @ qU j Re e ht @ðq Re @ @ Re þ Pht ¼ rht ðl þ lt Þ þ @t @xj @xj @xj
ð1Þ
ð2Þ
where q, Uj and xj stand for fluid density, velocity vector and Cartesian coordinates respectively. l and lt are molecular viscosity and turbulence eddy viscosity respectively. rc, rht are model constants e ht equations, and rc = 1.0, rht = 2.0. for c and Re Pc is the production term of c equation as shown below:
Pc ¼ F length ca1 qS½cF onset 0:5 ð1 ce1 cÞ
ð3Þ
e ht based on the flat plates T3A, T3B, where Flength is a correlation of Re and the test cases from Schubauer and Klebanof [20]. The function of Flength is used to control the length of transition zone and it is first published by Langtry and Menter [15]. S represents the strain-rate magnitude. ca1, ce1 are model constants, here ca1 = 2.0, ce1 = 1.0. The transition onset is controlled by the following functions: Rem F onset1 ¼ 2:193Re hc F onset2 ¼ min maxðF onset1 ; F 4onset1 Þ; 2:0 R 3 T F onset3 ¼ max 1 2:5 ;0
F onset ¼ max ðF onset2 F onset3 ; 0Þ
ð4Þ
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e ht and it is the critical Reynolds where Rehc is a correlation of Re number where the intermittency initially starts to increase in the boundary layer. Rev represents the vorticity Reynolds number:
Rem ¼ qSy l
2
or
qXy2 l
ð5Þ
Ec represents the dissipation term of c transport equation:
E ¼ ca2 qcF turb ðce2 c 1Þ F turb ¼ e
R 4 T 4
ð6Þ ð7Þ
where X is the vorticity magnitude. The source term Pht of the transition momentum thickness Reynolds number transport equation is defined as follows:
Pht ¼ cht
q t
~ ht ð1:0 F ht Þ Reht Re
ð8Þ
By using a blending function Fht, which equals one in the bound-
e ht to ary layer and zero in the freestream, Pht is designed to force Re match Reht outside the boundary layer. Reht is the transition onset momentum thickness Reynolds number, which is based on freestream conditions.
Reht ¼
qht U 0 l
ð10Þ
When coupling the transition equations with SST turbulence model, the transport equation of turbulence kinetic energy k is modified as follows:
@k @ @ @ ek D ek þP ðqkÞ þ ðquj kÞ ¼ l þ rk lt @t @xj @xj @xj e k ¼ c Pk ; D e k ¼ min maxðc ; 0:1Þ; 1:0 Dk P eff eff
l
ð12Þ
The correlation in Eq. (12), based on the observation of a Blasius boundary layer, is first proposed by Wilcox [21]. The application of the correlation can avoid additional nonlocal boundary layer parameters for transition prediction. This idea has also been used as a reference in other transition methods based on local variables. For example, Wang and Fu introduce the X in the effective length scale in the k-x-c model, which contains the information of boundary layer thickness. Fig. 1 shows the scaled vorticity Reynolds number Rev profile in a laminar boundary layer. The data comes from the computational results of a flat plate at Reynolds number 6 106/m. The freestream Mach numbers equal to 0.3 and 6 respectively. For the subsonic flow, the Rev,max is closed to the 2.193 times the magnitude of Reh, while the correlation is obviously unsuitable for hypersonic flow as shown in Fig. 1. A new correlation of Rev,max and Reh for hypersonic flow is presented as follows:
K ¼ 2:1 0:0895 Ma1
ð13Þ
D ¼ 300 þ 0:0000275 Re1
h i Reht ¼ 1173:51 589:428Tu þ 0:2196 Fðkht Þ; Tu 6 1:3 Tu2
qh2 dU kht ¼ t l ds
qUh Rem;max ¼ 2:193Reh ¼ 2:193
Rem;max ¼ KReh þ D ð9Þ
where ht is the transition onset momentum thickness and it can be obtained from an empirical correlation as follows:
Reht ¼ 331:50½Tu 0:56580:671 Fðkht Þ; Tu 6 1:3
transition onset. Among the three functions, Fonset1 is the most important term, which is based on a correlation as follows:
where Reh is the momentum thickness Reynolds number which is not a local flow variable. A boundary layer parameters identification method is used to obtain Reh accurately in the following test case. A detailed description of the boundary layer parameters identification method is presented in Ref. [22]. The location where Ue = 0.99U1 is defined as the boundary layer edge in this paper. In the new correlation, the Mach number and Reynolds number are introduced to reflect the compressible effect of hypersonic flow. Between the two parameters, the freestream Reynolds number is shown as follows:
Re1 ¼ ð11Þ
The equation of the specific turbulence dissipation rate x is the same with the original SST turbulence model. 2.2. Modification of the c-Reht model for hypersonic flow The original c-Reht model is proposed mainly based on experimental data in low speed flow, and the excellent performance of the method has been validated by Langtry and Menter especially in subsonic flow. So far, the c-Reht model has been widely used for general aviation applications. Langtry and Menter point out that the main missing extensions are crossflow instabilities and high speed flow correlation in the original method [15]. Many researchers have tried to extend the c-Reht model to hypersonic flow in recent years. Among those, You [18] and Frauholz [19] have successfully extended the original c-Reht model in predicting hypersonic boundary layer transition. They put emphasis on the modification with new correlations of Rehc and Flenght, however, the correlation about Rev,max and Reh which is unsuitable for hypersonic flow is ignored in their attempt. In this paper, the ignored correlation is taken into account and the detailed modification of the model for hypersonic flow is presented as follows. In the original c-Reht model, the function Fonset, composed of three functions (Fonset1, Fonset2 and Fonset3), is used to control the
q1 U 1 Lref l
ð14Þ
where Lref is the reference length and equals one in this paper. A flat plate case is used to obtain the relationship between Rev,max and Reh in hypersonic flow. The flow parameters are listed in Table 1. The wall temperature Tw equals 300K. Fig. 2 shows the comparison of the new and the original correlations with computational results in hypersonic flow. The new
Fig. 1. Scaled vorticity Reynolds number Rev profile in a laminar boundary layer.
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2 10 4 106 6 106 8 106
Ma1 4,6,8 4,6,8 4,6,8 4,6,8
correlation shows a good agreement with the computational results, however, the original correlation has an obvious deviation in Fig. 2. It can be concluded that the new correlation is more suitable for hypersonic flow than the original one. Both the new and the original correlations are linear. The obvious difference is that the intercept term of the original correlation is zero while it is a function of Re1 in the new correlation. When the new correlation is directly applied in the c-Reht method instead of the original one, the intermittency near the wall in turbulent region is obviously less than the normal level. That will lead to an insufficient viscosity in the region near wall. In order to solve the problem, a function Fr is constructed as follows:
Fr ¼ e
5Rev 2D2
ð15Þ
Fr is a function of Rev, similar to Fturb in the construction. The function Fturb can be found in the dissipation term of c transport equation. Finally, Fonset1 is modified as follows in the improved c-Reht model:
F onset1 ¼
Rem Dð1 F r Þ KRehc
ð16Þ
A flat plate is used to analyze the characteristic of function Fr. The length of the flat plate is 1500 mm and the freestream conditions are: Ma1 = 6.0, Re1 = 2.6 106/m. Fig. 3(a) shows the computational grid and Mach number contour within x = 1300 mm–1500 mm, which is in the turbulent region. Fig. 3 (b) shows the distribution of the function Fr. The value of Fr equals one near the wall and the region outside the boundary layer edge. It is zero in the most region within the boundary layer. Fig. 3 (c) and (d) show the comparison of the results with and without the application of Fr. Adopting Fr, the intermittency backs to the normal value in Fig. 3(d). Several relevant model parameters are also recalibrated to make the c-Reht model suitable for hypersonic flow as follows:
F length ¼ 20;
ce2 ¼ 20
ð17Þ
Other model parameters and a detailed formulation of the c-Reht transition model can be found in Refs. [13–15]. 3. Test cases Two cases have been selected to assess the performance of the improved c-Reht method for hypersonic flows in this section. One is a hypersonic flat plate and the other is the X-51A forebody configuration. The X-51A test case is analyzed with different wind tunnel noise levels and freestream Reynolds numbers respectively. Fig. 2. Comparison of the new and the original correlations with computational results in hypersonic flow.
3.1. A flat plate at Ma1 = 6 In this test case, hypersonic boundary layer flow over a flat plate is analyzed to evaluate the effectiveness of the improved c-Reht transition model. The flat plate model is the same as that shown in Fig. 3(a) and the flow conditions are consistent with the experimental study of Mee [23]. The wall temperature is Tw = 300K,
the freestream turbulence intensity is Tu = 0.4% and other detailed flow parameters are shown in Table 2. Grid convergence analysis is necessary for numerical study. Four grids with successively increasing resolutions are used to analyze the grid convergence of the improved method. The details are
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0.002
St
0.0015 0.001
Coarse Medium Fine Finest
0.0005 0
500
x(mm)
1000
1500
Fig. 4. Grid convergence analysis.
(a) Mach number contour and computational grid
(b) Distribution of the function Fr
(c) Intermittency factor without the application of Fr
(d) Intermittency factor with the application of Fr
Fig. 3. The test case for the function Fr.
3.2. X-51A forebody at different wind tunnel noise levels The test case is a 20% scale X-51A forebody model, which is tested in the Boeing/AFOSR Mach-6 wind tunnel [24]. The results reported here are under the conditions of Ma1 = 6 and a = 4°.
Table 2 Flow parameters for Mee’s flat plate model.
Ma Static pressure kPa Static temperature K Static density kg/m3 Unit Reynolds number m1
grid shows significant deviation from the others, while the results of the fine and finest grids coincide with each other very well. Here the position where the Stanton number departs from the laminar value is defined as the transition onset. The locations of transition onset are almost the same except the coarse grid. By the comparisons above, the grid convergence is established when the grid resolution reaches the fine grid or finest grid levels. In addition, the grids of the following test cases are all based on the finest grid resolution level. Fig. 5 compares the experimental [23] and computational Stanton numbers at three different Reynolds numbers. The locations of transition onset, predicted with the improved method, are almost consistent with the experimental results. Transition onset location moves downstream and the length of transition zone becomes longer with decreasing freestream Reynolds number. As to the original method, there is no transition onset in Fig. 5(a) and (b) and transition onset delays in Fig. 5(c) compared with the experiment. Therefore, the improved method is more excellent than the original one in predicting Stanton number and transition onset. While one point is needed to state that the Stanton numbers, predicted by both the improved and the original methods, are a little larger than the experimental result in the turbulent region. It should be on account of the performance of the turbulence model and some work about those will be conducted in the future study.
Low Re
Middle Re
High Re
6.3 2.8 570 0.017 1.7 106
6.2 5.4 690 0.027 2.6 106
6.1 12.1 800 0.053 4.9 106
Table 3 Grid resolutions for convergence analysis. Grids
Number of cells in axial direction
Number of cells in normal direction
y+
Coarse Medium Fine Finest
170 210 250 290
70 110 150 190
2.0 1.0 0.7 0.4
listed in Table 3 and y+ decreases from 2.0 to 0.4 with increasing resolutions. Fig. 4 shows the comparison of the Stanton number St obtained with four different grids: coarse, medium, fine and finest. The flow parameters are the middle Re condition. The result of the coarse
Fig. 5. Stanton number distributions at different Reynolds numbers.
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Fig. 7. Basic flow and the wall pressure distribution of the X-51A forebody.
Fig. 6. Computational domain of the X-51A forebody model.
Fig. 6 shows the computational domain. With y+ = 0.4 and 2.83 million cells (half model), the grid resolution is sufficient to obtain grid-converged results. The computational parameters are listed in Table 4, which are used to simulate the quiet and noisy wind tunnel conditions respectively. The basic flow and the wall pressure distribution of the X-51A forebody configuration are presented in Fig. 7. The flow separation occurs on the sideward and a shock wave generates near the compression corner on the windward side. Comparing the distribution of surface pressure, the pressure on the second compression ramp increases obviously and the pressure on the leeward side is relatively lower than that on the windward side. Figs. 8 and 9 compare the computational results obtained by the improved and the original methods with the experimental measurements of Borg [24]. Specifically, Fig. 8 shows the comparison under the quiet wind tunnel condition while Fig. 9 shows that under the noisy wind tunnel condition. The experimental measurements are presented using wall temperature, which is hard to be obtained accurately for current computations. Because the properties of the model material and radiation heat transfer should be taken into account when obtaining the wall temperature in computations. Therefore, obtaining the wall temperature would take more efforts than obtaining the heat transfer rate. Considering that, this paper puts emphasis on the prediction of transition onset. The heat transfer rate shows the same trend with the wall temperature when transition occurs, so heat transfer rate is adopted to identify the transition onset in this test case. For the quiet wind tunnel condition, the experimental measurement and the computational results of the two methods are all laminar as shown in Fig. 8. While for the noisy wind tunnel condition, the wall temperature increases obviously at x = 18 cm–24 cm in the experimental measurement and the transition onset is in a ‘‘V” shape as shown in Fig. 9(a). Comparing the results of the improved method in Fig. 9 (b) with the experimental measurement, the trends of the transition onset are almost the same. However, the result of the original method is still in a laminar state. Therefore, the improved model predicts the transition at different wind tunnel noise levels more accurately than the original c-Reht method. Fig. 10 shows the centerline wall temperatures of the experimental measurement under both quiet and noisy wind tunnel con-
(a) Wall temperature of experiment [24]
(b) Heat transfer rate of improved method
(c) Heat transfer rate of original method Fig. 8. Comparison of transition onset under quiet wind tunnel condition (Re1 = 6.6 106/m).
ditions. Fig. 11 shows the centerline heat transfer rate of the computational results. Under the quiet wind tunnel condition, no transition was predicted by the original method or the improved method, which is consistent with the experiment. For the noisy wind tunnel condition, the heat transfer rate predicted by the original method coincides with the laminar result, while the heat transfer rate predicted by the improved method changes at three different positions. The first position where the heat transfer rate departs from the laminar at x = 12.5 cm, the second position where the heat transfer rate has a sudden rise at about x = 18.0 cm, and the third position where there is a peak in the heat transfer rate at x = 26.0 cm. For the experimental measurement, the corresponding three positions are about at x = 13.1 cm, 17.8 cm and 26.8 cm as shown in Fig. 10. The first position of the experimental results is determined according to the measured data at different
Table 4 Computational parameters for X-51A at different wind tunnel noise levels.
Quiet tunnel Noise tunnel
Re1(1/m)
T0(K)
P0(kPa)
Tu
Tw(K)
6.6 106 7.4 106
418 424
586 621
0.05% 1.2%
300 300
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335
(a) Wall temperature of experiment [24]
(b) Heat transfer rate of improved method Fig. 11. Heat transfer rate distributions of computation along windward centerline under different wind tunnel conditions.
(c) Heat transfer rate of original method Fig. 9. Comparison of transition onset under noisy wind tunnel condition (Re1 = 7.4 106/m).
314
6
Re=6.6x10 /m,Quiet 6 Re=7.4x10 /m,Noisy
312
(a) Windward T(K)
310
26.8cm 17.8cm
308
13.1cm 306
(b) Leeward
304
302
12
14
16
18
20
22
24
26
28
30
32
34
Fig. 12. Distribution of surface streamlines and heat transfer rate under noisy wind tunnel condition.
x(cm) Fig. 10. Temperature distributions of experiment along windward centerline under different wind tunnel conditions [24].
Reynolds numbers in Ref. [24]. Compared with the experimental results, the relative errors of the prediction results by the improved method are less than 5%. Fig. 12 presents the distribution of surface streamlines and heat transfer rate under noisy condition. On the windward side, there is an obvious separation in the compression corner and the transition onset presents an ‘‘M” shape. Fig. 13 shows the amplified separation in the compression corner, and it can be seen that the separation is larger near the sideward and smaller near the centerline region. There is an obvious relationship between the size of separation and the transition onset. The size of the separation near the two peaks of the ‘‘M” shape is a little larger than other regions. As transition onset is greatly related to the vorticity Reynolds number in the c-Reht model, the vorticity Reynolds number distribution at x = 145 mm is shown in Fig. 14 for further analysis. The cross section location of x = 145 mm is shown in Fig. 15, which is downstream of the separation and upstream of the transition
Fig. 13. Separation in the compression corner.
Fig. 14. Distribution of vorticity Reynolds number at x = 145 mm.
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Fig. 15. Location of cross section at x = 145 mm.
onset. Compared with the shape of separation in Fig. 13, the vorticity Reynolds number distribution has an obvious relevance with the separation. The vorticity Reynolds number is larger following the large separation. It can be concluded that the large separation causes the transition onset to move forward.
3.3. X-51A forebody at different freestream Reynolds numbers In order to assess the performance of the improved c-Reht method at different Reynolds numbers, Fig. 16 compares the computational results with the experimental measurements on the leeward side of the forebody at Re1 = 3.2 106/m, 7.1 106/m and 11.7 106/m. The experimental results are presented using wall temperature and the computational results are shown with heat transfer rate. No transition is observed at Re1 = 3.2 106/m. As Re1 increases, transition onset moves forward on the leeward side of the forebody. The transition onset is in a ‘‘K” shape near the cen-
terline on the leeward side. The shape and the trend of the computational transition onsets show a good agreement with the experimental measurements. Therefore, the improved c-Reht method can effectively predict the transition onset at different Reynolds numbers. Figs. 17 and 18 present the transition distributions along the leeward centerline. The positions of transition onset and the peaks of heat transfer rate are marked by arrows. The transition distributions predicted by the improved c-Reht method show an agreement with the experiment at Re1 = 3.2 106/m and 11.7 106/m. However, there is a little delay of transition onset at Re1 = 7.1 106/m. The positions of computational transition onset are at x = 14.8 cm and 21.4 cm for Re1 = 11.7 106/m and Re1 = 7.1 106/m respectively, while the experimental results are about x = 15.1 cm and 18.4 cm respectively. As for the peaks positions at Re1 = 11.7 106/m, they are at about x = 24.2 cm and x = 24.0 cm for computation and experiment respectively. Figs. 19 and 20 show the heat transfer rate contours on the windward side and the heat transfer rate curves along the windward centerline at different Reynolds numbers respectively. Transition doesn’t take palce at Re1 = 3.2 106/m, that is the same as the result on leeward side. The transition onset is in a ‘‘V” shape distribution in the range of x = 18 cm 24 cm at Re1 = 7.1 106/ m, while in a ‘‘W” shape in the range of x = 13.5 cm 15 cm at Re1 = 11.7 106/m. Through the above comparisons, it can be concluded that the improved c-Reht method can successfully predict the boundary layer transition of hypersonic flow at different wind tunnel noise levels and freestream Reynolds numbers.
(a) Temperature of experiment, Re∞=3.2×106/m
(b) Heat transfer rate, Re∞=3.2×106/m
(c) Temperature of experiment, Re∞=7.1×106/m
(d) Heat transfer rate, Re∞=7.1×106/m
(e) Temperature of experiment, Re∞=11.7×106/m
(f)
Heat transfer rate, Re∞=11.7×106/m
Fig. 16. Comparison of transition onset on the leeward side at different Reynolds numbers. (left: experimental results [24], right: computational results with improved c-Reht method).
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307
6
Re= 3.2 x106/m Re= 7.1 x106/m Re=11.7x106/m
306.5
8
306
24cm
15.1cm
Q(kW/m2)
T(K)
305.5
Re= 3.2 x10 ,Transition Re= 7.1 x106,Transition Re=11.7x106,Transition Re= 3.2 x1066,Laminar Re= 7.1 x10 6,Laminar Re=11.7x10 ,Laminar
305 304.5
6
4
18.4cm 304
2 303.5 303 14
16
18
20
22
24
26
28
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34
0
x(cm) Fig. 17. Temperature distributions of experiment along leeward centerline at different Reynolds numbers [24].
12
14
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20
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24
26
28
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34
x(cm) Fig. 20. Heat transfer rate distributions of computation along windward centerline at different Reynolds numbers.
4. Conclusion In this paper, the the heat transfer of Two test cases have the improved c-Reht drawn as follows.
Fig. 18. Heat transfer rate distributions of computation along leeward centerline at different Reynolds numbers.
original c-Reht model is extended to predict hypersonic boundary layer transition flow. been selected to assess the performance of transition model. Some conclusions can be
(1) A new correlation of the momentum thickness Reynolds number based on the hypersonic flow is first presented in this paper. In the correlation, the Mach number and Reynolds number are introduced to reflect the compressible effect of hypersonic flow. The function Fonset1 used to control the transition onset and several relevant model parameters are also modified to make the c-Reht model suitable for hypersonic flow. (2) Compared with the original model, the improved c-Reht model can successfully predict the changes of the Stanton number and the heat transfer rate caused by boundary layer transition. The model can also reasonably reflect the effects of freestream Reynolds number and wind tunnel noise level on hypersonic boundary layer transition. (3) The heat transfer rate distributions of the X-51A forebody are analyzed in this paper. The windward side transition is found to be related to the separation in the compression corner and large separation can cause earlier transition onset in boundary layer. (4) Although the improved c-Reht model shows a good performance in predicting hypersonic boundary layer transition of a flat plate and the X-51A forebody, more test cases still need to be considered in the future work to further validate the performance of the improved c-Reht model.
Acknowledgement
Fig. 19. Distributions of heat transfer rate on the windward side at different Reynolds numbers.
The authors acknowledge the technical support provided by National Laboratory for Computational Fluid Dynamics in this investigation.
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