Tinxin ZHANG,Jinqing CHEN,Ytin ZHAO,Zijie LIU,Cho YAN,*

a School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China

b Computational Aerodynamics Institute, China Aerodynamics Research & Development Center, Mianyang 621000, China

c School of Aeronautics and Astronautics, Central South University, Changsha 410083, China

KEYWORDS Laminar kinetic energy;Mechanical consideration;Shear stress transport turbulence model;Transition model;V-model

Abstract A three-equation transition model based on the transition V-model is proposed for subsonic flows in this study. Considering the mechanical approximation of the generation process of the pre-transitional vorticities,the value of laminar Reynolds shear stress related to the mean shear deformation was calculated in the original transition V-model.Then a new transition model,named V-SA model,was proposed,which considered the phenomenological process of transition and presented great results for flows with and without pressure gradient.It is well-known that the baseline Shear Stress Transport (SST) turbulence model shows excellent performance of accuracy and robustness in plentiful flow cases,but it is important to predict boundary layer transition.The current model (V-SST) successfully couples the V-model to the SST turbulence model by introducing the effective turbulent viscosity and additional correction terms into the transport equations. A thorough evaluation of its ability to predict transition features is performed versus the welldocumented flat plate of ERCOFTAC, including T3A and T3B without pressure gradient, T3L2 and T3L3 with semi-circular leading edge, the three-dimensional 6:1 prolate-spheroid under two angles of attack,and the NLR-7301 airfoil under different Mach numbers.Numerical results show that the current model has an attractive and superior performance in the simulation of boundary layer transition processes

1. Introduction

Laminar-turbulent transition process is one of substantial considerations in aircraft design due to its significant influence on skin friction, heat transfer,and aerodynamic noise.Therefore,reliable and accurate transition prediction is greatly important for aerodynamic evaluation, thermal protection and flow control. However, owing to its high complexity, nonlinearity and multiple influencing factors,1boundary layer transition is still a challenging problem in Computational Fluid Dynamics(CFD).

In the past few decades, a great deal of researches on the boundary layer transition have been done. With the improvement of numerical simulation accuracy and in-depth study of the mechanism of turbulence-transition flows, various transition prediction methods have been proposed and developed.These numerical methods could be classified as the transition criteria, transport equation models under the Reynolds Averaged Navier-Stokes (RANS) framework, eNmethods based on stability theory and high-accuracy methods including Large Eddy Simulation (LES), and Direct Numerical Simulation(DNS).2,3

Based on abundant wind-tunnel and flight tests, the transition criteria have been universally utilized in engineering and have shown favorable performance in return module and shuttle aircraft.4Considering the streamwise and crossflow instability, Zhou et al.5developed a criteria-based transition model appropriate for engineering transition effectively.Recently, Yang et al.6proposed an efficient and simplified eNmethod coupled with Adjoint method, which can capture the development of the TS waves. Nevertheless, transition criteria only performed well in certain specific cases owing to the lack of universality and thought of physical mechanisms. The eNmethod, a semi-empirical and linear-stability-theory based transition prediction method, may be the most widely utilized transition predicting model before in engineering fields.7However, because of the fact that the eNmethod relies on the parallel flow hypothesis and solves nonlocal variables such as the integral of the growth of disturbance along stream line,8it is not appropriate for modern CFD simulation with complex grid systems and parallel programs. With the improvement of computation performance, high-accuracy methods have been developed rapidly to precisely reveal transition process and capture meticulous transition flow structures. DNS may be the most powerful research tool for discovering the new transition mechanisms.9However, restricted by the extremely high computational expense and time resolutions, it is still unrealistic for simulation of complicated flows in the next few decades.

Considering the physical mechanisms in the boundary layer, a new class of transition models based on the theory of Laminar Kinetic Energy (LKE) are developed under the RANS framework and show great advantages in engineering application. The concept of LKE initially presented by Mayle and Schulz10revealed a new mechanism to explain the production of fluctuations in the laminar region, and greatly promoted the awareness of transition flows. Bradshaw11proposed splat mechanism to explain the evolution and development of streamwise and wall-normal fluctuating velocity in pre-transitional region. Waters and Leylek12developed a new three-equation model (kT-kL-ε model) with the linear eddy-viscosity framework to solve the LKE, and then formulated the production and growth of laminar fluctuations.Compared with kT-kL-ε model, the kT-kL-ω model developed by Waters and Cokljat13improved the accuracy of boundary layer transition simulation using an additional equation of specific dissipation rate.14Unfortunately, the effective length scale λeffwas only based on the wall distance and turbulence length scaleλT, which means that different geometries shared the same λeffnear the wall. To avoid this deficiency,Qin et al.15revised the effective length scale with consideration of the physical phenomena in boundary layer. Following this improvement, Liu et al.1developed a four-equation transition model to predict distributed roughness induced transition.Applying it in a flat plate, sharp biconic configuration and hemisphere with roughness, the model had attraction and validity in transition prediction. More recently, a new transition V-model proposed by Vizinho et al.16under the LKE framework has gained much attention and presented good results for benchmark transition flows.Based on the modeling of laminar shear stress due to mean flow shear,this mechanical model focused on the pre-transition region and predicted the phenomenological effect of fluctuations appearing in the laminar region. Further, the V-model was combined with the Spalart-Allmaras turbulence model (marked by V-SA model)to simulate the turbulence-transition flows, and the performance of the integrated model was validated by ERCOFTAC flat-plate test cases. Then verification was performed to document the capability of the V-SA model to predict the transition features induced by crossflow, applying it in 6:1 prolatespheroid and transonic DLR-F5 3D test cases.17Abdollahzadeh et al.18assessed several RANS turbulence models including S-A, Low-Re SST and V-SA transition model in forced convection,and the results showed that the V-SA model can correctly predict the processes of transition flows.

The SST turbulence model, firstly proposed by Menter,19has been utilized to great numbers of typical test cases and has presented superior accuracy and robustness in predicting aerodynamic flows.Generally,it is regarded as one of the most outstanding turbulent models20and widely used by both turbulence model researchers and experienced CFD practitioners,21while the universality and accuracy of the S-A turbulence model in simulating complex flows and engineering geometries need to be further improved.Inspired by these facts and the researches of Vizinho16and Medida22et al.,this study develops a three-equation transition model, combining the Vmodel with SST-based turbulence model and constructing the transition model based on the work of Wang and Fu.23In the current model, laminar kinetic energy is solved by V-model and several modifications are made of SST transport equation for advancement of transition and turbulence prediction.Then the capability of the developed transition model will be verified by T3 flat plate,6:1 prolate-spheroid and NLR-7301 airfoil test cases. Moreover, the current transition model follows the Local-Correlation based Transition Modelling (LCTM) concept proposed by Menter et al.,24–27which means that no nonlocal variables are used all through the model. Hence, it is appropriate for the modern industrial CFD solvers.

The remaining of this paper is organized as follows: Section 2 provides the basic framework of the flow solver and numerical method. Section 3 gives the specific construction of the developed three-equation transition model. Section 4 presents the detailed computational results and discussion of several verification cases. Section 5 summarizes the main conclusions.

2. Numerical method

All flow simulations in this work are carried out on an inhouse CFD solver, which contains a mass of numerical calculation schemes28,29and transition/turbulence models.Its superior accuracy and reliability have been verified and validated through simulating different flow states and complex geometries.30–38The three-dimensional Navier-Stokes equations,including mass,momentum and energy conservation equation,are solved on multi-block structured meshes with the Finite Volume Method (FVM). Then the governing equations are outlined as.39

where μ is the molecular viscosity calculated using the Sutherland’s law,40and μTpresents the eddy viscosity,obtained from the transition/turbulence model. The laminar Prandtl number PrLequals 0.72 and the turbulent Prandtl number PrTis 0.9.

3. Transition model

3.1. Summary of V-model

Fig. 1 Evolution of angle of vortex deformation with ratio.

3.2. Coupling transition V-model to SST turbulence model

In the original V-SA model,transition V-model is coupled with the SA model by multiplying an exponential coefficient in the source term.The behavior of damp function related to‘‘ratio”value is actually similar to the turbulent intermittence, presented by Cho and Chuang.41That is, the mechanical V-SA model is limited to predict laminar flow when the ratio is large,and it switches to a fully turbulent simulation when the ratio reaches the threshold. Besides, due to the fact that the index of the exponential format is negative, the damp function is positive and no larger than one, which is presented in Fig. 2.Thus, referring to the concept of intermittency in the k-ω-γ transition model, this work couples transition V-model to the SST turbulence model. Modified transport equations for the TKE(marked as kT)and the specific turbulence dissipation rate ω are written as.

Fig. 2 Function curves with respect to ratio.

where F1is a blending equation42defined as follows.It is constructed to be one in near wall region and gradually approaches to zero far from the wall, which means that the original Wilcox model is activated in boundary region and the robustness k-ε model is conducted in the wake region and shear layers.

For brevity’s sake,other formulas and constant parameters which consist with those given in the original SST model43are omitted here.

Finally, in order to solve the government equations in current simulation, Roe upwind scheme and second-order MUSCL reconstruction with minmod limiter are utilized to discretize the inviscid fluxes.44For viscous fluxes, secondorder central difference scheme is applied to reduce the amount of calculation. The implicit Lower-Upper Symmetric Gauss-Seidel (LUSGS) method is employed for time marching.45

4. Results and discussion

In this section, the performance of the current three-equation transition model is evaluated using several incompressible flow cases that include Zero Pressure Gradient (ZPG) flat-plate with different freestream conditions, namely T3A and T3B,the T3L-series of flat-plate with a semi-circular leading edge,the three-dimensional 6:1 prolate-spheroid under two angles of attack and the NLR-7301 airfoil under different Mach numbers.It is worth noting that most of these flow cases are incompressible while the numerical methods are compressible. In order to be able to perform simulations under this framework,we selected a small Mach number(0.3)to satisfy the compressibility of the flow.At the same time,according to the flow similarity criterion,all the freestream conditions of incompressible flow cases,including pressure and density,are reset to guarantee that the Reynolds number is consistent with the experimental conditions. Therefore, the simulations can be realized by the in-house solver.

4.1. Zero-pressure-gradient flat plate

The T3 series experiment cases from ERCOFTAC database46are commonly used to validate the transitional behavior predicted by transition/turbulence models as a standard benchmark.47The first comparison is T3A flat plate with Tu∞= 3% and U∞= 5.4 m/s as shown in Table 1. Re∞stands for the freestream Reynolds number calculated by reference diameter which is unit meter. The flat plate has a tiny leading edge with curvature radius of Rn= 0.005 mm and the length of it is 10 m to avoid the influence of outlet disturbance.The simulation results predicted by the V-SA model are from the work of Vizinho et al.16

There is a general awareness that the prediction results of transition/turbulence models strongly depend on the grid scales especially in wall normal direction. Thus, grid independence tests are necessary before formal flow simulation. Four different grid sizes with cell number in the wall normal direc-tion ranging from 81 to 111 are generated. The details can be seen in Table 2.The y+values of four computational grids are less than one to guarantee adequate points inside the boundary layer.Further,the y+values of all the following test cases are assured lower than one for high gird resolution. The G3 grid and local magnification of the grid near the leading edge with boundary conditions for T3A plate are presented in Fig.3.Fig.4 shows the variation of skin-friction coefficients(Cf)with Reynolds number characterized by the distance from the leading edge of the surface (Rex). It can be noted that the prediction results of different grids in the laminar and full turbulence region are almost the same. Defining the position where Cfrapidly increases with the transition onset, it is manifest that the transition location moves upstream obviously,and the differences decrease gradually as grid size changes from G1 to G3.Considering that the G4 grid has the most grid cell number in the boundary layer corresponding to the lowest y+, but indicates a little changes in the distribution compared with the G3 grid, the G3 grid is adopted in the following numerical simulations.

Table 1 Freestream boundary sets for T3A and T3B.

Computation results of T3A flat-plate are presented in Fig. 5. In laminar region, it is obvious that both the results of V-SA and the current model(V-SST)show good agreement with the theoretical solution. Moreover, different models predict the transition onset rightly, except that the V-SA model obtains a slightly larger skin friction coefficient than the experiment. However, in the transitional region, the V-SA model simulates the process of transition flow turning to be fully turbulent flow too early, which means that the predicted transition region is too short and the result departs from experiment data obviously, while the current model predicts a transitional behavior successfully. As to the full turbulence region,two transition models and SST turbulence model compute a marginally larger value than the experiment data.Overall, the current model has a better performance in predicting transition behavior of T3A flat-plate compared with the VSA model.

As mentioned in the last section, the V-model predicts the transition onset though calculating the viscosity induced by the small negative u-′v′in the pre-transitional region.Therefore,comparisons of T3A experiment data of u-′v′value with these calculated by the V-SA and the V-SST model in three differentaxis positions d near the leading edge, corresponding to 0.095 m, 0.395 m and 0.495 m or Rexof 3.24 × 104,13.48 × 104, 16.92 × 105accordingly, are shown in Fig. 6.As can be noted, the V-SST transition model performs better in the laminar region although the peak values predicted by it in the transitional region are much lower than the results calculated by the V-SA model and the experiment data. Maybe the explanation for this difference is that the transitional process predicted by the current model is obviously delayed compared with the V-SA model, and thus the value of laminar kinetic energy simulated by the current model keeps in a lower level than the V-SA predicted, which is directly related to the calculation of u-′v′.

Table 2 Grid resolutions for convergence analysis.

Fig. 3 Computational grid for flat plate.

Fig. 4 Grid convergence analysis.

Fig. 5 Comparison of Cf distribution for T3A flat plate.

The second ZPG flat-plate test case is T3B and the details of far-field boundary conditions are listed in Table 1. In general, the transition onset will move upstream as the Reynolds number and the freestream turbulence intensity increase. The results of skin-friction coefficient calculated by different models are compared in Fig. 7. It is obvious that the V-SA model predicts a larger Cfin the transition onset region and switches to fully turbulent prediction too early. Furthermore, the peak at the end of the transition region predicted by the V-SA model is lower than the experiment data. Surprisingly, the profile of Cfcomputed by the V-SST model is in great agreement with the experiment data.As to SST model,it computes a fake transition behavior due to the introduction of the damping function near the wall area.

Therefore, the current transition model can predict transition onset mechanically and demonstrate a superior performance compared with the V-SA model for zero-pressureplate test cases.

Fig. 6 Comparison of u-′v′values at different axial positions for T3A plate.

Fig. 7 Comparison of distribution for T3B plate.

4.2. T3L flat plate

In this section, the current transition model is tested by simulating T3L-series plate with a semi-circular leading edge from ERCOFTAC database,48,49namely, T3L2 and T3L3. According to the database, the radius of the flat plate equals 5 mm,and the length of it in flow direction is set to be 4 m.The computational grid is well generated with 251 × 81(streamwise direction×wall–normal direction)and the detailed computed conditions of the T3L flat plate are listed in Table 3.Re∞is the freestream Reynolds number based on the leading edge diameter.

For the reason that the presence of the semi-circular leading edge induces strong adverse pressure gradient,flow separation occurs near the leading edge. It is well demonstrated that the transition induced by flow separation is greatly different from the natural and bypass transition.However,the V-model is initially developed to simulate bypass transition, and the exponential component in the destruction term of transport equation in Eq.(10)is performed to limit the destruction effect of separation flows. The Cvsconstant of the exponential coefficient is set as 16.0 for the T3L test cases, and no other extra revisions and special treatments in this work have been executed for prediction of separation flows. The applicability of the V-model in simulating transition induced by the separation would be researched in the succeeding work. Fig. 8 shows the symmetry cut plane of velocity U (up) and non-dimensional turbulent kinetic energy k/U2(down). As can be seen, boundary layer separates close to the end of leading edge but reattaches at different downstream positions for cases T3L2 and T3L3 due to the different freestream conditions. Reasonably,the turbulent kinetic energy begins to increase rapidly near the reattachment region, and then the flow develops into full turbulence for both T3L2 and T3L3 cases in Fig.8(a)and(b).

The profiles of skin-friction coefficient for three T3L test cases are shown in Fig. 9. Besides the experiment data (EXP)and the computed results of V-SA and V-SST transition models, the results of Suzen and Huang model,50applying a tran-sition onset correlation with an intermittency transport equation based on Menter’s SST model, are also plotted for comparison. As can be seen, the V-SST model provides superior results for T3L2 and T3L3 test cases in simulating reattachment point and the Cfvalues in turbulence region, while the V-SA model predicts a slightly delayed reattachment for T3L3 case and the model proposed by Suzen et al. is less robust and predicts an overshoot of Cfvalue near the reattachment region. Further comparison of the X-coordinates of the separation point (Xs), reattachment point (Xr) and the length of separation bubble(Ls)is shown in Table 4,where Ls=Xr--Xs. It is obvious that the current model presents better consistency with experiment data than the Suzen’s model.

Table 3 Freestream boundary sets for T3L flat plates.

For further validating the simulation performance of the current model,the comparisons of velocity profiles at different streamwise stations are chosen and the results are shown in Fig. 10. Due to the appearance of backflow downstream of the separation point, velocity near the wall region turns to be negative, the inflection point appears, and then the profile gradually turns to be turbulent velocity profile downstream of the reattachment point. Overall, the V-SA model and the V-SST model show little difference, and the current model obtains correct velocity profiles for all of T3L test cases,except for the little difference with the experimental results in the backflow region for case T3L2.

4.3. 6:1 Prolate spheroid

The third test case is the 6:1 prolate spheroid, which is an incompressible three-dimensional case and computed using V-SA, V-SST and γ-Reθttransition models. The simulation results of V-SA and γ-Reθtmodels are from the work of Vizinho et al.17and the experimental research is carried out in the low-speed wind tunnel at DLR in Go¨ttingen.51In this paper, the test conditions including Mach number 0.136, two different angles of attack α = 5°,10°, and the same Reynolds numberRe∞= 6.5 × 106are chosen for the computation.For α = 5° and Re = 6.5 × 106, transition is caused due to T-S and crossflow instabilities, while the transition is only dominated by crossflow instability for the condition with α = 10°.52Since the strength of the V-SA transition model has been validated in predicting transition under cross-flow effect,17the performance of the V-SST transition model in the same computed conditions will be verified and validated in this section.

The total amount of computed grid cells are 1.68 × 106in two cases. The grid is locally refined and conducted with 131 grid cells in wall-normal direction to ensure that there is enough solution accuracy inside the boundary layer, and the streamwise resolution is 231 points.Meanwhile,the grid topology is well constructed to ensure the matching with the object shape and the orthogonality with the wall.Fig.11 displays the mesh of the symmetry plane for the prolate spheroid.In order to obtain the approximate turbulent state with experiment in the farfield boundary of the spheroid during the simulations,the turbulence intensity for both computations is supposed as Tu∞= 1%.52

Fig. 8 Symmetry cut plane of velocity(up) and non-dimensional turbulent kinetic energy(down).

Fig. 9 Skin-friction coefficient distribution for T3L test cases.

Table 4 Comparison of computed and experimental separation, reattachment points and length of separation bubble expressed for T3L test cases.

The skin friction coefficient distribution of the test case with α=5°is shown in Fig.12.Since the experiment data near the prolate spheroid tips is not accessible,the value of skin friction coefficient on tips is assigned to be zero. As can be seen,the results predicted by the V-SST model slightly deviate from the V-SA model data while both models predict the transition onset position too earlier than the experimental measurement.Fig.13 shows the predicted skin friction coefficient on the central X-Z plane and the presented consequences are consistent with the above analysis. As a comparison, the original γ-Reθtmodel without cross-flow correction predicts a quite delayed transition onset due to the incorrect angle of transition line along the prolate spheroid surface.

Fig. 10 Velocity profiles for T3L test cases.

Fig. 11 Computational grid for prolate spheroid.

For the larger angle of attack α =15°test case, the transition process is purely controlled by crossflow instability. As presented in Fig. 14, the V-SA and V-SST models correctly predict the transition onset near the experiment values at the location marked with letters, but the results predicted by the V-SST model seem to be more reasonable considering the effects of crossflow and angle of attack.The γ-Reθtmodel fails to simulate transition process in most of surface regions, and the distribution of Cfis far from the experiment data especially in the windward side.It should be noted that the distributions of Cfpredicted by three models all experience a similar banded structure in the marked region.The predicted Cfalong the central X-Z plane are shown in Fig.15.As observed in Fig.15,the transition onset position predicted by the V-SA model and the V-SST model is very close to the experiment data except that the simulated peak value of Cfis a bit lower than the one in experiment. What’s more, the difference between two transition models based on the V-model is slight in the transition region.

4.4. NLR-7301 airfoil

The typical transition flows over the NLR-7301 airfoil under various Mach numbers are simulated. Based on adequate experimental and numerical researches, the NLR-7301 airfoil is often used as a standard test case to validate the ability of transition models in predicting subsonic and transonic flows.Studies have shown that the transition occurs on the NLR-7301 airfoil via different mechanisms with the variation of Mach number and angle of attack,including the T-S instability and flow separation. The computations have been conducted by Wang et al.52using a correlation-based transition model,named γ-Reθt, to research the influences of Mach number and angle of attack on transitional simulation of NLR-7301 airfoil.Thus,two typical Mach numbers(0.3 and 0.5)are chosen in this paper and comparisons are conducted among the results computed by the V-SA model, those by the V-SST model and the experiment data. An estimated value of 0.3%52for the freestream turbulent intensity is selected due to no exact mention in the experiment database. Besides, the Reynolds number (Re) based on the chord length is 2.2×106for all of three cases and the other freestream parameters including static pressure, density and static temperature are adjusted to ensure the uniformity of the Reynolds number.The details of them are shown in Table 5. Body-fitted C-type structured mesh contains about 287,600 grid cells with a near-wall clustering. In order to finely capture the flow structure,the mesh in the boundary layer is well generated.Besides,the grids of the leading and trailing edges of the airfoil are locally refined.The detailed computed grid is shown in Fig.16.

Fig. 12 Skin friction coefficient distribution with α = 5°.

Fig. 13 Skin friction coefficient along central line with α = 5°.

Fig. 14 Skin friction coefficient distribution with α = 15°.

Fig.15 Skin friction coefficient along central line with α =15°.

Table 5 Computed conditions for NLR-7301 test cases.

Fig. 16 Computational grid for NLR-7301 test case.

The predicted and measured wall pressure coefficient Cpand Cfdistributions around the airfoil for two test cases are presented in the left and right side in Fig. 17 respectively.The pressure distributions that both transition models calculated are well consistent with experiment data for two Mach numbers,while the results obtained by two models are slightly different near the trailing edge.As for the Cfdistributions,the V-SST model predicts superior transition process and exact skin-friction coefficient value both on the windward and leeward side. However, the transition simulated by the V-SA model is earlier on the windward side and delayed on the leeward side, which is related with the initial value of ～vTat the inlet boundary. And the V-SA model computes an overpredicting skin-friction coefficient value before the flow turns to be full turbulence for the Mach number 0.3 test case.Fig. 18 displays the symmetry cut plane of non-dimensional pressure divided by the static pressure and local distribution of the ratio calculated by the V-SST transition model on the leeward and windward sides of the airfoil respectively. As can be noted,the pressure peak formed near the front stationary point of the airfoil increases with the growth of Mach number. As defined in Eq. (14), the ratio presents a relative size between the vorticity deformation and mean shear. With the increase of shear deformation, the curvature along the vortex surface will change and the centrifugal force distribution of the pre-transitional vortex will increase accordingly. The mechanical transition model assumes that it represents an absolute turbulent vorticity when the local curvature radius increases to be infinite, and the ratio will decrease to be zero.Therefore, the ratio is larger in the laminar region and tends to be zero in the turbulence region,and transition onset occurs at the position where the ratio threshold is reached. Clearly,the ratio distributions presented in Fig. 18 are well consistent with the prediction of the transition processes.

5. Conclusions

In this work, a three-equation transition model through mechanical approximation is developed for predicting boundary layer transition in subsonic flows. The V-model, based on the concept of the LKE and the calculation of laminar shear stress u-′v′in the pre-transitional region,is successfully coupled to the SST model through introducing the effective turbulent viscosity and an appended source term in the transport equation. The performance of current transition model has been adequately assessed and validated by the in-house code MICFD through simulating various benchmark test cases.

Fig. 17 Wall pressure coefficient (left) and skin-friction coefficient (right) distributions around airfoil for NLR-7301 test cases with different Mach numbers.

Fig. 18 Symmetry cut plane of non-dimensional pressure and local distributions of the ratio.

Taken together, these results demonstrate that the current transition model can describe boundary layer transition behavior mechanically and predict the values of characteristic quantities related to turbulence transition correctly.Compared with the V-SA model, a superior performance of the current model is validated with experimental database for zero-pressure-plate test cases.For T3L test cases with the mechanism of transition induced by separation, the V-SST model successfully predicts the separation position and length of the separation bubble through adjusting the Cvsconstant slightly. For the prolate spheroid cases, the V-SST model shows similar strengths to the V-SA model in predicting transition induced by crossflow.Meanwhile,the proposed transition model displays its capabilities of predicting the transition onset, skin-friction and pressure distribution on the NLR-7301 airfoil under different Mach numbers. However, many other factors influencing the transition, including roughness and compressibility, are not investigated in this paper.All these issues call for further studies to improve the current V-SST transition model.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This research is supported by the National Natural Science Foundation of China (No. 11721202).