Ytin ZHAO, Jinqing CHEN, Rui ZHAO, Hongkng LIU

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

b State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang 621000, China

c Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China

d School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

e Key Laboratory of Traffic Safety on Track of Ministry of Education, School of Traffic & Transportation Engineering, Central South University, Changsha 410075, China

KEYWORDS Boundary layer transition;Intermittency;Local variables;Separation bubble;Transition model

Abstract The purpose of this work is to improve the k-ω-γ transition model for separationinduced transition prediction. The fundamental cause of the excessively small separation bubble predicted by k-ω-γ model is scrutinized from the perspective of model construction. On the basis,three rectifications are conducted to improve the k-ω-γ model for separation-induced transition.Firstly, a damping function is established via comparing the molecular diffusion timescale with the rapid pressure-strain timescale. The damping function is applied to prevent the effective length scale from incorrect distribution near the leading edge of the separation bubble.Secondly,the pressure gradient parameter λζ, is proposed as an indicator for local susceptibility to the separation instability.Additionally,λζ,-based separation intermittency γsep is constructed to accelerate the substantial growth of turbulent kinetic energy after flow separation. The improved model appropriate for both low- and high-speed flow has been calibrated against a variety of diverse and challenging experiments, including the subsonic T3L plate, Aerospatial A airfoil, transonic NLR-7301 airfoil and deformed hypersonic inflatable aerodynamic decelerator aeroshell. The improved model is strictly based on local variables and Galilean invariance. Besides, the proposed improvement for k-ω-γ model can be fairly convenient to incorporate into other existing intermittency-based transition models.

1. Introduction

Due to the high complexity in intrinsic physics,the laminar-toturbulent transition, considered to be one of the most outstanding challenges, possesses important theoretical significance. Meanwhile, the transition is closely associated with the civil and military applications,because of its great impacts on the aerodynamic force,1thermal load, inlet quality and many other performances of the air vehicle.2Therefore, predicting the boundary layer transition accurately has been a hot and difficult issue. There are multiple paths in transition process, e.g. natural, bypass, and separation induced. Among them, the separation-induced transition, ubiquitously existing in low-pressure turbine blades, unmanned aerial vehicles and hypersonic configurations to name a few, has attached a lot of attention.

The laminar boundary layer subjected to the sufficiently strong Adverse Pressure Gradient(APG)is prone to detaching from the surface. Downstream of the separation point, the inflectional velocity profiles appear and the Kelvin–Helmholtz(KH) instability following a perturbation of the separated shear layer is triggered.This facilitates the transition from laminar to turbulence and a Laminar Separation Bubble (LSB) is subsequently generated. Obviously, the transition process has comparable influence on the mean topology and the behavior of LSBs. LSBs can be categorized as either ‘short’ or ‘long’.3The former which affect the pressure distribution locally are exactly what we focus on.For the short LSBs,the external disturbances and the instability waves initiated at the leading edge of bubble can be amplified by KH waves, and then subsequently form a spiral and thereafter an elliptical vortex.4Besides acting as an amplifier, KH instability could also directly promote transition. Marxen et al.5demonstrated that the distortion of vortex and braid zone in KH are susceptible to the elliptical and hyperbolic instabilities, respectively.Another special point deserving attention on the LSBs is the feedback loop:6the mean flow deformation across the LSBs alters the pressure distribution, which leads to a stabilization of the bubble with respect to the collectively amplified disturbances.3The feedback loop is a primary global instability,reflecting the self-sustaining nature of LSBs. A more extensive discussion of instability mechanisms in LSBs can be found in the reviews of Wauters and Degroote.7

The above studies of the physical mechanisms governing separation in general and the transition process in particular,although not yet entirely understood, lay the foundation for separated-flow transition prediction. The current numerical approaches for boundary-layer transition prediction mainly include Direct Numerical Simulation(DNS),Large Eddy Simulation(LES),stability theory and transition models based on Reynolds-Averaged Navier-Stokes(RANS)equations.Among them, RANS-based transition models, benefiting from their superior economy and robustness,are reckoned to be the most potential method for the transition prediction of complicated engineering configurations. Early attempts to simulate transition process with a RANS framework are attributed to low-Reynolds number turbulence models. A successful instance is the study of Langtry and Sjolander,8where the source term for the turbulent kinetic energy transport equation in Shear-Stress Transport (SST) model is modified by a function of the pressure gradient to account for the influence of APG.The modification is inspired by the great sensitivity of transition onset in bubbles to APG.9This in turn could enable the accurate prediction of reattachment point for the PAK-B low-pressure turbine blade without introduction of empirical information about separation bubbles.

However,such low-Reynolds models are strongly geometry dependent and thus are called as pseudo transitional models,for which one of significant reasons is the ignorance of intermittency phenomenon in transition process. Accordingly, a series of intermittency-based transition models have been subsequently developed, defining the intermittency factor γ to quantify the probability of the flow being turbulent in a given spatial point. γ can be solved either algebraically or dynamically (through transport equation). Examples of the former are the prescribed unsteady intermittency model10and the model proposed by Fu¨rst et al.11Unfortunately, the algebraic correlations are impotent to describe the non-local and historical effects of intermittency nature. It motivates the approach for γ transport equation as proposed by Wang,12,13Walters14and Ge15et al.for natural transition or bypass transition.This type of model performs well in the transitional flow with favorable and zero pressure gradient flow,but fails under large APG and separation. To predict separated-flow transition, Vicedo et al.16further improved the intermittency transport equation,which takes into account the growth of disturbances amplified by KH in shear layer, through allowance of the diffusion for freestream turbulence into the boundary layer by means of modeling the entrainment of the surrounding fluid.

Nevertheless,the intermittency only simulates the length of transition area. In order to predict transition, a separate transition-onset criterion generally relying on nonlocal properties is necessary.The nonlocal properties imply the integration over the boundary layer,leading to implementation difficulties in unstructured codes. Condition turned a corner when Langtry and Menter17proposed the γ-Reθtransition model, which triggers transition locally by relating vorticity Reynolds number Revto critical momentum thickness Reynolds number Reθc. They also pointed out that the application of Revshows potential superiority in predicting separation-induced transition, since the standard definition of Reθc, strictly speaking,is no longer valid in separated flows, while the alternative Revovercomes this limitation. Meanwhile, the increase in Revwith increasing shape factor can actually imply the dominant contribution of shape factor to separation-induced transition. On this foundation, Langtry and Menter further extended the concept of intermittency to separation flows and thus constructed a separation intermittency factor γsepin order to capture the transition induced by LSBs. The factor γsepcan reach the value of 2 inside the LSB, accelerating the production of turbulent kinetic energy and forcing the bubble to an early reattachment. The model has been calibrated against the T3C4 flat plate under the effect of various APG,Aerospatial A airfoil,and DLR-F5 wing.Following this work,Chen et al.18employed the Michel criterion19into γ-Reθmodel framework to improve its stability and convergence. The downside is that the γ-Reθmodel is not Galilean invariant,indicating that it is only applicable to the situation where the surfaces are stationary relative to the coordinate system. To the end, Menter et al.20formulated a one-equation local correlation-based transition model by removing the Reθequation from γ-Reθmodel to avoid the lack of Galilean invariance.The new model, whose ability for transition prediction in a separation bubble is illustrated in a series of test cases, has been implemented in both ANSYS CFD solvers, ANSYS CFX 15.0 and ANSYS Fluent 15.0. Following the success,Liu et al.21recently proposed a Galilean-invariant γ-SA transition model, which also adopts the concept of γsepto predict separated flow transition. Regrettably, this correlation-based model is not established on the basis of physical mechanisms.Pascal et al.22presented the stability-based transition model incorporating the Gleyzes criterion by means of transport equations to account for transition in separation bubbles.Wang et al.23designed a timescale of separated instability to reflect the separation contribution on non-turbulent fluctuation. In addition, an elliptic equation for γ is supplemented to take into consideration not only the local influence by which the disturbances enter into the boundary layer namely convection and viscous diffusion, but also the nonlocal influence by pressure diffusion. Zhang et al.24predicted the separationinduced transition by considering the mechanical approximation of the generation process of the pre-transitional vorticities. Based on the analysis results of linear stability theory,Xu et al.25proposed the amplification factor transport equations and successfully applied them to transition prediction over DLR-F4 wing body configuration, where the transition is dominated by T-S instability, crossflow instability26and laminar-separation instability.

All of the above models in their own way have various degrees of success, especially in specific flows which comply with the assumptions under which they are derived. Unfortunately, they are restricted to separational transition in lowspeed flows. By comparison, the high-speed boundary layer transition has a completely different instability mechanism,mainly including the existence of generalized inflection point in velocity profile,the dominance of three-dimensional viscous disturbances, and the appearance of multiple Mack modes.Recently, great efforts have been made to improve models for high-speed transition. Krause et al.27firstly formulated the new empirical correlations of transition onset and length for high-speed transition flows based on the experimental data of hypersonic flat plate flows.You et al.28further applied them to the γ-Reθmodel, and improved γsepby accounting for the effect of pressure gradient,so as to achieve an acceptable level of hypersonic separation-induced transition prediction.Nevertheless, these models are only suitable for either low- or highspeed flows because of the empirical nature of γ-Reθ. By comparison, the k-ω-γ model proposed by Wang and Fu12,13can give a great prediction of natural transition at both low and high speeds by considering various instability modes. It can be inspired that if the separation extension is developed within the k-ω-γ model framework, it may be attempted to construct a wide-speed-range model for separation-induced transition.This attempt can be found in the recent work of Wang et al.29which however aims at the massive separation, quite different from the separation in the short LSBs. Furthermore, the understanding of detailed performance of k-ω-γ model for transition in LSBs is far from complete, hindering its further development and advancement. Besides, in light of the aforementioned content, very few transition models meet all of the following requirements:

(1) Applicable to both low- and high-speed flows undergoing separation-induced transition.

(2) Based on local variables.

(3) Based on Galilean invariance.

(4) Account for physical mechanism.

Therefore, considering the ubiquitous separation-induced transition in both low-speed and high-speed flows and its remarkable impacts on aerodynamics and aerothermodynamics, the main purposes of the current study are: (A) to assess and expound the elaborate performance of k-ω-γ model on separated-flow transition prediction to gain an improved comprehension of model construction and further give a guidance for the model advancement; (B) to extend the capability of kω-γ model in prediction of separation-induced transition and meanwhile enable it to satisfy the four requirements above,ensuring enough degree of universality.

The remaining of this paper is laid out as follows:Section 2 presents the underlying original k-ω-γ model.Then its capability for separated-flow transition prediction is assessed in Section 3. In Section 4, the separation-induced transition model formulation is introduced in detail. The numerical results of test cases are discussed in Section 5. Finally, the main conclusions are drawn in Section 6.

2. k-ω-γ transition models

Inspired by the local formulation of Langtry and Menter17and the idea of Warren and Hassan30who adopted the effective eddy viscosity μeffto account for the effect of non-turbulence fluctuation, Wang and Fu12,13proposed and developed the k-ω-γ transition model, which could be applied to predicting transition for subsonic,transonic and hypersonic flows.Specifically, they constructed a unified turbulent kinetic energy transport equation (k) to characterize both non-turbulent(kL) and turbulent (kT) contributions with intermittency γ as weight. The framework of model is as follows:

Herein,the total turbulent kinetic energy k=kL+kT.The formulations of turbulent kinetic energy k and specific turbulence dissipation rate ω transport equations are roughly consistent with those in the original SST turbulence model, except that the original eddy viscosity μtis replaced by μeff. μeffis defined as.

where μntis the non-turbulent fluctuation viscosity, reflecting the effects of various instability disturbance modes on transition. As conveyed from Eq. (4), non-turbulent and turbulent components are coupled using the weight. For the turbulent boundary layers, γ = 1 is obtained, so μeff= μtand the kω-γ transition model deteriorates to SST turbulence model,while for the non-turbulent fluctuation area,γ=0,and hence μeffis determined directly by μnt.By analogy with the formulation of μt, μntcan be interpreted as.

in which τntdenotes the characteristic timescale whose construction considers the contributions of the first-mode τnt1,second-mode τnt2and crossflow-mode τcrosson the nonturbulent perturbation. Based on the linear stability theory and experimental results, τnt1and τnt2take the following form respectively:

where the relative Mach number Marel= (U-cr)/c is defined to distinguish the dominant area of the second-mode instability.cris the phase velocity of disturbance propagation and c denotes the local sound speed. Then the total timescale τntis given by.

3.Assessment of original k-ω-γ model for predicting separationinduced transition

In this section, a detailed assessment of the original k-ω-γ model is conducted to obtain a clear comprehension of its performance in predicting the separated-induced transition.From the perspective of model construction,the original k-ω-γ model(labelled by k-ω-γ-orig)is assessed and scrutinized against two canonical transitional flow test cases(i.e.the T3L flat plate and Aerospatial A airfoil) to provide targeted clues for the advancement of k-ω-γ model in the next step. The flow solver utilized in this study is an in-house code,whose competence for the laminar, transitional and turbulent flow prediction has been validated in the previous work.33–35The grid sensitivity can also be found in previous work.34,35Specifically, some strategies for meshes like the non-dimensional wall distances y+≤1.0 and the ration coefficient Rs>1.0 proposed by Smirnov et al.36,37are selected in the following cases. Besides, the meshes are clustered and optimized carefully near the separation bubble.

3.1. T3L flat plate

The T3L case from the ERCOFTAC database38is a zeropressure gradient flat plate with a semicircular leading-edge,whose diameter d=10 mm.Once the flow passes the semicircular leading-edge, the boundary layer separates under APG effects.The separated laminar flow is easy to trigger transition due to high instability in shear layer, and then reattaches as turbulence downstream. Obviously, it is the transition onsets that determine directly the size of separation bubbles.The freestream conditions are U∞= 5 m/s, Re∞= 3.29× 105/m and Tu∞= 0.63%.

Fig.1 presents the separation bubble colored by streamwise velocity of the laminar,SST model and k-ω-γ-orig model.The reattachment point of experiment38(x = 27 mm) marked by dash-dotted line is also provided. As presented, the coming laminar flow separates immediately once it bypasses the leading edge. All numerical methods predict the nearly identical separation onsets of roughly x=4.8 mm,while obtain the evident discrepancies in separation extents. The bubble predicted by laminar is significantly larger than the experimental result.By contrary, separation bubbles characterized by SST and kω-γ model are relatively small due to strong turbulent mixing effects.Furthermore, the skin friction coefficient (Cf)distributions predicted with the three models are compared against experimental data in Fig. 2. The separation onsets where Cf=0 firstly occur around x=5 mm.Downstream,the recirculation area appears,accordingly Cf<0.It is noted that the plateau of Cf=0 in ERCOFTAC database actually represents the separation bubble. Observing curves in Fig. 2, we can notice that the size of separation bubble predicted by k-ωγ-orig model is even unreasonably smaller than that by SST model, as shown in Table 1, which compares the bubble size quantitatively.These anomalous phenomena indicate the incapability of k-ω-γ-orig model in capturing separated-flow transition accurately, reflecting its intrinsic defect in the construction, which will be discussed in depth below.

Fig. 3 compares the eddy viscosity (μt) distributions predicted by SST model and the effective eddy viscosity(μeff)predicted by k-ω-γ-orig, both of which are normalized by molecular viscosity (μ). It can be found that μtcomputed by SST model develops close to the middle of bubble(x=10 mm),while μeffby k-ω-γ-orig model increases rapidly just behind separation point (x = 6 mm). This behavior demonstrates that once the separation appears, the overpredicted μeffcould cause an earlier reattachment of the separated flow,resulting in the excessively small bubble.Obviously,the unreasonable μeffprovided by k-ω-γ-orig model around separation bubbles deserves a further discussion.

In Eq.(4),γ is close to zero in the non-turbulent fluctuation area,and thus μeffis approximately equal to μnt.Generally,the modelling of μnttakes multiple instability modes into consideration, including first, second and crossflow modes. Due to the nonexistence of the crossflow and second mode in the current flow, μeffcan be simplified as.

Fig. 1 Velocity U contours for laminar (above), SST model (middle) and k-ω-γ-orig model (below).

Fig. 2 Surface skin friction coefficient (Cf) distributions.

Fig. 3 Eddy viscosity μt predicted by SST (above) and effective eddy viscosity μeff predicted by k-ω-γ-orig (below).

Table 1 Separation bubble details for T3L.

Van Driest and Blumer39found that disturbances in a laminar boundary layer could be magnified in the high vorticity areas but are well away from the wall damping areas for attached flows. In Eq. (11), the disturbance development is characterized by d2Ω, which increases with the thickening boundary layer. Accordingly, ζeffbased on d2Ω embodying such a physical mechanism, enables the k-ω-γ-orig model to achieve good performance in natural transition prediction.However, it is inappropriate for the separated flow which involves high vorticity, and thus the feasibility should be doubted. To explore a more intuitive explanation, Fig. 4 displays the predicted vorticity magnitude and effective length scale distributions around the semicircular head. Under the effect of strong APG, Ω does not increase gradually with the boundary layer evolution, but amplifies evidently in the shear layer ahead the separation bubble.Consequently,the overpredicted Ω causes the upsurge of ζeff, and a premature μnt, and finally promotes the generation of turbulent kinetic energy.The local separated flow tends to be turbulent and thus reattaches rapidly,leading to an underpredicted separation bubble in turn.From another perspective,ζeffis deduced from dimensional analysis of the correlation Rev,max=(ρd2Ω)max=2.193-Reθ. Unfortunately, this correlation derives intrinsically from the incompressible boundary layer flows on the flat plate with zero pressure gradient(the Blasius boundary layer).Therefore,for separated flows under APG,this correlation might be invalid. Clearly, the analysis above demonstrates that there would appear some flaws of k-ω-γ-orig model in predicting separation-induced transition, which is closely related to the construction of ζeff. Interestingly, it is noteworthy that the transition onset and development described by the k-ωγ-orig model are also triggered by Ω in shear layer of separation region, possibly causing an abrupt elevation of nonturbulent fluctuation in high vorticity areas. Consequently, it is imperative to clarify the underlying effects of Ω utilized by the k-ω-γ-orig model in the separation-induced transition prediction.

3.2. Aerospatial A airfoil

Fig. 4 Vorticity magnitude (above) and effective length scale(below) predicted by k-ω-γ-orig.

To make clear the effect mentioned above, the Aerospatial A airfoil is considered for further analysis. The Aerospatial A is tested in ONERA F1 1.5 m × 3.5 m wind tunnel with the freestream conditions: Ma∞= 0.15, α = 13.1°, Re∞= 2.1× 106/m, Tu∞= 0.2%.40At a high angle-of-attack of 13.1°,the laminar boundary layer develops and terminates in a separation bubble at 12% of chord near the suction peak. The separation bubble promotes the transition of the laminar boundary layer. Then the turbulent boundary layer develops downstream until it separates again near the tailing edge due to the strong APG. The experimental results show that the extent of the LSB is only about 2%of the chord.However,this nearly negligible small bubble greatly changes the momentum thickness distribution over the entire airfoil.

Fig. 5 provides the dimensionless turbulent kinetic energy k/U∞2contours with streamlines around Aerospatial A airfoil predicted by k-ω-γ-orig transition model and SST turbulence model.Meanwhile,Fig.6 provides the laminar results for comparison. As presented, both k-ω-γ-orig and SST capture the separation bubble near the trailing edge. As for the k-ωγ-orig,the predicted bubble is located at 12%of chord,which is well consistent with the experimental data. In contrast, the SST cannot predict the leading edge bubble due to the strong resistance to APG for turbulent flows. Notably, the laminar also acquires the leading-edge separation and the separation onset is close to that of k-ω-γ-orig model. However, because of the nonexistence of boundary layer transition downstream of the separation point, the momentum transport of laminar flow is not enough to trigger reattachments and hence a large separation zone is formed downstream in leeward side. The remarkable discrimination of flow structures between the laminar and k-ω-γ-orig model certifies that k-ω-γ-orig model could characterize the tendency of separation bubble to a certain extent. As mentioned above, this capacity for k-ω-γ-orig model could be attributed to the introduction of Ω in the ζeffconstruction. But importantly, it should be noted that the kω-γ-orig model is incapable of predicting the transition characteristics accurately. To illustrate it, the Cfdistributions obtained by k-ω-γ-orig model and SST model are quantitatively compared with the experimental data in Fig. 7. Downstream of the reattachment onset, Cfpredicted by k-ω-γ-orig model is seriously underestimated, and even much less than that by the SST model. Obviously, the predicted boundary layer transition downstream of the separation bubble does not fully develop,which is inconsistent with the practical flow.

In summary, although the dramatical elevation of Ω in shear layer facilitates the development of turbulent kinetic energy after a bubble, differentiating the flow from pure laminar, the k-ω-γ-orig model cannot predict the separated-flow transition accurately only relying on the real-time responses of the high vorticity near bubbles. Consequently, it is imperative to improve the transition model to embody the physics of separation-induced transition accurately and thoughtfully,which is one of our purposes in the current work.

4. Separation-induced transition model formulation

In this section, three rectifications employed in k-ω-γ-orig model are outlined. Firstly, the effective length scale ζeffis ameliorated for rationalization in separation bubbles. Secondly, the local-variable-based pressure gradient parameter λζis proposed as an indicator of local receptivity to disturbances of transition in separated flows. Additionally, based on λζ, the separation intermittency γsepis established to reflect appropriately the substantial growth of turbulent kinetic energy after reattachment point.

Fig.5 Turbulent kinetic energy(k/U∞2 )contours with streamlines around Aerospatial A airfoil predicted by k-ω-γ-orig(above)and SST(below) model.

Fig. 6 Streamlines featured by Mach number around Aerospatial A airfoil of laminar flows.

Fig. 7 Surface skin friction coefficient (Cf) distributions for Aerospatial A airfoil.

4.1. Modification of effective length scale

As pointed out in Section 3.1,k-ω-γ-orig model fails to predict a precise separation bubble extent owing to the imperfection of ζeffformulation which facilitates a sharp rise of effective eddy viscosity μeffafter separation onsets. Therefore, here the main work is to revise the ζeffformulation, and four requirements are expected to meet:

(1) It can automatically identify the areas that need to be amended.

(2) It should not undermine the favorable performance of underlying k-ω-γ-orig model in attached flows.

(3) It is a strictly local-variable-based model, and thus can be applied to unstructured codes.

(4) It is a Galilean invariant, and can be applicable to the situation where surfaces move relative to the coordinate system.

According to the assessment in Section 3.1, the underperformance of ζeffin separated flows is rooted in Ω. But at the same time, it is also the peculiarity of k-ω-γ-orig model,that it uses Ω to connote the physical information of boundary layer thickness to realize the local-based formulation.Hence,it is not advisable to remove Ω from ζeffdue to its inevitable influence on predicting attached flows transition. Another strategy is to introduce an appropriate threshold to filter Ω value. This approach is simple to realize but confronted with two problems: (A) determining a threshold universally valid for any arbitrary flow is challenging and the empirical threshold would cause uncertainty; (B) the formulation of other instability modes (e.g. first-mode and Mack mode) might be polluted when the magnitude of Ω value in shear layer is the same order as that in the pre-transitional boundary layer.

To avoid those problems, a damping function (fss) is constructed here to rectify ζeffby analyzing the relation between the molecular diffusion timescale (τd) and rapid pressurestrain timescale (τr). This definition is inspired by the concept of shear-sheltering proposed by Jacobs and Durbin.41This concept is utilized to characterize the dampening of turbulence dynamics in thin high-vorticity regions, which obviously have a lot in common with the area where ζeffis overpredicted and needs to be modified in the current study. Specifically, when the rapid pressure-strain fails to re-distribute energy amongst the normal stresses, shear-sheltering occurs in a pretransitional boundary layer. Once transition begins, it is gradually limited to the viscous sublayer of turbulent boundary layer. Walters14suggested that this is caused by short τd. The relationship between fluctuating velocities (u′) and viscous length-scale (lv) is given as.

Herein, k is the kinetic energy computed from Eq. (1),where k=kLwhen 0<γ<1,and k=kTwhen γ=1.This implies that τdcan be further deduced as.

in which Cssis a constant.This distributions of fsswith various Cssaround T3L plate are presented in Fig. 8. As shown, the proposed fssrealizes the automatic identification of ζeffthat demands correction and has few impacts on modeling of other instability modes. Specifically, in the strong shear layer areas ahead of the separation bubble and the thin regions of viscous sublayer, fssis equal to 0, while in the freestream and most of the boundary layer, fsssharply recovers to 1. Meanwhile, fssrelies strictly on the local variables and Galilean invariance.Therefore, four requirements mentioned above are fulfilled.It is noted that fssis closely related to the value of Css. With the increase of Css, the fss= 0 area expands steadily near the leading edge.Based on a large number of experimental validations for the T3L case, a better agreement between predicted results (the length of LSB) with experiments can be found when Css= 0.1 (see Section 5.1), although its effects with different values are relatively small. And more importantly,Css= 0.1 is also applicable for the other test cases considered in this work, indicating its rationality.

Fig. 8 Damping function (fss) distributions around the semicircular head of T3L plate.

4.2. Formulation of pressure gradient parameter λζ

Several transition models for separated flow transition prediction have been proposed. Among them, the related work of Langtry and Menter17could be a good reference, in which the separation onset is determined through a relationship between momentum-thickness Reynolds number Reθcand vorticity Reynolds number Rev. Unfortunately, the solution of Reθcintroduces an extra transport equation, which is detrimental to computational efficiency. Therefore, the object of this section is to develop a more efficient method without a loss of accuracy or applicability, of which the priority is to construct an indicator to discriminate the separated flow instability. Similar to those in Section 4.1, the properties expected to be met for this indicator are listed as follows:

(1) It is susceptible to the response of local boundary layer to the separation instability.

(2) No additional transport equations need to be supplemented.

(3) It has simple structure and high portability.

(4) It is strictly based on local variables and Galilean invariance.

The splat mechanism indicates the impermeability condition at a wall deflects the fluctuations in the normal direction to the streamwise direction and generates the local pressure gradient to trigger the boundary layer transition.43Reed et al.44emphasized that the appearance of velocity inflection points in the boundary layer with APG would lead to the strong inflection point instability.In this situation,even weak disturbances could induce a dramatic alternation of the flow instability.The results of Saric45reveal that the decrease of surface pressure coefficient Cpby 1% corresponds to the increase of Falkner-Skan pressure gradient factor by 0.1,and thus the critical Reynolds number by the linear stability theory is reduced by 3 times compared with the zero pressure gradient flows. Moreover, the pressure gradient exerts the prominent influence on the position and shape of the separation bubble. With regard to the self-sustaining nature of bubble, as mentioned in Section 1,the mean flow deformation induced by the disturbances alters the pressure distribution across the bubble, causing a stabilization.6All of the work above demonstrates the significance of pressure gradient in the separated flow.Consequently,it is speculated here that the pressure gradient factor (λθ)would appear to be an indicator of separation-induced transition. In fact, λθhas been adopted in some transition models,which is typically utilized in empirical correlations for natural transition prediction and is defined as

where V and y are defined as the wall-normal velocity and distance to the wall.Intrinsically,the formulation of this equation in function of dV/dy is the simple application of continuity equation for 2D flows. The advantage of the above transformation is twofold: (A) to simplify the derivative solution and make it formulate locally; (B) to guarantee that the equation can be utilized inside the boundary layer. Strictly speaking,the relation in Eq. (25) is only correct outside the boundary layer. But there are still two caveats. Firstly, the momentum thickness θ in Eq. (26) is non-locally available. Besides, by referring to the previous studies about Kelvin-Helmholtz instability, it can be known that disturbances in separation bubble are amplified in areas which have high shear and are well away from the wall damping region. This feature should be embodied by the indicator.Therefore,a locally-determinable quantity is required to replace θ,and the effective length scale ζeffmight be a feasible attempt:

Fig.9 displays the λζprofiles at different x positions of T3L test case. Langtry and Sjolander8proved that within a pretransitional boundary layer, the peak value location of vorticity Reynolds number Revcorresponds well with the experimental location of the most rapid growth of the laminar fluctuations. Meanwhile, Revcould reflect the influence of shape factor on separated-flow transition. Thus, the profiles of Revare also provided to verify the reaction sensitivity of λζto the development of laminar fluctuations and pressure gradient. Herein, the negative values of λζare shown in order to keep consistent with Rev. From the leading edge, the flow is accelerated and both λζand Revremain small until they increase steeply near the separation onset (x = 5 mm). Then they increase further in the recirculation zone,where the velocities are negative and the inflectional velocity profiles appear.Accordingly, λζprofiles also have the inflection point, indicating that λζis susceptible to the response of the local boundary layer to the separated disturbances.Meanwhile,whether ahead or inside the separation bubble,the maximum values of λζand Revhave a coordinated growth, indicating that λζcan reflect the influence of laminar fluctuations and shape factor. Therefore, the proposed λζcould be a good indicator for the separated flow transition with explicit physical meaning and application convenience.

4.3. Formulation of separation intermittency γsep

In k-ω-γ-orig model, the various instability modes are represented by its corresponding characteristic timescales (see Eq.(5)and Eq.(12)).Herein,the effects of the separated instability on transition are characterized by the straightforward formulation of the separation intermittency γsep(see Ref. 17)**,and that is out of the following two considerations.

(1) The modeling of γ in k-ω-γ-orig model, especially its source term Pγ, is derived from the empirical correlation proposed by Dhawan and Narasimha,32according to the experimental data of attached boundary layer transition around the flat plate with zero pressure gradient. Hence, this γ is no longer applicable to the separated flow and an intermittency for separation is necessary.

Fig.9 Pressure gradient factor λζ(above)and vorticity Reynolds number Rev (below) profiles at different x positions of T3L plate.

(2) The concept of γ is widely used in most transition models. With γ, the proposed γsepcould be fairly convenient to incorporate into the existing transition models.

Similar to Ref. 17, Freattachdisables the modification once the viscosity ratio is large enough to induce reattachment and is expressed as.

where the value of s1is the same with that in Ref. 17, Csep3is used to satisfy γsep≤1.0.The constants Csep1and Csep2are calibrated to be 0.3 and 1.0 respectively through the verification of the selected four test cases with the corresponding experimental results in the current work. To enhance its generality,in fact, these parameters deserve a further study and a thorough assessment with more complex transitional flows.

In general,the main idea of γsepis to oblige the rapid amplification of γeffonce the laminar flow separates.In details,flow separations have an impact on the growth of γeff,and then the development of laminar kinetic energy k embodies the effects of instability in separated flows, which promotes the laminar-turbulent transition.

5. Results and discussion

Four typical test cases are considered to validate the performance of k-ω-γ-sep model in separation-induced transition flows. The subsonic T3L flat plate is firstly tested to assess the prediction of separation bubble size and the response to freestream turbulence intensity (Tu∞). Next, the subsonic Aerospatial A airfoil is considered to verify the prediction capability for short separation bubble induced transition.Additionally, to evaluate the applicability of the model for transonic flow,the transition simulation of the NLR-7301 airfoil is performed.Finally,a three-dimensional complex configuration called the Hypersonic Inflatable Aerodynamic Decelerator (HIAD) aeroshell is selected to demonstrate the model performance in hypersonic flows.

5.1.T3L flat plate with different freestream turbulence intensity

The details of freestream conditions for T3L plate can be found in Section 3.1. Herein, T3L plate exposed to varying Tu∞are computed with k-ω-γ-sep model. Three different Tu∞values, that is Tu∞= 0.63%, 2.39% and 5.34%, are considered.

Fig. 10 shows the predicted results by improved k-ω-γ-sep model under Tu∞=0.63%.Compared with Fig.1,the bubble simulated by k-ω-γ-sep model is nearly more than doubled in extent,and even greatly larger than that predicted by SST turbulence model. This is because, as a result of the introduction of damping function fss(in Eq. (23)), the nonphysical amplification of ζeff(see Fig. 4) is shielded and then the development of turbulent kinetic energy (k) is decayed. Specifically, the starting point of k growth simulated by k-ω-γ-sep model is located at x=10.6 mm,where the separation bubble has fully developed and thus the reattachment point is extraordinarily postponed. Apparently, the decay of turbulent kinetic energy preliminary verifies the rationality of fssfunction.

The quantitative comparison of Cfdistributions under varying Tu∞is depicted in Fig.11.As can be seen from the figure, the k-ω-γ-orig forecasts the excessively early reattachment, and moreover its variation with the increase of Tu∞is irrational. Meanwhile, the underestimation of eddy viscosity leads to much lower Cfdownstream of the bubble than the experimental results. In contrast, the proposed k-ω-γ-sep model displays fair consistency with the tunnel experiment under different Tu∞. With the increase of Tu∞, the predicted transition onsets move upstream and the width of separated zone is foreshortened steadily.The performance above demonstrates the proper response of k-ω-γ-sep model to Tu∞.are also provided for comparison. Clearly, the k-ω-γ-sep model obtains the result which is more consistent with the experiment.

Fig. 10 Effective length scale (above) and turbulent kinetic energy (below) contours predicted by k-ω-γ-sep.

Table 2 Reattachment points for T3L.

To further assess the k-ω-γ-sep model performance in capturing flow characteristics in LSBs,Fig.12 exhibits a sequence of velocity profiles predicted by both k-ω-γ-orig and k-ω-γ-sep models. δ represents the wall distance. As shown, two models capture the first signs of separation at x=6 mm and the close agreement with experimental data remains until the initial stage of bubble development(x=8 mm).Remarkable distinctions can be observed firstly at x = 11 mm, where k-ω-γ-sep model attains the well consistent results with the experiment.Comparatively, the velocity profiles predicted by k-ω-γ-orig model deviate evidently from the experimental observations in the boundary layer.

5.2. Aerospatial A airfoil

Fig. 11 Surface skin friction coefficient (Cf) distributions under different freestream turbulence intensity (Tu∞).

Fig. 12 Velocity profiles across separation bubble (Tu∞=5.34%).

Fig. 13 Surface skin friction coefficient (Cf) and intermittency distributions.

The freestream conditions for Aerospatial A airfoil are detailed in Section 3.2.The Cfdistributions in the leeward surface are drawn with the absolute value in Fig.13,in which separation and reattachment points are marked by dashed lines.As shown in the graph, up to the reattachment onset, both models show the well agreement with experimental data,which manifests that the modification of k-ω-γ-sep model does no damage to intrinsically good performance of k-ω-γ-orig model in short bubble prediction.The apparent differences are visible downstream of the reattachment point,where the growth of kω-γ-orig model is too weak.By comparison,Cfperceived by kω-γ-orig model augments steeply and reaches a peak of 0.0125 at x=157 mm,which is in good agreement with experimental results. Reasons for improved modeling of k-ω-γ-sep model can be traced to the accurate intermittency distributions shown in Fig. 13 and Fig. 14. It is obvious that the transition onsets obtained by two transition models are almost the same,roughly located at the separation point x=90 mm.However,once boundary layer separates, the formulated γsepof k-ωγ-sep increases rapidly from 0 to 1 in regions that possess high vorticity but are well away the wall, causing the development of effective intermittency γeffand thus its peak of 1 at around x=157 mm corresponding to the peak value of Cf.When the flow transits to turbulence, velocity profiles change with the thickening of boundary layer, and then the ddy viscosity near the wall increases gradually.In contrast,after the bubble,k-ωγ-orig model seemingly predicts a trend of relaminarization,that is, γ climbs to a peak value of 0.22 around reattachment point and then declines steadily. Combining with the discussion in Section 3.2, this phenomenon derives from Ω utilized in ζeff. In other words, due to the intrinsically high vorticity in separation bubble, the γ of k-ω-γ-orig model rises significantly, but the turbulent kinetic energy cannot sufficiently develop to the required degree and thus is easy to attenuate with diminishment of Ω values downstream. Eventually, the intermittency distributions of k-ω-γ-sep model are greatly different from those of k-ω-γ-orig model.

5.3. NLR-7301 airfoil

Fig. 14 γsep and γeff predicted by k-ω-γ-sep model and γ predicted by k-ω-γ-orig model around bubble.

In order to validate the capability of k-ω-γ-sep model in transonic flows, a benchmark case NLR-7301 airfoil, which is a typical thick supercritical airfoil with a rather blunt nose, is chosen. In terms of this airfoil, large amounts of wind-tunnel and numerical investigations have been made.Previous studies show that as Mach number varies, boundary layer transitions for the NLR-7301 airfoil are triggered through different mechanisms such as T-S waves and separated-flow transition.Herein, two representative Mach number conditions(Ma∞= 0.6 and 0.774) are selected with a fixed angle of attack αt= 8.5° and Reynolds number Re∞= 2.2 × 106/m,where transition is mainly dominated by the separated flow.

Fig. 15 Pressure (p/p∞) contours with the close-up of separation intermittency (γsep) or intermittency (γ) contours.

Dimensionless pressure (p/p∞) contours simulated by k-ωγ-sep model are given in Fig.15.For clarity,Fig.15 also provides the local enlarged view of intermittency around the transition onset. Herein, the transition regions dominated by the separation and T-S instabilities are shown with γsepand γ respectively, which are obtained through the transport equation. The comparison certifies that the transition onset on the upper surface varies significantly with freestream Mach number due to the alternation of flow structure. Under Ma∞=0.6, the large APG near suction peak causes a LSB and then triggers the separation-induced transition. Thus, the transition onset approaches to the leading edge. Under Ma∞= 0.774, however, noticeable shock waves occur at around x/c = 0.6 along the airfoil, where c denotes the chord length. The APG near the leading edge is replaced by the flat pressure distribution. At this moment, boundary layer transition is induced by the shock-induced separation, so does the case on the lower surface. Thus, the transition moves downstream to the shock position. As for the lower surface at Ma∞= 0.6, there is no separation and the transition is promoted by T-S waves.According to Fig.15,whatever the cause of boundary layer separation is, γseputilized to promote the separation-induced transition begins to develop near the separation bubble, which facilitates the rapid growth of turbulent kinetic energy and eventually triggers transition.The transition onsets predicted by the original and improved transition models are listed in Table 3 and compared with experimental data.Clearly,the detailed results confirm the well performance of kω-γ-sep model,while the prediction of k-ω-γ-orig model deviates far from experiments due to its incapability for the separated flow transition.

Table 3 Transition-onset locations.

5.4. Hypersonic inflatable aerodynamic decelerator aeroshell

To illustrate the effectiveness of the improved transition model in the hypersonic flow, a much complicated configuration called Hypersonic Inflatable Aerodynamic Decelerator(HIAD) is selected to conduct the numerical prediction. The HIAD technology provides an effective, mass-saving alternative to orthodox rigid decelerators.However,under the impact of aerodynamic force, its flexible surface will be largely deformed, which would induce boundary layer transition,causing a severe aeroheating augmentation to challenge the survivability. Wind tunnel tests are conducted by Langley Research Center to determine the transition along deformed HIAD.46Relying on these valuable experimental data, Zhao et al.47–49performed a numerical study to make a deep comprehension of the flow field characteristic for HIAD. The results demonstrate that it is the T-S,crossflow and separation instabilities that dominate the transition, challenging the prediction of boundary layer transition. The freestream details are listed in Table 4. Herein, the condition of angle of attack α=0°is selected in order to exclude the interference of crossflow instability. The freestream turbulence intensity Tu∞is determined as.

in which q represents flux.H0and H300Kare total enthalpy and enthalpy at 300 K, respectively. The reference heat-transfer film coefficient hFRis based on Fay-Riddell theory for the hemisphere of 9.525 × 10-3m radius and equals to 0.501 kg/m2.s. The experimental result is also provided in the top right-hand corner of Fig. 16 for reference. As presented,in virtue of the wall undulating deformation, the streamlines separate at crests then reattach on valleys, subsequently promoting the transition. A ring-like transition zone is formed,starting near the third crest and terminating near the fifth valley,which is quite in accordance with the experimental data.In addition,the notable discrepancy for the separation extent can be found with the flow regime alternation. Namely, the reattachment line moves upstream once flow transits to turbulence,due to its greater ability to resist the effect of APG.The quantitative comparison of h/hFRalong the surface centerline is plotted in Fig. 17. Not surprisingly, almost negligible differences among laminar, k-ω-γ-orig and k-ω-γ-sep remain until the transition onset. Downstream of the transition onset, h/hFRpredicted by k-ω-γ-sep model rises and a good correspondence with experimental data is achieved, indicating its superior performance in separated-flow transition around such a complex configuration. By contrast, the k-ω-γ-orig predicts a fallacious result, nearly consistent with the laminar.

Fig. 16 Surface intermittency (γeff) contours (left) and heat flux(h/hFR) contours with streamlines (right).

Fig. 17 Centerline heat flux (h/hFR) distribution.

Table 4 Freestream conditions.

6. Conclusions

An improved k-ω-γ transition model for separation-induced transition prediction (called k-ω-γ-sep) is developed and validated for several representative transitional flows with subsonic to hypersonic speeds. The k-ω-γ-sep model is formulated strictly based on local variables and the Galilean invariance. Based on an in-depth analysis on the deficiency of the original k-ω-γ model (marked by k-ω-γ-orig) for the separation-induced transition,three critical steps are proposed to improve this model.First,the damping function(fss)is constructed to revise the effective length scale (ζeff). Then, the pressure gradient parameter (λζ) is introduced as an indicator for identifying the separation instability. Finally, the separation intermittency(γsep)is formulated based on λζto characterize the influence of flow separation on boundary layer transition. Main conclusions are drawn as follows:

(1) The wrongly-predicted separation bubble length by the k-ω-γ-orig model is ascribed to the irrational introduction of vorticity Ω in ζeffin the strong shear layer at the front of the separated bubble.Besides,due to a lack of intrinsic physical mechanism responsible for the separation effect in the transition process in k-ω-γ-orig model, the boundary layer transition after bubble does not fully develop, confirmed by the underrated surface skin friction coefficient.

(2) In light of the irrational distribution of ζefffor k-ωγ-orig model in separation flow, a damping function fssis constructed here to rectify ζeffby analyzing the relation between the molecular diffusion timescale (τd) and rapid pressure-strain timescale (τr). fsscould automatically identify the regions necessary to be rectified to prevent ζefffrom unduly early amplification without undermining the k-ω-γ-orig model in attached flow.This improvement enables the model to obtain an accurate extent of separation bubble.

(3) The ζeff-based pressure gradient factor λζis developed to characterize the pressure-gradient influence on amplification of Kelvin-Helmholtz instability in high-vorticity regions of separation bubble.Then,the separation intermittency γsepis constructed based on λζto trigger separation-induced transition. Although rooted in the k-ω-γ model framework, the local-variable-based γsepwithout introducing any additional transport equation can be easily incorporated into other transition models involving the intermittency.

(4) The improved model is applied to several wide-speedrange test cases,including subsonic T3L flat plate at different freestream turbulence intensities, subsonic Aerospatial A airfoil,transonic NLR-7301 airfoil at different Mach numbers and hypersonic inflatable aerodynamic decelerator aeroshell. In all test cases, well agreement with the experimental results is observed,demonstrating its great performance in predicting transition to turbulence of a laminar boundary layer undergoing separation.

(5) The current study focuses on the transition induced by short laminar separation bubble, while the capacity for long bubble is still ambiguous. Besides, the k-ω-γ-sep model needs further verification for more complex situation where multiple instability disturbances coexist and interact with each other. Therefore, further studies for the k-ω-γ-sep model are demanded in the future.

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.

Acknowledgments

This research is supported by the National Natural Science Foundation of China(Nos.11902367 and 12002355),the State Key Laboratory of Aerodynamics, China (No. SKLA-20200202) and the National Natural Science Foundation of Hunan Province, China (No. S2021JJQNJJ2716). This work is supported in part by the High Performance Computing Center of Central South University.