Data-driven approaches have made preliminary inroads into the area of transition–turbulence modeling, but are still in their infancy with regard to widespread industrial adoption. This paper establishes an artificial neural network (ANN)-based transition model to enhance the capacity of capturing the crossflow (CF) transition phenomena, which are frequently identified over a wide range of aerodynamic problems. By taking a new CF-extended shear stress transport (SST) transition-predictive (SST-γ) model as the baseline, a mapping from mean flow variables to transition intermittency factor (γ) is constructed by ANN algorithm at various Mach and Reynolds numbers of an infinite swept wing. Generalizability of the resulting ANN-based (SST- γ ANN) model is fully validated in the same infinite swept wing, an inclined 6:1 prolate spheroid, and a finite swept wing in extensive experiment regimes, together with two effective a priori analysis strategies. Furthermore, the calculation efficiency, grid dependence, and performance of the present model in non-typical transitional flow are also assessed to inspect its industrial feasibility, followed by the elucidation of rationality behind the preliminary success and transferability of present framework. The results manifest that the SST- γ ANN model aligns well with the benchmark SST-γ model, and both can capture the CF transition accurately compared with their experiment counterpart, completely breaking through the disability of original SST-γ model without CF correction. In addition, good properties of efficiency, robustness, and generalizability are achieved for the ANN-alternative transition model, together with the usability of present framework across various transitional flows.

The laminar–turbulent transition is of practical relevance in a wide variety of aerodynamic applications such as low- and high-speed aircrafts, helicopter rotor, unmanned aerial vehicle, turbomachinery, and wind turbines. Accurate transition–turbulence prediction has great influence on a catalogue of aerodynamic properties.1 With the accelerating developments in computer hardware and high-performance computing technique, the heavy responsibilities of transition–turbulence simulation are carried by the computational fluid dynamics (CFD) methods in most cases.2 

Although expansive applications of high-fidelity (Hi-Fi) CFD techniques such as direct numerical simulation (DNS) and large-eddy simulation (LES) have been witnessed in the past two decades, applying these numerical methods to complex industrial transitional flows still remains unfeasible so far. By contrast, Reynolds-averaged Navier–Stokes (RANS) methods have become the backbone for transition prediction owing to their computational tractability, robustness, and accuracy.3,4 Nevertheless, it should be pointed out that only by coupling the transition model, i.e., the transition-predictive RANS model, may the capture of transition phenomena become a reality. Various research types have shown that the transition-enabled RANS model performs much better than full-turbulence RANS model in transitional flows, with the results being much more anastomotic to the real physical processes.5–11 

In practice, the employment of transition models merely opens up the possibility of precise transition prediction. The key to accurate capture of the transition phenomena depends on the correct consideration and inclusion of the transition criteria and mechanisms within the transition model.12 Over the past few decades, a local correlation-based transition model, i.e., the γ- R e ̃ θ t transition transport model, proposed by Langtry and Menter5,8 has received wide attention upon its publication. In this transition model, only local quantities are used in determining the onset of transition, which forms a framework for implementation of correlation-based models in general-purpose CFD methods, including those on unstructured grids.6,7 The accuracy of this model has been validated in various low-speed transitional flows, such as flat-plate boundary layer,7,8,13 aeronautical aircraft configurations,5,8,13 and turbomachinery flows.7,8,13 Efforts have further been made to extend the applicability of this model, including the compressible correction,14 incorporation of γ- R e ̃ θ t model into Spalart–Allmaras (SA) one-equation model,15 etc.

Despite the success it has achieved in CFD application, the aforementioned γ- R e ̃ θ t model is only capable of reflecting two-dimensional transition phenomena, such as transition due to Tollmien–Schlichting (TS) instability, separation-induced transition, and bypass transition.16 In many practical aerodynamic problems, e.g., the flow past swept wing of an aircraft or rotor blade of a wind turbine, the crossflow (CF) instability plays a critically enabling role, or can even be the dominant mechanism of transition phenomena.12 Against this backdrop, the original γ- R e ̃ θ t model fails to capture the CF instability-induced transition phenomenon and is inapplicable to many practical aeronautic flows.

In view of limitations of the γ- R e ̃ θ t model, a wealth of studies has been dedicated to improve this model for predicting CF instability-induced transition. One prominent avenue to the CF-extended transition model is to derive the crossflow transition criteria from the approximate solution of three-dimensional boundary layer equations, namely, Falkner–Skan–Cooke (FSC) equation.17 In analogy to the original γ- R e ̃ θ t model,5,8 a CF-extended γ- R e θ t- R e δ 2 t transition model is suggested by Grabe et al.18,19 However, this model is established only based on a semilocal approach, which does not inherit the advantage of previous γ- R e ̃ θ t model, and has intrinsic drawback in terms of non-wing-like geometry. In this regard, Choi and Kwon12 proposed a novel methodology to evaluate the criteria for CF-induced transition with only local variables, and incorporated the interaction between the TS instability and CF instability in the switching function of transition model20,21 in consideration of the observations of eN method.22 It is found that their new transition model (namely, γ- R e ̃ θ t-CF) performs well in general engineering problems while maintaining the local property of baseline γ- R e ̃ θ t model.

Although the aforementioned transition-enabled RANS models, extending the CF-induced transition mechanism or not, are widely used in engineering problems, more empirical and human-art intervention should be invoked to close various terms compared with fully turbulent RANS models,23 and more free parameters ought to be calibrated, thus importing more uncertainty to some extent.24 Meanwhile, it consumes a higher amount of computational time than fully turbulent RANS model.

Such deficiencies are difficult to overcome from the viewpoint of traditional approaches. Over the past few decades, artificial intelligence (AI) techniques and, in particular, the machine learning (ML) algorithms have thrived, and AI applications have entered a new stage due to the launch of ChatGPT recently. These developments in ML techniques have sparked new interest in solving the long-standing turbulence/transition model problems.23,25 In the community of ML-augmented turbulence modeling, Duraisamy et al.26,27 proposed a prospective data-driven framework comprised of Bayesian-based field inversion and machine learning-based function reconstitution (FIML) framework, mapping a relation between local flow quantities and the inferred discrepancy. Following such a practice, a suite of studies are conducted which have made modifications and enhancements to the original RANS model.28–30 Rather than adding a corrective multiplier or a direct corrective source term to rectify the existing models, Zhang et al.31–33 reconstructed an algebraic ML surrogate to approximate the turbulent viscosity of transport equation model and avoided solving the partial differential equations (PDEs), which opens up a door for AI-powered methodology to engineering practice, and is employed in several relevant studies.34,35 On the other hand, ML algorithms have also initiated a more disciplined and rational use of Hi-Fi data to develop more accurate turbulence models. Of particular relevance to the present discussion are the data-driven Reynolds stress,36–38 the discrepancy of Reynolds stress,39–41 and turbulent viscosity42–44 substitution models trained from abundant36–42 or sparse43,44 Hi-Fi databases. Obviously, these break the well-known bottleneck of conventional intuition-based turbulence model creations by virtue of the state-of-the-art ML techniques.

In comparison with the enormous boom in the field of ML-augmented full-turbulence models, the laminar–turbulent transition modeling based on ML approaches is still in its infancy. The first attempt to apply ML algorithm to transition model is made by Duraiasmy et al.,45 who introduce their FIML framework to bypass transitional flow, in which a relation between mean flow variables and the source term of intermittency factor transport equation is constructed to improve the accuracy of traditional transition model. Following this direction, Yang and Xiao46 apply the methodology of FIML to the improvement of kωγAr transition model, and the resultant new model can capture a more accurate transition location than the baseline model. Similar modeling methods can also be found in the latest study, which makes up for the gap in hypersonic boundary layer transition.47 Different from the research works mentioned above, Akolekar et al.48 proposed a novel ML architecture based on gene expression programming (GEP)37,38 and the CFD-driven ideology, and derived more precise expressions for the laminar kinetic energy (LKE) production and transfer terms within the LKE transition model.

The above-mentioned studies are concentrated on the correction45–47 or substitution48 of particular terms in the baseline transition models, aiming at improving the performance of the baseline models in predictions of a catalogue of flows. In fact, over a broad range of engineering problems of interest, there exists a continuously concerted effort from the transition modeling community to apply the existing model in hand to these circumstances with comparatively approving accuracy (e.g., the compressible correction,14 CF extension as mentioned above,16,18–21 and consideration for the distributed surface roughness effect49,50). In this regard, there is no urgent need to enhance these well-behaved transition models with ML techniques. In view of the limitations of traditional transition-predictive RANS models and the powerful ML framework, we have established an industrial-practical artificial neural network (ANN)-fully substituted transition model, which avoids solving the PDE of intermittency factor γ in Menter's shear stress transport (SST)-γ model51 and Spalart–Allmaras (SA)-γ model.15 Results manifest that the SST- γ ANN and SA- γ ANN models share an equivalent accuracy with the benchmark SST-γ and SA-γ models, and both closely coincide with Hi-Fi numerical results.52,53 The same conclusion can be drawn from the latest similar studies.54,55 However, due to the disability of baseline models in reflecting three-dimensional transition mechanisms, it is unsurprising that both SST- γ ANN and SA- γ ANN models have certain weakness in capturing the crossflow transition phenomenon, which extensively limits their engineering practicability. Against this backdrop, we devote particular attention to the extension of ANN-alternative transition model for crossflow instability, aiming to derive a precise, efficient, robust, and well-generalized ANN-fully substitution transition model with aeronautical applications in the true sense. Moreover, we would like to suggest a universal framework for engineering-practical ANN-alternative transition model development through this process.

The rest of this paper is organized as follows. First of all, the CF transition mechanism needs to be imbedded into the baseline SST-γ model to further improve its predictive capability for three-dimensional transitional flows, which is presented in Sec. II together with the framework of ANN-alternative SST-γ model. Furthermore, the functional form of transition intermittency factor γ is trained using the data from the new CF-extended SST-γ model. The detailed modeling strategy about the construction of input features, numerical methods, case setup for training and testing, and ANN architecture are introduced in Sec. III. Moreover, several testing cases are conducted in different configurations and flow regimes in Sec. IV to systematically validate the performances of the new SST- γ ANN model and the benchmark CF-enhanced SST-γ model. In addition, several explorations are implemented and summarized in Sec. V to verify the industrial practicality and universality of current ANN-alternative transition model paradigm. Finally, the main findings and remaining limitations of the present research work are highlighted in Sec. VI.

To consider the transition effect, Langtry and Menter5,8 propose a local correction-based γ- R e ̃ θ t model (with γ being the intermittency factor and R e ̃ θ t the momentum thickness Reynolds number). After that, a one-equation transition γ model is developed by Menter et al.51 in an attempt to achieve the property of Galilean invariance and simplify the correlations of model construction. The transport equation of intermittency factor γ takes the following form:51,
ρ γ t + x j ( ρ { u j } γ ) = x j [ ( μ L + μ T σ γ ) γ x j ] + P γ E γ ,
(1)
where ϕ represents the Reynolds-averaged quantity of ϕ and { ϕ } = ρ ϕ / ρ denotes the Favre-averaged quantity of ϕ, with ρ being the density. To avoid redundancy, the symbols · and { · } are omitted for variables hereafter. The intermittency factor γ is presumed to be zero in the laminar boundary layer upstream of transition and is unity in the turbulent region. Here, μ L and μ T represent the molecular viscosity and turbulent viscosity, respectively. The production and destruction terms of γ transport equation are denoted by P γ and E γ, which read
P γ = F length ρ S γ ( 1 γ ) F onset ,
(2)
E γ = C a 2 ρ Ω γ F turb ( C e 2 γ 1 ) ,
(3)
with S and Ω being the magnitudes of strain and rotation rate tensors, respectively. As mentioned in the original article,51 the production term is active once the local transition onset criterion is met, which is triggered by the F onset function. The formulation of F onset contains the ratio of local vorticity Reynolds number R e V to the critical Reynolds number R e θ c, which is controlled by the following functions:
F onset 1 = R e V 2.2 R e θ c , F onset 2 = min ( F onset 1 , 2.0 ) , F onset 3 = max [ 1 ( R T 3.5 ) 3 , 0 ] , F onset = max ( F onset 2 F onset 3 , 0 ) .
(4)
The combination between Menter's full-turbulence k-ω SST model56 and the one-equation γ model [see Eq. (1)] is implemented by modifying the production and destruction terms and introducing an additional production term P k lim in the turbulent kinetic energy equation of original SST turbulence model as follows:51,
( ρ k ) t + x j ( ρ u j k ) = x j [ ( μ L + σ k μ T ) k x j ] + γ τ i j F S i j + P k lim max ( γ , 0.1 ) · β * ρ ω k ,
(5)
( ρ ω ) t + x j ( ρ u j ω ) = x j [ ( μ L + σ ω μ T ) ω x j ] + C ω ρ μ T τ i j F S i j β ρ ω 2 + 2 ( 1 f 1 ) ρ σ ω 2 ω k x j ω x j .
(6)
Here, the transport equations of turbulent kinetic energy (TKE) k and specific turbulence dissipation rate ω are solved to determine the turbulent viscosity μ T in the form
min ( ρ k ω , a 1 ρ k f 2 Ω ) ,
(7)
and the unclosed Reynolds stress tensor τ i j F is modeled based on Boussinesq's hypothesis
τ i j F = ρ u i u j = μ T ( u i x j + u j x i ) 2 μ T 3 u k x k δ i j 2 3 ρ k δ i j .
(8)
For brevity and simplicity, readers are referred to the original articles51,56 for more details about the functions ( R e V , R e θ c , R T , F turb , P k lim, f1, and f2), constants ( σ γ , F length , C a 2 , C e 2, σk, β * , σ ω , C ω, β, σ ω 2, and a1), and formulations of the SST-γ model.

In the proposed SST-γ model, the TKE is generated consistent with the real physical process owing to the implantation of intermittency factor γ in Eq. (5), which provides the correct prediction of turbulent viscosity μ T and Reynolds stress τ i j F. In such endeavors, it is evident that the transition-enabled SST-γ model performs better than the fully turbulent SST model in transitional flows.9,51,52

As stated in Sec. I, to derive a CF-enhanced ANN-alternative transition model, the overarching business is to improve the capability of baseline SST-γ model to capture crossflow transition phenomenon. In consideration of the model accuracy, physical interpretability, and the form of empirical correlations, we intend to follow the modeling ideology by Choi and Kwon,12,20,21 who evaluate the crossflow transition criterion from FSC velocity profiles.17 Such a methodology does not miss the local property of original model and is applicable to arbitrarily shaped geometries.16 For completeness, the CF-enhanced one-equation γ model promoted from γ- R e ̃ θ t model by Choi et al.12,20,21 is briefly described below, and a detailed modeling philosophy can be found in the previous articles.12,20,21

Based on the two-dimensional boundary layer similarity velocity profiles,57 Cooke17 derives the following ordinary differential equations for three-dimensional boundary layer, i.e., the FSC equations:
f + f f + β H ( 1 f 2 ) = 0 , g + f g = 0 , B . C . f ( 0 ) = f ( 0 ) = g ( 0 ) = 0 , f ( ) = g ( ) = 1 ,
(9)
with β H being the Hartree parameter, f and g representing the dimensionless stream function and scaled spanwise velocity, respectively. The dimensionless streamwise and crosswise velocities are then given by
U = u U e = f cos 2 φ + g sin 2 φ , W = w U e = ( g f ) cos φ sin φ ,
(10)
where φ is the local sweep angle. Before the CF-extended transition model takes effect, three databases λ θ = F ( β H , φ ) , H 12 = G ( λ θ , φ ), and r = H ( β H ) are supposed to be constructed beforehand in the relevant parameter range of β H and φ, which are shown in Fig. 1. Here, λ θ donates the pressure gradient parameter, H 12 is the streamwise shape factor, and r represents the crossflow Reynolds number ratio, all of which can be obtained from the velocity profiles of Eq. (10). In this paper, the GEP method (see Zhao and coworkers37,38) is employed to obtain the following explicit forms of these databases:
β H = 0.657 λ θ { 17.2 λ θ + exp [ e 2.49 λ θ ( 8.35 λ θ + 3.38 φ 5.30 ) ] + 4.25 } , H 12 = λ θ 2 ( 1248 λ θ + 1343 ) ( 16.8 λ θ 6.29 ) ( 62.3 λ θ φ + 12.0 λ θ 1.02 φ + 30.6 ) 65.6 λ θ + 70.6 , r = 1.64 0.494 β H + 0.151 β H 2 3.48 β H 3 + 45.8 β H 4 201 β H 5 + 462 β H 6 643 β H 7 + 569 β H 8 323 β H 9 + 114 β H 10 22.9 β H 11 + 1.99 β H 12 .
(11)
FIG. 1.

Three databases in the CF-enhanced SST-γ model: (a) λ θ = F ( β H , φ ), (b) H 12 = G ( λ θ , φ ), and r = H ( β H ).

FIG. 1.

Three databases in the CF-enhanced SST-γ model: (a) λ θ = F ( β H , φ ), (b) H 12 = G ( λ θ , φ ), and r = H ( β H ).

Close modal
The functional forms in Eq. (11) are repeatedly employed to obtain β H , H 12, and r, which serve as the core variables for the crossflow transition criterion as follows:
F onset 1 , CF = R e ν , cross r · R e δ 2 t * , R e ν , cross = ρ d 2 | Ω | x μ L .
(12)
Here, d and | Ω | x denote the distance to the nearest wall and the magnitude of the vorticity component aligned with the external potential flow direction, respectively. The empirical C1 criterion suggested by Aenal et al.58 is adopted to obtain the crossflow displacement thickness Reynolds number at transition onset R e δ 2 t *, which depends upon the streamwise shape factor H 12.
In the above endeavors, the two-dimensional streamwise and crossflow transition phenomena are taken into account by functions F onset 1 and F onset 1 , CF in Eqs. (4) and (12), respectively. The rational intrusion of F onset 1 , CF plays an important role in the performance of a CF-enhanced transition model. Since the transition process may be accelerated due to considerable interaction between the TS and CF instabilities,22 a linear combination of F onset 1 and F onset 1 , CF is suggested as the function F onset 1 , inter, and the unified function F onset 1 , 3 D is proposed as follows:
F onset 1 , inter = { 34 37 F onset 1 + 20 37 F onset 1 , CF , F onset 1 > 0.5   and   F onset 1 , CF > 0.15 , 0 , F onset 1 0.5   or   F onset 1 , CF 0.15 ,
(13)
F onset 1 , 3 D = max ( F onset 1 , F onset 1 , CF , F onset 1 , inter ) .
(14)
Other transition onset functions are defined in the same manner as the original γ- R e ̃ θ t model,6,8 i.e.,
F onset 2 , 3 D = min [ max ( F onset 1 , 3 D , F onset 1 , 3 D 4 ) , 2.0 ] , F onset 3 , 3 D = max [ 1 ( R T 2.5 ) 3 , 0 ] , F onset , 3 D = max ( F onset 2 , 3 D F onset 3 , 3 D , 0 ) ,
(15)
because the type of instability only affects the initiation of transition phenomena.59 In addition, the governing equation and formulations of the production and destruction terms for the present CF-enhanced γ model still maintain their original forms as given in Eqs. (1)–(3). The transition length function F length is recalibrated to 1.7 by using the experimental data60 for an infinite swept NLF(2)–0415 wing at a Reynolds number of 3.72 × 10 6 and a Mach number of 0.239, and is termed as F length , 3 D, which significantly simplifies the correlations compared with that in γ- R e ̃ θ t model.12,20,21

By extending the effect of CF instability, it is manifested that the new CF-enhanced SST-γ model behaves more accurately than its original counterpart in Sec. II A. For brevity, the corresponding results will be presented in Sec. IV.

As is well recognized, the reasonable introduction of intermittency factor γ model brings crucial enhancement to the full-turbulence RANS model.9,10,12,16 Nevertheless, the traditional transition-enabled RANS models are plagued by their deficiencies as mentioned in Sec. I. Considering the imperative role of intermittency factor γ and state-of-the-art ML algorithms, the present study is intended to establish an ANN-based physical relation between the RANS mean flow variables q ( x ) and space-dependent intermittency factor γ ( x ) field. Specifically, the ANN transition model is first trained using the data from SST-γ model and then is fully substituted for solving the γ equation during CFD iteration, which is followed by some necessary inspections over its engineering usability. Admittedly, the precision of ANN-substituted SST- γ ANN model is restricted by the baseline SST-γ model itself. In this scenario, the property of accurately capturing the CF transition phenomena is taken as the essential requirement for baseline model with the goal of developing a practical ANN-substituted transition model, i.e., the SST- γ ANN model. The overall schematic diagram is illustrated in Fig. 2.

FIG. 2.

The overall schematic diagram of practically aeronautical ANN-substituted transition–turbulence model.

FIG. 2.

The overall schematic diagram of practically aeronautical ANN-substituted transition–turbulence model.

Close modal

The proper selection of input features plays a critically enabling role for the model performance. To date, there has been a concerted effort to construct the inputs with clear physical reasoning for various ML-augmented turbulence modeling issues.31,39,61,62 In this study, seven inputs are selected carefully to convey as much amount of turbulence (i.e., the intermittency factor γ) as possible in terms of physically relevant quantities, which are abbreviated as f1f7 in Table I. The interpretations for most inputs are provided with the brief descriptions in the table, but a few need further discussions. First, the function f d = 1 tanh ( r d ), i.e., the modification of term f d = 1 tanh ( 8 r d ) 3 in detached eddy simulation,63 is established to identify the boundary layer61 since γ behaves intricately in the near-wall region. Input f4 is an indicator to distinguish boundary layers from shear flows, and often serves as the wall function in traditional turbulence models. In addition, inputs f3 and f5 suggest the amount of turbulence to a large extent and exhibit similar distributions to the target γ by qualitative analysis (see, e.g., the contours of turbulence intensity and γ in Fig. 3), which is considered as a plain idea of constructing input features.32 

TABLE I.

Input features f1f7 and p1p9. Here, r d = ( ν L + ν T ) / κ 2 d 2 u i , j u i , j, in which ν L and ν T are the (kinematic) molecular and turbulent viscosities, respectively, and κ = 0.41 is the Karman constant. S and Ω are the strain rate and rotation tensors, respectively, and | | · | | represents the matrix norm. ωi and Sij are the corresponding components of vorticity vector and strain rate tensor. Other notations are the same as those in Sec. II. The relevant model constants β * , C ω, and β can be found in Mentor's kω SST model.56 

Input features Description Formula
f1  Boundary layer identification function61   1 tanh ( r d ) 
f2  Q-criterion39,62  | | Ω | | 2 | | S | | 2 | | Ω | | 2 + | | S | | 2 
f3  Turbulence intensity39,62  k k + 0.5 u i u i 
f4  Wall-distance-based Reynolds number39,62  min ( k d 50 ν L , 2 ) 
f5  Eddy viscosity ratio39,62  μ T μ T + 100 μ L 
f6  Ratio of pressure normal stresses to shear stresses39,62  p x i p x i p x j p x j + 1 2 ρ u k 2 x k 
f7  Streamwise velocity31   u 
p1  Wall distance function  exp 1 ( d ) 
p2  Vortex stretching  ω j ρ u i x j ω k ρ u i x k ω l ρ u n x l ω m ρ u n x m + | | Ω | | 2 ρ 
p3  Pure production of k equation  τ i j F S i j β * ρ ω k 
p4  Pure production of ω equation  C ω ρ μ T τ i j F S i j β ρ ω 2 
p5  Local turbulence intensity in γ model equation (divided by ten)  min ( 100 2 k / 3 ω d , 100 ) / 10 
p6  Critical momentum thickness Reynolds number in γ model equation (logarithmic form)  lg  R e θ ca 
p7  Model function in destruction term of γ equation  F turba 
p8  Model function in production term of CF-extended γ equation  F onset , 3 Db 
p9  Helicity Reynolds number (absolute-logarithmic form)  lg ( ρ d 2 μ L | u i ω i | u j u j ) 
Input features Description Formula
f1  Boundary layer identification function61   1 tanh ( r d ) 
f2  Q-criterion39,62  | | Ω | | 2 | | S | | 2 | | Ω | | 2 + | | S | | 2 
f3  Turbulence intensity39,62  k k + 0.5 u i u i 
f4  Wall-distance-based Reynolds number39,62  min ( k d 50 ν L , 2 ) 
f5  Eddy viscosity ratio39,62  μ T μ T + 100 μ L 
f6  Ratio of pressure normal stresses to shear stresses39,62  p x i p x i p x j p x j + 1 2 ρ u k 2 x k 
f7  Streamwise velocity31   u 
p1  Wall distance function  exp 1 ( d ) 
p2  Vortex stretching  ω j ρ u i x j ω k ρ u i x k ω l ρ u n x l ω m ρ u n x m + | | Ω | | 2 ρ 
p3  Pure production of k equation  τ i j F S i j β * ρ ω k 
p4  Pure production of ω equation  C ω ρ μ T τ i j F S i j β ρ ω 2 
p5  Local turbulence intensity in γ model equation (divided by ten)  min ( 100 2 k / 3 ω d , 100 ) / 10 
p6  Critical momentum thickness Reynolds number in γ model equation (logarithmic form)  lg  R e θ ca 
p7  Model function in destruction term of γ equation  F turba 
p8  Model function in production term of CF-extended γ equation  F onset , 3 Db 
p9  Helicity Reynolds number (absolute-logarithmic form)  lg ( ρ d 2 μ L | u i ω i | u j u j ) 
a

The detailed expressions are described in the original article by Menter et al.51 

b

The detailed expressions are described in Sec. II B.

FIG. 3.

Contours of (a) turbulence intensity and (b) intermittency factor around the leading edge on the upper surface of NLF(2)–0415 wing at the Reynolds number of 3.72 × 10 6 and the Mach number of 0.239.

FIG. 3.

Contours of (a) turbulence intensity and (b) intermittency factor around the leading edge on the upper surface of NLF(2)–0415 wing at the Reynolds number of 3.72 × 10 6 and the Mach number of 0.239.

Close modal

Moreover, nine additional input features are recruited to ensure a more reasonable physical relevance between RANS mean flow variables and intermittency factor, which are also listed in Table I and denoted by p1-p9. In analogy to feature f7, the wall distance function p1 is designed to highlight the weight of near-wall region, but from the geometric perspective of flow. In addition, the vortex stretching term p2 is bound up with the energy cascade phenomenon, which is an important characteristic of turbulence.64 Despite certain weaknesses encountered in conventional transition–turbulence modeling, the rationale and philosophy behind it is worth the cost of excavation. First, the pure production terms (i.e., the production term minus the destruction term) of TKE Pk and specific turbulence dissipation rate P ω (see p3p4 in Table I) are taken as the inputs as in previous research,52 in which the plus–minus characteristics of Pk and P ω reflect the laminar or turbulent flow state.65 With regard to the γ model equation, features p5 and p6 behave as the core arguments in the transition onset correlation of γ model,51 and in particular, p6 implies the location where the intermittency factor first starts to increase in the boundary layer. Similar principles can also be found in inputs p7 and p8. Meanwhile, the helicity Reynolds number R e He = ρ d 2 μ L u · ( × u ) | u |, which is usually used to quantify the strength of local crossflow,16 is also involved in the input family.

Since dimensionless governing equations are solved in the present in-house code, there is no absolute necessity to normalize these input features unless one is much larger than unity. As for these 16 inputs, only f2, f3, f5, f6, and p2 are normalized with the scheme proposed by Ling and Templeton.62 Moreover, several functional forms are selected to rescale the features p5, p6, and p9 to ease the updating of trainable network parameters.66 All of these additions are briefly illustrated in Table I.

The novel permutation feature importance (PFI) method is employed to quantify the relative importance of the 16 input features. PFI is a model inspection technique that can be used for measure of the sensitivity of a trained ANN model to the presence or absence of an input feature. If a feature is indeed significant, evaluating the model on perturbed values of that feature will cause an increase in the prediction error.67 Shown in Fig. 4 is the normalized PFI of present ANN model. It turns out that some inputs (e.g., f3, p1, p4, and p6) have relatively weak influence on the model formulation. Theoretically, this outcome can be used to inspect the redundancy of the 16 inputs and exclude some non-informative features. However, the a posteriori results (not shown here for brevity) manifest that the removal of some relatively unimportant inputs such as f3 spoils the model accuracy to some extent, which demonstrates a reasonable balance of the current input family in Table I between model conciseness and the level of reflected phenomenological details.

FIG. 4.

Normalized permutation feature importance of the ANN model.

FIG. 4.

Normalized permutation feature importance of the ANN model.

Close modal

To generate the dataset for training and testing, numerical simulations are carried out with an in-house code based on structured finite volume method, which integrates the three-dimensional Favre- and Reynolds-averaged Navier–Stokes equations and crossflow-enhanced SST-γ model equations. In this flow solver, inviscid fluxes are evaluated using Roe's flux-difference splitting scheme,68 and the viscous fluxes are discretized with traditional second-order central finite difference scheme. The lower-upper symmetric Gauss–Seidel (LU-SGS) method without sub-iterations is employed for time marching. To validate the model generalizability thoroughly, flow past the infinite swept NLF(2)–0415 wing, one of the simplest geometries concerned with CF transition, at several flow regimes are adopted as the training set (termed as cases N1, N3, N4, and N5 in Table II). The test-bed is composed of flows past the NLF(2)–0415 wing, non-wing-like 6:1 inclined prolate spheroid, and finite ONERA M6 swept wing. Such an arrangement represents the most ambitious exploration for generalizing the present data-driven transition model developed merely in canonical case to complex engineering practice. The expansive parameter regimes of the freestream angle of attack (AoA), the Mach number (Ma), and the Reynolds number (Re) are imposed as such due to the availability of experimental data,60,69,70 different from parameter space sampling by Latin hypercube sampling (LHS) method used in previous studies.52,53

TABLE II.

The training (bold symbols and numbers) and testing cases (all in the table).

Case Geometry AoA Ma Re ( × 10 6) Case Geometry AoA Ma Re ( × 10 6)
N1  NLF(2)–0415  −4°  0.123  1.92  S1        3.01 
N2      0.140  2.19  S2  Spheroid  29.5°  0.136  4.48 
N3      0.151  2.37  S3        8.52 
N4      0.174  2.73  M1    0°     
N5      0.209  3.27  M2  M6  5°  0.262  3.5 
N6      0.239  3.72  M3    15°     
Case Geometry AoA Ma Re ( × 10 6) Case Geometry AoA Ma Re ( × 10 6)
N1  NLF(2)–0415  −4°  0.123  1.92  S1        3.01 
N2      0.140  2.19  S2  Spheroid  29.5°  0.136  4.48 
N3      0.151  2.37  S3        8.52 
N4      0.174  2.73  M1    0°     
N5      0.209  3.27  M2  M6  5°  0.262  3.5 
N6      0.239  3.72  M3    15°     
In this paper, fully connected artificial neural network (ANN), a modern machine learning method with outstanding performance comparable to other state-of-the-art techniques,71 is employed to construct the data-driven transition model. The typical structure of ANN framework is depicted roughly in Fig. 2(b). Gradient descent methods based on backpropagation algorithms72 are used to adjust the weights w and bias b of each neuron in ANN until the deviation between model prediction and true value (i.e., the loss function) converges. In the present study, the Adam algorithm73 with its learning rate decreasing dynamically is adopted to compute the cost function gradient. The commonly used mean square error (MSE) is employed as the loss function. In view of the tremendous challenge for generalizing the present ANN-alternative transition model to multiple flows with different regimes, the L2-norm regularization [see the second term in Eq. (16)] is supplemented to suppress overfitting as follows:
M S E L 2 norm = 1 2 N i = 1 N | y i y i ̂ | 2 + λ 2 N | | w | | 2 ,
(16)
in which yi and y i ̂ represent the true and predicted outputs of the ith sample, respectively. Formally, λ is the regulation coefficient that offers a balance between the modeling accuracy in training set and generalization capability of data-driven transition model, which will be discussed in Sec. IV A. In addition, the LeakyReLU function,74,
ϕ ( x ) = { x if x 0 , a x if x < 0 ,
(17)
is adopted as the activation function of neurons with the constant a set to 0.05 in the present study. Similar to a previous investigation,52 the architecture of two hidden layers with 128 and 64 neurons is adopted, which also proves to hold for the current modeling problem.

By virtue of the physical implication and inlet value setting of intermittency factor γ,6,8 the non-trivial region of γ, where its value does not fall around unity, only occupies the concentrated near-wall area. This imbalance phenomenon of samples has been encountered in many FIML problems.30 Among several relevant approaches to obtain the training data, we choose to reserve the samples only inside a prescribed area containing the near-wall region and discard the others, which is demonstrated to be more effective and efficient compared with the downsampling and unbiased sampling methods.

As described in Sec. II C, the mutual iterative strategy between ANN transition model and CFD solver is employed. However, it consumes great amount of time to invoke the ANN in each iteration. For reducing the computation load, the concept of “duty cycle” is proposed, during which the ANN model is only invoked once, leading to the unique update of γ, and then, the following utilization of ANN shifts toward the next duty cycle. In this paper, a duty cycle of 50 iterations is adopted. However, it should be stressed that the concept of duty cycle may not apply to traditional transition modeling framework because γ is governed by compact PDE equation [Eq. (1)], and the computation will probably diverge due to the indolence in solving the model equation. In this sense, the present ANN alternative may be an enabler for more efficient operation of high-performance-computing (HPC) facilities, which will be further discussed in Sec. V A. It needs to be stressed that other than the mutual iterative strategy, the frozen coupling mode75 is also feasible in the present modeling framework with good numerical stability and convergence property (not shown in this paper for brevity). Nevertheless, it is unsuitable for engineering design applications due to the necessity for converged flow fields given in advance by corresponding baseline model.

Contemplating the inconclusive effect of regulation coefficient λ [see Eq. (16)] among the intricate testing cases, the a priori test process is conducted to obtain an overview of the impact of λ on the model predictive capability. In this paper, the coefficient of determination R2 that quantifies how well the regression predictions approximate the real data points is adopted in the form
R 2 = 1 RSS TSS = 1 i ( y i y i ̂ ) 2 i ( y i y ¯ ) 2 .
(18)
Here, RSS and TSS represent the sum of squares of residuals and the total deviation of sum of squares, respectively, and y ¯ denotes the average of true value.76 An R2 of one indicates that the regression predictions fit the data perfectly, and a better approximation to one leads to a better model performance.

To obtain an optimal regulation coefficient λ, several ANN models trained with λ set to zero, 0.001, 0.003, 0.005, and 0.01, separately, are evaluated by R2 score through the testing cases listed in Table II. As shown in Fig. 5, the intermittency factor γ from ANN transition model is in high agreement with that from baseline SST-γ model in NLF(2)–0415 configuration, especially for λ = 0. By contrast, the prediction ability of ANN models for spheroid and M6 wing is slightly inferior to that for NLF(2)–0415 wing. Actually, it is presumed to be reasonable from perspective of the constitution of training dataset. As for the specific value of λ for each configuration, it is manifested that the model trained with regulation coefficient λ of 0.003 outperforms others in all cases of the spheroid and M6 wing without any exception, different from λ = 0 being the best choice for NLF(2)–0415 wing, which confirms the overfitting phenomenon and the necessity of regularization term. In this regard, the model established without the regularization term is employed to implement the a posteriori test for NLF(2)–0415 wing, whereas the model with λ set to 0.003 is used for the spheroid and M6 wing. Based on the test study (not reported in this paper for brevity), these selections of regulation coefficient from a priori perspective are generally optimal in the a posteriori assessment.

FIG. 5.

Coefficient of determination R2 for different testing cases: (a) NL2(2)–0415 wing, (b) spheroid, and (c) M6 wing.

FIG. 5.

Coefficient of determination R2 for different testing cases: (a) NL2(2)–0415 wing, (b) spheroid, and (c) M6 wing.

Close modal

The infinite swept NLF(2)–0415 wing with a geometric sweep angle (Λ) of 45° is first employed to validate the present transition–turbulence model. Series of experiments have been conducted in the wind tunnel of Arizona State University,60 which are summarized in Table II. In this paper, a C-type grid is generated with 1201 and 181 points along the circumferential and wall-normal directions, respectively. The wing is stretched by one chord length in the spanwise direction with Λ = 45 ° and 4 grids, and periodic boundary conditions are imposed to represent an infinite wing. In addition, the first off-wall grid height is set to 2 × 10 6 chord length, corresponding to y + = 0.6 or so.

To date, relevant studies have reached a consensus that the transition process of the swept wing is completely dominated by crossflow instability for Reynolds numbers great than 2.3 × 10 6, and the transition location moves toward the leading edge of the wing as Reynolds number increases.12,16,60 Fig. 6 depicts distributions of the mean skin friction coefficient C f = 2 τ w / ( ρ u 2 ) obtained using original SST-γ [denoted as SST-γ model without CF (SST-γ-NoCF) hereafter], SST-γ, and SST- γ ANN models along the upper surface at the midspan of NLF(2)–0415 wing. For comparison, results from the fully turbulent SST model are also included. As can be seen, SST- γ ANN achieves good agreement with the baseline SST-γ model, both exhibiting a rational variation trend of Cf with the growth of the Reynolds number. Therefore, the crossflow transition phenomenon is generally captured by the present SST-γ and SST- γ ANN models. By contrast, due to the absence of crossflow mechanism, the SST-γ-NoCF model nearly remains stationary across the whole parameter regimes. Similar situation can also be observed from the curves for SST model, which even fails to capture the transition phenomenon.

FIG. 6.

Distributions of computed mean skin friction coefficient along the upper surface at the midspan of NLF(2)–0415 wing for cases: (a) N1, (b) N2, (c) N3, (d) N4, (e) N5, and (f) N6.

FIG. 6.

Distributions of computed mean skin friction coefficient along the upper surface at the midspan of NLF(2)–0415 wing for cases: (a) N1, (b) N2, (c) N3, (d) N4, (e) N5, and (f) N6.

Close modal

To further validate the performance of present transition–turbulence model, comparative study is conducted upon the transition onset locations (denoted by chordwise coordinate Xtr) at different Reynolds numbers. As depicted in Fig. 7, the results from SST- γ ANN coincide perfectly with those from SST-γ model, and both fall in vicinity of the locations given by experiment60 and eN method.77,78 The SST-γ-NoCF model, however, shows no correlation with the Reynolds number, failing to predict the transition locations except for the low-Reynolds-number case with weak crossflow effect.

FIG. 7.

Computed and measured transition onset locations on the upper surface of NLF(2)–0415 wing for different Reynolds numbers. In numerical results, the transition location refers to the intersection point of the laminar and transition regions of friction coefficient curve.54 The results from experiment60 and eN method77,78 are also exhibited for comparison.

FIG. 7.

Computed and measured transition onset locations on the upper surface of NLF(2)–0415 wing for different Reynolds numbers. In numerical results, the transition location refers to the intersection point of the laminar and transition regions of friction coefficient curve.54 The results from experiment60 and eN method77,78 are also exhibited for comparison.

Close modal

As seen in Fig. 7, relatively accurate transition locations are predicted by transition–turbulence models at a low Reynolds number for case N1, whether the CF mechanism is considered or not. However, apparent disparity can be observed from the skin friction coefficient distributions and the streamlines on the upper surface of NLF(2)–0415 wing, which are displayed in Fig. 8 to intuitively illustrate the effect of crossflow instability. For comparison purpose, we show on the right-hand-side column the corresponding results for another representative case N6 at a high Reynolds number. As depicted in Figs. 8(a)–8(c), a laminar separation bubble (LSB) is recognized from the SST-γ-NoCF model, leading to a separation-induced transition phenomenon, which differs from the natural transition due to the Tollmien–Schlichting instability in CF-enhanced SST-γ and SST- γ ANN models. Such difference also explains the discrepancy in absolute quantity of the minimal Cf between the SST-γ-NoCF and SST-γ/SST- γ ANN model [see Fig. 6(a)]. As for the high-Reynolds-number case N6, crossflow instability is well captured by SST-γ and SST- γ ANN models, and as a result, the high skin friction region moves farther upstream, in consistence with the pictures observed in Figs. 6(f) and 7.

FIG. 8.

Streamlines overlaid on contours of skin friction coefficient on the upper surface of NLF(2)–0415 wing with different Reynolds numbers and transition–turbulence models: (a) SST-γ-NoCF, (b) SST-γ, and (c) SST- γ ANN at R e = 1.92 × 10 6, (d) SST-γ-NoCF, (e) SST-γ, and (f) SST- γ ANN at R e = 3.72 × 10 6.

FIG. 8.

Streamlines overlaid on contours of skin friction coefficient on the upper surface of NLF(2)–0415 wing with different Reynolds numbers and transition–turbulence models: (a) SST-γ-NoCF, (b) SST-γ, and (c) SST- γ ANN at R e = 1.92 × 10 6, (d) SST-γ-NoCF, (e) SST-γ, and (f) SST- γ ANN at R e = 3.72 × 10 6.

Close modal

Due to the availability of experimental results, Fig. 9 presents the distributions of mean pressure coefficient C p = 2 ( p p ) / ( ρ u 2 ), where the legends “lower/upper” refer to the locations of pressure taps near the testing model. For brevity, only curves for case N6 are depicted since the difference is barely discernible across different cases. As can be seen, favorable pressure gradient extends broadly from the leading edge to the location of about 0.71 chord length, which creates a comfortable environment for the development of crossflow instability rather than TS instability. Meanwhile, a tiny plateau is observed near X / C = 0.75 of the curve for SST-γ-NoCF, which also verifies the existence of LSB and the resultant separation-induced transition.9,52,53 From the perspective of comparison, it is evident that data-driven SST- γ ANN model is in very good agreement with the traditional SST-γ model, and the predicted pressure distribution is much more consistent with the experimental data compared with the original SST-γ-NoCF model. Overall, it is ensured that the predictive nature of an ANN-substituted intermittency factor closure is well established in terms of NLF(2)–0415 wing based on our modeling strategy, thus achieving an accurate alternative to the traditional transition model.

FIG. 9.

Computed and measured distributions of mean pressure coefficient along the upper surface at the midspan of NLF(2)–0415 wing for case N6.

FIG. 9.

Computed and measured distributions of mean pressure coefficient along the upper surface at the midspan of NLF(2)–0415 wing for case N6.

Close modal

The present ANN-substituted transition–turbulence model is further validated in flows past a 6:1 inclined prolate spheroid, which appears to be a challenging configuration due to its non-wing-like geometry.16 In the present study, the experimental data reported by Kreplin et al.69 are included for comparison purpose. The computational grid is depicted in Fig. 10. The upstream and downstream boundaries are set at a location of ten spheroid lengths away from the spheroid edges to eliminate the effect of imposed boundary conditions. Meanwhile, the first off-wall grid height is about 5 × 10 6 spheroid length to fully meet the requirement of normal resolution in the near-wall region.

FIG. 10.

Mesh pattern and grid-point distribution of (a) whole computational domain and (b) near-wall region for the 6:1 inclined prolate spheroid.

FIG. 10.

Mesh pattern and grid-point distribution of (a) whole computational domain and (b) near-wall region for the 6:1 inclined prolate spheroid.

Close modal

All three testing cases listed in Table II correspond to the relevant experimental parameter regimes and are implemented using the present CF-enhanced SST-γ and ANN-substituted SST- γ ANN models. Shown in Fig. 11 are contours of the skin friction coefficient obtained using current SST-γ and SST- γ ANN models at three Reynolds numbers of 3.01 × 10 6 , 4.48 × 10 6, and 8.52 × 10 6, respectively. For clarity, the surface is unfolded from the windward symmetry plane to leeward symmetry plane [i.e., from ϕ = 0 ° to ϕ = 180 ° in y axis of Fig. 11, which is illustrated in Fig. 10(b)]. As seen in these panels, the results from SST- γ ANN model almost collapse onto those from the baseline SST-γ model. Recalling the training set displayed in Table II, it is highly encouraging that a high generalizability capability is exhibited across the non-wing-like geometry and various flow circumstances, despite the limited training cases (not only the simplest configuration but also the onefold flow regime).

FIG. 11.

Contours of the skin friction coefficient predicted by the present SST-γ and SST- γ ANN models at different Reynolds numbers for the 6:1 prolate spheroid. The experimentally measured transition locations69 are also plotted in each subfigure for comparison. For clarity, the corresponding information is labeled in each panel.

FIG. 11.

Contours of the skin friction coefficient predicted by the present SST-γ and SST- γ ANN models at different Reynolds numbers for the 6:1 prolate spheroid. The experimentally measured transition locations69 are also plotted in each subfigure for comparison. For clarity, the corresponding information is labeled in each panel.

Close modal

To assess the precision of present SST-γ and SST- γ ANN models, contours of the skin friction from the SST-γ-NoCF model are plotted in Fig. 12 for supplementary comparison. The experimentally measured transition locations are also displayed in Figs. 11 and 12. Evidently, all three models exhibit similar behaviors due to the dominant TS waves in transition phenomena at the flow regime of R e = 3.01 × 10 6. With the increase in the Reynolds number, the crossflow instability gradually dominates the flow development. In the case of R e = 4.48 × 10 6, the transition onset locations provided by the SST-γ/SST- γ ANN models tend to move toward the middle part of prolate spheroid, along with the enhancement of the skin friction magnitude. For this flow regime, the crossflow-induced transition is accurately captured by the present SST-γ/SST- γ ANN models, in sharp contrast to the SST-γ-NoCF model. Moreover, the crossflow instability becomes stronger as the Reynolds number increases to 8.52 × 10 6, at which the difference between the SST-γ-NoCF and SST-γ/SST- γ ANN models becomes more noticeable. Even though some deviation exists between the computed (based on the SST-γ/SST- γ ANN models) and measured transition locations, it is indicated that the CF-enhanced models yield a significant improvement over the original SST-γ-NoCF model, which is considered to be a preliminary success of the former over the challenging non-wing-like geometry.

FIG. 12.

Contours of the skin friction coefficient predicted by the SST-γ-NoCF model at Reynolds numbers of (a) 3.01 × 10 6, (b) 4.48 × 10 6, and 8.52 × 10 6 for the 6:1 prolate spheroid. The experimentally measured transition locations69 are also plotted in each subfigure for comparison.

FIG. 12.

Contours of the skin friction coefficient predicted by the SST-γ-NoCF model at Reynolds numbers of (a) 3.01 × 10 6, (b) 4.48 × 10 6, and 8.52 × 10 6 for the 6:1 prolate spheroid. The experimentally measured transition locations69 are also plotted in each subfigure for comparison.

Close modal

The performances of new SST-γ/SST- γ ANN models are also tested and evaluated via the finite ONERA M6 swept wing, which is considered to be a benchmark model with greater sophistication and is experimentally investigated in the ONERA S2Ch low-speed tunnel.70 For brevity, readers are referred to the related articles18–20 for more detailed geometrical details. In the present paper, a computational mesh is generated with nearly 8 × 10 6 grid cells to avoid the effects of mesh resolution. The minimum grid distance to the solid wall is set to 1.5 × 10 6 root chord to ensure the requirement of y + < 1. The flow regimes under consideration are in accordance with the reported experiments70 as listed in Table II.

Figure 13 depicts the laminar (blue-colored) and turbulent (red-colored) regions of the M6 wing predicted using SST-γ/SST- γ ANN models by cutting the intermittency factor distributions with a threshold value. As one can see, results from the data-driven SST- γ ANN model are generally consistent with those from the baseline SST-γ model, but the high degree of consistency observed for the NLF(2)–0415 wing and prolate spheroid is apparently reduced in the case of M6 wing. However, such degradation is not subtly reflected in advance through the a priori analysis in Sec. IV A, which is considered to be reasonable due to the intrication of CFD solver and variation of geometries. To our knowledge, there exists no systematic a priori protocol to foresee the a posteriori predictive characteristics finely, and a preliminary attempt will be made to tackle such issue in Sec. IV E.

FIG. 13.

Laminar and turbulent (colored by blue and red, respectively) regions recognized by the intermittency factor in the present SST-γ and SST- γ ANN models for the M6 wing. For simplicity, the corresponding parameters and descriptions are labeled in each panel. The experimental data70 (black solid circles, see also Fig. 14) are plotted for comparison.

FIG. 13.

Laminar and turbulent (colored by blue and red, respectively) regions recognized by the intermittency factor in the present SST-γ and SST- γ ANN models for the M6 wing. For simplicity, the corresponding parameters and descriptions are labeled in each panel. The experimental data70 (black solid circles, see also Fig. 14) are plotted for comparison.

Close modal

For further comparison, the experimentally measured laminar and turbulent (colored by light and dark, respectively) regions of the M6 wing by means of naphthalene distribution70 are presented in Fig. 14. The relevant results obtained using the original SST-γ-NoCF model are displayed in Fig. 15. As can be seen, the SST-γ-NoCF model totally fails to capture the physical process across the whole parameter flow regimes, especially on the CF-dominated lower surface of the wing. By sharp contrast, the SST-γ/SST- γ ANN models properly capture the crossflow instability, thus pushing the transition point further upstream for all three angles of attack over the lower surface of M6 wing, and the results are in good accordance with the experimental pictures. Although some visible inconsistency is discovered between the SST-γ/SST- γ ANN models and experiment, it bears repeating that the overarching goal of this paper is not to capture all cases in perfect agreement with the experiments, but to establish the fundamental guidelines for generalizing ANN-fully substituted transition model beyond the capacity of original model, i.e., the crossflow effect.

FIG. 14.

Laminar (light) and turbulent (dark) regions on the lower and upper surfaces of the M6 wing measured in previous experiments.70 

FIG. 14.

Laminar (light) and turbulent (dark) regions on the lower and upper surfaces of the M6 wing measured in previous experiments.70 

Close modal
FIG. 15.

Laminar (blue) and turbulent (red) regions on the lower and upper surfaces of the M6 wing predicted by the original SST-γ-NoCF model at different angles of attack: (a) 0°, (b) 5°, and (c) 15°. The experimental data70 (black solid circles, see also Fig. 14) are plotted for comparison.

FIG. 15.

Laminar (blue) and turbulent (red) regions on the lower and upper surfaces of the M6 wing predicted by the original SST-γ-NoCF model at different angles of attack: (a) 0°, (b) 5°, and (c) 15°. The experimental data70 (black solid circles, see also Fig. 14) are plotted for comparison.

Close modal

Overall, when the flow configuration shifts to the finite ONERA M6 wing, the deviations between the ANN-substituted SST- γ ANN model and its benchmark SST-γ model are merely enlarged slightly in comparison with the NLF(2)–0415 wing and spheroid. Both of the CF mechanism involved models exhibit transparent enhancement over the original SST-γ-NoCF model, which probably marks the possibility for generalizing data-driven transition model developed only with canonical case to complex engineering practices, as one often expects with advanced traditional transition models.

To address the question raised in Sec. IV D, i.e., the underlying causation of the deteriorative performance experienced by the SST- γ ANN model in the case of finite ONERA M6 wing, we make use of the state-of-the-art visualization strategy for large, high-dimensional datasets, i.e., t-distributed stochastic neighbor embedding (t-SNE) technique.79 With the aid of t-SNE technique, a suite of high-dimensional data points can be compressed into a two- or three-dimensional space, during which the pairwise similarity of the high-dimensional dataset is retained in a low-dimensional mainfold. A greater extent of “closeness” between the two data points in high-dimensional space contributes to a closer distance mutually in low-dimensional space. Therefore, the t-SNE technique is implemented to quantitatively assess the similarity among three concerning configurations, in which the input feature vectors are identified as the high-dimensional datasets.80 As depicted in Fig. 16, the points of prolate spheroid drop fully within the bounds of NLF(2)–0415 wing, which is considered to be sufficient for generalization of ANN-based model from NLF(2)–0415 wing to prolate spheroid. On the contrary, a large amount of input vectors of M6 wing are not supported by their counterpart of NLF(2)–0415 wing [emphasized by the dashed ellipses in Fig. 16(b)], indicating the disability to describe the finite M6 wing only with the infinite NLF(2)–0415 wing datasets. Consequently, the declining consistency between SST-γ and SST- γ ANN model in M6 wing may be attributed to the inadequate flow information in the training datasets, and the similarity goes for prolate spheroid. In this regard, the t-SNE technique provides an effectively descriptive methodology for beforehand evaluation and breaks through the limitation of commonly used metrics (e.g., the R2 score and MSE loss) of regression problems to some extent.

FIG. 16.

Two-dimensional t-SNE visualization of input vectors of (a) NLF(2)–0415 wing vs prolate spheroid and (b) NLF(2)–0415 wing vs M6 wing. For brevity, three representative cases (i.e., cases N3, S2, and M1 in Table II) are selected. The two t-SNE coordinates are denoted by “t-SNE Dim1” and “t-SNE Dim2,” respectively.

FIG. 16.

Two-dimensional t-SNE visualization of input vectors of (a) NLF(2)–0415 wing vs prolate spheroid and (b) NLF(2)–0415 wing vs M6 wing. For brevity, three representative cases (i.e., cases N3, S2, and M1 in Table II) are selected. The two t-SNE coordinates are denoted by “t-SNE Dim1” and “t-SNE Dim2,” respectively.

Close modal

Despite the encouraging performance of SST- γ ANN in substituting for the baseline SST-γ model, there still exist a series of questions to be answered or further verified for deriving an engineering-practical model. Of the primary relevance is the efficiency of SST- γ ANN model. Owing to the alternative strategy and ingenious adoption of duty cycle, the SST- γ ANN model is bound to be more computationally efficient than the traditional PDE-constrained transition model. As shown in Fig. 17, SST- γ ANN model consumes approximately the same amount of CPU time as the SST model, while exhibits an identical capability to the SST-γ model for transition prediction, even with the frequently dominant crossflow instability-induced transition in typical aeronautical engineering cases. Furthermore, the convergence property of SST- γ ANN is also identified to be superior to its baseline model (not shown in this paper for brevity). In this regard, the ANN-alternative strategy may further accelerate the numerical simulation of routine engineering design process without loss of precision.

FIG. 17.

Averaged CPU time for each iteration step with different models in simulations of flow past the NLF(2)–0415 wing.

FIG. 17.

Averaged CPU time for each iteration step with different models in simulations of flow past the NLF(2)–0415 wing.

Close modal

As is well accepted, a good property of grid independence is considered to be one of the basic essentials for advanced turbulence models.35 In this sense, bringing grid-independence verification into the present SST- γ ANN model is the logical next step. Overall, two aspects need to be verified. One is the grid convergence property, i.e., performance of the present SST- γ ANN model on coarser/finer grids compared with that on the training grid. The other is influence of the near-wall grid resolution. Against this backdrop, two corresponding validations are implemented based on case N6 of NLF(2)–0415 without loss of generality. The descriptions about the grids can be found in Tables III and IV.

TABLE III.

Cell distributions for a grid convergence study on the configuration of NLF(2)–0415 wing. The first off-wall grid height is set to 2 × 10 6 chord length for all four grids. Grid3 is taken as the training and testing grid in Sec. IV.

Grid Pressure side Suction side Wake region Wall-normal Spanwise Total
Grid1  120  400  41  81  601 × 81 × 4 
Grid2  160  600  61  121  881 × 121 × 4 
Grid3  200  800  101  181  1201 × 181 × 4 
Grid4  400  1600  201  361  2401 × 361 × 4 
Grid Pressure side Suction side Wake region Wall-normal Spanwise Total
Grid1  120  400  41  81  601 × 81 × 4 
Grid2  160  600  61  121  881 × 121 × 4 
Grid3  200  800  101  181  1201 × 181 × 4 
Grid4  400  1600  201  361  2401 × 361 × 4 
TABLE IV.

First off-wall grid height and the corresponding y+ for studying the influence of near-wall grid resolution on the configuration of NLF(2)–0415 wing. Here, C denotes the chord length. The cell distributions are the same as Grid3 in Table III for all six grids.

Grid First off-wall grid height y+ Grid First off-wall grid height y+
Grid5  2 × 10 6 C  0.6  Grid8  8 × 10 6 C  2.4 
Grid6  3 × 10 6 C  0.9  Grid9  16 × 10 6 C  4.8 
Grid7  4 × 10 6 C  1.2  Grid10  24 × 10 6 C  7.2 
Grid First off-wall grid height y+ Grid First off-wall grid height y+
Grid5  2 × 10 6 C  0.6  Grid8  8 × 10 6 C  2.4 
Grid6  3 × 10 6 C  0.9  Grid9  16 × 10 6 C  4.8 
Grid7  4 × 10 6 C  1.2  Grid10  24 × 10 6 C  7.2 

Depicted in Fig. 18 are skin friction distributions obtained using the SST-γ/SST- γ ANN models on different grids. An excellent grid convergence character is achieved by the present ANN-alternative SST-γ model, which accurately mimics the transitional flow all the way down to Grid2. By contrast, a convergent solution of skin friction is not obtained until Grid3 for traditional SST-γ model. In fact, as mentioned in related references,35 the outperformance over SST-γ model boils down to the circumvention of solving transition model equation in ANN-alternative SST-γ model, which observably reduces the discretization error of governing equations.

FIG. 18.

Distributions of the skin friction coefficient for case N6 of NLF(2)–0415 wing (on upper surface at the midspan only) on different grids obtained by using (a) SST-γ model and (b) SST- γ ANN model.

FIG. 18.

Distributions of the skin friction coefficient for case N6 of NLF(2)–0415 wing (on upper surface at the midspan only) on different grids obtained by using (a) SST-γ model and (b) SST- γ ANN model.

Close modal

With regard to the second validation for influence of the near-wall grid resolution, it is observed that no apparent disparity exists between the SST-γ and SST- γ ANN models (see Fig. 19). This indicates that a reasonable variation tendency of the skin friction coefficient is achieved by the ANN-substituted transition–turbulence model with the increase in near-wall grid resolution, in conformity with its traditional counterpart. Consequently, although the ANN transition model is established on a single set of grid, a remarkable generalizability over grid size and near-wall grid resolution is achieved even with relaxed restriction in grid size, which corresponds to a low grid dependence as one often expects with modern transition models.

FIG. 19.

Distributions of the skin friction coefficient for case N6 of NLF(2)–0415 wing (on upper surface at the midspan only) with different near-wall grid resolutions obtained by using (a) SST-γ model and (b) SST- γ ANN model.

FIG. 19.

Distributions of the skin friction coefficient for case N6 of NLF(2)–0415 wing (on upper surface at the midspan only) with different near-wall grid resolutions obtained by using (a) SST-γ model and (b) SST- γ ANN model.

Close modal

Among the main requirements for a fully CFD-compatible transition model proposed by Langtry and Menter,8 there exists a residual issue deserving further investigation, i.e., whether or not the transition model affects the underlying turbulence model in fully turbulent regimes. The case of ONERA M6 wing with AoA = 3.06 °, Ma = 0.8395, and R e = 11.72 × 10 6, which is not characterized by transition phenomenon, is selected to verify the corresponding property of present SST- γ ANN model. As can be seen in Fig. 20, shock waves are identified at different streamwise and spanwise locations. In this flow regime, the SST- γ ANN and SST-γ models are identical to each other, and almost retrieve the full-turbulence SST model, which demonstrates the same feasibility and reliability of SST- γ ANN model as its baseline SST-γ model even in non-typical transitional flow. Meanwhile, it should be pointed out that even though the parameter regime of this testing case is farther beyond our training set (see Table II for comparison), SST- γ ANN model also exhibits identical predictive nature to the benchmark SST-γ model, validating the satisfactory extrapolation capability as well as the great promise of engineering-reliable ANN-alternative transition model.

FIG. 20.

Distributions of mean pressure coefficient on the surface of M6 wing at (a) 20% span, (b) 65% span, and (c) 90% span: comparison among different models and experiments.81 

FIG. 20.

Distributions of mean pressure coefficient on the surface of M6 wing at (a) 20% span, (b) 65% span, and (c) 90% span: comparison among different models and experiments.81 

Close modal

So far, an accurate, efficient, robust, and well-generalized data-driven transition model has been established, which is oriented toward the industrial practice. Although the present model might be eclipsed in other specific transitional effects such as strong compressibility14 and surface roughness,49,50 the philosophy and formalism employed in this paper appear to be of a general nature. To further verify the transferability of current framework across these circumstances and underlying rationale behind the preliminary success of present model, two additional models (namely, SST- γ ANN-NoCF and SST- γ ANN-NoCF-Inputs) are constructed elaborately to investigate the necessities of enhancing baseline SST-γ model with CF physics and appending CF-correlated input features, respectively. The training sets for these two models are the same as listed in Table II. Without loss of generality, cases N1 and N6 of the NLF(2)–0415 wing are selected as the testing cases. The corresponding details are summarized in Table V.

TABLE V.

Essential details about the two additional data-driven models. The training cases remain the same as Table II. Cases N1 and N6 of the NLF(2)–0415 wing are selected as the testing cases.

Model Training data derivation Input features
SST- γ ANN-NoCF  SST-γ-NoCF  The same as Table I
SST- γ ANN-NoCF-Inputs  SST-γ  Remove the CF-correlated inputs in Table I.a 
Model Training data derivation Input features
SST- γ ANN-NoCF  SST-γ-NoCF  The same as Table I
SST- γ ANN-NoCF-Inputs  SST-γ  Remove the CF-correlated inputs in Table I.a 
a

Replace the input F onset , 3 D [see Eq. (15)] with F onset [see Eq. (4)] and remove the input p9 (i.e., helicity Reynolds number).

We show in Fig. 21 distributions of skin friction coefficient obtained using the SST- γ ANN-NoCF and its benchmark SST-γ-NoCF models, which coincide with each other and both fail to capture the crossflow transition phenomenon, even though the CF-correlated inputs (i.e., p8 and p9 in Table I) are retained as the input features in SST- γ ANN-NoCF model. This unsurprisingly demonstrates the strong dependency of the ANN-substituted model on the baseline model. By contrast, the model trained with the data from CF-enhanced SST-γ model is able to capture the crossflow effect with ease. As for the second issue, it is clearly seen from Fig. 22 that SST- γ ANN model is basically neck and neck with the one without CF-correlated input features, especially for case N6 in Fig. 22(b). However, this ANN model trained without CF-correlated input features cannot inherit the accuracy of SST- γ ANN model for more complicated geometries such as prolate spheroid (see Fig. 23), which reflects the fundamental role of primary input features as well as the critical importance of CF-correlated inputs.

FIG. 21.

Distributions of skin friction coefficient calculated using SST-γ-NoCF/SST- γ ANN-NoCF models for cases (a) N1 and (b) N6 of NLF(2)–0415 wing (on the upper surface at midspan only). The results from SST-γ/SST- γ ANN models in Sec. IV B are also depicted for comparison.

FIG. 21.

Distributions of skin friction coefficient calculated using SST-γ-NoCF/SST- γ ANN-NoCF models for cases (a) N1 and (b) N6 of NLF(2)–0415 wing (on the upper surface at midspan only). The results from SST-γ/SST- γ ANN models in Sec. IV B are also depicted for comparison.

Close modal
FIG. 22.

Distributions of skin friction coefficient calculated using SST- γ ANN/SST- γ ANN-NoCF-Inputs models as well as the baseline SST-γ model for cases (a) N1 and (b) N6 of NLF(2)–0415 wing (on the upper surface at midspan only).

FIG. 22.

Distributions of skin friction coefficient calculated using SST- γ ANN/SST- γ ANN-NoCF-Inputs models as well as the baseline SST-γ model for cases (a) N1 and (b) N6 of NLF(2)–0415 wing (on the upper surface at midspan only).

Close modal
FIG. 23.

Contours of skin friction coefficient obtained using (a) SST-γ, (b) SST- γ ANN, and (c) SST- γ ANN-NoCF-Inputs models for 6:1 prolate spheroid. The experimentally measured transition locations (black solid circles)69 are plotted in each subfigure for comparison.

FIG. 23.

Contours of skin friction coefficient obtained using (a) SST-γ, (b) SST- γ ANN, and (c) SST- γ ANN-NoCF-Inputs models for 6:1 prolate spheroid. The experimentally measured transition locations (black solid circles)69 are plotted in each subfigure for comparison.

Close modal

In light of these investigations, some fundamental guidelines can be mined for ANN-alternative transition model development. When we have a pressing need to accelerate the process of engineering-practical transition-sensitized simulations and have already derived a corrected/enhanced transition model with approving accuracy, the data-driven transition model proposed in this paper can be an attractive alternative. Owing to the physical consideration for intermittency factor, most of the input features are universal (such as f1p7 and F onset in Table I) and even supposed to be complete in relatively canonical flow regimes. With supplementary attention on specific inputs (such as the CF-correlated inputs p8 and p9 in this paper), the ANN-alternative transition model is deemed to generalize over a wide variety of cases. Meanwhile, analysis of first investigation indicates that the influence of specific inputs will be hidden from the irrelevant training data. In other words, for instance, there is no need to remove CF-correlated inputs even though attention is devoted to the distributed surface roughness transition flows, as long as the training data are surface roughness-related. This property exhibits a less intrusiveness over flows with different transition effects, more easily amenable to generalizability than their traditional counterpart. In addition, if the accuracy of baseline model still remains dissatisfactory even with significant correction or improvement effort, a modeling approach based on Hi-Fi data82 or a unified method of data assimilation and machine learning43 can be adopted to break through the precision restriction of baseline model, in which the machine learning process can be resorted by the present framework.

In this paper, a universal framework is established, which goes through the whole “enhancement-training-testing-verification” lifestyle for ANN-alternative transition modeling. Under this scenario, a CF-enhanced ANN transition model toward practical aeronautical applications is developed together with necessary validation and evaluation of its accuracy, efficiency, robustness, generalizability, and ease-of-use.

During the enhancement phase of lifestyle, significant effort is taken to improve the benchmark SST-γ model with crossflow instability mechanism, which is presumed to be the groundwork for remaining phases of the present framework. In the training phase, a suite of input features are raised to characterize their potential physical relevances to the target intermittency factor. In fact, subsequent verifications manifest that such inputs construction provides a guarantee for less intrusiveness to CFD solver and better robustness of ANN transition model over intrinsically different transitional flows. Considering the engineering practice, the ANN model is established only with data of canonical flows, while is tested across various complicated flow cases, which poses a great challenge to the new model's generalizability.

In the testing phase, the outcomes from CF-enhanced SST-γ/SST- γ ANN models are systematically evaluated against available experiments and original SST-γ-NoCF model among all three geometries with a wide range of flow regimes. On the whole, the results from SST- γ ANN model closely coincide with its benchmark SST-γ model, yielding a significant improvement over the SST-γ-NoCF model and exhibiting relatively consistent predictive nature with the real physical world. In view of different degrees of consistency between SST-γ and SST- γ ANN model across three geometry cases, two strategies from the a priori perspective are demonstrated to be effective for beforehand evaluation and further promotion. To investigate the industrial practicality of the present ANN-alternative transition model, several targeted study cases are inspected in the last phase of verification. As one often expects with engineering-usable data-driven model, the present SST- γ ANN model is also demonstrated to be highly robust and generalized in numerical grid resolution and non-typical transitional flow cases, with relaxed restriction of grid size and a more efficient calculation speed in comparison with its traditional counterpart. These properties are supposed to exert a transformative impact on the industrialization of ANN-alternative transition model.

In summary, this paper marks a significant step toward the practical use of data-driven transition model in the engineering environment in the near future, and outlines a universal framework for intrinsically diverse transitional flows with user-friendliness. Of course, this research work also bring to light many questions in need of further investigation. First of all, the present CF-enhanced/ANN-alternative transition models do not take into consideration the effect of surface roughness, which frequently excites stationary crossflow vortices to dominate the crossflow transition under low freestream turbulence condition. However, the quantitative influence of surface roughness can be readily involved in the modeling process of ANN-enhanced model by adding roughness-characterized input features. Second, although the a priori outcomes (both the R2 value and t-SNE methodology) may provide a preliminary foresight over the a posteriori calculation in this paper, more collaborative works are needed to bring the a priori analysis to a maturity level, including a more systematic criterion. Third, considering the apparent deviation over several testing cases, there seems to be a room for the optimization of network hyperparameters. Finally, certain weakness of baseline model is identified in some configurations (such as the ONERA M6 wing). In this regard, migrating the framework of current model to other reliable transition model, e.g., the two equation γ- R e ̃ θ t model, or supplementing the data assimilation approach might be generally adoptable strategy.

Numerical simulations were carried out on the Polaris computing platform of Peking University in Beijing. We thank Hanyu Zhou and Yaomin Zhao for many fruitful discussions. The authors acknowledge the financial supports provided by the National Natural Science Foundation of China (Grants Nos. 92152202 and 11988102). This work was also supported by the National Key Project (Grant No. GJXM92579).

The authors have no conflicts to disclose.

Lei Wu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing—original draft (equal). Bing Cui: Formal analysis (equal); Investigation (equal); Validation (equal). Rui Wang: Data curation (equal); Software (equal); Visualization (equal). Zuoli Xiao: Formal analysis (equal); Resources (equal); Supervision (equal); Writing—review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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