Turbulent Flow Model Research Articles

NN-augmented k-ω shear stress transport turbulence model for high-speed flows with shock-wave/boundary layer interaction

Menter k-ω shear stress transport (SST) turbulence model demonstrates excellent performance for incompressible, subsonic and transonic flows with mild separation but shows overprediction of the separation bubble of supersonic shock-wave/boundary layer interaction (SWBLI). Some efforts focus on the effect of the structure parameter in stress limiter in an ad-hoc way. Few studies attempt to construct the relation between the structure parameter and flow field variables. The motivation of this work is to construct such a relation to augment the prediction performance of the SST model by introducing a correction factor. Machine learning methods are used since the physical mechanism of SWBLI is complex and unclear. The simulation results show that the constructed relation enhances the structure parameter near the shock wave in the boundary layer when applied to the SST model. Compared with direct numerical simulation and experimental data, the pressure and skin friction coefficients along the wall and the velocity field are significantly improved. In addition, the introduced correction factor can automatically degrade for the subsonic benchmark case of NACA4412 airfoil and maintain the prediction accuracy of the original SST model, but delay the shock location of the transonic case.

Enhancing the performance of vertical axis hydrokinetic Savonius turbines using a novel cambered hydrofoil profile for rotor blades

A novel rotor blade configuration using a cambered hydrofoil profile is developed to enhance the output power. Several design variables are considered including camber angle, overlap ratio, and camber position, in addition to integrating endplates. To assess the performance parameters at different design conditions, a comprehensive three-dimensional unsteady incompressible turbulent flow model based on Reynolds-Averaged Navier-Stokes equations combined with k-ω model is developed. The predicted numerical results is validated using the available numerical and experimental data. The findings indicate that using Savonius rotor with hydrofoil profile results in a substantial increase in the maximum output power for all investigated camber angles compared to the conventional semi-circular rotor. At a flow velocity of 0.4 m/s and an overlap ratio of 0.15, the Savonius rotor with hydrofoil profile attains a maximum power coefficient of 0.21 at a camber angle of 105°. By increasing the camber angle to 140°, the maximum power coefficient reaches 0.26 whereas it is 0.13 for the conventional design. The camber position at 50 % of the chord length achieves the highest power coefficient for all studied cases. In addition, adding the endplates significantly enhances the power coefficient. The current findings provide a promising approach in developing Savonius rotor blades configuration by utilizing hydrofoil profile.

Mean Reattachment Length of Roof Separation Bubbles Using Proper Orthogonal Decomposition

Investigating flow separation regions on the surfaces of three-dimensional bluff bodies in turbulent flows is important because these regions can induce significant aerodynamic loads. Separation bubbles can generate extreme pressures, making the roof components of low-rise buildings vulnerable. In this study, proper orthogonal decomposition (POD) was applied to wind-induced roof pressures to elucidate the physical significance of the dominant modes. Based on the interpretation of the first mode from the POD, the mean reattachment length of the roof separation bubbles on a low-rise building model in turbulent flow was determined. The mean reattachment length derived from the POD was then compared with the length obtained from an aerodynamic database. For the centerline of the roof, the mean reattachment length based on the POD aligned well with that from the aerodynamic database, showing a difference of less than 5%. This study highlights the efficacy of POD as a powerful tool for estimating the reattachment length of separation bubbles on bluff bodies.

A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.

A priori tests of turbulence models for compressible flows

PurposeThe purpose of the paper is to analyse the performances of closures and compressibility corrections classically used in turbulence models when applied to highly-compressible turbulent boundary layers (TBLs) over flat plates.Design/methodology/approachA direct numerical simulation (DNS) database of TBLs, covering a wide range of thermodynamic conditions, is presented and exploited to perform a priori analyses of classical and recent closures for turbulent models. The results are systematically compared to the “exact” terms computed from DNS.FindingsThe few compressibility corrections available in the literature are not found to capture DNS data much better than the uncorrected original models, especially at the highest Mach numbers. Turbulent mass and heat fluxes are shown not to follow the classical gradient diffusion model, which was shown instead to provide acceptable results for modelling the vibrational turbulent heat flux.Originality/valueThe main originality of the present paper resides in the DNS database on which the a priori tests are conducted. The database contains some high-enthalpy simulations at large Mach numbers, allowing to test the performances of the turbulence models in the presence of both chemical dissociation and vibrational relaxation processes.

Large eddy simulation-bivariate sectional model for non-spherical TiO2 nanoparticle synthesis in flame

Numerical simulation of nanoparticle synthesis in flame necessitates a high level of fidelity for turbulent flame, as well as delivers more detail insight into the evolution of structure and size distribution of particles undergoing various dynamic events. In the present work, a new simulation approach coupling large-eddy simulation (LES) and bivariate sectional model is proposed to explore the synthesis of nanoparticles in turbulent flame. More specifically, the nonlinear LES-partially stirred reactor model (NLES-PaSR) is utilized to simulate the turbulent flame, wherein the gradient-type structural subgrid-scale (SGS) model for turbulent flow and partially stirred reactor (PaSR) model for turbulent combustion are considered. In addition, the volume-based bivariate sectional model is thoroughly coupled with the LES model for the first time to describe the spatiotemporally resolved formation and growth of nanoparticles using particle number concentration and surface area concentration as two sets of variables. Compared directly with the LES-monodisperse (LES-Mono) and LES-univariate sectional (LES-UniSe) models, the LES-bivariate sectional (LES-BiSe) model has significant advantages in its ability to analyze and predict the size, morphology, as well as polydispersed particle size distribution (PSD) evolution of nanoparticles at a moderate computational cost simultaneously. In the LES framework, the proposed model successfully simulates the spatiotemporally resolved formation and growth of non-spherical TiO2 nanoparticle in turbulent diffusion burner. The predicted results derived from the LES-BiSe model are in good agreement with the experimental measurements. Also, the aggregate and primary particle size distribution shows highly temporal fluctuation, which mainly results from the continuous variations of particle path induced by the unsteady turbulent combustion. Furthermore, the present work demonstrates the profound dependence of particle morphological on both spatiotemporal evolution and their intrinsic size distribution. In particular, the large aggregates in the higher sections are commonly found as irregular structure with fractal-like dimensions.

Generalization of the Penalized Wall Function Method for Modeling of Turbulent Flows with Adverse Pressure Gradient

Generalization of the Penalized Wall Function Method for Modeling of Turbulent Flows with Adverse Pressure Gradient

The Dimple Effect on Flettner Rotors for the Wind Power Drive System of a Sailing Merchant Ship

Wind-powered ship propulsion is investigated in relation to reducing ship exhaust emissions, such as the application of Flettner rotor systems. How to increase rotor thrust's contribution to ship speed is an intriguing research question. This paper investigates the influence of dimples on the surface of a Flettner rotor on its ability to generate thrust for a ship. The CFD code, which includes the Reynolds average Navier-Stokes equation and the k-ω model for turbulent flow, is utilized to analyse three rotor designs: one with a smooth surface and two with dimple surfaces of varying diameters. Lift and drag are computed by taking into account varying spin ratios and the incoming wind angle. The CFD results are validated by comparing them to experimental data gathered in a wind tunnel and determining that their error is less than 5%. The addition of dimpled surfaces to the rotor increases lift by 69.6% and decreases drag by 14.8% due to the smooth surface. For the case study, the two rotors are mounted on the sailing ship 900 DWT and can generate the most thrust to give the ship a speed of 8.5 knots with a spin ratio of 4 and apparent wind angles of 90o or 270o relative to its speed.

Numerical Simulation of Thermal Flow Model Under Fluid Mechanics

The internal flow field of vertical water vapor waste heat boiler was studied by numerical simulation. In this project, the RNGk-ε turbulent flow model and Lagrange stochastic trajectory model is used to numerically simulate the gas-solid flow in a direct pressure water vapor waste heat boiler. The influence of the inlet shape, flow rate and outlet clearance of cooling water on the flow and heat transfer performance of cooling water in the cooler was analysed. It is found that the use of three inlet design can improve the water side cooling efficiency, and make the flow rate and temperature distribution in the cooling chamber more uniform. The optimal cooling water volume is 200 tons/hour in the current 2-stage cooling inlet pipe type.

On the application of principal component transport for compression ignition of lean fuel/air mixtures under engine relevant conditions

Principal component transport-based data-driven reduced-order models (PC-transport ROM) are being increasingly adopted as a combustion model of turbulent reactive flows to mitigate the computational cost associated with incorporating detailed chemical kinetics. Previous studies were mainly limited to replicating relatively-simple chemistry in canonical configurations. The objective of the present study, therefore, is to further explore the accuracy of PC-transport ROM on more complex combustion phenomenon where, for example, large hydrocarbon fuel chemistry spanning a broad range of thermochemical space governs sequential multi-stage compression ignition processes. The cumulative error of PC-transport for this problem, and for others that depend upon sequential highly nonlinear physics, has to be minimal as the combustion phasing and heat release rate in internal combustion engines depends upon accurate predictions of minor ignition species whose concentrations start from ashes and grow orders of magnitude over the course of low- and high-temperature autoignition. Specifically, the PC-transport ROM is applied to predict the compression ignition characteristics of lean n-heptane/air and primary reference fuel (PRF)/air mixtures in a two-dimensional (2-D) constant volume computational domain initialized with a two-dimensional isotropic turbulence spectrum and temperature inhomogeneities. PCA is used to define the low-dimensional manifold that represents the original thermochemical state vector, and artificial neural network (ANN) models are adopted to tabulate chemical kinetics, transport, and thermodynamic properties. A series of 2-D pseudo-turbulent simulations are performed at engine pressures by varying the initial mean and r.m.s. of temperature, turbulence intensity, and the composition of fuel/air mixture. The results show that the PC-transport ROM accurately reproduces the instantaneous and statistical ignition characteristics of the fuel/air mixture, aided by pre-processing techniques including species subsetting, data clustering, and data transformation. It is found that PCs are not properly scaled with a power transformer if reactants are included in the species subset, which leads to a decrease in the accuracy of the PC-transport ROM. A separation of the reactants from the species subset ensures that the temporal evolution of the PCs starts from zero and spans orders of magnitude with time, and as such, this approach is found to effectively redistribute both PCs and their source terms with a power transformer. The computational speed-up factor of the PC-transport ROM ranges between 5.1 and 15.0 for the cases with n-heptane/air mixture and PRF/air mixture, respectively. Moreover, a potential further speed-up is anticipated through a combination of reduction in grid resolution requirements and in the stiffness of the chemical system. As an example, many of the pre-processing methods for inhomogeneous compression ignition may also apply to other complex intermittent combustion phenomena.Novelty and significance statement• The PCA-based reduced-order model (PC-transport ROM) has been applied to the multi-stage compression ignition of large hydrocarbon fuels under HCCI-relevant conditions. The present work presents a systematic procedure to accurately capture the two-stage ignition behavior of lean n-heptane/air or PRF50/air mixture.• The present work demonstrates an advantage of the PC-transport ROM in terms of computational speed-up. The computational speed-up factor for the ROM is up to 15, and moreover, a potential additional speed-up is anticipated through the reduction in the spatial and temporal resolution required.• A series of 2-D PC-transport ROMs are conducted to demonstrate the robustness of the ROM. A limitation of the ROM against different operating conditions is also discussed.

Towards optimal [formula omitted]-variational autoencoders combined with transformers for reduced-order modelling of turbulent flows

Variational autoencoders (VAEs) have shown promising potential as artificial neural networks (NN) for developing reduced-order models (ROMs) in the context of turbulent flows. In this study, we propose a method that combines β-VAEs for modal decomposition and transformer neural networks for temporal-dynamics prediction in the latent space to develop ROMs. We apply our method to an existing database of a turbulent flow around a wall-mounted square cylinder obtained by direct numerical simulation (DNS). A parametric study is performed to investigate the effects of the hyperparameters of the proposed β-VAEs and determine the optimal values. For the first time, we incorporate the consideration of the complexity of architecture into our studies, providing new insights into hyperparameter selection for β-VAEs, which remains a challenging problem for optimising model performance. Results regarding the influence of the different hyperparameters and guidelines to design these architectures are reported. Our optimal model achieves a reconstruction accuracy of 97.18% of the entire dataset using only ten modes. Subsequently, we employ the transformer models to identify latent-space temporal dynamics learned by the optimal β-VAE model and build ROMs to predict instantaneous fields. The resulting model achieves promising accuracy in temporal-dynamics predictions and yields energy reconstruction levels of 96.5% and 83% for a field 25 and 50 steps into the future, respectively, showcasing the potential of the transformer in predicting the temporal dynamics. Overall, the proposed method has potential applications in advanced flow control and fundamental studies of complex turbulent flows.

An inverse optimization of turbulent flow and heat transfer for a cooling passage with hierarchically arranged ribs in turbine blades

Owing to the limited cold-air amount and pressure in supply systems, high-efficient heat transfer with low-level friction loss is highly desired for cooling units of a turbine blade. To exploit the potential improvement of hierarchically arranged ribs in cooling passages proposed previously, multi-parameter optimizations for rib arrangements are implemented by integrating the simplified conjugate-gradient algorithm with the turbulent flow and heat transfer model. Rib heights as design variables are optimized with various performance indices as objective functions at a fixed Re. The optimizations confirm that using the wall temperature difference and Nu as the objective function, respectively, a limited heat transfer improvement is achieved with a greatly increased friction loss. Taking the overall performance factor as the objective function, different optimal designs at different constraint conditions possess hierarchical characteristics. A significant friction loss reduction of 52.1%, 54.7%, and 54.8%, is achieved with a moderate heat transfer loss of 10.9%, 7.0%, and 2.3%. Despite different thermal and friction performances, their overall performances are consistent with a remarkable increase of 13.9%, 21.2%, and 27.3%. Finally, the optimization strategy coupling the multi-parameter optimization and hierarchical scheme is confirmed as effective for enhancing the thermohydraulic performance of convective heat transfer systems with perturbation elements.

Invariance embedded physics-infused deep neural network-based sub-grid scale models for turbulent flows

In this paper, we present two novel physics-infused neural network (NN) architectures that satisfy various invariance conditions for constructing efficient and robust Deep Learning (DL)-based sub-grid scale (SGS) turbulence models for use in Large Eddy Simulation (LES) procedures widely used in various fluid engineering applications. The first architecture, called tensor basis neural networks (TBNN), recently proposed in the context of Reynolds-averaged Navier–Stokes (RANS) modeling, and introduced herein for SGS modeling of wall-bounded turbulence, embeds the analytical expansion of the SGS turbulence stresses into integrity basis tensors composed of the symmetric and anti-symmetric parts of the resolved velocity gradient tensor, thus incorporating the Galilean, rotational and reflectional invariances. In our second approach, a relatively simple yet powerful architecture, called Galilean Invariance embedded Neural Network (GINN) incorporates the Galilean invariance, and takes in as inputs the independent components of the integrity basis tensors in addition to the invariant inputs in a single input layer. Explicit filtering of data from direct simulations of the canonical channel flow at two friction Reynolds numbers Reτ≈395 and 590 provided accurate data for training and testing these models. Both sets of models are used to predict the SGS stresses for feature datasets generated with different filter widths, and at different Reynolds numbers. The GINN model yields less prediction error on test datasets (mean squared error ∼0.4) compared to the TBNN model (mean squared error ∼0.5). Upon comparison, it is revealed that GINN has better feature extraction capacity owing to its ability to establish relations between and extract information from cross-components of the integrity basis tensors and the SGS stresses. Based on their predictive performance, both models proposed herein have shown great promise for use in a posteriori actual LESs. The present work illustrates the significance of physics infusion as well as invariance embedding into NN architecture for constructing efficient and robust DL-based SGS turbulent models in achieving superior predictive efficacy.

Successful reduction of indoor radon activity concentration via cross-ventilation: experimental data and CFD simulations

ABSTRACT Advanced computational fluid dynamics (CFD) simulations are essential for predicting airflow in ventilated spaces and assessing indoor air quality. In this study, a focus was set on techniques for the reduction of indoor radon-222 activity concentration [Rn], and it is demonstrated how true-to-scale 3D CFD models can predict the evolution of complex ventilation experiments. A series of ventilation experiments in an unoccupied flat on the ground floor of a residential block in Bad Schlema (Saxony, Germany) were performed. Specifically, the ‘Cross-ventilation 100 %’ experiment resulted in room-specific [Rn] reductions from ∼3000 to ∼300 Bq m−3. We quantitatively interpreted the results of the ventilation experiment using a CFD model with a k–ϵ turbulent stationary flow model characterised by the used decentralised ventilation system. The model was coupled with a transient transport model simulating indoor [Rn]. In a first approach, the model overestimated the decrease in the starting of the experiment and the steady state. Adjusting the model parameters inflowing radon and inlet velocity the model results are in a good agreement with the experimental values. In conclusion, this paper demonstrates the potential of CFD modelling as a suitable tool in evaluating and optimising ventilation systems for an effective reduction of elevated [Rn].

Numerical and experimental study of inverse natural convection-conduction heat transfer in an inclined rectangular cavity

In this work, the inverse three-dimensional (3D) computational fluid dynamics (CFD) method combined with overdetermined experimental data and least squares method is applied to predict the unknown heat transfer rates and the suitable turbulent flow model in an inclined rectangular cavity. The conjugate heat transfer of two 3D natural convection systems at different temperatures separated by a cavity is solved by coupling. The effect of inclination angle θ on the appropriate flow model is investigated. The RMSE obtained by the appropriate flow model is smaller than that obtained by other flow models. The obtained Nusselt number on the hot wall also needs to be closer to the proposed correlation. An interesting finding is that the zero-equation model and RNG k-ε model are respectively more suitable for problems with 0≤θ ≤ 30° and 60°≤θ ≤ 90°, respectively. The number of natural circulation decreases with increasing θ. The obtained estimates of Nuh agree well with the proposed correlation and existing experimental results. A review of the literature revealed that these findings have not yet appeared in the open literature.

Dynamics of a data-driven low-dimensional model of turbulent minimal Couette flow

Because the Navier–Stokes equations are dissipative, the long-time dynamics of a flow in state space are expected to collapse onto a manifold whose dimension may be much lower than the dimension required for a resolved simulation. On this manifold, the state of the system can be exactly described in a coordinate system parameterising the manifold. Describing the system in this low-dimensional coordinate system allows for much faster simulations and analysis. We show, for turbulent Couette flow, that this description of the dynamics is possible using a data-driven manifold dynamics modelling method. This approach consists of an autoencoder to find a low-dimensional manifold coordinate system and a set of ordinary differential equations defined by a neural network. Specifically, we apply this method to minimal flow unit turbulent plane Couette flow at $Re=400$ , where a fully resolved solutions requires ${O}(10^5)$ degrees of freedom. Using only data from this simulation we build models with fewer than $20$ degrees of freedom that quantitatively capture key characteristics of the flow, including the streak breakdown and regeneration cycle. At short times, the models track the true trajectory for multiple Lyapunov times and, at long times, the models capture the Reynolds stress and the energy balance. For comparison, we show that the models outperform POD-Galerkin models with $\sim$ 2000 degrees of freedom. Finally, we compute unstable periodic orbits from the models. Many of these closely resemble previously computed orbits for the full system; in addition, we find nine orbits that correspond to previously unknown solutions in the full system.

A Diffusion-Inertia Model for the simulation of particulate pollutants dynamics inside a car cabin

Simulations of particle pollutants dynamics inside a car cabin were performed using the Diffusion-Inertia Model (DIM) coupled to the RANS (Reynolds-averaged Navier–Stokes equations) k−ω SST turbulence model for single-phase flow. The DIM model has been implemented in a commercial CFD solver and then validated on some reference flow configurations. The validation included particle dispersion in ventilated chambers of one and two compartments, particle deposition in circular 90°bends, and particle-laden jet flow. The DIM model has shown interesting capabilities in taking into account various transport mechanisms such as the transport by the mean flow, gravitational settling, centrifugal effect due to flow streamline curvatures, molecular and turbulent diffusion, and also turbophoresis transport mechanism due to turbulent energy gradients. Moreover, the application of the DIM model to the simulation of solid particles, characterizing particulate pollutants, has revealed characteristic dynamics that is mainly determined by the airflow topology, particularly for ultrafine and fine particles (0.1, 1 and 2.5μm). Inertial effects were found to become significant only for coarse particles (5 and 10μm), as the impact of gravitational and centrifugal effects gradually becomes more noticeable. Furthermore, the risk of exposure to high levels of particulate pollutants has been clearly established when considering the presence of occupants inside the vehicle cabin. These concentration levels are linked in particular to the complex interaction between occupants’ heads and the internal flow, as well as to the tendency of particles to accumulate in the upper part of the cabin and to the rapid homogenization observed throughout the whole interior volume.

Automated tuning for the parameters of linear solvers

Robust iterative methods for solving large sparse systems of linear algebraic equations often suffer from the problem of optimizing the corresponding tuning parameters. To improve the performance of the problem of interest, specific parameter tuning is required, which in practice can be a time-consuming and tedious task. This paper proposes an optimization algorithm for tuning the numerical method parameters. The algorithm combines the evolution strategy with the pre-trained neural network used to filter the individuals when constructing the new generation. The proposed coupling of two optimization approaches allows to integrate the adaptivity properties of the evolution strategy with a priori knowledge realized by the neural network. The use of the neural network as a preliminary filter allows for significant weakening of the prediction accuracy requirements and reusing the pre-trained network with a wide range of linear systems.The detailed algorithm efficiency evaluation is performed for a set of model linear systems, including the ones from the SuiteSparse Matrix Collection and the systems from the turbulent flow simulations. The obtained results show that the pre-trained neural network can be effectively reused to optimize parameters for various linear systems, and a significant speedup in the calculations can be achieved at the cost of about 100 trial solves. The hybrid evolution strategy decreases the calculation time by more than 6 times for the black box matrices from the SuiteSparse Matrix Collection and by a factor of 1.4–2 for the sequence of linear systems when modeling turbulent flows. This results in a speedup of up to 1.8 times for the turbulent flow simulations performed in the paper.

A SIMULATION OF EVAPORATION DISSOLUTION PROCESS FOR A LYOCELL SOLUTION IN A VERTICAL WIPED FILM EVAPORATOR (1: SIMULATION OF FLOW PROCESS FOR SOLUTION)

Vertical wiped film evaporators (VWFE) have found wide application in chemical, food and especially, fiber industries. In this paper, a mathematical simulation is represented for the vacuum evaporation dissolution process of Lyocell solution (reed cellulose, N-Methylmorpholine N-oxide (NMMO) solvent, water) in VWFE. The Turbulent Flow model in geometries with rotating parts and the frozen rotor approach in COMSOL Multi-physics 5.4 were used for simulation of the solution flow in VWFE. Comparing the obtained simulation results with experimental details, we show the effect of our simulation method.

Assessment of Compressibility Corrections on Spalart–Allmaras Turbulence Model for High-Mach-Number Flows

In this paper, various compressibility correction terms of the Spalart–Allmaras turbulence model are evaluated in hypersonic compressible flows. A unified transport equation containing six correction terms is equipped to investigate the separate and combined effects of the correction terms. The numerical results of the hypersonic flat-plate cases and corner cases are, respectively, calculated and compared with direct numerical simulation data and experimental data. By using the coarse-grained uncertainty metrics, errors on the profiles and wall coefficients are measured and analyzed in several cases. The results suggest that compressibility corrections should be applied when the strong cold-wall effect or the compressibility effect exists in the hypersonic flows. Catris and Aupoix’s correction (“Density Corrections for Turbulence Models,” Aerospace Science and Technology, Vol. 4, No. 1, 2000, pp. 1–11) and Séror, Rubin, Peigin, and Epstein’s correction (“Implementation and Validation of the Spalart–Allmaras Turbulence Model in Parallel Environment,” Journal of Aircraft, Vol. 42, No. 1, 2005, pp. 179–188) offer much more accurate predictions than other models in hypersonic corner cases. As the compressibility effect further increases, the conservation correction and Shur, Strelets, Zajkov, Gulyaev, Kozlov, and Sekundov’s compressibility correction (“Comparative Numerical Testing of One-and Two-Equation Turbulence Models for Flows with Separation and Reattachment,” 33rd Aerospace Sciences Meeting and Exhibit, AIAA Paper 1995-0863, 1995) drastically increase the prediction error of the profiles. The present work ends by emphasizing the importance of the individual and combined impacts of the correction terms for compressible flows at high Mach numbers, which may provide an understanding for developing further extensions of Spalart–Allmaras models.