Efficient and accurate simulation of unsteady flow presents a significant challenge that needs to be overcome in computational fluid dynamics. Temporal discretization method plays a crucial role in the simulation of unsteady flows. To enhance computational efficiency, we propose the Implicit-Explicit Two-Step Runge-Kutta (IMEX-TSRK) time-stepping discretization methods for unsteady flows, and develop a novel adaptive algorithm that correctly partitions spatial regions to apply implicit or explicit methods. The novel adaptive IMEX-TSRK schemes effectively handle the numerical stiffness of the small grid size and improve computational efficiency. Compared to implicit and explicit Runge-Kutta (RK) schemes, the IMEX-TSRK methods achieve the same order of accuracy with fewer first derivative calculations. Numerical case tests demonstrate that the IMEX-TSRK methods maintain numerical stability while enhancing computational efficiency. Specifically, in high Reynolds number flows, the computational efficiency of the IMEX-TSRK methods surpasses that of explicit RK schemes by more than one order of magnitude, and that of implicit RK schemes several times over.

To address the early separation problem in the Menter Shear-Stress Transport (SST) turbulence model, a correction for the Turbulent Kinetic Energy (TKE) production term, P_k, is introduced to account for the effect of the Adverse Pressure Gradient (APG). The correction is determined based on the distribution of P_k in the APG region before separation. When the friction coefficient C_f is decomposed, its direct dependence on P_k is clearly observed. However, with the introduction of Bradshaw’s assumption, P_k in the SST turbulence model is over-suppressed, resulting in a lower inner peak or no significant inner peak distribution at all. To address this problem, this paper proposes a Gaussian function, H_Gauss, which corrects the numerical values of P_k involved in the calculation of the Menter SST model by focusing on the inner peak region of P_k. The modified SST model is then applied to four cases with APGs. The modification leads to an increase in the wall friction coefficient C_f in the APG region and causes a downstream shift in the separation location, improving the model’s consistency with high-accuracy data and experimental results. It is demonstrated that this correction can improve the early separation problem in the Menter SST turbulence model.

Accurate prediction of Shock-Wave/Boundary Layer Interaction (SWBLI) flows has been a persistent challenge for linear eddy viscosity models. A major limitation lies in the isotropic representation of the Reynolds stress, as assumed under the Boussinesq approximation. Recent studies have shown promise in improving the prediction capability for incompressible separation flows by perturbing the Reynolds-stress anisotropy tensor. However, it remains uncertain whether this approach is effective for SWBLI flows, which involve compressibility and discontinuity. To address this issue, this study systematically quantifies the structural uncertainty of the anisotropy for oblique SWBLI flows. The eigenspace perturbation method is applied to perturb the anisotropy tensor predicted by the Menter Shear–Stress Transport (SST) model and reveal the impacts of anisotropy on the prediction of quantities of interest, such as separation and reattachment positions, wall static pressure, skin friction, and heat flux. The results demonstrate the potential and reveal the challenges of eigenspace perturbation in improving the SST model. Furthermore, a detailed analysis of turbulent characteristics is performed to identify the source of uncertainty. The findings indicate that eigenspace perturbation primarily affects turbulent shear stress, while the prediction error of the SST model is more related to turbulent kinetic energy.

The influence of local cooling/heating on two types of nonlinear instabilities of the high-speed boundary layer, namely, the First and Second Mode Oblique Breakdown (FMOB and SMOB), is studied using direct numerical simulations. Local cooling and heating are performed at the weak and strong nonlinear stages of the two types of nonlinear instabilities. It is found that for the FMOB, local cooling at the weak nonlinear region will suppress the increase of the fundamental mode, leading to transition delay. Opposite to local cooling, local heating at the weak nonlinear region of the FMOB will promote the growth of the fundamental mode, resulting in the occurrence of more upstream transition onset. However, if local cooling and heating are performed at the strong nonlinear region, the influence of both local cooling and heating on the FMOB can be neglected. Remarkably, both local heating and cooling can delay the SMOB for different mechanisms. Performing local cooling at the weak nonlinear region of the SMOB, the low amplitude of higher spanwise wavenumber steady mode caused by local cooling lies behind transition delay. When local cooling is set at the strong nonlinear region, the low amplitude of harmonic modes around the cooling area can cause transition delay. Additionally, local heating will suppress the SMOB for the slowing amplification rate of various modes caused by the local heating at both the weak and strong nonlinear stages of the SMOB.

Owing to the lack of physical knowledge of boundary layer transition, the γ-Re_θ transition model introduces closure parameters, which increase the uncertainty of transition prediction. The objective of this work is to quantify the uncertainties of closure parameters in the quantities of interests and identify the key parameters. The six closure parameters in the uncertainty intervals are used as input variables, and the uncertainties of the output results are propagated by a stochastic expansion based on the point-collocation nonintrusive polynomial chaos method. The relative contribution of each parameter to uncertainty is evaluated by the Sobol index. The computational cases include natural and bypass transitional flows on zero-pressure-gradient flat plates, and subsonic and transonic flows around airfoils. For most cases, c_e2, c_a2, and c_a1 dominate the uncertainty, and the influence of σ_θt is also significant when the history effects of flow are evident. The contribution of parameters in airfoils is more complex than that in flat plates. The transonic airfoil case shows that flow separation dramatically changes the distribution of Sobol indices, which poses a challenge to the accurate prediction of transition. Generally, c_e2 and c_a2 are the key parameters of the γ-Re_θ model.

The Shock Wave-Boundary Layer Interaction (SWBLI) flow generated by compression corner widely occurs in engineering. As one of the primary methods in engineering, the Reynolds Averaged Navier-Stokes (RANS) methods usually cannot correctly predict strong SWBLI flows. In addition to the defects of the eddy viscosity assumption, the uncertainty of the closure coefficients in RANS models often significantly impacts the simulation results. This study performs parametric sensitivity analysis and Bayesian calibration on the closure coefficients of the Menter k - ω Shear-Stress Transport (SST) model based on the SWBLI with different strengths. Firstly, the parametric sensitivity on prediction results is analyzed using the Sobol index. The results indicate that the Sobol indices of wall pressure and skin friction exhibited opposite fluctuation trends with the increase of SWBLI strength. Then, the Bayesian uncertainty quantification method is adopted to obtain the posterior probability distributions and Maximum A Posteriori (MAP) estimates of the closure coefficients and the posterior uncertainty of the Quantities of Interests (QoIs). The results indicate that the prediction ability for strong SWBLI of the SST model is significantly improved by using the MAP estimates, and the relative errors of QoIs are reduced dramatically.

Uncertainty is common in the life cycle of an aircraft, and Robust Aerodynamic Optimization (RAO) that considers uncertainty is important in aircraft design. To avoid the curse of dimensionality in surrogate-based optimization, this study proposes an adjoint RAO technique called “R-Opt”. Polynomial Chaos Expansion (PCE) is coupled with the R-Opt technique to quantify uncertainty in the responses of the target (including its mean and standard deviation). Only one process of PCE model construction is required in each iteration, and the gradients of uncertainty can be inferred via chain rules. The proposed method is more efficient than prevalent methods, and avoids the problem of a disagreement over the best PCE basis from among a number of PCE models (especially in case of sparse PCE). It also supports the application of sparse PCE. Two benchmark tests and two airfoil cases were used to verify R-Opt, and the optimal solutions were deemed to be robust. It improved the mean aerodynamic performance and reduced the standard deviation of the target.

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

A new physics-based model employing three transport equations is developed for the simulation of boundary layer transitions in a wide speed range. The laminar kinetic energy is used to represent pretransitional streamwise velocity fluctuations, taking account of different instability modes. The fluctuation velocity components normal to the streamwise direction are modeled by another transport equation. Transition is triggered automatically with the development of the pretransitional velocity fluctuations. In the fully turbulent region, the model reverts to the k-ω turbulence model. Different test cases, including subsonic, supersonic and hypersonic flows around flat plates, airfoils and straight cones, are numerically simulated to validate the performance of the model. The results demonstrate the excellent predictive capabilities of the model in different paths of transition. The model can serve as a basis for the extension of additional transition mechanisms, such as rotation and curvature effects, roughness-induced transition and crossflow-induced transition.

The Reynolds Averaged Navier-Stokes (RANS) models are still the workhorse in current engineering applications due to its high efficiency and robustness. However, the closure coefficients of RANS turbulence models are determined by model builders according to some simple fundamental flows, and the suggested values may not be applicable to complex flows, especially supersonic jet interaction flow. In this work, the Bayesian method is employed to recalibrate the closure coefficients of Spalart-Allmaras (SA) turbulence model to improve its performance in supersonic jet interaction problem and quantify the uncertainty of wall pressure and separation length. The embedded model error approach is applied to the Bayesian uncertainty analysis. Firstly, the total Sobol index is calculated by non-intrusive polynomial chaos method to represent the sensitivity of wall pressure and separation length to model parameters. Then, the pressure data and the separation length are respectively served as calibration data to get the posterior uncertainty of model parameters and Quantities of Interests (QoIs). The results show that the relative error of the wall pressure predicted by the SA turbulence model can be reduced from 14.99% to 2.95% through effective Bayesian parameter estimation. Besides, the calibration effects of four likelihood functions are systematically evaluated. The posterior uncertainties of wall pressure and separation length estimated by different likelihood functions are significantly discrepant, and the Maximum a Posteriori (MAP) values of parameters inferred by all functions show better performance than the nominal values. Finally, the closure coefficients are also estimated at different jet total pressures. The similar posterior distributions of model parameters are obtained in different cases, and the MAP values of parameters calibrated in one case are also applicable to other cases.