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To improve robustness and efficiency of automatic transition prediction in aerodynamic design, a reduced model of linear stability analysis is usually adopted, such as eN-envelope or eN-database me...

A set of several exact coherent states in plane Couette flow is computed under spanwise wall oscillation control, with a range of wall oscillation amplitudes and periods $({A_w}, T)$ . It is found that the wall oscillation generally stabilises the upper branch of the equilibrium solutions and achieves the corresponding drag reduction, while it influences modestly the lower branch. The stabilisation effect is found to increase with the oscillation amplitude with an optimal time period around ${T^{+}} \approx 100$ . The exact coherent states reproduce some key dynamical behaviours of streaks observed in previous studies, while exhibiting the rich coherent structure dynamics that cannot be extracted from a phase average of turbulent states. Visualisation of state portraits shows that the size of the state space supporting turbulent solution is reduced by the spanwise wall oscillation, and the upper-branch equilibrium solutions become less repelling, with many of their unstable manifolds being stabilised. This change of the state space dynamics leads to a significant reduction in lifetime of turbulence. Finally, the main stabilisation mechanism of the exact coherent states is found to be the suppression of the lift-up effect of streaks, explaining why previous linear analyses have been so successful for turbulence stabilisation modelling and the resulting drag reduction.

We compare in this paper the properties of Osher flux with O-variant and P-variant (Osher-O flux and Osher-P flux) in finite volume methods for the two-dimensional Euler equations and propose an entropy fix technique to improve their robustness. We consider both first-order and second-order reconstructions. For inviscid hypersonic flow past a circular cylinder, we observe different problems for different schemes: A first-order Osher-O scheme on quadrangular grids yields a carbuncle shock, while a first-order Osher-P scheme results in a dislocation shock for high Mach number cases. In addition, a second-order Osher scheme can also yield a carbuncle shock or be unstable. To improve the robustness of these schemes we propose an entropy fix technique, and then present numerical results to show the effectiveness of the proposed method. In addition, the influence of grid aspects ratio, relative shock position to the grid and Mach number on shock stability are tested. Viscous heating problem and double Mach reflection problem are simulated to test the influence of the entropy fix on contact resolution and boundary layer resolution.

Growing evidences show that the Symmetrical Conservative Metric Method (SCMM) is essential in preserving freestream conservation and orders of accuracy for high-order finite difference schemes to simulate flows with complex geometries. In this paper, a new family of Hybrid cell-edge and cell-node Dissipative Compact Schemes (HDCSs) has been developed for geometry-complex flows by fulfilling the SCMM as well as by introducing dissipation according to the concept adopted in the construction of the high-order Dissipative Compact Schemes (DCSs). The resolution and dissipation properties of HDCSs are investigated by the Fourier analysis, and the stability property of HDCSs is also investigated by asymptotic stability analysis and amplification factor analysis. HDCSs are validated by computing several benchmark test cases. The vortex convection test case demonstrates that the orders of accuracy of the HDCSs are preserved unless the GCL is satisfied. Although high resolution of HDCSs is observed in the test of acoustic wave scattering of multiple cylinders, the solutions can be contaminated if the GCL is not satisfied. Moreover, the numerical solutions of flow past a high lift trapezoidal wing demonstrate the promising ability of the newly developed HDCSs in solving complex flow problems.

The metrics and Jacobian in the fluid motion governing equations under curvilinear coordinate system have a variety of equivalent differential forms, which may have different discretization errors with the same difference scheme. The discretization errors of metrics and Jacobian may cause serious computational instability and inaccuracy in numerical results, especially for high-order finite difference schemes. It has been demonstrated by many researchers that the Geometric Conservation Law (GCL) is very important for high-order Finite Difference Methods (FDMs), and a proper form of metrics and Jacobian, which can satisfy the GCL, can considerably reduce discretization errors and computational instability. In order to satisfy the GCL for FDM, we have previously developed a Conservative Metric Method (CMM) to calculate the metrics [1] and the difference scheme @d^3 in the CMM is determined with the suggestion @d^[email protected]^2. In this paper, a Symmetrical Conservative Metric Method (SCMM) is newly proposed based on the discussions of the metrics and Jacobian in FDM from geometry viewpoint by following the concept of vectorized surface and cell volume in Finite Volume Methods (FVMs). Interestingly, the expressions of metrics and Jacobian obtained by using the SCMM with second-order central finite difference scheme are equivalent to the vectorized surfaces and cell volumes, respectively. The main advantage of SCMM is that it makes the calculations based on high-order WCNS schemes aroud complex geometry flows possible and somewhat easy. Numerical tests on linear and nonlinear problems indicate that the quality of numerical results may be largely enhanced by utilizing the SCMM, and the advantage of the SCMM over other forms of metrics and Jacobian may be more evident on highly nonuniform grids.

A modified transition model is given based on the shear stress transport (SST) turbulence model and an intermittency transport equation. The energy gradient term in the original model is replaced by flow strain rate to saving computational costs. The model employs local variables only, and then it can be conveniently implemented in modern computational fluid dynamics codes. The fifth-order weighted compact nonlinear scheme and the fourth-order staggered scheme are applied to discrete the governing equations for the purpose of minimizing discretization errors, so as to mitigate the confusion between numerical errors and transition model errors. The high-order package is compared with a second-order TVD method on simulating the transitional flow of a flat plate. Numerical results indicate that the high-order package give better grid convergence property than that of the second-order method. Validation of the transition model is performed for transitional flows ranging from low speed to hypersonic speed.

The geometric conservation law (GCL) includes the volume conservation law (VCL) and the surface conservation law (SCL). Though the VCL is widely discussed for time-depending grids, in the cases of stationary grids the SCL also works as a very important role for high-order accurate numerical simulations. The SCL is usually not satisfied on discretized grid meshes because of discretization errors, and the violation of the SCL can lead to numerical instabilities especially when high-order schemes are applied. In order to fulfill the SCL in high-order finite difference schemes, a conservative metric method (CMM) is presented. This method is achieved by computing grid metric derivatives through a conservative form with the same scheme applied for fluxes. The CMM is proven to be a sufficient condition for the SCL, and can ensure the SCL for interior schemes as well as boundary and near boundary schemes. Though the first-level difference operators ?3 have no effects on the SCL, no extra errors can be introduced as ?3=?2. The generally used high-order finite difference schemes are categorized as central schemes (CS) and upwind schemes (UPW) based on the difference operator ?1 which are used to solve the governing equations. The CMM can be applied to CS and is difficult to be satisfied by UPW. Thus, it is critical to select the difference operator ?1 to reduce the SCL-related errors. Numerical tests based on WCNS-E-5 show that the SCL plays a very important role in ensuring free-stream conservation, suppressing numerical oscillations, and enhancing the robustness of the high-order scheme in complex grids.