A machine learning-enhanced high-resolution weighted compact nonlinear scheme with unconditionally optimal high order

Freestream and vortex preservation properties of high-order WENO and WCNS on curvilinear grids

PurposeThe purpose of the present study is to develop a new numerical framework that can predict the supersonic base flow more accurately, including the development of axisymmetrically separated shear layer and recompression shock. To this end, two aspects are improved and combined, i.e. a newly self-adaptive turbulence eddy simulation (SATES) turbulence modeling method and a high-order discretization numerical scheme. Furthermore, the performance of the new numerical framework within a general-purpose PHengLEI software is assessed in detail.Design/methodology/approachSatisfactory prediction of the supersonic separated shear layer with unsteady wake flow is quite challenging. By using a unified turbulence model called SATES combining high-order accurate discretization numerical schemes, the present study first assesses the performance of newly developed SATES for supersonic axisymmetric separation flows. A high-order finite differencing-based compressible computational fluid dynamics (CFD) code called PHengLEI is developed and several different numerical schemes are used to investigate the effects on shock-turbulence interactions, which include the monotonic upstream-centered scheme for conservation laws (MUSCL), weighted compact nonlinear scheme (WCNS) and hybrid cell-edge and cell-node dissipative compact scheme (HDCS).FindingsCompared with the available experimental data and the numerical predictions, the results of SATES by using high-order accurate WCNS or HDCS schemes agree better with the experiments than the results by using the MUSCL scheme. The WCNS and HDCS can also significantly improve the prediction of flow physics in terms of the instability of the annular shear layer and the evolution of the turbulent wake.Research limitations/implicationsThe small deviations in the recirculation region can be found between the present numerical results and experimental data, which could be caused by the inaccurate incoming boundary layer condition and compressible effects. Therefore, a proper incoming boundary layer condition with turbulent fluctuations and compressibility effects need to be considered to further improve the accuracy of simulations.Practical implicationsThe present study evaluates a high-order discretization-based SATES turbulence model for supersonic separation flows, which is quite valuable for improving the calculation accuracy of aeronautics applications, especially in supersonic conditions.Originality/valueFor the first time, the newly developed SATES turbulence modeling method combining the high-order accurate WCNS or HDCS numerical schemes is implemented on the PHengLEI software and successfully applied for the simulations of supersonic separation flows, and satisfactory results are obtained. The unsteady evolutions of the supersonic annular shear layer are analyzed, and the hairpin vortex structures are found in the simulation.

This paper designs a high order semi-implicit weighted compact nonlinear scheme (WCNS) to solve 1D isentropic and full Euler equations. The Euler equations are split into stiff and non-stiff terms that are solved by the implicit and explicit time discretization method, respectively, in order to improve the computational efficiency for low Mach flows. The fifth-order WCNS is applied for the spatial discretization. Several numerical examples are given to demonstrate the performance of the designed method.

In this study, we propose a novel weighted compact nonlinear scheme that enhances the performance of the known third-order scheme and addresses the issue of inconsistent accuracy between its nonlinear reconstruction step and compact weighting step. To more accurately approximate steep gradients and high oscillations, an interpolation method based on exponential polynomials with a shape parameter is introduced. Specifically, a method for selecting locally optimised parameters that achieve fourth-order accuracy at non-critical points is proposed. Additionally, we introduce a smoothness indicator based on exponential polynomials, which increases the nonlinear weights of less smooth sub-stencils and improves the resolution of the scheme. To demonstrate the optimal numerical accuracy and the shock-capturing capabilities of the proposed scheme, numerical results are presented.

Calibration of a [formula omitted] transition model and its validation in low-speed flows with high-order numerical method

By introducing hybrid technique into high-order CPR (correction procedure via reconstruction) scheme, a novel hybrid WCNS-CPR scheme is developed for efficient supersonic simulations. Firstly, a shock detector based on nonlinear weights is used to identify grid cells with high gradients or discontinuities throughout the whole flow field. Then, WCNS (weighted compact nonlinear scheme) is adopted to capture shocks in these areas, while the smooth area is calculated by CPR. A strategy to treat the interfaces of the two schemes is developed, which maintains high-order accuracy. Convergent order of accuracy and shock-capturing ability are tested in several numerical experiments; the results of which show that this hybrid scheme achieves expected high-order accuracy and high resolution, is robust in shock capturing, and has less computational cost compared to the WCNS.

Constructing boundary schemes within finite difference methods that simultaneously provide numerical stability, robust shock capturing, high-order accuracy, and global integral conservation remains a formidable challenge. In this paper, based on the weighted compact nonlinear scheme (WCNS) framework, we devise a multi-block boundary treatment for block interfaces with grid distortion and possible shocks that (i) retains high order of discretization accuracy in smooth flows, (ii) maintains strong robustness across shock discontinuities, and (iii) rigorously enforces global integral conservation throughout the simulation. A series of strategies has been applied. First, we begin with the multi-resolution WCNS, which blends 1-, 3-, and 5-point stencils for shock capture. By redesigning the nonlinear weights, the scheme now regains full third-order accuracy whenever a discontinuity contaminates only the 5-point stencil, whereas the original formulation collapsed to first order in that situation. Second, exploiting the compact stencils near boundaries, we devise four nearly one-sided nonlinear interpolation schemes without directly using cross-interface data. The resulting boundary scheme is high-order accurate in smooth, distorted grids and stays sharp and non-oscillatory when a shock lies on the interface. Finally, to achieve global integral conservation of the conserved variables, we propose a hybrid conservative difference scheme that combines face-to-cell and cell-to-cell stencils, driving conservation errors to machine precision. At last, the proposed schemes are validated through a series of numerical tests. The results demonstrate that even with grid distortion across block interfaces, the proposed scheme ensures the desired order of accuracy in smooth flow and avoids oscillations near shock locations.

A multi-order candidates weighting framework for discontinuity capturing of hyperbolic conservation laws

For computational aeroacoustics (CAA), schemes with low dissipation and dispersion errors are a necessity. This is because of the small amplitude of the acoustic waves and because of the large distances over which they have to travel. Also, non-reflecting boundary conditions play a very important role, since spurious reflections could significantly reduce the quality of the solution. To damp out small spurious waves that might occur, Artificial Selective Damping (ASD) can be used. In this paper, the performance for CAA of a fourth order central compact finite volume scheme, in combination with ASD and two different sets of boundary conditions, is investigated.

A new compact scheme for parallel computing using domain decomposition

A compact fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations

Optimized prefactored compact schemes

Two solution methods for compressible flows and their application in Large-Eddy Simulation (LES) of turbulent flows are discussed. After a summary of the basic equations and subgrid scale models, the applied discretization methods, solution schemes and boundary conditions with their implementation on high-performance computers are presented. The two discretization methods for the advective terms used in the simulations are a second-order mixed central-upwind discretization based on a AUSM formulation and a higher-order compact finite difference scheme. First, results of the second-order scheme are shown for the case of a turbulent channel and pipe flow, which are only weakly influenced by subgrid scale models, but are in good agreement with DNS and experimental data. Higher-order compact schemes are then used for both laminar and turbulent flows and the results are compared to those from simulations with the second-order scheme. It is found that for the laminar flow and the LES of plane turbulent jets the higher-order compact scheme is not of principal advantage. This is primarily related to the sensitivity of the compact scheme to stretched and curvilinear grids, which requires explicit filtering steps to avoid unphysical oscillations in the solution. Further results are then presented, which are obtained with the computationally cheaper second-order scheme. The results include turbulent flows in pipe bends with different Reynolds and Dean numbers. The solutions show swirl switching with a distinct low frequency peak. Finally, turbulent flows in non-trivial geometries are considered. This includes a first step towards the LES of the flow in an internal combustion engine with four valves and a realistic geometry.

Most high-order computational fluid dynamics methods for compressible flows are based on Riemann solver for the flux evaluation and Runge-Kutta time stepping technique for temporal evolution. Different from these traditional high-order methods, the flux of the gas-kinetic scheme (GKS) is based on a time-dependent evolution solution of the kinetic equation, such as the Bhatnagar-Gross-Krook model, and it targets on the Euler and N-S solutions. High-order GKS (HGKS) has been systemically developed over the past decades. The recently proposed two-stage fourth-order HGKS is efficient, accurate and robust. In this thesis, we mainly focus on the efficient implementations of HGKS, especially on the construction of compact schemes. The first two parts of the thesis focus on the temporal and spatial discretizations of HGKS. By combining the second-order or third-order GKS flux functions with the multi-stage multi-derivative technique, a family of HGKSs can be constructed. As an extension of the previous two-stage fourth-order GKS, the fifth-order schemes with two and three stages have been developed. It provides a framework to construct HGKS with temporal accuracy towards to an arbitrary order. Then the weighted essentially non-oscillatory with adaptive order (WENO-AO) method is adopted for the initial reconstruction in HGKS. With the help of the new reconstruction method, HGKS can achieve uniform high-order for equilibrium and non-equilibrium states. Moreover, the new scheme becomes simpler in reconstruction, more flexible on numerical treatment at the Gaussian points, and more robust than the previous HGKS with standard WENO reconstruction. We present the fifth-order WENO-AO GKS for validation, while this reconstruction procedure can be easily extended to arbitrary order in space. Then, a fourth-order compact GKS is developed. It is composed of three ingredients, which are the two-stage fourth-order framework for temporal discretization, the higher-order gas evolution model for the evaluations of interface values and fluxes, and the fifth-order Hermite WENO reconstruction for the cell-averaged values updated through fluxes and their first order derivatives obtained through the differences of flow variables at the cell interfaces. These ingredients make the scheme to achieve higher-order of accuracy and better robustness with lower computational cost. The CFL number used in this compact GKS is on the order of 0.5 instead of 0.14 in the popular discontinuous Galerkin method. Furthermore, targeting on simulating more challenging flow problems, such as the aeroacoustic generation and propagation, and turbulence, up to eighth-order compact schemes have been developed. These higher-order schemes show spectral-like resolution for acoustic waves, and capture shocks crisply by the special designed WENO-Z reconstructions on a compact stencil. Lastly, the two-stage compact scheme has been successfully extended to unstructured meshes. Accurate solutions have been obtained for both inviscid and viscous flows without sensitive dependence on the quality of the mesh. The robustness of the scheme has been validated as well through many cases, including strong shocks in the hypersonic viscous flow simulations.

A new class of central compact schemes with spectral-like resolution II: Hybrid weighted nonlinear schemes