On the performance of WENO/TENO schemes to resolve turbulence in DNS/LES of high‐speed compressible flows

High‐speed compressible turbulent flows typically contain discontinuities and have been widely modeled using Weighted Essentially Non‐Oscillatory (WENO) schemes due to their high‐order accuracy and sharp shock capturing capability. However, such schemes may damp the small scales of turbulence and result in inaccurate solutions in the context of turbulence‐resolving simulations. In this connection, the recently developed Targeted Essentially Non‐Oscillatory (TENO) schemes, including adaptive variants, may offer significant improvements. The present study aims to quantify the potential of these new schemes for a fully turbulent supersonic flow. Specifically, DNS of a compressible turbulent channel flow with M = 1.5 and Re_τ = 222 is conducted using OpenSBLI, a high‐order finite difference computational fluid dynamics framework. This flow configuration is chosen to decouple the effect of flow discontinuities and turbulence and focus on the capability of the aforementioned high‐order schemes to resolve turbulent structures. The effect of the spatial resolution in different directions and coarse grid implicit LES are also evaluated against the WALE LES model. The TENO schemes are found to exhibit significant performance improvements over the WENO schemes in terms of the accuracy of the statistics and the resolution of the three‐dimensional vortical structures. The sixth‐order adaptive TENO scheme is found to produce comparable results to those obtained with nondissipative fourth‐ and sixth‐order central schemes and reference data obtained with spectral methods. Although the most computationally expensive scheme, it is shown that this adaptive scheme can produce satisfactory results if used as an implicit LES model.     

Simulation of multiple shock–shock interference using implicit anti‐diffusive WENO schemes

Accurate computations of two‐dimensional turbulent hypersonic shock–shock interactions that arise when single and dual shocks impinge on the bow shock in front of a cylinder are presented. The simulation methods used are a class of lower–upper symmetric‐Gauss–Seidel implicit anti‐diffusive weighted essentially non‐oscillatory (WENO) schemes for solving the compressible Navier–Stokes equations with Spalart–Allmaras one‐equation turbulence model. A numerical flux of WENO scheme with anti‐diffusive flux correction is adopted, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of first‐order dissipative methods. Experimental flow fields of type IV shock–shock interactions with single and dual incident shocks by Wieting are computed. By using the WENO scheme with anti‐diffusive flux corrections, the present solution indicates that good accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Computed surface pressure distribution and heat transfer rate are also compared with experimental data and other computational results and good agreement is found. Copyright © 2009 John Wiley & Sons, Ltd.     

A novel weighting switch function for uniformly high‐order hybrid shock‐capturing schemes

Hybrid schemes are very efficient for complex compressible flow simulation. However, for most existing hybrid schemes in literature, empirical problem‐dependent parameters are always needed to detect shock waves and hence greatly decrease the robustness and accuracy of the hybrid scheme. In this paper, based on the nonlinear weights of the weighted essentially non‐oscillatory (WENO) scheme, a novel weighting switch function is proposed. This function approaches 1 with high‐order accuracy in smooth regions and 0 near discontinuities. Then, with the new weighting switch function, a seventh‐order hybrid compact‐reconstruction WENO scheme (HCCS) is developed. The new hybrid scheme uses the same stencil as the fifth‐order WENO scheme, and it has seventh‐order accuracy in smooth regions even at critical points. Numerical tests are presented to demonstrate the accuracy and robustness of both the switch function and HCCS. Comparisons also reveal that HCCS has lower dissipation and less computational cost than the seventh‐order WENO scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

基于Level Set的多维双曲守恒律标量方程的激波追踪方法

结合四阶CWENO(Cemral Weighted Essentially Non-Oscillatory)格式、四阶NCE(Natural Continuous Extensions)Runge-Kutta法和Level Set方法，很好地处理了一维双曲守恒律标量方程的激波追踪问题。针对二维双曲守恒律标量方程，成功地用五阶WENO格式、非TVD格式的四阶Runge-Kutta方法和Level Set方法进行激波追踪。将所得的数值解与标准的高阶激波捕捉方法所得的数值解进行比较，说明基于Level Set的激波追踪方法的有效性与逐点收敛性。     

Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimized sixth‐order monotonicity‐preserving scheme by nonlinear spectral analysis

In this paper, sixth‐order monotonicity‐preserving optimized scheme (OMP6) for the numerical solution of conservation laws is developed on the basis of the dispersion and dissipation optimization and monotonicity‐preserving technique. The nonlinear spectral analysis method is developed and is used for the purpose of minimizing the dispersion errors and controlling the dissipation errors. The new scheme (OMP6) is simple in expression and is easy for use in CFD codes. The suitability and accuracy of this new scheme have been tested through a set of one‐dimensional, two‐dimensional, and three‐dimensional tests, including the one‐dimensional Shu–Osher problem, the two‐dimensional double Mach reflection, and the Rayleigh–Taylor instability problem, and the three‐dimensional direct numerical simulation of decaying compressible isotropic turbulence. All numerical tests show that the new scheme has robust shock capturing capability and high resolution for the small‐scale waves due to fewer numerical dispersion and dissipation errors. Moreover, the new scheme has higher computational efficiency than the well‐used WENO schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

Preventing numerical oscillations in the flux‐split based finite difference method for compressible flows with discontinuities,II

Problems in the characteristic‐wise flux‐split based finite difference method when compressible flows with contact discontinuities or material interfaces are computed were presented and analyzed. The current analysis showed the following: (i) Even with the local characteristic decomposition technique, numerical errors could be caused by point‐wise flux vector splitting (FVS) methods, such as the Steger–Warming FVS or the van Leer FVS. Therefore, the Lax–Friedrichs type FVS method is required. (ii) If the isobars of a material are vertical lines, the combination of using the local characteristic decomposition and the global Lax–Friedrichs FVS can avoid velocity and pressure oscillations of contact discontinuities in this material for weighted essentially non‐oscillatory (WENO) schemes. (iii) For problems with material interfaces, the quasi‐conservative approach can be realized using characteristic‐wise flux‐split based finite difference WENO schemes if nonlinear WENO schemes in genuinely nonlinear characteristic fields can be guaranteed to be the same and the decomposition equation representing material interfaces is discretized properly. Copyright © 2015 John Wiley & Sons, Ltd.     

A γ‐DGBGK scheme for compressible multi‐fluids

The paper presents a Discontinuous Galerkin γ‐BGK (γ‐DGBGK) method for compressible multicomponent flow simulations by coupling the discontinuous Galerkin method with a γ‐BGK scheme based on WENO limiters. In this γ‐DGBGK method, the construction of the flux in the DG method is based on the kinetic scheme which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous terms in the flux formulation at cell interfaces. WENO limiters are used to obtain uniform high‐order accuracy and sharp non‐oscillatory shock transition, and time accuracy obtained by integration for the flux function at the cell interface. Numerical examples in one and two space dimensions are presented to illustrate the robust and accuracy of the present scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

High‐order k‐exact WENO finite volume schemes for solving gas dynamic Euler equations on unstructured grids

This paper presents a family of High‐order finite volume schemes applicable on unstructured grids. The k‐exact reconstruction is performed on every control volume as the primary reconstruction. On a cell of interest, besides the primary reconstruction, additional candidate reconstruction polynomials are provided by means of very simple and efficient ‘secondary’ reconstructions. The weighted average procedure of the WENO scheme is then applied to the primary and secondary reconstructions to ensure the shock‐capturing capability of the scheme. This procedure combines the simplicity of the k‐exact reconstruction with the robustness of the WENO schemes and represents a systematic and unified way to construct High‐order accurate shock capturing schemes. To further improve the efficiency, an efficient problem‐independent shock detector is introduced. Several test cases are presented to demonstrate the accuracy and non‐oscillation property of the proposed schemes. The results show that the proposed schemes can predict the smooth solutions with uniformly High‐order accuracy and can capture the shock waves and contact discontinuities in high resolution. Copyright © 2011 John Wiley & Sons, Ltd.     

A numerical study for WENO scheme-based on different lattice Boltzmann flux solver for compressible flows

In this paper, the finite difference weighted essentially non-oscillatory (WENO) scheme is incorporated into the recently developed four kinds of lattice Boltzmann flux solver (LBFS) to simulate compressible flows, including inviscid LBFS I, viscous LBFS II, hybrid LBFS III and hybrid LBFS IV. Hybrid LBFS can automatically realize the switch between inviscid LBFS I and viscous LBFS II through introducing a switch function. The resultant hybrid WENO–LBFS scheme absorbs the advantages of WENO scheme and hybrid LBFS. We investigate the performance of WENO scheme based on four kinds of LBFS systematically. Numerical results indicate that the devopled hybrid WENO–LBFS scheme has high accuracy, high resolution and no oscillations. It can not only accurately calculate smooth solutions, but also can effectively capture contact discontinuities and strong shock waves.     

Equivalence conditions between linear Lagrangian finite element and node‐centred finite volume schemes for conservation laws in cylindrical coordinates

A complete set of equivalence conditions, relating the mass‐lumped Bubnov–Galerkin finite element (FE) scheme for Lagrangian linear elements to node‐centred finite volume (FV) schemes, is derived for the first time for conservation laws in a three‐dimensional cylindrical reference. Equivalence conditions are used to devise a class of FV schemes, in which all grid‐dependent quantities are defined in terms of FE integrals. Moreover, all relevant differential operators in the FV framework are consistent with their FE counterparts, thus opening the way to the development of consistent hybrid FV/FE schemes for conservation laws in a three‐dimensional cylindrical coordinate system. The two‐dimensional schemes for the polar and the axisymmetrical cases are also explicitly derived. Numerical experiments for compressible unsteady flows, including expanding and converging shock problems and the interaction of a converging shock waves with obstacles, are carried out using the new approach. The results agree fairly well with one‐dimensional simulations and simplified models. Copyright © 2013 John Wiley & Sons, Ltd.     

无粘可压缩流动的改进高精度方法

为更准确捕捉复杂流场的流动细节,通过对WENO格式的光滑因子进行改进,发展了一种新的五阶WENO格式。对三阶ENO格式进行加权可以得到五阶WENO格式,但是不同的加权处理,WENO格式在极值处保持加权基本无振荡的效果不同,本文构造了二阶精度的局部光滑因子,及不含一阶二阶导数的高阶全局光滑因子,从而实现WENO格式在极值处有五阶精度。基于改进五阶WENO格式,对一维对流方程、一维和二维可压缩无粘问题进行算例验证,并与传统WENO-JS格式和WENO-Z格式进行比较。计算结果表明,改进五阶WENO格式有较高的精度和收敛速度,有较低的数值耗散,能有效捕捉间断、激波和涡等复杂流动。     

Weighted Compact Scheme for Shock Capturing

In this paper a new class of finite difference schemes - the Weighted Compact Schemes are proposed. According to the idea of the WENO schemes, the Weighted Compact Scheme is constructed by a combination of the approximations of derivatives on candidate stencils with properly assigned weights so that the non-oscillatory property is achieve when discontinuities appear. The primitive function reconstruction method of ENO schemes is applied to obtain the conservative form of the Weighted Compact Scheme. This new scheme not only preserves the characteristic of standard compact schemes and achieves high order accuracy and high resolution using a compact stencil, but also can accurately capture shock waves and discontinuities without oscillation. Numerical examples show the new scheme is very promising and successful.     

Development of optimized weighted‐ENO schemes for multiscale compressible flows

The present paper deals with the development of optimized weighted–ENO schemes to improve the resolution of a class of compressible flows characterized by a wide disparity of scales, typical of compressible turbulence and/or aeroacoustic phenomena, and shock waves. The approach relies on a least square minimization of both the dispersion and dissipation error components together with the use of symmetric stencil support. Extensive numerical simulations of sound propagation, shock–sound interaction and isotropic compressible turbulence have been carried out, and the results confirm that the optimized schemes yield a resolution in wave number space greater than the non‐optimized ones. Copyright © 2003 John Wiley & Sons, Ltd.     

Convergence issues in using high‐resolution schemes and lower–upper symmetric Gauss–Seidel method for steady shock‐induced combustion problems

This paper reports numerical convergence study for simulations of steady shock‐induced combustion problems with high‐resolution shock‐capturing schemes. Five typical schemes are used: the Roe flux‐based monotone upstream‐centered scheme for conservation laws (MUSCL) and weighted essentially non‐oscillatory (WENO) schemes, the Lax–Friedrichs splitting‐based non‐oscillatory no‐free parameter dissipative (NND) and WENO schemes, and the Harten–Yee upwind total variation diminishing (TVD) scheme. These schemes are implemented with the finite volume discretization on structured quadrilateral meshes in dimension‐by‐dimension way and the lower–upper symmetric Gauss–Seidel (LU–SGS) relaxation method for solving the axisymmetric multispecies reactive Navier–Stokes equations. Comparison of iterative convergence between different schemes has been made using supersonic combustion flows around a spherical projectile with Mach numbers M = 3.55 and 6.46 and a ram accelerator with M = 6.7. These test cases were regarded as steady combustion problems in literature. Calculations on gradually refined meshes show that the second‐order NND, MUSCL, and TVD schemes can converge well to steady states from coarse through fine meshes for M = 3.55 case in which shock and combustion fronts are separate, whereas the (nominally) fifth‐order WENO schemes can only converge to some residual level. More interestingly, the numerical results show that all the schemes do not converge to steady‐state solutions for M = 6.46 in the spherical projectile and M = 6.7 in the ram accelerator cases on fine meshes although they all converge on coarser meshes or on fine meshes without chemical reactions. The result is based on the particular preconditioner of LU–SGS scheme. Possible reasons for the nonconvergence in reactive flow simulation are discussed.Copyright © 2012 John Wiley & Sons, Ltd.     

A high‐resolution method for compressible two‐fluid flows and simulation of three‐dimensional shock–bubble interactions

A high‐resolution method is developed to capture the material interfaces of compressible two‐fluid flows in multiple dimensions. A fluid mixture model system with single velocity and pressure is used, and viscous effect can also be taken into account. A consistent thermodynamic law based on the assumption of pressure equilibrium is employed to describe the thermodynamic behaviors of the pure fluids and mixture of two components. The splitting and unsplit Eulerian formulations of piecewise parabolic method are extended to numerically integrate the hyperbolic part of the model system, whereas the system of diffusion equations is solved using an explicit, central difference scheme. The block‐structured adaptive mesh refinement (AMR) capability is built in the hydrodynamic code to locally improve grid resolution. The resulting method is verified to be at least second‐order accurate in space. Numerical results show that the discontinuities, particularly contact discontinuities, can be resolved sharply. The use of AMR allows flow features at disparate scales to be resolved sufficiently. In addition, three‐dimensional shock–bubble interactions are simulated to investigate effects of Mach number on bubble evolution. The flow structures including those peculiar to three‐dimensional bubble are resolved correctly, and some physical phenomena with increasing Mach number are reported. Copyright © 2012 John Wiley & Sons, Ltd.     

A hybrid kinetic WENO scheme for inviscid and viscous flows

In this paper, we propose a high‐order finite volume hybrid kinetic Weighted Essentially Non‐Oscillatory (WENO) scheme for inviscid and viscous flows. Based on the WENO reconstruction technique, a hybrid kinetic numerical flux is introduced for the present method, which includes the mechanisms of both the free transfer and the collision of gas molecules. The collisionless free transfer part of the hybrid numerical flux is constructed from the conventional kinetic flux vector splitting treatment, and the collision contribution is considered by constructing an equilibrium gas state and calculating the corresponding numerical flux at the cell interface. The total variation diminishing Runge–Kutta methods are used for the temporal integration. The high‐order accuracy and good shock‐capturing capability of the proposed hybrid kinetic WENO scheme are validated by many numerical examples in one‐dimensional and two‐dimensional cases. Copyright © 2015 John Wiley & Sons, Ltd.     

A high‐order finite difference method for numerical simulations of supersonic turbulent flows

This paper describes a new class of three‐dimensional finite difference schemes for high‐speed turbulent flows in complex geometries based on the high‐order monotonicity‐preserving (MP) method. Simulations conducted for various 1D, 2D, and 3D problems indicate that the new high‐order MP schemes can preserve sharp changes in the flow variables without spurious oscillations and are able to capture the turbulence at the smallest computed scales. Our results also indicate that the MP method has less numerical dissipation and faster grid convergence than the weighted essentially non‐oscillatory method. However, both of these methods are computationally more demanding than the COMP method and are only used for the inviscid fluxes. To reduce the computational cost for reacting flows, the scalar equations are solved by the COMP method, which is shown to yield similar results to those obtained by the MP in supersonic turbulent flows with strong shock waves. Copyright © 2011 John Wiley & Sons, Ltd.     

High‐order shock capturing schemes for turbulence calculations

This work investigates high‐order central compact methods for simulating turbulent supersonic flows that include shock waves. Several different types of previously proposed characteristic filters, including total variation diminishing, monotone upstream‐centered scheme for conservation laws, and weighted essentially non‐oscillatory filters, are investigated in this study. Similar to the traditional shock capturing schemes, these filters can eliminate the numerical instability caused by large gradients in flow fields, but they also improve efficiency compared with classical shock‐capturing schemes. Adding the nonlinear dissipation part of a classical shock‐capturing scheme to a central scheme makes the method suitable for incorporation into any existing central‐based high‐order subsonic code. The amount of numerical dissipation to add is sensed by means of the artificial compression method switch. In order to improve the performance of the characteristic filters, we propose a hybrid approach to minimize the dissipation added by the characteristic filter. Through several numerical experiments (including a shock/density wave interaction, a shock/vortex interaction, and a shock/mixing layer interaction) we show that our hybrid approach works better than the original method, and can be used for future turbulent flow simulations that include shocks. Copyright © 2009 John Wiley & Sons, Ltd.     

Adaptive mesh refinement techniques for high‐order shock capturing schemes for multi‐dimensional hydrodynamic simulations

The numerical simulation of physical phenomena represented by non‐linear hyperbolic systems of conservation laws presents specific difficulties mainly due to the presence of discontinuities in the solution. State of the art methods for the solution of such equations involve high resolution shock capturing schemes, which are able to produce sharp profiles at the discontinuities and high accuracy in smooth regions, together with some kind of grid adaption, which reduces the computational cost by using finer grids near the discontinuities and coarser grids in smooth regions. The combination of both techniques presents intrinsic numerical and programming difficulties. In this work we present a method obtained by the combination of a high‐order shock capturing scheme, built from Shu–Osher's conservative formulation (J. Comput. Phys. 1988; 77 :439–471; 1989; 83 :32–78), a fifth‐order weighted essentially non‐oscillatory (WENO) interpolatory technique (J. Comput. Phys. 1996; 126 :202–228) and Donat–Marquina's flux‐splitting method (J. Comput. Phys. 1996; 125 :42–58), with the adaptive mesh refinement (AMR) technique of Berger and collaborators (Adaptive mesh refinement for hyperbolic partial differential equations. Ph.D. Thesis, Computer Science Department, Stanford University, 1982; J. Comput. Phys. 1989; 82 :64–84; 1984; 53 :484–512). Copyright © 2006 John Wiley & Sons, Ltd.