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.     

A high‐resolution scheme for compressible multicomponent flows with shock waves

A simple methodology for a high‐resolution scheme to be applied to compressible multicomponent flows with shock waves is investigated. The method is intended for use with direct numerical simulation or large eddy simulation of compressible multicomponent flows. The method dynamically adds non‐linear artificial diffusivity locally in space to capture different types of discontinuities such as a shock wave, contact surface or material interface while a high‐order compact differencing scheme resolves a broad range of scales in flows. The method is successfully applied to several one‐dimensional and two‐dimensional compressible multicomponent flow problems with shock waves. The results are in good agreement with experiments and earlier computations qualitatively and quantitatively. The method captures unsteady shock and material discontinuities without significant spurious oscillations if initial start‐up errors are properly avoided. Comparisons between the present numerical scheme and high‐order weighted essentially non‐oscillatory (WENO) schemes illustrate the advantage of the present method for resolving a broad range of scales of turbulence while capturing shock waves and material interfaces. Also the present method is expected to require less computational cost than popular high‐order upwind‐biased schemes such as WENO schemes. The mass conservation for each species is satisfied due to the strong conservation form of governing equations employed in the method. Copyright © 2010 John Wiley & Sons, Ltd.     

A high‐resolution hybrid scheme for hyperbolic conservation laws

A new hybrid scheme is proposed, which combines the improved third‐order weighted essentially non‐oscillatory (WENO) scheme presented in this paper with a fourth‐order central scheme by a novel switch. Two major steps have been gone through for the construction of a high‐performance and stable hybrid scheme. Firstly, to enhance the WENO part of the hybrid scheme, a new reference smoothness indicator has been devised, which, combined with the nonlinear weighting procedure of WENO‐Z, can drive the third‐order WENO toward the optimal linear scheme faster. Secondly, to improve the hybridization with the central scheme, a hyperbolic tangent hybridization switch and its efficient polynomial counterpart are devised, with which we are able to fix the threshold value introduced by the hybridization. The new hybrid scheme is thus formulated, and a set of benchmark problems have been tested to verify the performance enhancement. Numerical results demonstrate that the new hybrid scheme achieves excellent performance in resolving complex flow features, even compared with the fifth‐order classical WENO scheme and WENO‐Z scheme. Copyright © 2015 John Wiley & Sons, Ltd.     

Direct numerical simulation of compressible turbulent flows

This paper reviews the authors＇ recent studies on compressible turbulence by using direct numerical simulation （DNS）,including DNS of isotropic（decaying） turbulence, turbulent mixing-layer,turbulent boundary-layer and shock/boundary-layer interaction.Turbulence statistics, compressibility effects,turbulent kinetic energy budget and coherent structures are studied based on the DNS data.The mechanism of sound source in turbulent flows is also analyzed. It shows that DNS is a powerful tool for the mechanistic study of compressible turbulence.     

A new implicit factored scheme for the compressible Navier-Stokes equations

A new scheme for solving the compressible Navier-Stokes equations is developed. For the inviscid portion of the equations the single step scheme used by the authors is factored according to the sign of the eigenvalues of Jacobian matrix. For the viscous portion of the equations a scheme corrected with operator addition is factored too. The scheme obtained has second order accuracy in time and in space and is used to solve two-dimensional problem. The numerical results of 2-D shock wave-boundary layer interaction are compared with experimental data.     

Applications of a higher‐order bounded numerical scheme to turbulent flows

This paper applies the higher‐order bounded numerical scheme Weighted Average Coefficients Ensuring Boundedness (WACEB) to simulate two‐ and three‐dimensional turbulent flows. In the scheme, a weighted average formulation is used for interpolating the variables at cell faces and the weighted average coefficients are determined from a normalized variable formulation and total variation diminishing (TVD) constraints to ensure the boundedness of the solution. The scheme is applied to two turbulent flow problems: (1) two‐dimensional turbulent flow around a blunt plate; and (2) three‐dimensional turbulent flow inside a mildly curved U‐bend. In the present study, turbulence is evaluated by using a low‐Reynolds number version of the k–ω model. For the flow simulation, the QUICK scheme is applied to the momentum equations while either the WACEB scheme (Method 1) or the UPWIND scheme (Method 2) is used for the turbulence equations. The present study shows that the WACEB scheme has at least second‐order accuracy while ensuring boundedness of the solutions. The present numerical study for a pure convection problem shows that the ‘TVD’ slope ranges from 2 to 4. For the turbulent recirculating flow, two different mixed procedures (Method 1 and Method 2) produce a substantial difference for the mean velocities as well as for the turbulence kinetic energy. Method 1 predicts better results than Method 2 does, comparing the analytical solution and the experimental data. For the turbulent flow inside the mildly curved U‐bend, although the predictions of velocity distributions with two procedures are very close, a noticeable difference of turbulence kinetic energy is exhibited. It is noticed that the discrepancy exists between numerical results and the experimental data. The reason is the limit of the two‐equation turbulence model to such complex turbulent flows with extra strain‐rates. Copyright © 2001 John Wiley & Sons, Ltd.     

求解可压缩流的二维通量分裂格式

传统的一维通量分裂格式在计算界面数值通量时,只考虑网格界面法向的波系。采用传统的TV格式分别求解对流通量和压力通量。通过求解考虑了横向波系影响的角点数值通量来构造一种真正二维的TV通量分裂格式。在计算一维数值算例时,该格式与传统的TV格式具有相同的数值通量计算公式,因此其保留了传统的TV格式精确捕捉接触间断和膨胀激波的优点。在计算二维算例时,该格式比传统的TV格式具有更高的分辨率;在计算二维强激波问题时,消除了传统TV格式的非物理现象,表现出更好的鲁棒性;此外,该格式大大提高了稳定性CFL数,从而具有更高的计算效率。因此,本文方法是一种精确、高效并且具有强鲁棒性的数值方法,在可压缩流的数值模拟中具有广阔的应用前景。     

改进的五阶WENO-Z+格式

WENO-ZWENO-Z$+\!$格式的性能提升依赖于新增项的作用,该项的作用是在WENO-Z格式的基础上进一步增大欠光滑子模板上的权重. 系数$\lambda$被设置用来调控该项的作用, 以避免负耗散. 本文指出了WENO-Z$+\!$格式的缺陷,其所采用$\lambda $的取值方式既不能充分发挥格式的潜力, 也未完全消除负耗散;提出$\lambda $的值应随当地流场数据变化,方能充分发挥新增项在降低数值耗散、提高分辨率上的潜力. 基于此,本文重新设计了$\lambda $的计算公式,该公式能自适应地调控新增项的作用: 只在欠光滑子模板上的权重容易过度增大的地方削弱该项的作用,以避免负耗散; 在其他地方则充分发挥新增项的作用,最大限度增大欠光滑子模板上的权重, 提高格式的分辨率.将使用该系数公式的新格式命名为WENO-Z++, 并对其数值性能进行了系统的研究.理论分析表明, 新格式在间断处具有基本无振荡(essentially non-oscillatory,ENO)特性和更低的数值耗散. 对近似色散关系(approximate dispersion relation,ADR)的研究表明,新格式有效地避免了因过度增大欠光滑子模板上的权重而带来的负耗散,其频谱特性也得到了显著提升.本文还推导了使新格式在极值点处也能保持最优阶的精度的参数设置.一系列求解Euler方程的数值试验表明,新格式的激波捕捉能力和对复杂流场结构的分辨率都显著优于原WENO-Z$+\!$格式.}     

A hybrid scheme for the numerical simulation of shock/discontinuity problems

To address accuracy issues for direct numerical simulation, a hybrid scheme based on the weighted compact scheme (WCS) and weighted essentially non-oscillatory (WENO) scheme is developed. The new hybrid method incorporates the advantages of both schemes. Time integration is performed using the fourth-order total variation diminishing Runge–Kutta method with a characteristic filter. The accuracy of the scheme is assessed using several benchmark problems. Results show that the proposed scheme produces a more accurate solution for problems involving shocks and discontinuities in comparison with the traditional shock-capturing methods.     

Robustness of the hybrid DRP‐WENO scheme for shock flow computations

This paper describes a new variant of hybrid scheme that is constructed by a wave‐capturing scheme and a nonoscillatory scheme for flow computations in the presence of shocks. The improved fifth‐order upwind weighted essentially nonoscillatory scheme is chosen to be conjugated with the seven‐point dispersion‐relation‐preserving scheme by means of an adaptive switch function of grid‐point type. The new hybrid scheme can achieve a better resolution than the hybrid scheme which is based on the classical weighted essentially scheme. Ami Harten's multiresolution analysis algorithm is applied to density field for detecting discontinuities and setting point values of the switch function adaptively. Moreover, the tenth‐order central filter is applied in smooth part of the flow field for damping dispersion errors. This scheme can promote overall computational efficiency and yield oscillation‐free results in shock flows. The resolution properties and robustness of the new hybrid scheme are tested in both 1D and 2D linear and nonlinear cases. It performs well for computing flow problems with rich structures of weak/strong shocks and large/small vortices, such as the shock‐boundary layer interaction problem in a shock tube, which illustrates that it is very robust and accurate for direct numerical simulation of gas‐dynamics flows. Copyright © 2011 John Wiley & Sons, Ltd.     

A finite volume scheme for two-phase compressible flows

In gas-particle two-phase flows, when the concentration of the disperesed phase is low, certain assumptions may be made which simplify considerably the equations one has to solve. The gas and particle flows are then linked only via the interaction terms. One may therefore uncouple the full system of equations into two subsystems: one for the gas phase, whose homogeneous part reduces to the Euler equations; and a second system for the particle motion, whose homogeneous part is a degenerate hyperbolic system. The equations governing the gas phase flow may be solved using a high-resolution scheme, while the equations describing the motion of the dispersed phase are treated by a donor-cell method using the solution of a particular Riemann problem. Coupling is then achieved via the right-hand-side terms. To illustrate the capabilities of the proposed method, results are presented for a case specially chosen from among the most difficult to handle, since it involves certain geometrical difficulties, the treatment of regions in which particles are absent and the capturing of particle fronts.     

基于特征分裂改进的WENO格式与Roe格式对比研究

双曲性守恒方程组采用高阶、高分辨率的WENO格式时有两类分裂方法,即逐点分裂和特征分裂。本文基于后者,对特征分裂重构时强间断和接触间断位置出现的振荡情况进行研究,对重构变量加以改进,发现改进后的WENO格式克服了间断处的振荡,然后以LU-SGS为子迭代的双时间步法求解Euler方程,选用一维Sod、二维前台阶和双马赫反射算例,并与Roe格式计算结果进行对比,发现WENO格式分辨率更高,耗散更小。     

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。     

Compact finite difference schemes on non‐uniform meshes. Application to direct numerical simulations of compressible flows

In this paper, the development of a fourth‐ (respectively third‐) order compact scheme for the approximation of first (respectively second) derivatives on non‐uniform meshes is studied. A full inclusion of metrics in the coefficients of the compact scheme is proposed, instead of methods using Jacobian transformation. In the second part, an analysis of the numerical scheme is presented. A numerical analysis of truncation errors, a Fourier analysis completed by stability calculations in terms of both semi‐ and fully discrete eigenvalue problems are presented. In those eigenvalue problems, the pure convection equation for the first derivative, and the pure diffusion equation for the second derivative are considered. The last part of this paper is dedicated to an application of the numerical method to the simulation of a compressible flow requiring variable mesh size: the direct numerical simulation of compressible turbulent channel flow. Present results are compared with both experimental and other numerical (DNS) data in the literature. The effects of compressibility and acoustic waves on the turbulent flow structure are discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

一种提高极值点处收敛精度的三阶WENO-Z格式

为了提高三阶WENO-Z格式在极值点处的计算精度,通过理论推导给出三阶WENO格式满足收敛精度的充分条件。采用泰勒级数展开的方式,推导给出所构造格式非线性权重的计算公式,并综合权衡计算精度和计算稳定性确定所构造格式的参数。通过两个典型的精度测试,验证了改进格式在光滑流场极值点区域逼近三阶精度。进一步选用激波与熵波相互作用和Richtmyer-Meshkov不稳定性等经典算例,证实了本文提出的改进格式WENO-PZ3相较其他格式（WENO-JS3和WENO-Z3）不仅具有较高的精度,而且降低了格式的耗散,提高了对流场结构的分辨率。     

Numerical simulation of compressible turbulent flow via improved gas‐kinetic BGK scheme

In order to solve compressible turbulent flow problems, this study focuses on incorporating the Spalart–Allmaras turbulence model into gas‐kinetic BGK (Bhatnagar–Gross–Krook) scheme. The Spalart–Allmaras turbulence model is solved using finite difference discretization. The variables on the cell interface are interpolated via the van Leer limiter in the reconstruction stage. Simulation of subsonic and transonic flow over a NACA0012 airfoil has been implemented using two‐dimensional body‐fitted grids. The numerical results obtained appear in good agreement with the AGARD results, demonstrating the effectiveness and usefulness of the strategy of coupling the Spalart–Allmaras turbulence model with the BGK scheme for compressible turbulent flow simulation. Copyright © 2010 John Wiley & Sons, Ltd.     

An analytical skin friction and heat transfer model for compressible, turbulent, internal flows

An analytical skin friction model for compressible, turbulent, internal, fully developed flow involving adiabatic and non-adiabatic, smooth and rough flows has been developed by extending the incompressible law-of-the-wall relation to compressible cases. The formula recovers Prandtl's incompressible law of friction for pipes (within 2%) for incompressible flow. The model also shows good correlation with available data for compressible, adiabatic flows and flows involving cold wall heat transfer (within 15%). Comparison with hot wall data is only moderate (15–30%). Finally, using Reynold's analogy, the Stanton number and Nusselt numbers may be estimated.     

A weighted least square scheme for compressible flows

The article describes the development of a high order finite volume method for the solution of transonic flow problems. The method is based on a reconstruction procedure similar to the weighted essentially non-oscillatory scheme (WENO). The analysis of accuracy and stability of the method is carried out for the case of smooth data and for simple discontinuity. The computational results demonstrate the performance of the WLSQR method for the solution of several flow problems in 2D and 3D using both structured and unstructured meshes.     

An implicit mixed finite-volume–finite-element method for solving 3D turbulent compressible flows

The development of new aeronautic projects require accurate and efficient simulations of compressible flows in complex geometries. It is well known that most flows of interest are at least locally turbulent and that the modelling of this turbulence is critical for the reliability of the computations. A turbulence closure model which is both cheap and reasonably accurate is an essential part of a compressible code. An implicit algorithm to solve the 2D and 3D compressible Navier–Stokes equations on unstructured triangular/tetrahedral grids has been extended to turbulent flows. This numerical scheme is based on second-order finite element–finite volume discretization: the diffusive and source terms of the Navier–Stokes equations are computed using a finite element method, while the other terms are computed with a finite volume method. Finite volume cells are built around each node by means of the medians. The convective fluxes are evaluated with the approximate Riemann solver of Roe coupled with the van Albada limiter. The standard k–ϵ model has been introduced to take into account turbulence. Implicit integration schemes with efficient numerical methods (CGS, GMRES and various preconditioning techniques) have also been implemented. Our interest is to present the whole method and to demonstrate its limitations on some well-known test cases in three-dimensional geometries. © 1997 John Wiley & Sons, Ltd.     

A new smoothness indicator for third‐order WENO scheme

We present a new reference smoothness indicator for third‐order weighted essentially non‐oscillatory scheme to recover its design‐order convergence at critical points. This reference smoothness indicator, which involves both the candidate and global smoothness indicators in the weighted essentially non‐oscillatory framework, is devised according to a sufficient condition on the weights for third‐order convergence. The recovery of design‐order is verified by standard tests. Meanwhile, numerical results demonstrate that the present reference smoothness indicator produces sharper representation of the discontinuity owing to the combined effects of larger weight assignment to the discontinuous stencils and convergence rate recovery. Copyright © 2015 John Wiley & Sons, Ltd.