An efficient algorithm for compressible flows with real gases

An efficient algorithm is presented for the solution of the Euler equations of gas dynamics with a general convex equation of state. The scheme is based on solving linearized Riemann problems approximately, and in more than one dimension incorporates operator splitting. In particular, only one function evaluation in each computational cell is required by using a local parametrization of the equation of state. The scheme is applied to two standard test problems in gas dynamics for some specimen equations of state.     

Numerical solutions of Euler equations by using a new flux vector splitting scheme

A new flux vector splitting scheme has been suggested in this paper. This scheme uses the velocity component normal to the volume interface as the characteristic speed and yields the vanishing individual mass flux at the stagnation. The numerical dissipation for the mass and momentum equations also vanishes with the Mach number approaching zero. One of the diffusive terms of the energy equation does not vanish. But the low numerical diffusion for viscous flows may be ensured by using higher-order differencing. The scheme is very simple and easy to be implemented. The scheme has been applied to solve the one dimensional (1D) and multidimensional Euler equations. The solutions are monotone and the normal shock wave profiles are crisp. For a 1D shock tube problem with the shock and the contact discontinuities, the present scheme and Roe scheme give very similar results, which are the best compared with those from Van Leer scheme and Liou–Steffen's advection upstream splitting method (AUSM) scheme. For the multidimensional transonic flows, the sharp monotone normal shock wave profiles with mostly one transition zone are obtained. The results are compared with those from Van Leer scheme, AUSM and also with the experiment.     

An efficient upwind/relaxation algorithm for the Euler and Navier–Stokes equations

An efficient Euler and full Navier–Stokes solver based on a flux splitting scheme is presented. The original Van Leer flux vector splitting form is extended to arbitrary body-fitted co-ordinates in the physical domain so that it can be used with a finite volume scheme. The block matrix is inverted by Gauss–Seidel iteration. It is verified that the often used reflection boundary condition will produce incorrect flux crossing the wall and cause too large numerical dissipation if flux vector splitting is used. To remove such errors, an appropriate treatment of wall boundary conditions is suggested. Inviscid and viscous steady transonic internal flows are analysed, including the case of shock-induced boundary layer separation.     

An efficient implicit mesh‐less method for compressible flow calculations

A dual‐time implicit mesh‐less scheme is presented for calculation of compressible inviscid flow equations. The Taylor series least‐square method is used for approximation of spatial derivatives at each node which leads to a central difference discretization. Several convergence acceleration techniques such as local time stepping, enthalpy damping and residual smoothing are adopted in this approach. The capabilities of the method are demonstrated by flow computations around single and multi‐element airfoils at subsonic, transonic and supersonic flow conditions. Results are presented which indicate good agreements with other reliable finite‐volume results. The computational time is considerably reduced when using the proposed mesh‐less method compared with the explicit mesh‐less and finite‐volume schemes using the same point distributions. Copyright © 2010 John Wiley & Sons, Ltd.     

Non-oscillatory shock-capturing finite element methods for the one-dimensional compressible Euler equations

A class of shock-capturing Petrov–Galerkin finite element methods that use high-order non-oscillatory interpolations is presented for the one-dimensional compressible Euler equations. Modified eigenvalues which employ total variation diminishing (TVD), total variation bounded (TVB) and essentially non-oscillatory (ENO) mechanisms are introduced into the weighting functions. A one-pass Euler explicit transient algorithm with lumped mass matrix is used to integrate the equations. Numerical experiments with Burgers' equation, the Riemann problem and the two-blast-wave interaction problem are presented. Results indicate that accurate solutions in smooth regions and sharp and non-oscillatory solutions at discontinuities are obtainable even for strong shocks.     

Kinetic meshless method for compressible flows

We present a grid‐free or meshless approximation called the kinetic meshless method (KMM), for the numerical solution of hyperbolic conservation laws that can be obtained by taking moments of a Boltzmann‐type transport equation. The meshless formulation requires the domain discretization to have very little topological information; a distribution of points in the domain together with local connectivity information is sufficient. For each node, the connectivity consists of a set of nearby nodes which are used to evaluate the spatial derivatives appearing in the conservation law. The derivatives are obtained using a modified form of the least‐squares approximation. The method is applied to the Euler equations for inviscid flow and results are presented for some 2‐D problems. The ability of the new scheme to accurately compute inviscid flows is clearly demonstrated, including good shock capturing ability. Comparisons with other grid‐free methods are made showing some advantages of the current approach. Copyright © 2007 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。     

Numbering techniques for preconditioners in iterative solvers for compressible flows

We consider Newton–Krylov methods for solving discretized compressible Euler equations. A good preconditioner in the Krylov subspace method is crucial for the efficiency of the solver. In this paper we consider a point‐block Gauss–Seidel method as preconditioner. We describe and compare renumbering strategies that aim at improving the quality of this preconditioner. A variant of reordering methods known from multigrid for convection‐dominated elliptic problems is introduced. This reordering algorithm is essentially black‐box and significantly improves the robustness and efficiency of the point‐block Gauss–Seidel preconditioner. Results of numerical experiments using the QUADFLOW solver and the PETSc library are given. Copyright © 2007 John Wiley & Sons, Ltd.     

An immersed boundary method for simulation of inviscid compressible flows

In this paper, an immersed boundary method for simulating inviscid compressible flows governed by Euler equations is presented. All the mesh points are classified as interior computed points, immersed boundary points (interior points closest to the solid boundary), and exterior points that are blanked out of computation. The flow variables at an immersed boundary point are determined via the approximate form of solution in the direction normal to the wall boundary. The normal velocity is evaluated by applying the no‐penetration boundary condition, and therefore, the influence of solid wall in the inviscid flow is taken into account. The pressure is computed with the local simplified momentum equation, and the density and the tangential velocity are evaluated by using the constant‐entropy relation and the constant‐total‐enthalpy relation, respectively. With a local coordinate system, the present method has been extended easily to the three‐dimensional case. The present work is the first endeavor to extend the idea of hybrid Cartesian/immersed boundary approach to compressible inviscid flows. The tedious task of handling multi‐valued points can be eliminated, and the overshoot resulting from the extrapolation for the evaluation of flow variables at exterior points can also be avoided. In order to validate the present method, inviscid compressible flows over fixed and moving bodies have been simulated. All the obtained numerical results show good agreement with available data in the literature. Copyright © 2014 John Wiley & Sons, Ltd.     

An efficient algorithm for hypersonic viscous flows

The CSCM-S algorithm proposed by Lombard et al. is a very attractive tool for solving multidimensional Euler and Navier-Stokes equations. However, it is not economical due to the use of global sweeps in the whole computational domain. In this paper we suggest a modified strategy, which combines a single-marching technique for supersonic dominated region with a multi-sweep procedure for pure subsonic and complex flowfield. The new algorithm may save significantly CPU time and is more suitable for engineering applications. The project supported by National Natural Science Foundation of China     

Level Set方法和多介质可压缩流

多介质可压缩流问题计算的关键是如何精确的捕获不同时刻物质界面的位置,从而将多介质问题分解成多个单介质问题去处理.Level Set方法的优点是不用显示的追踪物质界面,而用距离函数就能精确定位界面.同时,用Level Set方法追踪界面运动易于处理界面拓扑结构的变化、易于处理大变形问题.本文成功地将Level Set方法应用在二维多介质可压缩流计算.     

An agglomeration‐based adaptive discontinuous Galerkin method for compressible flows

In this work, we exploit the possibility to devise discontinuous Galerkin discretizations over polytopic grids to perform grid adaptation strategies on the basis of agglomeration coarsening of a fine grid obtained via standard unstructured mesh generators. The adaptive agglomeration process is here driven by an adjoint‐based error estimator. We investigate several strategies for converting the error field estimated solving the adjoint problem into an agglomeration factor field that is an indication of the number of elements of the fine grid that should be clustered together to form an agglomerated element. As a result the size of agglomerated elements is optimized for the achievement of the best accuracy for given grid size with respect to the target quantities. To demonstrate the potential of this strategy we consider problem‐specific outputs of interest typical of aerodynamics, eg, the lift and drag coefficients in the context of inviscid and viscous flows test cases.     

A new,high-resolution shock-capturing hybrid scheme of flux vector splitting-Harten's TVD

The paper presents a new high-resolution hybrid scheme combining implicit flux vector splitting with Harten's TVD, which is proved suitable for shock-capturing calculation in gasdynamics. Fluxsplitting procedures are applied to discretize the implicit part of the Euler equations whereas Harten's numerical fluxes are used to calculate the residual of steady-state solutions. It ensures good shock-capturing properties and produces sharp numerical discontinuities without oscillations. It excludes expansion shocks and leads only to physically relevant solutions. The block-line-Gauss-Seidel relaxation procedure (block-LGS) is used to solve the resulting difference equations. The time step and the CFL number are much larger than those in the linearized block-alternating-direction-implicit approximate factorization method (block-ADI). Numerical experiments suggest that the hybrid scheme not only has a fairly rapid convergence rate, but also can generate a highly resolved approximation to the steady-state solution. Hence scheme seems to lead to an effective nonoscillatory shock capturing method for steady transonic flow. Project Supported by National Natural Science Foundation of China     

Simulation of thermoacoustic waves by a pressure-based algorithm for compressible flows

A modified SIMPLEC method which can solve compressible flows at low Mach number is introduced and used to study thermoacoustic waves induced by a rapid change of temperature at a solid wall and alternatingdirection flows generated by thermoacoustic effects in a tapered resonator.The results indicate that the algorithm adopted in this paper can be used for calculating compressible flows and thermoacoustic waves.It is found that the pressure and velocity in the resonator behave as standing waves,and the tapered resonator can suppress highfrequency harmonic waves as observed in a cylindrical resonator.     

An immersed boundary solver for inviscid compressible flows

In this paper, a simple and efficient immersed boundary (IB) method is developed for the numerical simulation of inviscid compressible Euler equations. We propose a method based on coordinate transformation to calculate the unknowns of ghost points. In the present study, the body‐grid intercept points are used to build a complete bilinear (2‐D)/trilinear (3‐D) interpolation. A third‐order weighted essentially nonoscillation scheme with a new reference smoothness indicator is proposed to improve the accuracy at the extrema and discontinuity region. The dynamic blocked structured adaptive mesh is used to enhance the computational efficiency. The parallel computation with loading balance is applied to save the computational cost for 3‐D problems. Numerical tests show that the present method has second‐order overall spatial accuracy. The double Mach reflection test indicates that the present IB method gives almost identical solution as that of the boundary‐fitted method. The accuracy of the solver is further validated by subsonic and transonic flow past NACA2012 airfoil. Finally, the present IB method with adaptive mesh is validated by simulation of transonic flow past 3‐D ONERA M6 Wing. Global agreement with experimental and other numerical results are obtained.     

A method for computing compressible,highly stratified flows in astrophysics based on operator splitting

A new numerical approach based on consistent operator splitting is presented for computing compressible, highly stratified flows in astrophysics. The algorithm is particularly designed to search for steady or almost steady solutions for the time-dependent Navier–Stokes equations, describing viscous flow under the influence of a strong gravitational field. The algorithm proposed is multidimensional and works in Cartesian, cylindrical or spherical co-ordinates. It uses a second-order finite volume scheme with third-order upwinding and a second-order time discretization. An adaptive time step control and monotonic multilevel grid distribution has been incorporated to speed up convergence. This method has been incorporated into a hydrodynamical code by which, for the first time, for two-dimensional models the dynamics of the boundary layer in the accretion disk around a compact star could be computed over the whole viscous time scale. © 1998 John Wiley & Sons, Ltd.     

Modelling far-field entrainment in compressible flows

A procedure that generates steady-states with accurate far-field entrainment is described here. It can be applied to any existing code originally designed for unsteady simulations. Such steady-states are obtained with physical-time damping, introduced through a dual time stepping methodology to effectively eliminate all transient fluctuations without affecting spatial resolution. Far-field entrainment is guaranteed by using properly selected local boundary conditions. Hence, reference profiles with accurate far-field entrainment are generated from the very same code that will eventually use them in unsteady simulations. This new procedure is tested on an unsteady code designed for compressible planar mixing-layer simulations at arbitrary Mach numbers. It is numerically stable for a wide range of Mach numbers and velocity ratios.     

Approximation of shallow water equations by Roe's Riemann solver

The inviscid shallow water equations provide a genuinely hyperbolic system and all the numerical tools that have been developed for a system of conservation laws can be applied to them. However, this system of equations presents some peculiarities that can be exploited when developing a numerical method based on Roe's Riemann solver and enhanced by a slope limiting of MUSCL type. In the present paper a TVD version of the Lax-Wendroff scheme is used and its performance is shown in 1D and 2D computations. Then two specific difficulties that arise in this context are investigated. The former is the capability of this class of schemes to handle geometric source terms that arise to model the bottom variation. The latter analysis pertains to situations in which strict hyperbolicity is lost, i.e. when two eigenvalues collapse into one.     

An extension of the TV-HLL scheme for multi-dimensional compressible flows

This paper investigates a very simple method to numerically approximate the solution of the multi-dimensional Riemann problem for gas dynamics, using the literal extension of the Toro Vazquez-Harten Lax Leer (TV-HLL) scheme as its basis. Indeed, the present scheme is obtained by following the Toro–Vazquez splitting, and using the HLL algorithm with modified wave speeds for the pressure system. An essential feature of the TV-HLL scheme is its simplicity and its accuracy in computing multi-dimensional flows. The proposed scheme is carefully designed to simplify its eventual numerical implementation. It has been applied to numerical tests and its performances are demonstrated for some two-dimensional and three-dimensional test problems.     

Optimal control of unsteady compressible viscous flows

The control of complex, unsteady flows is a pacing technology for advances in fluid mechanics. Recently, optimal control theory has become popular as a means of predicting best case controls that can guide the design of practical flow control systems. However, most of the prior work in this area has focused on incompressible flow which precludes many of the important physical flow phenomena that must be controlled in practice including the coupling of fluid dynamics, acoustics, and heat transfer. This paper presents the formulation and numerical solution of a class of optimal boundary control problems governed by the unsteady two‐dimensional compressible Navier–Stokes equations. Fundamental issues including the choice of the control space and the associated regularization term in the objective function, as well as issues in the gradient computation via the adjoint equation method are discussed. Numerical results are presented for a model problem consisting of two counter‐rotating viscous vortices above an infinite wall which, due to the self‐induced velocity field, propagate downward and interact with the wall. The wall boundary control is the temporal and spatial distribution of wall‐normal velocity. Optimal controls for objective functions that target kinetic energy, heat transfer, and wall shear stress are presented along with the influence of control regularization for each case. Copyright © 2002 John Wiley & Sons, Ltd.