A segregated method for compressible flow computation Part I: isothermal compressible flows

Traditionally, coupled methods have been employed for the computation of compressible flows, whereas segregated methods have been preferred for the computation of incompressible flows. Compared to coupled methods, segregated solvers present the advantage of reduced computer memory and CPU time requirements, although at the cost of an inferior robustness. Therefore, in a series of papers we present unified computational techniques to compute compressible and incompressible flows with segregated stabilized methods. The proposed algorithms have an increased robustness compared to existing techniques, while possessing additional benefits such as employing standard pressure boundary conditions. In this first part, the thermodynamics of isothermal, thermally perfect compressible flows is set up in the framework of symmetric systems and the corresponding segregated algorithms are introduced. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite element method for compressible viscous fluid flows

This work presents a mixed three‐dimensional finite element formulation for analyzing compressible viscous flows. The formulation is based on the primitive variables velocity, density, temperature and pressure. The goal of this work is to present a ‘stable’ numerical formulation, and, thus, the interpolation functions for the field variables are chosen so as to satisfy the inf–sup conditions. An exact tangent stiffness matrix is derived for the formulation, which ensures a quadratic rate of convergence. The good performance of the proposed strategy is shown in a number of steady‐state and transient problems where compressibility effects are important such as high Mach number flows, natural convection, Riemann problems, etc., and also on problems where the fluid can be treated as almost incompressible. Copyright © 2010 John Wiley & Sons, Ltd.     

Spectral-finite element method for compressible fluid flow

In this paper, a combined Fourier spectral-finite element method is proposed for solving n-dimensional (n=2, 3), semi-periodio compressible fiuid flow problems. The strict error estimation as well as the convergence rate, is presented.     

Stabilized finite‐element computation of compressible flow with linear and quadratic interpolation functions

The unsteady compressible flow equations are solved using a stabilized finite‐element formulation with C⁰ elements. In 2D, the performance of three‐noded linear and six‐noded quadratic triangular elements is compared. In 3D, the relative performance is evaluated for 6‐noded linear and 18‐noded quadratic wedge elements. Results are compared for the solutions to Euler, laminar, and turbulent flows at different Mach numbers for several flow problems. The finite‐element meshes considered for comparison have same location of nodes for the linear and quadratic interpolations. For the turbulent flow, the Spalart–Allmaras model is used for closure. It is found that the quadratic elements yield better performance than the linear elements. This is attributed to accurate representation of the stabilization terms that involve second‐order derivatives in the formulation. When these terms are dropped from the formulation with quadratic interpolation, the numerical results are similar to those obtained with linear interpolation. The absence of these terms result in added numerical diffusion that seems to give the effect of a relatively reduced Reynolds number. For the same location of nodes, the computations with the linear triangular and wedge elements are approximately 20% and 100% faster than those with quadratic triangular and wedge elements, respectively. However, if the same quadrature rule for numerical integration is used for both interpolations, the computations with quadratic elements are approximately 20% and 45% faster in 2D and 3D, respectively. Copyright © 2014 John Wiley & Sons, Ltd.     

A stabilized finite element formulation for high‐speed inviscid compressible flows using finite calculus

This paper aims at the development of a new stabilization formulation based on the finite calculus (FIC) scheme for solving the Euler equations using the Galerkin FEM on unstructured triangular grids. The FIC method is based on expressing the balance of fluxes in a space–time domain of finite size. It is used to prevent the creation of instabilities typically present in numerical solutions due to the high convective terms and sharp gradients. Two stabilization terms, respectively called streamline term and transverse term, are added via the FIC formulation to the original conservative equations in the space–time domain. An explicit fourth‐order Runge–Kutta scheme is implemented to advance the solution in time. The presented numerical test examples for inviscid flows prove the ability of the proposed stabilization technique for providing appropriate solutions especially near shock waves. Although the derived methodology delivers precise results with a nearly coarse mesh, a mesh refinement technique is coupled to the solution process for obtaining a suitable mesh particularly in the high‐gradient zones. Copyright © 2014 John Wiley & Sons, Ltd.     

A finite point method for adaptive three‐dimensional compressible flow calculations

The finite point method (FPM) is a meshless technique, which is based on both, a weighted least‐squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a broader investigation into the capabilities of the FPM to deal with 3D applications concerning real compressible fluid flow problems. In the first part of this work, the upwind‐biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting the meshless capabilities, an h‐adaptive methodology for 2D and 3D compressible flow calculations is developed. This adaptive technique applies a solution‐based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are encouraging. Several numerical examples are provided in order to illustrate the good performance of the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd.     

可压缩流体流动多尺度混合有限元数值方法研究

可压缩流体是天然油藏中广泛存在的一种流体,研究其在多孔介质中的渗流规律对于油藏开发具有重要意义。本文采用多尺度混合有限元方法,对可压缩流体渗流问题进行了研究。考虑流体的可压缩性以及介质形变,推导得到了可压缩流体渗流问题的多尺度计算格式。数值计算结果表明,多尺度混合有限元适于求解非均质性和可压缩流问题,具有节省计算量、计算精度高等优势,对于实际大规模油藏模拟具有重要意义。     

A finite volume method for two-dimensional transonic potential flow through turbomachinery blade rows

To predict inviscid transonic flow through turbomachinery blade rows, the exact transonic potential flow equation is solved on a mesh constructed from small area elements. A transformation is introduced through which distorted squares of the physical plane are mapped into computational squares. Two sets of overlapping elements are used; while the thermodynamic properties are calculated at the primary element centres, the flux balance is established on the secondary elements. For transonic flows an artificial compressibility term (upwind density gradient) is added to density in order to produce the desired directional bias in the hyperbolic region. while the entropy does not increase across mass conservative shock jump regions. Comparisons withexperiments and with other numerical and analytical solutions for various turbomachinery configurations show that this approach is comparatively accurate, reliable, and fast.     

Simplex space-time meshes in compressible flow simulations

Employing simplex space-time meshes enlarges the scope of compressible flow simulations. The simultaneous discretization of space and time with simplex elements extends the flexibility of unstructured meshes from space to time. In this work, we adapt a finite element formulation for compressible flows to simplex space-time meshes. The method obtained allows, for example, flow simulation on spatial domains that change topology with time. We demonstrate this with the two-dimensional simulation of compressible flow in a valve that fully closes and opens again. Furthermore, simplex space-time meshes facilitate local temporal refinement. A three-dimensional transient simulation of blow-by past piston rings is run in parallel on 120 cores. The timings point out savings of computation time gained from local temporal refinement in four-dimensional space-time meshes.     

A CWENO ghost fluid method for compressible multimaterial flow

In this work a new ghost fluid method (GFM) is introduced for multimaterial compressible flow with arbitrary equation of states. In previous researches, it has been shown that accurate wave decomposition at the interface by solving a Riemann problem alleviates the shortcomings of the standard GFM in dealing with the impingement of strong waves onto the interface but these Riemann‐based GFM are not consistent with the framework of the central WENO scheme in which the emphasis is to avoid solving Riemann problems at control volume faces and enjoy the black box property (being independent of equation of state). The aim of this work is to develop a new GFM that is completely consistent with the methodology behind central schemes; that is, it enjoys a black box property. The capabilities of the proposed GFM method is shown by solving various types of multimaterial compressible flows including gas–gas, gas–water and fluid–solid interfaces interacting with strong shock waves in one and two space dimensions. Copyright © 2013 John Wiley & Sons, Ltd.     

A lattice Boltzmann model for simulation of compressible flows

This paper presents a new model of lattice Boltzmann method for full compressible flows. On the basis of multi‐speed model, an extra potential energy distribution function is introduced to recover the full compressible Navier–Stokes equations with a flexible specific‐heat ratio and Prandtl number. The Chapman–Enskog expansion of the kinetic equations is performed, and the two‐dimension‐seventeen‐velocity density equilibrium distribution functions are obtained. The governing equations are discretized using the third order monotone upwind scheme for scalar conservation laws finite volume scheme. The van Albada limiter is used to avoid spurious oscillations. In order to verify the accuracy of this double‐distribution‐function model, the Riemann problems, Couette flows, and flows around a NACA0012 airfoil are simulated. It is found that the proposed lattice Boltzmann model is suitable for compressible flows, even for strong shock wave problem, which has an extremely large pressure ratio, 100,000. Copyright © 2014 John Wiley & Sons, Ltd.     

A hybrid building‐block and gridless method for compressible flows

A hybrid building‐block Cartesian grid and gridless method is presented to compute unsteady compressible flows for complex geometries. In this method, a Cartesian mesh based on a building‐block grid is used as a baseline mesh to cover the computational domain, while the boundary surfaces are represented using a set of gridless points. This hybrid method combines the efficiency of a Cartesian grid method and the flexibility of a gridless method for the complex geometries. The developed method is used to compute a number of test cases to validate the accuracy and efficiency of the method. The numerical results obtained indicate that the use of this hybrid method leads to a significant improvement in performance over its unstructured grid counterpart for the time‐accurate solution of the compressible Euler equations. An overall speed‐up factor from six to more than one order of magnitude and a saving in storage requirements up to one order of magnitude for all test cases in comparison with the unstructured grid method are demonstrated. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite volume method for three-dimensional transonic potential flow through turbomachinery blade rows

A finite-volume method has been developed for the calculation of transonic, potential flows through 3-D turbomachinery blades with complex geometries. The exact transonic potential flow equation is solved on a mesh constructed from small volume elements. A transformation is introduced through which cuboids of the physical plane are mapped into computational cubes. Two sets of overlapping volumes are used. While the thermodynamic properties are calculated at the primary volume centres, the flux balance is established on the secondary volumes. For transonic flows an artificial compressibility term (upwind density gradient) is added to density to produce the necessary directional bias in the hyperbolic region. The successive point over-relaxation Gauss-Seidel method has been used to solve the non-linear partial differential equations. Comparisons with experiments and/or other numerical solutions for various turbomachinery configurations show that the 3-D finite-volume approach is a relatively accurate, reliable and fast method for inviscid, transonic flow predictions through turbomachinery blade rows     

可压流数值模拟中当地DFD方法的改进和应用

阐述了求解守恒型Euler 方程的当地DFD (Domain-Free Discretization) 方法的改进和应用。DFD 离散策略的核心,是解域内点上控制方程的离散形式可与解域外的一些点相关。通过边界附近解的近似形式,外部相关点上的流动变量值得到确定并强加相应的边界条件。与最初的当地DFD方法不同,在解的近似形式构建中,采用了CCST技术 (Curvature-Corrected Symmetry Technique),因此外部相关点上的密度和切向速度分别由等熵和等总焓关系确定。空间离散采用Galerkin 有限体积格式。最后,给出了固定和运动物体可压缩绕流的数值模拟结果,以验证改进的当地DFD方法的可靠性和数值解精度的提高。     

Evaluation of adaptive mesh refinement and coarsening for the computation of compressible flows on unstructured meshes

The compressible gas flows of interest to aerospace applications often involve situations where shock and expansion waves are present. Decreasing the characteristic dimension of the computational cells in the vicinity of shock waves improves the quality of the computed flows. This reduction in size may be accomplished by the use of mesh adaption procedures. In this paper an analysis is presented of an adaptive mesh scheme developed for an unstructured mesh finite volume upwind computer code. This scheme is tailored to refine or coarsen the computational mesh where gradients of the flow properties are respectively high or low. The refinement and coarsening procedures are applied to the classical gas dynamic problems of the stabilization of shock waves by solid bodies. In particular, situations where oblique shock waves interact with an expansion fan and where bow shocks arise around solid bodies are considered. The effectiveness of the scheme in reducing the computational time, while increasing the solution accuracy, is assessed. It is shown that the refinement procedure alone leads to a number of computational cells which is 20% larger than when alternate passes of refinement and coarsening are used. Accordingly, a reduction of computational time of the same order of magnitude is obtained. Copyright © 2005 John Wiley & Sons, Ltd.     

Multidomain collocation techniques for a viscous compressible flow

This paper presents a viscous compressible flow problem to which an equilibrium solution, in terms of density and velocity, can be given implicitly by elementary functions. The corresponding initial boundary value problem is solved by time discretization by the Crank-Nicolson method, Newton linearization and space discretization using multidomain Chebyshev collocation techniques. The physical interval is covered by subintervals of equal length. Each subinterval utilizes the same number of collocation points and each interface consists of one or two points. Six ways of patching are tested. All of them yield solutions with spectral accuracy for a few time steps, but only three are stable in the long run. Details of the density evolution are illustrated.     

A ghost-cell immersed boundary method for large-eddy simulations of compressible turbulent flows

A methodology to perform a ghost-cell-based immersed boundary method (GCIBM) is presented for simulating compressible turbulent flows around complex geometries. In this method, the boundary condition on the immersed boundary is enforced through the use of ‘ghost cells’ that are located inside the solid body. The computations of variables on these ghost cells are achieved using linear interpolation schemes. The validity and applicability of the proposed method is verified using a three-dimensional (3D) flow over a circular cylinder, and a large-eddy simulation of fully developed 3D turbulent flow in a channel with a wavy surface. The results agree well with the previous numerical and experimental results, given that the grid resolution is reasonably fine. To demonstrate the capability of the method for higher Mach numbers, supersonic turbulent flow over a circular cylinder is presented. While more work still needs to be done to demonstrate higher robustness and accuracy, the present work provides interesting insights using the GCIBM for the compressible flows.     

High-order ghost-point immersed boundary method for viscous compressible flows based on summation-by-parts operators

A high-order immersed boundary method is devised for the compressible Navier-Stokes equations by employing high-order summation-by-parts difference operators. The immersed boundaries are treated as sharp interfaces by enforcing the solid wall boundary conditions via flow variables at ghost points. Two different interpolation schemes are tested to compute values at the ghost points and a hybrid treatment is used. The first method provides the bilinearly interpolated flow variables at the image points of the corresponding ghost points and the second method applies the boundary condition at the immersed boundary by using the weighted least squares method with high-order polynomials. The approach is verified and validated for compressible flow past a circular cylinder at moderate Reynolds numbers. The tonal sound generated by vortex shedding from a circular cylinder is also investigated. In order to demonstrate the capability of the solver to handle complex geometries in practical cases, flow in a cross-section of a human upper airway is simulated.     

Jump-reduced immersed boundary method for compressible flow

The main challenge of the immersed boundary approach is the proper enforcement of boundary conditions on the body interface without any spurious oscillations, which are induced by the nongrid-conforming boundary configuration. In this study, a new sharp interface ghost-cell immersed boundary method (IBM) is proposed for obtaining solutions near the immersed boundary with a high order of accuracy. The main idea is “jump-reduction” instead of jump-correction across the boundary interface by combining the ghost-cell method with the flow reconstruction method. In the proposed IBM, the unknown values at the three points, that is, boundary points, ghost cell, and flow field reconstruction point are solved simultaneously using equations formulated by the moving least-squares interpolation method. It is a hybrid of ghost-cell and flow reconstruction methods, correlated with interface values, which result in a reduced jump-discontinuity. In addition, a discontinuity-distinguishing algorithm is introduced so that the low-order method is applied only to the discontinuous or non smooth region, while the current high-order method is applied elsewhere. Reduced jump-discontinuity of the proposed IBM has been verified in both subsonic and supersonic flow using fundamental benchmark problems. We observed that the reduced jump-discontinuity does not hamper the mass conservation and shows even better conservation property than conventional methods due to the nonoscillatory performance in smooth regions. The numerical results further confirm the ability of the proposed IBM to solve complex flow physics with high-order accuracy and improved stability.