Efficient computation of compressible and incompressible flows

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Abstract

The combination of explicit Runge–Kutta time integration with the solution of an implicit system of equations, which in earlier work demonstrated increased efficiency in computing compressible flow on highly stretched meshes, is extended toward conditions where the free stream Mach number approaches zero. Expressing the inviscid flux Jacobians in terms of Mach number, an artificial speed of sound as in low Mach number preconditioning is introduced into the Jacobians, leading to a consistent formulation of the implicit and explicit parts of the discrete equations. Besides extension to low Mach number flows, the augmented Runge–Kutta/Implicit method allowed the admissible Courant–Friedrichs–Lewy number to be increased from O(1 0 0) to O(1 0 0 0). The implicit step introduced into the Runge–Kutta framework acts as a preconditioner which now addresses both, the stiffness in the discrete equations associated with highly stretched meshes, and the stiffness in the analytical equations associated with the disparity in the eigenvalues of the inviscid flux Jacobians. Integrated into a multigrid algorithm, the method is applied to efficiently compute different cases of inviscid flow around airfoils at various Mach numbers, and viscous turbulent airfoil flow with varying Mach and Reynolds number. Compared to well tuned conventional methods, computation times are reduced by half an order of magnitude.

Introduction

Over the last decades, numerical simulation of fluid flow has evolved from a topic only addressed in basic research toward a tool routinely used for research and design. Today’s maturation of computational fluid dynamics (CFD) has enabled the computation of flow around complex aerospace configurations such as complete airplanes, helicopters, and spacecraft [1], [2], [3], [4]. Furthermore, in the last years the combination of CFD with other disciplines like structural mechanics and flight mechanics [5], [6], [7] has found widespread attention and application. This rapid development of numerical flow simulation was driven by both, the successful research and subsequent advancement in efficient solution algorithms, and the continuous increase in available computational power. Having reached today’s level of maturity in numerical algorithms, it is tempting to assume that further progress in the applicability of numerical methods may be guaranteed by solely relying on the sustained development of computer technology. However, since relevant problem size will continue to increase as fast as available hardware permits, a number of severe challenges in the development of numerical methods for flow simulation remain. If the requirements of future complex, multidisciplinary applications shall be met, these challenges have to be addressed rigorously, despite the progress in algorithmic development achieved so far. Globally, without going into detail, these challenges may be summarized by the terms efficiency, robustness, and accuracy.

With respect to efficiency, one of the major breakthroughs in numerical methods for flow simulation was the introduction of multigrid [8], [9], and for the solution of the inviscid equations, numerical methods may now be considered as fairly effective, without stating that no further improvements may be necessary or possible [10]. In contrast to that, methods for the computation of viscous flow can not be considered as equally mature. This is mainly caused by the inadequacy of today’s methods to efficiently take into account the stiffness of the discrete system of equations. Discrete stiffness is provoked by two distinct sources [11]: the first results from the use of a scalar time step which is unable to cope with the disparity in the propagation speeds of convective and acoustic modes. The second source of discrete stiffness is introduced by the highly stretched computational meshes required for economical resolution of boundary layers in high Reynolds number flows. This second source is of far more serious concern than the first, since the corresponding high cell aspect ratios increase discrete stiffness by several orders of magnitude in large portions of the computational domain, resulting in severe convergence problems and very high computation times. One of the first successful attempts to address discrete geometrical stiffness is represented by the work of Martinelli [12], where coefficients of the implicit residual smoothing used in combination with explicit Runge–Kutta time integration were formulated as functions of cell aspect ratio. Subsequently, preconditioning of the discrete equations was used to mitigate the problem of discrete stiffness [11], [13], [14]. In Ref. [15], the approach which proved to be highly successful to solve the 2D-Euler equations [10] was extended to viscous flows, and favorable convergence rates were obtained for laminar flow with Reynolds numbers as high as 80,000.

Recently, a Runge–Kutta/Implicit method was proposed where the widespread strategy of combining multigrid with Runge–Kutta time integration [9], [12], [16] was augmented by replacing the implicit residual smoothing with the solution of an implicit system using Symmetric Gauss–Seidel iteration [17]. Here, turbulent flow around airfoils for Reynolds numbers in the order of 6 × 106 was computed, and the approach reduced the number of iterations required for convergence by about a factor of 8, where CPU-time was more than halved. Further work in Ref. [18] demonstrated the ability of the Runge–Kutta/Implicit method to efficiently compute flows with Reynolds numbers up to the order of 108 and corresponding cell aspect ratios of about 50,000.

Besides the efficiency in computing flow on meshes with very high cell aspect ratios, another matter of concern in algorithmic research is the robustness of numerical methods with respect to the applicability to both, compressible and incompressible flows. The difficulty here is caused by the disparity in the eigenvalues of the convective flux Jacobians in the compressible equations when approaching the incompressible limit: at low speeds, the largest eigenvalue tends toward the speed of sound, whereas the smallest eigenvalue approaches zero. Thus, the condition number of the system of equations tends to infinity and the stiffness of the system increases. Since this stiffness is not tied to a particular discretization, but solely associated with the analytic equations, it was characterized as analytical stiffness in Ref. [11]. To overcome analytical stiffness when computing low-speed flows with compressible methods, usually the system of compressible equations is ‘preconditioned’ for easier solution with iterative methods by multiplying the time derivatives with a suitable matrix [19], [20], [21], [22]. However, these low-speed preconditioning matrices may become singular at stagnation points or in recirculation regions [23], and robustness of the methods may be impaired [24].

As an alternative to preconditioning the compressible equations for low Mach number problems, codes primarily designed for incompressible flows are extended toward the compressible flow regime [25], [26], [27]. When computing incompressible flows, such methods respect the constraint of a divergence-free velocity field by solving a Poisson equation for pressure, which is not directly accounted for in the preconditioning approach [28]. Using this pressure based approach, in general the strong conservation form of the governing equations is not strictly respected [25], [26], [27], and up to now such methods did not gain widespread acceptance in the aerospace community.

In the present work, the discrete stiffness associated with high aspect ratio cells and the analytic stiffness at low Mach numbers shall be addressed by a unified approach. The Runge–Kutta/Implicit scheme introduced in Ref. [17], where the solution of an implicit system of equations was embedded into the framework of Runge–Kutta time integration, will serve as the basis for coping with discrete geometrical stiffness. For extending compressible codes to the incompressible flow regime, in Refs. [28], [29] a suitable equation for pressure was derived by exploiting principles of pressure based methods, and the role of an artificial speed of sound to appropriately scale terms of the numerical dissipation was discussed. To address analytical stiffness in the Runge–Kutta/Implicit method, this artificial speed of sound will be introduced into the implicit system similarly to the formulation of the pressure equation in Refs. [28], [29] to allow efficient computation of low-speed flows. The performance of the proposed method will be assessed by computing steady inviscid flow at various Mach numbers, and steady viscous flows at different Reynolds and Mach numbers.

Section snippets

Governing equations

We consider the two-dimensional Navier–Stokes equations for compressible flow. For a control volume fixed in time and space, the system of partial differential equations in integral form is given byVolWtdV+SF·ndS=0,where W represents the vector of conservative variables, F is the flux-density tensor, and Vol, S, and n denote volume, surface, and outward facing normal of the control volume. The flux density tensor F may be split into an inviscid, convective part Fc and a viscous part Fv:F=Fc

Basic solution scheme

The basic solution scheme employs a cell centered, finite volume space discretization on structured meshes [30], and time integration is achieved by combining an explicit Runge–Kutta scheme with the solution of an implicit system of equations [17]. Using the finite volume technique for space discretization, a semi-discrete form of Eq. (1) may be written asVolWt+all facesF·nS=0,where Vol now represents the volume of a computational cell and S is the area of a cell face. Eq. (5) can be

Extension of the basic approach to low Mach number flow

To use compressible methods for the computation of nearly incompressible flows where M  0, it is well known that the artificial dissipation of the spatial discretization needs to be scaled appropriately [19], since the discrete equations support pressure disturbances of O(M), in contrast to the analytical equations which only support O(M2) disturbances [42], [43]. For the FDS-discretization of Eq. (7) this scaling can easily be achieved by replacing the speed of sound c in Table 1 by an

Computational results

For all computations presented in this study, a 4-level W-cycle is employed in the multigrid algorithm, and the corresponding coarse meshes are created by successively omitting every second grid line. On the finest mesh, a second order space discretization is employed, which on coarse meshes is reduced to first order. For time integration generally a 5 stage Runge–Kutta scheme with coefficients of Ref. [44] for second order spatial accuracy is employed, combined with solution of the implicit

Concluding remarks

A computational approach was derived which addresses two sources of stiffness when solving fluid dynamic equations, namely discrete stiffness associated with high aspect ratio cells, and analytical stiffness occurring when the Mach number approaches zero. The Runge–Kutta/Implicit method, where an explicit Runge–Kutta time integration was combined with the solution of an implicit system of equations, had already demonstrated its efficiency for computations on highly stretched meshes in earlier

Acknowledgement

The author likes to express his thanks to Charles Swanson, NASA LaRC, and Eli Turkel, Tel Aviv University, for many helpful discussions and e-mail conversations.

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