Fluent公司大佬Kim关于LES大涡模拟计算方法的3篇文章
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This paper is concerned with an efficient, second-order-accurate time-advancement scheme applied to an unstructured mesh based Navier-Stokes solver that employs a co-located, finite-volume discretization. An implicit, three-level second-order scheme is used for temporal discretization. The solutions to the resulting system of discretized Navier-Stokes equations including the continuity, momentum, and additional scalar transport equations are obtained using a fractional-step method (FSM) wherein the momentum equations are decoupled from the continuity equation with the aid of an approximate factorization technique. By making the factorization error commensurate with the leading truncation error arising from the second-order temporal discretization, the fractional-step method preserves a second-order temporal accuracy without costly global iterations per each time-step. Several validations are presented for a number of laminar and turbulent flows.
17th AIAA Computational Fluid Dynamics Conference 6 - 9 June 2005, Toronto, Ontario Canada
AIAA 2005-5253
An Implicit Fractional-Step Method for Efficient Transient Simulation of Incompressible Flows
each time step, the entire fractional steps comprising the solutions of the momentum equations and the pressure (or pressure-correction) equation are executed repeatedly in an outer loop, until the solutions satisfy certain convergence criteria, which often requires tens of iterations. Providing the most accurate way of accounting for the nonlinearity in and the coupling among the individual equations, this iterative FSM incurs a considerable computational cost. The computational cost of the FSM can be substantially reduced without compromising the temporal accuracy, if one can obviate the need for the costly outer iterations and preserve the formal order of temporal accuracy. This is possible if the splitting error can be made commensurate with - the same order as or higher order than - the leading truncation error arising from the time-discretization, insofar as the overall temporal accuracy would then be preserved. In light of this, it is arguable that the outer iterations in the iterative FSM aimed at “eliminating” the splitting error is, computationally, an overkill. This line of reasoning produced several “non-iterative” methods in the general framework of the FSM.2, 3, 4, 5, 6, 7, 8, 9, 10 All of these non-iterative FSM’s are designed to ascertain second-order temporal accuracy, which requires that the splitting error be of second-order ( O´∆t 2 µ) or higher. And yet, they differ from one another in terms of the spatial discretization (e.g., staggered or co-located), the time-discretization schemes (e.g., implicit or semiimplicit), the choice of the variable for the continuity equation (e.g., pressure, pressure correction, pseudo-pressure), and the operator splitting employed. This study is concerned with the application of a non-iterative, second-order-accurate FSM to a finite-volumebased Navier-Stokes solver. The finite-volume solver employs a co-located (cell-centered), multi-dimensional linear (second-order-accurate) reconstruction scheme for spatial discretization that permits use of unstructured meshes. For temporal discretization, an implicit, three-level second-order scheme is adopted. The solutions are advanced in time by solving the momentum equations and the pressure-correction equations sequentially in a decoupled manner. The resulting non-iterative FSM closely resembles, in many aspects, the SIMPLEC algorithm 11 widely adopted in the today’s CFD community. Yet the present FSM, due to its non-iterative approach, has been found to significantly speed up the computations of transient flows compared to the fully-iterative schemes. As such, it lends itself as a highly cost-effective method for such CPU-intensive transient computations as direct numerical simulation (DNS), large eddy simulation (LES), and unsteady Reynolds-averaged Navier-Stokes (URANS) computations, for complex industrial flows. The paper briefly describes the finite-volume based spatial discretization schemes first, which will be followed by a somewhat detailed description of the non-iterative fractional-step method. Several validations for laminar and turbulent flows are presented.
I.
Introduction
ITH computationally intensive calculations such as the ones based on unsteady Reynolds-averaged Navier-Stokes (URANS) equations, large eddy simulation (LES) and, to a lesser degree, direct numerical simulation (DNS) beginning to be deployed for industrial applications nowadays, the cost of computing transient flows has become a pacing item in those endeavors. The computational cost of transient flow simulations is largely determined by the efficiency of the algorithms employed to advance numerical solutions in time. Explicit time-marching schemes have been widely used since the early days of computational fluid dynamics (CFD) for their low computational cost. However, the overall efficacy of explicit schemes is severely limited by the restriction on the allowable time-step size, especially for industrial applications mostly involving widely-varying mesh resolution and flow speed. In addition to the restriction on the time-step size imposed by the Courant-Friedrichs-Lewy (CFL) condition, in computations of viscous flows using explicit schemes, the time-step size is further limited by the restriction imposed by the diffusion time-scale of the flows which is typically more severe than the CFL-based one. Implicit time-advancement schemes allow one to use a much larger time-step size than explicit schemes, which makes them an attractive choice for industrial applications. With implicit schemes, the time-step size can be determined solely by the temporal resolution required to achieve a desired level of time accuracy. Implicit time-advancement schemes, however, incur a significant increase in the computational cost in both memory and CPU-time, inasmuch as the system of nonlinear, coupled implicit equations has to be solved at every time-step. For incompressible flows, the fractional-step method (FSM) - or projection method as it was originally called pioneered by Chorin 1 and its variants have been widely adopted. In the FSM, the momentum equations are decoupled from the continuity equation, in one way or another, with the aid of the mathematical technique called operatorsplitting or approximate factorization. Making numerical solutions of the Navier-Stokes equations more tractable, the FSM inevitably introduces a splitting error whose order and magnitude vary with the specific schemes employed in the splitting of the operators. In order to preserve the formal order of temporal accuracy innate in the time-discretization adopted (e.g., second-order), the FSM has often been employed in an iterative manner. In this iterative FSM, at
Downloaded by HUAZHONG UNIVERSITY OF SCIENCE on November 7, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2005-5253
Sung-Eun Kim£ anBiblioteka Baidu Boris Makarov†
Fluent Inc, Lebanon, New Hampshire, 03766, U.S.A.
£ Principal
† Principal
W
Engineer, Fluent Inc., Lebanon N.H., Senior Member AIAA Engineer, Fluent Inc., Lebanon N.H.
1 of 12 American Institute of Aeronautics and Astronautics
Copyright © 2005 by Fluent Inc. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
Downloaded by HUAZHONG UNIVERSITY OF SCIENCE on November 7, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2005-5253
17th AIAA Computational Fluid Dynamics Conference 6 - 9 June 2005, Toronto, Ontario Canada
AIAA 2005-5253
An Implicit Fractional-Step Method for Efficient Transient Simulation of Incompressible Flows
each time step, the entire fractional steps comprising the solutions of the momentum equations and the pressure (or pressure-correction) equation are executed repeatedly in an outer loop, until the solutions satisfy certain convergence criteria, which often requires tens of iterations. Providing the most accurate way of accounting for the nonlinearity in and the coupling among the individual equations, this iterative FSM incurs a considerable computational cost. The computational cost of the FSM can be substantially reduced without compromising the temporal accuracy, if one can obviate the need for the costly outer iterations and preserve the formal order of temporal accuracy. This is possible if the splitting error can be made commensurate with - the same order as or higher order than - the leading truncation error arising from the time-discretization, insofar as the overall temporal accuracy would then be preserved. In light of this, it is arguable that the outer iterations in the iterative FSM aimed at “eliminating” the splitting error is, computationally, an overkill. This line of reasoning produced several “non-iterative” methods in the general framework of the FSM.2, 3, 4, 5, 6, 7, 8, 9, 10 All of these non-iterative FSM’s are designed to ascertain second-order temporal accuracy, which requires that the splitting error be of second-order ( O´∆t 2 µ) or higher. And yet, they differ from one another in terms of the spatial discretization (e.g., staggered or co-located), the time-discretization schemes (e.g., implicit or semiimplicit), the choice of the variable for the continuity equation (e.g., pressure, pressure correction, pseudo-pressure), and the operator splitting employed. This study is concerned with the application of a non-iterative, second-order-accurate FSM to a finite-volumebased Navier-Stokes solver. The finite-volume solver employs a co-located (cell-centered), multi-dimensional linear (second-order-accurate) reconstruction scheme for spatial discretization that permits use of unstructured meshes. For temporal discretization, an implicit, three-level second-order scheme is adopted. The solutions are advanced in time by solving the momentum equations and the pressure-correction equations sequentially in a decoupled manner. The resulting non-iterative FSM closely resembles, in many aspects, the SIMPLEC algorithm 11 widely adopted in the today’s CFD community. Yet the present FSM, due to its non-iterative approach, has been found to significantly speed up the computations of transient flows compared to the fully-iterative schemes. As such, it lends itself as a highly cost-effective method for such CPU-intensive transient computations as direct numerical simulation (DNS), large eddy simulation (LES), and unsteady Reynolds-averaged Navier-Stokes (URANS) computations, for complex industrial flows. The paper briefly describes the finite-volume based spatial discretization schemes first, which will be followed by a somewhat detailed description of the non-iterative fractional-step method. Several validations for laminar and turbulent flows are presented.
I.
Introduction
ITH computationally intensive calculations such as the ones based on unsteady Reynolds-averaged Navier-Stokes (URANS) equations, large eddy simulation (LES) and, to a lesser degree, direct numerical simulation (DNS) beginning to be deployed for industrial applications nowadays, the cost of computing transient flows has become a pacing item in those endeavors. The computational cost of transient flow simulations is largely determined by the efficiency of the algorithms employed to advance numerical solutions in time. Explicit time-marching schemes have been widely used since the early days of computational fluid dynamics (CFD) for their low computational cost. However, the overall efficacy of explicit schemes is severely limited by the restriction on the allowable time-step size, especially for industrial applications mostly involving widely-varying mesh resolution and flow speed. In addition to the restriction on the time-step size imposed by the Courant-Friedrichs-Lewy (CFL) condition, in computations of viscous flows using explicit schemes, the time-step size is further limited by the restriction imposed by the diffusion time-scale of the flows which is typically more severe than the CFL-based one. Implicit time-advancement schemes allow one to use a much larger time-step size than explicit schemes, which makes them an attractive choice for industrial applications. With implicit schemes, the time-step size can be determined solely by the temporal resolution required to achieve a desired level of time accuracy. Implicit time-advancement schemes, however, incur a significant increase in the computational cost in both memory and CPU-time, inasmuch as the system of nonlinear, coupled implicit equations has to be solved at every time-step. For incompressible flows, the fractional-step method (FSM) - or projection method as it was originally called pioneered by Chorin 1 and its variants have been widely adopted. In the FSM, the momentum equations are decoupled from the continuity equation, in one way or another, with the aid of the mathematical technique called operatorsplitting or approximate factorization. Making numerical solutions of the Navier-Stokes equations more tractable, the FSM inevitably introduces a splitting error whose order and magnitude vary with the specific schemes employed in the splitting of the operators. In order to preserve the formal order of temporal accuracy innate in the time-discretization adopted (e.g., second-order), the FSM has often been employed in an iterative manner. In this iterative FSM, at
Downloaded by HUAZHONG UNIVERSITY OF SCIENCE on November 7, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2005-5253
Sung-Eun Kim£ anBiblioteka Baidu Boris Makarov†
Fluent Inc, Lebanon, New Hampshire, 03766, U.S.A.
£ Principal
† Principal
W
Engineer, Fluent Inc., Lebanon N.H., Senior Member AIAA Engineer, Fluent Inc., Lebanon N.H.
1 of 12 American Institute of Aeronautics and Astronautics
Copyright © 2005 by Fluent Inc. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
Downloaded by HUAZHONG UNIVERSITY OF SCIENCE on November 7, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2005-5253