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1.
Semi-Implicit Interior Penalty Discontinuous Galerkin Methods for Viscous Compressible Flows
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We deal with the numerical solution of the Navier-Stokes equations describing a motion of viscous compressible fluids. In order to obtain a sufficiently stable
higher order scheme with respect to the time and space coordinates, we develop a
combination of the discontinuous Galerkin finite element (DGFE) method for the space
discretization and the backward difference formulae (BDF) for the time discretization.
Since the resulting discrete problem leads to a system of nonlinear algebraic equations at each time step, we employ suitable linearizations of inviscid as well as viscous
fluxes which give a linear algebraic problem at each time step. Finally, the resulting
BDF-DGFE scheme is applied to steady as well as unsteady flows and achieved results
are compared with reference data. 相似文献
2.
A High-Order Central ENO Finite-Volume Scheme for Three-Dimensional Low-Speed Viscous Flows on Unstructured Mesh
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Marc R. J. Charest Clinton P. T. Groth & Pierre Q. Gauthier 《Communications In Computational Physics》2015,17(3):615-656
High-order discretization techniques offer the potential to significantly reduce
the computational costs necessary to obtain accurate predictions when compared
to lower-order methods. However, efficient and universally-applicable high-order
discretizations remain somewhat illusive, especially for more arbitrary unstructured
meshes and for incompressible/low-speed flows. A novel, high-order, central essentially
non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for
the solution of the conservation equations of viscous, incompressible flows on three-dimensional
unstructured meshes. Similar to finite element methods, coordinate transformations
are used to maintain the scheme's order of accuracy even when dealing
with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied
to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes
equations and the resulting discretized equations are solved with a parallel implicit
Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach
is adopted and the resulting temporal derivatives are discretized using the family of
high-order backward difference formulas (BDF). The proposed finite-volume scheme
for fully unstructured mesh is demonstrated to provide both fast and accurate solutions
for steady and unsteady viscous flows. 相似文献
3.
Numerical Simulation of Compressible Vortical Flows Using a Conservative Unstructured-Grid Adaptive Scheme
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Giuseppe Forestieri Alberto Guardone Dario Isola Filippo Marulli & Giuseppe Quaranta 《Communications In Computational Physics》2012,12(3):866-884
A two-dimensional numerical scheme for the compressible Euler equations
is presented and applied here to the simulation of exemplary compressible vortical
flows. The proposed approach allows to perform computations on unstructured moving grids with adaptation, which is required to capture complex features of the flow-field. Grid adaptation is driven by suitable error indicators based on the Mach number
and by element-quality constraints as well. At the new time level, the computational
grid is obtained by a suitable combination of grid smoothing, edge-swapping, grid
refinement and de-refinement. The grid modifications—including topology modification due to edge-swapping or the insertion/deletion of a new grid node—are interpreted at the flow solver level as continuous (in time) deformations of suitably-defined
node-centered finite volumes. The solution over the new grid is obtained without explicitly resorting to interpolation techniques, since the definition of suitable interface
velocities allows one to determine the new solution by simple integration of the Arbitrary Lagrangian-Eulerian formulation of the flow equations. Numerical simulations
of the steady oblique-shock problem, of the steady transonic flow and of the start-up
unsteady flow around the NACA 0012 airfoil are presented to assess the scheme capabilities to describe these flows accurately. 相似文献
4.
A Jacobian-Free Newton Krylov Implicit-Explicit Time Integration Method for Incompressible Flow Problems
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We have introduced a fully second order IMplicit/EXplicit (IMEX) time integration technique for solving the compressible Euler equations plus nonlinear heat conduction problems (also known as the radiation hydrodynamics problems) in Kadioglu et al., J. Comp. Physics [22,24]. In this paper, we study the implications when this method is applied to the incompressible Navier-Stokes (N-S) equations. The IMEX method is applied to the incompressible flow equations in the following manner. The hyperbolic terms of the flow equations are solved explicitly exploiting the well understood explicit schemes. On the other hand, an implicit strategy is employed for the non-hyperbolic terms. The explicit part is embedded in the implicit step in such a way that it is solved as part of the non-linear function evaluation within the framework of the Jacobian-Free Newton Krylov (JFNK) method [8,29,31]. This is done to obtain a self-consistent implementation of the IMEX method that eliminates the potential order reduction in time accuracy due to the specific operator separation. We employ a simple yet quite effective fractional step projection methodology (similar to those in [11,19,21,30]) as our preconditioner inside the JFNK solver. We present results from several test calculations. For each test, we show second order time convergence. Finally, we present a study for the algorithm performance of the JFNK solver with the new projection method based preconditioner. 相似文献
5.
Kinetic Energy Preserving and Entropy Stable Finite Volume Schemes for Compressible Euler and Navier-Stokes Equations
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Praveen Chandrashekar 《Communications In Computational Physics》2013,14(5):1252-1286
Centered numerical fluxes can be constructed for compressible Euler equations
which preserve kinetic energy in the semi-discrete finite volume scheme. The essential
feature is that the momentum flux should be of the form where are any consistent approximations to the
pressure and the mass flux. This scheme thus leaves most terms in the numerical
flux unspecified and various authors have used simple averaging. Here we enforce
approximate or exact entropy consistency which leads to a unique choice of all the
terms in the numerical fluxes. As a consequence, a novel entropy conservative flux that
also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed.
These fluxes are centered and some dissipation has to be added if shocks are
present or if the mesh is coarse. We construct scalar artificial dissipation terms which
are kinetic energy stable and satisfy approximate/exact entropy condition. Secondly,
we use entropy-variable based matrix dissipation flux which leads to kinetic energy
and entropy stable schemes. These schemes are shown to be free of entropy violating
solutions unlike the original Roe scheme. For hypersonic flows a blended scheme is
proposed which gives carbuncle free solutions for blunt body flows. Numerical results
for Euler and Navier-Stokes equations are presented to demonstrate the performance
of the different schemes. 相似文献
6.
An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations
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The computation of compressible flows becomes more challenging when the
Mach number has different orders of magnitude. When the Mach number is of order
one, modern shock capturing methods are able to capture shocks and other complex
structures with high numerical resolutions. However, if the Mach number is small, the
acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus
demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes
equations that is uniformly stable and accurate for all Mach numbers. Our idea is to
split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a
linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection
system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a
projection method-like incompressible solver. We present numerical results in one and
two dimensions in both compressible and incompressible regimes. 相似文献
7.
On the Stability and CPU Time of the Implicit Runge-Kutta Schemes for Steady State Simulations
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Implicit time integration schemes are popular because their relaxed stability
constraints can result in better computational efficiency. For time-accurate unsteady
simulations, it has been well recognized that the inherent dispersion and dissipation
errors of implicit Runge-Kutta schemes will reduce the computational accuracy for
large time steps. Yet for steady state simulations using the time-dependent governing
equations, these errors are often overlooked because the intermediate solutions are of
less interest. Based on the model equation dy/dt = (µ+iλ)y of scalar convection diffusion
systems, this study examines the stability limits, dispersion and dissipation errors
of four diagonally implicit Runge-Kutta-type schemes on the complex (µ+iλ)∆t
plane. Through numerical experiments, it is shown that, as the time steps increase,
the A-stable implicit schemes may not always have reduced CPU time and the computations
may not always remain stable, due to the inherent dispersion and dissipation
errors of the implicit Runge-Kutta schemes. The dissipation errors may decelerate the
convergence rate, and the dispersion errors may cause large oscillations of the numerical
solutions. These errors, especially those of high wavenumber components, grow
at large time steps. They lead to difficulty in the convergence of the numerical computations,
and result in increasing CPU time or even unstable computations as the time
step increases. It is concluded that an optimal implicit time integration scheme for
steady state simulations should have high dissipation and low dispersion. 相似文献
8.
An Implicit Algorithm for High-Order DG/FV Schemes for Compressible Flows on 2D Arbitrary Grids
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Laiping Zhang Ming Li Wei Liu & Xin He 《Communications In Computational Physics》2015,17(1):287-316
A Newton/LU-SGS (lower-upper symmetric Gauss-Seidel) iteration implicit
method was developed to solve two-dimensional Euler and Navier-Stokes equations
by the DG/FV hybrid schemes on arbitrary grids. The Newton iteration was employed
to solve the nonlinear system, while the linear system was solved with LU-SGS iteration.
The effect of several parameters in the implicit scheme, such as the CFL number,
the Newton sub-iteration steps, and the update frequency of Jacobian matrix, was investigated
to evaluate the performance of convergence history. Several typical test
cases were simulated, and compared with the traditional explicit Runge-Kutta (RK)
scheme. Firstly the Couette flow was tested to validate the order of accuracy of the
present DG/FV hybrid schemes. Then a subsonic inviscid flow over a bump in a channel
was simulated and the effect of parameters was alsoinvestigated. Finally, the implicit
algorithm was applied to simulate a subsonic inviscid flow over a circular cylinder
and the viscous flow in a square cavity. The numerical results demonstrated that
the present implicit scheme can accelerate the convergence history efficiently. Choosing
proper parameters would improve the efficiency of the implicit scheme. Moreover,
in the same framework, the DG/FV hybrid schemes are more efficient than the same
order DG schemes. 相似文献
9.
A Scalable Numerical Method for Simulating Flows Around High-Speed Train Under Crosswind Conditions
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Zhengzheng Yan Rongliang Chen Yubo Zhao & Xiao-Chuan Cai 《Communications In Computational Physics》2014,15(4):944-958
This paper presents a parallel Newton-Krylov-Schwarz method for the numerical
simulation of unsteady flows at high Reynolds number around a high-speed
train under crosswind. With a realistic train geometry, a realistic Reynolds number,
and a realistic wind speed, this is a very challenging computational problem. Because
of the limited parallel scalability, commercial CFD software is not suitable for
supercomputers with a large number of processors. We develop a Newton-Krylov-Schwarz
based fully implicit method, and the corresponding parallel software, for the
3D unsteady incompressible Navier-Stokes equations discretized with a stabilized finite
element method on very fine unstructured meshes. We test the algorithm and
software for flows passing a train modeled after China's high-speed train CRH380B,
and we also compare our results with results obtained from commercial CFD software.
Our algorithm shows very good parallel scalability on a supercomputer with over one
thousand processors. 相似文献
10.
Variable High-Order Multiblock Overlapping Grid Methods for Mixed Steady and Unsteady Multiscale Viscous Flows,Part II: Hypersonic Nonequilibrium Flows
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Andrea Lani Bjö rn Sjö green H. C. Yee & William D. Henshaw 《Communications In Computational Physics》2013,13(2):583-602
The variable high-order multiblock overlapping (overset) grids method of
Sjögreen & Yee [CiCP, Vol. 5, 2009] for a perfect gas has been extended to nonequilibrium flows. This work makes use of the recently developed high-order well-balanced
shock-capturing schemes and their filter counterparts [Wang et al., J. Comput. Phys.,
2009, 2010] that exactly preserve certain non-trivial steady state solutions of the chemical nonequilibrium governing equations. Multiscale turbulence with strong shocks
and flows containing both steady and unsteady components is best treated by mixing of numerical methods and switching on the appropriate scheme in the appropriate
subdomains of the flow fields, even under the multiblock grid or adaptive grid refinement framework. While low dissipative sixth- or higher-order shock-capturing filter
methods are appropriate for unsteady turbulence with shocklets, second- and third-order shock-capturing methods are more effective for strong steady or nearly steady
shocks in terms of convergence. It is anticipated that our variable high-order overset
grid framework capability with its highly modular design will allow for an optimum
synthesis of these new algorithms in such a way that the most appropriate spatial discretizations can be tailored for each particular region of the flow. In this paper some of
the latest developments in single block high-order filter schemes for chemical nonequilibrium flows are applied to overset grid geometries. The numerical approach is validated on a number of test cases characterized by hypersonic conditions with strong
shocks, including the reentry flow surrounding a 3D Apollo-like NASA Crew Exploration Vehicle that might contain mixed steady and unsteady components, depending
on the flow conditions. 相似文献
11.
A Compact Third-Order Gas-Kinetic Scheme for Compressible Euler and Navier-Stokes Equations
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In this paper, a compact third-order gas-kinetic scheme is proposed for the
compressible Euler and Navier-Stokes equations. The main reason for the feasibility
to develop such a high-order scheme with compact stencil, which involves only
neighboring cells, is due to the use of a high-order gas evolution model. Besides the
evaluation of the time-dependent flux function across a cell interface, the high-order
gas evolution model also provides an accurate time-dependent solution of the flow
variables at a cell interface. Therefore, the current scheme not only updates the cell
averaged conservative flow variables inside each control volume, but also tracks the
flow variables at the cell interface at the next time level. As a result, with both cell averaged
and cell interface values, the high-order reconstruction in the current scheme
can be done compactly. Different from using a weak formulation for high-order accuracy
in the Discontinuous Galerkin method, the current scheme is based on the strong
solution, where the flow evolution starting from a piecewise discontinuous high-order
initial data is precisely followed. The cell interface time-dependent flow variables can
be used for the initial data reconstruction at the beginning of next time step. Even with
compact stencil, the current scheme has third-order accuracy in the smooth flow regions,
and has favorable shock capturing property in the discontinuous regions. It can
be faithfully used from the incompressible limit to the hypersonic flow computations,
and many test cases are used to validate the current scheme. In comparison with many
other high-order schemes, the current method avoids the use of Gaussian points for
the flux evaluation along the cell interface and the multi-stage Runge-Kutta time stepping
technique. Due to its multidimensional property of including both derivatives of
flow variables in the normal and tangential directions of a cell interface, the viscous
flow solution, especially those with vortex structure, can be accurately captured. With
the same stencil of a second order scheme, numerical tests demonstrate that the current
scheme is as robust as well-developed second-order shock capturing schemes, but
provides more accurate numerical solutions than the second order counterparts. 相似文献
12.
A Preliminary Calculation of Three-Dimensional Unsteady Underwater Cavitating Flows Near Incompressible Limit
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Recently, cavitated flows over underwater submerged bodies have attracted
researchers to simulate large scale cavitation. Comparatively Computational Fluid
Dynamics (CFD) approaches have been used widely and successfully to model developed cavitation. However, it is still a great challenge to accurately predict cavitated
flow phenomena associated with interface capturing, viscous effects, unsteadiness and
three-dimensionality. In this study, we consider the preconditioned three-dimensional
multiphase Navier-Stokes equations comprised of the mixture density, mixture momentum and constituent volume fraction equations. A dual-time implicit formulation
with LU Decomposition is employed to accommodate the inherently unsteady physics.
Also, we adopt the Roe flux splitting method to deal with flux discretization in space.
Moreover, time-derivative preconditioning is used to ensure well-conditioned eigenvalues of the high density ratio two-phase flow system to achieve computational efficiency. Validation cases include an unsteady 3-D cylindrical headform cavitated flow
and an 2-D convergent-divergent nozzle channel cavity-problem. 相似文献
13.
Adaptive Fully Implicit Simulator with Multilevel Schwarz Methods for Gas Reservoir Flows in Fractured Porous Media
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Large-scale reservoir modeling and simulation of gas reservoir flows in fractured porous media is currently an important topic of interest in petroleum engineering. In this paper, the dual-porosity dual-permeability (DPDP) model coupled with
the Peng-Robinson equation of state (PR-EoS) is used for the mathematical model of
the gas reservoir flow in fractured porous media. We develop and study a parallel and
highly scalable reservoir simulator based on an adaptive fully implicit scheme and
an inexact Newton type method to solve this dual-continuum mathematical model.
In the approach, an explicit-first-step, single-diagonal-coefficient, diagonally implicit
Runge–Kutta (ESDIRK) method with adaptive time stepping is proposed for the fully
implicit discretization, which is second-order and L-stable. And then we focus on the
family of Newton–Krylov methods for the solution of a large sparse nonlinear system
of equations arising at each time step. To accelerate the convergence and improve the
scalability of the solver, a class of multilevel monolithic additive Schwarz methods is
employed for preconditioning. Numerical results on a set of ideal as well as realistic
flow problems are used to demonstrate the efficiency and the robustness of the proposed methods. Experiments on a supercomputer with several thousand processors
are also carried out to show that the proposed reservoir simulator is highly scalable. 相似文献
14.
A DDG Method with a Residual-Based Artificial Viscosity for the Transonic/Supersonic Compressible Flow
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Xiaofeng He Kun Wang Yiwei Feng Tiegang Liu & Xiaojun Wang 《Communications In Computational Physics》2022,31(4):1134-1161
In this work, a direct discontinuous Galerkin (DDG) method with artificial
viscosity is developed to solve the compressible Navier-Stokes equations for simulating the transonic or supersonic flow, where the DDG approach is used to discretize
viscous and heat fluxes. A strong residual-based artificial viscosity (AV) technique is
proposed to be applied in the DDG framework to handle shock waves and layer structures appearing in transonic or supersonic flow, which promotes convergence and robustness. Moreover, the AV term is added to classical BR2 methods for comparison.
A number of 2-D and 3-D benchmarks such as airfoils, wings, and a full aircraft are
presented to assess the performance of the DDG framework with the strong residual-based AV term for solving the two dimensional and three dimensional Navier-Stokes
equations. The proposed framework provides an alternative robust and efficient approach for numerically simulating the multi-dimensional compressible Navier-Stokes
equations for transonic or supersonic flow. 相似文献
15.
Shuang Tan Wenjun Sun Kun Xu Junxia Wei & Guoxi Ni 《Communications In Computational Physics》2020,28(3):1189-1218
In this paper, a time implicit unified gas kinetic scheme (IUGKS) for 3D
multi-group neutron transport equation with delayed neutron is developed. The
explicit scheme, implicit 1st-order backward Euler scheme, and 2nd-order Crank-Nicholson scheme, become the subsets of the current IUGKS. In neutron transport,
the microscopic angular flux and the macroscopic scalar flux are fully coupled in an
implicit way with the combination of dual-time step technique for the convergence acceleration of unsteady evolution. In IUGKS, the computational time step is no longer
limited by the Courant-Friedrichs-Lewy (CFL) condition, which improves the computational efficiency in both steady and unsteady simulations with a large time step.
Mathematically, the current scheme has the asymptotic preserving (AP) property in
recovering automatically the diffusion solution in the continuum regime. Since the
explicit scanning along neutron traveling direction within the computational domain
is not needed in IUGKS, the scheme can be easily extended to multi-dimensional and
parallel computations. The numerical tests demonstrate that the IUGKS has high computational efficiency, high accuracy, and strong robustness when compared with other
schemes, such as the explicit UGKS, the commonly used finite difference, and finite
volume methods. This study shows that the IUGKS can be used faithfully to study
neutron transport in practical engineering applications. 相似文献
16.
A Parallel,Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Arbitrary Grids
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Hong Luo Luqing Luo Amjad Ali Robert Nourgaliev & Chunpei Cai 《Communications In Computational Physics》2011,9(2):363-389
A reconstruction-based discontinuous Galerkin method is presented for the
solution of the compressible Navier-Stokes equations on arbitrary grids. In this method,
an in-cell reconstruction is used to obtain a higher-order polynomial representation
of the underlying discontinuous Galerkin polynomial solution and an inter-cell reconstruction
is used to obtain a continuous polynomial solution on the union of two
neighboring, interface-sharing cells. The in-cell reconstruction is designed to enhance
the accuracy of the discontinuous Galerkin method by increasing the order of the underlying
polynomial solution. The inter-cell reconstruction is devised to remove an
interface discontinuity of the solution and its derivatives and thus to provide a simple,
accurate, consistent, and robust approximation to the viscous and heat fluxes
in the Navier-Stokes equations. A parallel strategy is also devised for the resulting
reconstruction discontinuous Galerkin method, which is based on domain partitioning
and Single Program Multiple Data (SPMD) parallel programming model. The
RDG method is used to compute a variety of compressible flow problems on arbitrary
meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The
numerical results demonstrate that this RDG method is third-order accurate at a cost
slightly higher than its underlying second-order DG method, at the same time providing
a better performance than the third order DG method, in terms of both computing
costs and storage requirements. 相似文献
17.
Kinetic Slip Boundary Condition for Isothermal Rarefied Gas Flows Through Static Non-Planar Geometries Based on the Regularized Lattice-Boltzmann Method
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Jean-Michel Tucny David Vidal Sé bastien Leclaire & Franç ois Bertrand 《Communications In Computational Physics》2022,31(3):816-868
The simulation of rarefied gas flows through complex porous media is challenging due to the tortuous flow pathways inherent to such structures. The Lattice
Boltzmann method (LBM) has been identified as a promising avenue to solve flows
through complex geometries due to the simplicity of its scheme and its high parallel
computational efficiency. It has been proposed to model the stress-strain relationship
with the extended Navier-Stokes equations rather than attempting to directly solve
the Boltzmann equation. However, a regularization technique is required to filter out
non-resolved higher-order components with a low-order velocity scheme. Although
slip boundary conditions (BCs) have been proposed for the non-regularized multiple
relaxation time LBM (MRT-LBM) for planar geometries, previous slip BCs have never
been verified extensively with the regularization technique. In this work, following
an extensive literature review on the imposition of slip BCs for rarefied flows with the
LBM, it is proven that earlier values for kinetic parameters developed to impose slip
BCs are inaccurate for the regularized MRT-LBM and differ between the D2Q9 and
D3Q15 schemes. The error was eliminated for planar flows and good agreement between analytical solutions for arrays of cylinders and spheres was found with a wide
range of Knudsen numbers. 相似文献
18.
Shuai Shao Ming Li Nianhua Wang & Laiping Zhang 《Communications In Computational Physics》2020,27(3):725-752
A new hybrid reconstruction scheme DDG/FV is developed in this work
by combining the DDG method and DG/FV hybrid scheme developed in the authors' previous work [1–4] to simulate three-dimensional compressible viscous flow on tetrahedral grids. The extended von Neumann stencils are used in the reconstruction process to ensure the linear stability, and the L2 projection and the least-squares method
are adopted to reconstruct higher order distributions for higher accuracy and robustness. In addition, a quadrature-free L2 projection based on orthogonal basis functions
is implemented to improve the efficiency of reconstruction. Three typical test cases,
including the 3D Couette flow, laminar flows over an analytical 3D body of revolution
and over a sphere, are simulated to validate the accuracy and efficiency of DDG/FV
method. The numerical results demonstrate that the DDG scheme can accelerate the
convergence history compared with widely-used BR2 scheme. More attractively, the
new DDG/FV hybrid method can deliver the same accuracy as BR2-DG method with
more than 2 times of efficiency improvement in solving 3D Navier-Stokes equations on
tetrahedral grids, and even one-order of magnitude faster in some cases, which shows
good potential in future realistic applications. 相似文献
19.
Keiichi Kitamura Eiji Shima Keiichiro Fujimoto & Z. J. Wang 《Communications In Computational Physics》2011,10(1):90-119
In low speed flow computations, compressible finite-volume solvers are
known to a) fail to converge in acceptable time and b) reach unphysical solutions.
These problems are known to be cured by A) preconditioning on the time-derivative
term, and B) control of numerical dissipation, respectively. There have been several
methods of A) and B) proposed separately. However, it is unclear which combination
is the most accurate, robust, and efficient for low speed flows. We carried out a
comparative study of several well-known or recently-developed low-dissipation Euler
fluxes coupled with a preconditioned LU-SGS (Lower-Upper Symmetric Gauss-Seidel)
implicit time integration scheme to compute steady flows. Through a series of numerical
experiments, accurate, efficient, and robust methods are suggested for low speed
flow computations. 相似文献
20.
S. C. Fu R. M. C. So & W. W. F. Leung 《Communications In Computational Physics》2011,9(5):1257-1283
The objective of this paper is to seek an alternative to the numerical simulation
of the Navier-Stokes equations by a method similar to solving the BGK-type
modeled lattice Boltzmann equation. The proposed method is valid for both gas and
liquid flows. A discrete flux scheme (DFS) is used to derive the governing equations
for two distribution functions; one for mass and another for thermal energy. These
equations are derived by considering an infinitesimally small control volume with a
velocity lattice representation for the distribution functions. The zero-order moment
equation of the mass distribution function is used to recover the continuity equation,
while the first-order moment equation recovers the linear momentum equation. The
recovered equations are correct to the first order of the Knudsen number (Kn); thus,
satisfying the continuum assumption. Similarly, the zero-order moment equation of
the thermal energy distribution function is used to recover the thermal energy equation.
For aerodynamic flows, it is shown that the finite difference solution of the DFS
is equivalent to solving the lattice Boltzmann equation (LBE) with a BGK-type model
and a specified equation of state. Thus formulated, the DFS can be used to simulate a
variety of aerodynamic and hydrodynamic flows. Examples of classical aeroacoustics,
compressible flow with shocks, incompressible isothermal and non-isothermal Couette
flows, stratified flow in a cavity, and double diffusive flow inside a rectangle are used
to demonstrate the validity and extent of the DFS. Very good to excellent agreement
with known analytical and/or numerical solutions is obtained; thus lending evidence
to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid
flow simulations. 相似文献