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1.
High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation
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Sebastiano Boscarino Seung-Yeon Cho Giovanni Russo & Seok-Bae Yun 《Communications In Computational Physics》2021,29(1):1-33
In this paper, we present a conservative semi-Lagrangian finite-difference
scheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large time steps, for all the
range of Knudsen number [17, 27, 30]. Unfortunately, however, such schemes are not
conservative. Lack of conservation is analyzed in detail, and two main sources are
identified as its cause. Firstly, when using classical continuous Maxwellian, conservation error is negligible only if velocity space is resolved with sufficiently large number
of grid points. However, for a small number of grid points in velocity space such error
is not negligible, because the parameters of the Maxwellian do not coincide with the
discrete moments. Secondly, the non-linear reconstruction used to prevent oscillations
destroys the translation invariance which is at the basis of the conservation properties
of the scheme. As a consequence, the schemes show a wrong shock speed in the limit
of small Knudsen number. To treat the first problem and ensure machine precision
conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian
introduced in [22]. The second problem is treated by implementing a conservative correction procedure based on the flux difference form as in [26]. In this way we can construct conservative semi-Lagrangian schemes which are Asymptotic Preserving (AP)
for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness of
the proposed scheme is demonstrated by extensive numerical tests. 相似文献
2.
V. A. Titarev 《Communications In Computational Physics》2012,12(1):162-192
The paper is devoted to the development of an efficient deterministic framework for modelling of three-dimensional rarefied gas flows on the basis of the numerical solution of the Boltzmann kinetic equation with the model collision integrals. The
framework consists of a high-order accurate implicit advection scheme on arbitrary
unstructured meshes, the conservative procedure for the calculation of the model collision integral and efficient implementation on parallel machines. The main application
area of the suggested methods is micro-scale flows. Performance of the proposed approach is demonstrated on a rarefied gas flow through the finite-length circular pipe.
The results show good accuracy of the proposed algorithm across all flow regimes and
its high efficiency and excellent parallel scalability for up to 512 cores. 相似文献
3.
Florian Bernard Angelo Iollo & Gabriella Puppo 《Communications In Computational Physics》2014,16(4):956-982
The solution of complex rarefied flows with the BGK equation and the Discrete Velocity Method (DVM) requires a large number of velocity grid points leading
to significant computational costs. We propose an adaptive velocity grid approach exploiting the fact that locally in space, the distribution function is supported only by a
sub-set of the global velocity grid. The velocity grid is adapted thanks to criteria based
on local temperature, velocity and on the enforcement of mass conservation. Simulations in 1D and 2D are presented for different Knudsen numbers and compared to a
global velocity grid BGK solution, showing the computational gain of the proposed
approach. 相似文献
4.
A Gas-Kinetic Unified Algorithm for Non-Equilibrium Polyatomic Gas Flows Covering Various Flow Regimes
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Wen-Qiang Hu Zhi-Hui Li Ao-Ping Peng & Xin-Yu Jiang 《Communications In Computational Physics》2021,30(1):144-189
In this paper, a gas-kinetic unified algorithm (GKUA) is developed to investigate the non-equilibrium polyatomic gas flows covering various regimes. Based
on the ellipsoidal statistical model with rotational energy excitation, the computable
modelling equation is presented by unifying expressions on the molecular collision relaxing parameter and the local equilibrium distribution function. By constructing the
corresponding conservative discrete velocity ordinate method for this model, the conservative properties during the collision procedure are preserved at the discrete level
by the numerical method, decreasing the computational storage and time. Explicit
and implicit lower-upper symmetric Gauss-Seidel schemes are constructed to solve
the discrete hyperbolic conservation equations directly. Applying the new GKUA,
some numerical examples are simulated, including the Sod Riemann problem, homogeneous flow rotational relaxation, normal shock structure, Fourier and Couette flows,
supersonic flows past a circular cylinder, and hypersonic flow around a plate placed
normally. The results obtained by the analytic, experimental, direct simulation Monte
Carlo method, and other measurements in references are compared with the GKUA
results, which are in good agreement, demonstrating the high accuracy of the present
algorithm. Especially, some polyatomic gas non-equilibrium phenomena are observed
and analysed by solving the Boltzmann-type velocity distribution function equation
covering various flow regimes. 相似文献
5.
A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods
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The Boltzmann equation (BE) for gas flows is a time-dependent nonlinear
differential-integral equation in 6 dimensions. The current simplified practice is to linearize the collision integral in BE by the BGK model using Maxwellian equilibrium
distribution and to approximate the moment integrals by the discrete ordinate method
(DOM) using a finite set of velocity quadrature points. Such simplification reduces
the dimensions from 6 to 3, and leads to a set of linearized discrete BEs. The main
difficulty of the currently used (conventional) numerical procedures occurs when the
mean velocity and the variation of temperature are large that requires an extremely
large number of quadrature points. In this paper, a novel dynamic scheme that requires only a small number of quadrature points is proposed. This is achieved by
a velocity-coordinate transformation consisting of Galilean translation and thermal
normalization so that the transformed velocity space is independent of mean velocity and temperature. This enables the efficient implementation of Gaussian-Hermite
quadrature. The velocity quadrature points in the new velocity space are fixed while
the correspondent quadrature points in the physical space change from time to time
and from position to position. By this dynamic nature in the physical space, this new
quadrature scheme is termed as the dynamic quadrature scheme (DQS). The DQS was
implemented to the DOM and the lattice Boltzmann method (LBM). These new methods with DQS are therefore termed as the dynamic discrete ordinate method (DDOM)
and the dynamic lattice Boltzmann method (DLBM), respectively. The new DDOM
and DLBM have been tested and validated with several testing problems. Of the same
accuracy in numerical results, the proposed schemes are much faster than the conventional schemes. Furthermore, the new DLBM have effectively removed the incompressible and isothermal restrictions encountered by the conventional LBM. 相似文献
6.
Jaw-Yen Yang Bagus Putra Muljadi Zhi-Hui Li & Han-Xin Zhang 《Communications In Computational Physics》2013,14(1):242-264
An accurate and direct algorithm for solving the semiclassical Boltzmann
equation with relaxation time approximation in phase space is presented for parallel
treatment of rarefied gas flows of particles of three statistics. The discrete ordinate
method is first applied to discretize the velocity space of the distribution function to
render a set of scalar conservation laws with source term. The high order weighted essentially non-oscillatory scheme is then implemented to capture the time evolution of
the discretized velocity distribution function in physical space and time. The method
is developed for two space dimensions and implemented on gas particles that obey
the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics. Computational examples in one- and two-dimensional initial value problems of rarefied gas flows are
presented and the results indicating good resolution of the main flow features can be
achieved. Flows of wide range of relaxation times and Knudsen numbers covering
different flow regimes are computed to validate the robustness of the method. The
recovery of quantum statistics to the classical limit is also tested for small fugacity
values. 相似文献
7.
This paper develops a high-order accurate gas-kinetic scheme in the framework
of the finite volume method for the one- and two-dimensional flow simulations,
which is an extension of the third-order accurate gas-kinetic scheme [Q.B. Li, K. Xu, and
S. Fu, J. Comput. Phys., 229(2010), 6715-6731] and the second-order accurate gas-kinetic
scheme [K. Xu, J. Comput. Phys., 171(2001), 289-335]. It is formed by two parts: quartic
polynomial reconstruction of the macroscopic variables and fourth-order accurate flux
evolution. The first part reconstructs a piecewise cell-center based quartic polynomial
and a cell-vertex based quartic polynomial according to the "initial" cell average approximation
of macroscopic variables to recover locally the non-equilibrium and equilibrium
single particle velocity distribution functions around the cell interface. It is in
view of the fact that all macroscopic variables become moments of a single particle velocity
distribution function in the gas-kinetic theory. The generalized moment limiter
is employed there to suppress the possible numerical oscillation. In the second part,
the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means
of the simple particle transport mechanism in the microscopic level, i.e. free transport
and the Bhatnagar-Gross-Krook (BGK) collisions. In other words, the fourth-order
flux evolution is based on the solution (i.e. the particle velocity distribution function)
of the BGK model for the Boltzmann equation. Several 1D and 2D test problems are
numerically solved by using the proposed high-order accurate gas-kinetic scheme. By
comparing with the exact solutions or the numerical solutions obtained the second-order
or third-order accurate gas-kinetic scheme, the computations demonstrate that
our scheme is effective and accurate for simulating invisid and viscous fluid flows,
and the accuracy of the high-order GKS depends on the choice of the (numerical) collision
time. 相似文献
8.
Jaw-Yen Yang Li-Hsin Hung & Yao-Tien Kuo 《Communications In Computational Physics》2011,10(2):405-421
Computations of microscopic circular pipe flow in a rarefied quantum gas
are presented using a semiclassical axisymmetric lattice Boltzmann method. The
method is first derived by directly projecting the Uehling-Uhlenbeck Boltzmann-BGK
equations in two-dimensional rectangular coordinates onto the tensor Hermite polynomials
using moment expansion method and then the forcing strategy of Halliday
et al. [Phys. Rev. E., 64 (2001), 011208] is adopted by adding forcing terms into the
resulting microdynamic evolution equation. The determination of the forcing terms
is dictated by yielding the emergent macroscopic equations toward a particular target
form. The correct macroscopic equations of the incompressible axisymmetric viscous
flows are recovered through the Chapman-Enskog expansion. The velocity profiles
and the mass flow rates of pipe flows with several Knudsen numbers covering different
flow regimes are presented. It is found the Knudsen minimum can be captured in
all three statistics studied. The results also indicate distinct characteristics of the effects
of quantum statistics. 相似文献
9.
Comparison of Fifth-Order WENO Scheme and Finite Volume WENO-Gas-Kinetic Scheme for Inviscid and Viscous Flow Simulation
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The development of high-order schemes has been mostly concentrated on
the limiters and high-order reconstruction techniques. In this paper, the effect of the
flux functions on the performance of high-order schemes will be studied. Based on the
same WENO reconstruction, two schemes with different flux functions, i.e., the fifth-order WENO method and the WENO-Gas-Kinetic scheme (WENO-GKS), will be compared. The fifth-order finite difference WENO-SW scheme is a characteristic variable
reconstruction based method which uses the Steger-Warming flux splitting for inviscid terms, the sixth-order central difference for viscous terms, and three stages Runge-Kutta time stepping for the time integration. On the other hand, the finite volume
WENO-GKS is a conservative variable reconstruction based method with the same
WENO reconstruction. But it evaluates a time dependent gas distribution function
along a cell interface, and updates the flow variables inside each control volume by
integrating the flux function along the boundary of the control volume in both space
and time. In order to validate the robustness and accuracy of the schemes, both methods are tested under a wide range of flow conditions: vortex propagation, Mach 3
step problem, and the cavity flow at Reynolds number 3200. Our study shows that
both WENO-SW and WENO-GKS yield quantitatively similar results and agree with
each other very well provided a sufficient grid resolution is used. With the reduction of mesh points, the WENO-GKS behaves to have less numerical dissipation and
present more accurate solutions than those from the WENO-SW in all test cases. For
the Navier-Stokes equations, since the WENO-GKS couples inviscid and viscous terms
in a single flux evaluation, and the WENO-SW uses an operator splitting technique, it
appears that the WENO-SW is more sensitive to the WENO reconstruction and boundary treatment. In terms of efficiency, the finite volume WENO-GKS is about 4 times
slower than the finite difference WENO-SW in two dimensional simulations. The current study clearly shows that besides high-order reconstruction, an accurate gas evolution model or flux function in a high-order scheme is also important in the capturing of physical solutions. In a physical flow, the transport, stress deformation, heat conduction, and viscous heating are all coupled in a single gas evolution process. Therefore,
it is preferred to develop such a scheme with multi-dimensionality, and unified treatment of inviscid and dissipative terms. A high-order scheme does prefer a high-order
gas evolution model. Even with the rapid advances of high-order reconstruction techniques, the first-order dynamics of the Riemann solution becomes the bottleneck for
the further development of high-order schemes. In order to avoid the weakness of the
low order flux function, the development of high-order schemes relies heavily on the
weak solution of the original governing equations for the update of additional degree
of freedom, such as the non-conservative gradients of flow variables, which cannot be
physically valid in discontinuous regions. 相似文献
10.
Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations
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Haiyang Gao Z. J. Wang & H. T. Huynh 《Communications In Computational Physics》2013,13(4):1013-1044
A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR
approach employs the differential form of the equation and accounts for the jumps
in flux values at the cell boundaries by a correction procedure. In addition to being
simple and economical, it unifies several existing methods including discontinuous
Galerkin, staggered grid, spectral volume, and spectral difference. To discretize the diffusion terms, we use the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG),
and I-continuous approaches. The first three of these approaches, originally derived
using the integral formulation, were recast here in the CPR framework, whereas the
I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a
triangular mesh. Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out. Finally, results for the Navier-Stokes
equations are shown to compare the various schemes as well as to demonstrate the
capability of the CPR approach. 相似文献
11.
High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions
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Liang Pan & Kun Xu 《Communications In Computational Physics》2020,28(4):1321-1351
The high-order gas-kinetic scheme (HGKS) has achieved success in simulating compressible flows with Cartesian meshes. To study the flow problems in general
geometries, such as the flow over a wing-body, the development of HGKS in general
curvilinear coordinates becomes necessary. In this paper, a two-stage fourth-order gas-kinetic scheme is developed for the Euler and Navier-Stokes solutions in the curvilinear coordinates from one-dimensional to three-dimensional computations. Based on
the coordinate transformation, the kinetic equation is transformed first to the computational space, and the flux function in the gas-kinetic scheme is obtained there and
is transformed back to the physical domain for the update of flow variables inside
each control volume. To achieve the expected order of accuracy, the dimension-by-dimension reconstruction based on the WENO scheme is adopted in the computational domain, where the reconstructed variables are the cell averaged Jacobian and the
Jacobian-weighted conservative variables. In the two-stage fourth-order gas-kinetic
scheme, the point values as well as the spatial derivatives of conservative variables at
Gaussian quadrature points have to be used in the evaluation of the time dependent
flux function. The point-wise conservative variables are obtained by ratio of the above
reconstructed data, and the spatial derivatives are reconstructed through orthogonalization in physical space and chain rule. A variety of numerical examples from the
accuracy tests to the solutions with strong discontinuities are presented to validate the
accuracy and robustness of the current scheme for both inviscid and viscous flows.
The precise satisfaction of the geometrical conservation law in non-orthogonal mesh is
also demonstrated through the numerical example. 相似文献
12.
With discretized particle velocity space, a multi-scale unified gas-kinetic
scheme for entire Knudsen number flows has been constructed based on the kinetic
model in one-dimensional case [J. Comput. Phys., vol. 229 (2010), pp. 7747-7764]. For
the kinetic equation, to extend a one-dimensional scheme to multidimensional flow is
not so straightforward. The major factor is that addition of one dimension in physical
space causes the distribution function to become two-dimensional, rather than axially
symmetric, in velocity space. In this paper, a unified gas-kinetic scheme based on the
Shakhov model in two-dimensional space will be presented. Instead of particle-based
modeling for the rarefied flow, such as the direct simulation Monte Carlo (DSMC)
method, the philosophical principal underlying the current study is a partial-differential-equation (PDE)-based modeling. Since the valid scale of the kinetic equation and
the scale of mesh size and time step may be significantly different, the gas evolution in a discretized space is modeled with the help of kinetic equation, instead of
directly solving the partial differential equation. Due to the use of both hydrodynamic and kinetic scales flow physics in a gas evolution model at the cell interface,
the unified scheme can basically present accurate solution in all flow regimes from
the free molecule to the Navier-Stokes solutions. In comparison with the DSMC and
Navier-Stokes flow solvers, the current method is much more efficient than DSMC in
low speed transition and continuum flow regimes, and it has better capability than
NS solver in capturing of non-equilibrium flow physics in the transition and rarefied
flow regimes. As a result, the current method can be useful in the flow simulation
where both continuum and rarefied flow physics needs to be resolved in a single computation. This paper will extensively evaluate the performance of the unified scheme
from free molecule to continuum NS solutions, and from low speed micro-flow to high
speed non-equilibrium aerodynamics. The test cases clearly demonstrate that the unified scheme is a reliable method for the rarefied flow computations, and the scheme
provides an important tool in the study of non-equilibrium flow. 相似文献
13.
Convergence Study of Moment Approximations for Boundary Value Problems of the Boltzmann-BGK Equation
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Manuel Torrilhon 《Communications In Computational Physics》2015,18(3):529-557
The accuracy of moment equations as approximations of kinetic gas theory
is studied for four different boundary value problems. The kinetic setting is given by
the BGK equation linearized around a globally constant Maxwellian using one space
dimension and a three-dimensional velocity space. The boundary value problems include
Couette and Poiseuille flow as well as heat conduction between walls and heat
conduction based on a locally varying heating source. The polynomial expansion of
the distribution function allows for different moment theories of which two popular
families are investigated in detail. Furthermore, optimal approximations for a given
number of variables are studied empirically. The paper focuses on approximations
with relatively low number of variables which allows to draw conclusions in particular
about specific moment theories like the regularized 13-moment equations. 相似文献
14.
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
15.
Liang Wang Zhaoli Guo Baochang Shi & Chuguang Zheng 《Communications In Computational Physics》2013,13(4):1151-1172
A comparative study is conducted to evaluate three types of lattice Boltzmann equation (LBE) models for fluid flows with finite-sized particles, including the
lattice Bhatnagar-Gross-Krook (BGK) model, the model proposed by Ladd [Ladd AJC,
J. Fluid Mech., 271, 285-310 (1994); Ladd AJC, J. Fluid Mech., 271, 311-339 (1994)], and
the multiple-relaxation-time (MRT) model. The sedimentation of a circular particle in
a two-dimensional infinite channel under gravity is used as the first test problem. The
numerical results of the three LBE schemes are compared with the theoretical results
and existing data. It is found that all of the three LBE schemes yield reasonable results in general, although the BGK scheme and Ladd's scheme give some deviations
in some cases. Our results also show that the MRT scheme can achieve a better numerical stability than the other two schemes. Regarding the computational efficiency,
it is found that the BGK scheme is the most superior one, while the other two schemes
are nearly identical. We also observe that the MRT scheme can unequivocally reduce
the viscosity dependence of the wall correction factor in the simulations, which reveals
the superior robustness of the MRT scheme. The superiority of the MRT scheme over
the other two schemes is also confirmed by the simulation of the sedimentation of an
elliptical particle. 相似文献
16.
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
17.
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
18.
Are extensions to continuum formulations for solving fluid dynamic problems in the transition-to-rarefied regimes viable alternatives to particle methods? It
is well known that for increasingly rarefied flow fields, the predictions from continuum
formulation, such as the Navier-Stokes equations lose accuracy. These inaccuracies are
attributed primarily to the linear approximations of the stress and heat flux terms in the
Navier-Stokes equations. The inclusion of higher-order terms, such as Burnett or high-order moment equations, could improve the predictive capabilities of such continuum
formulations, but there has been limited success in the shock structure calculations, especially for the high Mach number case. Here, after reformulating the viscosity and heat
conduction coefficients appropriate for the rarefied flow regime, we will show that the
Navier-Stokes-type continuum formulation may still be properly used. The equations
with generalization of the dissipative coefficients based on the closed solution of the
Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation, are solved using the
gas-kinetic numerical scheme. This paper concentrates on the non-equilibrium shock
structure calculations for both monatomic and diatomic gases. The Landau-Teller-Jeans
relaxation model for the rotational energy is used to evaluate the quantitative difference
between the translational and rotational temperatures inside the shock layer. Variations
of shear stress, heat flux, temperatures, and densities in the internal structure of the
shock waves are compared with, (a) existing theoretical solutions of the Boltzmann solution, (b) existing numerical predictions of the direct simulation Monte Carlo (DSMC)
method, and (c) available experimental measurements. The present continuum formulation for calculating the shock structures for monatomic and diatomic gases in the
Mach number range of 1.2 to 12.9 is found to be satisfactory. 相似文献
19.
A projection-based reduced order model (ROM) based on the Fourier collocation method is proposed for compressible flows. The incorporation of localized
artificial viscosity model and filtering is pursued to enhance the robustness and accuracy of the ROM for shock-dominated flows. Furthermore, for Euler systems, ROMs
built on the conservative and the skew-symmetric forms of the governing equation are
compared. To ensure efficiency, the discrete empirical interpolation method (DEIM)
is employed. An alternative reduction approach, exploring the sparsity of viscosity
is also investigated for the viscous terms. A number of one- and two-dimensional
benchmark cases are considered to test the performance of the proposed models. Results show that stable computations for shock-dominated cases can be achieved with
ROMs built on both the conservative and the skew-symmetric forms without additional stabilization components other than the viscosity model and filtering. Under
the same parameters, the skew-symmetric form shows better robustness and accuracy
than its conservative counterpart, while the conservative form is superior in terms of
efficiency. 相似文献
20.
Guo-Quan Shi Huajun Zhu & Zhen-Guo Yan 《Communications In Computational Physics》2022,31(4):1215-1241
A priori subcell limiting approach is developed for high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) methods on two-dimensional unstructured quadrilateral meshes. Firstly, a modified indicator based on
modal energy coefficients is proposed to detect troubled cells, where discontinuities
exist. Then, troubled cells are decomposed into nonuniform subcells and each subcell has one solution point. A second-order finite difference shock-capturing scheme
based on nonuniform nonlinear weighted (NNW) interpolation is constructed to perform the calculation on troubled cells while smooth cells are calculated by the CPR
method. Numerical investigations show that the proposed subcell limiting strategy on
unstructured quadrilateral meshes is robust in shock-capturing. 相似文献