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1.
The generalized Riemann problem (GRP) scheme for the Euler equations and gas-kinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a slightly better computational efficiency, and has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solvers. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an equilibrium one through particle collisions. The different mechanism in the flux evaluation deviates their numerical performance. Through this study, we may conclude scientifically that it may NOT be valid to use the Euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution. To adapt the Navier–Stokes (NS) equations is NOT valid either because the NS equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity. This fact alludes to the consistency of the Euler and Navier–Stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high Mach number flows. The development of numerical algorithm is similar to the modeling process in deriving the governing equations, but the control volume here cannot be shrunk to zero. 相似文献
2.
We develop a numerical method to simulate a two-phase compressible flow
with sharp phase interface on Eulerian grids. The scheme makes use of a level set to
depict the phase interface numerically. The overall scheme is basically a finite volume
scheme. By approximately solving a two-phase Riemann problem on the phase interface,
the normal phase interface velocity and the pressure are obtained, which is used
to update the phase interface and calculate the numerical flux between the flows of
two different phases. We adopt an aggregation algorithm to build cell patches around
the phase interface to remove the numerical instability due to the breakdown of the
CFL constraint by the cell fragments given by the phase interface depicted using the
level set function. The proposed scheme can handle problems with tangential sliping
on the phase interface, topological change of the phase interface and extreme contrast
in material parameters in a natural way. Though the perfect conservation of the mass,
momentum and energy in global is not achieved, it can be quantitatively identified in
what extent the global conservation is spoiled. Some numerical examples are presented
to validate the numerical method developed. 相似文献
3.
G. Th?mmes J. Becker M. Junk A.K. Vaikuntam D. Kehrwald A. Klar K. Steiner A. Wiegmann 《Journal of computational physics》2009,228(4):1139-1156
We consider the lattice Boltzmann method for immiscible multiphase flow simulations. Classical lattice Boltzmann methods for this problem, e.g. the colour gradient method or the free energy approach, can only be applied when density and viscosity ratios are small. Moreover, they use additional fields defined on the whole domain to describe the different phases and model phase separation by special interactions at each node. In contrast, our approach simulates the flow using a single field and separates the fluid phases by a free moving interface. The scheme is based on the lattice Boltzmann method and uses the level set method to compute the evolution of the interface. To couple the fluid phases, we develop new boundary conditions which realise the macroscopic jump conditions at the interface and incorporate surface tension in the lattice Boltzmann framework. Various simulations are presented to validate the numerical scheme, e.g. two-phase channel flows, the Young–Laplace law for a bubble and viscous fingering in a Hele-Shaw cell. The results show that the method is feasible over a wide range of density and viscosity differences. 相似文献
4.
5.
The lattice Boltzmann method (LBM) for two-phase flow simulation is often hindered by insufficient resolution at the interface. As a result, the LBM simulation of bubbles in bubbling flows is commonly limited to spherical or slightly deformed bubble shapes. In this study, the adaptive mesh refinement method for the LBM is developed to overcome such a problem. The approach for this new method is based on the improved interaction potential model, which is able to maintain grid-independent fluid properties in the two-fluid phases and at the interface. The LBM–AMR algorithm is described, especially concerning the LBM operation on a non-uniform mesh and the improved interaction potential model. Numerical simulations have been performed to validate the method in both single phase and multiphase flows. The 2D and 3D simulations of the buoyant rise of bubbles are conducted under various conditions. The agreement between the simulated bubble shape and velocity with experiments illustrates the capability of the LBM–AMR approach in predicting bubble dynamics even under the large bubble deformation conditions. Further, the LBM–AMR technique is capable of simulating a complex topology change of the interface. Integration of LBM with AMR can significantly improve the accuracy and reduce computation cost. The method developed in this study may appreciably enhance the capability of LBM in the simulation of complex multiphase flows under realistic conditions. 相似文献
6.
A Discontinuous Galerkin Method Based on a BGK Scheme for the Navier-Stokes Equations on Arbitrary Grids
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A discontinuous Galerkin Method based on a Bhatnagar-Gross-Krook
(BGK) formulation is presented for the solution of the compressible
Navier-Stokes equations on arbitrary grids. The idea behind this
approach is to combine the robustness of the BGK scheme with the
accuracy of the DG methods in an effort to develop a more accurate,
efficient, and robust method for numerical simulations of viscous
flows in a wide range of flow regimes. Unlike the traditional
discontinuous Galerkin methods, where a Local Discontinuous Galerkin
(LDG) formulation is usually used to discretize the viscous fluxes
in the Navier-Stokes equations, this DG method uses a BGK scheme to
compute the fluxes which not only couples the convective and
dissipative terms together, but also includes both discontinuous and
continuous representation in the flux evaluation at a cell interface
through a simple hybrid gas distribution function. The developed
method is used to compute a variety of viscous flow problems on
arbitrary grids. The numerical results obtained by this BGKDG method
are extremely promising and encouraging in terms of both accuracy
and robustness, indicating its ability and potential to become not
just a competitive but simply a superior approach than the current
available numerical methods. 相似文献
7.
Ghost Fluid方法与双介质可压缩流动计算 总被引:1,自引:1,他引:0
应用带有Isobaric修正的GhostFluid方法配合LevelSet方法计算可压缩双介质无粘流动.该方法可以消除计算流体界面时所产生的数值跳动和耗散,且编程上比界面跟踪法简单.应用WENO格式数值求解欧拉方程和LevelSet方程,对由刚性气体状态方程所支配的一二维双介质流动进行数值计算,得到了分辨率较高的计算结果. 相似文献
8.
9.
An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity 总被引:1,自引:0,他引:1
A formally second-order accurate immersed boundary method is presented and tested in this paper. We apply this new scheme to simulate the flow past a circular cylinder and study the effect of numerical viscosity on the accuracy of the computation by comparing the numerical results with those of a first-order method. The numerical evidence shows that the new scheme has less numerical viscosity and is therefore a better choice for the simulation of high Reynolds number flows with immersed boundaries. 相似文献
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12.
A second-order volume-of-fluid method (VOF) is presented for interface tracking and sharp interface treatment on general structured grids. Central to the new method is a second-order distance function construction scheme on a general structured grid based on the reconstructed interface. A novel technique is developed for evaluating the interface normal vector using the distance function. With the normal vector, the interface is reconstructed from the volume fraction function via a piecewise linear interface calculation (PLIC) scheme on the computational domain. Several numerical tests are conducted to demonstrate the accuracy and efficiency of the present method. In general, the new VOF method is more efficient than both the high-order level set and the coupled level set and volume-of-fluid (CLSVOF) methods. The results from the new method are better than those from the benchmark VOF method, particularly in the under-resolved regions, and are comparable to those from the CLSVOF method. Breaking waves over a submerged bump and around a wedge-shaped bow are simulated to demonstrate the application of the new method and sharp interface treatment in a two-phase flow solver on curvilinear grids. The computational results are in good agreement with the available experimental measurements. 相似文献
13.
In order to simulate multiscale problems such as turbulent flows effectively, the high-order accurate reconstruction based on mini- mized dispersion and controllable dissipation (MDCD) is implemented in the second-order accurate gas-kinetic scheme (GKS) to improve the accuracy and resolution. MDCD is firstly extended to non-uniform grids through the modification of dissipation and dispersion coefficients for uniform grids based on the local stretch ratio. Remarkable improvements in accuracy and resolution are achieved on general grids. Then a new scheme, MDCD-GKS is constructed, with the help of MDCD reconstruction, not only for conservative variables, but also for their gradients. MDCD-GKS shows good accuracy and efficiency in typical numerical tests. MDCD-GKS is also coupled with the improved delayed detached-eddy simulation (IDDES) hybrid model and applied in the fine simulation of turbulent flow around a cylinder, and the prediction is in good agreement with experiments when using the relatively coarse grid. The high accuracy and resolution of the developed GKS guarantee its high efficiency in practical applications. 相似文献
14.
This paper presents efficient second-order kinetic schemes on unstructured meshes for both compressible unsteady and incompressible steady flows. For compressible unsteady flows, a time-dependent gas distribution function with a discontinuous particle velocity space at a cell interface is constructed and used for the evaluations of both numerical fluxes and conservative flow variables. As a result, a compact scheme on the unstructured meshes is developed. For incompressible steady flows, a continuous second-order gas-kinetic BGK type scheme is presented, for which the time-dependent gas distribution function with a continuous particle velocity is used on unstructured meshes. The efficiency of the schemes lies in the fact that the slopes of the flow variables inside each cell can be constructed using values of the flow variables within that cell only without involving neighboring cells. Therefore, even with the stencil of a first-order scheme, a high resolution method is constructed. Numerical examples are presented which are compared with the benchmark solutions and the experimental measurements. 相似文献
15.
We present a new numerical method for calculating an evolving 2D Hele-Shaw interface when surface tension effects are neglected. In the case where the flow is directed from the less viscous fluid into the more viscous fluid, the motion of the interface is ill-posed; small deviations in the initial condition will produce significant changes in the ensuing motion. This situation is disastrous for numerical computation, as small roundoff errors can quickly lead to large inaccuracies in the computed solution. Our method of computation is most easily formulated using a conformal map from the fluid domain into a unit disk. The method relies on analytically continuing the initial data and equations of motion into the region exterior to the disk, where the evolution problem becomes well-posed. The equations are then numerically solved in the extended domain. The presence of singularities in the conformal map outside of the disk introduces specific structures along the fluid interface. Our method can explicitly track the location of isolated pole and branch point singularities, allowing us to draw connections between the development of interfacial patterns and the motion of singularities as they approach the unit disk. In particular, we are able to relate physical features such as finger shape, side-branch formation, and competition between fingers to the nature and location of the singularities. The usefulness of this method in studying the formation of topological singularities (self intersections of the interface) is also pointed out. 相似文献
16.
复杂地表条件下快速推进法地震波走时计算 总被引:3,自引:0,他引:3
为获得计算复杂地表条件下地震波走时的方法,对常规快速推进法(Fast marching method,简写为FMM)做了两点改进:①引入不等距差分格式,用于地表和界面处的局部走时计算;②增加新的网格节点类型,用于实现不规则边界条件下的窄带技术.通过对算法的计算精度、效率及实例的分析可得,算法计算精度高,其中反射波走时计算的精度高于初至波;不会因为处理不规则边界而引入过多额外的计算量;能灵活稳定地处理各种强起伏复杂地形、近地表及地下复杂介质等问题.计算结果满足复杂地表条件下地震波的传播规律. 相似文献
17.
《Journal of computational physics》2002,175(2):674-701
In this paper we deal with the construction of hybrid flux-vector-splitting (FVS) schemes and flux-difference-splitting (FDS) schemes for a two-phase model for one-dimensional flow. The model consists of two mass conservation equations (one for each phase) and a common momentum equation. The complexity of this model, as far as numerical computation is concerned, is related to the fact that the flux cannot be expressed in terms of its conservative variables. This is the motivation for studying numerical schemes which are not based on (approximate) Riemann solvers and/or calculations of Jacobian matrix. This work concerns the extension of an FVS type scheme, a Van Leer type scheme, and an advection upstream splitting method (AUSM) type scheme to the current two-phase model. Our schemes are obtained through natural extensions of corresponding schemes studied by Y. Wada and M.-S. Liou (1997, SIAM J. Sci. Comput.18, 633–657) for Euler equations. We explore the various schemes for flow cases which involve both fast and slow transients. In particular, we demonstrate that the FVS scheme is able to capture fast-propagating acoustic waves in a monotone way, while it introduces an excessive numerical dissipation at volume fraction contact (steady and moving) discontinuities. On the other hand, the AUSM scheme gives accurate resolution of contact discontinuities but produces oscillatory approximations of acoustic waves. This motivates us to propose other hybrid FVS/FDS schemes obtained by removing numerical dissipation at contact discontinuities in the FVS and Van Leer schemes. 相似文献
18.
Jiangxing Wang Ziqing Xie & Chuanmiao Chen 《advances in applied mathematics and mechanics.》2015,7(6):796-817
An implicit discontinuous Galerkin method is introduced to solve the time-domain
Maxwell's equations in metamaterials. The Maxwell's equations in metamaterials
are represented by integral-differential equations. Our scheme is based on discontinuous
Galerkin method in spatial domain and Crank-Nicolson method in temporal
domain. The fully discrete numerical scheme is proved to be unconditionally stable.
When polynomial of degree at most $p$ is used for spatial approximation, our scheme is
verified to converge at a rate of $\mathcal{O}(τ^2+h^{p+1/2})$. Numerical results in both 2D and 3D
are provided to validate our theoretical prediction. 相似文献
19.
Simulations of the pinch-off of an inviscid fluid column are carried out based upon a potential flow model with capillary forces. The interface location and the time evolution of the free surface boundary condition are both approximated by means of level set techniques on a fixed domain. The interface velocity is obtained via a Galerkin boundary integral solution of the 3D axisymmetric Laplace equation. A short-time analytical solution of the Raleigh–Taylor instability in a liquid column is available, and this result is compared with our numerical experiments to validate the algorithm. The method is capable of handling pinch-off and after pinch-off events, and simulations showing the time evolution of the fluid tube are presented. 相似文献
20.
Satoshi Ii Kazuyasu Sugiyama Shintaro Takeuchi Shu Takagi Yoichiro Matsumoto Feng Xiao 《Journal of computational physics》2012,231(5):2328-2358
An interface capturing method with a continuous function is proposed within the framework of the volume-of-fluid (VOF) method. Being different from the traditional VOF methods that require a geometrical reconstruction and identify the interface by a discontinuous Heaviside function, the present method makes use of the hyperbolic tangent function (known as one of the sigmoid type functions) in the tangent of hyperbola interface capturing (THINC) method [F. Xiao, Y. Honma, K. Kono, A simple algebraic interface capturing scheme using hyperbolic tangent function, Int. J. Numer. Methods Fluids 48 (2005) 1023–1040] to retrieve the interface in an algebraic way from the volume-fraction data of multi-component materials. Instead of the 1D reconstruction in the original THINC method, a multi-dimensional hyperbolic tangent function is employed in the present new approach. The present scheme resolves moving interface with geometric faithfulness and compact thickness, and has at least the following advantages: (1) the geometric reconstruction is not required in constructing piecewise approximate functions; (2) besides a piecewise linear interface, curved (quadratic) surface can be easily constructed as well; and (3) the continuous multi-dimensional hyperbolic tangent function allows the direct calculations of derivatives and normal vectors. Numerical benchmark tests including transport of moving interface and incompressible interfacial flows are presented to validate the numerical accuracy for interface capturing and to show the capability for practical problems such as a stationary circular droplet, a drop oscillation, a shear-induced drop deformation and a rising bubble. 相似文献