A gridless boundary condition method for the solution of the Euler equations on embedded Cartesian meshes with multigrid

Kirshman, David; Liu, F.

doi:10.1016/j.jcp.2004.05.006

Cited by 41 publications

(28 citation statements)

References 26 publications

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“…In this way, we expect to arrive at a general solution method that is flexible, efficient, and accurate for problems with complex geometries. A similar approach to the present work was reported by Kirshman and Liu,29 who used a finite difference scheme with Van Leer flux-splitting technique. In the present work, a finite volume formulation with central differencing for the Euler equations is used, and an alternative simpler gridless approach is employed in discretizing the surface.…”

Section: Introductionmentioning

confidence: 81%

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Euler Solution Using Cartesian Grid with a Gridless Least-Squares Boundary Treatment

2005

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An approach that uses gridless or meshless methods to address the problem of boundary implementation associated with the use of Cartesian grid is discussed. This method applies the gridless concept only at the interface, whereas a standard structured grid method is used everywhere else. The Cartesian grid is used to specify and distribute the computational points on the boundary surface but not to define the geometrical properties. Euler fluxes for the neighbors of cut cells are computed using the gridless method involving a local least-squares curve fit of a "cloud" of grid points. The boundary conditions implemented on the surface points are automatically satisfied in the process of evaluating the surface values in a similar least-squares fashion. The present method does not require the use of halo points. Subsonic, transonic, and supersonic flows are computed for the NACA 0012 and RAE 2822 airfoils, and the results compare well with solutions obtained by a standard Euler solver on body-fitted grids. The method is also used to calculate the flow over a three-element airfoil configuration, and the result is compared with the exact solution for this configuration obtained by conformal mapping.

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Section: Introductionmentioning

confidence: 81%

“…Use of locally refined embedded mesh as adopted in Ref. 29 would improve both convergence and accuracy of the computations. Nevertheless, this three-element test case demonstrates the flexibility offered by the Cartesian method with the present gridless boundary condition implementation.…”

Section: Three-element Airfoilmentioning

confidence: 99%

Euler Solution Using Cartesian Grid with a Gridless Least-Squares Boundary Treatment

2005

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“…Advantages of this method include the grid generation being a simple task, the ease of implementing high-order schemes as well as the low memory requirements [1,2]. Furthermore, with the Cartesian grid method, the quality of the mesh can be maintained as no mesh movement or remeshing is required during the process.…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately, this will cause difficulties when enforcing the boundary conditions directly on the grid lines (or volume surfaces). This problem has been addressed by many investigators, and great effort has been made to develop effective methods to overcome this deficiency [1,2,[11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

A Cartesian ghost‐cell multigrid Poisson solver for incompressible flows

Qian

Causon

et al. 2010

Numerical Meth Engineering

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SUMMARYIn this paper, a simple Cartesian ghost-cell multigrid Poisson solver is proposed for simulating incompressible fluid flows. The flow field is discretized efficiently on a rectangular mesh, in which solid bodies are immersed. A small number of ghost mesh cells and their symmetric image cells are distributed in the vicinity of the solid boundary. With the aid of the ghost and image cells, the Dirichlet and Neumann boundary conditions can be implemented effectively. Chorin's fractional-step projection method is adopted for the coupling of velocity and pressure for the solution of the Navier-Stokes equations. Point-wise Gauss-Seidel iteration is used to solve the pressure Poisson equation. To speed up the convergence of the solution to the corresponding linear system, sub-level coarse meshes embedded with ghost and image cells are also introduced and operated in a sequential V-cycle. Several test cases including the classical ideal incompressible flow around a cylinder, a lid-driven cavity flow and viscous flow past a fixed/rotating cylinder are presented to demonstrate the accuracy and efficiency of the current approach.

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“…If meshless technique is employed then the diffusive flux can be obtained as the product of point fluxes and least square (LSQ) coefficients along coordinate directions prior to time integration to steady state solution. Irregular boundaries and fine broken cells at the wall, which were constraints to computational accuracy of Cartesian mesh solvers, are now implemented efficiently by meshless wall boundary techniques [3]. The accuracy with which shock waves reflecting at the wall boundaries can be captured depends both on the computational schemes used and the shock structure model adopted.…”

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confidence: 99%

Development of meshless method boundary conditions for high speed flows

2017

IJLTET

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SHOCK STRUCTURE MODELDuring numerical flow simulation, diffusion can be induced through artificial viscosity term to control development of flow. Early researchers included the dissipation term in the formulation of finite volume Euler equations to suppress odd-even point oscillations arising from pressure-velocity coupling. It was found that second difference diffusion terms with second and fourth difference pressure coefficients rendered wiggle-free computation. The highorder central differencing schemes produce accurate results in smooth flow regions but give rise to oscillations at shocks or discontinuities. This problem has been resolved through special upwinding methods and by Roe linearization technique [1], with or without characteristic decomposition. The development of Euler codes has paralleled the development of positivity or monotonicity preserving schemes that satisfy Jameson's local extremum diminishing principle (LED criterion) [2] in order that physically meaningful values of pressure and density could be computed. Customarily, the residual is evaluated as convective and diffusive fluxes separately in order to improve the accuracy of computation of flow properties. If meshless technique is employed then the diffusive flux can be obtained as the product of point fluxes and least square (LSQ) coefficients along coordinate directions prior to time integration to steady state solution. Irregular boundaries and fine broken cells at the wall, which were constraints to computational accuracy of Cartesian mesh solvers, are now implemented efficiently by meshless wall boundary techniques [3]. The accuracy with which shock waves reflecting at the wall boundaries can be captured depends both on the computational schemes used and the shock structure model adopted. Even after a century old research, a concrete shock wave development model does not exist. Among various models suggested by researchers, the three-shock theory [4, 5] (3ST) putforth by von Neumann stands out as the most realistic one to describe the commonly encountered shock wave phenomena. In situations where only a regular shock reflection is expected as in ducted flows, a Mach wave reflection too may occur. A consequence of this effect is the appearance of Mach stem at the wall accompanied by a regular shock wave. Now, according to 3ST, non-regular shock reflection called Mach reflection (MR) is produced by three intersecting shock waves: incident wave, reflected wave and Mach stem. Their point of intersection called triple point (T) may move toward, away or parallel to the reflecting surface, giving rise to direct, inverse or stationary MR as shown in the Figures 1. These occurrences are termed as Mach stem and it forms the subject matter of this paper.

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A gridless boundary condition method for the solution of the Euler equations on embedded Cartesian meshes with multigrid

Cited by 41 publications

References 26 publications

Euler Solution Using Cartesian Grid with a Gridless Least-Squares Boundary Treatment

Euler Solution Using Cartesian Grid with a Gridless Least-Squares Boundary Treatment

A Cartesian ghost‐cell multigrid Poisson solver for incompressible flows

Development of meshless method boundary conditions for high speed flows

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