An efficient second-order projection method for viscous incompressible flow

Bell, John B.; Howell, Louis H.; Colella, Phillip

doi:10.2514/6.1991-1560

Cited by 69 publications

(92 citation statements)

References 8 publications

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“…We discretize the low Mach number equation set derived in the previous section using an extension of the second-order accurate projection methodology developed for incompressible flows ( Bell et al 1989( Bell et al , 1991Almgren et al 1996Almgren et al , 1998Bell & Marcus 1992) and extended to low Mach number combustion ( Pember et al 1998;Day & Bell 2000) and to small-scale reacting flow for SNe Ia (Bell et al 2004). We refer the reader to the above references for numerical examples demonstrating the second-order accuracy of the overall methodology and for many of the details of projection methods.…”

Section: Numerical Methodology For the Low Mach Number Modelmentioning

confidence: 99%

Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics

Almgren

Bell

Rendleman

et al. 2006

ApJ

125

135

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We introduce a low Mach number equation set for the large-scale numerical simulation of carbon-oxygen white dwarfs experiencing a thermonuclear deflagration. Since most of the interesting physics in a Type Ia supernova transpires at Mach numbers from 0.01 to 0.1, such an approach enables both a considerable increase in accuracy and a savings in computer time compared with frequently used compressible codes. Our equation set is derived from the fully compressible equations using low Mach number asymptotics, but without any restriction on the size of perturbations in density or temperature. Comparisons with simulations that use the fully compressible equations validate the low Mach number model in regimes where both are applicable. Comparisons to simulations based on the more traditional anelastic approximation also demonstrate the agreement of these models in the regime for which the anelastic approximation is valid. For low Mach number flows with potentially finite amplitude variations in density and temperature, the low Mach number model overcomes the limitations of each of the more traditional models and can serve as the basis for an accurate and efficient simulation tool.

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Section: Numerical Methodology For the Low Mach Number Modelmentioning

confidence: 99%

Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics

Almgren

Bell

Rendleman

et al. 2006

ApJ

125

135

View full text Add to dashboard Cite

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“…Here µ is the dynamic viscosity coefficient, k is the diffusive coefficient for c, and H c is the source term for c. In general one could advect an arbitrary number of scalars, either passively or conservatively. The development of the single grid second-order projection methodology for the incompressible Navier-Stokes equations is discussed in a series of papers by Bell, Colella, and Glaz [6], Bell, Colella, and Howell [7], and Almgren, Bell, and Szymczak [4]. The method discussed here is an adaptive version of the algorithm presented by Almgren et al [4], generalized to include finite amplitude density variation as originally discussed in Bell and Marcus [8].…”

Section: Introductionmentioning

confidence: 99%

“…In this approach all grid levels are advanced with the same time step which is determined by the data at the finest level. Minion uses the treatment of the convection terms discussed in Bell, Colella, and Howell [7] in which a MAC projection is used as an intermediate step in the convection algorithm in order to enforce incompressibility at the half-time level. He also uses an approximate cell-centered projection based on the MAC projection to enforce the divergence constraint at the end of the time step.…”

Section: Introductionmentioning

confidence: 99%

A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier–Stokes Equations

Almgren

Bell

Colella

et al. 1998

Journal of Computational Physics

Self Cite

461

473

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In this paper we present a method for solving the equations governing timedependent, variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids. The method is based on a projection formulation in which we first solve advection-diffusion equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergence-free vector fields. Our treatment of the first step uses a specialized second-order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high Reynolds number flow. Density and other scalars are advected in such a way as to maintain conservation, if appropriate, and free-stream preservation. Our approach to adaptive refinement uses a nested hierarchy of logically-rectangular girds with simultaneous refinement of the girds in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time, fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized. The single grid algorithm is described briefly, but the emphasis here is on the time-stepping procedure for the adaptive hierarchy. Numerical examples are presented to demonstrate the algorithms's accuracy and convergence properties, and illustrate the behavior of the method. An additional example demonstrates the performance of the method on a more realistic problem, namely, a three-dimensional variable density shear layer.

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“…This splitting is known as the projection method, after Bell et al [1991], because the predicted velocities are projected onto a divergence-free subspace (required by (2.5c)). Provided the discrete divergence and Laplacian operators commute, this splitting method is also exact away from boundaries.…”

Section: Algorithm Overviewmentioning

confidence: 99%

Simulation of the Navier–Stokes equations in three dimensions with a spectral collocation method

Subich

Lamb

Stastna

2013

Numerical Methods in Fluids

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SUMMARYThis paper describes a nonlinear, three‐dimensional spectral collocation method for the simulation of the incompressible Navier–Stokes equations under the Boussinesq approximation, motivated by geophysical and environmental flows. These flows are driven by the interaction of stratified fluid with topography, which this model accurately accounts for by using a mapped coordinate system. The spectral collocation method is implemented with both a Fourier trigonometric expansion and the Chebyshev polynomials, as appropriate for the domain boundary conditions. The coordinate mapping prohibits the use of existing, fast solution methods that rely on the separation of variables, so a preconditioner based on the approximate solution of a corresponding finite‐difference problem with geometric multigrid is used. The model is parallelized with the Message Passing Interface library, and it runs effectively on shared and distributed‐memory systems. Copyright © 2013 John Wiley & Sons, Ltd.

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An efficient second-order projection method for viscous incompressible flow

Cited by 69 publications

References 8 publications

Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics

Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics

A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier–Stokes Equations

Simulation of the Navier–Stokes equations in three dimensions with a spectral collocation method

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