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Systematic derivation of jump conditions for the immersed interface method in three-dimensional flow simulation
- SIAM J. Sci. Comput
"... Abstract. In this paper, we systematically derive jump conditions for the immersed interface ..."
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Abstract. In this paper, we systematically derive jump conditions for the immersed interface
A second order virtual node method for elliptic problems with interfaces and . . .
- JOURNAL OF COMPUTATIONAL PHYSICS
, 2011
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Locally corrected semi-Lagrangian methods for Stokes flow with moving elastic interfaces
, 2007
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A well-conditioned augmented system for solving Navier–Stokes equations in irregular domains
- JOURNAL OF COMPUTATIONAL PHYSICS
, 2009
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A Velocity Decomposition Approach for Moving Interfaces in Viscous Fluids
"... We present a second-order accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the Navier-Stokes equations, with a singular force due to the stretching of the moving interface. We decompose the veloc ..."
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We present a second-order accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the Navier-Stokes equations, with a singular force due to the stretching of the moving interface. We decompose the velocity into a “Stokes ” part and a “regular ” part. The first part is determined by the Stokes equations and the singular interfacial force. The Stokes solution is obtained using the immersed interface method, which gives second-order accurate values by incorporating known jumps for the solution and its derivatives into a finite difference method. The regular part of the velocity is given by the Navier-Stokes equations with a body force resulting from the Stokes part. The regular velocity is obtained using a time-stepping method that combines the semi-Lagrangian method with the backward difference formula. Because the body force is continuous, jump conditions are not necessary. For problems with stiff boundary forces, the decomposition approach can be combined with fractional time-stepping, using a smaller time step to advance the interface quickly by Stokes flow, with the velocity computed using boundary integrals. The small time steps maintain numerical stability, while the overall solution is updated on a larger time step to reduce computational cost.
A Second Order Virtual Node Algorithm for Stokes Flow Problems with Interfacial Forces and Discontinuous Material Properties
"... We present a numerical method for the solution of the Stokes equations that handles interfacial discontinuities due to both singular forces and discontinuous fluid properties such as viscosity and density. The discretization couples a Lagrangian representation of the material interface with an Euler ..."
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We present a numerical method for the solution of the Stokes equations that handles interfacial discontinuities due to both singular forces and discontinuous fluid properties such as viscosity and density. The discretization couples a Lagrangian representation of the material interface with an Eulerian representation of the fluid velocity and pressure. The method is efficient, easy to implement and yields discretely divergence-free velocities that are second order accurate. No knowledge of the jumps on the fluid variables and their derivatives is required along the interface. We discretize the equations using an embedded approach on a uniform MAC grid employing virtual nodes and duplicated cells at the interfaces. These additional degrees of freedom allow for accurate resolution of discontinuities in the fluid stress at the material interface but require a Lagrange multiplier term to enforce continuity of the fluid velocity. We provide a novel discretization of this term that accurately resolves the constant pressure null modes. We show that the accurate resolution of these modes accelerates the overall speed of our simulations. Interfaces are represented with a hybrid Lagrangian/level set method. The discrete coupled equations for the velocity, pressure and Lagrange multipliers are in the form of a symmetric KKT system. Numerical results indicate second order accuracy for the velocities and first order accuracy for the pressure (in L ∞).
1 A High-Order Cut-Cell Method for Numerical Simulation of Hypersonic-Boundary Transition with Arbitrary Surface Roughness
"... Hypersonic boundary-layer transition can be affected significantly by surface roughness. Many important mechanisms which involve transition induced by arbitrary roughness are not well understood. In this paper, we propose a new high-order cut cell method which combined the non-uniform finite differe ..."
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Hypersonic boundary-layer transition can be affected significantly by surface roughness. Many important mechanisms which involve transition induced by arbitrary roughness are not well understood. In this paper, we propose a new high-order cut cell method which combined the non-uniform finite difference method for discrete points near the curvilinear boundary and shock-fitting method for the bow shock. The receptivity process induced by interaction of Mach 5.92 flow over flat plate under the combination effect of two-dimensional surface roughness and blow-suction is investigated. Both steady state solutions and unsteady solutions have been obtained by using the new method. For steady flow with roughness, there is significant change inside the boundary layer with flow separation before and after the roughness element. For unsteady flow, the results for flow instability induced by both blow-suction slot and roughness are obtained and analyzed by Linear Stability Theory (LST). These results show that the roughness element with height to be half the boundary layer thickness may delay the hypersonic transition. I.
A Second Order Virtual Node Method for Poisson Interface Problems on Irregular Domains
"... We present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities on an irregular domain. We discretize the equations using an embedded approach on a uniform Cartesian grid employing virtual ..."
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We present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities on an irregular domain. We discretize the equations using an embedded approach on a uniform Cartesian grid employing virtual nodes at interfaces and boundaries. A variational method is used to define numerical stencils near these special virtual nodes and a Lagrange multiplier approach is used to enforce jump conditions and Dirichlet boundary conditions. Our combination of these two aspects yields a symmetric positive definite discretization. In the general case, we obtain the standard 5-point stencil away from the interface. For the specific case of interface problems with continuous coefficients, we present a discontinuity removal technique that admits use of the standard 5-point finite difference stencil everywhere in the domain. Numerical experiments indicate second order accuracy in L∞.
