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**1 - 4**of**4**### A class of Cartesian grid embedded boundary algorithms for incompressible flow with time-varying complex geometries

"... We present a class of numerical algorithms for simulating viscous fluid problems of incom-pressible flow interacting with moving rigid structures. The proposed Cartesian grid embedded boundary algorithms employ a slightly different idea from the traditional direct-forcing Im-mersed Boundary Methods; ..."

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We present a class of numerical algorithms for simulating viscous fluid problems of incom-pressible flow interacting with moving rigid structures. The proposed Cartesian grid embedded boundary algorithms employ a slightly different idea from the traditional direct-forcing Im-mersed Boundary Methods; i.e. the proposed algorithms calculate and apply the force-density in the extended solid domain to uphold the solid velocity and hence the boundary condition at the rigid-body surface. The principle of the embedded boundary algorithm allows us to solve the fluid equations on a Cartesian grid with a set of external forces spread onto the grid points occupied by the rigid structure. The proposed algorithms use MAC (Marker And Cell) algorithm for solving the incompressible Navier-Stokes equations. Unlike projection methods, the MAC scheme incorporates the gradient of force-density in solving the pressure Poisson equation, so that the dipole force, due to the jump of pressure across the solid-fluid interface, is directly balanced by the gradient of force-density. We validate the proposed algorithms via the classical benchmark problem of flow past cylinder. Our numerical experiments show that numerical solutions of the velocity field obtained by using the proposed algorithms are smooth across the solid-fluid interface. Finally, we consider the problem of cylinder moving between two parallel plane walls. Numerical solutions of this problem obtained by using the proposed algorithms are compared with the classical asymptotic solutions. We show that the two solutions are in good agreement.

### A Second Order Virtual Node Algorithm for Navier-Stokes Flow Problems with Interfacial Forces and Discontinuous Material Properties

"... We present a numerical method for the solution of the Navier-Stokes equations in three dimensions that handles interfacial discontinuities due to singular forces and discontinuous fluid properties such as viscosity and density. We show that this also allows for the enforcement of normal stress and v ..."

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We present a numerical method for the solution of the Navier-Stokes equations in three dimensions that handles interfacial discontinuities due to singular forces and discontinuous fluid properties such as viscosity and density. We show that this also allows for the enforcement of normal stress and velocity boundary conditions on irregular domains. The method improves on results in [1] (which solved the Stokes equations in two dimensions) by providing treatment of fluid inertia as well as a new discretization of jump and boundary conditions that accurately resolves null modes in both two and three dimensions. We discretize the equations using an embedded approach on a uniform MAC grid to yield discretely divergence-free velocities that are second order accurate. We maintain our interface using the level set method or, when more appropriate, the particle level set method. We show how to implement Dirichlet (known velocity), Neumann (known normal stress), and slip velocity boundary conditions as special cases of our interface representation. The method leads to a discrete, symmetric KKT system for velocities, pressures, and Lagrange multipliers. We also present a novel simplification to the standard combination of the second order semi-Lagrangian and BDF schemes for discretizing the inertial terms. Numerical results indicate second order spatial accuracy for the velocities (L ∞ and L2) and first order for the pressure (in L∞, second order in L2). Our temporal discretization is also second order accurate.

### DEVELOPMENT OF A TWO-DIMENSIONAL NAVIER-STOKES SOLVER

, 2011

"... Approval of the thesis: ..."

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