A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions

Abstract

A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original (Peskin's) immersed boundary method. Unlike Peskin's method, boundary forces are regarded as Lagrange multipliers that are used to satisfy the no-slip condition. The incompressible Navier-Stokes equations are discretized on an unbounded staggered Cartesian grid and are solved in a finite number of operations using lattice Green's function techniques. These techniques are used to automatically enforce the natural free-space boundary conditions and to implement a novel block-wise adaptive grid that significantly reduces the run-time cost of solutions by limiting operations to grid cells in the immediate vicinity and near-wake region of the immersed surface. These techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half-explicit Runge-Kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation, incompressibility constraint, and no-slip constraint. Linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation-free nested projection technique. The algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems, e.g. discrete Poisson problems, that are compatible with recently developed parallel fast multipole methods for difference equations. Numerical experiments on low-aspect-ratio flat plates and spheres at Reynolds numbers up to 3,700 are used to verify the accuracy and physical fidelity of the formulation.

Immersed Boundary Lattice Green Function methods for External Aerodynamics

“…a pitching airfoil or a rotating rotor), it is convenient to rewrite the IB formulation of the incompressible Navier-Stokes equations (1) in an accelerating frame of reference, but then using the velocity in the inertial frame of reference. The main reason behind this manipulation is due to the source terms arising from the accelerating frame of reference.…”

“…Then the source terms can be incorporated into the pressure gradient and nonlinear terms and the boundary condition on the velocity at large distance from the body tends to zero, that is u(x, t) → 0 for |x| → ∞. The system of equations (1) in the new frame of reference and with the change of variables for the velocity becomes…”

“…The operators δ(∂B(t), f ∂B , x) and δ(∂B(t), u, ξ) denote the δ-convolutions in equation (1). Finally, the vectors X a (ξ, t) = X(ξ, t) − R(t) and u ∂B,a (ξ, t) = u ∂B (ξ, t) − u r (X a (ξ, t), t) represent the position and velocity of a point on ∂B(t) in the accelerating frame of reference.…”

“…A schematic representation of a cell of the mesh is depicted in figure [1]. The notation adopted to define the mesh object Q whose a given grid function g belong to is g ∈ Q := {V, E, F, C}, where V denotes vertices, E denotes edges, F denotes faces and C denotes cells.…”

“…The notation adopted in this case for a generic operator T that operates on a grid function g 1 ∈ C and produces a grid function g 2 ∈ F is T : R C → R F . In table [1], we report the definition of the differential continuous and discrete counterparts of the operators necessary to define system (2). By Operator Continuous Discrete using the conventions just defined, the semi-discrete form of equation (2) becomes…”