Modeling a No-Slip Flow Boundary with an External Force Field

Abstract

A novel technique related to Peskin's immersed boundary approach is used to introduce solid surfaces into a simulated flow field. The Navier-Stokes equations permit the presence of an externally imposed body force that may vary in space and time. Forces are chosen to lie along a desired surface and to have a magnitude and direction opposing the local flow such that the flow is brought to rest on an element of the surface. For unsteady viscous flow the direct calculation of the needed force is facilitated by a feedback scheme in which the velocity is used to iteratively determine the desired value. In particular, we determine the surface body force from the relation f(x 5, t ) = α ∫ t 0 U(x 5, t′ ) dt′ + βU(x 5, t) for surface points x 5, velocity U, time t, and negative constants α and β. Examples are presented which include 2D flow around cylinders, 3D turbulent channel flow where one boundary is simulated with a force field, and turbulent channel flow over a riblet-covered surface. While the new method may be applied to complex geometries on a non-Cartesian mesh, we have chosen to use a simple Cartesian grid. All simulations are done with a spectral code in a single computational domain without any mapping of the mesh.