Analysis of an immersed boundary method for three-dimensional flows in vorticity formulation

Abstract

a b s t r a c tThis article presents numerical analysis and practical considerations for three-dimensional flow computation using an implicit immersed boundary method. The Euler equations, or half a step of the Navier-Stokes equations when using fractional step algorithms, are investigated in their vorticity formulation. The context of flow computation around an arbitrarily shaped body is especially investigated.In conventional immersed boundary methods using vorticity, singular vortex are dispatched over the body surface. In the present study, one prefers using sources of potential velocity field, dispatched on the body, whose nature is not vorticity. Such a formulation is compatible to the Euler equations. In practice, these sources of potential flow produce a velocity through this surface, aiming in practice at cancelling a flow-through velocity.This article focuses on the use of the source-to-flow-through linear application, its properties being the key points for fast convergence. Its self-adjointness, or lack thereof, conditioning and preconditioning aspects are investigated. It follows that computing a velocity field with no-flow-through conditions in complex geometry, when using the source-toflow-through linear application, can be achieved for 4/3 of the computational cost of standard Poisson equation in a Cartesian box.The robustness of immersed boundaries is especially interesting when used together with vortex-in-cell methods, well known for their robustness in time and their ability to compute accurately convective effects. A few examples, based on real-world geometries, illustrate the method capabilities.

Mucus dynamics subject to air and wall motion

“…Lagrangian velocity-vorticity methods, in three-dimensional context, have been proved to be robust, efficient and linearly scalable in complex geometry [15] under complex boundary conditions [16] with spatial [12] or temporal [12] strong variations, and even in multi-scale context [14]. Consequently, these methods offer a good framework for future work and have been chosen for the present study, in a simplified context.…”

“…Initial mucus location is a film of height H = L/3 at the bottom of the computational box. First, we introduce a vorticity-to-velocity operator in the spirit of [15] : on a domain Ω =]0, A[×]0, L[, one takes at each time a vorticity field ω : Ω → R and gets a velocity field u : Ω → R 2 . This linear application is denoted u = Aω, and it is defined by means of the two following embedded Poisson equations :…”

“…For the Laplacian as well as for the derivations, centered second order finite difference schemes are used, except for fluxes at boundaries for which second order one-sided schemes are used. Using such a vorticity-to-velocity operation A, one can use time splitting algorithm in order to split transport, gravity and momentum diffusion in equation (15). These three steps, for which initial condition is the final value of the step performed before, are as follows :…”

Vorticity vector-potential method for 3D viscous incompressible flows in time-dependent curvilinear coordinates

“…One of the numerical methods used to simulate such flows is the immersed boundary method, which has been applied to vorticity based formulation [11]. Another approach is to construct time-dependent curvilinear coordinates.…”

Hybrid grid–particle methods and Penalization: A Sherman–Morrison–Woodbury approach to compute 3D viscous flows using FFT

“…FFT-based solvers provide excellent efficiency even for very large flows [6], particularly for unbounded or periodic domains with no obstacles [7]. Kernel methods (such as Biot-Savard laws for the Poisson equation or Stokeslets for the Stokes equation) are usually less efficient, but they can handle boundaries naturally using integral equations on surfaces [32], and are well fitted for infinite domain problems [30]. Their main drawback is that the expression of the kernel for variable flow parameters (such as viscosity) does not exist, and therefore these methods cannot be applied.…”