“Curl- q”: A vorticity damping artificial viscosity for essentially irrotational Lagrangian hydrodynamics calculations

Abstract

The bane of Lagrangian hydrodynamics calculations in multi-dimensions is the appearance of vorticity that causes tangling of the mesh and consequent run termination. This vorticity may be numerical or physical in origin, and is in addition to the spurious ‘‘hourglass’’ modes associated with quadrilateral or hexahedral zones that in pure form have both zero curl and divergence associated with their velocity field. The purpose of this note is to introduce a form of vorticity damping, based on a previously published edgecentered artificial viscosity [1], that extends the runtime and range of calculations over which a pure Lagrangian code can compute. Since the explicit inclusion of an artificial viscosity into the fluid equations is often referred to as the ‘‘q term’’, we denote this new term as the ‘‘curl-q’’, because it is a function of the curl of the velocity field in a zone. It is formulated in the context of the ‘‘discrete, compatible formulation of Lagrangian hydrodynamics’’ [2,3]. This employs a staggered placement of variables in space (velocity and position at nodes, with density and stresses in zones), but a predictor/corrector time integration scheme so that all variables are known at the same time level, allowing total energy to be exactly conserved [2]. This new ‘‘curl-q’’ does not resolve shock waves and is always to be utilized with an artificial viscosity that performs this task. In order to set the stage for the introduction of the new curl-q force, the edge-centered artificial viscosity given in [1] is briefly reviewed in a slightly simplified form; after this the curl-q force is formulated as an analogy to this edge-centered artificial viscosity. Numerical results are given in both 2D and 3D that display its effectiveness. In particular, results are contrasted between this new term and a recently published tensor artificial viscosity [4]. It is shown that these two forms give quite similar results in 2D. We end with a brief discussion concerning the validity of the use of this type of numerical device. In Fig. 1 is shown a quadrilateral zone with its defining points, i = 1–4, and associated median mesh vectors ~Si. These vectors point in the indicated direction and have a magnitude of the surface area that lies between their defining points in 2D or in 3D [2,5]. In terms of the median mesh vector~S1, the force exerted by the edgecentered artificial viscosity between points ‘‘1’’ and ‘‘2’’ from zone ‘‘z’’ is given by