Derivation of the velocity divergence constraint for low-Mach flow solvers R. McDermott ∗ , K. McGrattan, and W. E. Mell Building and Fire Research Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899-8663, USA Abstract For low-Mach flows the continuity equation and equation of state together imply a constraint on the divergence of the velocity field. This constraint is en- forced by the projection scheme employed by the Fire Dynamics Simulator (FDS) for time advancement of the primitive flow variables. The divergence constraint itself is a complex mathematical expression with many terms that account for a multitude of underlying physical processes. The present work documents the derivation of this expression and accounts for fire-specific subgrid physics such as water-droplet evaporation and heat transfer from unresolved solid objects like fire brands. 1 Introduction The purpose of this work is to document the derivation of the velocity divergence constraint used in the Fire Dynamics Simulator (FDS) projection algorithm [3, 4, 5]. This constraint is similar to that used by Bell et al. [1]. A key difference in the formulation presented here is that we consider the introduction of bulk sources of mass, momentum, and energy, which emanate from unresolved solid fuel elements (due to evaporation of water droplets [from a sprinkler, for instance], pyrolysis, drag, and convective heat transfer). In this work, our treatment of these bulk sources is general. More detail can be found in [5]. * Corresponding author. Email: randall.mcdermott@nist.gov NIST Technical Note 1487 6 November 2007