On the geometric conservation law in transient flow calculations on deforming domains

This note revisits the derivation of the ALE form of the incompressible Navier-Stokes equations in order to retain insight into the nature of geometric conservation. It is shown that the flow equations can be written such that time derivatives of integrals over moving domains are avoided prior to discretisation. The geometric conservation law is introduced into the equations and the resulting formulation is discretised in time and space without loss of stability and accuracy compared to the fixed grid version. There is no need for temporal averaging remaining. The formulation applies equally to different time integration schemes within a finite element context.