On the relationship between the pressure and the projection function in the numerical computation of viscous incompressible flow - Pozrikidis, C (European journal of mechanics, B, Fluids, 2003)
The relationship between the pressure p and the projection function φ employed in the numerical computation of viscous incompressible flow using the fractional-step method is discussed. To leading order, the difference p− φ is proportional to the divergence of the intermediate velocity, u ∗ , computed by integrating the equation of motion in the absence of the pressure gradient. Previous authors have shown that the intermediate rate of expansion, α ∗≡∇· u ∗ is supported by numerical boundary-layers of thickness δ≃( νΔ t) 1/2, where Δ t is the time step and ν is the kinematic viscosity. We demonstrate that, in the absence of singularities due to discontinuous boundary velocity, the magnitude of α ∗ changes by an amount of order δ across the boundary layers and of higher order in the bulk of the flow, and argue that adding a computable correction to the projection function allows us to recover the pressure with temporal accuracy whose order matches that of the method used for carrying out the convection–diffusion step. When the boundary velocity is discontinuous, the normal derivative of the pressure exhibits strong singularities, and the computation of the pressure using finite-difference methods on non-staggered grids is notably sensitive to the numerical implementation. In contrast, in spite of the singular behavior of the intermediate rate of expansion, the solution of the Poisson equation for the projection function subject to the homogeneous Neumann boundary condition is less sensitive to the numerical method. Computing the projection function thus emerges as a preferred venue of approximating the pressure on non-staggered grids even under demanding conditions.