A 3D second‐order accurate projection‐based Finite Volume code on non‐staggered, non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows

Abstract

SUMMARYIt is well known that exact projection methods (EPM) on non-staggered grids su er for the presence of non-solenoidal spurious modes. Hence, a formulation for simulating time-dependent incompressible ows while allowing the discrete continuity equation to be satisÿed up to machine-accuracy, by using a Finite Volume-based second-order accurate projection method on non-staggered and non-uniform 3D grids, is illustrated. The procedure exploits the Helmholtz-Hodge decomposition theorem for deriving an additional velocity ÿeld that enforces the discrete continuity without altering the vorticity ÿeld. This is accomplished by ÿrst solving an elliptic equation on a compact stencil that is by performing a standard approximate projection method (APM). In such a way, three sets of divergence-free normalto-face velocities can be computed. Then, a second elliptic equation for a scalar ÿeld is derived by prescribing that its additional discrete gradient ensures the continuity constraint based on the adopted linear interpolation of the velocity. Characteristics of the double projection method (DPM) are illustrated in details and stability and accuracy of the method are addressed. The resulting numerical scheme is then applied to laminar buoyancy-driven ows and is proved to be stable and e cient.

What does Finite Volume-based implicit filtering really resolve in Large-Eddy Simulations?

“…When a non-staggered collocation is chosen, solving the Navier-Stokes equations for incompressible flows at second order of accuracy in space requires some care in order to produce both divergence-free and oscillation-free solutions [12,15,17,26]. Moreover, setting the central reconstruction compels us to choose between two possible methods to evaluate the mean value of the convective flux.…”

“…Hence, according to (10), (26) and (28), one the NSF II -based discrete equation for the ith velocity component can be written…”

“…Indeed, staggering the velocity components in a second order accurate method allows the divergence-free velocity constraint to be accomplished, in a discrete sense, up to machine-precision. This method is also traditionally referred to as the MAC method [12,15,26,40]. This scheme will be denoted as SV (Staggered Velocity) discretization.…”

“…It must therefore be taken into account that the presence of the pressure gradient and the effect of the projection methods in terms of the (time-space) filtering cannot be disregardable [26,29,[31][32][33]. Moreover, the practical use of FV-based LES is often devoted to the simulation of turbulent flow problems that are not homogeneous in some or all of 13 A reviewer confirmed that better results are achieved by using a spectral LES code with a 48 Â 65 Â 96 grid supplied by a standard dynamic Smagorinsky model.…”

“…An FV-based second order accurate [17,26,42] code was run without using any turbulence modeling, on a 64 Â 64 Â 80 computational grid (details are reported in the results section), sufficiently coarse to ensure that we had in fact a no-model LES. Being the unresolved terms simply disregarded, only the smoothing implicitly induced by the FV method is present, therefore such a simulation is expected to produce a clear wrong solution.…”