Mathematical derivation of a finite volume formulation for laminar flow in complex geometries

Abstract

SUMMARYThis paper treats the mathematical derivation of a novel formulation of the Navier-Stokes equation for general non-orthogonal curvilinear co-ordinates. The covariant velocity components are solved in this FVM formulation, which leads to the pressure-velocity coupling becoming relatively easy to handle at the expense of a more complicated expression of the convective and diffusive fluxes. When a velocity component is solved at a point P, the neighbouring velocities are projected in the direction of the velocity component at the point P. Thus the base vectors are changed at the neighbouring points. This renders a simpler expression for the covariant derivatives. Neither the Cristoffel symbol nor its derivatives need be computed. This contributes to the accuracy of the formulation. The procedure of changing the base vectors affects only the convected velocity. The convecting term (dot product of velocity and area) is calculated without any change of the base vectors. The same is true for the operator on the covariant velocity in the diffusion term.It is shown that when using upwind differencing the use of projected velocities gives better results than when curvature effects are included in the source term. The discretized equations are written in a form which enables the use of the tridiagonal matrix algorithm (TDMA). The equations can be solved using either the SIMPLEC or the PIS0 procedure.Two examples of laminar flows are given.

A Unified Method for Computing Incompressible and Compressible Flows in Boundary-Fitted Coordinates

“…10). A laminar computation was performed for a Reynolds number of 4 × 10 5 . The resulting Mach number contours are shown in Fig.…”

“…However, for incompressible flows they are attractive, because no artificial measures need to be taken to avoid spurious pressure oscillations and the physical boundary conditions suffice. Furthermore, for the incompressible case recently accurate staggered discretizations in general coordinates have appeared; some references, apart from those just quoted, are [5,16,24,34,42,52,53]. Given an accurate staggered scheme in general coordinates, inclusion of compressibility is quite feasible along the lines already laid out by Harlow and Amsden.…”

Covariant Velocity-Based Calculation Procedure With Nonstaggered Grids for Computation of Pulsatile Flows

“…Depending on the choice of the velocity vector components (Cartesian, covariant, contravariant) and the grid configuration (staggered or nonstaggered), there exist various forms of forrnulation. For example, Karki [I], Karki and Patankar 121, and Davidson and Hedberg [3] used the covariant components of velocity together with a staggered grid system. Dernirdzic and co-workers [4, 51 employed the contravariant components of velocity as the dependent variables in a staggered grid system, while in another study, Demirdzic and Peric [6] used the Cartesian velocity components as dependent variables together with a collocated grid arrangement.…”

Numerical simulation of vortex shedding past triangular cylinders at high Reynolds number using a k‐ε turbulence model

“…In Figure 13 the (V)-velocity at a point below the flameholder is presented as a function of time. As in the previous case, a well-defined vortex-shedding frequency exists, The Strouhal number Su (see equation (15)) is 0.26 here, which should be compared with a value of 0.19 from experiments. The Strouhal number calculated here is approximately the same as in the previous case.…”