Computational fluid dynamics for building air flow prediction—current status and capabilities

Pressure-based finite-volume techniques have emerged as the methods of choice for a wide variety of industrial applications involving incompressible fluid flow. In this paper, we trace the evolution of this class of solution techniques. We review the basics of the finite-volume method, and trace its extension to unstructured meshes through the use of cell-based and control-volume finite-element schemes. A critical component of the solution of incompressible flows is the issue of pressure-velocity storage and coupling. The development of staggered-mesh schemes and segregated solution techniques such as the SIMPLE algorithm are reviewed. Co-located storage schemes, which seek to replace staggered-mesh approaches, are presented. Coupled multigrid schemes, which promise to replace segregated-solution approaches, are discussed. Extensions of pressure-based techniques to compressible flows are presented. Finally, the shortcomings of existing techniques and directions for future research are discussed.

Pressure-Based Finite-Volume Methods in Computational Fluid Dynamics

Benchmark problems are solved with the steady incompressible Navier-Stokes equations discretized with a finite volume method in general curvilinear co-ordinates on a staggered grid. The problems solved are skewed driven cavity problems, recently proposed as non-orthogonal grid benchmark problems. The system of discretized equations is solved efficiently with a non-linear multigrid algorithm, in which a robust line smoother is implemented. Furthermore, another benchmark problem is introduced and solved in which a 90 o change in grid line direction occurs

/pdf/benchmark-solutions-for-the-incompressible-navier-stokes-193d9nv131.pdf

Benchmark solutions for the incompressible Navier–Stokes equations in general co‐ordinates on staggered grids

Calculations of unsteady turbulent flow around and behind triangular-shaped flameholders using a finite volume code with a k-e model of turbulence are presented.

The flow behind the flameholders is found to be unsteady (a von Karman vortex street appears) with a well defined Strouhal frequency (predicted Sr=0·27 compared with an experimental value of 0·25). The predicted profiles of velocity and fluctuating kinetic energy agree well with experiments. The periodic motions in the vortex street are shown to be far more important than the turbulent stochastic motions in exchanging momentum in the transversal direction.

The pressure-velocity coupling is handled with the SIMPLEC pressure correction procedure. The discretization in time is fully implicit and 90 times steps are used to resolve one time cycle. It was found that to capture the vortex street it is very important that the grid spacing is sufficiently fine (180 × 100).

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

The discretization of the incompressible Navier-Stokes equation on boundary-fitted curvilinear grids is considered. The discretization is based on a staggered grid arrangement and the Navier-;Stokes equations in tensor formulation including Christoffel symbols. It is shown that discretization accuracy is much enhanced by choosing the velocity variables in a special way. The time-dependent equations are solved by a pressure-correction method in combination with a GMRES method. Special attention is paid to the discretization of several types of boundary conditions. It is shown that fairly non-smooth grids may be used using our approach.

/pdf/invariant-discretization-of-the-incompressible-navier-stokes-2yc2bsi37g.pdf

Invariant discretization of the incompressible Navier‐Stokes equations in boundary fitted co‐ordinates

This 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 PISO procedure.

Two examples of laminar flows are given.

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

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