In this proposal, we use the UNIC-UNS code, developed by Y.S. Chen et al, to simulate an unsteady compressible flow. It uses Navier-Stokes solver with finite volume method. The governing equation, boundary condition, numerical methods, algorithm and so on will be discussed below.
2.1 Governing Equations
The general form of mass conservation, energy conservation, Navier-Stokes
equation and other transport equations can be written in Cartesian tensor form:
( )
ρφ(
ρ φ)
μφ φ φ momentum, total energy and turbulence equation, respectively.2.2 Spatial Discretization
The cell-centered scheme is employed here then the control volume surface can be represented by the cell surfaces and the coding structure can be much simplified. The transport equations can also be written in integral form as:
∫ ∫ ∫
normal in outward direction. The flux function Frconsists of the inviscid and the viscous parts:
The finite volume formulation of flux integral can be evaluated by the summation of the flux vectors over each face,
∫ ∑
( )where k(i) is a list of faces of cell i, Fi,j represents convection and diffusion fluxes
through the interface between cell i and j, ΔΓj is the cell-face area.
The viscous flux for the face e between control volumes P and E as shown in
Figure 2.1 can be approximated as:
( )
⎟⎟ That is based on the consideration that(
E P)
The inviscid flux is evaluated through the values at the upwind cell and a linear reconstruction procedure to achieve second order accuracy
(
e u)
where the subscript u represents the upwind cell and Ψe is a flux limiter used to prevent from local extrema introduced by the data reconstruction. The flux limiter proposed by Barth [Barth, T.J., 1993] is employed in this work. Defining
(
φu φj)
φ(
φu φj)
2.3 Time Integration
A general implicit discretized time-marching scheme for the transport equations
can be written as:
where NB means the neighbor cells of cell P. The high order differencing terms and cross diffusion terms are treated using known quantities and retained in the source term and updated explicitly.
The Δ-form used for time-marching in this work can be written as:
φ φ
where θ is a time-marching control parameter which needs to specify. θ = 1 and θ = 0.5 are for implicit first-order Euler time-marching and second-order time-centered time-marching schemes. The above derivation is good for non-reacting flows. For general applications, a dual-time sub-iteration method is now used in UNIC-UNS for time-accurate time-marching computations.
2.4 Pressure-Velocity-Density Coupling
In an extended SIMPLE [Chen, Y.S., 1989] family pressure-correction algorithm, the pressure correction equation for all-speed flow is formulated using the perturbed equation of state, momentum and continuity equations. The simplified formulation can be written as:
where Du is the pressure-velocity coupling coefficient. Substituting Eq. (12) into Eq. (13), the following all-speed pressure-correction equation is obtained,
(
D p)
n( )
u nFor the cell-centered scheme, the flux integration is conducted along each face and its contribution is sent to the two cells on either side of the interface. Once the integration loop is performed along the face index, the discretization of the governing equations is completed. First, the momentum equation (9) is solved implicitly at the predictor step. Once the solution of pressure-correction equation (14) is obtained, the velocity, pressure and density fields are updated using Eq. (12). The entire corrector step is repeated 2 and 3 times so that the mass conservation is enforced. The scalar equations such as turbulence transport equations, species equations etc. are then solved sequentially. Then, the solution procedure marches to the next time level for transient calculations or global iteration for steady-state calculations. Unlike for incompressible flow, the pressure-correction equation, which contains both convective and diffusive terms is essentially transport-like. All treatments for inviscid and the viscous fluxes described above are applied to the corresponding parts in Eq.
(14).
2.5 Linear Matrix Solver
The discretized finite-volume equations can be represented by a set of linear algebra equations, which are non-symmetric matrix system with arbitrary sparsity patterns. Due to the diagonal dominant for the matrixes of the transport equations, they can converge even through the classical iterative methods. However, the
coefficient matrix for the pressure-correction equation may be ill conditioned and the classical iterative methods may break down or converge slowly. Because satisfaction of the continuity equation is of crucial importance to guarantee the overall convergence, most of the computing time in fluid flow calculation is spent on solving the pressure-correction equation by which the continuity-satisfying flow field is enforced. Therefore the preconditioned Bi-CGSTAB [Van Der Vorst, H.A., 1992] and GMRES [Saad, Y. and Schultz, M.H., 1986] matrix solvers are used to efficiently solve, respectively, transports equation and pressure-correction equation.
2.6 Parallelization
Compared with a structured grid approach, the unstructured grid algorithm is more memory and CPU intensive because “links” between nodes, faces, cells, needs to be established explicitly, and many efficient solution methods developed for structured grids such as approximate factorization, line relaxation, SIS, etc. cannot be used for unstructured methods.
As a result, numerical simulation of three-dimensional flow fields remains very expensive even with today’s high-speed computers. As it is becoming more and more difficult to increase the speed and storage of conventional supercomputers, a parallel architecture wherein many processors are put together to work on the same problem
unlimited. It is reasonable to claim that parallel computing can provide the ultimate throughput for large-scale scientific and engineering applications. It has been demonstrated that performance that rivals or even surpasses supercomputers can be achieved on parallel computers.