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Flow Model
The two-dimensional Reynolds-averaged Navier-Stokes (RANS) equations with the Spalart & Allmaras one-equation turbulence model [4] are used to model the unsteady fbw. The ft>w is assumed to be fully turbulent and no transition model is included. The turbulence model was chosen by the author and previous researchers [2, 3] because of its ability to accurately model the ft>w at a reasonable computational effort [5]. Better turbulence models do exist, but the effort required to solve their equations is significantly greater and prohibits their use for design calculations. The unsteady ft>w equations for a moving control volume can be written
where U is the vector of conserved variables, # is the control volume, S is the control surface, H is the fhx of conserved variables at the control surface, t is time, and S is the vector of source terms due to the turbulence model.
The ft>w domain is discretised by a multi-block structured grid, and the ft>w equations are discretised using a cell-centered finite-volume scheme. The discretised ft>w equations can be written
d 4
-[U + = s#,
i=1
or
where U is the vector of the cell-averaged conserved variables, # is the volume of the cell, Si is the area of a cell interface between two cells, and Hi is the average flix through a cell interface. The inviscid flix is determined by reconstructing the ft>w states at both sides (typically called left and right) of the cell interface using MUSCL interpolation with a modified van Albada limiter
. The flix at the interface can be estimated from the left and right states using one of the following upwind schemes: Roe’s approximate Riemann solver
, AUSMDV [9], an exact Riemann solver [10], and EFM [11]. The spatial derivatives required for the viscous fluxes are calculated by an application of the divergence theorem [12].
The boundary conditions applied at the wall, on the periodic boundaries, and at the far-field boundaries are the same as applied by the previous researchers [2, 3]. It should be stated that the exact (for uniform ft>w) non-refecting boundaries described by Giles [6] has been implemented at the far-field boundaries. The implementation and verification of this boundary condition required significant effort.
