Overview of the forced response methodology Flow model

The unsteady, compressible Reynolds-averaged Navier-Stokes equations for a 3D bladerow can be cast in terms of absolute velocity u but solved in a rela­tive reference frame rotating with angular velocity u. This system of equations, written in an arbitrary Eulerian-Lagrangian conservative form for a control vol­ume D with boundary Г, take the form:

where n represents the outward unit vector of the control volume boundary. The viscous term G on the left hand side of (1) has been scaled by the reference Reynolds number for non-dimensionalization purposes. The solution vector U of conservative variables is given by:

P

pu

pe

The inviscid fhx vector F can be written as:

where Stj represents the Kronecker delta function and v is the velocity in the relative frame of reference. The pressure p and the total enthalpy h are re­lated to density p, absolute velocity u and internal energy e by two perfect gas

where 7 is the constant specific heat ratio. The viscous flix vector G has the following components

0

Where ^ represents the molecular viscosity given by the Sutherland’s formula, denotes the turbulent eddy viscosity, which must be determined by a suitable turbulence model. Therefore, ^ is the total viscosity of the fluid. The

value of A is given by the Stokes relation A = — | ц while the laminar Prandtl number, Prj, is taken as 0.7 for air. The turbulent Prandtl number, Prt, is taken as 0.9.

The term S in (1) is given by:

S = [ 0 0 ршп2 pum3 0 ]T (6)

Equation (1) is discretised on unstructured mixed-element grids via a finite volume method, the details of which are given by Sayma et al. (2000a) . As described by Sbardella et al. (2000), the blades are discretized using a semi­structured mesh which consists of brick elements in the boundary layer and triangular prisms further away in the blade passage. To achieve further compu­tational efficiency, the mesh is characterized via an edge-based data structure, i. e. the grid is presented to the solver as a set of node pairs connected by edges, a feature that allows the solver to have a unified data structure. Furthermore, the edge-based formulation has the advantage of computing and storing the edge weights prior to the main unsteady flow calculation, hence reducing the CPU effort. The central differencing scheme is stabilized using a mixture of second – and fourth-order matrix artificial dissipation. In addition, a pressure switch, which guarantees that the scheme is total variation diminishing (TVD) and reverts to a first-order Roe scheme in the vicinity of discontinuities, is used for numerical robustness. The resulting semi-discrete system of equa­tions is advanced in time using a point-implicit scheme with Jacobi iterations and dual time stepping. Such an approach allows relatively large time steps for the external Newton iteration. For steady-state calculations, solution ac­celeration techniques, such as residual smoothing and local time stepping are employed. For unsteady computations, an outer Newton iteration procedure is used where the time steps are dictated by the physical restraints and fixed

through the solution domain. Within the Newton iteration, the solution is ad­vanced to convergence using the traditional acceleration techniques described previously.

For the sliding planes that occur at bladerow boundaries, the solution is up­dated at the interface by interpolating the variables in the stator computational domain to obtain rotor fluxes, and in the rotor computational domain to obtain the stator flixes (Rai 1986, Sayma et al. 2000b). The flixes are computed using a characteristic technique, which allows the correct propagation of the information. In other words, fl»w data are exchanged between the two grids via specially formulated boundary conditions at the interface.