Cascade flow calculations by a multigrid Euler method

An explicit finite-volume method for the Euler equations with multigrid is presented for calculating cascade flows. The method is validated against theoretical test cases of the Hobson shock-free impulse cascade and a supersonic wedge cascade. Results for a VKI turbine cascade and a low-pressure turbine cascade are presented and compared with experimental data. Isentropic Mach number distributions on blade surfaces show good agreement with experiments at design conditions, whereas, discrepancy exists at off-design conditions due to flow separation. With equal efficiency the method is also able to capture qualitative features of secondary flow due to inlet side-wall boundary layers. Nomenclature c = sound speed D = artificial dissipation E = stagnation energy E, F, G = flux vectors in the Euler equations H = stagnation enthalpy (i> j, k) = grid point indices k — summation index P = forcing function in the multigrid method p. = pressure q = total flow speed Q = flux balance R = residual for the Euler equations R = averaged residual for the Euler equations S = entropy (u, v, w) = velocity components in x, y, z directions Vijk = cell volume W = conserved flow variables in the Euler equations (x, y, z) = Cartesian coordinates (a, j3, y) = inlet flow directional angles aq = time-stepping coefficient for the qth stage of the multistage time-stepping scheme y = ratio of specific heats AS = cell surface area 8 = central difference operator, 8X(-) = (')i+l/2- OX--1/2 e = smoothing parameter p = density fl = control volume Subscripts h, 2h = coarse and fine meshes i = interior cell values n = outer normal direction to a boundary q = stages in the multistage time stepping 00 = far-stream conditions