Calculation of Transonic Internal Flows Using an Efficient High-Resolution Upwind Scheme

Abstract

A new efficient upwind scheme based on the concept of convective upwind and split pressure is developed. The upwinding of the convective term and the pressure split are consistent with their characteristic directions. The scheme has low diffusion to resolve accurately wall boundary layers and is able to capture crisp shock waves and exact contact discontinuities. The accuracy of the scheme is compared with other popularly used schemes including the Roe scheme, the Liou's latest advection upstream splitting method scheme (AUSM + + ), the Van Leer scheme, and the Van Leer-Hänel scheme. The new scheme is tested for the one-dimensional Sod shock tube problem, one-dimensional slowly moving contact surface, supersonic flat plate laminar boundary layer, a transonic nozzle with oblique shock waves and reflections that do not align with the mesh lines, and a transonic inlet diffuser with shock wave/turbulent boundary-layer interaction. The test cases show that the new scheme is accurate, robust, and efficient. Nomenclaturea = speed of sound D = numerical dissipation vector E = inviscid flux of quasi-one-dimensional Euler equations E , F , G = inviscid flux vectors in ξ, η, and ζ directions e = total energy per unit mass F = inviscid flux of quasi-one-dimensional Euler equations per unit area H = source term of quasi-one-dimensional Euler equations, total enthalpy J = transformation Jacobian l = control volume interface area vectors, l x i + l y j + l z k M = Mach number p = static pressure Q = conservative variable vector R , S , T = viscous flux vectors in ξ, η, and ζ directions S = cross-sectional area of one-dimensional duct T = right eigenvector matrix of the Jacobian matrix U = normal contravariant velocity in ξ direction U = conservative variable vector of quasi-one-dimensional Euler equations multiplied by duct area u, v, w = velocity in x, y, and z directions V = velocity vector, ui + vj + wk x, y, z = spatial coordinates in the Cartesian coordinate system γ = specific heat ratio = eigenvalue matrix of the Jacobian matrix ξ, η, ζ = spatial coordinates in the generalized coordinate system ρ = density Subscripts i = cell index L = left value R = right value t = time 1 2 = control interface position Superscripts c = convective n, n + 1 = time level index p = pressure + = upwind direction − = downwind directioñ = Roe's average = flux vectors of H convective upwind and split pressure schemes, flux vectors in three-dimensional generalized coordinates

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

“…For hydrodynamics, the characteristic analysis is given as the foundation to construct the E-CUSP [32,33,35,36]. The basic idea is to split the flux E to the convective flux F c and the pressure flux F p .…”

“…The Liou's AUSM family schemes [24][25][26][27][28], Van Leer-Hänel scheme [29], and Edwards's LDFSS schemes [30,31] belong to the H-CUSP group. The schemes developed by Zha et al [32][33][34][35][36] belong to the E-CUSP group.…”

Low diffusion E-CUSP scheme with implicit high order WENO scheme for preconditioned Navier–Stokes equations

“…In [13,14,16,17], the characteristic analysis is given as the foundation to construct the E-CUSP scheme. The basic idea is to split the flux E to the convective flux E c and the pressure flux E p .…”

“…The different formulations for U + , U À , P + and P À can be found in [13,14,16,17] with different behavior. The preconditioning of the LDE scheme needs to satisfy two conditions when the flow velocity approaching zero: (1) the eigenvalues of the Jacobian matrices should be at the same order of magnitude of the velocity and (2) the numerical dissipation should diminish.…”

“…The Liou's AUSM family schemes [4][5][6][7][8][9], van Leer-Hänel scheme [10], and Edwards's LDFSS schemes [11,12] belong to the H-CUSP group. The schemes developed by Zha et al [13][14][15][16][17] belong to the E-CUSP group.…”