Three-Dimensional Distributed Mass Weighting for Noninverted Convective Skew Upwinding

Abstract

Numerical studies of skew upstream differencing are presented for three-dimensional convective heat transfer problems. Two new types of noninverted skew upwind schemes are developed and compared, so that local inversion of influence coefficient matrices is not required. Unlike previous schemes, skew upwind values of temperature are expressed explicitly in terms of surrounding nodal variables. Different mass weighting alternatives, including 3-node/3-point and 4-node/8-point formulations are developed and evaluated. Results are presented for three application problems, that is, convective step change of temperature, tank inflow/outflow, and radial heat flow in a rotating hollow sphere. Although the effects of upwind nodal asymmetry appear minor in the first and second problems, noticeable improvement with 4-node/8-point interpolation is observed in the rotating sphere problem. Additional reduction of CPU time due to noninverted convective upwinding is reported. NomenclatureC up = convective length scale in upstream direction c p = specific heat, J/kgK D = diffusion length scale e * = error at the center of the spherê i,ĵ,k = unit vectors in the x, y, and z directions J = Jacobian determinant k = thermal conductivity, W/mK M = combined mass flow rate, kg/ṡ m = mass flow rate, kg/s N i = interpolation shape function at local node i n = normal direction Pe = Peclet number Q = flow of φ r 1 , r 2 = inner, outer radius of sphere, m S = source term s = streamwise coordinate T 1 , T 2 = inner, outer temperature of sphere, • C t = time, s u, v, w = fluid velocity components in the x, y, and z directions v = fluid velocity vector, m/s v = average velocity magnitude, m/s x, y, z = global Cartesian coordinates, m = diffusion coefficient n, l, m = normal and tangential length scales ρ = fluid density, kg/m 3 φ = scalar dependent variable ω = angular velocity of sphere about the x axis, rad/s Subscripts c = convection d = diffusion

Slip-flow irreversibility of dissipative kinetic and internal energy exchange in microchannels

“…This paper develops a numerical formulation of microchannel flows with a new mass-weighted convection scheme called NISUS (non-inverted skew upwind scheme). The method extends past finite element advances (Ogedengbe and Naterer [19]) to a SIMPLEC finite volume formulation of microchannel flows. Past numerical studies of microchannel flows have typically used either a continuum [20] or molecular approach [21].…”