Question

3. Some algorithms for the numerical solution of two-dimensional flows use variables know as the stream function and the vorticity. These algorithms eliminate the need to solve for the pressure as one of the variables. In this problem we will consider two dimensional flows with constant properties. (a) Show that the continuity and momentum equations for two-dimensional, constant property flows with no body forces are given by the equations below, where v = μ/ρ is the kinematic viscosity. ou a ox y

Answer 1

a) The constant property momentum equation can be found, partial differential delta u_j/ partial differential t + partial differential delta u_i u_j/ partial differential x_i = - partial differential delta/ partial differential x_i + partial differential/ partial differential x_i [mu (partial differential u_i/ partial differential x_j + partial differential u_j/ partial differential x_i)] + partial differential/ partial differential x_j [(k - 2/3 mu) Delta] + delta beta_j for constant density flows the Delta = 0. and for no body force BJ = 0, further more If the density and viscosity are constant we can remove them from the derivative operations, divide the equation through by the density and use the function v = mu/delta d u_j/ partial differential t + partial differential u_i u_j/ partial differential x_i = -1/delta partial differential p/ partial differential x_j + v[partial differential/ partial differential x_i . partial differential v_i/ partial differential x_j + partial differential/ partial differential x_i . partial differential u_j/ partial differential x_i] The terms multiplied by kinematic viscosity may be re arranged as follows to show the in compressible