On the solution for steady state and time dependent flows in certain channels with small wall curvature
Abstract
An asymptotic scheme is presented for the solution of the steadystate and timedependent stream functions for flows in symmetric curved walled channels. In this scheme a class of nonlinear JefferyHamel solutions appear at 0(1), and thus provide the first approximation to the steadystate stream function. This class of JefferyHamel solutions are evaluated by using a simple perturbation about Poiseuille flow. The classic OrrSommerfeld eigenproblem appears at 0(1) in the asymptotic development of the timedependent stream function, but here there is a slow streamwise dependence. This eigenvalue problem, for a complex wave number, is solved using an algorithm which automatically provides an initial guess which is then used to iterate to the correct eigenvalue. Higherorder terms in the asymptotic development, for both the steadystate and timedependent stream functions, are evaluated to provide a solution for the total stream function.
 Publication:

International Journal for Numerical Methods in Fluids
 Pub Date:
 February 1985
 DOI:
 10.1002/fld.1650050206
 Bibcode:
 1985IJNMF...5..169G
 Keywords:

 Channel Flow;
 Computational Fluid Dynamics;
 Flow Geometry;
 Steady Flow;
 Stream Functions (Fluids);
 Asymptotic Methods;
 Curvature;
 Differential Equations;
 Incompressible Fluids;
 Laminar Flow;
 NavierStokes Equation;
 OrrSommerfeld Equations;
 Time Dependence;
 Viscous Fluids;
 Fluid Mechanics and Heat Transfer