Incompressible spectralelement method: Derivation of equations
Abstract
A fractionalstep splitting scheme breaks the full NavierStokes equations into explicit and implicit portions amenable to the calculus of variations. Beginning with the functional forms of the Poisson and Helmholtz equations, we substitute finite expansion series for the dependent variables and derive the matrix equations for the unknown expansion coefficients. This method employs a new splitting scheme which differs from conventional threestep (nonlinear, pressure, viscous) schemes. The nonlinear step appears in the conventional, explicit manner, the difference occurs in the pressure step. Instead of solving for the pressure gradient using the nonlinear velocity, we add the viscous portion of the NavierStokes equation from the previous time step to the velocity before solving for the pressure gradient. By combining this 'predicted' pressure gradient with the nonlinear velocity in an explicit term, and the CrankNicholson method for the viscous terms, we develop a Helmholtz equation for the final velocity.
 Publication:

NASA STI/Recon Technical Report N
 Pub Date:
 April 1993
 Bibcode:
 1993STIN...9326554D
 Keywords:

 Calculus Of Variations;
 Computational Fluid Dynamics;
 Incompressible Flow;
 NavierStokes Equation;
 Spectral Methods;
 CrankNicholson Method;
 Helmholtz Equations;
 Poisson Equation;
 Pressure Gradients;
 Velocity Distribution;
 Fluid Mechanics and Heat Transfer