Numerical solution of 3D NavierStokes equations in vorticityvelocity form for incompressible flows
Abstract
The three dimensional NavierStokes equations are formulated in vorticityvelocity variables for incompressible flows. The attractive features of this formulation are that the pressure is eliminated from the essential part of the governing equations and the form of the equations remains invariant even in noninertial reference frames. The system of equations consists of the vorticitytransport equation and the kinematic velocity problem. The velocity problem is a constrained system of firstorder CauchyRiemann type equations in three dimensions. Formulated in generalized curvilinear orthogonal coordinates, the equations are discretized using a finite difference method on a staggered grid. The vorticitytransport equation is integrated by an efficient ADI procedure, whereas the velocity problem is solved by a multigrid method. The pressure, if desired, is obtained by solving the pressure Poisson equation, also with a multigrid method. A sheardriven cavity flow is selected for code validation and for a study of the flow itself. Good agreement with available experimental results is obtained. The unsteady structure of the flow is examined in detail. The vorticityvelocity methodology is then applied to the solution of three dimensional flow over an axisymmetric afterbody. A novel analysis is developed for the governing equations at the centerline in the cylindrical coordinate to treat the 'coordinate singularity.' The evolution of streamwise vorticity, its interaction with the wall, and the wake pressure distribution are analyzed in detail. The results show that the streamwise vortices are attenuated significantly at the wall via interaction with the boundary layer. Further, the circumferential vorticity is strengthened by this interaction. The pressure distribution shows that the streamwise vorticity is instrumental in modifying the pressure field. The flow simulation and analysis have shown that the present vorticityvelocity methodology for three dimensional flows is capable of accurately and efficiently predicting the vortex dynamics of complex flows.
 Publication:

Ph.D. Thesis
 Pub Date:
 1992
 Bibcode:
 1992PhDT........76H
 Keywords:

 CauchyRiemann Equations;
 Computational Fluid Dynamics;
 Incompressible Flow;
 NavierStokes Equation;
 Three Dimensional Flow;
 Vorticity;
 Cavity Flow;
 Poisson Equation;
 Unsteady Flow;
 Vortices;
 Fluid Mechanics and Heat Transfer