Numerical Solution to the Navier-Stokes Equations in Primitive Variables

Abstract

In this thesis, a hybrid approach combining second-order finite differences with a spectral method, in particular Chebyshev collocation, is considered to solve the Navier-Stokes equations in primitive variables. First, we compute the numerical solution to the test problem of the 2D Stokes equations which is a special case of the Navier-Stokes equations when considering large viscosity, using staggered and non-staggered grids. The numerical solution to the original problem by having the pressure fixed at one point is not accurate and unstable, as the mesh sizes in both directions converge to zero. Therefore, two correction methods are proposed to improve the numerical solution. The first correction method is implemented using a staggered grid, which results in obtaining an accurate and stable numerical solution using only even number of the Chebyshev points. The second method gives an accurate and stable numerical solution using both even and odd number of the Chebyshev points. The eigenvalue problem of the original problem is solved to investigate the inf-sup stability condition and the existence of pressure modes. Then, the hybrid approach with the second correction method are implemented to solve the test problem of the 2D Navier-Stokes equations using a staggered grid. The numerical solution obtained is accurate and stable, as the mesh sizes in both directions converge to zero. After that, the lid-driven cavity problem is solved using the same approach and the numerical solution is obtained at high Reynolds number, from Re = 100 to Re =20000, in which the stream function, vorticity and pressure contours are presented at several values of the Reynolds number. The numerical results compare well with the previous literature. Finally, the test problem of the 3D Stokes equations is solved using staggered and non-staggered grids. The second-order finite differences is implemented to approximate the derivatives in one direction and Chebyshev collocation is implemented to approximate the derivatives in the other directions. The numerical solution is accurate using a fine grid; however, it is not completely stable as the mesh sizes in all directions converge to zero.

Date of Award	1 Aug 2019
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Catherine Powell (Supervisor) & Jitesh Gajjar (Supervisor)