Anastasios Liakos

Discretization of the Navier‐Stokes equations with slip boundary condition

We propose and analyze a two-level method of discretizing the nonlinear Navier-Stokes equations with slip boundary condition. The slip boundary condition is appropriate for problems that involve free boundaries, flows past chemically reacting walls, and other examples where the usual no-slip condition u = 0 is not valid. The two-level algorithm consists of solving a small nonlinear system of equations on the coarse mesh and then using that solution to solve a larger linear system on the fine mesh. The two-level method exploits the quadratic nonlinearity in the Navier-Stokes equations. Our error estimates show that it has optimal order accuracy, provided that the best approximation to the true solution in the velocity and pressure spaces is bounded above by the data.

Two-level finite element discretization of viscoelastic fluid flow

Abstract We propose and analyze a two-level method of discretizing the equations of steady-state flow of a viscoelastic fluid obeying an Oldroyd-type constitutive equation with no-slip boundary condition. The two-level algorithm consists of solving a small non-linear system of equations on the coarse mesh and then using that solution to solve a larger linear system on the fine mesh. Specifically, following Najib and Sandri [Numer. Math. 72 (1995) 223], we linearize the Oldroyd-type constitutive equation about the coarse mesh solution thus nullifying the difficulties brought by the advection term. Our theoretical error estimates show that it has optimal order accuracy provided the true solution is smooth and its norm is sufficiently small. In addition, our computational error estimates exhibit the validity of our analysis.

Direct numerical simulation of steady state, three dimensional, laminar flow around a wall mounted cube

The topology and evolution of flow around a surface mounted cubical object in three dimensional channel flow is examined for low to moderate Reynolds numbers. Direct numerical simulations were performed via a home made parallel finite element code. The computational domain has been designed according to actual laboratory experiment conditions. Analysis of the results is performed using the three dimensional theory of separation. Our findings indicate that a tornado-like vortex by the side of the cube is present for all Reynolds numbers for which flow was simulated. A horseshoe vortex upstream from the cube was formed at Reynolds number approximately 1266. Pressure distributions are shown along with three dimensional images of the tornado-like vortex and the horseshoe vortex at selected Reynolds numbers. Finally, and in accordance to previous work, our results indicate that the upper limit for the Reynolds number for which steady state results are physically realizable is roughly 2000.

Three-Dimensional, Laminar Flow Past a Short, Surface-Mounted Cylinder

The topology and evolution of three-dimensional flow past a cylinder of slenderness ratio SR=1 mounted in a wind tunnel is examined for 0.1≤Re≤325 (based on the diameter of the cylinder) where steady-state solutions have been obtained. Direct numerical simulations were computed using an in-house parallel finite element code. The three-dimensional theory of separation is used to analyze and interpret the flow phenomena. Results indicate that symmetry breaking occurs at Re=1, while the first prominent structure is a horseshoe vortex downstream from the cylinder. At Re=150, two foci are observed, indicating the formation of two tornadolike vortices downstream. Concurrently, another horseshoe vortex is formed upstream from the cylinder. For higher Reynolds numbers, the flow downstream is segmented to upper and lower parts, whereas the topology of the flow on the solid boundaries remains unaltered. Pressure distributions show that pressure, the key physical parameter in the flow, decreases everywhere except im...