Lafayette K. Taylor

Multigrid algorithm for three-dimensional incompressible high-Reynolds number turbulent flows

A robust multigrid algorithm is presented for solving three-dimensional incompressible high-Reynolds number turbulent flows on high aspect ratio grids. The artificial compressibility form of the Navier-Stokes equations is discretized in a cell-centered finite volume form on a time-dependent curvilinear coordinate system, and the so-called discretized Newton-relaxation scheme is used as the iterative procedure for the solution of the system of equations. A nonlinear multigrid scheme (full approximation scheme [FAS]) is applied to accelerate the convergence of the time-dependent equations to a steady state. Two methods for constructing the coarse grid operator, the Galerkin coarse grid approximation and the discrete coarse grid approximation have also been investigated and incorporated into the FAS. A new procedure, called implicit correction smoothing that leads to high efficiency of the multigrid scheme by allowing large Courant-Friedrichs-Lewy numbers, is introduced in this work. Numerical solutions of high-Reynolds number turbulent flows for practical engineering problems are presented to illustrate the efficiency and accuracy of the current multigrid algorithm.

A time accurate calculation procedure for flows with a free surface using a modified artificial compressibility formulation

Abstract A time accurate, three-dimensional, finite volume, implicit scheme has been proposed for the calculations of flows with a free surface on unsteady Eulerian grids, which uses a modified form of the artificial compressibility formulation. The modification comes from the addition of the gravitational potential to the pressure term appearing in the continuity equation of the original formulation. The main advantage of this approach comes from the fact that the analysis of the inviscid flux Jacobians in terms of their eigenvalues, eigenvectors, etc., turns out to be of the same form as that of the original formulation without the body force. Also the present formulation allows one to represent the gravitational potential as a third-order accurate term by the use of Roe variables in an implicit manner, which is not possible otherwise. Thus, the formal accuracy of the original scheme is preserved. The gravitational potential appears explicitly only in the free surface boundary condition in the present formulation, which means minimal code modification to a code based on the original artificial compressibility method. The boundary conditions used at the free surface as well as at the radiating boundary differ from other formulations and are also discussed. An unsteady test case of a wave channel has been chosen to demonstrate the capabilities of the method, and comparisons are made with available experimental results. The results clearly demonstrate the time accuracy as well as the feasibility of the current method for solving flows with a free surface.

Computation of steady and unsteady flows with a free surface around the Wigley hull

The previously reported Modified Artificial Compressibility approach to solve unsteady, two dimensional free surface flows using Euler equations has been extended to solve unsteady, three dimensional free surface flows using Navier-Stokes equations. This approach provides a natural way of incorporating the body force term due to gravity in a higher order scheme. The numerical scheme, UNCLE, is a finite volume, implicit, upwind scheme that uses the Roe variables for computing the fluxes at the cell interfaces and the MUSCL approach for obtaining higher order corrections. The flux Jacobians are obtained using numerical (Frechet) differentiation. The free surface is tracked using the moving grid approach which results in an accurate way of applying the boundary conditions preventing leaks through the free surface. Both Euler and Navier-Stokes computations have been performed. Turbulence is accounted using the Baldwin-Lomax model. This scheme is used to study steady and unsteady flows around a Wigley hull. Steady flow is studied by keeping the body stationary in a uniform flow. Unsteady aspect of the flow is induced by heaving the body using a sinusoidal function. The amplitude of heaving was 0.4 times the draft. Comparisons have been made with available experimental results for the case of the steady motion and the agreement is quite good. No data is available to make comparisons with the unsteady motion.

Computation of Vortex-Intensive Incompressible Flow Fields

The objective of this study is to use UNCLE, an unstructured, parallel, Reynolds averaged Navier-Stokes solver, to compute vortex intensive flow fields and compare the results with experimental data. Vortex intensive flow fields arise under a variety of conditions. The ones considered in this study are those arising due to rotating propellers and due to flow past submarine hulls. The examples considered here involve computations at design as well as offdesign conditions for the propellers and the SUBOFF model at various angles of drift. Results are presented for propeller P-4381 (overall thrust and torque coefficients) as well as for propeller P-5168 (velocity profiles downstream of the propeller). Forces and moments are also compared for the SUBOFF hull at various angles of drift to experimental data. Good agreement with experimental data is obtained for the various cases.

Unsteady multigrid method for simulating 3-D incompressible Navier-Stokes flows on dynamic relative motion grids

An unsteady multiblock multigrid method is presented for simulating three-dimensional incompressible turbulent flows on dynamic grids, where the blocks may be in motion relative to one another. The artificial compressibility form of the three-dimensional incompressible Navier-Stokes equations is solved using a second-order time-accurate discretized Newton-relaxation scheme. The unsteady multiblock multigrid method is developed to accelerate the convergence of time-accurate solutions. The localized grid distortion technique is used to treat the relative motion of dynamic multiblock grids. Applications include an impeller, an isolated propeller, and a fully appended submarine with a rotating propulsor, which are presented to demonstrate the accuracy, efficiency, robustness, and applicability of the current method. (Author)

Towards computations of ocean flows using Navier-Stokes equations

The equations for mass, momentum, temperature and salinity are formulated in a non-inertial general curvilinear coordinate system to be solved numerically. A flux-difference split type of numerical flux formulation is used for the convective fluxes and the equations are numerically solved in a strongly coupled manner using the discretized Newton-relaxation scheme. The addition of artificial viscosity is not required for stability. Test cases were considered that included laminar flow over a flat plate with prescribed wall temperature and salinity as well as laminar flow with heat transfer over a backward facing step. These computations compare well with existing results. The present numerical scheme was also applied to a body-fitted grid that approximates the entire Atlantic Ocean and rotates with the Earth. These results, though encouraging, must be considered preliminary.<<ETX>>