John Bell

A second-order projection method for the incompressible navier-stokes equations

In this paper we describe a second-order projection method for the time-dependent, incomxad pressible Navier-Stokes equations. As in the original projection method developed by Chorin, we first solve diffusion-convection equations to predict intermediate velocities which are then projected onto the space of divergence-free vector fields. By introducing more coupling between the diffusion--{;onvection step and the projection step we obtain a temporal discretizaxad tion that is second-order accurate. Our treatment of the diffusion-convection step uses a specialized higher order Godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high Reynolds number. The Godunov procedure is second-order accurate for smooth flow and remains stable for discontinuous initial data, even in the zero-viscosity limit. We approximate the projection directly using a Galerkin procedure that uses a local basis for discretely divergence-free vector fields. Numerical results are presented validating the convergence properties of the method. We also apply the method to doubly periodic shear-layers to assess the performance of the method on more difficult

A Second-Order Projection Method for Variable- Density Flows*

This paper describes a second-order projection method for variabledensity incompressible flows. The method is suitable for both finite amplitude density variations and for fluids that are modeled using a Boussinesq approximation. It is based on a second-order fractional step scheme in which diffusion-convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields. The nonlinear convection terms are treated using a Godunov-type procedure that is second order for smooth flow and remain stable and non-oscillatory for nonsmooth flows with low fluid viscosities. The method is described for finite-amplitude density variation and the simplifications for a Boussinesq approximation are sketched. Numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems.

Higher order Godunov methods for general systems of hyperbolic conservation laws

Abstract We describe an extension of higher order Godunov methods to general systems of hyperbolic conservation laws. This extension allow the method to be applied to problems that are not strictly hyperbolic and exhibit local linear degeneracies in the wave fields. The method constructs an approximation of the Riemann problem from local wave information. A generalization of the Engquist-Osher flux for systems is then used to compute a numerical flux based on this approximation. This numerical flux replaces the Godunov numerical flux in the algorithm, thereby eliminating the need for a global Riemann problem solution. The additional modifications to the Godunov methodology that are needed to treat loss of strict hyperbolicity are described in detail. The method is applied to some simple model problems for which the global analytic structure is known. The method is also applied to the black-oil model for multiphase flow in petroleum reservoirs.

AN EFFICIENT SECOND-ORDER PROJECTION METHOD FOR VISCOUS INCOMPRESSIBLE FLOW

In this paper we describe a second-order projection method for the time-dependent, incompressible Navier-Stokes equations. The method is a second-order fractional step scheme in which one first solves diffusion-convection equations to determine intermediate velocities which are then projected onto the space of divergence-free vector fields. The diffusion-convection step uses a specialized second-order Godunov method for differencing the nonlinear convective terms that is conservative and free-stream preserving and provides a robust treatment of the nonlinearities at high Reynolds number. The projection is based on cell-centered centered difference approximations to divergence and gradient operators with the resulting linear system solved using a multigrid relaxation scheme. We apply the method to vortex spindown in a box to validate the numerical convergence of the method and to measure its overall performance. 13 refs., 2 figs.

Mathematical Structure of Compositional Reservoir Simulation

In this paper multicomponent two-phase isothermal fluid flow in petroleum reservoirs is described. The fluid-flow model consists of component conservation equations, Darcys law for the volumetric flow rates, balance between the fluid volume and the rock void, and the conditions of thermodynamic equilibrium that determine the distribution of the chemical components into phases. Thermodynamic equilibrium is described by means of a mathematical model for the chemical potentials of each component in each phase of the fluid. The flow equations are manipulated to form a pressure equation and a modified component-conservation equation; these form the basis for the sequential method. It is shown that the pressure equation is parabolic under reasonable assumptions on the thermodynamic equilibrium model, and that the component-conservation equations are hyperbolic in the absence of diffusive forces such as capillary pressure and mixing. A numerical method based on the sequential formulation of the flow equations is outlined and used to illustrate the kinds of flow behavior that occur during miscible gas injection.

Vorticity intensification and transition to turbulence in the three-dimensional Euler equations

The evolution of a perturbed vortex tube is studied by means of a second-order projection method for the incompressible Euler equations. We observe, to the limits of grid resolution, a nonintegrable blowup in vorticity. The onset of the intensification is accompanied by a decay in the mean kinetic energy. Locally, the intensification is characterized by tightly curved regions of alternating-sign vorticity in a 2n-pole structure. After the firstL∞ peak, the enstrophy and entropy continue to increase, and we observe reconnection events, continued decay of the mean kinetic energy, and the emergence of a Kolmogorov (k−5/3) range in the energy spectrum.

Conservative Front-Tracking for Inviscid Compressible Flow

In this paper we describe a front-ncking algo-rithm for modeling the propagation of discontinuous waves in two space dimensions. The algorithm uses a volume-of-fluid representation of the front in which the local frontal geometry is reconstructed from the state information on either side of the discontinuity and the Rankine·Hugoniot relations. The algorithm is coupled co an unsplit second-order Godunov algorithm and is fully conservative, maintaining conservation at the fronL The combination of a volume-of-fluid representation of the front and a fully conservative algorithm leads to a robust high resolution method that easily acomodates changes in the topology of the front as well as kinks arising when a tracked front interacts with a captured discontinuity. The Godunov/ll2.Cking integration scheme is coupled to a local adaptive mesh refinement algorithm that selectively refines regions of the computational grid to achieve a desired level of accuracy. An example showing the combination of tracking and local refinement is presented.

Adaptive Mesh Refinement on Moving Quadrilateral Grids

This paper describes an adaptive mesh refinement algorithm for unsteady gas dynamics. The algorithm is based on an unsplit, second-order Godunov integration scheme for logically-rectangular moving quadrilateral grids. The integration scheme is conservative and provides a robust, high resolution discretization of the equations of gas dynamics for problems with strong nonlinearities. The integration scheme is coupled to a local adaptive mesh refinement algorithm that dynamically adjusts the location of refined grid patches to preserve the accuracy of the solution, preserving conservation at interfaces between coarse and fine grids while adjusting the geometry of the fine grids so that grid lines remain smooth under refinement. Numerical results are presented illustrating the performance of the algorithm. 5 refs., 3 figs.

A Second-Order Projection Method for the Incompressible Navier Stokes Equations on Quadrilateral Grids

This paper describes a second-order projection method for the incompressible Navier-Stokes equations on a logically-rectangular quadrilateral grid. The method uses a second-order fractional step scheme in which one first solves diffusion-convection equations to predict intermediate velocities which are then projected onto the space of divergence-free vector fields. The spatial discretization of the diffusion-convection equations is accomplished by formally transforming the equations to a uniform computational space. The diffusion terms are then discretized using standard finite-difference approximations. The convection terms are discretized using a second-order Godunov method that provides a robust discretization of these terms at high Reynolds number. The projection is approximated using a Galerkin procedure that uses a local basis for discretely divergence-free vector fields. Numerical results are presented illustrating the performance of the method.

An adaptive multifluid interface-capturing method for compressible flow in complex geometries

We present a numerical method for solving the multifluid equations of gas dynamics using an operator-split second-order Godunov method for flow in complex geometries in two and three dimensions. The multifluid system treats the fluid components as thermodynamically distinct entities and correctly models fluids with different compressibilities. This treatment allows a general equation-of-state (EOS) specification and the method is implemented so that the EOS references are minimized. The current method is complementary to volume-of-fluid (VOF) methods in the sense that a VOF representation is used, but no interface reconstruction is performed. The Godunov integrator captures the interface during the solution process. The basic multifluid integrator is coupled to a Cartesian grid algorithm that also uses a VOF representation of the fluid-body interface. This representation of the fluid-body interface allows the algorithm to easily accommodate arbitrarily complex geometries. The resulting single grid multifluid-Cartesian grid integration scheme is coupled to a local adaptive mesh refinement algorithm that dynamically refines selected regions of the computational grid to achieve a desired level of accuracy. The overall method is fully conservative with respect to the total mixture. The method will be used for a simple nozzle problem in two-dimensional axisymmetric coordinates.