Second-order Finite Volume Scheme Research Articles

A method for computing compressible, highly stratified flows in astrophysics based on operator splitting

A new numerical approach based on consistent operator splitting is presented for computing compressible, highly stratified flows in astrophysics. The algorithm is particularly designed to search for steady or almost steady solutions for the time-dependent Navier-Stokes equations, describing viscous flow under the influence of a strong gravitational field. The algorithm proposed is multidimensional and works in Cartesian, cylindrical or spherical co-ordinates. It uses a second-order finite volume scheme with third-order upwinding and a second-order time discretization. An adaptive time step control and monotonic multilevel grid distribution has been incorporated to speed up convergence. This method has been incorporated into a hydrodynamical code by which, for the first time, for two-dimensional models the dynamics of the boundary layer in the accretion disk around a compact star could be computed over the whole viscous time scale.

Kinetic Flux–Vector Splitting for the Navier–Stokes Equations

Before a hybrid scheme can be developed combining the direct simulation Monte Carlo (DSMC) method and a Navier–Stokes (NS) representation, one must have access to compatible kinetic-split fluxes from the NS portion of the hybrid scheme. The kinetic theory basis is given for the development of the required fluxes from the Chapman–Enskog velocity distribution function for a simple gas; and these are then extended to a polyatomic gas by use of the Eucken approximation. The derived fluxes are then used to implement boundary conditions at solid surfaces that are based on concepts associated with kinetic theory and the DSMC method. This approach is shown to lead to temperature slip and velocity slip as a natural outcome of the new formulation, a requirement for use in the near-continuum regime where DSMC and NS must be joined. Several different flows, for which solid boundaries are not present, are computed using the derived fluxes, together with a second-order finite-volume scheme, and the results are shown to agree well with several established numerical schemes for the NS equations.

AOn WAF-Type Schemes for Multidimensional Hyperbolic Conservation Laws

We explore how the weighted average flux approach can be used to generate first- and second-order accurate finite volume schemes for the linear advection equatons in one, two, and three space dimensions. The derived schemes have multidimensional upwinding aspects and good stability properties. From the two-dimensional methods, we construct a scheme for nonlinear systems of hyperbolic conservation laws that is second-order accurate in smooth flow. Spurious oscillations are controlled by making use of one-dimensional TVD limiter functions. Numerical results are presented for the shallow water equations in two space dimensions. The equivalent schemes are derived for nonlinear systems in three space dimensions.

Direct simulation of wave interactions in unsteady natural convection in a cavity

Numerical solutions for unsteady natural convection flow in a square cavity with differentially heated side walls are obtained using an implicit second-order time-accurate finite volume scheme, and briefly compared to experimental data reported elsewhere. The results predict the occurrence of a cavity scale seiche, the presence of waves in the vertical thermal boundary layer that travel from the walls into the horizontal intrusions that form on the horizontal boundaries, and a region of strong divergence at the upstream end of the intrusions. These three mechanisms are observed to interact at a Rayleigh number of 5 × 10 9 to produce a mixing patch in the intrusion, suggestive of a transition to turbulence. The net heat transfer and the approach to steady state are strongly influenced by the presence of the waves.