AMR for Incompressible Flow Modeling

The algorithmic complexity of AMR algorithms is significantly higher for low-speed as opposed to fully compressible flows. The equations of inviscid gas dynamics are systems of conservation laws, purely hyperbolic in character. The equations governing incompressible flow, by contrast, include hyperbolic equations governing advection, parabolic equations for the Crank-Nicolson discretization of diffusing quantities, and elliptic equations to enforce the velocity divergence constraint. The method we use to solve these equations is based on a projection formulation in which we first solve advection-diffusion equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergence-free vector fields. Our treatment of the first step uses a specialized second-order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high Reynolds number flow. Density and other scalars are advected in such a way as to maintain conservation, if appropriate, and free-stream preservation.

Inviscid Turbulent Plane Jet

This simulation is an inviscid calculation of a spatially evolving turbulent plane jet. The flow enters the left-hand side of the domain with a high-speed stream in the center with low-speed streams above and below. Turbulent jets have applications in such fields as propulsion, combustion and environmental flows. In the case of combustion, many engines are designed to have the fuel and oxidizer introduced into the combustion chamber as co-flowing jets. Mixing on the molecular level of these two streams is essential for combustion to occur. Near the inflow, the mixing process is dominated by the large-scale structures in the flow. Further downstream, as the jet develops, the mixing is influenced more by small-scale turbulence.

Mpeg animations are available of the evolution of the vorticity and passive scalar fields in this jet. The vorticity animation ( mpeg 14MB) shows development of strong two-dimensional vortical structures in the shear layers near the nozzle lips (left-hand-side of the image) followed by the downstream breakdown to three-dimensional turbulence. The animation of the passive scalar field ( mpeg 2MB ) shows the evolution of an isosurface of the scalar. Near the inflow, two-dimensional structures can be seen due to the large-scale engulfment of co-flow fluid by the two-dimensional vortical structures mentioned above. Downstream, the isosurface of the passive scalar is more three-dimensional and random due to small-scale turbulent mixing.

Using adaptive mesh refinement, a resolution equivalent to 8 million data points per time step was achieved in this simulation using only approximately 2-3 million data points per time step for 740 time steps. This simulation was performed using 38 wall clock hours on 32 processors of the NERSC Cray T3E-900 and generated 50 gigabytes of data.

Three-dimensional Variable Density Incompressible Shear Layer

Here we show a "snapshot" in time from the calculation of a time-evolving three-dimensional variable density shear layer calculation. The figure here is a three-dimensional rendering of vorticity; green is the maximum value, and the background black is the absence of vorticity. The problem specification models the classic experiments performed by Brown and Roshko (1974) to study the effects of density variation on low speed shear layers. We can see the essentially two-dimensional roll-up of the vortex sheet as well as the spanwise structure induced by a three-dimensional perturbation of the inflow data designed to mimic a mild "flutter" of the splitter plate used in experiments.

Although the grids are not shown in this image, this calculation was performed adaptively, on a DEC Alpha workstation. In this calculation the density interface and subsequent roll-up are captured at the finest resolution, while the regions near the boundary are at coarser resolution. The computational results were in agreement with experimental data for both the visual spreading rate and the mean profiles of velocity.

Reference

A. S. Almgren, J. B. Bell, P. Colella, L. H. Howell, and M. L. Welcome, ``A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier-Stokes Equations,'', J. Comp. Phys., 142, pp. 1-46, 1998. [ps.gz]

The IAMR code which generated the results discussed above is available as part of the CCSE Applications Suite. For more about the incompressible adaptive methodology, or about these calculations, contact Ann Almgren of CCSE.