Elsewhere we describe the application of AMR to low-speed fluid flows and to inviscid high-speed (compressible) fluid flows. In this section, we consider high-speed flows for which viscosity is important. Frequently high-speed flows are insensitive to the presence of viscous forces in the fluid. However, even weak viscous forces can be important if the fluid flow is sensitive to the dynamics at very short length scales. That is the case in the examples considered in this section.
The first example which will be considered is a relatively direct application of the AMR methodology to the problem of strong shocks diffracting over rigid ramps in argon gas. This example is of particular interest because it resolves a long-standing paradox between the von Neumann theory of shock reflection and experiments.
The second example describes Adaptive Mesh and Algorithm Refinement. This elaborates the basic AMR algorithm by allowing different levels of refinement to use different numerical methods to describe the fluid flow. On very small length scales, the continuum description of a fluid is not valid, and it is necessary to use a particle method which simulates the motion of individual gas particles.However, such particle methods are too computationally expensive to use over the entire problem domain. This example uses the particle method only where necessary to resolve the small scale particle dynamics, and uses a continuum method in the rest of the problem domain.
This calculation simulates experiments in which strong shocks in argon diffract over steel ramps. It has been known for many years that such shocks will diffract over the ramp with one of two distinct patterns, called regular reflection and Mach reflection. The transition between the two patterns is determined by the angle of the ramp. It has been apparent for many years that the transition angle determined experimentally is in conflict with the theoretical predictions of inviscid gas dynamics, sometimes called the von Neumann shock theory. This discrepancy is sometimes called the von Neumann paradox. This calculation has removed the contradiction between theory and experiment by accurately simulating the diffraction in viscous gas dynamics. This calculation would not have been possible without the ability to resolve a wide range of length scales in a calculation. This calculation spanned length scales differing by a factor of 105, from an experimental scale of 4 centimeters, to the grid spacing of 0.4 micron on the finest level of refinement. The fine scale is necessary to resolve the viscous boundary layer that forms on the ramp behind the shock. Because viscous forces are relatively weak, the boundary layer is thin. But even a thin boundary layer can have an effect near the transition angle between regular and Mach reflection.
The following figure shows the temperature in one of our calculations near the transition angle. The bottom surface of each graph is located at the top surface of the ramp. The shock impacts the ramp surface, arriving from the upper left. The density on the far right of color plot (a) is the temperature of the ambient argon. Each of the color plots shows the calculation at different levels of refinement, from coarse to fine. The progression may be considered as "zooming in" on the calculation. Each succeeding level is refined by a factor of four compared with the coarser level.
The boxes that are superimposed upon the color temperature plots show the locations of the adaptively refined grids at each level of refinement. It should be understood that this is an unsteady calculation in which the shock propagates from left to right as the calculation progresses. The refined grids dynamically adjust to follow the physically interesting region and maintain the accuracy of the calculation.
In the finest color plot (e), the Mach stem is clearly apparent, although it is less than 10 microns tall at that point. The blue regions forming on the ramp surface behind the plot is the viscous boundary layer. The gas in the layer is cooled by contact with the steel ramp. Resolving the boundary layer is crucial to bringing theoretical calculations into agreement with experiment. Experiments that confirm the predictions of this calculation are described in "Experiments with the persistence of regular reflection in strong shock diffraction over rigid ramps," by L. F. Henderson, K. Takayama, W.Y. Crutchfield, and S. Itabashi, accepted for publication in J. Fluid Mechanics.
This figure also illustrates the capability of computational fluid dynamics to resolve important dynamical regions that are difficult to access experimentally. In this case, the dynamics is driven by a corner signal propagating in the boundary layer. As can be seen in the coarsest plot, (a), the boundary layer is quite invisible at ordinary length scales. In order to probe the interesting region, an experimenter must be able to resolve micron scale features, such as those shown in plot (e).
The code for modeling the compressible Navier-Stokes equations is an extension of HyperCLaw, but is not available for release. For more about these calculations, contact William Crutchfield of CCSE.
L. F. Henderson, W. Y. Crutchfield, and R. J. Virgona, The Effects of Heat Conductivity and Viscosity of Argon on Shock Waves Diffracting over Rigid Ramps, J. Fluid Mech., 331, pp. 1-36, 1997. [ps.gz]
As illustrated above, adaptive mesh refinement enables the use of extremely fine mesh spacing in regions that require high resolution. However, hydrodynamic formulations break down as the grid spacing approaches the molecular scale (e.g., the mean free path in a gas). We have developed a hybrid Adaptive Mesh and Algorithm Refinement (AMAR), in which a continuum algorithm, such as a Navier-Stokes solver, is replaced by a particle algorithm, namely Direct Simulation Monte Carlo (DSMC), at the finest grid scale.
As an illustration, consider the flow of a gas through a microscopic channel, such as between the head and platter in a disk drive. The continuum description of the flow and the quantities derived from it, such as wall drag, are not accurate whenever the Knudsen number, the ratio of the mean free path to the width of the channel, is sufficiently large. Kinetic theory extensions to the continuum equations (e.g., Burnett expansion) have had limited success. Another approach is to introduce kinetic corrections to the boundary conditions but these are often not accurate and can even give wrong qualitative features of the flow. We have developed a new methodology that allows us to model these types of flows accurately and efficiently.
Rigorously, a kinetic formulation is required at microscopic scales; unfortunately, using a kinetic approach is too expensive for problems where we need to model larger hydrodynamic scales. However, at hydrodynamic scales the continuum approximation is valid. AMAR capitalizes on this characteristic by using a particle method in regions of a flow requiring microscopic resolution and a continuum method, with varying levels of refinement, to evaluate the flow at larger scales. Thus, AMAR provides an effective methodology that can span a broad range of length scales while retaining the advantages of a kinetic formulation where required.
The AMR framework we have developed allows us to use continuum methods for solving the compressible Navier-Stokes equations at coarser levels while coupling to a kinetics-based particle method, direct simulation Monte Carlo (DSMC), at the finest resolution. DSMC provides an accurate discretization of the kinetics behavior and is several orders of magnitude more efficient than molecular dynamics for the simulation of gases.
The new hybrid methodology has been validated on a number of sample problems. The methodology correctly predicts the details of viscous shock wave profiles and Knudsen velocity slip past a moving wall (the Rayleigh problem). Both of these test problems cannot be accurately predicted with a continuum solver alone, but are accurately predicted by the AMAR algorithm.
As an additional test of the hybrid algorithm we computed a Mach 2 flow about a microscopic sphere, small enough so that Knudsen effects are important. The sphere has a diameter of five free stream mean free paths, about 0.3 microns. The computation uses two continuum levels with a third level for the DSMC region. Figure 1 shows the density on a plane cutting through the center of the domain. A black box surrounds the DSMC region. The microscopic sphere is not shown in this density plot. Note the shock emanating from the sphere and the rarefaction region behind the sphere. A continuum representation of the density in the DSMC region is presented in Figure 2. This is in effect a closeup of Figure 1, but only the DSMC region is shown. Three dimensional effects are shown using a cross-planes display. Note that the color contours are not smooth, which is a reflection of the discrete, stochastic nature of the DSMC algorithm. Figure 3 shows the location of the individual DSMC particles around the sphere that have hit the sphere. The DSMC portion of the calculation uses only about 100,000 particles.
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The hybrid algorithm was able to successfully predict the same drag on the sphere as computed with a pure DSMC approach. In comparison, a purely continuum calculation predicts a drag coefficient a factor of two larger, in disagreement with DSMC calculations and experiment.
The AMAR code which generated the above results is not available for release. For more about these calculations, or the AMAR methodology, contact John Bell of CCSE.
A. L. Garcia, J. B. Bell, W. Y. Crutchfield, B. J. Alder, ``Adaptive Mesh and Algorithm Refinement,'' J. Comp. Phys., 154, pp. 134-155, 1999. [ps.gz]