Solution Adaptive Methods for Low-Speed and All–Speed Flows

The goal of this work was to design new fast algorithms that could be used to soIve fluid flows at all speeds by building upon the best approaches now available for solving very low speed flows and high speed flows. Furthermore the algorithms developed must be appropriate for use on complex moving geometries and for use with adaptivern=h refmernmx The algorithms must also be extendible to chemically reacting (combustion) flows. To this end we have developed new methods for efllcientiy computing fluid problems that involve low-speed flows and problems that are a mixture of low-speed and high-speed flows. The algorithms have been implemented in 2D and 3D on moving overlapping grids and will be a fundamental component of the chemically reacting flow solvers that we are now developing for industrial applications.


and All-Speed F'IOWS
The goal of this work was to design new fast algorithms that could be used to soIve fluid flows at all speeds by building upon the best approaches now available for solving very low speed flows and high speed flows. Furthermore the algorithms developed must be appropriate for use on complex moving geometries and for use with adaptive rn=h refmernmx The algorithms must also be extendible to chemically reacting (combustion) flows.
To this end we have developed new methods for efllcientiy computing fluid problems that involve low-speed flows and problems that are a mixture of low-speed and high-speed flows. The algorithms have been implemented in 2D and 3D on moving overlapping grids and will be a fundamental component of the chemically reacting flow solvers that we are now developing for industrial applications.

Background and Research Objectives
The computation of low-speed (i.e., slightly compressible) flow is a significant computational challenge. It is of great importance since almost all industrial flow problems and many flows found in nature are low speed. The difficulty is directly related to the fact that a slightly compressible flow is, to first approximation, a combination of an incompressible flow together with rapidly moving sound waves. To compute the fine spatial structure of the vortices in an incompressible flow requires high spatial resolution but a relatively large time step, while computing the rapidly moving sound waves requires a very small time step but not as fine spatial resolution. Applying standard methods to the combined problem requires both a fme grid and a very small time step--thus making it infeasible to solve many problems of interest.
There are a number of approaches that have been tried to treat all-speed flows. help for moderately low Mach numbers although it will be quite expensive to apply to the full three-dimensional Navier-Stokes equations. Straight forward implicit schemes further suffer from the fact that the implicit matrix that needs to be inverted becomes highly skewed (far from symmetric), which can lead to loss of accuracy. To overcome this problem to some extent it is possible to "precondition" the matrix [1]. This has the effect of artificially slowing down the fast sound waves. However even the best preconditioners still have problems for very small Mach numbers.
Another approach to solving Iow Mach number flows is the method of artificial Starting from the works of [6], [7] and extended by others [4], [12], [13] it became apparent that it was not necessary to treat the entire system of equations implicitly (in 3D the compressible Navier-Stokes equations area system of five coupled partial differential equations). Since only two of the five wave speeds associated with this system are large (corresponding to sound waves) it was possible to derive a nonlinear equation for the pressure (a sort of wave equation) that could be solved implicitly. This method worked fairly well although there could still be difficulties for very low Mach number and Mach numbers of order one (since the methods were not optimal when shocks were present).
The previous approaches have started from the fully compressible equations.
Another class of methods use asymptotic expansions to derive equations valid for low Mach number. The basic idea here is that a slightly compressible flow consists of an incompressible piece (M=O) with a correction that is proportional to the Mach number.
Thus one can compute corrections to incompressible flow by solving extra equations that are derived by formal asymptotic expansions, treating the Mach number as a small parameter [11],[5], [10].
In recent years, some very good numerical procedures have been developed for computing the solution to problems that are either truly "high-speed" (i.e., compressible) or truly incompressible (in which case the sound waves have been removed entirely). There are many problems, however, that are a combination of low-speed and high-speed flow.
The high-speed flow may be restricted to a part of the domain or it may only appear periodically in time. In an internal combustion engine. or pulsed combustor. for example, the flow is slightly compressible, except for part of a cycle in part of the domain.
The purpose of the work done here was to devise, analyze and implement techniques for the efilcient computation of problems involving low-speed (slightly compressible) fluid flows and for problems that contain regions of both low-speed and high-speed (compressible) flow. We have built upon the efilcient and accurate methods that have been developed for incompressible flows, such as the projection methods of Bell, Collela and Glaz [3] and the fourth-order accurate method of Henshaw, Kreiss and Reyna [9]. Incompressible flow methods such as these usually incorporate an efficient To tackle the problem that involves both low-and high-speed flow we have chosen the approach of using a "solution-adaptive" hybrid method. We will adaptively choose the most appropriate method for the current state of the solution. In regions in space and time when the flow is low speed we will use the slow-speed method proposed here and described in more detail below; in regions of high-speed flow we will smoothly change over to a state-of-the-art method for compressible flows. In the past it was probably too difficult to take this solution adaptive approach since the programming details became overwhelming. We have taken advantage of our experience with new object-oriented programming techniques using new computer languages (C++) to allow us to implemented much more complicated programs that not only solve different equations on complicated three-dimensional geometries but also allow for moving and adaptive grids. Moreover since we use the Overture (see publication 2) framework the codes will also be able to run on both serial and parallel machines. 3

Importance to LANL's Science and Technology Base and National R&D Needs
The algorithms developed in this work will be a fundamental component of the ailspeed chemically reacting flow solver that we are now developing for a follow-on DOE funded project in collaboration with Caterpillar and Ford. This solver will handle complicated moving geometries using overlapping grids and the Overture objected-oriented framework. Adaptive mesh refinement will allow the accurate and efficient computation of complicated flows. A major impact will be to provide state-of-the-art algorithms and numerical methods to indust~.
These solvers will have many industrial and weapons technology applications beside these combustion computations. They will likely be used by one or more of the ASCI funded university initiatives. Researchers at the University of Utah are planning to use the work in their modeling of accidental fires and explosions while the University of Illinois has plans for utilizing the work in their rocket modeling.

Scientific Approach and .Accomplishments
We

A Projection Method for Low-Speed Flows
To some degree of approximation a low-speed flow has three basic components- The system of flow equations with time evolution equations for the density, velocity and pressure is thus extended to a system for the density, the nearlyincompressible velocity, a potentiaI velocity, a potential pressure, and an acoustic pressure.
The key to developing an efficient scheme for this new system is based on the knowledge of the characteristics of the different components. The potential velocity and acoustic pressure contain the very fast times scales associated with the speed of sound and since we are not interested in resolving this part of the solution we can use a time discretization (backward-Euler) that will damp these fast sound waves. This allows us to compute with a much bigger time step. The equations are coupled, but they can be advanced in a particular order so that the appropriate information is known when it is required. Care must be taken

A Solution Adaptive Approach for All-Speed Flows
The second approach we have developed couples a partially implicit method for low This solution adaptive approach has been incorporated into a general purpose solver for overlapping grids with moving geometries in two-and three-space dimensions. The solver uses the Overture class libraries that we have developed, which provide a high-level interface to solving PDEs on moving overlapping grids (see publication 3). This means that there is significant reuse of code between the solvers and that future solvers will be easier to write. The requirements of the all-speed solvers have also influenced the design and capabilities of the Overture library. Those capabilities that are more generic and that