Temperature in a
cross section of a 3-D Space Shuttle Solid
The Fluid Dynamics group works on system-scale solid rocket motor core flow model development as well as subscale model development relevant to the turbulent dynamics of the combustion interface, dispersion and combustion of Al particulates in the core flow, and crack flow. A major focus is the development of appropriate large eddy simulation (LES) models for the core flow using insights gained from direct numerical simulation (DNS) and particle imaging velocimetry (PIV) laboratory experiments.
The fluid flows in the core, from the injector and in the nozzle and plume of the rocket, are high-temperature, high-speed, highly turbulent flows, with entrained chemically reacting aluminum particles from the propellant. To predict unsteady pressure conditions in the combustion region, the aeroacoustic forcing of acoustic instabilities, heat and mass transfer to the propellant and nozzle, and erosion of the nozzle by the particles, a three-dimensional, time-dependent simulation of the large scales of turbulent motion is needed. In addition, the fluid simulation must account for mass and heat injection from the propellant, the roughness of the interface, and compressibility and shocks in the nozzle and plume.
There are three interrelated issues to be addressed in performing the turbulent fluid flow simulations: (1) formulation and modeling of the equations governing the large scales of turbulence, (2) treatment of the particles in the flow and (3) numerical representation, discretization, and gridding.
The simulation of only the largest scales of turbulence is generally called Large Eddy Simulation (LES). The formulation of LES equations and models is a difficult and unsettled aspect of turbulence simulation research , and so there are several LES approaches. We will pursue three different approaches here. First, we will pursue the "classical approach" in which the Navier-Stokes (or Euler) equations are formally filtered and "subgrid models" for unknown terms are employed. The best currently available models (e.g. the dynamic model ) will be used, and model development will continue as a Center activity (see below). This is expected to be the most reliable approach to the turbulence simulation, and will therefore likely be the primary one. Second, a "no-model" implementation will be pursued, in which numerical dissipation plays the role of the model. This is a simple approach, which will be used initially for early system integration. Finally, the unsteady RANS approach, in which the Reynolds averaged equations and models are solved in three-dimensions with time dependence, will be pursued. Unsteady RANS is expected to be less reliable than classical LES, but it will require significantly less computation, and thus it is well suited to abnormal scenarios in which the fluids are not critical (e.g,. a detonation caused by mechanical damage). It is also a technique being pursued by industry, allowing us to interact with these external efforts.
Unfortunately, even the best current LES modeling techniques do not provide reliable and accurate quantitative predictions in the complicated flow situations encountered here. But models for these and many other flow situations can be developed by formally minimizing the mean square error in the evolution of the resolved field [18,19]. Given sufficient data from idealized flows, this minimization can be done, resulting in an "optimum" LES formulation and a measure of the uncertainty (i.e. the mean square error). Further, these techniques can be used to optimize and evaluate any of the various styles of LES (e.g., no-model, RANS, or classical). This approach is being developed for simple incompressible flows under an NSF grant, and will be applied to the rocket flow problem as part of the ASCI effort. The required data will be obtained from direct numerical simulations of turbulence in several subscale simulations. These include DNS of a compressible wall-bounded flow with strong wall blowing and modeled roughness to obtain data for the core-flow conditions, particle evolution in a DNS of a simple wall-bounded turbulent flow for modeling particle dispersion by subgrid turbulence, and the propellant combustion simulation for data on the reacting boundary. In addition, to circumvent the Reynolds number limitations of DNS, these simulations will be supplemented by limited small laboratory fluid flow experiments, using particle image velocimetry (PIV) diagnostic techniques. The subscale simulations and laboratory data will also be used for validation of fluid component simulations.
The internal flow resulting from combustion of metallized solid propellants is a two-phase flow problem, including propellant combustion gases and metal (typically aluminum) particulates as large as 100 Ám. These particles are injected at the grain burn front, combust to Al2O3 throughout the cavity, and are strongly coupled to the mass, momentum, energy, and species concentration budgets of the gas. In addition, failure modes are attributable to the particle phase, such as nozzle erosion and slag ejection.
Because the particle dynamics are ballistic in some regions, a hybrid Euler-Lagrangian treatment is required. Following , we will develop a hybrid model for two-phase reactive flow in the flow cavity. The Eulerian LES model for the gas phase is discussed above, and will include the Al2O3 "smoke," which is the dominant source of opacity in the gas. Radiative heat fluxes arising from the distributed combustion of the Al particles will, in a first approximation, be modeled by flux-limited radiation diffusion using the product formula algorithm of . The particle phase will be modeled with Lagrangian particle-mesh N-body methods, supplemented by empirical laws for particle drag and combustion. These will be developed from Al particle combustion subsimulations. Turbulent dispersion of the particles is computed directly from our LES model, although a subgrid dispersion contribution will still be required as discussed earlier. A particle agglomeration model will be developed to model slag accumulation.
In implementing this model, we will draw heavily on our extensive experience developing hybrid Eulerian-Lagrangian codes for simulation of cosmological structure formation on massively parallel computers, particularly the hybrid Euler-Lagrangian SAMR cosmological code developed by Bryan and Norman.
The nature of turbulence, in which small-scale features exist everywhere, rather than being concentrated at fronts or boundaries, makes numerical representation particularly challenging . Flexibility and local refinement capabilities will be needed due to the complicated and evolving geometry of the flow cavity, while extremely good resolution will be needed for the turbulence.
The requirements for high spatial resolution and high-order accuracy make unstructured meshes unattractive, since CFD algorithms implemented using them are generally of low-order accuracy. Instead, body-fitted, structured meshes will be used in the main flow cavity, and resolution flexibility will be attained using structured adaptive mesh refinement (SAMR ), which allows selective refinement of regions requiring higher spatial resolution. To apply SAMR to a body-fitted grid, we will use an object-oriented system for SAMR on parallel computers (AMR++) developed by Balsara and Quinlan  in conjunction with the Overture framework ([25,26] for body-fitted grids. To handle extreme geometric complexity (e.g., propellant cracks), the structured mesh will interface with an unstructured mesh representing the complicated feature. Thus, an unstructured mesh will be used between the relatively simple boundary of the structured mesh and the more complicated physical boundary. Furthermore, the unsteady RANS formulation does not require such high spatial resolution, so, for this formulation, unstructured grids can be used throughout the flow volume.
The gridding and adaptation technology discussed above can be used with an expanding class of flow solvers, and high-resolution methods appropriate for the turbulent simulation will be used. For unstructured grids, a control volume finite-element method will be used . In addition, two promising approaches to improve resolution properties for turbulence simulation in the context of such adaptive grids will be pursued. One is the use of spline representations, which provide the high resolution of high-order compact schemes while allowing for resolution embedding as occurs in AMR grids. The second is the use of advanced nodal integral methods, which allow parts of the relevant operators to be treated analytically, thus improving both resolution and accuracy.
 M. Germano, U. Piomelli, P. Moin, and W. H. Cabot. A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A, 3:1760, 1991.
 R. J. Adrian. Stochastic estimation of sub-grid scale motions. Appl. Mech. Rev., 43:214, 1990.
 T. G. Bagwell. Stochastic estimation of near-wall closure in turbulence models. PhD thesis, University of Illinois at Urbana-Champaign, Urbana, IL, 1994.
 J. Sabnis, F. de Jong, and J. Gibeling. AIAA paper 92-3509. In 28th Joint Propulsion Conference and Exhibit. AIAA/SAE/ASME/ASEE, 1992.
 F. Graziani. The product formula algorithm applied to linear and radiation diffusion. J. Comp. Phys., 118:9-23, 1995.
 S. Ghosal. An analysis of numerical errors in large-eddy simulations of turbulence. J. Comp. Phys., 125:187, 1996.
 M. J. Berger and P. Colella. Local adaptive mesh refinement for shock hydrodynamics. J. Comp. Phys., 82:6484, 1989.
 D. S. Balsara and D. J. Quinlan. Parallel object-oriented adaptive mesh refinement. In 8th SIAM Conference of Parallel Processing for Scientific Computing, 1997.
 D. L. Brown, G. S. Chesshire, W. D. Henshaw, and D. Quinlan. Overture: An object-oriented software system for solving partial differential equations in serial and parallel environments. In Proc. 8th SIAM Conf. Parallel Processing, 1997.
 W. D. Henshaw. Grid, GridFunction and Interpolant classes for Overture, AMR++ and CMP-GRD, User Guide, version 1.00. Technical Report 96-3464, Los Alamos National Laboratory, 1996.
 R. Jyotsna and S. P. Vanka. Multigrid calculation of steady viscous flow in a triangular cavity. Journal of Computational Physics, 122:107-117, 1995.