James J. Gottlieb

Assessment of Riemann solvers for unsteady one-dimensional inviscid flows for perfect gases

Abstract The solution of Riemann problems for the one-dimensional Euler equations with polytropic gases usually involves a numerical iterative solution procedure, and more efficient Riemann solvers can reduce computational times and costs by factors of up to 25, Riemann solvers that have been used in past computational fluid dynamics, those that are used in current numerical work, and a new and more efficient one reported in this paper are all assessed in terms of their relative computational performance. This assessment includes the type of shock and rarefaction-wave equations, iterative procedures, and initial guesses used by Godunov, Chorin, Van Leer, Smoller, and others. Various aspects of the Riemann problem and its solution for unsteady flows are also discussed in terms of the pressure-velocity diagram, both for completeness and to add some new practical insights for improving computer codes.

Review of some adaptive node-movement techniques in finite-element and finite-difference solutions of partial differential equations

Abstract This article summarizes the results of a literature search for adaptive numerical methods of solving partial differential equations; the methods discussed involve the adaptive movement of nodes, so as to obtain a low level of solution truncation error while minimizing the number of nodes used in the calculation. Such methods are applicable to the solution of nonstationary flow problems that contain moving regions of rapid change in the flow variables, surrounded by regions of relatively smooth variation. Flows with shock waves, contact surfaces, slip streams, phase-change interfaces, and boundary layers can be modelled with great precision by these methods. It will be shown that significant economies of execution can be attained if nodes are moved so that they remain concentrated in regions of rapid variation of the flow variables.

A parallel solution-adaptive scheme for multi-phase core flows in solid propellant rocket motors

The development of a parallel adaptive mesh refinement (AMR) scheme is described for solving the governing equations for multi-phase (gas–particle) core flows in solid propellant rocket motors (SRMs). An Eulerian formulation is used to describe the coupled motion between the gas and particle phases. A cell-centred upwind finite-volume discretization and the use of limited linear reconstruction, Riemann solver based flux functions for the gas and particle phases, and explicit multi-stage time-stepping allows for high solution accuracy and computational robustness. A Riemann problem is formulated for prescribing boundary data at the burning surface and a mesh adjustment algorithm has been implemented to adjust the multi-block quadrilateral mesh to the combustion interface. A flexible block-based hierarchical data structure is used to facilitate automatic solution-directed mesh adaptation according to physics-based refinement criteria. Efficient and scalable parallel implementations are achieved with domain decomposition on distributed memory multi-processor architectures. Numerical results are described to demonstrate the capabilities of the approach for predicting SRM core flows.

Analytical study of sonic boom from supersonic projectiles

Abstract Whithams first-order theory for steady flow at moderate supersonic Mach numbers around slender axisymmetric bodies is reviewed and applied to determine sonic boom overpressure signatures from bodies of various shapes, particularly those of projectiles in steady and rectilinear flight. Based on his theory, certain closed-form solutions are derived, including extended first-order decay laws for signature evolution with increasing distance from the flight path. Also, for the case of arbitrary axisymmetric bodies of simple and smooth shape, whose axial profile can be defined piecewise by segments of polynomial curves, a new analytical solution is presented for Whithams ‘smooth-body’ F-function integral. An alternate formulation for the F-function that was derived by Lighthill for non-smooth bodies is shown to be better for actual smooth and non-smooth projectile shapes. A computer code was developed to implement the analysis with various F-functions and compute sonic boom signatures for specified body shapes. These numerical studies were conducted to investigate and illustrate important features of these signatures and their evolution with increasing miss distance. The effects of body shape such as surface protuberances and afterbody flow are two examples.

Parallel AMR Scheme for Turbulent Multi-Phase Rocket Motor Core Flows

The development of a parallel adaptive mesh refinement (AMR) scheme is described for solving the governing equations for turbulent multi-phase (gas-particle) core flows in solid propellant rocket motors (SRMs). The Favre-Averaged Navier-Stokes equations are solved for the gas-phase. Turbulence closure is achieved by using a two equation turbulence model. An Eulerian formulation is used to describe the motion of the inert, dilute, and disperse particle-phase. A cell-centred upwind finite-volume discretization and the use of limited solution reconstruction, Riemann solver based flux functions to determine the inviscid flux for the gas and particle phases at cell interfaces. Green-Gauss integration over the diamond-path defined at cell interfaces is used to determine the primitive-variable gradients for evaluation of the viscous fluxes. A parallel multigrid method coupled with an explicit optimally-smoothing multi-stage time-stepping scheme is used to obtain steady state solutions. Unsteady calculations are achieved through the use of a dual time-stepping approach. The propagation of the propellant-core flow interface is tracked using the level set method and a mesh adjustment scheme is used to fit the computational mesh to the location of the burning interface. Application of block-based AMR accurately resolves the multiple solution scales of the fluid flow and enables efficient and scalable parallel implementations on distributed memory multi-processor architectures. High-scalability of the model has been achieved on a parallel cluster computer consisting of 276 processors. Various numerical test cases are presented to verify the validity of the scheme as well as demonstrate the capabilities of the approach for predicting SRM core flows.

Staggered and nonstaggered grids with variable node spacing and local time step ping for the random choice method

Abstract The staggered grid and random sampling procedure used currently in the random choice method for solving hyperbolic equations like those for one-dimensional unsteady flows and two-dimensional axisymmetric and planar steady supersonic flows are reviewed and extended to include variable node spacing. A nonstaggered grid with variable node spacing, more convenient random sampling process, and local time stepping are then introduced, and there use in significantly reducing computational time and costs in solving certain problems is discussed. These methods are easily applied in the random choice method for problems with more dimensions.

Gas density and particle concentration measurements in shock‐induced dusty‐gas flows

The concept of obtaining both particle concentration and gas density measurements of steady and unsteady gas‐particle flows by means of two complementary instruments is introduced. The two instruments‐the light extinctiometer and beta‐ray densitometer‐and their theory are described, and new experimental techniques for accurately calibrating these devices are presented. These two instruments are combined for the first time to successfully separate the densities of the gas and particulate phases when both properties are varying with time.

PARALLEL SOLUTION-ADAPTIVE SCHEME FOR MULTI-PHASE CORE FLOWS IN ROCKET MOTORS

The development of a parallel adaptive mesh refinement (AMR) scheme is described for solving the governing equations for multi-phase (gas-particle) core flows in solid propellant rocket motors (SRM). An Eulerian formulation is used for both the gas and particle phases, which leads to a degenerate hyperbolic system of partial differential equations. The cause and effect of the degeneracy is examined. A cell-centered upwind finite-volume discretization and the use of limited solution reconstruction, Riemann solver based flux functions for the gas and particle phases, and explicit multi-stage timestepping allows for high solution accuracy and computational robustness. A Riemann problem is formulated for prescribing boundary data at the burning surface of the propellant grain and an iterative solver is proposed for its solution. Efficient and scalable parallel implementations are achieved with domain decomposition on distributed memory multi-processor architectures. High-scalability of the model has been achieved on a Beowulf-class cluster consisting of 104 processors. Numerical results are described to demonstrate the capabilities of the approach for predicting SRM core flows.

A new finite-difference solution-adaptive method

A new moving finite difference (MFD) method has been developed for solving hyperbolic partial differential equations and is compared with the moving finite element (MFD) method of K. Miller and R. N. Miller. These methods involve the adaptive movement of nodes so as to reduce the number of nodes needed to solve a problem; they are applicable to the solution of non-stationary flow problems that contain moving regions of rapid change in the flow variables, surrounded by regions of relatively smooth variation. Both methods solve simultaneously for the flow variables and the node locations at each time-step, and they move the nodes so as to minimize an ‘error’ measure that contains a function of the time derivatives of the solution. This error measure is manipulated to obtain a matrix equation for node velocities. Both methods make use of penalty functions to prevent node crossing. The penalty functions result in extra terms in the matrix equation that promote node repulsion by becoming large when node separation becomes small. Extensive work applying the MFE and MFD methods to one-dimensional gasdynamic problems has been conducted to evaluate their performance. The test problems include Burgers’ equation, ideal viscid planar flow within a shock-tube, propagation of shock and rarefaction waves through area changes in ducts, and viscous transition through a contact surface and a shock.

Numerical evaluation of Whitham's F-function for supersonic projectiles

dimensional cases difficult and in some cases impossible by other methods. An additional advantage is that the procedure works as well in the multidimensional case, including nonlinear terms, which are then written as sums of the appropriate Adomian polynomials as discussed in Ref. 1. The procedure is also easily extended to Vu =f(x,y,z) + ku or even Vu + Nu =f(x,y,z) + ku where Nu is an analytic term.