Eswar Josyula

Deterministic solution of the spatially homogeneous Boltzmann equation using discontinuous Galerkin discretizations in the velocity space

Abstract We present a new deterministic approach for the solution of the spatially homogeneous Boltzmann kinetic equation based on nodal discontinuous Galerkin (DG) discretizations in the velocity space. In the new approach the collision operator has the form of a bilinear operator with a pre-computed kernel; its evaluation requires O ( n 5 ) operations at every point of the phase space where n is the number of degrees of freedom in one velocity dimension. The method is generalized to any molecular potential. Results of numerical simulations are presented for the problem of spatially homogeneous relaxation for the hard spheres potential. Comparison with the method of Direct Simulation Monte Carlo showed excellent agreement.

Direct Simulation Monte Carlo with Octree Cartesian Mesh

We describe an implementation of a new Direct Simulation Monte Carlo (DSMC) code with adaptive octree Cartesian mesh in the Unified Flow Solver (UFS) framework. UFS combines a variety of coupled Boltzmann and NS solvers, and addition of the DSMC solvers allows one to further expand its capabilities. The UFS-DSMC code utilizes a single mesh for (i) particle collision and (ii) statistics collection/visualization and particle movement. Such a single mesh can be easily built using AMR capabilities based on local properties (mean free path and/or gradients of flow parameters) for flows around complex shapes, and can be used to efficiently perform some of the same functions as transient subcells or other collision partner selection options available in modern DSMC codes. The quad/octree data structure allows straightforward and efficient data management during dynamic grid refinement/coarsening and makes possible seamless parallelization of the code. The capabilities of UFS-DSMC are illustrated for benchmark cases of steady-state flows past blunt objects. Results of UFS-DSMC are compared with solutions of HAP and MONACO DSMC codes and good agreement is found. UFS-DSMC was observed to show similar efficiency compared to the baseline HAP code.

Hypersonic nonequilibrium flow computations using the Roe flux-difference split scheme

The Roe flux difference splitting scheme is investigated for accuracy in simulating hypersonic reacting flows. The extension of the Roe scheme to include the finite rate chemical kinetic equations follows the approach of Grossman and Cinnella. Formal second-order accuracy is obtained by employing the monotonic upstream schemes for conservation laws (MUSCL) approach in conjunction with the minmod limiter to degenerate the solution to first-order accuracy in the vicinity of strong shock waves. The full Navier-Stokes equations are solved with finite rate chemistry for the flow past an axisymmetric blunt body at zero incidence at several Mach numbers. The vibrational energy is assumed to be in thermodynamic equilibrium with the other internal energy modes

Automated Aerodynamic Optimization for Lifting Hypersonic Vehicles at High Altitude

Automated design optimization procedures are coupled to three-dimensional direct simulation Monte Carlo (DSMC) calculations for deformation of waverider-derived hypersonic vehicle shapes, in order to maximize the lift-to-drag ratio (L/D) under high altitude reentry conditions. For a lifting reentry vehicle, increased L/D under such conditions should permit a reduction in aerothermal heating, body forces and ionization effects, as well as an increase in range and other potential benefits related to mission cost and safety. In the optimization procedure employed in this paper, a new parallel Cartesian adaptive grid implementation of the DSMC method is integrated with surface deformation and threshold accepting algorithms for probabilistic multivariable optimization. Several waverider geometries, created through an inverse design technique, were used as starting points for optimization calculations, and multiple optimization runs were carried out for each starting shape. Including all iterations for each run, optimization calculations performed as part of this work involve 2880 independent three-dimensional DSMC simulations. A maximum increase in L/D of approximately 65% is demonstrated at Mach 14 and 90 km altitude, for a 5 m long vehicle with a global Knudsen number of 0.0033. Additional simulations were performed for this optimized geometry at design conditions, in order to assess the extent of continuum breakdown, flowfield characteristics, surface heat flux and temperatures, and the effect of leading edge bluntness on both heat flux and L/D values. To the authors’ knowledge, this work represents the first integration of DSMC within an automated design optimization scheme, and demonstrates the potential efficacy of DSMC based aerodynamic optimization in reentry vehicle design.

Efficient Direct Simulation Monte Carlo Modeling of Very Low Knudsen Number Gas Flows

Although the direct simulation Monte Carlo (DSMC) method can in theory provide high fidelity simulation results for any dilute gas flow, in practice this method has generally been restricted by considerations of computational expense from modeling gas flows with very low global Knudsen number (Kn). In this work, a series of DSMC algorithm modifications are proposed for efficient simulation of very low Kn flows, based on domain decomposition between regions where viscous effects can be neglected, quasi-one-dimensional boundary layer regions, and other portions of a flowfield where viscous effects are important and local flow characteristics are inherently multidimensional. Modifications to DSMC collision limiter procedures are presented for improved accuracy in DSMC-based inviscid flow simulation, by addressing numerical diffusion errors associated with finite collision separation, free-molecular advection and incomplete collisional equilibration. Locally onedimensional flow approximations are imposed within portions of the boundary layer in order to satisfy DSMC guidelines while greatly reducing the local particle population, and binary collision probabilities are modified through a density interpolation technique which permits much larger collision cells than traditional DSMC procedures. Applicable conditions include low Kn (uf03c 10) laminar steady state gas flows where viscous effects are negligible within large portions of the flowfield, and where the boundary layer thickness tends to be small relative to the local surface curvature radius. Proposed modifications are implemented in the recently developed HAP DSMC code, and a series of test cases are presented to assess numerical error sources. DSMC simulation results are compared with analytical solutions or available Euler/Navier-Stokes CFD results for a simple free shear flow, an inviscid shock interaction flow, and a viscous hypersonic flow around a blunted wedge.

Numerical Simulation of Hypersonic Blunt Body and Nozzle Flows using Master Equation

Numerical simulations of hypersonic flow past blunt body and expanding nozzles were conducted using an operator splitting approach for coupling the master equation consisting of state‐to‐state kinetics with the fluid dynamic equations. The vibrational‐translational (V‐T) and vibrational‐vibrational (V‐V) energy transfer processes were included in the master equation. The resulting stiff system of equations were solved satisfactorily with the operator splitting approach. Separate nozzle simulations were performed with pure nitrogen and pure oxygen in the temperature ranges of 5000 K–6000 K and 2200 K–3500 K, respectively. Highly nonequilibrium (i.e. non‐Boltzmann) distributions were predicted at the nozzle exit for the selected range of temperatures. The contribution of vibrational‐vibrational (V‐V) transition rates to the overall vibrational relaxation process was found to be higher at the lower nozzle throat temperatures.

Multiquantum Vibrational Energy Exchanges in Nonequilibrium Hypersonic Flows

Numerical simulations are presented of steady state, hypersonic blunt body flows and the influence of multi-quantum vibrational energy exchange assesed for vibrational heating and cooling conditions. The objective is to understand the thermodynamic nonequilibrium phenomena encountered along the trajectory of hypersonic aerospace vehicles. The nonequilibrium vibrational energy distributions were modeled by the master kinetic equation and the population distributions in the quantum energy states of the diatomic molecule evaluated under multiple-quantum vibrational-translational (VT) and vibration-vibration (VV) energy exchanges. The competing effects of resonant and non-resonant VV exchanges and VT de-excitation rates were studied for Treanor distributions that exist for conditions typically encountered in expanding nozzles. For vibrational cooling flows, additional uppumping of energy in the intermediate quantum levels due to enhanced VV exchanges of the double quantum transitions was captured in the numerical simulation. Results from a newly developed dissociation model compared with existing experiments in pure nitrogen suggest that the ladder model with strong vibrational bias is suitable at temperatures 10,000 K and below; at much higher temperatures a weak bias is favored.

Sensitivity Analysis and Uncertainty Quantification for a Set of Hypersonic Shock Interaction Flows

Hypersonic gas flows around double cone and double wedge geometries tend to involve complex shock-shock and shock-boundary layer interactions, and are characterized by the formation of several flow structures which are highly sensitive to physical models and input parameter values utilized in flow simulation. This work is intended to clarify the influence of several modeling parameters and approximations on output quantities of interest by means of a more global and systemic approach than has previously been employed for these types of flows. A Monte Carlo approach for sensitivity analysis (SA) and uncertainty quantification (UQ) is integrated with a new direct simulation Monte Carlo (DSMC) gas flow simulation code, and a large number of DSMC simulations are performed for a representative hypersonic flow over a double cone. Simulation results are analyzed to determine the relative sensitivity of local and global output quantities to several input uncertainties, and probability distributions are computed for these output quantities. Aleatory and epistemic uncertainties are considered independently with different sampling techniques, and both uncertainty types are included in UQ and SA calculations. In addition to DSMC-based SA/UQ calculations, double cone calculations are presented for a NavierStokes computational fluid dynamics (CFD) code. DSMC and CFD simulation results are compared with other computational and/or experimental data to ensure simulation accuracy under nominal flow conditions, and UQ results are employed to help explain discrepancies with published data.

Gas-kinetic scheme for rarefied flow simulation

For increasingly rarefied flowfields, the predictions from continuum formulation, such as the Navier-Stokes equations lose accuracy. These inaccuracies are attributed primarily to the linear approximations of the stress and heat flux terms in the Navier-Stokes equations. The inclusion of higher order terms, such as Burnett, or high-order moment equations, could improve the predictive capabilities of such continuum formulations, but there has been limited success in the shock structure calculations, especially in the high Mach number case. Here, after reformulating the viscosity and heat conduction coefficients appropriate for the rarefied flow regime, we will show that the extended Navier-Stokes-type continuum formulation may still be properly used. The equations with generalized dissipative coefficients based on the closed solution of the Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation, are solved using the gas-kinetic numerical scheme.

Assessment of Vibrational Nonequilibrium for State Resolved Simulation of a Hypersonic Flow

Recent developments in internal energy nonequilibrium modeling for hypersonic gas flow simulations indicate a strong need for improved fidelity, and efforts have consequently focused on discrete state kinetic relaxation (DSKR) techniques which independently model quantum state populations. High computational requirements for DSKR modeling of large scale flows have to date limited the applicability of this type of model, and further work is needed to greatly improve model efficiency without compromising accuracy. With an ultimate goal of developing models for DSKR methods with locally adaptive (model) fidelity or other strategies to achieve large efficiency gains, in the present work we investigate various means of quantifying and predicting vibrational nonequilibrium for a representative hypersonic flow. We assess a series of nonequilibrium metrics based on the normalized difference between translational and vibrational temperatures, Kolmogorov-Smirnov (KS) statistics and population ratios which directly measure deviation from equilibrium state populations. Additional metrics are based on time scale ratios that indicate relative magnitudes of collisional equilibration effects and gradient-driven nonequilibrium forcing for individual vibrational states. We consider a simple two-dimensional case of inviscid and N2 flowing over a cylindrical forebody with a freestream Mach number of 6.5. The forced harmonic oscillator model is used to determine rate coefficients for single quantum vibrational-translational energy exchange, while dissociation and vibration-vibration transfer effects are neglected. In observing trends among different nonequilibrium metrics, we find areas of agreement as well as significant discrepancies. These observations point to a need for careful interpretation of nonequilibrium assessments and choice of nonequilibrium metrics in the use of state-to-state kinetics in CFD flow solvers.