Christopher J. Roy

GRID CONVERGENCE ERROR ANALYSIS FOR MIXED-ORDER NUMERICAL SCHEMES

New developments are presented in the area of grid convergence error analysis and error estimation for mixed-order numerical schemes. A mixed-order scheme is defined here as a numerical method where the order of the local truncation error varies either spatially (e.g., at a shock wave) or for different terms in the governing equations (e.g., first-order convection with second-order diffusion). The case examined herein is the Mach 8 laminar flow of a perfect gas over a sphere-cone geometry. This flowfield contains a strong bow shock wave where the formally second-order numerical scheme is reduced to first order via a flux limiting procedure. The mixedorder error analysis method allows for non-monotone behavior in the solutions variables as the mesh is refined. Non-monotonicity in the local solution variables is shown to arise from a cancellation of first- and second-order error terms for the present case. The proposed error estimator, which is based on the mixed-order analysis, is shown to provide good estimates of the actual error. Furthermore, this error estimator nearly always provides conservative error estimates, in the sense that the actual error is less than the error estimate, for the case examined.

Review of Discretization Error Estimators in Scientific Computing

Discretization error occurs during the approximate numerical solution of differential equations. Of the various sources of numerical error, discretization error is generally the largest and usually the most difficult to estimate. The goal of this paper is to review the different approaches for estimating discretization error and to present a general framework for their classification. The first category of discretization error estimator is based on estimates of the exact solution to the differential equation which are higher-order accurate than the underlying numerical solution(s) and includes approaches such as Richardson extrapolation, order refinement, and recovery methods from finite elements. The second category of error estimator is based on the residual (i.e., the truncation error) and includes discretization error transport equations, finite element residual methods, and adjoint method extensions. Special attention is given to Richardson extrapolation which can be applied as a post-processing step to the solution from any discretization method (e.g., finite different, finite volume, and finite element). Regardless of the approach chosen, the discretization error estimates are only reliable when the numerical solution, or solutions, are in the asymptotic range, the demonstration of which requires at least three systematically refined meshes. For complex scientific computing applications, the asymptotic range is often difficult to achieve. In these cases, it is appropriate to treat the numerical error estimates as an uncertainty. Issues related to mesh refinement are addressed including systematic refinement, the grid refinement factor, fractional refinement, and unidirectional refinement. Future challenges in discretization error estimation are also discussed.

Assessment of One- and Two-Equation Turbulence Models for Hypersonic Transitional Flows

Hypersonic transitional flows over a flat plate and a sharp cone are studied using four turbulence models: the one-equation eddy viscosity transport model of SpalartAllmaras, a low Reynolds number k-ε model, the Menter k-ωmodel, and the Wilcox k-ωmodel. A framework is presented for the assessment of turbulence models that includes documentation procedures, solution accuracy, model sensitivity, and model validation. The accuracy of the simulations is addressed, and the sensitivities of the models to grid refinement, freestream turbulence levels, and wall y+ spacing are presented. The flat plate skin friction results are compared to the well-established laminar and turbulent correlations of Van Driest. Correlations for the sharp cone are discussed in detail. These correlations, along with recent experimental data, are used to judge the validity of the simulation results for skin friction and surface heating on the sharp cone. The Spalart-Allmaras performs the best with regards to model sensitivity and model accuracy, while the Menter k-ω model also performs well for these zero pressure gradient boundary layer flows. Nomenclature Cf skin friction coefficient Cμ turbulence modeling constant (= 0.09) f generic solution variable H total enthalpy, J/kg h specific enthalpy, J/kg k specific turbulent kinetic energy, m2/s2  1 † Senior Member of Technical Staff, MS 0835, E-mail: cjroy@sandia.gov, Member AIAA ‡ Distinguished Member of Technical Staff, MS 0825, E-mail: fgblott@sandia.gov, AIAA Fellow * Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000. This paper is declared a work of the U. S. Government and is not subject to copyright protection in the United States. M Mach number m Van Driest correlation parameter Pr Prandtl number (= 0.71) PrT turbulent Prandtl number (= 1.0) p pressure, N/m2 q heat flux, W/m2 Re Reynolds number rf recovery factor St Stanton number s surface distance, m T temperature, K Tu freestream turbulence intensity percent t time, s u axial velocity, m/s V velocity magnitude, m/s x axial coordinate, m y wall normal direction, m y+ wall normal direction in wall coordinates ratio of specific heats specific dissipation rate, m2/s3 θ momentum thickness, m absolute viscosity, Ns/m2 kinematic viscosity, m2/s density, kg/m3 τ wall shear stress, N/m2 specific turbulent frequency, 1/s Subscripts aw adiabatic wall value c cone value e edge condition fp flat plate value k mesh level RE Richardson Extrapolated value t turbulent quantity w wall value ∞ freestream value γ ε

RANS simulations of a simplified tractor/trailer geometry.

Steady-state Reynolds-Averaged Navier-Stokes (RANS) simulations are presented for the three-dimensional flow over a simplified tractor-trailer geometry at zero degrees yaw angle. The simulations are conducted using the SACCARA multi-block, structured CFD code. Two turbulence closure models are employed: the one-equation Spalart-Allmaras model and the two-equation k-ω model of Menter. The discretization error is estimated by employing two grid levels: a fine mesh of approximately 20 million grid points and a coarse mesh of approximately 2.5 million grid points. Simulation results are compared to the experimental data obtained at the NASA-Ames 7×10 ft wind tunnel. Quantities compared include: surface pressures on the tractor/trailer, vehicle drag, and time-averaged velocities in the base region behind the trailer. The results indicate that both turbulence models are able to accurately capture the surface pressure on the vehicle, with the exception of the base region. The Menter k-ω model does a reasonable job of matching the experimental data for base pressure and velocities in the near wake, and thus gives an accurate prediction of the drag. The Spalart-Allmaras model significantly underpredicted the base pressure, thereby overpredicting the vehicle drag.

Preconditioned multigrid methods for two-dimensional combustion calculations at all speeds

The development of an effective implicit integration strategy for two-dimensional (axisymmetric) combustion calculations at all speeds is presented. A time-derivative preconditioning technique is first combined with an implicit line relaxation algorithm to yield an approach capable of removing the acoustic time step restriction at low flow speeds while handling stiff chemical kinetics in a fully implicit fashion. Numerical performance is further improved through the addition of a full multigrid/full approximation storage (FMG-FAS) convergence acceleration strategy. Numerical simulations of a subsonic reacting shear layer (finite rate hydrogen-air chemistry), a subsonic bluff-body stabilized flame (mixing-limited methane-air chemistry), and a supersonic jet diffusion flame (finite rate hydrogen-air chemistry) are presented to test the basic attributes of the algorithm. Comparisons with experimental data are presented for all cases, and a detailed examination of the computational efficiency of the new procedure is conducted. The strengths and weaknesses of multigrid ideas for fully coupled combustion calculations are particularly highlighted.

Development of a One-Equation Transition/Turbulence Model

This paper reports on the development of a unified one-equation model for the prediction of transitional and turbulent flows. An eddy viscosity - transport equation for non-turbulent fluctuation growth based on that proposed by Warren and Hassan (Journal of Aircraft, Vol. 35, No. 5) is combined with the Spalart-Allmaras one-equation model for turbulent fluctuation growth. Blending of the two equations is accomplished through a multidimensional intermittence function based on the work of Dhawan and Narasimha (Journal of Fluid Mechanics, Vol. 3, No. 4). The model predicts both the onset and extent of transition. Low-speed test cases include transitional flow over a flat plate, a single element airfoil, and a multi-element airfoil in landing configuration. High-speed test cases include transitional Mach 3.5 flow over a 5{degree} cone and Mach 6 flow over a flared-cone configuration. Results are compared with experimental data, and the spatial accuracy of selected predictions is analyzed.

Numerical Simulation of a Three-Dimensional Flame/Shock Wave Interaction

A three-dimensional Navier-Stokes solver for chemically reacting flows is used to study the structure of a supersonic hydrogen-air flame stabilized in a Mach 2.4 rectangular cross-section wind tunnel. The numerical model uses a 9-species, 21-reaction hydrogen oxidation mechanism and employs Menters hybrid κ-ω/κ-e turbulence model. An assumed probability density function is used to account for the effects of turbulent temperature fluctuations on the ensemble-averaged chemical reaction rates. Results are presented for a configuration studied at the University of Michigan in which the effects of wedge-generated shock waves on flame stability were determined. Computed pitot and static pressure profiles are compared with experimental measurements, and axial density gradient contour plots are compared with experimental schlieren photographs. The highly three-dimensional structure of the flame is described in detail, and stabilization mechanisms are discussed

Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier-Stokes simulations

Despite their well-known limitations, Reynolds-Averaged Navier-Stokes (RANS) models are still the workhorse tools for turbulent flow simulations in todays engineering application. For many practical flows, the turbulence models are by far the largest source of uncertainty. In this work we develop an open-box, physics-informed Bayesian framework for quantifying model-form uncertainties in RANS simulations. Uncertainties are introduced directly to the Reynolds stresses and are represented with compact parameterization accounting for empirical prior knowledge and physical constraints (e.g., realizability, smoothness, and symmetry). An iterative ensemble Kalman method is used to assimilate the prior knowledge and observation data in a Bayesian framework, and to propagate them to posterior distributions of velocities and other Quantities of Interest (QoIs). We use two representative cases, the flow over periodic hills and the flow in a square duct, to evaluate the performance of the proposed framework. Simulation results suggest that, even with very sparse observations, the posterior mean velocities and other QoIs have significantly better agreement with the benchmark data compared to the baseline results. At most locations the posterior distribution adequately captures the true model error within the developed model form uncertainty bounds. The framework is a major improvement over existing black-box, physics-neutral methods for model-form uncertainty quantification, and has potential implications in many fields in which the model uncertainty comes from unresolved physical processes. A notable example is climate modeling, where high-consequence decisions are made based on predictions (e.g., projected temperature rise) with major uncertainties originating from closure models that are used to account for unresolved or unknown physics including radiation, cloud, and boundary layer processes.

A Complete Framework for Verification, Validation, and Uncertainty Quantification in Scientific Computing (Invited)

This paper gives a broad overview of a complete framework for assessing the predictive uncertainty of scientific computing applications. The framework is complete in the sense that it treats both types of uncertainty (aleatory and epistemic) and inco rporates uncertainty due to the form of the model and any numerical approximations used. Aleatory (or random) uncertainties in model inputs are treated using cumulative distribution functions, while epistemic (lack of knowledge) uncertainties are treated a s intervals. Approaches for propagating both types of uncertainties through the model to the system response quantities of interest are discussed. Numerical approximation errors (due to discretization, iteration, and round off) are estimated using verifica tion techniques, and the conversion of these errors into epistemic uncertainties is discussed. Model form uncertainties are quantified using model validation procedures, which include a comparison of model predictions to experimental data and then extrapolation of this uncertainty structure to points in the application domain where experimental data do not exist. Finally, methods for conveying the total predictive uncertainty to decision makers are presented.

Strategies for Driving Mesh Adaptation in CFD (Invited)

This paper examines different approaches for driving mesh adaptation and provides theoretical developments for understanding the relationship between discretization error, the numerical scheme, and the mesh. Discrete and continuous equations governing the transport of discretization error are developed and it is shown that the truncation error acts as the local source for these equations. Examination of the truncation error in generalized coordinates provides insight into the role of mesh quality (mesh stretching for the 1D case) in the discretization error. Numerical results are presented for 1D steady-state Burgers equation at Reynolds numbers of 32 and 128. Four different approaches for driving mesh adaption are implemented for this case: solution gradients, solution curvature, discretization error, and truncation error. The truncation-error based adaption is shown to provide superior results for both cases. Finally, two approaches for estimating the truncation error are also discussed which would allow truncation error-based adaption to be implemented for complex numerical methods.