Christopher A. Kennedy

Additive Runge-Kutta schemes for convection-diffusion-reaction equations

Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, (N = 2), additive Runge-Kutta (ARK2) methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms of the partitioned method are of equal order to those of the elemental methods. Derived ARK2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z[I] → -∞, and retain high stability efficiency in the absence of stiffness, z[I] → 0. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK2 error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK2 methods.

Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations

The derivation of low-storage, explicit Runge-Kutta (ERK) schemes has been performed in the context of integrating the compressible Navier-Stokes equations via direct numerical simulation. Optimization of ERK methods is done across the broad range of properties, such as stability and accuracy efficiency, linear and nonlinear stability, error control reliability, step change stability, and dissipation/dispersion accuracy, subject to varying degrees of memory economization. Following van der Houwen and Wray, 16 ERK pairs are presented using from two to five registers of memory per equation, per grid point and having accuracies from third- to fifth-order. Methods have been assessed using the differential equation testing code DETEST, and with the 1D wave equation. Two of the methods have been applied to the DNS of a compressible jet as well as methane-air and hydrogen-air flames. Derived 3(2) and 4(3) pairs are competitive with existing full-storage methods. Although a substantial efficiency penalty accompanies use of two- and three-register, fifth-order methods, the best contemporary full-storage methods can be nearly matched while still saving two to three registers of memory.

Improved boundary conditions for viscous, reacting, compressible flows

Previous studies on physical boundary conditions for flame-boundary interactions of an ideal, multicomponent, compressible gas have neglected reactive source terms in their boundary condition treatments. By combining analyses of incompletely parabolic systems with those based on the hyperbolic Euler equations, a rational set of boundary conditions is determined to address this shortcoming. Accompanying these conditions is a procedure for implementation into a maltidimensional code. In the limits of zero reaction rate or one species, the boundary conditions reduce in a predictable way to cases found in the literature. Application is made to premixed and nonpremixed flames in one and two dimensions to establish efficacy. Inclusion of source terms in boundary conditions derived from characteristic analysis is essential to avoid unphysical generation of pressure and velocity gradients as well as flow reversals. Minor deficiencies in the boundary conditions are attributed primarily to the diffusive terms. Imposing vanishing diffusive boundary-normal flux gradients works better than imposing vanishing fluxes but neither is entirely satisfactory.

Reduced aliasing formulations of the convective terms within the Navier-Stokes equations for a compressible fluid

The effect on aliasing errors of different formulations describing the cubically nonlinear convective terms within the discretized Navier-Stokes equations is examined in the presence of a non-trivial density spectrum. Fourier analysis shows that the existing skew-symmetric forms of the convective term result in reduced aliasing errors relative to the conservation form. Several formulations of the convective term, including a new formulation proposed for cubically nonlinear terms, are tested in direct numerical simulation (DNS) of decaying compressible isotropic turbulence both in chemically inert (small density fluctuations) and reactive cases (large density fluctuations) and for different degrees of resolution. In the DNS of reactive turbulent flow, the new cubic skew-symmetric form gives the most accurate results, consistent with the spectral error analysis, and at the lowest cost. In marginally resolved DNS and LES (poorly resolved by definition) the new cubic skew-symmetric form represents a robust convective formulation which minimizes both aliasing and computational cost while also allowing a reduction in the use of computationally expensive high-order dissipative filters.

The structure of variable property, compressible mixing layers in binary gas mixtures

We present the results of a study of the structure of a parallel compressible mixing layer in a binary mixture of gases. The gases included in this study are hydrogen (H2), helium (He), nitrogen (N2), oxygen (O2), neon (Ne) and argon (Ar). Profiles of the variation of the Lewis and Prandtl numbers across the mixing layer for all 30 combinations of gases are given. It is shown that the Lewis number can vary by as much as a factor of 8 and the Prandtl number by a factor of 2 across the mixing layer. Thus assuming constant values for the Lewis and Prandtl numbers of a binary gas mixture in the shear layer, as is done in many theoretical studies, is a poor approximation. We also present profiles of the velocity, mass fraction, temperature and density for representative binary gas mixtures at zero and supersonic Mach numbers. We show that the shape of these profiles is strongly dependent on which gases are in the mixture as well as on whether the denser gas is in the fast stream or the slow stream.

Mean flow effects on the linear stability of compressible planar jets

An analytical solution is derived for the two-dimensional, laminar, compressible, planar free jet. The solution assumes constant pressure, specific heats, and unity Prandtl number and accounts for the effects of heat conduction and viscous dissipation in a self-consistent fashion. Exact closed-form expressions are provided for the streamwise and transverse velocities, temperature, vorticity, and dilatation. Temporal instability analyses of these high Reynolds number mean flows indicate that jet-to-ambient temperature ratio exerts a far greater effect on instability growth rates than compressibility. Relative to isothermal conditions, a hot jet flowing into cold ambient fluid is an order of magnitude more unstable and is unstable over a far greater range of wavenumbers. For this hot jet both symmetric and antisymmetric modes are equally amplified whereas isothermal jets have relatively stronger amplification of their antisymmetric modes. A cold jet issuing into a hot fluid is very stable relative to isothe...