Wai-Sun Don

The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: a study of the boundary error

The conventional method of imposing time dependent boundary conditions for Runge-Kutta (RK) time advancement reduces the formal accuracy of the space-time method to first order locally, and second order globally, independently of the spatial operator. This counter intuitive result is analyzed in this paper. Two methods of eliminating this problem are proposed for the linear constant coefficient case: 1) impose the exact boundary condition only at the end of the complete RK cycle, 2) impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives. The first method, while retaining the RK accuracy in all cases , results in a scheme with much reduced CFL condition, rendering the RK scheme less attractive. The second method retains the same allowable time step as the periodic problem. However it is a general remedy only for the linear case. For non-linear hyperbolic equations the second method is effective only for for RK schemes of third order accuracy or less. Numerical studies are presented to verify the efficacy of each approach.

Numerical Convergence Study of Nearly Incompressible, Inviscid Taylor-Green Vortex Flow

A spectral method and a fifth-order weighted essentially non-oscillatory method were used to examine the consequences of filtering in the numerical simulation of the three-dimensional evolution of nearly-incompressible, inviscid Taylor–Green vortex flow. It was found that numerical filtering using the high-order exponential filter and low-pass filter with sharp high mode cutoff applied in the spectral simulations significantly affects the convergence of the numerical solution. While the conservation property of the spectral method is highly desirable for fluid flows described by a system of hyperbolic conservation laws, spectral methods can yield erroneous results and conclusions at late evolution times when the flow eventually becomes under-resolved. In particular, it is demonstrated that the enstrophy and kinetic energy, which are two integral quantities often used to evaluate the quality of numerical schemes, can be misleading and should not be used unless one can assure that the solution is sufficiently well-resolved. In addition, it is shown that for the Taylor–Green vortex (for example) it is useful to compare the predictions of at least two numerical methods with different algorithmic foundations (such as a spectral and finite-difference method) in order to corroborate the conclusions from the numerical solutions when the analytical solution is not known.

A high-order WENO-Z finite difference based particle-source-in-cell method for computation of particle-laden flows with shocks

A high-order particle-source-in-cell (PSIC) algorithm is presented for the computation of the interaction between shocks, small scale structures, and liquid and/or solid particles in high-speed engineering applications. The improved high-order finite difference weighted essentially non-oscillatory (WENO-Z) method for solution of the hyperbolic conservation laws that govern the shocked carrier gas flow, lies at the heart of the algorithm. Finite sized particles are modeled as points and are traced in the Lagrangian frame. The physical coupling of particles in the Lagrangian frame and the gas in the Eulerian frame through momentum and energy exchange, is numerically treated through high-order interpolation and weighing. The centered high-order interpolation of the fluid properties to the particle location is shown to lead to numerical instability in shocked flow. An essentially non-oscillatory interpolation (ENO) scheme is devised for the coupling that improves stability. The ENO based algorithm is shown to be numerically stable and to accurately capture shocks, small flow features and particle dispersion. Both the carrier gas and the particles are updated in time without splitting with a third-order Runge-Kutta TVD method. One and two-dimensional computations of a shock moving into a particle cloud demonstrates the characteristics of the WENO-Z based PSIC method (PSIC/WENO-Z). The PSIC/WENO-Z computations are not only in excellent agreement with the numerical simulations with a third-order Rusanov based PSIC and physical experiments in [V. Boiko, V.P. Kiselev, S.P. Kiselev, A. Papyrin, S. Poplavsky, V. Fomin, Shock wave interaction with a cloud of particles, Shock Waves, 7 (1997) 275-285], but also show a significant improvement in the resolution of small scale structures. In two-dimensional simulations of the Mach 3 shock moving into forty thousand bronze particles arranged in the shape of a rectangle, the long time accuracy of the high-order method is demonstrated. The fifth-order PSIC/WENO-Z method with the fifth-order ENO interpolation scheme improves the small scale structure resolution over the third-order PSIC/WENO-Z method with a second-order central interpolation scheme. Preliminary analysis of the particle interaction with the flow structures shows that sharp particle material arms form on the side of the rectangular shape. The arms initially shield the particles from the accelerated flow behind the shock. A reflected compression wave, however, reshocks the particle arm from the shielded area and mixes the particles.

Spectral simulation of an unsteady compressible flow past a circular cylinder

Abstract An unsteady compressible viscous wake flow past a circular cylinder has been successfully simulated using spectral methods. A new approach in using the Chebyshev collocation method for a periodic problem is introduced. We have further proved that the eigenvalues associated with the differentiation matrix are purely imaginary, reflecting the periodicity of the problem. It has been shown that the solution of a model problem has exponential growth in time if an ‘improper’ boundary conditions procedure is used. A characteristic boundary conditions, which is based on the characteristics of the Euler equations of gas dynamics, has been derived for the spectral code. The primary vortex shedding frequency computed agrees well with the results in the literature for Mach = 0.4, Re = 80. No secondary frequency is observed in the power spectrum analysis of the pressure data.

A High-Order Dirac-Delta Regularization with Optimal Scaling in the Spectral Solution of One-Dimensional Singular Hyperbolic Conservation Laws

A regularization technique based on a class of high-order, compactly supported piecewise polynomials is developed that regularizes the time-dependent, singular Dirac-delta sources in spectral approximations of hyperbolic conservation laws. The regularization technique provides higher-order accuracy away from the singularity. A theoretical criterion that establishes a lower bound on the support length (optimal scaling) has to be satisfied to achieve optimal order of convergence. The optimal scaling parameter has been shown to be, instead of a fixed constant value, a function of the smoothness of the compactly supported piecewise polynomial and the number of vanishing moment of the polynomial when integrated with respect to mononomial of degree up to some order (moment). The effectiveness of the criterion is illustrated in the solutions of a linear and a nonlinear (Burgers) scalar hyperbolic conservation law with a singular source, as well as the nonlinear Euler equations with singular sources, a system of ...

High-Fidelity Eulerian-Lagrangian Methods for Simulation of Three Dimensional, Unsteady, High-Speed, Two-Phase Flows in High-Speed Combustors

A preliminary study of a three-dimensional numerical model for determining the flow and particle developments through a shock tube containing 10 bronze particles is presented here. The numerical experiment is initialized as a hexahedral cloud of particles immediately adjacent to a right running normal shock. Flow characteristics are computed using a three-dimensional high-order Eulerian-Lagrangian method, which solves the Euler equations governing the gas dynamics with an improved high order weighted essentially non-oscillatory (WENO-Z) scheme, while individual particle trajectories are traced in the Lagrangian frame using high-order time integration schemes. Two way coupling between the carrier gas and the particles is modeled, using a high-order ENO interpolation, via the exchange of the momentum and energy at the particle positions through the use of an additional source term in the Euler equations. A high-order central weighing deposits the particle influence on the carrier phase. The preliminary solution as computed in the 3D model is then compared to similar experiments analyzed with 2D models.

Computation of Normal Shocks Running into a Cloud of Particles using a High-Order Particle-Source-in-Cell Method

In this paper, the two-dimensional particle-laden flow developments are studied with bronze particle cloud in the accelerated flow behind a running shock. The forty thousands particle clouds are arranged initially in a rectangular, triangular and circular shape. The flows are computed with a recently developed high-order Eulerian-Lagrangian method, that approximates the Euler equations governing the gas dynamics with the improved high order weighted essentially non-oscillatory (WENO-Z) scheme, while individual particles are traced in the Lagrangian frame using high-order time integration schemes. A high-order ENO interpolation determines the carrier phase properties at the particle location. A high-order central weighing deposits the particle influence on the carrier phase. Reflected shocks form ahead of all the cloud shapes. The detached shock in front of the triangular cloud is weakest. At later times the wake behind the cloud becomes unstable and a twodimensional vortex-dominated wake forms. Separated shear layers at the edges of the clouds pulls particles initially out of the clouds that are consequently transported along the shear layers. Since flows separated trivially at sharp corners, particles are mostly transported out of the cloud into the flow at the sharp front corner of the rectangular cloud, and the trailing corner of the triangular cloud. Particles are transported smoothly out of the circular cloud, since it lacks sharp corners. At late times, the accelerated flow behind the running shock disperses the particles in cross-stream direction the most for the circular cloud, followed by the rectangular cloud and the triangular cloud.