Joseph Shang

Practical aspects of higher-order numerical schemes for wave propagation phenomena

This paper examines practical issues related to the use of compact-difference-based fourth- and sixth-order schemes for wave propagation phenomena with focus on Maxwells equations of electromagnetics. An outline of the formulation and scheme optimization is followed by an assessment of the error accruing from application on stretched meshes with two approaches: transformed plane method and physical space differencing. In the first technique, the truncation error expansion for the sixth-order compact scheme confirms that the order of accuracy is preserved if a consistent mesh refinement strategy is followed and further that metrics should be evaluated numerically even if analytic expressions are available. Physical space-differencing formulas are derived for the five-point stencil by expressing the coefficients in terms of local spacing ratios. The order of accuracy of the reconstruction operator is then verified with a numerical experiment on stretched meshes. To ensure stability for a broad range of problems, Fourier analysis is employed to develop a single-parameter family of up to tenth-order tridiagonal-based spatial filters. The implementation of these filters is discussed in terms of their effect on the interior scheme as well as in a 1-D cavity where they are employed to suppress a late-time instability. The paper concludes after demonstrating the application of the scheme to several 3-D canonical problems utilizing Cartesian as well as curvilinear meshes. Published in 1999 by John Wiley & Sons, Ltd. This article is a U.S. Government work and is in the public domain in the United States.

Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach

A compact central-difference approximation in conjunction with the Yee (1966) grid is used to compute the spatial derivatives in Maxwells equations. To advance the semi-discrete equations, the four-stage Runge-Kutta (RK) integrator is invoked. This combination of spatial and temporal differencing leads to a scheme that is fourth-order accurate, conditionally stable, and highly efficient. Moreover, the use of compact differencing allows one to apply the compact operator in the vicinity of a perfect conductor-an attribute not found in other higher order methods. Results are provided that quantify the spectral properties of the method. Simulations are conducted on problem spaces that span one and three dimensions and whose domains are of the open and closed type. Results from these simulations are compared with exact closed-form solutions; the agreement between these results is consistent with numerical analysis.

Characteristic-based algorithms for solving the Maxwell equations in the time domain

Several numerical algorithms, developed in the computational-fluid-dynamics community for solving the Euler equations, are found to be equally effective for solving the Maxwell equations in the time domain. The basic approach of these numerical procedures is to achieve the Riemann approximation to the time-dependent, three-dimensional problem in each spatial direction. The three-dimensional equations are then solved by a sequence of one-dimensional problems. This approach is referred to as a characteristic-based method. The basic algorithm can be implemented for both finite-difference and finite-volume procedures, and has the potential to eliminate the spurious-wave reflections from the numerical boundaries of the computational domain. The formulation and relative merit of the finite-difference and the finite-volume approximations are presented, together with numerical results from these procedures. >

Characteristic-based, time-dependent Maxwell equation solvers on a general curvilinear frame

Four cell-centered, finite-volume characteristic-based procedures for solving time-dependent, three-dimensional Maxwell equations on a general curvilinear frame are developed and implemented. On a spherical coordinate system, these methods generate simulations or an oscillating electric dipole without wave reflection from the truncated computational domain. Through a comparative study, it is established that an upwind-biased third-order scheme demonstrates less dispersive and dissipative errors than second-order schemes

Numerical simulation of leading-edge vortex flows

Steady flowfields describing respectively the distinguished structure for subsonic, sonic, and supersonic leading-edge flow about a thin delta wing at angle of attack in a supersonic freestream are calculated numerically. Solutions of the steady three-dimensional compressible laminar Navier-Stokes equations are obtained by time integration. Details of these solutions demonstrate that the essential physical behavior of such flows, including both primary and secondary vortex motions, has been simulated. For purposes of comparison, a corresponding inviscid numerical solution was generated for the case of a subsonic leading edge. It is shown that, although the secondary features are absent, the gross dominant characteristics of the flowfield have been reproduced by the Euler equations. Effects of turbulence are assessed by incorporating a simple closure model in the viscous computation. Comparison between the numerical solutions and experimental data is provided for all flow regimes.

Hypersonic Flow Control Using Surface Plasma Actuator

Plasma-fluid-dynamic interaction has been shown to be a viable mechanism for hypersonic flow control. An effective and verified flow control process using direct current surface discharge is summarized. The operating principle is based on a small electromagnetic perturbation to the growth rate of the displacement thickness of a shear layer that is strongly amplified by a subsequent pressure interaction. The aerodynamic control is delivered in less than a millisecond time frame and produces no parasitic effect when deactivated. The magnitude of the resultant aerodynamic force and moment can be significant and does not require a large amount of power for plasma generation to overcome the inefficient ionizing process, thus reducing the weight of a high-speed vehicle. The electromagnetic perturbation is derived from a surface gas discharge with or without an externally applied magnetic field. An embedded plasma actuator near the leading edge of a flat plate has produced high surface pressure equivalent to more than a 5 deg flow deflection at Mach 5, and the flow control effectiveness will increase with an increasing oncoming Mach number. The detailed flow structure of weakly ionized airstreams has been investigated by a combination of experimental effort and computational simulation solving the magneto-fluid-dynamic equations in the low magnetic Reynolds number limit with a drift-diffusion plasma model. The identical plasma actuator is investigated as a variable geometry cowl of a hypersonic inlet. All phenomena are replicated by computational results and are fully validated by experimental observations.

Magnetoaerodynamic actuator for hypersonic flow control

The classic pressure interaction over a sharp leading edge is a unique feature of hypersonic flow in which the boundary layer interacts strongly with the oncoming stream. The present magnetoaerodynamic actuator is based on the fact that the plasma ignited on a surface can modify the boundary-layer displacement thickness. This electromagnetic perturbation to the flowfield is further amplified by the hypersonic viscous-inviscid interaction to produce a high-surface-pressure plateau on a microsecond scale. The induced surface pressure is an effective hypersonic flow control mechanism. In essence, the minuscule magnetoaerodynamic interaction adjacent to the solid surface is introduced as an added physical mechanism for flow control. The electromagnetic force enhanced inviscid-viscous interaction generates a greater high-pressure regime over the solid surface than that of the classic hypersonic pressure interaction. This plasma actuator is demonstrated by solving the two-dimensional magnetoaerodynamic equation including the Lorentz force and Joule heating. For plasma generated by secondary electronic emission, the well-known ionization formulation for thermal excitation is inapplicable. The drift-diffusion plasma model is adopted to describe the detailed electrodynamic structure of the glow discharge. The idea of a plasma actuator has been supported by preliminary experimental data at a Mach number of five.

Thin-layer full Navier-Stokes simulations over a supersonic delta wing

Steady flowfields describing the flow over a 75-deg swept delta wing at M ∞=1.95 and Re=4.48×10 6 were simulated at α=20 and 30 deg using two computer codes. Comparisons were made between calculations using 1) the laminar and turbulent thin-layer Navier-Stokes equations, and 2)laminar thin-layer and Navier-Stokes equations.

On the structure of an unsteady type IV interaction at Mach 8

Abstract The unsteady laminar flow due to a Type IV shock-on-shock interaction at Mach 8 is examined with a flux-split scheme based on a modified Steger-Warming method. A sequence of meshes is utilized to perform a grid resolution study and the results are compared with available experimental data. The formulation is cell-centered finite-volume with second order accuracy obtained with the MUSCL approach in conjunction with a limiter to prevent unphysical spatial oscillations. Viscous terms are centrally differenced. The solution is advanced with implicit Gauss-Seidel line relaxation. The calculations indicate unsteady behavior on all meshes. The coherent structure of the asymptotic solution is examined. The impinging supersonic jet limit cycle exhibits a relatively regular oscillatory behavior with a dominant frequency of about 30 kHz. This is accompanied by sizeable movement of the terminating jet shock and associated variation in surface pressure and heat transfer rates.

Characteristic Based Methods for the Time-Domain Maxwell Equations

Abstract : Numerical procedures for solving the time-domain Maxwell equations based on the theory of characteristics were successfully developed. Both explicit and implicit methods were formulated by the time-central and spatial-windward algorithm to better describe wave motion. A new trapezoidal consistent implicit scheme was shown to be unconditionally stable for the linear initial value system and was able to generate numerical solutions comparable to those of the established explicit method. The formulation of the three-dimensional system including generalized coordinate system was completed but not explored. The present 2-D results on Cartesian frame demonstrated a potential for numerical efficiency improvement. Time-domain Maxwell equation, Trapezoidal consistent implicit scheme, Cartesian frame.