Fang Q. Hu

On the three families of instability waves of high-speed jets

In this paper the normal-mode small-amplitude waves of high-speed jets are investigated analytically and computationally. Three families of instability waves, each having a distinct wave pattern and propagation characteristics, have been found. One of the families of waves is the familiar Kelvin-Helmholtz instability wave. The other two families of waves do not appear to have been clearly identified and systematically studied before. Waves of one of the new wave family propagate with supersonic phase velocities relative to the ambient gas. They are, therefore, referred to as supersonic instability waves. Waves of the other family have subsonic phase velocities. Accordingly they are called subsonic waves. The subsonic waves have the unusual property that they are unstable only for jets with finite thickness mixing layers. They are neutral waves when calculated by a vortex-sheet jet model. Earlier Oertel (1979, 1980, 1982) using a novel optical technique observed in a series of experiments three sets of waves in high-speed jets. The origin of these waves, however, remains so far unexplained and a theory has yet to be developed. In the present study it will be shown that the computed wave patterns and propagation characteristics of the Kelvin-Helmholtz, the supersonic and the subsonic instability waves match essentially those observed by Oertel. The physical mechanisms which give rise to the three families of waves as well as some of the most salient characteristic features of each set of waves are discussed and reported here.

Absorbing Boundary Conditions

Absorbing boundary conditions for computational aeroacoustics (CAA) are reviewed. Commonly used absorbing zonal techniques, such as sponge layers and buffer zones, as well as perfectly matched layers (PML) are discussed. The basic ideas and central results of these methods are surveyed and summarized. Special attention will be given to the recently emerged PML technique and its application to CAA. Numerical examples are presented for PML in duct acoustics. A comparison of PML and non-PML absorbing boundary conditions will also be given.

The instability and acoustic wave modes of supersonic mixing layers inside a rectangular channel

At high supersonic convective Mach numbers the familiar Kelvin-Helmholtz instability of a thin unconfined two-dimensional shear layer becomes neutrally stable. In this paper, it is shown that when the same shear layer is put inside a rectangular channel the coupling between the motion of the shear layer and the acoustic modes of the channel produces new two-dimensional instability waves. The instability mechanism of these waves is examined. Extensive numerical computation of the properties of these new instability waves has been carried out. Based on these results two classes of these waves are identified. Some of the important characteristic features of these waves are reported in this paper. In addition to the unstable waves, a thorough analysis of the normal modes of a supersonic shear layer inside a rectangular channel reveals that there are basically two other families of neutral acoustic waves. Examples of some of the prominent characteristics of these neutral acoustic waves are also provided in this paper. The new instability waves are the dominant instabilities of a confined mixing layer at high supersonic convective Mach number. As such they are very relevant to the supersonic mixing and combustion processes inside a ramjet engine combustion chamber.

Absorbing boundary conditions for nonlinear Euler and Navier-Stokes equations based on the perfectly matched layer technique

Absorbing boundary conditions for the nonlinear Euler and Navier-Stokes equations in three space dimensions are presented based on the perfectly matched layer (PML) technique. The derivation of equations follows a three-step method recently developed for the PML of linearized Euler equations. To increase the efficiency of the PML, a pseudo mean flow is introduced in the formulation of absorption equations. The proposed PML equations will absorb exponentially the difference between the nonlinear fluctuation and the prescribed pseudo mean flow. With the nonlinearity in flux vectors, the proposed nonlinear absorbing equations are not formally perfectly matched to the governing equations as their linear counter-parts are. However, numerical examples show satisfactory results. Furthermore, the nonlinear PML reduces automatically to the linear PML upon linearization about the pseudo mean flow. The validity and efficiency of proposed equations as absorbing boundary conditions for nonlinear Euler and Navier-Stokes equations are demonstrated by numerical examples.

Multi-component reactive transport modeling of natural attenuation of an acid groundwater plume at a uranium mill tailings site

Natural attenuation of an acidic plume in the aquifer underneath a uranium mill tailings pond in Wyoming, USA was simulated using the multi-component reactive transport code PHREEQC. A one-dimensional model was constructed for the site and the model included advective-dispersive transport, aqueous speciation of 11 components, and precipitation-dissolution of six minerals. Transport simulation was performed for a reclamation scenario in which the source of acidic seepage will be terminated after 5 years and the plume will then be flushed by uncontaminated upgradient groundwater. Simulations show that successive pH buffer reactions with calcite, Al(OH)3(a), and Fe(OH)3(a) create distinct geochemical zones and most reactions occur at the boundaries of geochemical zones. The complex interplay of physical transport processes and chemical reactions produce multiple concentration waves. For SO4(2-) transport, the concentration waves are related to advection-dispersion, and gypsum precipitation and dissolution. Wave speeds from numerical simulations compare well to an analytical solution for wave propagation.

On the construction of PML absorbing boundary condition for the non-linear Euler equations

A Perfectly Matched Layer (PML) absorbing boundary condition for the nonlinear Euler equations in two space dimensions is presented. The derivation of PML equation follows athree-step method recently developed for the PML of linearized Euler equations. Two versions of time-domain PML equations are given. One uses unsplit physical variables and the other uses the split equation in the derivation but requires fewer auxiliary variables. Both versions are given for the nonlinear Euler equation written in the conservation form, so they can be implemented easily in most existing codes. To increase the efficiency of the PML, a pseudo mean-flow is introduced in the derivation of the PML equations. The proposed PML absorbs exponentially the difference between the nonlinear total variable and a prescribed pseudo mean flow. Moreover, the non-linear PML reduces to the linearized PML upon linearization about the pseudo mean-flow. The validity and efficiency of PML as an absorbing boundary condition for non-linear Euler equations are demonstrated by numerical examples, including the absorption of an isentropic vortex, a nonlinear pressure pulse and roll-up vortices of shear flows. Satisfactory computational results are reported.

Nonuniform time-step Runge-Kutta discontinuous Galerkin method for Computational Aeroacoustics

With many superior features, Runge-Kutta discontinuous Galerkin method (RKDG), which adopts Discontinuous Galerkin method (DG) for space discretization and Runge-Kutta method (RK) for time integration, has been an attractive alternative to the finite difference based high-order Computational Aeroacoustics (CAA) approaches. However, when it comes to complex physical problems, especially the ones involving irregular geometries, the time step size of an explicit RK scheme is limited by the smallest grid size in the computational domain, demanding a high computational cost for obtaining time accurate numerical solutions in CAA. For computational efficiency, high-order RK method with nonuniform time step sizes on nonuniform meshes is developed in this paper. In order to ensure correct communication of solutions on the interfaces of grids with different time step sizes, the values at intermediate-stages of the Runge-Kutta time integration on the elements neighboring such interfaces are coupled with minimal dissipation and dispersion errors. Based upon the general form of an explicit p-stage RK scheme, a linear coupling procedure is proposed, with details on the coefficient matrices and execution steps at common time-levels and intermediate time-levels. Applications of the coupling procedures to Runge-Kutta schemes frequently used in simulation of fluid flow and acoustics are given, including the third-order TVD scheme, and low-storage low dissipation and low dispersion (LDDRK) schemes. In addition, an analysis on the stability of coupling procedures on a nonuniform grid is carried out. For validation, numerical experiments on one-dimensional and two-dimensional problems are presented to illustrate the stability and accuracy of proposed nonuniform time-step RKDG scheme, as well as the computational benefits it brings. Application to a one-dimensional nonlinear problem is also investigated.

An efficient solution of time domain boundary integral equations for acoustic scattering and its acceleration by Graphics Processing Units

The present paper is aimed at developing a fast numerical solution of the time domain boundary integral equation (TDBIE) reformulated from the convective wave equation for large scale wave scattering and propagation problems. Historically, numerical solutions of boundary integral equation in the time domain have encountered two major difficulties. The first is the intrinsic numerical instability in the early time domain boundary integral equation formulations. And the second is the formidably high computational cost associated with the direct solution of the time-domain boundary integral equation. In this paper, both issues are addressed. A stable Burton-Miller type formulation is proposed for the time domain boundary integral equation in the presence of a mean flow. A justification for stability through the energy equation associated with the convective wave equation is given. A comparison of the current formulation with a previous one in literature is also offered. The boundary integral equation is solved by a time domain boundary element method (TDBEM), using high-order basis functions and unstructured surface elements. To significantly reduce the computational cost, a Time Domain Propagation and Distribution (TDPD) algorithm is proposed, making use of the delayand amplitude-compensated field with a mean flow. Implemented in multi-level interactions, the current algorithm shows a computational cost of O(N) per time step where N is the total number of unknowns on surface elements. Furthermore, GPU computing has been utilized to speedup the computation. Numerical aspects of the GPU computing for boundary element solutions are discussed. Comparison with CPU executions is also given. Numerical examples that demonstrate the capabilities of the proposed method are presented.

INSTABILITIES OF SUPERSONIC MIXING LAYERS INSIDE A RECTANGULAR CHANNEL

At high supersonic convective Mach numbers the fami 1 i ar Kelvin-Helmhol tz instabi 1 ity of a thin unconfined two dimensional shear layer becomes neutrally stable. In this paper, it is shown that when the same shear layer is put inside a rectangul ar channel the coup1 i ng between the motion of the shear layer and the acoustic modes of the channel produces new instability waves. The instability mechanism of these waves is examined. E xtensive numerical c omputation of the properties of these new instability waves has been carried out. Based on these results two classes of these waves are identified. Some of the characteristic features of these waves are reported in this paper. In addition to the unstable waves, a thorough analysis of the normal modes of a supersonic shear layer inside a rectangular channel reveals that there are basically two other families of neutral acoustic waves. Examples of some of the important characteristics of these neutral acoustic waves are also provided in this paper. The new instability waves are the dominant instabilities of a confined mixing layer at high supersonic convective Mach number. As such they are very relevant to the supersonic mixing and combustion processes inside a ramjet engine combustion chamber.

DGM-FD: A finite difference scheme based on the discontinuous Galerkin method applied to wave propagation

In this paper we formulate a numerical method that is high order with strong accuracy for numerical wave numbers, and is adaptive to non-uniform grids. Such a method is developed based on the discontinuous Galerkin method (DGM) applied to the hyperbolic equation, resulting in finite difference type schemes applicable to non-uniform grids. The schemes will be referred to as DGM-FD schemes. These schemes inherit naturally some features of the DGM, such as high-order approximations, applicability to non-uniform grids and super-accuracy for wave propagations. Stability of the schemes with boundary closures is investigated and validated. Proposed scheme is demonstrated by numerical examples including the linearized acoustic waves and solutions of non-linear Burgers equation and the flat-plate boundary layer problem. For non-linear equations, proposed flux finite difference formula requires no explicit upwind and downwind split of the flux. This is in contrast to existing upwind finite difference schemes in the literature.