Scale-selective time integration for Long-Wave Linear Acoustics

In this note, we present a new method for the numerical integration of one dimensional linear acoustics with long time steps. It is based on a scale-wise decomposition of the data using standard multigrid ideas and a scale-dependent blending of basic time integrators with different principal features. This enables us to accurately compute balanced solutions with slowly varying short-wave source terms. At the same time, the method effectively filters freely propagating compressible short-wave modes. The selection of the basic time integrators is guided by their discrete-dispersion relation. Furthermore, the ability of the schemes to reproduce balanced solutions is shortly investigated. The method is meant to be used in semi-implicit finite volume methods for weakly compressible flows

A Scale-selective Multilevel Method for Long-Wave Linear Acoustics

A new method for the numerical integration of the equations for one-dimensional linear acoustics with large time steps is presented. While it is capable of computing the "slaved" dynamics of short-wave solution components induced by slow forcing, it eliminates freely propagating compressible short-wave modes, which are under-resolved in time. Scale-wise decomposition of the data based on geometric multigrid ideas enables a scale-dependent blending of time integrators with different principal features. To guide the selection of these integrators, the discrete-dispersion relations of some standard second-order schemes are analyzed, and their response to high wave number low frequency source terms are discussed. The performance of the new method is illustrated on a test case with "multiscale" initial data and a problem with a slowly varying high wave number source term

Regime of Validity of Sound-Proof Atmospheric Flow Models

Ogura and Phillips (1962) derived their original anelastic model through systematic formal asymptotics using the flow Mach number as the expansion parameter. To arrive at a reduced model which would simultaneously represent internal gravity waves and the effects of advection, they had to adopt a distinguished limit stating that the dimensionless stability of the background state be of the order of the Mach number squared. For typical flow Mach numbers of M = 1/30 this amounts to total variations of potential temperature across the troposphere of less than one Kelvin, i.e., to unrealistically weak stratication. Various generalizations of Ogura and Phillips' anelastic model have been proposed to remedy this issue, e.g., by Dutton & Fichtl (1969), and Lipps & Hemler (1982). Following the same goals, but a somewhat different route of argumentation, Durran proposed the pseudoincompressible model in 1989. The present paper provides a scale analysis showing that the regime of validity of two of these extended models covers stratification strengths of order of the Mach number to the power 2/3, which corresponds to realistic variations of potential temperature across the pressure scale height of about 30 K

Improved Thin Tube Methods for Slender Vortex Simulations

This paper explores three numerical schemes for efficient simulation of slender vortex filaments. The schemes defeat the spatial and temporal stiffness of the equations of motion by requiring only adequate resolution of the filament centerline and allowing large integration time steps. In order to correctly capture the self-induced filament velocity, the first scheme uses an explicit velocity correction method, the second scheme relies on a logarithmic extrapolation of two velocity predictions, and the third scheme employs a local refinement algorithm. The performances of the three schemes are contrasted in light of unsteady computations of a perturbed vortex ring with small core to radius ratio

Asymptotic Vorticity Structure and Numerical Simulation of Slender Vortex Filaments

A new asymptotic analysis of slender vortices in three dimensions, based solely on the vorticity transport equation and the non-local vorticity-velocity relation gives new insight into the structure of slender vortex filaments. The approach is quite different from earlier analyses using matched asymptotic solutions for the velocity field and it yields additional information. This insight is used to derive three different modifications of the thin-tube version of a numerical vortex element method. Our modifications remove an O(1) error from the node velocities of the standard thin-tube model and allow us to properly account for any prescribed physical vortex core structure independent of the numerical vorticity smoothing function. We demonstrate the performance of the improved models by comparison with asymptotic solutions for slender vortex rings and for perturbed slender vortex filaments in the Klein-Majda regime, in which the filament geometry is characterized by small-amplitude-short-wavelength displacements from a straight line. These comparisons represent a stringent mutual test for both the proposed modified thin-tube schemes and for the Klein-Majda theory. Importantly, we find a convincing agreement of numerical and asymptotic predictions for values of the Klein-Majda expansion parameter E as large as 1/2. Thus, our results support their findings in earlier publications for realistic physical vortex core sizes

Effect of equivalence ratio on premixed flame response to unsteady strain-rate and curvature

The detailed dynamical response of flames in turbulent reacting flow involves a complex interaction between unsteady flow structures and flame chemistry. Two essential features of this interaction are the unsteady strain-rate and curvature disturbances to the reaction zone. In this work, the authors focus on a single flow length/time scale feature in two dimensions (2D), and its effect on a premixed flame for a range of mixture conditions. In particular, they study the interaction of a premixed freely propagating methane-air flame with a 2D counter-rotating vortex pair in an unbounded domain. In earlier work, they studied this flow using C{sub 1} kinetics, at stoichiometric conditions. Notable observations include the shift of the reaction zone into the products on the vortex-pair centerline, leading to depletion of H, O and OH, and the consequent general drop in reaction rates on the centerline flame segment. Curvature-induced focusing/defocusing effects were observed at the positively curved flame cusp, leading to modifications in internal transport fluxes of various species and radicals in the flame, and associated effects on H production and fuel consumption rates. These results were extended to more detailed kinetics, using other C{sub 1} and C{sub 2} mechanisms, which demonstrated the effect of choice of chemical mechanism on the observed transient flame response. The present study focuses on the dependence of the transient flame response on reactants mixture equivalence ratio. Two reactants mixture conditions are studied: case 1 is a stoichiometric conditions - equivalence ratio {Phi} = 1.0, and case 2 is rich at {Phi} = 1.2. In both cases, the reactants are 20% N{sub 2}-diluted

Interaction of a Slender Vortex Filament with a Rigid Sphere: Dynamics and far-field noise.

Interactions between a slender vortex filament and a stationary rigid sphere are analyzed using a vortex element scheme which tracks the motion of the filament centerline. The filament velocity is expressed as the sum of a self-induced velocity and potential velocity due to the presence of the sphere. The self-induced velocity is estimated numerically using a line Biot–Savart integral which is carefully desingularized so as to reflect the correct asymptotic behavior of the core vorticity distribution under the influence of stretching and viscous diffusion. Meanwhile, the potential velocity is evaluated from a recently derived formula, which expresses it as a line integral along the image of the filament centerline in the sphere with regular weight functions. From the far-field behavior of an unsteady vortical flow outside a stationary sphere, formulas for the acoustic far field are obtained. It is shown that the interaction between the slender vortex filament and the sphere generates dipoles and quadrupoles in addition to the quadrupoles generated by the filament alone in space. The strengths and orientations of the dipoles and quadrupoles are completely determined by the time evolution of the weighted first and second moments of vorticity. The formulas are applied to compute the far-field sound generated by the passage of a slender vortex ring over the sphere. Both coaxial and noncoaxial passage events are analyzed in the computations, as well as the effects of initial core size and asymmetric perturbations

Numerical Simulation of a Thermo-Acoustic Refrigerator

A thermoacoustic device consists of two main components: (1) a resonance tube where the flow is characterized by length scales of the order of the acoustic wavelength, and (2) a stack of plates which are separated by distances much smaller than the acoustic wavelength. This effort focuses on the development of numerical schemes which overcome this scale disparity in an efficient manner. Two approaches are discussed in the paper. The first is a multiple-pressure-variable approach that is suitable for the simulation of resonance tube acoustics and for analyzing interactions between heat addition and long waves. The second is a multi-dimensional model of the stack region which is based on fast solution of the zero-Mach-number conservation equations

Representation of Core Dynamics in Slender Vortex Filament Simulations,

The numerical description of slender vortex motion faces several major obstacles: (i) The stiffness induced by the rapid rotatory motion in the vortex core, where peak velocities are an order of magnitude larger than the filament velocity. In a vorticity-velocity formulation, this stiffness is reflected by the singular behavior of the line-Biot-Savart integral as one approaches the vortex geometry. Regularization occurs physically by viscous smoothing of the vorticity. (ii) The vortex core vorticity distribution has a crucial influence on the vortex filament motion. Thus, an accurate description of the core structure evolution due to vortex stretching and vorticity diffusion is necessary. We propose a numerical scheme that allows an accurate description of the effects of axial flow in the core, viscosity and vortex stretching on slender vortex filament motion. The approach is based on incorporating the detailed asymptotic analyses of the vortex core structure evolution by Callegari and Ting [SIAM J. Appl. Math. 15, 148 (1978)] and Klein and Ting [Appl. Math. Lett. 8, 45 (1995)] for stretched viscous slender vortices into the improved thin-tube vortex element schemes of Klein and Knio (1995). The resulting schemes overcome the difficulties mentioned above except for the issue of temporal stiffness, which we leave for future work

Vortex Dominated Flows: Analysis and Computation for Multiple Scales

This monograph provides in-depth analyses of vortex dominated flows via matched and multiscale asymptotics, and demonstrates how insight gained through these analyses can be exploited in the construction of robust, efficient, and accurate numerical techniques. The book explores the dynamics of slender vortex filaments in detail, including fundamental derivations, compressible core structure, weakly non-linear limit regimes, and associated numerical methods. Similarly, the volume covers asymptotic analysis and computational techniques for weakly compressible flows involving vortex-generated sound and thermoacoustics. The book is addressed to both graduate students and researchers