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Register a new userOn Linear and Nonlinear Acoustics in Stratified, Variable-Area Ducts and Atmospheres, and Lighthills Proposition
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Added by
Christopher D. Matzner
Publication date
2020
fields
Physics
and research's language is
English
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We consider linear and nonlinear waves in a stratified hydrostatic fluid within a channel of variable area, under the restriction of one-dimensional flow. We derive a modified version of Riemanns invariant that is related to the wave luminosity. This quantity obeys a simple dynamical equation in linear theory, from which the rules of wave reflection are easily discerned; and it is adiabatically conserved in the high-frequency limit. Following a suggestion by Lighthill, we apply the linear adiabatic invariant to predict mildly nonlinear waves. This incurs only moderate error. We find that Lighthills criterion for shock formation is essentially exact for leading shocks, and for shocks within high-frequency waves. We conclude that approximate invariants can be used to accurately predict the self-distortion of low-amplitude acoustic pulses, as well as the dissipation patterns of weak shocks, in complicated environments such as stellar envelopes. We also identify fully nonlinear solutions for a restricted class of problems.
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Kraichnan seminal ideas on inverse cascades yielded new tools to study common phenomena in geophysical turbulent flows. In the atmosphere and the oceans, rotation and stratification result in a flow that can be approximated as two-dimensional at very large scales, but which requires considering three-dimensional effects to fully describe turbulent transport processes and non-linear phenomena. Motions can thus be classified into two classes: fast modes consisting of inertia-gravity waves, and slow quasi-geostrophic modes for which the Coriolis force and horizontal pressure gradients are close to balance. In this paper we review previous results on the strength of the inverse cascade in rotating and stratified flows, and then present new results on the effect of varying the strength of rotation and stratification (measured by the ratio $N/f$ of the Brunt-Vaisala frequency to the Coriolis frequency) on the amplitude of the waves and on the flow quasi-geostrophic behavior. We show that the inverse cascade is more efficient in the range of $N/f$ for which resonant triads do not exist, $1/2 le N/f le 2$. We then use the spatio-temporal spectrum, and characterization of the flow temporal and spatial scales, to show that in this range slow modes dominate the dynamics, while the strength of the waves (and their relevance in the flow dynamics) is weaker.
To investigate the formation mechanism of energy spectra of internal waves in the oceans, direct numerical simulations are performed. The simulations are based on the reduced dynamical equations of rotating stratified turbulence. In the reduced dynamical equations only wave modes are retained, and vortices and horizontally uniform vertical shears are excluded. Despite the simplifications, our simulations reproduce some key features of oceanic internal-wave spectra: accumulation of energy at near-inertial waves and realistic frequency and horizontal wavenumber dependencies. Furthermore, we provide evidence that formation of the energy spectra in the inertial subrange is dominated by scale-separated interactions with the near-inertial waves. These findings support oceanographers intuition that spectral energy density of internal waves is the result of predominantly wave-wave interactions.
The 2D second-mode is a potent instability in hypersonic boundary layers (HBLs). We study its linear and nonlinear evolution, followed by its role in transition and eventual breakdown of the HBL into a fully turbulent state. Linear stability theory (LST) is utilized to identify the second-mode wave through FS-synchronization, which is then recreated in linearly and nonlinearly forced 2D direct numerical simulations (DNSs). The nonlinear DNS shows saturation of the fundamental frequency, and the resulting superharmonics induce tightly braided ``rope-like patterns near the generalized inflection point (GIP). The instability exhibits a second region of growth constituted by the fundamental frequency, downstream of the primary envelope, which is absent in the linear scenario. Subsequent 3D DNS identifies this region to be crucial in amplifying oblique instabilities riding on the 2D second-mode ``rollers. This results in lambda vortices below the GIP, which are detached from the ``rollers in the inner boundary layer. Streamwise vortex-stretching results in a localized peak in length-scales inside the HBL, eventually forming haripin vortices. Spectral analyses track the transformation of harmonic peaks into a turbulent spectrum, and appearance of oblique modes at the fundamental frequency, which suggests that fundamental resonance is the most dominant mechanism of transition. Bispectrum reveals coupled nonlinear interactions between the fundamental and its superharmonics leading to spectral broadening, and traces of subharmonic resonance as well.
A mean-field theory of differential rotation in a density stratified turbulent convection has been developed. This theory is based on a combined effect of the turbulent heat flux and anisotropy of turbulent convection on the Reynolds stress. A coupled system of dynamical budget equations consisting in the equations for the Reynolds stress, the entropy fluctuations and the turbulent heat flux has been solved. To close the system of these equations, the spectral tau approach which is valid for large Reynolds and Peclet numbers, has been applied. The adopted model of the background turbulent convection takes into account an increase of the turbulence anisotropy and a decrease of the turbulent correlation time with the rotation rate. This theory yields the radial profile of the differential rotation which is in agreement with that for the solar differential rotation.
Practical application of Gauss law in acoustics is not a very well known method. However, any inverse square law behavior can be formulated in the way similar to Gauss law, which allows us to extend the same principle to sound waves propagation. We show in this paper how the acoustic power of sound source can be related to the sound intensity flow through a given surface by means of the Gauss law. Several different sound-source shapes, important in practical applications, are analyzed by means of the Gauss law. A suitable choice of the Gaussian surface allows us to obtain the simple and straightforward method for calculating the sound intensity distribution in space.
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