Session E36: Waves I

Wavemaking by a vortex pair in stratified flow

Recent simulations of a vortex pair in a stratified fluid show that for small Froude number $W/Nb$ the vortices disintegrate into internal waves, where $W$ is the vortex strength, $b$ is the vortex spacing, and $N$ is the buoyancy frequency. The kinetic energy loss from the vortex pair in this regime can be remarkably fast, essentially annihilating the coherent vortex pair before any noticeable propagation. If the Froude number is large the vortices remain coherent and propagate as they would in constant density flow. The transition in behavior occurs near a Froude number of unity, but is apparently not a sharp transition, as some wave-making appears to happen for Froude numbers above unity. Here we quantify the wave-making with an integral of the momentum flux around a sequence of circles centered on the vortex pair and moving with it. Numerical solutions are obtained using a spectral method, the flow is treated as Boussinesq and viscous, and the initial conditions are approximately the flow due to a line vortex. The results confirm that the transition is gradual, although the complexity of the wavy flow makes interpretation difficult. These results are related to vortex roll-up in a stratified fluid.   

Resonance Van Hove Singularities in Weak Wave Turbulence

Wave kinetic theory has been developed by Hasselmann, Benney, and others to describe turbulence of weakly nonlinear, dispersive waves. However, systems which are generally dispersive can have resonant sets of wave modes with identical group velocities, leading to a local breakdown of dispersivity. This shows up as a geometric singularity of the resonant manifold and possible divergence of the phase measure in the collision integral. Such singularities occur widely, including acoustical waves, Rossby waves, helical waves in rotating fluids, etc. These singularities are the exact analogue of the critical points found by Van Hove in 1953 for phonon dispersion relations in crystals. The importance of these singularities in wave kinetics depends on the dimension of phase space D$=$(N-2)d (d physical space dimension, N the number of waves in resonance) and the degree of degeneracy of the critical points. Following Van Hove, we show that point (non-degenerate) singularities produce no divergences for D\textgreater 2 but lead to logarithmic divergences when D$=$2 and possible breakdown of wave kinetics. These divergences are not removed by nonlinear broadening of the resonances. We discuss an example of Michel et al. (2010) for optical turbulence (N$=$4,d$=$1). When D\textgreater 2, breakdown can occur for degenerate critical points which live on higher-dimensional lines and surfaces. We give examples for inertial waves in rotating fluids (N$=$3,d$=$3) and the dilute electron-hole plasma in graphene (N$=$4, d$=$2).   

Numerical study of nonlinear full wave acoustic propagation

With the aim of describing nonlinear acoustic phenomena, a form of the conservation equations for fluid dynamics is presented, deduced using slightly less restrictive hypothesis than those necessary to obtain the well known Westervelt equation. This formulation accounts for full wave diffraction, nonlinearity, and thermoviscous dissipative effects. A CLAWPACK based, 2D finite-volume method using Roe's linearization has been implemented to obtain numerically the solution of the proposed equations. In order to validate the code, two different tests have been performed: one against a special Taylor shock-like analytic solution, the other against published results on a HIFU system, both with satisfactory results. The code is written for parallel execution on a GPU and improves performance by a factor of over 50 when compared to the standard CLAWPACK Fortran code. This code can be used to describe moderate nonlinear phenomena, at low Mach numbers, in domains as large as 100 wave lengths. Applications range from modest models of diagnostic and therapeutic HIFU, parametric acoustic arrays, to acoustic wave guides. A couple of examples will be presented showing shock formation and oblique interaction.   

Wave propagation in a viscous fluid with a pipeline shear mean flow and application for ultrasonic flow meter

This paper deals with the problem of wave propagation in a compressible viscous fluid confined by a rigid-walled circular pipeline in the presence of a shear mean flow. On the assumption of isentropic and axisymmetric wave propagation, the convected acoustic equations are mathematically deduced from the conservations of continuity and momentum, leading to a set of coupled second-order differential equations with respect of the acoustic pressure and velocity components in radial and axial directions. A solution based on the Fourier-Bessel theory, which is complete and orthogonal in Lebesgue space, is introduced to transform the differential equations to an infinite set of homogeneous algebraic equations, thus the wave number can be calculated due to the existence condition of a non-trivial solution. After the discussion of the method's convergence, the cut-off frequency of the wave mode is theoretically analyzed. Furthermore, wave attenuation of the first four wave modes due to fluid viscosity is numerically studied in the presence of the laminar and turbulent flow profiles. Meanwhile, the measurement performance of an ultrasonic flow meter based on the difference of downstream and upstream wave propagations is parametrically addressed.   

Forced Convective Thermal Transport and Flow Stability Characteristics in Near-Critical Supercritical Fluid

Forced convective thermal transport characteristics of supercritical carbon dioxide in vertical flow are numerically investigated. A tube with a circular cross-section and heated side-wall is considered. A real-fluid model for representing the thermo-physical properties of the supercritical fluid along with the fully compressible form of the Navier--Stokes equations and an implicit time-marching scheme is used to solve the problem. Thermo-physical properties of near-critical supercritical fluids show diverging characteristics. Large variations of density of near-critical supercritical fluid in forced convective flow can induce thermo-hydraulic instability similar to \textit{density wave oscillations}. The developed numerical model is used for studying the effect of geometrical parameters of the tube, wall heat flux and pressure on steady-state convective thermal transport as well as the stability behavior of the supercritical fluid near its critical point. The enhancement or deterioration of heat transfer caused by the temperature-induced variation of physical properties (especially specific heat) is also investigated, as well as the effect of buoyancy on the forced convective flow.