Stratified wake of a tilted cylinder. Part 2. Lee internal waves

2012 ◽  
Vol 699 ◽  
pp. 198-215 ◽  
Author(s):  
Patrice Meunier
Keyword(s):  
Circular Cylinder ◽  
Internal Waves ◽  
Bluff Body ◽  
Theoretical Study ◽  
Horizontal Cylinder ◽  
Quantitative Prediction ◽  
Dispersive Waves ◽  
Lee Waves ◽  
The Waves ◽  
Good Agreement

AbstractThis experimental, numerical and theoretical study considers the lee internal waves generated by the wake of a circular cylinder, whose axis is tilted with respect to a stable density gradient. The main difference with the case of a horizontal cylinder is that the lee waves contain a large axial velocity, which are located in a row of lobes extending downstream from the cylinder. At small tilt angles, the wavelength is equal to $2\lrm{\pi} U/ N$, $U$ being the velocity of the cylinder and $N$ the Brunt–Väisälä frequency, which can be explained by the fact that the group velocity of the waves is small. The amplitude of the waves can be predicted using the Lighthill theory for dispersive waves applied to the case of a tilted bluff body. The flow around the cylinder is modelled empirically in order to reach a quantitative prediction in good agreement with the experimental and numerical results. The spatial structure of the predicted internal waves is qualitatively correct although some discrepancies arise because the advection by the flow around the cylinder is neglected.

scholarly journals The spin-up of a linearly stratified fluid in a sliced, circular cylinder

2016 ◽  
Vol 806 ◽  
pp. 254-303
Author(s):  
R. J. Munro ◽  
M. R. Foster
Keyword(s):  
Circular Cylinder ◽  
Rossby Waves ◽  
Rotation Axis ◽  
A Priori ◽  
Propagation Speed ◽  
Stratified Fluid ◽  
Decay Rates ◽  
The Waves ◽  
Western Half ◽  
Good Agreement

A linearly stratified fluid contained in a circular cylinder with a linearly sloped base, whose axis is aligned with the rotation axis, is spun-up from a rotation rate $\unicode[STIX]{x1D6FA}-\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}$ to $\unicode[STIX]{x1D6FA}$ (with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}\ll \unicode[STIX]{x1D6FA}$) by Rossby waves propagating across the container. Experimental results presented here, however, show that if the Burger number $S$ is not small, then that spin-up looks quite different from that reported by Pedlosky & Greenspan (J. Fluid Mech., vol. 27, 1967, pp. 291–304) for $S=0$. That is particularly so if the Burger number is large, since the Rossby waves are then confined to a region of height $S^{-1/2}$ above the sloped base. Axial vortices, ubiquitous features even at tiny Rossby numbers of spin-up in containers with vertical corners (see van Heijst et al.Phys. Fluids A, vol. 2, 1990, pp. 150–159 and Munro & Foster Phys. Fluids, vol. 26, 2014, 026603, for example), are less prominent here, forming at locations that are not obvious a priori, but in the ‘western half’ of the container only, and confined to the bottom $S^{-1/2}$ region. Both decay rates from friction at top and bottom walls and the propagation speed of the waves are found to increase with $S$ as well. An asymptotic theory for Rossby numbers that are not too large shows good agreement with many features seen in the experiments. The full frequency spectrum and decay rates for these waves are discussed, again for large $S$, and vertical vortices are found to occur only for Rossby numbers comparable to $E^{1/2}$, where $E$ is the Ekman number. Symmetry anomalies in the observations are determined by analysis to be due to second-order corrections to the lower-wall boundary condition.

Linearly stratified flow past a horizontal circular cylinder

1989 ◽  
Vol 328 (1601) ◽  
pp. 501-528 ◽  
Keyword(s):  
Circular Cylinder ◽  
Parameter Space ◽  
Scaling Analysis ◽  
Maximum Width ◽  
Wide Range ◽  
Lee Waves ◽  
Streamwise Distance ◽  
Horizontal Circular Cylinder ◽  
Good Agreement ◽  
Internal Froude Number

The flow of a linearly stratified fluid past a long circular cylinder in a channel is considered experimentally. The characteristics of the flow depend on the internal Froude number F i the Reynolds number Re and the cylinder diameter to fluid depth ratio, d/H . A wide range of characteristic flow fields are observed in the parameter space investigated; i.e. 0.02 ⩽ F i ⩽ 13 , 5 ⩽ Re ⩽ 4000 and 0.03 ⩽ d / H ⩽ 0.20 . A flow regime diagram of F i against Re for these characteristic flows is developed. Some of the lower F i Re experiments are compared with numerical experiments. A theory is advanced which indicates that the dimensionless length, x b ∗ = x b / d of the blocked region ahead of the cylinder should scale as x b ∗ ≈ ( δ / d ) 5 R e F i − 2 , where δ is the thickness of the shear layer between the external flow and the approximately stagnant blocked region; the results of an experimental programme that support this scaling are presented. Measurements are made which indicate that for the range of parameter space in which lee waves occur, the lee wavelengths are predicted to a good approximation by linear theory. A scaling analysis is carried out which suggests that the height of the rotors above the streamwise centreline, Z r ∗ = Z r / d , scales with F i experiments aire in good agreement with this prediction. For conditions under which the wake of the cylinder is turbulent, scaling arguments suggest that the dimensionless maximum width of the wake, γ m ∗ = γ m / d , and the dimensionless streamwise distance at which this maximum occurs, β m ∗ = β m / d , scale as F i 1 2 Experiments are presented which support this scaling.

Internal waves generated by a stratified wake: experiment and theory

2018 ◽  
Vol 846 ◽  
pp. 752-788 ◽  
Author(s):  
P. Meunier ◽  
S. Le Dizès ◽  
L. Redekopp ◽  
G. R. Spedding
Keyword(s):  
Phase Velocity ◽  
Internal Waves ◽  
Bluff Body ◽  
Stokes Equations ◽  
Buoyancy Frequency ◽  
Correct Prediction ◽  
Double Row ◽  
Von Karman ◽  
Lee Waves ◽  
Von Kármán

This paper presents experimental and theoretical results on the internal waves emitted by a bluff body moving horizontally in a linearly stratified fluid. Three different bluff bodies (a sphere, a spheroid and a cylinder) have been used in order to study the effect of the shape of the bluff body, although most of the results are obtained for the sphere. Two types of internal waves have been observed experimentally: large wavelength lee waves generated by the bluff body itself and small wavelength coherent wake waves generated by the turbulent wake. First, the lee waves are separated from the wake waves by averaging the experimental measurements in the frame moving with the bluff body. The velocity amplitude of the lee waves scales as the inverse of the Froude number$F=2U_{B}/(ND)$for$F>2$(where$U_{B}$is the towing velocity,$D$the diameter and$N$the buoyancy frequency). This scaling proves that the internal waves are related to the drag of the bluff body which is due to the separation of the flow behind the bluff body. This separation is usually not taken into account in the classical models which assume that the flow is dipolar. The drag can be modelled as a point force in the Navier–Stokes equations, which gives a correct prediction of the structure and the amplitude of the lee waves. Second, the wake waves have been separated from the lee waves by averaging the velocity fields in the frame moving at the phase velocity of the waves. The phase velocity and the wavelength scale as$F^{-2/3}$and$F^{1/3}$respectively which correspond to the velocity and distance between same sign vortices of the von Kármán vortex street. A simplified model is derived for the internal waves emitted by the double row of moving point vortices of the von Kármán street. The amplitude of the wake waves is measured experimentally and seems to depend on the Reynolds number.

scholarly journals The energy flux spectrum of internal waves generated by turbulent convection

2018 ◽  
Vol 854 ◽  
Author(s):  
Louis-Alexandre Couston ◽  
Daniel Lecoanet ◽  
Benjamin Favier ◽  
Michael Le Bars
Keyword(s):  
Internal Waves ◽  
Energy Flux ◽  
Wave Energy ◽  
Three Dimensional ◽  
Turbulent Convection ◽  
Reynolds Stresses ◽  
Wave Energy Flux ◽  
Flux Spectrum ◽  
The Waves ◽  
Good Agreement

We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of mixed convective and stably stratified fluids. We demonstrate that in the limit of large Rayleigh number ($Ra\in [4\times 10^{7},10^{9}]$) and large stratification (Brunt–Väisälä frequencies$f_{N}\gg f_{c}$, where$f_{c}$is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region as described in Lecoanet & Quataert (Mon. Not. R. Astron. Soc., vol. 430 (3), 2013, pp. 2363–2376). Specifically, we demonstrate that the wave energy flux spectrum scales like$k_{\bot }^{4}\,f^{-13/2}$for weakly damped waves (with$k_{\bot }$and$f$the waves’ horizontal wavenumbers and frequencies, respectively), and that the total wave energy flux decays with$z$, the distance from the convective region, like$z^{-13/8}$.

The phase configuration of internal waves around a body moving in a density stratified fluid

1973 ◽  
Vol 60 (4) ◽  
pp. 759-767 ◽  
Author(s):  
T. N. Stevenson
Keyword(s):  
Internal Waves ◽  
Vertical Plane ◽  
Wave Theory ◽  
Horizontal Cylinder ◽  
Stratified Fluid ◽  
Circular Path ◽  
Amplitude Wave ◽  
Special Cases ◽  
The Waves ◽  
Theory And Experiment

A body is started impulsively from rest and moves in a curved path in a stably stratified fluid. The phase configuration of the internal waves which are generated is studied using small amplitude wave theory. The theory is compared with experiment for a few special cases which include a horizontal cylinder moving in a circular path in a vertical plane, oscillating through a large amplitude in a horizontal plane, and moving with constant velocity past stationary plates. Theory and experiment show reasonable agreement except where the waves produced by the wake dominate the flow.

Explanation of the Influence of Inflow Air Content on the Lift of Cavitating Hydrofoils

2018 ◽  
Vol 140 (8) ◽  
Author(s):  
Eduard Amromin
Keyword(s):  
Experimental Data ◽  
Theoretical Study ◽  
Lift Coefficient ◽  
Cavity Surface ◽  
Air Bubbles ◽  
Incoming Flow ◽  
Fluid Density ◽  
Air Content ◽  
Theoretical Results ◽  
Good Agreement

According to several known experiments, an increase of the incoming flow air content can increase the hydrofoil lift coefficient. The presented theoretical study shows that such increase is associated with the decrease of the fluid density at the cavity surface. This decrease is caused by entrainment of air bubbles to the cavity from the surrounding flow. The theoretical results based on such explanation are in a good agreement with the earlier published experimental data for NACA0015.

Axisymmetric internal waves generated by a travelling oscillating body

1969 ◽  
Vol 35 (2) ◽  
pp. 219-224 ◽  
Author(s):  
T. N. Stevenson
Keyword(s):  
Internal Waves ◽  
Salt Solution ◽  
Reynolds Numbers ◽  
Mean Velocity ◽  
Dispersive Waves ◽  
The Mean ◽  
Oscillating Sphere

Experiments are presented in which axisymmetric internal waves are generated by an oscillating sphere moving vertically in a stably stratified salt solution. The Reynolds numbers for the sphere based on the diameter and the mean velocity are between 10 and 200. Lighthill's theory for dispersive waves is used to calculate the phase configuration of the internal waves. The agreement between experiment and theory is reasonably good.

Global Stability Analysis of the Wake behind a Circular Cylinder Based on the POD-Chiba Method

2011 ◽  
Vol 137 ◽  
pp. 72-76
Author(s):  
Wei Zhang ◽  
Xian Wen ◽  
Yan Qun Jiang
Keyword(s):  
Stability Analysis ◽  
Circular Cylinder ◽  
Global Stability ◽  
Dimensional Model ◽  
Global Stability Analysis ◽  
The Stability ◽  
Low Dimensional ◽  
Proper Orthogonal ◽  
Calculated Amount ◽  
Good Agreement

A proper orthogonal decomposition (POD) method is applied to study the global stability analysis for flow past a stationary circular cylinder. The flow database at Re=100 is obtained by CFD software, i.e. FLUENT, with which POD bases are constructed by a snapshot method. Based on the POD bases, a low-dimensional model is established for solving the two-dimensional incompressible NS equations. The stability of the flow solution is evaluated by a POD-Chiba method in the way of the eigensystem analysis for the velocity disturbance. The linear stability analysis shows that the first Hopf bifurcation takes place at Re=46.9, which is in good agreement with available results by other high-order accurate stability analysis methods. However, the calculated amount of POD is little, which shows the availability and advantage of the POD method.

Internal waves in a viscous atmosphere

1975 ◽  
Vol 72 (4) ◽  
pp. 773-786 ◽  
Author(s):  
W. L. Chang ◽  
T. N. Stevenson
Keyword(s):  
Energy Source ◽  
Incompressible Fluid ◽  
Internal Waves ◽  
Infinite Number ◽  
Similarity Solution ◽  
Small Amplitude ◽  
Horizontal Plane ◽  
Two Dimensional ◽  
The Waves ◽  
Isothermal Atmosphere

The way in which internal waves change in amplitude as they propagate through an incompressible fluid or an isothermal atmosphere is considered. A similarity solution for the small amplitude isolated viscous internal wave which is generated by a localized two-dimensional disturbance or energy source was given by Thomas & Stevenson (1972). It will be shown how summations or superpositions of this solution may be used to examine the behaviour of groups of internal waves. In particular the paper considers the waves produced by an infinite number of sources distributed in a horizontal plane such that they produce a sinusoidal velocity distribution. The results of this analysis lead to a new small perturbation solution of the linearized equations.

Dissipating and reflecting internal waves

2021 ◽  
Author(s):  
Callum J. Shakespeare ◽  
Brian K. Arbic ◽  
Andrew McC. Hogg
Keyword(s):  
Numerical Simulations ◽  
Linear Theory ◽  
Internal Waves ◽  
Energy Flux ◽  
Internal Tide ◽  
Internal Tides ◽  
Linear Wave ◽  
Lee Waves ◽  
Lee Wave ◽  
Upper Boundary

AbstractInternal waves generated at the seafloor propagate through the interior of the ocean, driving mixing where they break and dissipate. However, existing theories only describe these waves in two limiting cases. In one limit, the presence of an upper boundary permits bottom-generated waves to reflect from the ocean surface back to the seafloor, and all the energy flux is at discrete wavenumbers corresponding to resonant modes. In the other limit, waves are strongly dissipated such that they do not interact with the upper boundary and the energy flux is continuous over wavenumber. Here, a novel linear theory is developed for internal tides and lee waves that spans the parameter space in between these two limits. The linear theory is compared with a set of numerical simulations of internal tide and lee wave generation at realistic abyssal hill topography. The linear theory is able to replicate the spatially-averaged kinetic energy and dissipation of even highly non-linear wave fields in the numerical simulations via an appropriate choice of the linear dissipation operator, which represents turbulent wave breaking processes.

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