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Internal wave energy flux from density perturbations in nonlinear stratifications
- Frank M. Lee, Michael R. Allshouse, Harry L. Swinney, Philip J. Morrison
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- Journal:
- Journal of Fluid Mechanics / Volume 856 / 10 December 2018
- Published online by Cambridge University Press:
- 12 October 2018, pp. 898-920
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Internal gravity wave energy contributes significantly to the energy budget of the oceans, affecting mixing and the thermohaline circulation. Hence it is important to determine the internal wave energy flux $\boldsymbol{J}=p\,\boldsymbol{v}$, where $p$ is the pressure perturbation field and $\boldsymbol{v}$ is the velocity perturbation field. However, the pressure perturbation field is not directly accessible in laboratory or field observations. Previously, a Green’s function based method was developed to calculate the instantaneous energy flux field from a measured density perturbation field $\unicode[STIX]{x1D70C}(x,z,t)$, given a constant buoyancy frequency $N$. Here we present methods for computing the instantaneous energy flux $\boldsymbol{J}(x,z,t)$ for an internal wave field with vertically varying background $N(z)$, as in the oceans where $N(z)$ typically decreases by two orders of magnitude from the pycnocline to the deep ocean. Analytic methods are presented for computing $\boldsymbol{J}(x,z,t)$ from a density perturbation field for $N(z)$ varying linearly with $z$ and for $N^{2}(z)$ varying as $\tanh (z)$. To generalize this approach to arbitrary $N(z)$, we present a computational method for obtaining $\boldsymbol{J}(x,z,t)$. The results for $\boldsymbol{J}(x,z,t)$ for the different cases agree well with results from direct numerical simulations of the Navier–Stokes equations. Our computational method can be applied to any density perturbation data using the MATLAB graphical user interface ‘EnergyFlux’.
Propagating and evanescent internal waves in a deep ocean model
- M. S. Paoletti, Harry L. Swinney
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- Journal:
- Journal of Fluid Mechanics / Volume 706 / 10 September 2012
- Published online by Cambridge University Press:
- 13 July 2012, pp. 571-583
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We present experimental and computational studies of the propagation of internal waves in a stratified fluid with an exponential density profile that models the deep ocean. The buoyancy frequency profile (proportional to the square root of the density gradient) varies smoothly by more than an order of magnitude over the fluid depth, as is common in the deep ocean. The non-uniform stratification is characterized by a turning depth , where is equal to the wave frequency and . Internal waves reflect from the turning depth and become evanescent below the turning depth. The energy flux below the turning depth is shown to decay exponentially with a decay constant given by , which is the horizontal wavenumber at the turning depth. The viscous decay of the vertical velocity amplitude of the incoming and reflected waves above the turning depth agree within a few per cent with a previously untested theory for a fluid of arbitrary stratification (Kistovich and Chashechkin, J. Appl. Mech. Tech. Phys., vol. 39, 1998, pp. 729–737).
Sedimenting sphere in a variable-gap Hele-Shaw cell
- ANDREW T. LEE, EDUARDO RAMOS, HARRY L. SWINNEY
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- Journal:
- Journal of Fluid Mechanics / Volume 586 / 10 September 2007
- Published online by Cambridge University Press:
- 14 August 2007, pp. 449-464
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We have measured the trajectory and visualized the wake of a single sphere falling in a fluid confined between two closely spaced glass plates (a Hele-Shaw cell). The position of a sedimenting sphere was measured to better than 0.001d, where d is the sphere diameter, for Reynolds numbers (based on the terminal velocity) between 20 and 330, for gaps between the plates ranging from 1.014d to 1.4d. For gaps in the range 1.01d–1.05d, the behaviour of the sedimenting sphere is found to be qualitatively similar to that of an unconfined cylinder in a uniform flow, but our sedimenting sphere begins to oscillate and shed von Kármán vortices for Re>200, which is far greater than the Re = 49 for the onset of vortex shedding behind cylinders in an open flow. When the gap is increased to 1.10d–1.40d, the vortices behind the sphere are different – they are qualitatively similar to those behind a sphere sedimenting in the absence of confining walls. Our precision measurements of the velocity of a sedimenting sphere and the amplitude and frequency of the oscillations provide a benchmark for numerical simulations of the sedimentation of particles in fluids.
Statistical mechanics of two-dimensional turbulence
- SUNGHWAN JUNG, P. J. MORRISON, HARRY L. SWINNEY
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- Journal:
- Journal of Fluid Mechanics / Volume 554 / 10 May 2006
- Published online by Cambridge University Press:
- 24 April 2006, pp. 433-456
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The statistical mechanical description of two-dimensional inviscid fluid turbulence is reconsidered. Using this description, we make predictions about turbulent flow in a rapidly rotating laboratory annulus. Measurements on the continuously forced, weakly dissipative flow reveal coherent vortices in a mean zonal flow. Statistical mechanics has two crucial requirements for equilibrium: statistical independence of macro-cells (subsystems) and additivity of invariants of macro-cells. We investigate these requirements in the context of the annulus experiment. The energy invariant, an extensive quantity, should thus be additive, i.e. the interaction energy between a macro-cell and the rest of the system (reservoir) should be small, and this is verified experimentally. Similarly, we use additivity to select the appropriate Casimir invariants from the infinite set available in vortex dynamics, and we do this in such a way that the exchange of micro-cells within a macro-cell does not alter an invariant of a macro-cell. A novel feature of the present study is our choice of macro-cells, which are continuous phase-space curves based on mean values of the streamfunction. Quantities such as energy and enstrophy can be defined on each curve, and these lead to a local canonical distribution that is also defined on each curve. The distribution obtained describes the anisotropic and inhomogeneous properties of a flow. Our approach leads to the prediction that on a mean streamfunction curve there should be a linear relation between the ensemble-averaged potential vorticity and the time-averaged streamfunction, and our laboratory data are in good accord with this prediction. Further, the approach predicts that although the probability distribution function for potential vorticity in the entire system is non-Gaussian, the distribution function of micro-cells should be Gaussian on the macro-cells, i.e. for curves defined by mean values of the streamfunction. This prediction is also supported by the data. While the statistical mechanics approach used was motivated by and applied to experiments on turbulence in a rotating annulus, the approach is quite general and is applicable to a large class of Hamiltonian systems, including drift-wave plasma models, Vlasov–Poisson dynamics, and kinetic theories of stellar dynamics.
Flow regimes in a circular Couette system with independently rotating cylinders
- C. David Andereck, S. S. Liu, Harry L. Swinney
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- Journal:
- Journal of Fluid Mechanics / Volume 164 / March 1986
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- 21 April 2006, pp. 155-183
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Our flow-visualization and spectral studies of flow between concentric independently rotating cylinders have revealed a surprisingly large variety of different flow states. (The system studied has radius ratio 0.883, aspect ratios ranging from 20 to 48, and the end boundaries were attached to the outer cylinder.) Different states were distinguished by their symmetry under rotation and reflection, by their azimuthal and axial wavenumbers, and by the rotation frequencies of the azimuthal travelling waves. Transitions between states were determined as functions of the inner- and outer-cylinder Reynolds numbers, Ri and Ro, respectively. The transitions were located by fixing Ro and slowly increasing Ri. Observed states include Taylor vortices, wavy vortices, modulated wavy vortices, vortices with wavy outflow boundaries, vortices with wavy inflow boundaries, vortices with flat boundaries and internal waves (twists), laminar spirals, interpenetrating spirals, waves on interpenetrating spirals, spiral turbulence, a flow with intermittent turbulent spots, turbulent Taylor vortices, a turbulent flow with no large-scale features, and various combinations of these flows. Some of these flow states have not been previously described, and even for those states that were previously described the present work provides the first coherent characterization of the states and the transitions between them. These flow states are all stable to small perturbations, and the transition boundaries between the states are reproducible. These observations can serve as a challenge and test for future analytic and numerical studies, and the map of the transitions provides several possible codimension-2 bifurcations that warrant further study.
Spatial and temporal characteristics of modulated waves in the circular Couette system
- M. Gorman, Harry L. Swinney
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- Journal:
- Journal of Fluid Mechanics / Volume 117 / April 1982
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- 20 April 2006, pp. 123-142
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We have used flow-visualization and spectral techniques to study the spatial and temporal properties of the flow that precedes the onset of weak turbulence in a fluid contained between concentric cylinders with the inner cylinder rotating (the circular Couette system). The first three flow regimes encountered as the Reynolds number is increased from zero are well-known – Couette flow, Taylor-vortex flow, and wavyvortex flow. The present study concerns the doubly periodic regime that follows the (singly periodic) wavy-vortex-flow regime. Wavy-vortex flow is characterized by a single frequency f1, which is the frequency of travelling azimuthal waves passing a point of observation in the laboratory. The doubly periodic regime was discovered in studies of power spectra several years ago, but the fluid motion corresponding to the second frequency f2 was not identified. We have found that f2 corresponds to a modulation of the azimuthal waves; the modulation can be observed visually as a periodic flattening of the wavy-vortex outflow boundaries. Moreover, in addition to the previously observed doubly periodic flow state, we have discovered 11 more doubly periodic flow states. Each state can be labelled with two integers m and k, which are simply related to physical characteristics of the flow: m is the number of azimuthal waves, and k is related to the phase angle between the modulation of successive azimuthal waves by Δϕ = 2πk/m. This expression for the phase angle was first conjectured from the flow-visualization measurements and then tested to an accuracy of 0·01π in spectral measurements. Recently Rand (1981) has used dynamical-systems concepts and symmetry considerations to derive predictions about the space–time symmetry of doubly periodic flows in circularly symmetric systems. He predicted that only flows with certain space–time symmetries should be allowed. The observed flow states are in agreement with this theory.
Wave speeds in wavy Taylor-vortex flow
- Gregory P. King, Y. Li, W. Lee, Harry L. Swinney, Philip S. Marcus
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- Journal:
- Journal of Fluid Mechanics / Volume 141 / April 1984
- Published online by Cambridge University Press:
- 20 April 2006, pp. 365-390
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The speed of travelling azimuthal waves on Taylor vortices in a circular Couette system (with the inner cylinder rotating and the outer cylinder at rest) has been determined in laboratory experiments conducted as a function of Reynolds number R, radius ratio of the cylinders η, average axial wavelength $\overline{\lambda}$, number of waves m1 and the aspect ratio Γ (the ratio of the fluid height to the gap between the cylinders). Wave speeds have also been determined numerically for axially periodic flows in infinite-length cylinders by solving the Navier-Stokes equation with a pseudospectral technique where each Taylor-vortex pair is represented with 32 axial modes, 32 azimuthal modes (in an azimuthal angle of 2π/m1) and 33 radial modes. Above the onset of wavy-vortex flow the wave speed for a given η decreases with increasing R until it reaches a plateau that persists for some range in R. In the radius-ratio range examined in our experiments we find that the wave speed in the plateau region increases monotonically from 0.14Ω at η = 0.630 to 0.45Ω at η = 0.950 (where the wave speed is expressed in terms of the rotation frequency Ω of the inner cylinder). There is a much weaker dependence of the wave speed on $\overline{\lambda}$, m1 and Γ. For three sets of parameter values (R, $\overline{\lambda}$, η and m1) the wave speeds have been measured, extrapolated to infinite aspect ratio, and compared with the numerically computed values. For each of these three cases the agreement is within 0.1 %.
Dynamical instabilities and the transition to chaotic Taylor vortex flow
- P. R. Fenstermacher, Harry L. Swinney, J. P. Gollub
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- Journal:
- Journal of Fluid Mechanics / Volume 94 / Issue 1 / 11 September 1979
- Published online by Cambridge University Press:
- 19 April 2006, pp. 103-128
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We have used the technique of laser-Doppler velocimetry to study the transition to turbulence in a fluid contained between concentric cylinders with the inner cylinder rotating. The experiment was designed to test recent proposals for the number and types of dynamical regimes exhibited by a flow before it becomes turbulent. For different Reynolds numbers the radial component of the local velocity was recorded as a function of time in a computer, and the records were then Fourier-transformed to obtain velocity power spectra. The first two instabilities in the flow, to time-independent Taylor vortex flow and then to time-dependent wavy vortex flow, are well known, but the present experiment provides the first quantitative information on the subsequent regimes that precede turbulent flow. Beyond the onset of wavy vortex flow the velocity spectra contain a single sharp frequency component and its harmonics; the flow is strictly periodic. As the Reynolds number is increased, a previously unobserved second sharp frequency component appears at R/Rc = 10·1, where Rc is the critical Reynolds number for the Taylor instability. The two frequencies appear to be irrationally related; hence this is a quasi-periodic flow. A chaotic element appears in the flow at R/Rc ≃ 12, where a weak broadband component is observed in addition to the sharp components; this flow can be described as weakly turbulent. As R is increased further, the component that appeared at R/Rc= 10·1 disappears at R/Rc = 19·3, and the remaining sharp component disappears at R/Rc = 21·9, leaving a spectrum with only the broad component and a background continuum. The observance of only two discrete frequencies and then chaotic flow is contrary to Landau's picture of an infinite sequence of instabilities, each adding a new frequency to the motion. However, recent studies of nonlinear models with a few degrees of freedom show a behaviour similar in most respects to that observed.
Experimental and numerical studies of an eastward jet over topography
- YUDONG TIAN, ERIC R. WEEKS, KAYO IDE, J. S. URBACH, CHARLES N. BAROUD, MICHAEL GHIL, HARRY L. SWINNEY
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- Journal:
- Journal of Fluid Mechanics / Volume 438 / 10 July 2001
- Published online by Cambridge University Press:
- 05 July 2001, pp. 129-157
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Motivated by the phenomena of blocked and zonal flows in Earth's atmosphere, we conducted laboratory experiments and numerical simulations to study the dynamics of an eastward jet flowing over wavenumber-two topography. The laboratory experiments studied the dynamical behaviour of the flow in a barotropic rotating annulus as a function of the experimental Rossby and Ekman numbers. Two distinct flow patterns, resembling blocked and zonal flows in the atmosphere, were observed to persist for long time intervals.
Earlier model studies had suggested that the atmosphere's normally upstream- propagating Rossby waves can resonantly lock to the underlying topography, and that this topographic resonance separates zonal from blocked flows. In the annulus, the zonal flows did indeed have super-resonant mean zonal velocities, while the blocked flows appear subresonant. Low-frequency variability, periodic or irregular, was present in the measured time series of azimuthal velocity in the blocked regime, with dominant periodicities in the range of 6–25 annulus rotations. Oscillations have also been detected in zonal states, with smaller amplitude and similar frequency. In addition, over a large region of parameter space the two flow states exhibited spontaneous, intermittent transitions from the one to the other.
We numerically simulated the laboratory flow geometry in a quasi-geostrophic barotropic model over a similar range of parameters. Both flow regimes, blocked and zonal, were reproduced in the simulations, with similar spatial and temporal characteristics, including the low-frequency oscillations associated with the blocked flow. The blocked and zonal flow patterns are present over wide ranges of forcing, topographic height, and bottom friction. For a significant portion of parameter space, both model flows are stable. Depending on the initial state, either the blocked or the zonal flow is obtained and persists indefinitely, showing the existence of multiple equilibria.
Long-wavelength surface-tension-driven Bénard convection: experiment and theory
- STEPHEN J. VANHOOK, MICHAEL F. SCHATZ, J. B. SWIFT, W. D. MCCORMICK, HARRY L. SWINNEY
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- Journal:
- Journal of Fluid Mechanics / Volume 345 / 25 August 1997
- Published online by Cambridge University Press:
- 25 August 1997, pp. 45-78
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Surface-tension-driven Bénard (Marangoni) convection in liquid layers heated from below can exhibit a long-wavelength primary instability that differs from the more familiar hexagonal instability associated with Bénard. This long-wavelength instability is predicted to be significant in microgravity and for thin liquid layers. The instability is studied experimentally in terrestrial gravity for silicone oil layers 0.007 to 0.027 cm thick on a conducting plate. For shallow liquid depths (<.017 cm for 0.102 cm2 s−1 viscosity liquid), the system evolves to a strongly deformed long-wavelength state which can take the form of a localized depression (‘dry spot’) or a localized elevation (‘high spot’), depending on the thickness and thermal conductivity of the gas layer above the liquid. For slightly thicker liquid depths (0.017–0.024 cm), the formation of a dry spot induces the formation of hexagons. For even thicker liquid depths (>0.024 cm), the system forms only the hexagonal convection cells. A two-layer nonlinear theory is developed to account properly for the effect of deformation on the interface temperature profile. Experimental results for the long-wavelength instability are compared to our two-layer theory and to a one-layer theory that accounts for the upper gas layer solely with a heat transfer coefficient. The two-layer model better describes the onset of instability and also predicts the formation of localized elevations, which the one-layer model does not predict. A weakly nonlinear analysis shows that the bifurcation is subcritical. Solving for steady states of the system shows that the subcritical pitchfork bifurcation curve never turns over to a stable branch. Numerical simulations also predict a subcritical instability and yield long-wavelength states that qualitatively agree with the experiments. The observations agree with the onset prediction of the two-layer model, except for very thin liquid layers; this deviation from theory may arise from small non-uniformities in the experiment. Theoretical analysis shows that a small non-uniformity in heating produces a large steady-state deformation (seen in the experiment) that becomes more pronounced with increasing temperature difference across the liquid. This steady-state deformation becomes unstable to the long-wavelength instability at a smaller temperature difference than that at which the undeformed state becomes unstable in the absence of non-uniformity.