Turbulence In Stratified Fluid Research Articles

Numerical simulations of compressible Rayleigh–Taylor turbulence in stratified fluids

We present the results of our numerical simulations of the Rayleigh–Taylor turbulence, performed using a recently proposed (Sbragaglia et al 2009 J. Fluid Mech. 628 299, Scagliarini et al 2010 Phys. Fluids 22 055101) lattice Boltzmann method that can describe consistently a thermal compressible flow subjected to an external forcing. The method allowed us to study the system in both the nearly Boussinesq regime and the strongly compressible regime. Moreover, we show that when the stratification is important, the presence of the adiabatic gradient causes the arrest of the mixing process.

Self-organization of decaying quasi-two-dimensional turbulence in stratified fluid in rectangular containers

Laboratory experiments on decaying quasi-two-dimensional turbulence have been performed in stratified fluid inside containers with length-to-width ratios . The numerically obtained final states appeared to agree with the experimental observations. Furthermore, they indicate a clear difference between the predictions of quasi-stationary final states from statistical-mechanical theories and the final states as found in the numerical simulations.

Environmental Stratified Flows. Topics in Environmental Fluid Mechanics, Vol. 3

9R72. Environmental Stratified Flows. Topics in Environmental Fluid Mechanics, Vol 3. - R Grimshaw (Dept of Math Sci, Loughborough Univ, UK). Kluwer Acad Publ, Dordrecht, Netherlands. 2002. 284 pp. ISBN 0-7923-7605-6. $135.00.Reviewed by O Phillips (Dept of Earth and Planet Sci, Johns Hopkins Univ, Charles and 34th St, Baltimore MD 21218-2681).This book is a collection of technical essays on some topics of current interest in environmental fluid mechanics. Though numerous references are given at the end of each essay, the individual chapter is generally not a broad review of some aspect of the field, but a detailed exposition of some part of it. The authors are all well known in the field, with cumulatively impressive authority, but the casual reader, even though educated in fluid mechanics, may find the book to be hard going. The first three chapters are all on various aspects of solitary internal waves. The first, by the editor Grimshaw himself, is concerned largely with developing the variable-coefficient, extended Kortweg-de Vries (KdV) equation, and its two-dimensional extension, with a fair degree of rigor, but with an eye towards application to unsteady solitary waves and non-dissipative undular bores. Peter Holloway and Efin Pelinovsky apply the KdV equation with added bottom stress and Coriolis terms, to the evolution of internal tides and long internal waves as they propagate over variable bottom topography of the continental slope and shelf. A good database from the northwest Australian shelf allows detailed comparison with numerical simulations. James Rottman and Roger Grimshaw consider the atmospheric counterparts, particularly the Morning Glory, a spectacular, but unthreatening low-level solitary wave that sweeps across the southern coast of the Gulf of Carpentaria in Northern Australia. Similar events seem to occur in the eastern United States, though stronger topographic effects may complicate observation. James Rottman and Paul Linden do give a good review of the basics of gravity currents, quasi-horizontal flows of one fluid into surrounding fluid of another density, and include good field and laboratory photographs that are worth quite a few thousand words. Ron Smith surveys stratified flows over topography, giving a bibliography of over 100 useful references. Harindra Fernando’s essay on turbulence in stratified fluids can hardly do justice to the intricacies of entrainment, turbulent “blini” and fossil turbulence, but some of the themes are picked up by Don Boyer and A Srdic-Mitrovic on laboratory observations of flow past obstacles. The final chapter by Larry Redekopp sketches the various cases of flow instability, referring the reader to Turner’s well-known monograph, older but more physical and more comprehensive in scope. The index of the book is cursory. There is valuable content in Environmental Stratified Flows, but it is marred by so many word-processing glitches, that this reviewer wonders whether page proofs still exist. If they do, it is almost certain that nobody looked at them. In the editor’s preface, authors’ names occur randomly in italic and roman script. The sub-heading and text at the bottom of page 129 appear also at the top of page 130. The font size changes for no apparent reason in the middle of page 162. Blank half pages occur mysteriously on pages 205 and 206. There are many more examples. It is a great pity that the careful work of the authors has been undermined by such sloppy presentation of the book.

Self-organization of quasi-two-dimensional turbulence in stratified fluids in square and circular containers

Laboratory experiments on decaying quasi-2D (two-dimensional) turbulence have been performed in stratified fluids in both square and circular containers. The turbulence was generated by towing an array of vertical cylinders through the container, which was filled with either a two-layer or a linearly stratified fluid. By varying the grid configuration a different amount of angular momentum could be added to the initial flow. The evolution of the flow was visualized by 2D particle tracking velocimetry. The observed decay scenario has been investigated with emphasis on the evolution of the kinetic energy and the enstrophy of the horizontal flow, vortex statistics and the angular momentum of the flow. In particular it is shown that the experiments in both the square and the circular container support the observations from numerical simulations of decaying 2D turbulence in bounded domains with no-slip walls. Two striking examples are the experimental observation of the spontaneous spin-up phenomenon (in the square-container experiments) and the confirmation that the angular momentum of the flow in the circular-container experiment is better conserved than the total kinetic energy of the flow. The role of the initial nonzero net angular momentum on the decay of quasi-2D turbulence is investigated for both geometries and indications for an acceleration of the self-organization process are presented.

A newly developed low drift fast response salinity meter

Here a new conductivity probe, which is basically a stainless steel tube with a plastic plug with a hole, plugged in one end, is described. The tube is externally insulated and the other electrode is (usually earthed) a metallic slab fixed in the fluid. The electronic used is of a conventional type and uses a 1 kHz signal whose amplitude is modulated by the conductance across the probes terminals. The signal is amplified, rectified, and filtered. The probe is different from the conventional probes, in that the fluid is sucked through the small hole in the probe tip, and only through the fluid in the tip can an electric current flow. Hence, the output and the dynamic response depend, respectively, on the conductivity of the small volume of the fluid in the tip, and how fast it can be replaced. Tests on this and other types of probes, show that this probe has almost no drift and has a satisfactory dynamic response. Probes with different tip geometries have also been tested and show that the frequency response of the probe (typically 200 Hz) does not change significantly. Possible adaptation of such a probe for oceanographic measurements will be discussed. The author has successfully used such a system for laboratory study of turbulence in stratified fluids.

Homogeneous turbulence evolution in stably stratified flow—I. Internal gravity waves at low inverse Froude numbers

Abstract The evolution of homogeneous grid-generated turbulence in stratified fluid is investigated analytically. A small parameter decomposition on inverse Froude number is applied to the theory of nearly homogeneous turbulence developed earlier in refs. [1] and [2]. Applying the multiple scale method we calculate the frequency and amplitude of internal gravity waves and formulate differential system describing the wave-averaged behaviour of turbulence characteristics. The comparison with the numerical results published in [2] and with known experimental data shows a very good agreement as to internal waves as to small scale turbulence in collapsed state. The long-term turbulence evolution in weakly stratified media can be regarded as a singular perturbation problem permitting us to gain an insight into the interaction between gravity waves and active turbulence.

The effect of curvature on turbulence in stratified fluids

The influence of streamline curvature on small‐scale turbulence and vertical mixing in stratified fluids is the subject of this study. The roles of curvature and stratification in enhancing and suppressing turbulent mixing are explored using second‐moment closure for turbulence. Governing equations for second moments are expressed in generalized orthogonal curvilinear coordinates, from which, through a series of approximations, simplified expressions are derived for second moments in the limit of small streamline curvature. The governing equations are then used to obtain a quasi‐equilibrium turbulence model suited for application to atmospheric and oceanic mixed layers. A typical model application is illustrated by simulation of stratified flows over two‐dimensional, idealized mountains and valleys. The limit of local equilibrium is further invoked to derive semi‐analytical results for the enhancement and suppression of vertical turbulent mixing under the combined influence of stratification and curvature. It is shown that stabilizing curvature can drastically suppress turbulence even when the stratification is strongly destabilizing. Conversely, under strong stable stratification that would otherwise lead to total suppression of turbulence, destabilizing curvature can keep turbulence alive. Streamline curvature is also shown to significantly modify the Monin‐Obukhov similarity laws for momentum and heat fluxes in the constant flux region of the atmospheric boundary layer. Finally, the need for observational data on curvature effects on mixing in stratified flows either in the laboratory or in flows over topography in the oceans and the atmosphere is highlighted.

Effect of rotation on vertical mixing and associated turbulence in stratified fluids

Combined effects of stratification and rotation on vertical mixing and the characteristics of associated small‐scale turbulence are explored using second‐moment closure methodology; the rotational terms in the equations for Reynolds stresses and turbulent heat fluxes are retained, not ignored as in earlier works. Semianalytical results valid for arbitrary values of rotation and stratification are derived by further invoking the local equilibrium limit of closure. Two cases are considered: nonzero vertical rotation and nonzero meridional rotation; the latter case is of more general interest in geophysics because of its potential application to equatorial mixed layers. In both cases the influence of rotation on mixing coefficients and Monin‐Obukhov constant flux layer similarity relations is investigated for arbitrary values of rotation and stratification. In both cases, turbulent mixing coefficients assume tensorial properties. However, meridional rotation appears to have a stronger influence on vertical mixing and turbulence characteristics than does vertical rotation. These results, along with perturbation expansions for weak rotation, suggest that for geophysical flows, in most cases, the direct effect of rotation on vertical turbulent mixing itself is but a small correction, a few tens of percent at best. It is seldom large, although it might not be negligible in some particular cases. Nevertheless, the study of rotational effects on small‐scale turbulence provides a fascinating insight into the direct impact of rotation on the characteristics of small‐scale turbulence and mixing in stratified fluids; the results are also of interest in other fields such as engineering.

非定常交換流におけるＫ‐Ｈ不安定波の３次元破壊過程の可視化

The breaking process of the K-H wave in an unsteady exchange flow is investi-gated through the experiment by using a simple visualization technique. Two-dimensional vortex pairing, convective rolles and knots which are mentioned by S.A. Thorpe as transitional phenomena of turbulence in stratified fluids are con-firmed in the flow of the low Richardson number. On the other hand, the flow condition of the high Richardson number brings several uninvestigated three-dimensional breaking process.

Turbulence in stratified fluids: A review

Buoyancy length scale and internal disturbance Froude number criteria for turbulence collapse in stratified fluid are examined in the light of recent laboratory experiments, oceanographie observations, and numerical simulations. From these it emerges that the onset of collapse occurs when the turbulence integral length scale is of the order of the buoyancy length scale, giving a disturbance Froude number close to 1. Complete collapse of three‐dimensional turbulence, characterized by nearly zero vertical mass flux, occurs at a lower value of the disturbance Froude number. In particular, a spatially inhomogeneous turbulence undergoes a drastic change in structure at a value of the disturbance Froude number of 0.2–0.3. In the collapsed state, internal wave motion and quasi two‐dimensional turbulence coexist on different time scales, results supported also by the rotating turbulence analogy. Vertical diffusion coefficients and mixing efficiency, which are of direct practical importance, are also examined.

Transitional phenomena and the development of turbulence in stratified fluids: A review

The processes which lead to vertical mixing in stratified fluids operate in a multiparameter space, much of which has not been explored theoretically or in laboratory experiments. This review focuses on the transitional phenomena which precede, and may eventually lead to, the broadband spectra associated with turbulent flow. Conditions of static instability are known to be produced in wave‐wave, wave‐current, and wave‐boundary interactions, but little is known of the subsequent stages of transition. Whilst there has been considerable progress in describing transition in Rayleigh‐Bènard and Taylor‐Couette flows, that in Kelvin‐Helmholtz instability is relatively unexplored, and in consequence, our knowledge of energy exchange and of density and momentum fluxes is extremely sparse. Attention is drawn here to the flow disturbance caused by sidewalls in laboratory experiments, and some recently discovered examples of transitional phenomena in Kelvin‐Helmholtz instability are described. More information is available about fluxes in double‐diffusive convection, but here too the transitional processes are as yet poorly known. Applying this information to the natural environment (and emphasis is given to the oceans) poses the fundamental difficulty of estimating the probability of conditions favoring the development of any particular kind of instability. Knowledge of transition in conditions in which the background field varies in space and time is deficient.

87:6152 Transitional phenomena and the development of turbulence in stratified fluids: a review: Thorpe, S.A., 1987. J. geophys. Res., 92(C5):5231–5248

87:6152 Transitional phenomena and the development of turbulence in stratified fluids: a review: Thorpe, S.A., 1987. J. geophys. Res., 92(C5):5231–5248

Laminated structures in the final stage in the decay of turbulence in stratified fluids

Laminated structures in the final stage in the decay of turbulence in stratified fluids

On the spectrum of shear-induced turbulence in stratified fluids

A numerical analysis of the instability range of stratified sheared flows is carried out. Attention is devoted to cases in which a variation of the Brunt-Vaisala frequency exists far from a localized shear layer. It is shown that such BV jumps significantly affect the spectrum of turbulence induced by a localized shear layer.

A theory of turbulence in stratified fluids

An effort is made to understand turbulence in fluid systems like the oceans and atmosphere in which the Richardson number is generally large. Toward this end, a theory is developed for turbulent flow over a flat plate which is moved and cooled in such a way as to produce constant vertical fluxes of momentum and heat. The theory indicates that in a co-ordinate system fixed in the plate the mean velocity increases linearly with heightzabove a turbulent boundary layer and the mean density decreases asz3, so that the Richardson number is large far from the plate. Near the plate, the results reduce to those of Monin & Obukhov.Thecurvatureof the density profile is essential in the formulation of the theory. When the curvature is negative, a volume of fluid, thoroughly mixed by turbulence, will tend to flatten out at a new level well above the original centre of mass, thereby transporting heat downward. When the curvature is positive a mixed volume of fluid will tend to fall a similar distance, again transporting heat downward. A well-mixed volume of fluid will also tend to rise when the density profile is linear, but this rise is negligible on the basis of the Boussinesq approximation. The interchange of fluid of different, mean horizontal speeds in the formation of the turbulent patch transfers momentum. As the mixing in the patch destroys the mean velocity shear locally, kinetic energy is transferred from mean motion to disturbed motion. The turbulence can arise in spite of the high Richardson number because the precise variations of mean density and mean velocity mentioned above permit wave energy to propagate from the turbulent boundary layer to the whole region above the plate. At the levels of reflexion, where the amplitudes become large, wave-breaking and turbulence will tend to develop.The relationship between the curvature of the density profile and the transfer of heat suggests that the density gradient near the level of a point of inflexion of the density curve (in general cases of stratified, shearing flow) will increase locally as time goes on. There will also be a tendency to increase the shear through the action of local wave stresses. If this results in a progressive reduction in Richardson number, an ultimate outbreak of Kelvin–Helmholtz instability will occur. The resulting sporadic turbulence will transfer heat (and momentum) through the level of the inflexion point. This mechanism for the appearance of regions of low Richardson number is offered as a possible explanation for the formation of the surfaces of strong density and velocity differences observed in the oceans and atmosphere, and for the turbulence that appears on these surfaces.

Spectra of Internal Waves and Turbulence in Stratified Fluids: 2. Experiments on the Breaking of Internal Waves in a Two‐Fluid System

The breaking of internal waves on a sloping surface in a two‐fluid system is studied experimentally in a wave tank. The characteristics of breaking internal waves are examined through their frequency spectra for cases of different frequency, amplitude, and wavelength. Waveheight measurements are made with capacitance probes in conjunction with a recently developed capacitance bridge and are processed by computer utilizing the Cooley‐Tukey fast Fourier transform algorithm. By using a direct, on‐line computer system spectra can be computed with a high degree of accuracy and a very narrow bandwidth. Spectra measured in the breaking region of the tank reveal a series of sharp peaks in the low frequency range representing the different harmonics of the internal waves. These higher modes, which possess little or no energy in the incident wave, gain energy at the expense of the lowest mode as the wave train propagates over the sloping surface. The result is a transfer of energy to higher frequencies through growth of the harmonics, each mode gaining energy at the expense of lower modes and transferring it to higher modes. At higher frequencies the spectra no longer possess these distinct harmonics. The high frequency end of the spectrum in the breaking region has a slope of approximately −3 for all cases considered.

Spectra of Internal Waves and Turbulence in Stratified Fluids: 1. General Discussion and Indications from Measurements in Stably Stratified Atmosphere and Ocean

A unified spectral description of internal waves and turbulence in a stably stratified atmosphere or ocean is proposed. We envisage that the fluctuating motions in a free atmosphere with scales smaller than the synoptic scale consist of internal waves and turbulence. At low wavenumbers, internal waves predominate; this range of wavenumbers may be called the internal wave subrange. The internal waves can be identified from the characteristics of velocity‐scalar cospectra and quadrature spectra, as shown recently by Pao [1969b] in a laboratory experiment. When the internal waves are sufficiently strong and distinct, the different harmonics of internal waves may appear in the autospectra of velocity and temperature where the peaks and valleys are clearly identifiable. This, we believe, explains the presence of peaks and valleys in some velocity autospectra of clear air turbulence measured in strongly stratified regions of the atmosphere. At intermediate wavenumbers (buoyant subrange) the buoyancy effect is still strong but the turbulent scrambling process also becomes important; the higher‐order harmonics of internal waves are no longer distinct and cannot be detected in the autospectra; the velocity autospectra are steeper than k−5/3. At high wavenumbers (inertial and viscous subranges) turbulence predominates. The characteristics of fluctuating motion in this range of wavenumbers can be described well with Kolmogorov's concept of local isotropy and are not affected by the stratification. The proposed spectral behavior for the free atmosphere is described in detail and supported with existing measurements in stably stratified regions of the atmosphere and ocean in part 1. It will be further supported by our laboratory experiments on the breaking of internal waves in a two‐fluid system [Hall and Pao, 1969] in part 2.

Turbulence in Stratified Fluids

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