Experimental study of the Ekman layer instability in steady or oscillating flows

To investigate the instabilities of steady and oscillating Ekman layers, an 8 m×2 m horizontal plate was moved at controlled speed in homogeneous water at rest in solid body rotation in the “Coriolis” 13 m diameter rotating tank. For a steady Ekman layer two distinct wave types were found, in agreement with previous experimental or numerical studies. Type I was stationary, was oriented positively with respect to the flow direction and had a wavelength of about 10 times the Ekman layer thickness. Type II was oriented negatively with respect to the flow direction and had a wavelength which was more than 20 times the Ekman layer thickness and a phase-speed between 0.3 and 0.5 the forcing interior velocity. The growth rates of both type I and type II waves for various Reynolds numbers Re (computed with the Ekman layer thickness) were estimated and their Re-variations qualitatively agree with previous numerical results. For an oscillating Ekman layer, experimental results depended strongly on Ro_t, the temporal Rossby number: only when Ro_t<1 was it possible to observe either type I or type II instabilities. Moreover, for all Ro_t and average to high Re, there was a noticeable upward turbulent transport occurring during each cycle between the flow maximum and the flow reversal. Such an upward turbulent transport is consistent with observations in the English Channel where maximum upward benthic movements and maximum turbidity were recorded at the flow reversal, hence Ekman layer instabilities and transition to turbulence are likely to occur in shallow tidal seas where they may be relevant for sediment resuspension and transport as well as for some biological processes.     

Mechanisms for magnetic field generation in precessing cubes

ABSTRACT

It is shown that flows in precessing cubes develop at certain parameters large axisymmetric components in the velocity field which are large enough to either generate magnetic fields by themselves, or to contribute to the dynamo effect if inertial modes are already excited and acting as a dynamo. This effect disappears at small Ekman numbers. The critical magnetic Reynolds number also increases at low Ekman numbers because of turbulence and small-scale structures.     

Dynamics and spectra of cumulant update closures for two-dimensional turbulence

Abstract

Non-Markovian closure theories are compared with ensemble averaged direct numerical simulations (DNS) for decaying two-dimensional turbulence at large scale Reynolds numbers ranging from ≈ 50 to ≈ 4000. The closures, as well as DNS, are formulated for discrete wave numbers relevant to flows on the doubly periodic domain and are compared with the results of continuous wave number closures. The direct interaction approximation (DIA), self-consistent field theory (SCFT) and local energy-transfer theory (LET) closures are also compared with cumulant update versions of these closures (CUDIA, CUSCFT, CULET). The cumulant update closures are shown to have comparable performance to the standard closures but are much more efficient allowing long time integrations.

The discrete wave number closures perform considerably better than continuous wave number closures as far as evolved energy and transfer spectra and skewness are concerned. The discrete wave number closures are in reasonable agreement with DNS in the energy containing range of the large scales for Reynolds numbers ranging from ≈ 50 to ≈ 4000. The closures tend to underestimate the enstrophy flux to high wave numbers, increasingly so with increasing Reynolds number, resulting in underestimation of small-scale kinetic energy.     

On the basic structure of oceanic gravity currents

Results from numerical simulations of idealised, 2.5-dimensional Boussinesq, gravity currents on an inclined plane in a rotating frame are used to determine the qualitative and quantitative characteristics of such currents. The current is initially geostrophically adjusted. The Richardson number is varied between different experiments. The results demonstrate that the gravity current has a two-part structure consisting of: (1) the vein, the thick part that is governed by geostrophic dynamics with an Ekman layer at its bottom, and (2) a thin friction layer at the downslope side of the vein, the thin part of the gravity current. Water from the vein detrains into the friction layer via the bottom Ekman layer. A self consistent picture of the dynamics of a gravity current is obtained and some of the large-scale characteristics of a gravity current can be analytically calculated, for small Reynolds number flow, using linear Ekman layer theory. The evolution of the gravity current is shown to be governed by bottom friction. A minimal model for the vein dynamics, based on the heat equation, is derived and compares very well to the solutions of the 2.5-dimensional Boussinesq simulations. The heat equation is linear for a linear (Rayleigh) friction law and non-linear for a quadratic drag law. I demonstrate that the thickness of a gravity current cannot be modelled by a local parameterisation when bottom friction is relevant. The difference between the vein and the gravity current is of paramount importance as simplified (streamtube) models should model the dynamics of the vein rather than the dynamics of the total gravity current. In basin-wide numerical models of the ocean dynamics the friction layer has to be resolved to correctly represent gravity currents and, thus, the ocean dynamics.     

Turbulence avoidance and the wind-driven transport of plankton in the surface Ekman layer

Observations of turbulence avoidance in zooplankton are compared to estimates of the wind-driven turbulence in the upper ocean. Plankton that avoid wind-driven turbulence by moving deeper are no longer transported by the wind-driven Ekman currents near the surface because they are no longer near the surface. Here, a threshold level of turbulence that triggers an avoidance response is estimated, and is used to infer the wind speed and water column stratification conditions that would lead to zooplankton leaving the Ekman layer. Turbulence avoidance is argued to lead to near-shore retention in wind-driven upwelling systems, and to a reduction of the delivery of zooplankton to Georges Bank from the deeper waters of the Gulf of Maine.     

The Regularized Dia Closure for Two-Dimensional Turbulence

A regularized version of the direct interaction approximation closure (RDIA) is compared with ensemble averaged direct numerical simulations (DNS) for decaying two-dimensional turbulence at large-scale Reynolds numbers ranging between low (≈?50) and high (≈?4000). The regularization localizes transfer by removing the interaction between large-scale and small-scale eddies depending on a specified cut-off ratio α. It thus eliminates spurious convection effects of small-scale eddies by large-scale eddies in the Eulerian direct interaction approximation (DIA) that causes the underestimation of small-scale kinetic energy by the DIA. Cumulant update versions of the RDIA closure that have comparable performance but are much more efficient computationally have also been analyzed. Both the closures and DNS use discrete wavenumber representations relevant to flows on a doubly periodic domain. This means that any differences between them are intrinsic and not partly due to using continuous wavenumber formulation for the closures.

Comparisons between the regularized closures and DNS have focused on evolved kinetic energy and palinstrophy spectra and as well on enstrophy flux spectra and on the evolution of skewness which depends sensitively on small-scale differences. All of these diagnostics compare quite well when α = 6. And this is the case for runs started from each of three initial spectra, for the range of evolved large-scale Reynolds numbers ranging from ≈?50 to ≈?4000 and for regularized DIA closures with, and particularly without, cumulant update restarts. The performance of the RDIA compared with quasi-Lagrangian closure models is discussed.     

Modeling preasymptotic transport in flows with significant inertial and trapping effects – The importance of velocity correlations and a spatial Markov model

We study solute transport in a periodic channel with a sinusoidal wavy boundary when inertial flow effects are sufficiently large to be important, but do not give rise to turbulence. This configuration and setup are known to result in large recirculation zones that can act as traps for solutes; these traps can significantly affect dispersion of the solute as it moves through the domain. Previous studies have considered the effect of inertia on asymptotic dispersion in such geometries. Here we develop an effective spatial Markov model that aims to describe transport all the way from preasymptotic to asymptotic times. In particular we demonstrate that correlation effects must be included in such an effective model when Péclet numbers are larger than O(100) in order to reliably predict observed breakthrough curves and the temporal evolution of second centered moments. For smaller Péclet numbers correlation effects, while present, are weak and do not appear to play a significant role. For many systems of practical interest, if Reynolds numbers are large, it may be typical that Péclet numbers are large also given that Schmidt numbers for typical fluids and solutes can vary between 1 and 500. This suggests that when Reynolds numbers are large, any effective theories of transport should incorporate correlation as part of the upscaling procedure, which many conventional approaches currently do not do. We define a novel parameter to quantify the importance of this correlation. Next, using the theory of CTRWs we explain a to date unexplained phenomenon of why dispersion coefficients for a fixed Péclet number increase with increasing Reynolds number, but saturate above a certain value. Finally we also demonstrate that effective preasymptotic models that do not adequately account for velocity correlations will also not predict asymptotic dispersion coefficients correctly.     

A note on Stokes production of turbulence kinetic energy in the oceanic mixed layer: observations in the Baltic Sea

Shear- and convection-driven turbulence coexists with wind-generated surface gravity waves in the upper ocean. The turbulent Reynolds stresses in the oceanic mixed layer can therefore interact with the shear of the wave-generated Stokes drift velocity to extract energy from the surface waves and inject it into turbulence, thus augmenting the mean shear-driven turbulence. Stokes production of turbulence kinetic energy (TKE) is difficult to measure in the field, since it requires simultaneous measurement of the turbulent stress and the Stokes drift profiles in the water column. However, it is readily inferred using second moment closure models of the oceanic mixed layer provided: (1) wave properties are available, along with the usual water mass properties, and radiative and air–sea fluxes needed to drive the mixed layer model and (2) the model skill can be assessed by comparing the model results against the observed dissipation rates of TKE. Comprehensive measurements made during the Reynolds 2002 campaign in the Baltic Sea have made the estimation of Stokes production possible, and in this paper, we report on the effort and the conclusions reached. Measurements of air–sea exchange parameters and water mass properties during the campaign allowed a mixed layer model to be run and the turbulent stress in the water column to be inferred. Simultaneous wave spectrum measurements enabled Stokes drift profile to be deduced and wave breaking to be included in the model run, and the Stokes production of TKE in the water column estimated. Direct measurements of the TKE dissipation rate from an upward traversing microstructure profiler were used to assure that the model could reproduce the turbulent dissipation rate in the water column. The model results indicate that the Stokes production of TKE in the mixed layer is of the same order of magnitude as the shear production and must therefore be included in mixed layer models.     

Wake effects on turbulent transport in the convective boundary layer

The effect of the downstream propagation of a wake on the transport of momentum, energy and scalars (such as humidity) in the convective boundary layer (CBL) is studied using a direct numerical simulation. The incompressible Navier–Stokes and energy equations are integrated under neutral and unstable thermal stratification conditions in a rotating coordinate frame with the Ekman layer approximation. Wake effects are introduced by modifying the mean velocity field as an initial condition on a converged turbulent Ekman layer flow. With this initial velocity distribution, the governing equations are integrated in time to determine how turbulent transport in the CBL is affected by the wake. Through the use of Taylor’s hypothesis, temporal evolution of the flow field in a doubly periodic computational domain is transformed into a spatial evolution. The results clearly indicate an increase in the scalar flux at the surface for the neutrally stratified case. An increase in wall scalar and heat flux is also noted for the CBL under unstable stratification, though the effects are diminished given the enhanced buoyant mixing associated with the hot wall.     

Experimental investigation of stratorotational instability using a thermally stratified system: instability,waves and associated momentum flux

Stratorotational instability (SRI) has been proposed as a mechanism for outward angular momentum transport in Keplerian accretion disks. A particular designed Taylor–Couette laboratory experiment with axial stratification is suitable for studying the instability. Bottom endplate is cooled and top endplate is heated to achieve axial stratification. Due to constructive constraints, endplates are visually unamenable and quantitative measurement techniques in the co-rotating frame can only be done by looking through the outer cylinder. For this purpose, we built a co-rotating mini-PIV (Particle Image Velocimetry) system with a camera having a tilted viewing angle regarding the horizontal laser sheet. The aim of this study is (i) to quantify the uncertainty of the mini-PIV together with the used calibration technique and (ii) to compare experimental findings on SRI with theoretical predictions.

We perform measurements of the azimuthal and radial component of the velocity in axial stably stratified Taylor–Couette flows, consider velocity profiles and do frequency-filtering and flow decomposition. The absolute error of the mini-PIV system is 2% and we realised that stratified Taylor–Couette flows have smaller Ekman endwall effects than homogeneous ones. Still, Ekman pumping has an impact of the flow and might be responsible for differences between the data and theoretical models ignoring the endwalls. Here we focus on the flow structure during transition to SRI, the drift rate of SRI modes and the radial momentum flux as a function of the Reynolds number. Whereas the structure in form of trapped boundary Kelvin modes and the drift rate corresponds well with earlier predictions, the momentum flux shows a nonlinear dependency with respect to the Reynolds number. Away from the region of transition, theoretical models show a linear relationship. Several possible reasons for the mismatch between the experimental and theoretical models are discussed. Most important, we experimentally demonstrated that in the Rayleigh stable flow regime the SRI can provide a significant amount of outward momentum flux which makes this instability interesting in the context of accretion disks and also of atmospheric vortices where rotation and stratification also play a significant role.     

Modeling the effect of saltation on surface layer turbulence

Particle–turbulence interaction has been a research focus in the field of pneumatic transport, especially in aeolian environments. However, knowledge regarding the effect of saltating particles on the turbulence characteristics is very limited. In this article, a process of sand-laden flow from forming sand streamers to stability is investigated via a coupled mathematical model of wind-blown sand that includes the spatiotemporal development. The variations in the turbulence characteristics, such as the mean velocity and turbulence intensity in clean air or sand-laden flow field, are analyzed. The results show that the splash process of sand grains near the wall decrease the wind speed in the saltation layer and destroy the low-speed streaks. Moreover, the profiles of streamwise turbulence intensity exhibit a transition from ‘decreasing’ to ‘increasing’ and approximately intersect at an ‘intensity focus’, which is presented for the first time. Furthermore, it was found that saltating particles could enhance the Reynolds stress. Meanwhile, it was also noticed that the shear stress at the wall surface is greater than the impact threshold and that there is a tendency towards the impact threshold. Therefore, saltation makes the particle Reynolds number of sand-laden flow higher than that under non-saltation conditions, thus changing the particles’ effect on the turbulence intensity. Gravity-dominated saltation is probably the most essential difference between wind-blown sand and other traditional two-phase flows. © 2020 John Wiley & Sons, Ltd.     

Flow over a rounded backward-facing step,using a <Emphasis Type="Italic">z</Emphasis>-coordinate model and a <Emphasis Type="Italic">σ</Emphasis>-coordinate model

Homogeneous, nonrotating flow over a backward-facing rounded step is simulated using the 2D vertical version of two general circulation models, a z-coordinate model—the Massachusetts Institute of Technology general circulation model (MITgcm)—and a σ-coordinate model—the Bergen Ocean Model (BOM). The backward-facing step is a well-known testcase since it is geometrically simple but still embodies important flow characteristics such as separation point, reattachment length, and recirculation of the flow. The study compares the core of the two models and uses constant eddy viscosities and diffusivities. The Reynolds numbers ranges from 2·10² to 2·10⁶. The results correspond with previously published results having a relatively stationary separation point and a fluctuating reattachment length due to downslope propagating eddies released from the reattachment zone for Reynolds numbers higher than or equal to 2 · 10⁴. For Reynolds number within the laminar regime, the flow is stationary. The discrepancies between the models increase by enhancing Reynolds numbers. The σ-coordinate model experiences a reduction in eddy sizes with increasing resolution and Reynolds numbers in correspondence with published experiments, while the size of the eddies are independent of the Reynolds number using the MITgcm. Due to mixing generated by the staircase topography, the z-coordinate model gives a better convergence of the separation point and reattachment length compared with the BOM; however, this conclusion might change with the inclusion of a relevant turbulence scheme.     

Geophysical flows with anisotropic turbulence and dispersive waves: flows with stable stratification

The quasi-normal scale elimination (QNSE) is an analytical spectral theory of turbulence based upon a successive ensemble averaging of the velocity and temperature modes over the smallest scales of motion and calculating corresponding eddy viscosity and eddy diffusivity. By extending the process of successive ensemble averaging to the turbulence macroscale one eliminates all fluctuating scales and arrives at models analogous to the conventional Reynolds stress closures. The scale dependency embedded in the QNSE method reflects contributions from different processes on different scales. Two of the most important processes in stably stratified turbulence, internal wave propagation and flow anisotropization, are explicitly accounted for in the QNSE formalism. For relatively weak stratification, the theory becomes amenable to analytical processing revealing just how increasing stratification modifies the flow field via growing anisotropy and gravity wave radiation. The QNSE theory yields the dispersion relation for internal waves in the presence of turbulence and provides a theoretical reasoning for the Gargett et al. (J Phys Oceanogr 11:1258–1271, 1981) scaling of the vertical shear spectrum. In addition, it shows that the internal wave breaking and flow anisotropization void the notion of the critical Richardson number at which turbulence is fully suppressed. The isopycnal and diapycnal viscosities and diffusivities can be expressed in the form of the Richardson diffusion laws thus providing a theoretical framework for the Okubo dispersion diagrams. Transitions in the spectral slopes can be associated with the turbulence- and wave-dominated ranges and have direct implications for the transport processes. We show that only quasi-isotropic, turbulence-dominated scales contribute to the diapycnal diffusivity. On larger, buoyancy dominated scales, the diapycnal diffusivity becomes scale independent. This result underscores the well-known fact that waves can only transfer momentum but not a scalar and sheds a new light upon the Ellison–Britter–Osborn mixing model. It also provides a general framework for separation of the effects of turbulence and waves even if they act on the same spatial and temporal scales. The QNSE theory-based turbulence models have been tested in various applications and demonstrated reliable performance. It is suggested that these models present a viable alternative to conventional Reynolds stress closures.     

A 3-D coastal tidal rectification process: observations, theory and experiments

In homogeneous rotating fluid, when there is an oscillating forcing in the interior fluid with a period long enough for an Ekman layer to develop, there is an interaction between the oscillatory Ekman layer and the vertical wall, since the latter imposes an alternating adjustment flow confined near the wall. As a result, this coastal rectification process leads to a Lagrangian transport along the coast. The Ekman number, the Rossby number and the temporal Rossby number of the forcing flow are the governing parameters of that mechanism which can be described by a simplified analytical model taking into account both the vertical time-dependent structure of the current and the presence of the wall. The model shows that the residual (rectified) current flowing with the coast to its right results from the strong nonlinear interaction between along- and cross-shore tidal currents leading to asymmetrical momentum exchanges between the Ekman bottom layer and the coastal boundary layer. The model provides simple scaling laws for the maximum intensity and width of the residual current. The latter is significantly larger than the friction (Stokes) lateral boundary layer of the forcing flow. A comprehensive set of experiments is performed in the 13 m diameter rotating tank by oscillating an 8 m×2 m horizontal plate and vertical wall in a homogeneous fluid at rest in solid-body rotation and measuring the two horizontal components of the current at several locations and depths above the central part of the plate. The predicted and experimentally measured maximum intensity and width of the residual current are in very good agreement, within the range of validity of the model, i.e. when the Ekman number is sufficiently small. However experiments also show that the residual current still occurs when the Ekman layer thickness is of the same order as the fluid depth, but it is then confined to a narrower band along the vertical wall. The flow structure found experimentally is also correctly described by a numerical model developed by Zhang et al. (1994). Current measurements in the Eastern part of the English Channel near the French coast reveal a significant coastal residual current flowing Northward and the coastal rectification process described here may account for part of it.     

Computational modeling of flow and sediment transport and deposition in meandering rivers

A computational modeling analysis of the flow and sediment transport, and deposition in meandering-river models was performed. The Reynolds stress transport model of the FLUENT^TM code was used for evaluating the river flow characteristics, including the mean velocity field and the Reynolds stress components. The simulation results were compared with the available experimental data of the river model and discussed. The Lagrangian tracking of individual particles was performed, and the transport and deposition of particles of various sizes in the meandering river were analyzed. Particular attention was given to the sedimentation patterns of different size particles in the river-bend model. The flow patterns in a physical river were also studied. A Froude number based scale ratio of 1:100 was used, and the flow patterns in the physical and river models are compared. The result shows that the mean-flow quantities exhibit dynamic similarity, but the turbulence parameters of the physical river are different from the model. More strikingly, the particle sedimentation features in the physical and river models do not obey the expected similarity scaling.     

An analytical model for ice-edge upwelling

Abstract

This paper presents an analytical, two-dimensional model of the wind-induced homogeneous circulation near the edge of an ice pack floating on the ocean surface. It is shown that a vertical shear layer arises under the ice edge, by which the wind-driven geostrophic motion in the open ocean is matched to the flow region underneath the ice. As in coastal upwelling models, this shear layer consists of a thin E ^1/2-layer inside a thicker E ^1/4-layer (E being the Ekman number). Under certain conditions the shear layer produces a vertical mass flux from the bottom to the surface Ekman layer. Near the surface this upwelling flux is concentrated in the narrow E ^1/2-layer. Comparison with observations of upwelling at the edge of a polar ice pack shows good agreement.     

Three-layer model for vertical velocity distribution in open channel flow with submerged rigid vegetation

Based on the detailed laboratory experiments and theoretical analysis, a new three-layer model is proposed to predict the vertical velocity distribution in an open channel flow with submerged vegetation. The time averaged velocity and turbulence behaviour of a steady uniform flow with fully submerged artificial rigid vegetation was measured using a 3D Micro ADV, and the vertical distribution of velocity and Reynolds shear stress at different vegetation height, vegetation density and measuring positions were obtained. The results show that the velocity profile consists of three hydrodynamic regimes (i.e. the upper non-vegetated layer, the outer and bottom layer within vegetation); accordingly different methods had been adopted to describe the vertical velocity distribution. For the upper non-vegetated layer, a modified mixing length theory combined with the concept of ‘the new vegetation boundary layer’ was adopted, and an analytical model was presented to predict the vertical velocity distribution in this region. For the bottom layer within vegetation, the depth average velocity was obtained by numerically solving the momentum equations. For the upper layer within vegetation, the analytical solution was presented by expressing the shear stress as a formula fitted to the experimental data. Finally, the analytical predictions of the vertical velocity over the whole flow depth were compared with the results obtained by other researchers, and the good agreement proved that the three-layer model can be used to predict the velocity distribution of the open channel flow with submerged rigid vegetation.     

Turbulence fields in the lee of two‐dimensional transverse dunes simulated in a wind tunnel

The turbulence field of airflow in the lee of a dune has significant impacts on dune dynamics and related processes. We used particle image velocimetry in a wind tunnel simulation to obtain detailed velocity measurements in the lee of two‐dimensional transverse dune models, then used the results to analyse their turbulence fields. The dune models used in this study had a single lee angle of 30°, and a total of six stoss angles: 3°, 5°, 10°, 15°, 20° and 25°. We used vorticity, turbulence intensity, Reynolds stress and turbulent kinetic energy to characterize the turbulence fields. These parameters were functions of stoss angle, wind velocity, distance from the dune crest and height above the ground surface. The stoss angles could generally be divided into two groups based on the profiles of mean velocity, turbulence and Reynolds stress. Stoss angles of 3° and 5° usually had similar profiles, and angles of 15°, 20° and 25° formed a second group with similar profiles. The profiles for the stoss angle of 10° were usually transitional and were intermediate between the two groups. Vorticity, Reynolds stress and turbulent kinetic energy increased monotonically with increasing free‐stream wind velocity, but their variations with respect to the stoss angle were complex. The stoss angles of 15° and 20° had the maximum values of these three parameters, thus these angles may have special significance in dune development given the characteristics of the mean velocity fields and turbulence fields they produce within the lee airflow. It is the streamwise velocity component and its turbulence that determine the surface shear stress. Copyright © 2008 John Wiley and Sons, Ltd.     

Double-averaging methodology and its application to turbulent flow in and above vegetation canopies

Double averaged equations for atmospheric boundary layer flows are introduced as natural extensions of single averaged Reynolds equations. We show that in circumstances where double averaged equations are needed, the two fundamental properties of Reynolds averaging are violated. First, we consider double-averaging in free air turbulence, where the aim is to separate coherent motions from background turbulence. We illustrate the different properties of the main operators that have been used and the physical meaning of the terms that result. Second, in canopy flows, the multiply connected nature of the canopy airspace leads to a different set of departures from the standard Reynolds equations. We establish the physical meaning of the extra terms that arise. Finally we briefly discuss the problems, both practical and theoretical, that arise when we use double averaged equations to interpret real data.