Experiments on entrainment in an annulus with and without velocity gradient across the density interface

The entrainment process in a two layer density stratified fluid column was studied experimentally by imposing external shear stress on one or both layers. The experiments have been conducted in an annular tank containing two water layers of different salt concentration and the shear stress was applied by means of rotating screens. The following quantities were measured: the screen velocity (which was kept constant during each experiment), the stress at the upper screen, and vertical profiles of circumferential velocity and density at different radial locations. When equal stress was imposed at the surface of the upper layer and at the bottom of the lower layer, entrainment took place from the two sides of the density interface at equal rate so that the interface was stationary in the central position between the two screens and there was no velocity gradient across the interface. The dependence of the entrainment coefficient on Richardson number obtained in these experiments was similar in form to that obtained in the shear-free experiments with an oscillating grid (e.g. Nokes 1988). When a shear stress was applied at the upper surface only, the upper layer depth increased with time and a velocity gradient existed at the interface. The influence of the interfacial velocity gradient on the entrainment rate was studied by comparing the rates obtained with and without this velocity gradient. The entrainment rates were approximately the same for high values of the Richardson number while at low Richardson number the entrainment rate was much larger when a velocity gradient existed across the interface. The main results of this work are as follows:

1. Despite the curved geometry of the annular system, the dependence of the entrainment coefficient on Richardson number for shear-free interface experiments was found to be similar in form to that obtained for oscillating grid experiments.
2. The entrainment across the interface is due to turbulent energy generated at some distance from the interface by an external source (i.e. shear stress induced by a screen) and due to turbulence produced locally at the interface by a velocity gradient. The relative contribution of each turbulence source to the total entrainment was found to depend on the stability of the interface.

     

稳定分层流密度界面处湍流混合与分形结构

采用室内混合箱研究稳定分层流(上层淡水、下层盐水) 无剪切密度界面处的湍流混合与分形结构. 湍流通过浸没在盐水层中的振动格栅产生, 密度界面结构通过在盐水层中添加荧光剂或染料可视化, 共进行12 组实验. 实验观测并记录了:(1) 淡、盐水密度界面距混合箱底部的平均高程(h);(2) 淡、盐水层的密度(ρ₀,ρ), (3) 淡、盐水密度界面. 其中, 淡、盐水密度界面通过照片、录像进行记录. 观测结果用于计算:(1) 盐水层密度;(2) 卷挟速度, (3) 整体理查孙数(Ri_o), (4) 二维、三维密度界面, (5) 二维、三维密度界面的分形维度. 结果分析发现:(1) 湍流卷挟率随Rio 增大而减小, 并且满足Ri_o的-3=2 或-7=4 幂律;表明随着湍流强度的减弱, 混合的速度也越来越缓慢;(2) 二维密度界面分形维度大于1, 三维密度界面分形维度大于2;表明二维、三维密度界面存在分形结构;(3) 分形维度随Ri_o的增大而减小;表明随着湍流强度的减弱, 密度界面也越来越趋于光滑.      

A Direct Numerical Study of Transport and Anisotropy in a Stably Stratified Turbulent Flow with Uniform Horizontal Shear

Direct numerical simulations of homogeneous turbulence in stably stratified shear flow have been performed to aid the understanding of turbulence and turbulent mixing in geophysical flow. Two cases are compared. In the first case, which has been studied in the past, the mean velocity has vertical shear and the mean density is vertically stably stratified. In the second case, which has not been studied systematically before, the mean velocity has horizontal shear and the mean density is again vertically stably stratified. The critical value of the gradient Richardson number, for which a constant turbulence level is obtained, is found to be an order of magnitude larger in the horizontal shear case. The turbulent transport coefficients of momentum and vertical mass transfer are also an order of magnitude larger in the horizontal shear case. The anisotropy of the turbulence intensities are found to be in the range expected of flows with mean shear with no major qualitative change in the range of Richardson numbers studied here. However, the anisotropy of the turbulent dissipation rate is strongly affected by stratification with the vertical component dominating the others. This revised version was published online in July 2006 with corrections to the Cover Date.     

An investigation of thermally stratified turbulent channel flow with temperature oscillation on the bottom wall by large eddy simulation

Thermally stratified shear turbulent channel flow with temperature oscillation on the bottom wall of the channel is calculated to investigate the behavior of turbulent flow and heat transfer by use of large eddy simulation (LES) approach coupled with dynamic subgrid-scale (SGS) models. The objective of this study is to deal with the effect of the temperature oscillation on turbulent behavior of thermally stratified turbulent channel flow and to examine the effectiveness of the LES technique for predicting statistically unsteady turbulent flow driven by time-varying buoyancy force. To validate the present calculation, thermally stratified shear turbulent channel flow is computed and compared with available data obtained by direct numerical simulation (DNS), which confirm that the present approach can be used to predict thermally stratified turbulent channel flow satisfactorily. Further, to illustrate the effect of the temperature oscillation with different Richardson numbers and periods of the oscillation on turbulence characteristics, the phase-averaged mean value and fluctuation of the resolved velocities and temperature, and instantaneous velocity fluctuation structures are analyzed.     

Flow field investigation in rotating rib-roughened channel by means of particle image velocimetry

The turbulent velocity field over the rib-roughened wall of an orthogonally rotating channel is investigated by means of two-dimensional particle image velocimetry (PIV). The flow direction is outward, with a bulk Reynolds number of 1.5 × 10⁴ and a rotation number ranging from 0.3 to 0.38. The measurements are obtained along the wall-normal/streamwise plane at mid-span. The PIV system rotates with the channel, allowing to measure directly the relative flow velocity with high spatial resolution. Coriolis forces affect the stability of the boundary layer and free shear layer. Due to the different levels of shear layer entrainment, the reattachment point is moved downstream (upstream) under stabilizing (destabilizing) rotation, with respect to the stationary case. Further increase in rotation number pushes further the reattachment point in stabilizing rotation, but does not change the recirculation length in destabilizing rotation. Turbulent activity is inhibited along the leading wall, both in the boundary layer and in the separated shear layer; the opposite is true along the trailing wall. Coriolis forces affect indirectly the production of turbulent kinetic energy via the Reynolds shear stresses and the mean shear. Two-point correlation is used to characterize the coherent motion of the separated shear layer. Destabilizing rotation is found to promote large-scale coherent motions and accordingly leads to larger integral length scales; on the other hand, the spanwise vortices created in the separating shear layer downstream of the rib are less organized and tend to be disrupted by the three-dimensional turbulence promoted by the rotation. The latter observation is consistent with the distributions of span-wise vortices detected in instantaneous flow realizations.     

Influence of a strong density stratification on the entrainment of a turbulent axisymmetric plume

The effect of entrainment and the role of the interface during the interaction between an axisymmetric turbulent mass plume and a strong stratified layer are investigated. We describe mainly the characteristics of the plume: the change in the profiles of the density, the horizontal component of the velocity and the corresponding intensity of turbulence, the change in the entrainment co-efficient, when the plume goes through the impingement interface, assuming a self-similar Gaussian property of the axial velocity component and of the density difference. The influence of the stratification on the plume angle coefficient is studied, and compared with the results related to a homogeneous environment, obtained elsewhere. Experimental correlation on the mean entrainment coefficient in a given plume cross-section, is formulated.     

Shear flow induced interface instability

 A system of two stratified layers at a free surface, consisting of distilled water above a layer of salty water separated by an interface, is studied under laboratory conditions involving uniform temperature heating from below. Shadowgraph and particle images have been used with temperature and salt concentration measurements to investigate the interface instability induced by convection when it is developing in the upper and lower layer. It is found that the interface is governed by local shear flow that induces a Kelvin–Helmholtz instability. Moreover, the entrainment interface is subject to a combination of two closely related effects: (1) double diffusion and convective motion and (2) double diffusion and Kelvin–Helmholtz instability. Received: 22 December 1999/Accepted: 31 October 2000     

Turbulence Budgets in Buoyancy-affected Vertical Backward-facing Step Flow at Low Prandtl Number

The paper investigates buoyancy impact on the vertical flow over a backward-facing step at low Prandtl number by Direct Numerical Simulation. In particular, the very low Prandtl number of liquid sodium, 0.0088, is considered in the regime of mixed convection, i.e. for Richardson numbers below unity. The effects of buoyancy on mean flow, heat transfer and turbulence are assessed. Buoyancy is found to attenuate recirculation and, consequently, increase heat transfer. Turbulence is decreased in the attached boundary layer for moderate buoyancy impact but surpasses the levels found in forced convection at the largest Richardson number investigated. Beyond the mean flow and second moments, the budgets of turbulent kinetic energy, Reynolds shear stress, temperature variance, and turbulent heat flux components are studied and related to the alterations in the mean field quantities. Due to scale separation, production and dissipation nearly balance for temperature variance while this is not the case for turbulent kinetic energy. Similar findings for the turbulent heat fluxes show that the correlation between temperature and pressure gradient is the most important contribution to the budget aside from production and dissipation. In addition to the physical insight into this flow, the data presented may be used for the validation and improvement of turbulence models for liquid metal flows.     

Turbulent diffusion of a buoyant layer at a wall

Summary When a light fluid is injected at a steady rate at the roof of a tunnel in which there is a turbulent main flow of a heavier fluid, the turbulent diffusion of the light layer may be considerably reduced due to buoyancy. For large Richardson numbers turbulent mixing ceases altogether.The equations of motion and diffusion were solved by introducing an eddy diffusivity which is dependant on the Richardson number. Experiments were made on brine (floor) layers in a water flow, and on methane (roof) layers in an air flow. Results were essentially in agreement with theory.The motion and mixing of the layers depend mainly on the inclination of the tunnel and on a dimensionless combination of main-flow velocity, gravity, relative density difference, volume input rate of layer fluid, and tunnel width. Values of the dimensionless parameter are suggested to overcome the effects of buoyancy on mixing, and to prevent layers from moving up a slope against the main flow.     

Laboratory observations of interactions of forced plumes with stratified shear layers

Plumes of fluid are often observed in nature to interact with stratified shear layers. Examples of this include chimney plumes hitting inversion-layer ceilings; sewage plumes impinging on unmixed fresh/saltwater interfaces; descending plumes of cold water formed at ice-leads interacting with the oceanic thermocline; and volcano plumes interacting with atmospheric interfaces. Controlled laboratory studies of these phenomena have not previously been described in the literature, and as a result there is a lack of understanding regarding their morphology and dynamics. Thus, a novel set of experiments is described here in which the behaviour of a turbulent plume is observed in the presence of a two-layer ambient. The lower layer, into which the plume initially emerges, is quiescent and at a relatively high density. The upper layer is forced to flow uniformly across the top of the lower layer, and has a lower density. The flow of the resulting plume is characterised by (a) its vertical and lateral spreading in the lower layer; (b) the nature of its extension upstream and downstream at the interface; and (c) the extent to which it penetrates into the upper layer. The behaviour is found to be governed by three non-dimensional parameters: the initial gradient Richardson number of the interface Ri_G, the ratio of the upper layer crossflow speed to the speed of the plume when it first impinges on the interface U_F/U_PI, and the ratio of the plume Monin–Obukhov lengthscale to the lower layer depth L_MO/H_L. Regime diagrams are presented showing the effects of changing these parameters on the plume flow, quantitative relationships are determined, and practical applications of the results are considered.     

Modeling Of A Stable Boundary Layer With A Low-Level Jet

An improved three-parameter model of turbulence is applied to study the characteristics of a flow of a stably stratified boundary layer with formation of a jet in its lower part. The results obtained show that the turbulent Prandtl number increases with increasing thermal stability, which agrees with the results of laboratory experiments and actual observations in the atmosphere. It is also shown that the gradient Richardson number is nonmonotonic in time in the range below the maximum jet velocity.     

The flow structure in the front of a moving surface layer

Summary Behind the frontal wedge of a moving surface layer there is a zone of entrainment in which the flow is turbulent due to the onset of a Kelvin-Helmholtz instability. The entrainment coefficient is inversely proportional to the Richardson number. In the turbulent region the gradient Richardson number approached a constant value less than 1/4. The layer thickness and the mean velocity depend on the salt concentration of the underlying fluid. Finally the turbulence decayed leaving a stable interface and the layer behaved much like a free homogeneous flow over a solid boundary.

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Übersicht Hinter der keilförmigen Front einer fließenden Schicht gibt es eine Mischungszone, in der die Strömung turbulent ist. Die Turbulenz entsteht durch eine Kelvin-Helmholtz Instabilität. Der Ansaugbeiwert der turbulenten Strömung verhält sich umgekehrt zur Richardsonschen Zahl. In der turbulenten Strömung nähert sich die Richardsonsche Zahl einer konstanten Zahl kleiner als 1/4. Die Dicke und die mittlere Geschwindigkeit der Schicht hängt sehr von der Dichte der unteren Flüssigkeit ab. Schließlich stirbt die turbulente Strömung ab, und die Schicht fließt wie eine homogene Strömung längs einer ebenen Platte.

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The work described herein was carried out as part of a research programme of the Hydraulics Research Station, and the paper is published by permission of the Director of Hydraulics Research. The author wishes also to express his thanks to Mr. J. A. Weller for his help in measurement, and to a referee for his helpful comments.     

Experiments on the spreading of shear-free turbulence under the influence of confinement and rotation

From Lie-group (symmetry) analysis of the multi-point correlation equation Oberlack and Günther (Fluid Dyn Res 33:453–476, 2003) found three different solutions for the behavior of shear-free turbulence: (i) a diffusion like solution, in which turbulence diffuses freely into the adjacent calm fluid, (ii) a deceleration wave like solution when there is an upper bound for the integral length scale and (iii) a finite domain solution for the case when rotation is applied to the system. This paper deals with the experimental validation of the theory. We use an oscillating grid to generate turbulence in a water tank and Particle Image Velocimetry (PIV) to determine the two-dimensional velocity and out-of-plane vorticity components. The whole setup is placed on a rotating table. After the forcing is initiated, a turbulent layer develops which is separated from the initially irrotational fluid by a sharp interface, the so-called turbulent/non-turbulent interface (TNTI). The turbulent region grows in time through entrainment of surrounding fluid. We measure the propagation of the TNTI and find quantitative agreement with the predicted spreading laws for case one and two. For case three (system rotation), we observe that there is a sharp transition between a 3D turbulent flow close to the source of energy and a more 2D-like wavy flow further away. We measure that the separation depth becomes constant and in this sense, we confirm the theoretical finite domain solution.     

A von Neumann–Smagorinsky turbulent transport model for stratified shear flows

A simple subgrid turbulent diffusion model based on an analogy to the von Neumann–Richtmyer artificial viscosity is explored for use in modelling mixing in turbulent stratified shear flow. The model may be more generally applicable to multicomponent turbulent hydrodynamics and to subgrid turbulent transport of momentum, composition and energy. As in the case of the von Neumann artificial viscosity and many subgrid-scale models for large-eddy simulation, the turbulent diffusivity explicitly depends on the grid size and is not based on a quantitative model of the unresolved turbulence. In order to address the issue that it is often not known a priori when and where a flow will become turbulent, the turbulent diffusivity is set to zero when the flow is expected to be stable on the basis of a Richardson/Rayleigh–Taylor stability criterion, in analogy to setting the von Neumann artificial viscosity to zero in expanding flows. One-dimensional predictions of this model applied to a simple shear flow configuration are compared to those obtained using a K–ε model. The density and velocity profiles predicted by both models are shown to be very similar.     

The turbulent/non-turbulent interface in an adverse pressure gradient turbulent boundary layer

The turbulent/non-turbulent interface (TNTI) in an adverse pressure gradient (APG, β = 1.45) turbulent boundary layer (TBL) is explored here by using direct numerical simulation (DNS) data; β is the Clauser pressure gradient parameter. For comparison, the DNS data for a zero pressure gradient (ZPG) TBL is included. The interface is extracted with an approach based on enstrophy criteria. Depending on the enstrophy, the outer boundary layer flow can be classified into the free stream, boundary layer wake, and intermittent flow regimes. The fractal dimension of the interface is obtained by using the box-counting algorithm, and was found to be constant over a long range of box sizes. The TNTI shows a monofractal behavior. The geometric complexity of a TNTI can be determined in terms of the genus, which is defined as the number of handles in a geometric object. We examine the volume and projection area of the genus of the TNTI to analyze the entrainment process. The geometric complexity of the APG TBL interface and the local entrainment are greater than those of the ZPG TBL, as is evident in the increases in the genus near the interface. The local entrainment velocity is dominantly affected by the viscous diffusion at the interface.     

Experiments to study interactions between baroclinic lower flows and a stably stratified upper layer

Experiments were conducted on a rotating fluid annulus to study the basic interactions between baroclinic lower flows and a stably stratified upper layer. Sufficiently stable stratification is necessary for steady flows to emerge in the lower layer. Upward fluid motions make the baroclinic flows permeate into the upper layer. The stable stratification, however, suppresses upward motions so that zonal fluid velocities decrease with height. In fact, their maximum appears at the top level of the baroclinic lower layer and the sign of the radial temperature gradient changes there; namely, it is warmer on the inner side of the annulus in the upper layer. This temperature profile is reflected in a meridional fluid circulation mixing both layers. In the upper layer of the wave flow, there exists a critical level below and above which the zonal fluid velocities have opposite directions for the wave to have a phase shift of half a wavelength in appearance. The experimental results correspond to real atmospheric phenomena.     

速度分布对分层流中底部边界混合层结构的影响

对分层流中海底边界混合层发展问题提出了一个新的垂直积分模型。在这个模型里，新建立的能量方程中出现了一个表征速度大小随高度变化规律的参数，从而得出了被Ｐｏｌｌａｒｄ等人文章中应用的但被本文修正过的Ｒｉｃｈａｎｄｓｏｎ数封闭模型，对在水平乎底上的混合层成长问题，给出了分析解。从而把以往的公式都统一到本文的结果中，而且给出了以往海底边界混合层公式的理论依据与适用范围，这个结果与Ｗｅａｔｈｅｒｌｙ与Ｍａｒｔｉｎ（１９７８）用Ｍｅｌｌｏｒ与Ｙａｍａｄａ（１９７４）的高阶模型所得的数值解结果是基本一致的，并与现场实验相符。     

Turbulent breaking of overturning gravity waves below a critical level

The interaction of an internal gravity wave with its evolving critical layer and the subsequent generation of turbulence by overturning waves are studied by three-dimensional numerical simulations. The simulation describes the flow of a stably stratified Boussinesq fluid between a bottom wavy surface and a top flat surface, both without friction and adiabatic. The amplitude of the surface wave amounts to about 0.03 of the layer depth. The horizontal flow velocity is negative near the lower surface, positive near the top surface with uniform shear and zero mean value. The bulk Richardson number is one. The flow over the wavy surface induces a standing gravity wave causing a critical layer at mid altitude. After a successful comparison of a two-dimensional version of the model with experimental observations (Thorpe [21]), results obtained with two different models of viscosity are discussed: a direct numerical simulation (DNS) with constant viscosity and a large-eddy simulation (LES) where the subgrid scales are modelled by a stability-dependent first-order closure. Both simulations are similar in the build-up of a primary overturning roll and show the expected early stage of the interaction between wave and critical level. Afterwards, the flows become nonlinear and evolve differently in both cases: the flow structure in the DNS consists of coherent smaller-scale secondary rolls with increasing vertical depth. On the other hand, in the LES the convectively unstable primary roll collapses into three-dimensional turbulence. The results show that convectively overturning regions are always formed but the details of breaking and the resulting structure of the mixed layer depend on the effective Reynolds number of the flow. With sufficient viscous damping, three-dimensional turbulent convective instabilities are more easily suppressed than two-dimensional laminar overturning.     

The effect of sidewalls on rectangular jets

An experimental study is presented regarding the influence of sidewalls on the turbulent free jet flow issuing from a smoothly contracting rectangular nozzle of aspect ratio 15. “Sidewalls” are two parallel plates, flush with each of the slots’ short sides, practically establishing bounding walls extending the nozzle sidewalls in the downstream direction. Measurements of the streamwise and lateral velocity mean and turbulent characteristics have been accomplished, with an x-sensor hot wire anemometer, up to an axial distance of 35 nozzle widths, for jets with identical inlet conditions with and without sidewalls. Centreline measurements for both configurations have been collected for three Reynolds numbers, Re_D = 10,000, 20,000 and 30,000. For Re_D = 20,000 measurements in the transverse direction were collected at 13 different downstream locations in the range, x = 0–35 nozzle widths, and in the spanwise direction at three different downstream locations, x = 2, 6 and 25 nozzle widths.Results indicate that, the two jet configurations (with and without sidewalls) produce statistically different flow fields. Sidewalls do not lead to the production of a 2D flow field as undulations in the spanwise mean velocity distribution indicate. They do increase the two-dimensionality of the jet increasing the longevity of 2D spanwise rollers structures formed in the initial stages of entrainment, which are responsible for the convection of longitudinal momentum towards the outer field, establishing larger streamwise mean velocities at the jet edges. In the near field, up to 25 nozzle widths, lower outward lateral velocities in the presence of the sidewalls are held responsible for the decrease of turbulent terms including rms of velocity fluctuations and Reynolds stresses. Skewness factors increase monotonically across the shear layers from negative values to positive forming sharp peaks at the outer edges of the jet, illustrative of the presence of well defined 2D roller structures in the jet with sidewalls.     

Some properties of multi-fluid stokes flows

The solution of Stokes' equations for a rotating axisymmetric body which possesses reflection symmetry about a planar interface between two infinite immiscible quiescent viscous fluids is shown to be independent of the viscosities of the fluids and identical with the solution when the fluids have the same viscosity. The result is generalized to a rotating axisymmetric system of bodies which possesses reflection symmetry about each interface of a plane stratified system of fluids. An analogous result for two-fluid systems with a nonplanar static interface is also derived. The effect on torque reduction produced by the presence of a second fluid layer adjacent to a rotating axisymmetric body is considered and explicit calculations are given for the case of a sphere. A proof of uniqueness for unbounded multi-fluid Stokes' flow is given and the asymptotic far field structure of the velocity field is determined for axisymmetric flow caused by the rotation of axisymmetric bodies.