Linear Dynamics of Perturbations in Flows with Constant Horizontal Shear

The dynamics of perturbations in shallow water and incompressible stratified fluid flows with constant horizontal shear are described using the nonmodal analysis. It is shown that the shear flow perturbations can be divided into two classes on the basis of the potential vorticity: rapidly oscillating wave perturbations with zero potential vorticity and slow vortex perturbations with nonzero potential vorticity. In the cases of weak and strong shear the main features of the dynamics of wave and vortex perturbations are studied analytically (using the WKBG method) and numerically. It is shown that for large times the wave perturbation energy increases linearly, i.e., the shear flow is algebraically unstable due to the growth of rapid wave perturbations. This instability can be of importance in processes of turbulence development and surface and internal wave generation.     

Flow pattern transition,pressure gradient,hold-up predictions in gas/non-Newtonian power-law fluid stratified flow

This work focuses on gas/non-Newtonian power-law fluid stratified pipe flow. Two different theoretical approaches to obtain pressure gradient and hold-up predictions are presented: the steady fully developed two-fluid model and the pre-integrated model. The theoretical predictions are compared with experimental data available for horizontal and for slightly downward inclined air/shear thinning fluid stratified flow taken from literature. The predictions of the pre-integrated model are validated showing a good agreement when compared with experimental data. The criteria for the transition from the stratified flow pattern are applied to gas/non-Newtonian stratified flow. The neutral stability analysis (smooth/wavy stratified flow) and the well-posedness (existence region of stratified flow) of governing equations are carry out. The predicted transition boundaries are obtained using the steady fully developed two-fluid model and the pre-integrated model, where the shape factors and their derivatives are accounted for. A comparison between the predicted boundaries and experimental flow pattern maps is presented and shows a good agreement. A comment on the shear stress modeling by the pre-integrated model is provided.     

Wave/vortex decompositions in stably stratified flows

The slowly evolving vortex part of stably stratified flows, both homogeneous and of shear-layer type, is extracted using a diagnostic decomposition of the velocity field based upon the potential vorticity. Comparisons with a decomposition theoretically valid for the linear regime are also made. For the homogeneous flows considered here, the vortex part apparently strongly interacts with the wave field, whatever the decomposition in use. A numerical wave filtering process is applied to simulate the flow dynamics driven by the vortex part only. The resulting vortex flow is the same for each decomposition. In the shear layers, by contrast, only the decomposition based upon the potential vorticity is able to extract the vortex part of the flow, whether the shear layer is moderately or strongly stratified. We propose an argument to account for the fact that a highly energetic vortex part is more likely to be found in a strongly stratified shear layer.This work has been supported by EDF (Direction des Etudes et Recherches, Laboratoire National d'Hydraulique) under Contract No. 2J6773.     

Non-newtonian liquid-air stratified flow through horizontal tubes—ii

Non-Newtonian liquid gas stratified flow data were obtained using 0.052 and 0.025 m dia horizontal circular ducts. Unless the liquid velocity was very low, the flow pattern generally observed was non-uniform stratified flow having an interfacial level gradient between the two phases. The Heywood-Charles model is valid for predicting the pressure drop and liquid holdup in pseudoplastic (shear thinning) non-Newtonian liquid-gas uniform stratified flow. Two-phase drag reduction, which is predicted by the Heywood-Charles model did not occur because there was a transition to semi-slug flow before the model criteria were reached. Interfacial liquid and gas shear stresses were compared.     

Direct Numerical Simulation of Turbulence in a Stably Stratified Fluid and Wave-Shear Interaction

Turbulence decay in a strongly stratified medium is simulated by a direct pseudo-spectral code solving the three-dimensional equations of motion under the Boussinesq approximation. The results are compared to non-stratified simulations results. We focus on the production of mean shear energy observed in the stratified case. We then simulate the decay of stratified turbulence when affected by an initial horizontal mean flow and show that this mean flow is the major component remaining at large t. Next, we give some analytical elements on wave-shear interaction by using a simple refraction calculation with WKB hypothesis. This calculation is illustrated by simulating the interaction between one monochromatic internal wave and a vertical shear profile. We conclude that the existence of singularities in the mean shear production term in the presence of internal gravity waves may be one of the possible mechanisms involved within stratified turbulent shear flows.     

Interfacial instabilities in a stratified flow of two superposed fluids

Here we shall present a linear stability analysis of a laminar, stratified flow of two superposed fluids which are a clear liquid and a suspension of solid particles. The investigation is based upon the assumption that the concentration remains constant within the suspension layer. Even for moderate flow-rates the base-state results for a shear induced resuspension flow justify the latter assumption. The numerical solutions display the existence of two different branches that contribute to convective instability: long and short waves which coexist in a certain range of parameters. Also, a range exists where the flow is absolutely unstable. That means a convectively unstable resuspension flow can be only observed for Reynolds numbers larger than a lower, critical Reynolds number but still smaller than a second critical Reynolds number. For flow rates which give rise to a Reynolds number larger than the second critical Reynolds number, the flow is absolutely unstable. In some cases, however, there exists a third bound beyond that the flow is convectively unstable again. Experiments show the same phenomena: for small flow-rates short waves were usually observed but occasionally also the coexistence of short and long waves. These findings are qualitatively in good agreement with the linear stability analysis. Larger flow-rates in the range of the second critical Reynolds number yield strong interfacial waves with wave breaking and detached particles. In this range, the measured flow-parameters, like the resuspension height and the pressure drop are far beyond the theoretical results. Evidently, a further increase of the Reynolds number indicates the transition to a less wavy interface. Finally, the linear stability analysis also predicts interfacial waves in the case of relatively small suspension heights. These results are in accordance with measurements for ripple-type instabilities as they occur under laminar and viscous conditions for a mono-layer of particles.     

Direct numerical simulation of stable and unstable turbulent thermal boundary layers

This paper presents direct numerical simulations (DNS) of stable and unstable turbulent thermal boundary layers. Since a buoyancy-affected boundary layer is often encountered in an urban environmental space where stable and unstable stratifications exist, exploring a buoyancy-affected boundary layer is very important to know the transport phenomena of the flow in an urban space. Although actual observation may qualitatively provide the characteristics of these flows, the relevant quantitative turbulent quantities are very difficult to measure. Thus, in order to quantitatively investigate a buoyancy-affected boundary layer in detail, we have here carried out for the first time time- and space-developing DNS of slightly stable and unstable turbulent thermal boundary layers. The DNS results show the quantitative turbulent statistics and structures of stable and unstable thermal boundary layers, in which the characteristic transport phenomena of thermally stratified boundary layers are demonstrated by indicating the budgets of turbulent shear stress and turbulent heat flux. Even though the input of buoyant force is not large, the influence of buoyancy is clearly revealed in both stable and unstable turbulent boundary layers. In particular, it is found that both stable and unstable thermal stratifications caused by the weak buoyant force remarkably alter the structure of near-wall turbulence.     

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.     

Stability of stratified shear flows in channels with variable cross sections

For the instability problem of density stratified shear flows in sea straits with variable cross sections, a new semielliptical instability region is found. Furthermore, the instability of the bounded shear layer is studied in two cases: (i) the density which takes two different constant values in two layers and (ii) the density which takes three different constant values in three layers. In both cases, the dispersion relation is found to be a quartic equation in the complex phase velocity. It is found that there are two unstable modes in a range of the wave numbers in the first case, whereas there is only one unstable mode in the second case.     

Numerical simulation of the onset of slug initiation in laminar horizontal channel flow

Results are presented for the initiation of slug-type structures from stratified 2D, two-layer pressure-driven channel flow. Good agreement is obtained with an Orr–Sommerfeld-type stability analysis for the growth rate and wave speed of very small disturbances. The numerical results elucidate the non-linear evolution of the interface shape once small disturbances have grown substantially. It is shown that relatively short waves (which are the most unstable according to linear theory) saturate when the length of the periodic domain is equally short. In longer domains, coalescence of short waves of small-amplitude is shown to lead to large-amplitude long waves, which subsequently exhibit a tendency towards slug formation. The non-uniform distribution of the interfacial shear stress is shown to be a significant mechanism for wave growth in the non-linear regime.     

Shear flow disturbances due to interaction with convective rolls

The two-dimensional stationary problem of convection in a two-component stratified medium (for example, humid air) over a nonuniformly heated horizontal surface is solved in the linear approximation. In the case in which the surface temperature is harmonically dependent on the horizontal coordinate, convective rolls are formed above the surface. Depending on the background temperature stratification, situations characterized by temperature response sign reversal can occur (for example, a decrease in the temperature of the medium as a response to heating from beneath, or “negative” heat capacity of the medium). The disturbances arising in the background horizontal flow with vertical shear directed along the convective rolls are also calculated.     

The solitary waves in stratified shear flow

In this paper, the basic equation of internal long waves in stratified shear flow is derived under Boussinesq assumption, the first order approximation solution is given for solitary waves with the effects of slowly varying topograph at the sea bottom, weak stratification and basic shear flow. The Project Supported by the National Natural Science Foundation of China.     

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.     

Unstable parallel flows triggered by a fast chemical reaction

We present an experimental investigation of an instability triggered by a fast chemical reaction in a low inertia parallel flow in a small channel. Two fluids evenly injected in a straight channel react creating a strongly stratified distribution of viscosity near their interface, which destabilizes the flow. Depending on the flow rates and the aspect ratio of the flow channel, several unstable regimes are observed: mixing regime, weakly stratified regime and a stable stratified regime. In channels of 1:1 aspect ratio we find most efficient mixing (and highest flow resistance) at an intermediate pressure drops, as the flow transitions between a fully 3D regime and more stratified regime. For channels of small aspect ratio the non-monotonicity is less evident: higher pressure drops lead to increased mixing.     

The stability of two concentric non-Newtonian fluids in circular pipe flow

The flow of two concentric non-Newtonian fluids, under constant pressure gradient in a circular tube, is studied by linear stability analysis. The viscosities of the two fluids are different and their dependence on shear stress is described by the Ellis model. It is found that the steady state flow can be unstable, depending on certain combinations of the values of physical parameters, to infinitesimal axisymmetric disturbances of large wavelengths, for any Reynolds number however small. The flow is predominantly stable if the inner fluid is more viscous and predominantly unstable if the outer fluid is more viscous. Stronger dependence of viscosity on shear stress can both stabilize and destabilize the flow. Interfacial tension is also destabilizing when the Weber number is small than about 10⁴.     

Unstable capillary flow of reinforced polymer melts

The injection capillary flow of various unfilled and glass fibre or calcium carbonate filled polypropylene and nylon 6.6 melts is studied using either a single capillary of five capillaries in series, separated by small reservoirs. Only unfilled nylon 6.6 yields instability during flow through a single capillary due to mechanochemical degradation in the capillary at extremely high shear rates above 5 × 10⁵ s^?1. It is found that only short glass fibre reinforced polypropylene yields high frequency oscillations in the reservoir pressure and extrudate diameter and has discontinuity in the flow curve when the apparent shear rate is above 4 × 10⁵ s^?1 and the flow is through multiple capillaries. Further increase in the shear rate restores the stable flow. The intensity of the oscillations and the range of shear rate during which unstable flow occurs are increased with increasing melt temperature. The mechanism of this unstable flow is investigated by studying fibre orientation at the capillary entrance and exit using mouldings simulating capillary entry-exit flows.     

Mathematical characteristics of the pom-pom model

 Recently, in order to describe the complex rheological behavior of polymer melts with long side branches like low density polyethylene, new constitutive equations called the pom-pom equations have been derived in the integral/differential form and also in the simplified differential type by McLeish and Larson on the basis of the reptation dynamics with simplified branch structure taken into account. In this study, mathematical stability analysis under short and high frequency wave disturbances has been performed for these constitutive equations. It is proved that the differential model is globally Hadamard stable, as long as the orientation tensor remains positive definite or the smooth strain history in the flow is previously given. However both versions of the model are Hadamard unstable if we neglect the arm withdrawal in the case of maximum backbone stretch. It is also dissipatively unstable, since the steady shear flow curves exhibit non-monotonic dependence on shear rate. Additionally, in the flow regime of creep shear flow where the applied constant shear stress exceeds the maximum achievable value in the steady flow curves, the constitutive equations exhibit severe instability that the solution possesses strong discontinuity at the moment of change of chain dynamics mechanisms. Received: 14 August 2001 Accepted: 18 October 2001     

A microstructural investigation of the nonlinear response of electrorheological suspensions

In the preceding paper, Part 1, the transition from linear to nonlinear behavior for electrorheological (ER) suspensions under start-up of steady shear flow was found to first arise from the slight rearrangement of unstable structures. In this paper, we investigate the transition to nonlinear behavior for ER suspensions under oscillatory shear flow, focusing on the role of the rearrangement of unstable structures, and employing experimental and simulation results. Again, we find that nonlinear deformation first arises from these rearrangements, as opposed to the gross rearrangement or rupture of particulate chains. The Fourier transform of the simulated time-dependent shear stress is employed to quantify the dependence of the critical strain on the deformation frequency and electric field strength. The predicted behavior is consistent with experimental trends. Methods for verifying the predictions are discussed, as well as possible avenues for exploiting this information in improved operating strategies and improved ER fluids.     

Structural and interfacial stability of multiple solutions for stratified flow

Prediction of the liquid level in stratified two-phase upwards flow shows that one may have multiple solutions. In this case it is necessary to determine which solutions will actually occur and whether hysteresis is possible, namely whether it is possible to have two or more solutions for the same operating conditions. In this work the stability of the solutions for stratified flow is considered using two types of stability analyses: (1) structural stability analysis; and (2) interfacial stability analysis (Kelvin—Helmholtz, K—H). For the K—H stability analysis we used two methods: an approximate simplified method suggested by Taitel & Dukler; and a more rigorous method suggested by Barnea, which is based on a combination of the viscous K—H and inviscid K—H analyses. The results show that whenever three solutions exist only the first, i.e. the solution with the thinnest liquid level, is stable. The middle solution is always structurally unstable (linearly), whereas the third solution is structurally unstable to large disturbances (non-linear stability). The third solution is usually also unstable to the K—H type of instability. As a result it is concluded that hysteresis is not possible and that only the thinnest solution will be observed practically.     

Elastic flow-front fingering instability in flowing polymer solutions

An experimental investigation of the flow-front behavior of dilute and semi-dilute polymer solutions has been carried out to gain a better understanding of the underlying mechanisms leading to the occurrence of an unstable flow at the advancing flow-front during the filling of a rectangular Hele-Shaw cell. Our experimental results have revealed the existence of an elastic finger-like instability at the advancing flow-front that develops in semi-dilute solutions of high molecular weight polymers, with an onset time of approximately a few hundred milliseconds. Although at shear rates above critical, narrow finger patterns develop at the flow-front, their amplitude and number remain roughly constant throughout the flowing. At critical condition, no secondary flow was observed in the vicinity of the front region where the unstable flow develops. Transient response of the normal stress difference and the shear stress in the plate-and-plate geometry at shear rate above critical (for the elastic fingering instability in the Hele-Shaw cell) did not reveal any anomalous that could lead to the formation of such finger-like instabilities. These instabilities were observed for both the ideal elastic Boger fluids and shear thinning viscoelastic fluids.