Homogeneous Shear Flow Research Articles

An explicit algebraic Reynolds stress model in turbulence

A new algebraic Reynold stress model is constructed with recourse to the realizability constraints. Model coefficients are made a function of strain and vorticity invariants through calibration by reference to homogeneous shear flow data. The anisotropic production in near-wall regions is accounted for substantially by modifying the model constants Ce(1, 2) and adding a secondary source term in the e equation. Hence, it reduces the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. To facilitate the evaluation of the turbulence model, some extensively used benchmark cases in the turbulence modelling are convoked. The comparisons demonstrate that the new model maintains qualitatively good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2006 John Wiley & Sons, Ltd.

On the prediction of equilibrium states in stably stratified homogeneous sheared turbulence

An assessment of second-order closure models in the prediction of stratified homogeneous shear flows has been conducted. The emphasis is placed on analyzing the effects of three modeled dissipation rate equations on the prediction of equilibrium states. It is found that second-order closure models are unable to reproduce the main features of stratified homogeneous shear flow. In particular, the models considered cannot predict an important experimental observation, namely, the existence of a stationary gradient Richardson number, Rs, below which the turbulent kinetic energy, K, shows exponential growth and, above which, K, decays exponentially. This result is partly due to deficiencies in the modeled dissipation rate equation. It is shown, herein, that the performance of second-order closure models will substantially improve if stationary conditions of the turbulent kinetic energy K and the dissipation rate ε are satisfied.

The effect of rotation on rapidly sheared homogeneous turbulence and passive scalar transport. Linear theory and direct numerical simulation

The effect of rotation on a homogeneous turbulent shear flow has been studied by means of a series of direct numerical simulations with different rotation numbers. The evolution of passive scalar fields with mean gradients in each of the three orthogonal directions in the flow was investigated in order to elucidate the effect of rotation on turbulent scalar transport. Conditions of the near-wall region of a boundary layer were approached by using a rapid shear and therefore, comparisons could be made with rapid distortion theory based on the linearized equations of the flow and scalar transport. Reynolds stresses, pressure–strain correlations and two-point velocity correlations were computed and turbulent structures were visualized. It is shown that rotation has a strong influence on the time development of the turbulent kinetic energy, the anisotropy of the flow and on the turbulent structures. Furthermore, rotation significantly affects turbulent scalar transport. The transport rate of the scalar and the direction of the scalar flux vector show large variations with different rotation numbers, and a strong alignment was observed between the scalar flux and the principal axes of the Reynolds stress tensor. The ratio of the turbulent and scalar time scales is influenced by rotation as well. The predictions of the linear theory of the turbulent one-point statistics and the scalar flux agreed fairly well with direct numerical simulation (DNS) results based on the full nonlinear governing equations. Nonetheless, some clear and strong nonlinear effects are observed in a couple of cases which significantly influence the development of the turbulence and scalar transport.

Determination of the interfacial tension of low density difference liquid–liquid systems containing surfactants by droplet deformation methods

The interfacial tension, σ , between two low density difference liquids containing a surfactant was determined using drop deformation method with a computer-controlled parallel bands apparatus. This device applies a linear homogeneous shear flow field to a fluid matrix in which a droplet of another immiscible and buoyancy-free fluid is immersed. The flow induces topological changes on the initially spherical drop, which deforms into an ellipsoid and orients respect to the flow direction. The time evolution of the sheared droplets was recorded with two CCD cameras (placed along the x and z directions) for extended times, allowing the steady state of the drops to be achieved accurately, and further digitally analyzed. Appropriate theories corresponding to each flow-induced mechanism of the droplet under shear (steady-state deformation and orientation) were employed for determining the interfacial tension, under the basis of reaching equilibrium states of deformation of the sheared drop. The calculated σ values via the drop deformation theories were checked for a wide range of viscosity ratios of binary systems conformed by silicon oils as droplets and a water solution of polyvinylpyrrolidone as continuous phase (both Newtonian), being found that σ is independent on λ . These values were compared with measurements carried out in a conventional tensiometer, using the drop volume method. The comparison showed a very good agreement between both techniques.

Shear flow of viscoelastic fluids as a control problem

We consider a homogeneous parallel shear flow of a viscoelastic fluid with a single relaxation time. This problem results in a set of ordinary differential equations for the stresses. In this system, we view the shear rate as a control and consider the problem of steering the system to a given state of stress. For several popular constitutive models, we characterize the stresses which can be reached.

Explicit algebraic stress modelling of homogeneous and inhomogeneous flows

AbstractCapability of the explicit algebraic stress models to predict homogeneous and inhomogeneous shear flows are examined. The importance of the explicit solution of the production to dissipation ratio is first highlighted by examining the algebraic stress models performance at purely irrotational strain conditions. Turbulent recirculating flows within sudden expanding pipes are further simulated with explicit algebraic stress model and anisotropic eddy viscosity model. Both models predict better stress–strain interactions, showing reasonable shear layer developments. The anisotropic stress field are also accurately predicted by the models, though the anisotropic eddy viscosity model of Craft et al. returns marginally better results. Copyright © 2005 John Wiley & Sons, Ltd.

On the normalized turbulent energy dissipation rate

This paper revisits values of the normalized energy dissipation rate (Cϵ) in different flows (two-dimensional wakes, grid turbulence, and homogeneous shear flow). Previously published as well as new data are considered over a relatively wide range of the Taylor-microscale Reynolds number Rλ. Cϵ exhibits wide scatter (in the range 0.5–2.5 for Rλ>50) although, for a given flow and initial conditions, it is independent of Rλ when the latter is sufficiently large. An alternative definition [B. R. Pearson, P.-Å. Krogstad, and W. van de Water, “Measurements of the turbulent energy dissipation rate,” Phys. Fluids 14, 1288 (2002)] of Cϵ has been checked in the same flows but has failed to yield a universal value for the coefficient.

Self-similar turbulence evolution and the dissipation rate transport equation

The dissipation rate transport equation is analyzed in the setting of time-dependent isotropic turbulence driven by a statistically unsteady force. In the limit of slow spectral variation, the balance between vortex stretching and enstrophy destruction postulated by Tennekes and Lumley [A First Course in Turbulence (MIT Press, Cambridge, MA, 1972)] is verified and spectral closure is used to identify the O(Re0) difference between them; however, no definite formulation of an ϵ-equation results. The ϵ equation is usually calibrated to predict self-similar unit flows such as decaying turbulence and homogeneous shear flow. The limitations of this approach are shown by constructing classes of self-similar states of forced isotropic turbulence. Any choice of constants in the ϵ equation yields a model that is consistent with some self-similar states, but not with all possible states: two-equation models of the standard form select a class of admissible self-similar states and rule out the others.

Exponential shear flow of branched polyethylenes in rotational parallel-plate geometry

Exponential shear flow, as a strong flow with the potential to generate a high degree of molecular stretching, has attracted considerable interest in recent years. So far, exponential shear flow has been realized by either sliding-plate or cone-and-plate (CP) geometry. Both geometries guarantee homogeneous shear flow. Here, we present experimental data on exponential shear flow of several long-chain branched polyethylene melts with different degrees of strain hardening obtained by using parallel-plate (PP) geometry in a rotational rheometer. This type of geometry, which is standard in linear-viscoelastic characterization of polymer materials, produces inhomogeneous shear flow. A comparison of exponential shear flow data obtained by PP and CP geometry is made. Additionally, the experimental data are compared to predictions of the rubber-like liquid (RLL) and the molecular stress function (MSF) theories. For this purpose, the relaxation spectra of the polymer melts considered were obtained by standard linear-viscoelastic characterization. In addition, two irrotational parameters and one rotational parameter are required by the MSF theory. While the irrotational parameters were obtained from fitting to elongational viscosity data, the value of the rotational parameter was used as given in the literature. It can be concluded that viable experimental data in exponential shear flow can be obtained by PP geometry. For finite linear-viscoelasticity (RLL theory), predictions of reduced shear stress for CP and PP geometry coincide, but nonlinear material behavior (as modeled by the MSF theory) leads to small differences between both geometries. Furthermore, it is shown that the MSF predictions are in excellent agreement with the experimental data in exponential shear flow and that this type of flow leads to much less chain stretching than elongational flow.

An explicit algebraic Reynolds stress model based on a nonlinear pressure strain rate model

The use of a pressure strain rate model including terms nonlinear in the mean strain and rotation rate tensors in an explicit algebraic Reynolds stress model (EARSM) is considered. For 2D mean flows the nonlinear contributions can be fully accounted for in the EARSM formulation. This is not the case for 3D mean flows and a suggestion of how to modify the nonlinear terms to make the EARSM formulation in 3D mean flows consistent with its 2D counterpart is given. The corresponding EARSM is derived in conjunction with the use of streamline curvature corrections emanating from the advection of the Reynolds stress anisotropy. The proposed model is tested for rotating homogeneous shear flow, rotating channel flow and rotating pipe flow and the nonlinear contributions are shown to have a significant effect on the predicted flow characteristics. In cases where the 3D effects are strong, the approximations of the production to dissipation ratio made in the EARSM formulation for 3D mean flows must be made carefully and a 3D mean flow correction is considered. For the rotating pipe flow at the highest rotation rate investigated, the standard formulation even prevented convergence, while inclusion of the 3D correction gives reasonable results.

Scaling Properties in the Production Range of Shear Dominated Flows

In large Reynolds number turbulence, isotropy is recovered as the scale is reduced and homogeneous-isotropic scalings are eventually observed. This picture is violated in many cases, e.g., wall bounded flows, where, due to the shear, different scaling laws emerge. This effect has been ascribed to the contamination of the inertial range by the larger anisotropic scales. The issue is addressed here by analyzing both numerical and experimental data for a homogeneous shear flow. In fact, under strong shear, the alteration of the scaling exponents is not induced by the contamination from the anisotropic sectors. Actually, the exponents are universal properties of the isotropic component of the structure functions of shear dominated flows. The implications are discussed in the context of turbulence near solid walls, where improved closure models would be advisable.

Extension of the Launder, Reece and Rodi model on compressible homogeneous shear flow

This article describes the second order closure progress that was made to calculate compressible homogeneous shear flow with significant compressibility. Several DNS results show that compressibility has an important effect on the pressure-strain correlation. The term recognized as the principal responsible for the change in the magnitude of Reynolds-stress anisotropies. Thus, the pressure-strain incompressible models do not correctly predict compressible turbulence at high-speed shear flow. A method of including compressibility effects in the pressure-strain correlation is the subject of the present study. The concept of the growth rate of turbulent kinetic energy can be used to construct a compressible correction to the Launder, Reece and Rodi model for the pressure-strain correlation. This correction concerns essentially the C 1,C 3 and C 4 coefficients which become in a compressible turbulence situation a function of the turbulent Mach number. The application of the new model shows good agreement with DNS results of Sarkar for cases A 1, A 2 and A 3. These cases correspond to a moderate mean shear rate, so that nonlinear effects are important.

Numerical simulation data for assessment of particle-laden turbulent flow models

This paper presents a collection of numerical simulation data which provides a reference for the assessment of various statistical/stochastic models in incompressible homogeneous particle-laden turbulent flows. Four different homogeneous flow configurations are studied, namely, homogeneous shear flow, homogeneous plane strain flow, homogeneous axisymmetric expansion and contraction. An Eulerian–Lagrangian formulation is used for the two-phase flow simulation. A Fourier pseudospectral method is used for the solution of the Eulerian carrier-phase equations without resorting to any turbulence model. The Lagrangian equations for the dispersed phase are integrated using a modified Stokes drag. For the shear flow, both monodispersed and polydispersed particles have been considered. In this paper, only the results that are relevant for assessment of various statistical models for both the fluid and dispersed phases are presented.

Umbrella sampling in non-equilibrium computer simulations

We describe the application of the umbrella sampling technique well known from equilibrium Monte Carlo to dynamical, non-equilibrium simulations. This method is used specifically to calculate the nucleation barrier of Brownian Yukawa particles in a homogeneous shear flow.

A Lagrangian stochastic model for dispersion in stratified turbulence

In this paper we discuss the development of a Lagrangian stochastic model (LSM) for turbulent dispersion of a scalar (species). Given any tensorally linear second-moment closure (SMC) turbulence model we show how to derive a mathematically equivalent set of stochastic differential equations (SDEs), i.e., the second-moment equations constructed from these SDEs are exactly the same (within a realizability constraint) as the given SMC. This set of equations forms the LSM. Both turbulence anisotropy and buoyancy effects are incorporated by this method. In order to achieve the correct critical Richardson number and to obtain the simplest Lagrangian formulation, a revised set of constants for the isotropization of production model is proposed. They improve agreement with experiments. The LSM is applied to homogeneous shear flow with varying degrees of stratification. Our model is shown to capture important physics associated with buoyant flows and we also parametrize our results. Finally the form of the current model is compared with a few of the well-known Lagrangian stochastic models.

Drag and Lift Forces Acting on a Spherical Droplet in Homogeneous Shear Flow

Drag and lift forces acting on a spherical droplet in a homogeneous linear shear flow were numerically studied by means of a three dimensional direct numerical simulation based on a marker and cell (MAC) method. The effects of fluid shear rate and particle Reynolds number on drag and lift forces acting on a spherical droplet were compared with those on a rigid sphere. The results show that the drag force acting on a spherical droplet in a linear shear flow increases with shear. The lift force acts on a droplet from high-speed side to low-speed side in the linear shear flow with high particle Reynolds number of Rep≥100. The direction of the lift force is different between low and high particle Reynolds number flows. The behavior of the lift force on a droplet is quite similar to that on a rigid sphere, but the effects of fluid shear rate on the lift force acting on a spherical droplet in the linear shear flow become smaller than that acting on a rigid sphere in the low Reynolds number region of Rep<10.

Shear Flow Analysis of Nanosize Particle Flow Behavior Using Kinetic Theory

Nanosize particle flow is significantly affected by inter-particle force. Due to the inter-particle force, the most significant characteristic of nanosize particle flow may become the formation of agglomerates or clusters which considerably affects the flow patterns. The formation of agglomerates or clusters results in a reduction in the number and an increase in the size of particles, both of which directly affect the frequency of inter-particle collisions and, in turn, the particle phase properties such as viscosity and pressure, as well as gas/particle drag force in gas/particle flow systems. In this present work, we focus our attention on the verification of nanosize particle flow behavior due to the formation of agglomerates or clusters under different fluctuation of flow and inelasticity of particle collision. By extending the application of the cohesive model using kinetic theory to nanosize particle flow system, we performed the homogeneous simple shear flow analysis using various fluctuation energy and restitution coefficient. The predicted flow properties, such as particle diameter growth, agreed well with the expected trends.

A compressible turbulence model for the pressure–strain correlation

Recent DNS (Direct Numerical Simulation) results indicate that the main compressibility effect comes from the reduced pressure–strain term due to reduced pressure fluctuations. Using the concept of moving equilibrium in compressible homogeneous shear flow, we develop a new compressible pressure–strain model by modifying the incompressible linear pressure–strain model, which is characterized by a decrease in the primary redistribution term but an increase in the secondary redistribution term. The model was applied to a compressible mixing layer. The predicted results show that the growth rate is reduced with increasing convective Mach number and the Reynolds stresses are reduced. All the components of the pressure–strain term and the shear stress anisotropy are reduced with the increase of the compressibility, whereas the normal stress anisotropy is increased—then results are in accord with experimental and DNS results.

Molecular orientation of a commercial thermotropic liquid crystalline polymer in simple shear and complex flow

In-situ X-ray scattering methods have been used to measure the average degree of molecular orientation in the commercial thermotropic copolyesteramide, Vectra B. Experiments were conducted in both homogeneous shear flow and in extrusion-fed channel flows that provided mixed shear/extensional deformations. In the channel flows, extension has a dramatic effect on the average orientation state in the vicinity of stagnation points or expansions/contractions in cross-sectional area. Of particular note, a temporary increase and subsequent decay in orientation observed in a 4:1 slit-contraction flow provides additional indirect evidence supporting the hypothesis that Vectra B exhibits director tumbling. This is consistent with results from other fully aromatic copolyesters but contrasts with findings in “model” thermotropes incorporating flexible spacers. Thus, it seems that the stiffer backbone of commercial main chain LCPs is the main feature which, apparently, leads to tumbling. Measurements of average molecular orientation in transient shear flows show some connections with the corresponding mechanical response, but fail to show the distinctive characteristics that have previously been associated with either tumbling or aligning in LCPs using similar procedures. These experiments might be adversely affected by the comparatively slow rate of data acquisition, which leads to lengthy experiments in which the sample is more prone to degradation.

The footprint concept in complex terrain

The use of footprint functions in complex flows has been questioned because of anomalous behaviour reported in recent model studies. We show that the concentration footprint can be identified with the Green's function of the scalar concentration equation or the transition probability of a Lagrangian formulation of the same equation and so is well behaved and bounded by 0 and 1 in both simple and complex flows. The flux footprint in contrast is not a Green's function but a functional of the concentration footprint and is not guaranteed to be similarly well behaved. We show that in simple, homogeneous shear flows, the flux footprint, defined as the vertical eddy flux induced by a unit point source, is bounded by 0 and 1 but that this is not true in more general flow fields. Analysis of recent model studies also shows that the negative flux footprints reported in homogeneous plant canopy flows are an artefact of reducing a canopy with a complex source–sink distribution in the vertical to a single layer but that in canopies on hilly topography, the problems are more fundamental. Finally we compare footprint inversion with the direct mass-balance method of measuring surface exchange. We conclude first that the flux footprint is an appropriate measure of the area influencing both eddy and advective fluxes on a tower but that the concentration footprint is the correct measure when the storage term is important. Second, we deduce that there are serious obstacles to inverting flux footprints in complex terrain.