Homogeneous Turbulent Shear Flow Research Articles

Plane-Couette flow between smooth and rough walls

A novel approach is utilized to acquire new experimental information on plane-Couette flow. To this end a rigid plate, functioning as the moving wall, is propelled through air by the carriage of a towing channel, and the fixed wall is a stationary bench. It is shown that, irrespective of the state of the walls, the critical Reynolds number, expressed in terms of relative wall velocity and wall spacing, is about 1,200. Findings support the notion of universality of turbulent flow structure in the wall region, but shed doubt on the conjecture of homogeneous shear-flow turbulence in the core.

Direct simulation of compressible turbulence in a shear flow

The purpose of this study is to investigate compressibility effects on the turbulence in homogeneous shear flow. Physical experiments have not been and perhaps cannot be performed for homogeneous shear flows at flow speeds which are sufficiently high to introduce compressible effects on the turbulence. However, direct numerical simulation (DNS) of this problem could provide meaningful data, especially since DNS for the incompressible problem ([1], [2]) has been successful in giving realistic flow fields akin to those obtained in physical experiments. The present simulations of the homogeneous shear problem augment and extend our previous results ([3], [4]) for compressible isotropic turbulence. The compressible Navier-Stokes equations are written in a frame of reference moving with the mean flow u. We use a Fourier collocation method for the spatial discretization of the governing equations. A third order, low storage Runge-Kutta scheme is used for advancing the solution in time. Up to 1283 points are used for the simulations. The simulated turbulent fields have Taylor microscale Reynolds numbers Re? up to 35 and turbulent Mach numbers Mt up to 0.6. The results are summarized below. We find that an increase in initial compressibility level, either due to increased turbulent Mach number or increased dilatational fraction of the velocity field decreases the growth of turbulent kinetic energy (and all its components) in the case of homogeneous shear flow. This is reminiscent of the reduction in turbulent velocity intensities observed in experiments on supersonic free shear

The structure of a passive scalar field with a uniform mean gradient in rapidly sheared homogeneous turbulent flow

The effect of an arbitrarily oriented mean passive scalar gradient on one-point passive scalar statistics is studied in homogeneous turbulent shear flow in the limit of rapid shearing. By neglecting the nonlinear inertial transfer to small scales an analytical solution for individual Fourier modes is obtained for the case of unity Prandtl number. This solution is used to compute the development of one-point statistics for the velocity and scalar fields in the inviscid limit. Comparisons to direct numerical simulations of the full nonlinear equations for the same flow show that in addition to describing the early time response to the imposed shear, the linear solution gives reasonable estimates of several correlation coefficients for the developed shear flow.

A critical comparison of turbulence models for homogeneous shear flows in a rotating frame

A variety of turbulence models, including five second-order closures and four two-equation models, are tested for the problem of homogeneous turbulent shear flow in a rotating frame. The model predictions for the time evolution of the turbulent kinetic energy and dissipation rate, as well as those for the equilibrium states, are compared with the results of physical and numerical experiments. Most of the two-equation models predict the same results for all rotation rates, in which there is an exponential time growth of the turbulent kinetic energy and dissipation rate. The second-order closures are qualitatively superior since, consistent with physical and numerical experiments, they only predict this type of unstable flow for intermediate rotation rates in the range −0.1≤Ω/S≤0.6. For rotation rates outside this range, there is an exchange of stabilities with a solution whose kinetic energy and dissipation rate decay with time. Although the second-order closures are superior to the two-equation models, there are still problems with the quantitative accuracy of their predictions.

Effects of organized turbulence structures on the phase distortion in a coherent optical beam propagating through a turbulent shear flow

Effects of organized turbulence structures on the propagation of an optical beam in a turbulent shear flow have been analyzed. An instantaneous passive-scalar field in a computed homogeneous turbulent shear flow is used to represent index-of-refraction fluctuations, and phase distortion induced in a coherent optical beam by turbulent fluctuations is calculated. The organized vortical structures (‘‘hairpin-shaped’’ eddies) in the turbulent flow give rise to a scalar distribution with elongated regions of intense fluctuation, which have an inclination (about 30°) with respect to the mean flow, similar to that of the characteristic ‘‘hairpin’’ eddies. Two-point correlations of vorticity and scalar fluctuations support a proposed physical model in which the regions of intense scalar fluctuation are produced primarily by hairpin vortices. It is found that the spatial distribution of the phase distortion has a substantial variation with the direction of propagation. A highly localized distribution of intense phase distortions is produced when the optical beam propagates at an angle (45°) close to the inclination of hairpin vortices; at larger angles of propagation the distribution shows an elongated pattern with smaller phase distortions. It is also found that the root-mean-square phase distortion depends significantly on the propagation direction, and the phase distortion can be minimized at an angle of propagation approximately normal to the inclination of hairpin eddies. This study shows how the characteristics of an optical beam propagating through a turbulent shear flow are affected by the geometrical configurations of organized vortical structures.

On the prediction of equilibrium states in homogeneous turbulence

A comparison of several commonly used turbulence models (including the K–ε model and three second-order closures) is made for the test problem of homogeneous turbulent shear flow in a rotating frame. The time evolution of the turbulent kinetic energy and dissipation rate is calculated for these models and comparisons are made with previously published experiments and numerical simulations. Particular emphasis is placed on examining the ability of each model to predict equilibrium states accurately for a range of the parameter Ω/S (the ratio of the rotation rate to the shear rate). It is found that none of the commonly used second-order closure models yield substantially improved predictions for the time evolution of the turbulent kinetic energy and dissipation rate over the somewhat defective results obtained from the simpler K–ε model for the unstable flow regime. There is also a problem with the equilibrium states predicted by the various models. For example, the K–ε model erroneously yields equilibrium states that are independent of Ω/S while the Launder, Reece & Rodi model and the Shih-Lumley model predict a flow relaminarization when Ω/S > 0.39 - a result that is contrary to numerical simulations and linear spectral analyses, which indicate flow instability for at least the range 0 [les ] Ω/S [les ] 0.5. The physical implications of the results obtained from the various turbulence models considered herein are discussed in detail along with proposals to remedy the deficiencies based on a dynamical systems approach.

An algebraic model for the turbulent flux of a passive scalar

The behaviour of passive-scalar fields resulting from mean scalar gradients in each of three orthogonal directions in homogeneous turbulent shear flow has been studied using direct numerical simulations of the unsteady incompressible Navier-Stokes equations with 128 × 128 × 128 grid points. It is found that, for all orientations of the mean scalar gradient, the sum of the pressure-scalar gradient and velocity gradient-scalar gradient terms in the turbulent scalar flux balance equation are approximately aligned with the scalar flux vector itself. In addition, the time derivative of the scalar flux is also approximately aligned with the flux vector for the developed fields (corresponding to roughly constant correlation coefficients). These alignments lead directly to a gradient transport model with a tensor turbulent diffusivity. The simulation results are used to fit a dimensionless model coefficient as a function of the turbulence Reynolds and Péclet numbers. The model is tested against two different passive-scalar fields in fully developed turbulent channel flow (also generated by direct numerical simulation) and is found to predict the scalar flux quite well throughout the entire channel.

A solution of the E-? model for nearly homogeneous turbulence with a mean shear

An analytical solution of the E-e model for the downstream evolution of a stationary and nearly homogeneous turbulent shear flow is presented. In case that the turbulent time scale has adjusted itself to the time scale imposed by the shear, an asymptotic solution can be derived from the full solution, which shows that both E and e increase downstream exponentially. By comparing this asymptotic solution with experimental data a value for the unknown constant c le , in the e-equation, is derived. Moreover, we find an expression for the downstream development of the variance of a scalar, which is also compared with experimental data. The analytical solution shows that a homogeneous shear flow with a uniform velocity gradient can only be obtained if the shear is sufficiently small. In the experiments this condition is not always satisfied. A discussion is given on how a nearly homogeneous shear flow can be obtained over a limited downstream interval by changing the initial conditions in E and e, and a comparison is made with experimental data. Finally it is shown that better transverse homogeneity can be obtained by taking an exponential velocity profile instead of a linear profile.

Direct numerical simulation of stratified homogeneous turbulent shear flows

The exact time-dependent three-dimensional Navier-Stokes and temperature equations are integrated numerically to simulate stably stratified homogeneous turbulent shear flows at moderate Reynolds numbers whose horizontal mean velocity and mean temperature have uniform vertical gradients. The method uses shear-periodic boundary conditions and a combination of finite-difference and pseudospectral approximations. The gradient Richardson number Ri is varied between 0 and 1. The simulations start from isotropic Gaussian fields for velocity and temperature both having the same variances.The simulations represent approximately the conditions of the experiment by Komori et al. (1983) who studied stably stratified flows in a water channel (molecular Prandtl number Pr = 5). In these flows internal gravity waves build up, superposed by hot cells leading to a persistent counter-gradient heat-flux (CGHF) in the vertical direction, i.e. heat is transported from lower-temperature to higher-temperature regions. Further, simulations with Pr = 0.7 for air have been carried out in order to investigate the influence of the molecular Prandtl number. In these cases, no persistent CGHF occurred. This confirms our general conclusion that the counter-gradient heat flux develops for strongly stable flows (Ri ≈ 0.5–1.0) at sufficiently large Prandtl numbers (Pr = 5). The flux is carried by hot ascending, as well as cold descending turbulent cells which form at places where the highest positive and negative temperature fluctuations initially existed. Buoyancy forces suppress vertical motions so that the cells degenerate to two-dimensional fossil turbulence. The counter-gradient heat flux acts to enforce a quasi-static equilibrium between potential and kinetic energy.Previously derived turbulence closure models for the pressure-strain and pressure-temperature gradients in the equations for the Reynolds stress and turbulent heat flux are tested for moderate-Reynolds-number flows with strongly stable stratification (Ri = 1). These models overestimate the turbulent interactions and underestimate the buoyancy contributions. The dissipative timescale ratio for stably stratified turbulence is a strong function of the Richardson number and is inversely proportional to the molecular Prandtl number of the fluid.

Scaling laws for homogeneous turbulent shear flows in a rotating frame

The scaling properties of plane homogeneous turbulent shear flows in a rotating frame are examined mathematically by a direct analysis of the Navier–Stokes equations. It is proved that two such shear flows are dynamically similar if and only if their initial dimensionless energy spectrum E*(k*,0), initial dimensionless shear rate SK0/ε0, initial Reynolds number K20/νε0, and the ratio of the rotation rate to the shear rate Ω/S are identical. Consequently, if universal equilibrium states exist at high Reynolds numbers, they will only depend on the single parameter Ω/S. The commonly assumed dependence of such equilibrium states on Ω/S through the Richardson number Ri=−2(Ω/S)(1−2Ω/S) is proved to be inconsistent with the full Navier–Stokes equations and to constitute no more than a weak approximation. To be more specific, Richardson number similarity is shown to only rigorously apply to certain low-order truncations of the Navier–Stokes equations (i.e., to certain second-order closure models) wherein closure is achieved at the second-moment level by assuming that the higher-order moments are a small perturbation of their isotropic states. The physical dependence of rotating turbulent shear flows on Ω/S is discussed in detail, along with the implications for turbulence modeling.

The evolution equation of joint Pdf of turbulent velocity and dissipation

According to the hypothesis that the dissipation of turbulent kinetic energy satisfies log-normal distribution, a stochastic model of dissipation is provided and the Langevin model[6] of velocity is modified. Then a joint Pdf equation of turbulent velocity and dissipation is derived. We solve numerically the joint Pdf equation using Monte Carlo method and obtain satisfactory results for decaying turbulence and homogeneous turbulent shear flow. The preliminary results show that the model is well working.

Stochastic estimation of organized turbulent structure: homogeneous shear flow

The large-scale organized structures of turbulent flow can be characterized quantitatively by a conditional eddy, given the local kinematic state of the flow as specified by the conditional average of u(x’, t) given the velocity and the deformation tensor at a point x: 〈u(x’, t)|u(x, t), d(x, t)〉. By means of linear mean-square stochastic estimation, 〈u’|u, d〉 is approximated in terms of the two-point spatial correlation tensor, and the conditional eddy is evaluated for arbitrary values of u(x, t) and d(x, t), permitting study of the turbulent field for a wide range of local kinematic states. The linear estimate is applied to homogeneous turbulent shear flow data generated by direct numerical simulation. The joint velocity-deformation probability density function is used to obtain conditions corresponding to those events that contribute most to the Reynolds shear stress. The primary contributions to the second-quadrant and fourth-quadrant Reynolds-stress events in homogeneous shear flow come from flow induced through the ‘legs’ and close to the ‘heads’ of upright and inverted ‘hairpins’, respectively.The equation governing the joint probability density function of fu,d (u, d) is derived. It is shown that this equation contains 〈u’/u, d〉 and that the equations for second-order closure can be derived from it. Closure requires approximation of 〈u’/u, d〉.

The structure of the vorticity field in homogeneous turbulent flows

The structure of the vorticity fields in homogeneous turbulent shear flow and various irrotational straining flows is examined using results from direct numerical simulations of the unsteady, incompressible Navier-Stokes equations with up to 128 × 128 × 128 grid points. In homogeneous shear flow, the distribution of the inclination angle of the vorticity vectors and contour plots of two-point correlations of both velocity and vorticity are consistent with the existence of persistent vortical structures inclined with respect to the flow direction. Early in the development of these shear flows, the angle of inclination at which most of these structures are found is near 45°; after the flow develops, this angle lies between 35°–40°. Instantaneous vorticity-vector and vortex-line plots confirm the presence of hairpin vortices in this flow at the two Reynolds numbers simulated. These vortices are formed by the roll-up of sheets of mean spanwise vorticity. The average hairpin leg spacing decreases with increasing Reynolds number but increases relative to the Taylor microscale for developed shear flows. Examination of irrotational axisymmetric contraction, axisymmetric expansion, and plane strain flows shows, as expected, that the vorticity tends to be aligned with the direction of positive strain. For example, the axisymmetric contraction flow is dominated by coherent longitudinal vortices. Without the presence of mean shear, however, hairpin structures do not develop. The simulations strongly indicate that the vorticity occurs in coherent filaments that are stretched and strengthened by the mean strain. When compressed, these filaments appear to buckle rather than to decrease in strength.

The effect of contraction on a homogeneous turbulent shear flow

A homogeneous turbulent shear flow in its asymptotic stage of development was subjected to an additional (longitudinal) strain by passing the flow through gradual contraction in the direction perpendicular to that of the mean shear. Two contractions, of area ratio 1.4 and 2.6, were used. Mean velocity and turbulent stress (both normal and shear) distributions were measured at several streamwise locations in the contraction region. The mean velocity distributions agree quite well with calculations based on the (inviscid) Bernoulli equation. Until at least half-way down the contraction with the larger area ratio, the rapid-distortion calculations considering only the streamwise acceleration were found to be reasonably successful in predicting the turbulent intensities. For the smaller-area-ratio contraction, corrections for the ‘natural development’ of the shear flow become important nearly everywhere. Similar calculations considering the shear as the only straining mechanism are generally less successful, although the shear strain rate is at least as rapid as, or even more so than, the longitudinal one. The pressure-rate-of-strain covariance terms estimated from the approximate component energy balance were used to test the adequacy of three models with varying degrees of complexity. Although none of these models appears general enough, their performance is generally adequate for the lower-area-ratio contraction; perhaps not surprisingly, the more complex the model the better its performance.

Asymptotic laws for transversely homogeneous turbulent shear flows

An analysis predicts an exponential downstream development of turbulent kinetic energy in the asymptotic development range of transversely homogeneous, uniformly sheared, turbulent flows. The analysis is found to be in good agreement with available experimental results and numerical simulations.

Effects of shear on the turbulent diffusivity tensor

Earlier measurements of some components of the turbulent diffusivity tensor D ij in a nearly homogeneous turbulent shear flow with constant mean temperature gradient parallel to the mean shear are now supplemented with measurements of additional components in the same flow but with a mean temperature gradient perpendicular to both mean velocity and mean shear. A quasi-Lagrangian analysis provides expressions for all components of D ij and confirms the expectation that D ij is non-diagonal and nonsymmetric. The analysis also suggests the possibility that D 11 and D 12 might become positive at sufficiently large values of mean shear. Independent estimates of D ij based on a numerical simulation of homogeneous shear flow as well as on a rapid distortion type analysis are also reported.

Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. Part 1

A reasonably uniform mean temperature gradient has been superimposed upon a nearly homogeneous turbulent shear flow in a wind tunnel. The overheat is small enough to have negligible effect on the turbulence. Away from the wind-tunnel entrance, the transverse statistical homogeneity is good and the temperature fluctuations and their integral scales grow monotonically like the corresponding velocity fluctuations (Harris, Graham & Corrsin 1977). Measurements of several moments, one- and two-point correlation functions, spectra, integral scales, microscales, probability densities, and joint probability densities of the turbulent velocities, temperature fluctuations, and temperature-velocity products are reported. The heat-transport characteristics are much like those of momentum transport, with the turbulent Prandtl number nearly 1. The temperature fluctuation is better correlated with the streamwise than the transverse velocity component, and the cross-componentD12of the turbulent diffusivity tensor has sign opposite to and about twice the magnitude of the diagonal componentD22. Some resemblance of directional properties (relative magnitudes of correlation functions, integral scales, microscales) of the temperature with those of the streamwise velocity is also observed. Comparisons of the present data with measurements in the inner part of a heated boundary layer and a fully turbulent pipe flow (x2/d= 0·25) show comparable magnitudes of temperature-velocity correlation coefficients, turbulent Prandtl numbers and ratios of turbulent diffusivities, and show similar shapes of two-point correlation functions.

Experiments in nearly homogeneous turbulent shear flow with a uniform mean temperature gradient. Part 2. The fine structure

Previous measurements in nearly homogeneous sheared turbulence with a uniform mean temperature gradient are here supplemented with data on the fine structure of the velocity and temperature fluctuation fields. The statistics of signal derivatives and of band-passed signals show that neither field is locally isotropic in the spectral range covered, possibly because of the insufficiently large turbulent Reynolds and Péclet numbers. Observed skewnesses of both velocity and temperature derivatives are explained qualitatively with the use of a kind of ‘mixing-length’ model. The flatness factors of the derivatives and of band-passed, high-frequency signals indicate appreciable departures from normality, consistent with the spatially ‘spotty’ fine structure. The temperature flatnesses are a bit larger than those of the streamwise velocity. The homogeneous shear flow data are compatible with measurements in turbulent boundary layers at comparable RΛ and PΛθ.

Analysis of a strongly sheared, nearly homogeneous turbulent shear flow

Harris, Graham & Corrsin (1977) have measured the properties of the quasi-homo-geneous turbulence field induced by a large mean shear, and in analysing their measurements they neglect the diagonal components of the mean-field contribution to the pressure–strain correlation. Their measurements are re-analysed using the recommendations of Gibson & Launder (1978) for modelling this correlation, the imbalance between production and dissipation being allowed for by the algebraic modelling technique of Rodi (1976): the experimental data give strong support to Rodi's basic assumption.The data suggest that the only substantial defect in the Gibson–Launder model is its failure to predict the anisotropy measured in the 23 plane normal to the mean flow. The longitudinal predictions are good, and those for the shear (12) component are much improved when measured values of the anisotropy are substituted into the calculation. This analysis does not suggest any clear need for a nonlinear representation of the pressure–strain correlation. However, the most general linear representation of the mean-field term is even more complex than the analysis of Launder, Reece & Rodi (1975) would suggest: their model is disproved by an example.Attempts to deduce the dissipation directly from the experiments, rather than by energy balance, are not very successful.

Further experiments in nearly homogeneous turbulent shear flow

The experiment of Champagne, Harris & Corrsin in generating and studying a nearly homogeneous turbulent shear flow has been extended to larger values of the dimensionless downstream time or strain by the use of a larger mean velocity gradient in the same wind tunnel. The system appears to reach an asymptotic state in which scales and turbulent energy grow monotonically. Two-point covariances and tensor structure of one-point ‘Reynolds stress’ and ‘pressure/strain-rate covariance’ agree with the earlier case. However, the linear intercomponent energy exchange hypothesis due to Rotta, very roughly confirmed by the earlier experiment, is contradicted by the present data.