Anisotropic pressure correlation spectra in turbulent shear flow

2012 ◽  
Vol 694 ◽  
pp. 50-77 ◽  
Author(s):  
Yoshiyuki Tsuji ◽  
Yukio Kaneda
Keyword(s):  
Mixing Layer ◽  
Inertial Subrange ◽  
Turbulent Shear ◽  
Pressure Probe ◽  
Anisotropic Pressure ◽  
Turbulent Shear Flow ◽  
Velocity Correlation ◽  
Mean Velocity ◽  
Correlation Spectrum ◽  
The Mean

AbstractWe measured the correlation spectrum ${\hat {Q} }_{p} (\mathbi{k})$ of pressure fluctuations in a driving mixing layer with a Taylor-scale Reynolds number ${R}_{\lambda } $ up to ${\simeq }700$ by a newly developed pressure probe with spatial and temporal resolutions that are sufficient to analyse inertial-subrange statistics. The influence of the mean velocity gradient tensor ${S}_{ij} $ in the mixing layer, which is almost constant near its centreline, is studied using an idea similar to that underlying the linear response theory developed in statistical mechanics for systems at or near thermal equilibrium. If we write the spectrum ${\hat {Q} }_{p} (\mathbi{k})$ as ${\hat {Q} }_{p} (\mathbi{k})= { \hat {Q} }_{p}^{(0)} (\mathbi{k})+ \mrm{\Delta} {\hat {Q} }_{p} (\mathbi{k})$, where ${ \hat {Q} }_{p}^{(0)} (\mathbi{k})$ is the isotropic Kolmogorov spectrum in the absence of mean shear, then for small ${S}_{ij} $ the deviation $ \mrm{\Delta} {\hat {Q} }_{p} (\mathbi{k})$ due to the shear is approximately linear and is determined by a few non-dimensional universal constants in addition to ${S}_{ij} $, $k$ and the mean energy dissipation rate. We also measured the pressure–velocity and velocity–velocity correlation spectra. Deviations from isotropy due to shear are shown to be approximately proportional to ${S}_{ij} $ at large ${R}_{\lambda } $.

The maintenance of Reynolds stress in turbulent shear flow

1967 ◽  
Vol 27 (1) ◽  
pp. 131-144 ◽  
Author(s):  
O. M. Phillips
Keyword(s):  
Shear Flow ◽  
Velocity Profile ◽  
Reynolds Stress ◽  
Mixing Layer ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Mean Flow ◽  
Mean Velocity ◽  
Velocity Fluctuations ◽  
The Mean

A mechanism is proposed for the manner in which the turbulent components support Reynolds stress in turbulent shear flow. This involves a generalization of Miles's mechanism in which each of the turbulent components interacts with the mean flow to produce an increment of Reynolds stress at the ‘matched layer’ of that particular component. The summation over all the turbulent components leads to an expression for the gradient of the Reynolds stress τ(z) in the turbulence\[ \frac{d\tau}{dz} = {\cal A}\Theta\overline{w^2}\frac{d^2U}{dz^2}, \]where${\cal A}$is a number, Θ the convected integral time scale of thew-velocity fluctuations andU(z) the mean velocity profile. This is consistent with a number of experimental results, and measurements on the mixing layer of a jet indicate thatA= 0·24 in this case. In other flows, it would be expected to be of the same order, though its precise value may vary somewhat from one to another.

Outline of a theory of turbulent shear flow

1956 ◽  
Vol 1 (5) ◽  
pp. 521-539 ◽  
Author(s):  
W. V. R. Malkus
Keyword(s):  
Shear Flow ◽  
Velocity Profile ◽  
Boundary Conditions ◽  
Momentum Transport ◽  
Turbulent Shear ◽  
Spectral Structure ◽  
Turbulent Shear Flow ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean

In this paper the spatial variations and spectral structure of steady-state turbulent shear flow in channels are investigated without the introduction of empirical parameters. This is made possible by the assumption that the non-linear momentum transport has only stabilizing effects on the mean field of flow. Two constraints on the possible momentum transport are drawn from this assumption: first, that the mean flow will be statistically stable if an Orr-Sommerfeld type equation is satisfied by fluctuations of the mean; second, that the smallest scale of motion that can be present in the spectrum of the momentum transport is the scale of the marginally stable fluctuations of the mean. Within these two constraints, and for a given mass transport, an upper limit is sought for the rate of dissipation of potential energy into heat. Solutions of the stability equation depend upon the shape of the mean velocity profile. In turn, the mean velocity profile depends upon the spatial spectrum of the momentum transport. A variational technique is used to determine that momentum transport spectrum which is both marginally stable and produces a maximum dissipation rate. The resulting spectrum determines the velocity profile and its dependence on the boundary conditions. Past experimental work has disclosed laminar, ‘transitional’, logarithmic and parabolic regions of the velocity profile. Several experimental laws and their accompanying constants relate the extent of these regions to the boundary conditions. The theoretical profile contains each feature and law that is observed. First approximations to the constants are found, and give, in particular, a value for the logarithmic slope (von Kármán's constant) which is within the experimental error. However, the theoretical boundary constant is smaller than the observed value. Turbulent channel flow seems to achieve the extreme state found here, but a more decisive quantitative comparison of theory and experiment requires improvement in the solutions of the classical laminar stability problem.

Experimental investigation of turbulent shear flow with quadratic mean-velocity profiles

1976 ◽  
Vol 73 (1) ◽  
pp. 165-188 ◽  
Author(s):  
H. K. Richards ◽  
J. B. Morton
Keyword(s):  
Flow Direction ◽  
Correlation Coefficients ◽  
Velocity Profiles ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Mean Velocity ◽  
Quadratic Mean ◽  
The Mean

Three turbulent shear flows with quadratic mean-velocity profiles are generated by using an appropriately designed honeycomb and parallel-rod grids with adjustable rod spacing. The details of two of the flow fields, with quadratic mean-velocity profiles with constant positive mean-shear gradients ($\partial^2\overline{U}_1/\partial X^2_2 >0$), are obtained, and include, in the mean flow direction, the development and distribution of mean velocities, fluctuating velocities, Reynolds stresses, microscales, integral scales, energy spectra, shear correlation coefficients and two-point spatial velocity correlation coefficients. A third flow field is generated with a quadratic mean velocity profile with constant negative mean-shear gradient ($\partial^2\overline{U}_1/\partial X^2_2 < 0$), to investigate in the mean flow direction the effect of the change in sign on the resulting field. An open-return wind tunnel with a 2 × 2 × 20 ft test-section is used.

The effect of contraction on a homogeneous turbulent shear flow

1985 ◽  
Vol 154 ◽  
pp. 187-213 ◽  
Author(s):  
K. R. Sreenivasan
Keyword(s):  
Shear Flow ◽  
Area Ratio ◽  
Turbulent Shear ◽  
Shear Strain Rate ◽  
Turbulent Shear Flow ◽  
Turbulent Stress ◽  
Mean Velocity ◽  
Stage Of Development ◽  
Flow Through ◽  
The Mean

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.

Pressure spectra in turbulent free shear flows

1984 ◽  
Vol 148 ◽  
pp. 155-191 ◽  
Author(s):  
William K. George ◽  
Paul D. Beuther ◽  
Roger E. A. Arndt
Keyword(s):  
Pressure Fluctuations ◽  
Mixing Layer ◽  
Inertial Subrange ◽  
Pressure Probe ◽  
Spectral Measurements ◽  
Axisymmetric Jet ◽  
Spectral Models ◽  
Turbulent Pressure ◽  
The Mean ◽  
Wavenumber Region

Spectral models for turbulent pressure fluctuations are developed by directly Fourier transforming the integral solution to the Poisson equation for a homogeneous constantmean-shear flow. The turbulence-turbulence interaction is seen to possess the well-known k−7/3 inertial subrange and to dominate the high-wavenumber region. The turbulence–mean-shear contribution is seen to be dominant in the energy-containing range and falls off as $k^{-\frac{11}{3}}$ in the inertial subrange. The subrange constants and the mean-square pressure fluctuation are evaluated using a spectral model for the velocity. A spectral analysis of the velocity contamination of a pressure probe is also presented. Results are compared with spectral measurements with a static-pressure probe in the mixing layer of an axisymmetric jet.

On the Separation, Reattachment, and Redevelopment of Incompressible Turbulent Shear Flow

1964 ◽  
Vol 86 (2) ◽  
pp. 221-225 ◽  
Author(s):  
T. J. Mueller ◽  
H. H. Korst ◽  
W. L. Chow
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Shear Flow ◽  
Roughness Element ◽  
Single Step ◽  
Parameter Family ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Mean Velocity ◽  
The Mean

An experimental and theoretical investigation is presented which describes the character of the mean motion and the structure of turbulence for the separation, reattachment, and redevelopment of the incompressible turbulent shear flow downstream of a single step-type roughness element. For the redeveloping turbulent boundary layer downstream of reattachment, it is shown that the mean velocity profiles constitute a one-parameter family and that as far as the shape parameters are concerned, this one-parameter family is essentially the same as for a boundary layer developing toward separation. This similarity between developing (toward separation) and redeveloping (after reattachment) turbulent shear layers is utilized to establish an integral method for calculating the redeveloping turbulent boundary layer at essentially zero pressure gradient.

Local transport equations for turbulent shear flow

1972 ◽  
Vol 330 (1583) ◽  
pp. 495-516 ◽  
Keyword(s):  
Shear Stress ◽  
Kinetic Energy ◽  
Transport Equations ◽  
Wave Number ◽  
Turbulent Shear ◽  
Spatial Average ◽  
Turbulent Shear Flow ◽  
Mean Velocity ◽  
The Mean ◽  
Integral Transport

The nonlinear integral transport equations for turbulent flow are expanded in a series of homogeneous kernels, the expansion being in the centroid coordinate. This yields equations for the mean velocity, kinetic energy and shear stress which are non-local in wave number and differential in the centroid variables. A recently developed method is used to approximate the integral equations for the kinetic energy and shear stress. It is shown that the spatial average of the kinetic energy spectrum is given by the Kolmogoroff distribution. If we take this result as an approximation to the general kinetic energy function, a simple solution for the mean velocity profile results, which compares fairly well with experimental results as does the distribution of kinetic energy and shear stress.

Scaling laws for fully developed turbulent shear flows. Part 1. Basic hypotheses and analysis

1993 ◽  
Vol 248 ◽  
pp. 513-520 ◽  
Author(s):  
G. I. Barenblatt
Keyword(s):  
Reynolds Number ◽  
Velocity Distribution ◽  
Scaling Laws ◽  
Scaling Law ◽  
Turbulent Shear ◽  
Power Exponent ◽  
Turbulent Shear Flow ◽  
Mean Velocity ◽  
Friction Law ◽  
The Mean

The present work consists of two parts. Here in Part 1, a scaling law (incomplete similarity with respect to local Reynolds number based on distance from the wall) is proposed for the mean velocity distribution in developed turbulent shear flow. The proposed scaling law involves a special dependence of the power exponent and multiplicative factor on the flow Reynolds number. It emerges that the universal logarithmic law is closely related to the envelope of a family of power-type curves, each corresponding to a fixed Reynolds number. A skin-friction law, corresponding to the proposed scaling law for the mean velocity distribution, is derived.In Part 2 (Barenblatt & Prostokishin 1993), both the scaling law for the velocity distribution and the corresponding friction law are compared with experimental data.

The turbulent mixing layer with an asymmetrical distribution of temperature

1978 ◽  
Vol 89 (3) ◽  
pp. 561-587 ◽  
Author(s):  
Claude Béguier ◽  
Louis Fulachier ◽  
James F. Keffer
Keyword(s):  
Mixing Layer ◽  
Low Frequency ◽  
Passive Scalar ◽  
Frequency Component ◽  
Temperature Fields ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Turbulent Mixing Layer ◽  
The Mean ◽  
Large Eddies

An experimental programme has been carried out to examine the spread of heat as a passive scalar contaminant in a turbulent shear flow. The situation involves a slightly heated two-dimensional jet expanding into a quiescent medium on one side and a uniform stream with velocity equal to that of the warm jet on the other. Thus the developed flow is a typical mixing layer with an asymmetric mean temperature profile superimposed on it. Measurements of the mean and fluctuating velocity and temperature fields show the existence of a region where the production of temperature fluctuations is negative. Spectral analysis in this zone indicates a separation of large and small wavenumber components of the cospectrum into two regimes. The sign of the high-frequency portion is consistent with gradient-transport concepts while the low-frequency component is of opposite sign. From this it is inferred that the large eddies are mainly responsible for the negative production. A mathematical model has been developed to describe the transport within this region.

An experimental examination of the large-eddy equilibrium hypothesis

1966 ◽  
Vol 24 (1) ◽  
pp. 89-98 ◽  
Author(s):  
Ian S. Gartshore
Keyword(s):  
Reynolds Shear Stress ◽  
Turbulent Shear ◽  
Two Dimensional ◽  
Turbulent Shear Flow ◽  
Mean Velocity ◽  
Large Eddy ◽  
Energy Equilibrium ◽  
The Mean ◽  
Large Eddies ◽  
Made In

The large-eddy energy equilibrium hypothesis states that the largest eddies of a turbulent shear flow are in approximate energy equilibrium throughout a significant part of their lives. This hypothesis leads to a relationship between the mean rate of shear strain and the Reynolds shear stress which involves the scale of the large eddies. By assuming that the large-eddy scale is proportional to the standard deviation of the free turbulent boundary, or laminar superlayer, the validity of this hypothesis may be checked experimentally. Intermittency and mean velocity measurements made in five different two-dimensional shear flows are presented and these results, together with values calculated from Townsend's measurements in a two-dimensional wake, support the form of relationship suggested by the energy equilibrium hypothesis.

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