A simple turbulence closure hypothesis for the triple-velocity correlation functions in homogeneous isotropic turbulence

1984 ◽  
Vol 140 ◽  
pp. 45-61 ◽  
Author(s):  
J. A. Domaradzki ◽  
G. L. Mellor
Keyword(s):  
Experimental Data ◽  
Correlation Functions ◽  
Eddy Viscosity ◽  
Isotropic Turbulence ◽  
Initial Period ◽  
Laboratory Data ◽  
Reynolds Numbers ◽  
Isotropic Case ◽  
Velocity Correlation ◽  
Viscosity Function

A simple two-point closure scheme for homogeneous axisymmetric turbulence is developed. For the isotropic case it is essentially an eddy-viscosity assumption in real space for the Kármán-Howarth equation. The eddy-viscosity function for large internal Reynolds numbers is derived from Kolmogoroff's 1941 theory. For moderate Reynold's numbers of order 102, approximately the same expression for the eddy-viscosity function is determined from experimental data. The resulting closed equation for the double-correlation function is solved numerically for both large and moderate Reynolds numbers, and the results are compared with experimental data. Self-similar solutions of the basic equation predict turbulent energy decay inversely proportional to time. It is shown that the departure from this ‘initial-period decay law’ observed in laboratory data is due to the behaviour of grid-produced correlation functions for large separation distances.

A comparison of heisenberg's spectrum of turbulence with experiment

1951 ◽  
Vol 47 (1) ◽  
pp. 158-176 ◽  
Author(s):  
Ian Proudman ◽  
G. K. Batchelor
Keyword(s):  
Reynolds Number ◽  
Correlation Functions ◽  
Isotropic Turbulence ◽  
Reynolds Numbers ◽  
Velocity Correlation ◽  
Approximate Form ◽  
Velocity Correlation Functions ◽  
Limiting Case

AbstractIn this paper, the theoretical double and triple velocity correlation functions, f(r), g(r) and h(r), which correspond to Heisenberg's spectrum of isotropic turbulence, are obtained numerically for two Reynolds numbers. One set of these correlations is for the limiting case of infinite Reynolds number. In addition, a method is developed for deriving the approximate form of the double correlations for any Reynolds number, which is not too small, from the corresponding correlations for infinite Reynolds number. These theoretical correlations are then compared with the results of experiment.

Triple velocity correlations in isotropic turbulence

1951 ◽  
Vol 47 (1) ◽  
pp. 146-157 ◽  
Author(s):  
R. W. Stewart
Keyword(s):  
Reynolds Number ◽  
Kinematic Viscosity ◽  
Isotropic Turbulence ◽  
Initial Period ◽  
Reynolds Numbers ◽  
Velocity Correlation ◽  
Mean Wind Speed ◽  
The Mean ◽  
Mesh Grid ◽  
Working Section

AbstractThe triple velocity correlation, in turbulence produced by inserting a square-mesh grid near the beginning of the working section of a wind tunnel, has been measured for mesh Reynolds numbers of RM = 5300, 21,200 and 42,400 (RM = UM/ν, where U is the mean wind speed in the working section of the tunnel and M is the centre to centre spacing of the rods making up the grid; ν is the kinematic viscosity of air). At the lowest Reynolds number the correlation has been measured at distances downstream of the grid varying from 20 to 120M. This range covers practically all of the initial period of the decay of turbulence, where the turbulent intensity varies as t−1.

Recirculation within a fluid sphere at moderate Reynolds numbers

2002 ◽  
Vol 465 ◽  
pp. 293-300 ◽  
Author(s):  
D. A. BARRY ◽  
J.-Y. PARLANGE
Keyword(s):  
Experimental Data ◽  
Vertical Velocity ◽  
Laboratory Data ◽  
Major Axis ◽  
Reynolds Numbers ◽  
Fluid Sphere ◽  
Stagnation Region ◽  
Data Set ◽  
Different Levels

Motion of a single fluid sphere is described by two theories, each characterized by different levels of Hill's vortex circulation within the sphere. An existing experimental data set giving measurements of vertical velocity along the major axis of the sphere is re-examined. Contrary to published discussions of that experiment, we find that the theory of Parlange agrees better with the laboratory data than that of Harper & Moore. This agreement supports the key difference between the two theories, i.e. that the fluid within the sphere is unlikely to have a singular (infinite) velocity as it moves upwards towards the stagnation region at the top of the sphere.

Similarity and self-preservation in isotropic turbulence

1951 ◽  
Vol 243 (867) ◽  
pp. 359-386 ◽  
Keyword(s):  
Energy Spectrum ◽  
Isotropic Turbulence ◽  
Initial Period ◽  
Reynolds Numbers ◽  
Wave Number ◽  
Inertial Subrange ◽  
Local Similarity ◽  
Mean Flow ◽  
Spectrum Function ◽  
Energy Spectrum Function

Measurements of the double and triple velocity correlation functions and of the energy spectrum function have been made in the uniform mean flow behind turbulence-producing grids of several shapes at mesh Reynolds numbers between 2000 and 100000. These results have been used to assess the validity of the various theories which postulate greater or less degrees of similarity or self-preservation between decaying fields of isotropic turbulence. It is shown that the conditions for the existence of the local similarity considered by Kolmogoroff and others are only fulfilled for extremely small eddies at ordinary Reynolds numbers, and that the inertial subrange in which the spectrum function varies as k -35 ( k is the wave-number) is non-existent under laboratory conditions. Within the range of local similarity, the spectrum function is best represented by an empirical function such as k -a log k , and it is concluded that all suggested forms for the inertial transfer term in the spectrum equation are in error. Similarity of the large scale structure of flows of differing Reynolds numbers at corresponding times of decay has been confirmed, and approximate measurements of the Loitsianski invariant in the initial period have been made. Its value, expressed non-dimensionally, decreases slowly with grid Reynolds number within the range of observation. Turbulence-producing grids of widely different shapes are found to produce flows identical in energy decay and in structure of the smaller eddies. The largest eddies depend markedly on the grid shape and are, in general, significantly anisotropic. Within the initial period of decay, the greater part of the energy spectrum function is self-preserving, and this part has a shape independent of the shape of the turbulence-producing grid. The part that is not self-preserving contains at least one-third of the total energy, and it is concluded that theories postulating quasi-equilibrium during decay must be considered with great caution.

scholarly journals Decay of isotropic turbulence in the initial period

1948 ◽  
Vol 193 (1035) ◽  
pp. 539-558 ◽  
Keyword(s):  
Correlation Function ◽  
Isotropic Turbulence ◽  
Initial Period ◽  
Rate Of Change ◽  
Velocity Correlation ◽  
Integral Scale ◽  
Direct Measurements ◽  
Theoretical Predictions ◽  
History Of ◽  
Velocity Correlation Functions

In a previous paper the authors described direct measurements of all the terms in the equation for the rate of change of mean square vorticity in isotropic turbulence. The present paper is concerned with developments arising from the earlier work and with the experimental verification of some recent theoretical investigations. The results of measurements of the turbulent intensity u ' and of λ are presented; these establish that u' -2 and λ 2 are each proportional to the time of decay provided that the time is not too large. Within this initial period of the decay, the double and triple velocity correlation functions are found to maintain their form, i.e. to be self-preserving, for small values of the distance r between the two points at which the correlations are taken. For larger separations the double velocity correlation function changes its form slightly during decay and direct measurements of λ and of the integral scale L show that λ/ L increases during the decay. Theoretical predictions about the shape of the correlation function, for limited ranges of r , at high and at low Reynolds numbers are compared with measurements. Theory has shown that the above decay law cannot persist indefinitely, and the present experiments confirm that the decay law changes in the expected direction when the time is large. A division of the life-history of the turbulence into initial, transition and final periods is suggested; within the initial period, a classification based on the Reynblds number is also possible. Some speculations on the interpretation of the initial period are presented.

Experimental evidence for the theory of local isotropy

1948 ◽  
Vol 44 (4) ◽  
pp. 560-565 ◽  
Author(s):  
A. A. Townsend ◽  
Geoffrey Taylor
Keyword(s):  
Isotropic Turbulence ◽  
Reynolds Numbers ◽  
Velocity Correlation ◽  
Integral Scale ◽  
Local Isotropy ◽  
High Reynolds Numbers ◽  
Wide Range ◽  
Skewness Factor ◽  
High Reynolds ◽  
Image Position

Some new measurements of isotropic turbulence produced behind a biplane grid have been made at high Reynolds numbers, and these results are compared with the predictions of the theory of local isotropy developed by A. N. Kolmogoroff. The transverse double-velocity correlation has been measured at mesh Reynolds numbers up to 3·2 × 105, and the observed form agrees well with the predicted form. Measurements of the skewness factor of velocity differences over finite intervals have also been made, and the factor is nearly constant and equal to −0·38, if the interval is small compared with the integral scale. The invariance of dimensionless functions of the velocity derivatives has been confirmed for the flattening factor of ∂u/∂x, namely,which is nearly constant over a wide range of conditions. It is concluded that the theory of local isotropy is substantially correct for isotropic turbulence of high Reynolds number.

scholarly journals Decay of turbulence in the final period

1948 ◽  
Vol 194 (1039) ◽  
pp. 527-543 ◽  
Keyword(s):  
Correlation Coefficient ◽  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Initial Period ◽  
Energy Decay ◽  
Transitional Period ◽  
Asymptotic State ◽  
Time Interval ◽  
Velocity Correlation ◽  
Time Intervals

The final period of decay of a turbulent motion occurs when the effects of inertia forces are negligible. Under these conditions the instantaneous velocity distribution in the turbulence field may be solved as an initial value problem. It is shown that homogeneous turbulence tends to an asymptotic statistical state which is independent of the initial conditions. In this asymptotic state the energy of turbulence is proportional to t -5/2 and the longitudinal double-velocity correlation coefficient for two points distance r apart is e -r2/svt , where t is the time of decay. The asymptotic time-interval correlation coefficient is found to be different from unity for very large time intervals only, showing the aperiodic character of the motion. The whole field of motion comes gradually to rest, smaller eddies decaying more rapidly than larger eddies, and the above stable eddy distribution is established when only the largest eddies of the original turbulence remain. Relevant measurements have been made in the field of isotropic turbulence downstream from a grid of small mesh. The above energy decay and space-interval correlation relations are found to be valid at distances from the grid greater than 400-mesh lengths and at a mesh Reynolds number of 650. The duration of the transitional period, in which the energy decay law is changing from that appropriate to the initial period of decay to the above asymptotic law, increases very rapidly with R M . There is a brief discussion of the criterion for the existence of final period decay, although clarification must wait until the existence and termination of the initial period of decay are better understood.

The diffusion behind a line source in homogeneous turbulence

1954 ◽  
Vol 224 (1159) ◽  
pp. 487-512 ◽  
Keyword(s):  
Turbulent Flow ◽  
Decay Time ◽  
Molecular Diffusion ◽  
Initial Period ◽  
Reynolds Numbers ◽  
Particle Diffusion ◽  
Velocity Correlation ◽  
Rate Of Spread ◽  
Heated Wire ◽  
The Mean

The theory of turbulent diffusion by continuous movements relates the mean particle diffusion from a fixed source to the Lagrangian velocity correlation function, and measure­ments of the diffusion of heat behind a thin heated wire in a uniform turbulent flow have been used to compute this correlation, assuming that the processes of diffusion by con­tinuous movements and molecular conduction are statistically independent. A series of measurements both of the mean temperatures and the temperature fluctuations in the wake of a thin heated wire has been made in the uniform turbulent flow behind bi-plane grids, for grid Reynolds numbers between 2700 and 21000 and within the initial period of decay of the turbulence. In these measurements, the rate of spread of the heat wake was determined in two ways, directly from measurements of the turbulent transport of heat and by numerical differentiation of widths computed from observations of mean temperature. The extent of the accelerated diffusion of heat, which is caused by intensification of the temperature gradients by the turbulent motion, can be computed from the measurements of lateral temperature correlations in the flow, and was found to be comparable with the total diffusion. Both the total diffusion process and the process of accelerated molecular dif­fusion are very nearly self-preserving during decay in the initial period, with a time scale that varies as the decay time and a velocity scale varying inversely as the square root of the decay time, which is consistent with the observed self-preservation of Eulerian correlations. The rate of spread of the heat wake is simply related to the particle diffusion only for short diffusion times at ordinary Reynolds numbers, and the mean-square particle accelera­tion can be computed. The results are significantly larger than those found by other workers who have neglected the additional spread of the wake by accelerated molecular diffusion.

scholarly journals Three-dimensional Lagrangian Voronoï analysis for clustering of particles and bubbles in turbulence

2012 ◽  
Vol 693 ◽  
pp. 201-215 ◽  
Author(s):  
Yoshiyuki Tagawa ◽  
Julián Martínez Mercado ◽  
Vivek N. Prakash ◽  
Enrico Calzavarini ◽  
Chao Sun ◽  
...  
Keyword(s):  
Experimental Data ◽  
Isotropic Turbulence ◽  
Response Times ◽  
Three Dimensional ◽  
Voronoi Cell ◽  
Reynolds Numbers ◽  
Point Particle ◽  
Heavy Particles ◽  
Data Set ◽  
Light Particles

AbstractThree-dimensional Voronoï analysis is used to quantify the clustering of inertial particles in homogeneous isotropic turbulence using data sets from numerics in the point particle limit and one experimental data set. We study the clustering behaviour at different density ratios, particle response times (i.e. Stokes numbers $\mathit{St}$) and two Taylor–Reynolds numbers (${\mathit{Re}}_{\lambda } = 75$ and 180). The probability density functions (p.d.f.s) of the Voronoï cell volumes of light and heavy particles show different behaviour from that of randomly distributed particles, i.e. fluid tracers, implying that clustering is present. The standard deviation of the p.d.f. normalized by that of randomly distributed particles is used to quantify the clustering. The clustering for both light and heavy particles is stronger for higher ${\mathit{Re}}_{\lambda } $. Light particles show maximum clustering for $\mathit{St}$ around 1–2 for both Taylor–Reynolds numbers. The experimental data set shows reasonable agreement with the numerical results. The results are consistent with previous investigations employing other approaches to quantify the clustering. We also present the joint p.d.f.s of enstrophy and Voronoï volumes and their Lagrangian autocorrelations. The small Voronoï volumes of light particles correspond to regions of higher enstrophy than those of heavy particles, indicating that light particles cluster in higher vorticity regions. The Lagrangian temporal autocorrelation function of Voronoï volumes shows that the clustering of light particles lasts much longer than that of heavy or neutrally buoyant particles. Due to inertial effects arising from the density contrast with the surrounding liquid, light and heavy particles remain clustered for much longer times than the flow structures which cause the clustering.

The modified cuinulant expansion for two-dimensional isotropic turbulence

1981 ◽  
Vol 110 ◽  
pp. 475-496 ◽  
Author(s):  
Tomomasa Tatsumi ◽  
Shinichiro Yanase
Keyword(s):  
Numerical Solutions ◽  
Isotropic Turbulence ◽  
Initial Period ◽  
Three Dimensional ◽  
Critical Time ◽  
Reynolds Numbers ◽  
Inviscid Limit ◽  
Inertial Subrange ◽  
Quasi Equilibrium ◽  
Two Dimensional

The two-dimensional isotropic turbulence in an incompressible fluid is investigated using the modified zero fourth-order cumulant approximation. The dynamical equation for the energy spectrum obtained under this approximation is solved numerically and the similarity laws governing the solution in the energy-containing and enstrophy-dissipation ranges are derived analytically. At large Reynolds numbers the numerical solutions yield the k−3 inertial subrange spectrum which was predicted by Kraichnan (1967), Leith (1968) and Batchelor (1969) assuming a finite enstrophy dissipation in the inviscid limit. The energy-containing range is found to satisfy an inviscid similarity while the enstrophy-dissipation range is governed by the quasi-equilibrium similarity with respect to the enstrophy dissipation as proposed by Batchelor (1969). There exists a critical time tc which separates the initial period (t < tc) and the similarity period (t > tc) in which the enstrophy dissipation vanishes and remains non-zero respectively in the inviscid limit. Unlike the case of three-dimensional turbulence, tc is not fixed but increases indefinitely as the viscosity tends to zero.

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