John L. Lumley

THE DYNAMICS OF COHERENT STRUCTURES IN THE WALL REGION OF A TURBULENT BOUNDARY LAYER

We have modelled the wall region of a turbulent boundary layer by expanding the instantaneous field in so-called empirical eigenfunctions, as permitted by the proper orthogonal decomposition theorem (Lumley 1967, 1981). We truncate the representation to obtain low-dimensional sets of ordinary differential equations, from the Navier-Stokes equations, via Galerkin projection. The experimentally determined eigenfunctions of Herzog (1986) are used ; these are in the form of streamwise rolls. Our model equations represent the dynamical behaviour of these rolls. We show that these equations exhibit intermittency, which we analyse using the methods of dynamical systems theory, as well as a chaotic regime. We argue that this behaviour captures major aspects of the ejection and bursting events associated with streamwise vortex pairs which have been observed in experimental work (Kline et al. 1967). We show that although this bursting behaviour is produced autonomously in the wall region, and the structure and duration of the bursts is determined there, the pressure signal from the outer part of the boundary layer triggers the bursts, and determines their average frequency. The analysis and conclusions drawn in this paper appear to be among the first to provide a reasonably coherent link between low-dimensional chaotic dynamics and a realistic turbulent open flow system.

Computational Modeling of Turbulent Flows

Publisher Summary This chapter discusses that in many situations of practical importance, “second-order modeling” technique makes possible computations that often agree with what data is available. Inevitably, the technique is also applied in many situations in which data does not exist, which must be regarded as a dangerous practice as the limitations of the technique are not known with any precision. It is primarily the possibility of practical computation that has been responsible for the great interest in this method. Even in its most stripped-down form, it results in general in the simultaneous solution of four partial differential equations in the domain of interest; more elaborate models in a three-dimensional situation might require the simultaneous solution of as many as 36 partial differential equations to obtain the mechanical field only. This is within the capabilities of present computers at a reasonable price, which cannot be said of any other technique. The chapter explores that direct simulation is not an alternative for practical computation. The various sophisticated closures suffer from essentially the same problems as the direct simulations and therefore, are limited to homogeneous situations. Thus, the second-order modeling is the only possibility for practical computation.

Turbulence measurements in axisymmetric jets of air and helium. I - Air jet. II - Helium jet

A turbulent round jet of helium was studied experimentally using a composite probe consisting of an interference probe of the Way-Libby type and an x-probe. Simultaneous measurements of two velocity components and helium mass fraction concentration were made in the x/d range 50-120. These measurements are compared with measurements in an air jet of the same momentum flux reported in Part 1. The jet discharge Froude number was 14000 and the measurement range was in the intermediate region between the non-buoyant jet region and the plume region

Some measurements of particle velocity autocorrelation functions in a turbulent flow

Particle velocity autocorrelations of single spherical beads (46·5 μhollow glass, 87 μ glass, 87 μ corn pollen, and 46·5 μ copper) were measured in a grid-generated turbulence. The hollow glass beads were small and light enough to behave like fluid points; the other types had significant inertia and ‘crossing trajectories’ effects. The autocorrelations decreased much faster for heavier particles, in contradiction to previous experimental results. The integral scale for the copper beads was 1/3 of that for the hollow glass beads. The particle velocity correlations and the Eulerian spatial correlation were coincident within experimental error when the separation was non-dimensionalized by the respective integral scale. The data generated by the hollow glass beads can be used to estimate Lagrangian fluid properities. The Lagrangian time integral scale is approximated by L / u ′, where L is the Eulerian integral scale and u ′ is the turbulence intensity.

The return to isotropy of homogeneous turbulence

Three types of homogeneous anisotropic turbulence were produced by the plane distortion, axisymmetric expansion and axisymmetric contraction of grid-generated turbulence, and their behaviour in returning to isotropy was experimentally studied using hot-wire anemometry. It was found that the turbulence trajectory after the plane distortion was highly nonlinear, and did not follow Rottas linear model in returning to isotropy. The turbulence wanted to become axisymmetric even more than it wanted to return to isotropy. In order to show the rate of return to isotropy of homogeneous turbulence, a map of the ratio of the characteristic time scale for the decay of turbulent kinetic energy to that of the return to isotropy was constructed. This demonstrated that the rate of return to isotropy was much lower for turbulence with a greater third invariant of the anisotropy tensor. The invariant technique was then applied to the experimental results to develop a new turbulence model for the return-to-isotropy term in the Reynolds stress equation which satisfied the realizability conditions. The effect of the Reynolds number on the rate of return to isotropy was also investigated and the results incorporated in the proposed model.

An experimental study of the decay of temperature fluctuations in grid-generated turbulence

Previous measurements of the decay rate of the fluctuation intensity of passive scalars in grid-generated turbulence show large variation. New results presented here show that the decay rate of passive temperature fluctuations produced by heating the grid is a function of the initial temperature fluctuation intensity. Although a full reason for this is wanting, spectra of the temperature fluctuations show that, by varying the heat applied to the grid, the wavenumber of the maximum in the temperature spectrum changes, indicating that the geometry of the thermal fluctuations is being altered in some way. In these experiments the one-dimensional temperature spectrum shows an anomalous

Viscous Sublayer and Adjacent Wall Region in Turbulent Pipe Flow

-\frac{5}{3}

Interpretation of Time Spectra Measured in High‐Intensity Shear Flows

slope. In order to eliminate the dependence of the decay rate of the temperature fluctuations on their intensity, we describe a new way of generating temperature fluctuations by means of placing a heated parallel array of fine wires (a mandoline ) downstream from the unheated grid. Results of this experiment show that the decay rate of passive thermal fluctuations is uniquely determined by the wave-number of the initial temperature fluctuations. In this type of flow there appears to be no equilibrium value for the thermal fluctuation decay rate and hence for the mechanical/thermal time-scale ratio since the thermal fluctuation decay rate does not change within the tunnel length, which is the equivalent of nearly one turbulence decay time.

Coherent Structures in Turbulence

The boundary‐layer research facility utilizing the highly viscous fluid, glycerine, was constructed to permit detailed experimental investigation of the viscous sublayer. At a pipe Reynolds number of 8700 the sublayer thickness corresponding to a nondimensional distance from the wall of yuτ/ν = 5.00 was 0.110 in. Detailed measurements of the streamwise fluctuating velocities were obtained with hot‐film anemometers within the viscous sublayer as well as in the transition region between the linear and logarithmic mean velocity profiles. These data were used to form the space‐time correlation function of the streamwise fluctuating velocities. An eigenfunction decomposition of the streamwise fluctuating velocity into a sum of products of eigenfunctions in the inhomogeneous coordinate direction, with random coefficients dependent on the homogeneous and stationary variables, was obtained from the correlation data. One dominant eigenfunction with a structure nearly identical to the mean velocity in the wall regi...

Toward a turbulent constitutive relation

Fisher and Davies have suggested several mechanisms which can prevent the direct interpretation of time spectra as space spectra by the frozen pattern hypothesis. It is shown that, when various criteria such as that of Lin are satisfied, all but one of these mechanisms, the fluctuation of convection velocity, can be disregarded at high frequencies. An analytical model of this mechanism is constructed and several of its properties examined. On the basis of this model, a correction formula is obtained for the high‐frequency part of the one‐dimensional streamwise spectrum. This is applied to experimental data taken in a curved turbulent mixing layer. It is found that even under extreme circumstances corrections from this source are not likely to exceed 30%.