Grid Reynolds Number Research Articles

Decay of vorticity in homogeneous turbulence.

We report on observations of turbulent behavior made without requiring the use of Taylor's ``frozen turbulence'' hypothesis. Initially, a towed grid generates homogeneous turbulence of grid Reynolds number of order ${10}^{5}$ within a stationary channel filled with helium II. The subsequent decay in time t of the line density of quantum vortices is measured by second sound attenuation, and the associated rms vorticity \ensuremath{\omega} follows the behavior expected of a classical fluid with \ensuremath{\omega}\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}3/2}$, consistent with the notion of a coupled turbulent state of helium II. This technique also yields the time dependence of the Kolmogorov microscale.

Micromixing in grid-generated turbulence: Theoretical analysis and experimental study

The properties of the decaying turbulence downstream from a grid depend upon its design (diameter and spacing of the rods). Over a certain range the turbulent energy dissipation rate decays as the inverse square of the distance from the grid. The local rate of micromixing, which is due to the engulfing action of energy-dissipating vortices, then decreases linearly with increasing distance. It is first shown how micromixing in this inhomogeneous field can be calculated. Semi-batch reactor operation with a single, concentric feed tube, situated in grid turbulence, is considered. Modeling encompasses micromixing, self-engulfment and radial turbulent dispersion of the feed. The product distribution of four competitive-consecutive reactions is calculated by integrating three ordinary differential equations. Experimental results (30 runs), using various flow rates and flow rate ratios, feed locations, concentrations and feed pipes, are reported and compared with the model. The principal conclusion is that the feed tube was a source of additional turbulence, which also decays rapidly. A correlation, which includes the grid Reynolds number, the feed point location and the bore of the feed tube, accounting for this additional turbulence was established.

A compact unconditionally stable finite-difference method for transient one-dimensional advection-diffusion

A new compact second-order finite-difference method for solving the time-dependent one-dimensional constant-coefficient advection—diffusion equation is developed. It involves a spread of only two grid points in space and two levels in time. It may be evaluated in an explicit marching manner and requires the availability of only upstream boundary values. It is unconditionally stable and has the property of giving ‘non-negative’ values for a large range of Courant and diffusion numbers. For all grid Reynolds numbers greater than 2 it is shown to be more accurate than commonly used second-order methods, all of which involve a spread of three grid points in space.

On the Importance of Vertical Resolution in Certain Ocean General Circulation Models

Abstract In centered difference models of ocean circulation, two grid-point computational modes can be excited if grid Reynolds and Peclet numbers are greater than two. The Bryan-Cox General Circulation Model (GCM) is used to show the dramatic effect that this instability has on the equatorial thermohaline circulation. In many recent numerical calculations researchers have used 12 vertical levels. It is shown that this resolution produces an artificial cell at the equator when typical values of the vertical diffusivity and viscosity parameters are used. This artifical cell rotates counter to the primary cell driven by deep water formation at high latitudes, is driven by downwelling at the eastern boundary near the equator and is 40% the strength of the primary cell for the parameters used in the present study. When the vertical resolution is increased the cell vanishes. It is suggested therefore that higher vertical resolution should be used in Bryan-Cox GCM deep-ocean modeling studies when current values...

Fully coupled solution of the equations for incompressible recirculating flows using a penalty-function finite-difference formulation

This paper describes two new solution algorithms for steady recirculating flows that use a penalty formulation to eliminate the pressure from the finite difference form of the governing equations. One algorithm uses successive substitution to linearize the equations, while the other employs the Newton-Raphson linearization. In both cases, the equations are solved in a fully coupled manner using a sparse matrix form of LU decomposition. The D'Yakonov iteration is used to avoid unnecessary factorizations of the coefficient matrix, significantly improving the computational efficiency. The Newton-Raphson linearization leads to faster convergence, but the execution times of the two methods are comparable. The algorithms converge rapidly and are robust to changes in grid size and Reynolds number. In a number of laminar two-dimensional flows, the new methods proved to be two to ten times faster than some conventional iterative methods.

Transformation invariance of the longitudinal velocity correlation in grid-generated turbulence at high Reynolds numbers

For grid Reynolds numbers from 12800 to 81000, the Frenkiel-Klebanoff-Huang data for approximately isotropic homogeneous grid-generated turbulence shows that the longitudinal correlation function is given by the simple empirical expression f = [1 + (r/2L)]−3, where r(≥ 0.01M) is the separation distance between two points in the fluid flow and L = L(t) is the integral scale. It follows that the longitudinal velocity correlation 〈u1(x + re,t)u1(x,t)〉 = u2f with e = (1,0,0) is invariant under the separation-distance time-contraction transformations for all positive parameter values λ ≤ 1. Conversely, if the longitudinal correlation function is prescribed to have the form f = J (r/L(t)), then the indicated transformation invariance holds if and only if [Fscr ](ξ) = (1 + ½ξ)−3. It is also shown that a Gaussian normal probability distribution at t = 0 and the Karman-Howarth equation for all t > 0 are compatible with the transformation invariance and associated expression for f.

Large-scale symmetry in the longitudinal correlation function of isotropic homogeneous fluid turbulence.

It is observed that the Frenkiel-Klebanoff-Huang turbulence data for the longitudinal correlation function at grid Reynolds numbers from 12 800 to 81 000 are subsumed by the simple empirical expression f=[1+(r/2L${)]}^{\mathrm{\ensuremath{-}}3}$, where r denotes the separation distance between two points in the turbulent fluid flow and L=L(t) is the integral scale. Hence, the longitudinal velocity correlation is rescaled by the self-similarity dilatation factor ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}3}$ under the separation-distance transformations r\ensuremath{\rightarrow}\ensuremath{\lambda}r+(\ensuremath{\lambda}-1)2L at fixed t.

Simulation of Flow in Rectangular Clarifiers

A numerical model has been developed to simulate flow in rectangular clarifiers operating at neutral density conditions. The equations of motion in the ψ-ω format are solved using a finite‐difference method. A two‐step ADI three time level, weighted upwind‐centered difference scheme is used to solve the ω‐equation. The centered‐difference analog of the ψ‐equation is solved using the SOR method. A truncation convergence criterion was derived where the local Courant Number has to be kept below one. Computational stability depends on both Courant number and grid Reynolds number. A variable size mesh reduces both execution time and storage requirements without loss of solution accuracy. A partial slip boundary condition improves on the turbulence model. The finite difference and experimental flow fields are in good agreement.

Development of a second order approximation for the Navier-Stokes equations

The purpose of this paper is the development of a 2nd order finite difference approximation to the steady state Navier-Stokes equations governing flow of an incompressible fluid in a closed cavity. The approximation leads to a system of equations that has proved to be very stable. In fact, numerical convergence was obtained for Reynolds numbers up to 20,000. However, it is shown that extremely small mesh sizes are needed for excellent accuracy with a Reynolds number of this magnitude. The method uses a nine point finite difference approximation to the convection term of the vorticity equation. At the same time it is capable of avoiding values at corner nodes where discontinuities in the boundary conditions occur. Figures include level curves of the stream and vorticity functions for an assortment of grid sizes and Reynolds numbers.

Resolution of downstream boundary layers in the Chebyshev approximation to viscous flow problems

Steady solutions characterized by downstream boundary layers are sought to the one-dimensional linearized Burgers equation for a variety of computational—that is, spatial and temporal differencing—schemes. When well resolved, the presence of these boundary layers does not seriously affect the accuracy of the interior solutions. For narrow outflow boundary layers, however, the Chebyshev collocation method may be unstable, unlike the finite-difference technique, for grid Reynolds numbers less than a certain critical value. Although the spectral solutions are improved by using fractional step time-differencing methods in which the viscous and advective effects are separately treated, the stability of the resulting multistep technique need not be guaranteed by the stability of its component steps. Analogous computational restrictions on the use of the Chebyshev collocation method are shown to hold for the nonlinear Burgers equation.

The accuracy of finite difference schemes for the numerical solution of the Navier-Stokes equations

Comparisons are made between various finite difference algorithms used for the numerical solution of two-dimensional viscous flow. Central difference algorithms recently developed by the authors are compared with the well established upwind difference algorithms. Some analytical solutions to the Navier-Stokes equations have been produced in order for meaningful comparisons to be made. On the basis of the results, the new central difference algorithms are recommended as being more accurate whilst requiring little additional computational effort. There is an upper limit on the grid Reynolds number for which these new methods will converge, but this will usually be when the flow would in practice be turbulent.

Unified Analysis of Grid Turbulence

The deficiencies of the conventional power laws for the decay of grid turbulence, i.e., the discontinuous change from an initial to a final zone and the undefined range of validity, have been successfully overcome by analyzing grid flow as the plane-source counterpart of flows past point and line sources of turbulence energy. A comparison of the analytical results with experimental data on turbulence intensity, dissipation length, and integral scale for a great variety of grid geometries and Reynolds numbers reveals good agreement for all down-stream zones. The data used include those of Baines and Peterson, Batchelor and Townsend, Comte-Bellot and Corrsin, Dryden, Stewart and Townsend, Uberoi, van der Hegge-Zijnen, von Karman, and Wyatt. Functional relationships between the constants of the resulting analytical expressions, including the position of the virtual origin, and the grid-flow parameters have been deduced to facilitate prediction of turbulence characteristics and their downstream development for any given grid flow.

Functional Integration Theory for Incompressible Fluid Turbulence. II

Two iterative solutional procedures are reported for the nonlinear integro-differential dynamical equation obtained previously by the stationary functionality method and supplemented here by a subsidiary dynamical condition. Rapidly convergent for grid Reynolds numbers between about 100 and 300, both methods yield general expressions for the two-point equal-time velocity-correlation tensor in approximate agreement with experiment for the initial period of decay.

SIMULATION OF HORIZONTAL TURBULENT DIFFUSION OF PARTICLES UNDER WAVES

By oscillation of an array of turbulence-generating grids in still water, the turbulent fluid velocity field m shoaling waves near the bottom is simulated in a laboratory channel. Solid particles with fall velocities varying between 1 and 40 mm/sec are introduced into the test volume from above. Multiple- image photography using ultraviolet lighting techniques and a suitably placed mirror allow recording of the gram trajectories as functions of time and three space dimensions simultaneously. The Lagrangian intensities of turbulence and diffusion coefficients are then directly measured from the photographic data. The scale times, scale lengths, and the frequencies of the power spectra modes can then be calculated. Properties of the fluid turbulence are inferred from the quasi-neutral particles. The analysis, which is restricted to the component of diffusion in the hori2ontal direction normal to the grid motion, shows that the turbulent velocity distributions of both fluid and heavy particles are Gaussian, and that their standard deviations (intensities of turbulence) increase regularly with increasing grid Reynolds numbers (grid speeds). Diffusion coefficients likewise generally increase with increasing grid Reynolds numbers. Diffusivities of the heavy particles relative to the fluid are a function of both particle fall velocity and the structure of the fluid turbulence itself.

Grid turbulence at large Reynolds numbers

Measurements of grid turbulence have been obtained for grid Reynolds numbers ranging from 2·4 × 106 to 1·2 × 105. The decay law and the effect of Reynolds number on the turbulence level are established. The measured power spectra of the turbulence are consistent with Kolmogoroff scaling for kη > 0·1 (k is the wave-number and η is the Kolmogoroff length); but for kη < 0·1, the spectra of the stream-wise turbulence velocity component and of the cross-stream component do not appear to be isotropically related. However, the stream-wise spectrum does display a region, which increases in extent with increasing Reynolds number.

The diffusion behind a line source in homogeneous turbulence

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.

Similarity and self-preservation in isotropic turbulence

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.

Decay of vorticity in isotropic turbulence

The equation describing the rate of change of the mean square vorticity in homogeneous isotropic turbulence is obtained and the terms occurring therein are discussed. A negative contribution to d͞ω 2 / dt arises from the effect of viscosity, while a positive contribution is produced by the tendency for the random diffusive motion to extend the vortex lines. This latter contribution can be related to the skewness of the probability distribution of the rate of extension of line elements of the fluid aligned in any given direction. The results of direct measurements of each of the factors appearing in the vorticity equation are then described. The measurements were made by analyzing electrically the output from a hot-wire anemometer placed downstream from a grid in a uniform stream. Both U 2 / u͞ 2 and ʎ 2 are found to increase approximately linearly with time during decay of the turbulence and their rates of change are consistent with the energy equation. The skewness factor mentioned above is approximately constant during decay, with the same value at all Reynolds numbers. It follows that the rate of increase of ͞ω 2 ¯ due to vortex extension is proportional to ( ͞ω 2 ) 3/2 , and further measurements show that the effect of viscosity has a similar dependence, so that the ratio of the two contributions to d͞ω 2 / dt remains the same throughout the decay. The viscous contribution is always the greater but the contributions tend to equality as the grid Reynolds number increases. The measurements of all terms in the vorticity equation are shown to satisfy the equation with sufficient accuracy. One of the deductions from the measurements is that the double velocity correlation function tends to a cusp at the origin as the Reynolds number increases indefinitely.