Z. Warhaft

On the onset of high-Reynolds-number grid-generated wind tunnel turbulence

Using an active grid devised by Makita (1991), shearless decaying turbulence is studied for the Taylor-microscale Reynolds number, R λ , varying from 50 to 473 in a small (40 x 40 cm 2 cross-section) wind tunnel. The turbulence generator consists of grid bars with triangular wings that rotate and flap in a random way. The value of R λ is determined by the mean speed of the air (varied from 3 to 14 m s -1 ) as it passes the rotating grid, and to a lesser extent by the randomness and rotation rate of the grid bars. Out main findings are as follows. A weak, not particularly well-defined scaling range (i.e. a power-raw dependence of both the longitudinal (u) and transverse (v) spectra, F 11 (k 1 ) and F 22 (k 1 ) respectively, on wavenumber k 1 ) first appears at R λ ∼ 50, with a slope, n 1 , (for the u spectrum) of approximately 1.3. As R λ was increased, n 1 increased rapidly until R λ ∼ 200 where n ∼ 1.5. From there on the increase in n 1 was slow, and even by R λ = 473 it was still significantly below the Kolmogorov value of 1.67. Over the entire range, 50 ≤ R λ ≤ 473, the data were well described by the empirical fit : n 1 = 5/3(1-3.15R λ -2/3 ). Using a modified form of the Kolmogorov similarity law : F 11 (k 1 ) = C 1 .e 2/3 k 1 -5/3 (k 1 η) 5/3-n1 where e is the turbulence energy dissipation rate and η is the Kolmogorov microscale, we determined a linear dependence between n 1 and C 1 . : C 1 . = 4.5 - 2.4n 1 . Thus for n 1 = 5/3 (which extrapolation of out results suggests will occur in this flow for R λ ∼ 10 4 ), C 1 . = 0.5, the accepted high-Reynolds-number value of the Kolmogorov constant. Analysis of the p.d.f. of velocity differences Δu(r) and Δv(r) where r is an inertial subrange interval, conditional dissipation, and other statistics showed that there was a qualitative difference between the turbulence for R λ 200 (strong turbulence). For the latter, the p.d.f.s of Δu(r) and Δv(r) had super Gaussian tails and the dissipation (both of the u and v components) conditioned on Δu(r) and Δv(r) was a strong function of the velocity difference. For R λ 200 are consistent with the predictions of the Kolmogorov refined similarity hypothesis (and make a distinction between the dynamical and kinematical contributions to the conditional statistics). They have much in common with similar statistics done in shear flows at much higher R λ , with which they are compared.

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

Passive scalar statistics in high-Péclet-number grid turbulence

-\frac{5}{3}

The effect of a passive cross-stream temperature gradient on the evolution of temperature variance and heat flux in grid turbulence

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.

Probability distribution, conditional dissipation, and transport of passive temperature fluctuations in grid‐generated turbulence

The statistics of a turbulent passive scalar (temperature) and their Reynolds number dependence are studied in decaying grid turbulence for the Taylor-microscale Reynolds number, R λ , varying from 30 to 731 (21[les ] Pe λ [les ]512). A principal objective is, using a single (and simple) flow, to bridge the gap between the existing passive grid-generated low-Peclet-number laboratory experiments and those done at high Peclet number in the atmosphere and oceans. The turbulence is generated by means of an active grid and the passive temperature fluctuations are generated by a mean transverse temperature gradient, formed at the entrance to the wind tunnel plenum chamber by an array of differentially heated elements. A well-defined inertial–convective scaling range for the scalar with a slope, n θ , close to the Obukhov–Corrsin value of 5/3, is observed for all Reynolds numbers. This is in sharp contrast with the velocity field, in which a 5/3 slope is only approached at high R λ . The Obukhov–Corrsin constant, C θ , is estimated to be 0.45–0.55. Unlike the velocity spectrum, a bump occurs in the spectrum of the scalar at the dissipation scales, with increasing prominence as the Reynolds number is increased. A scaling range for the heat flux cospectrum was also observed, but with a slope around 2, less than the 7/3 expected from scaling theory. Transverse structure functions of temperature exist at the third and fifth orders, and, as for even-order structure functions, the width of their inertial subranges dilates with Reynolds number in a systematic way. As previously shown for shear flows, the existence of these odd-order structure functions is a violation of local isotropy for the scalar differences, as is the existence of non-zero values of the transverse temperature derivative skewness (of order unity) and hyperskewness (of order 100). The ratio of the temperature derivative standard deviation along and normal to the gradient is 1.2±0.1, and is independent of Reynolds number. The refined similarity hypothesis for the passive scalar was found to hold for all R λ , which was not the case for the velocity field. The intermittency exponent for the scalar, μ θ , was found to be 0.25±0.05 with a possible weak R λ dependence, unlike the velocity field, where μ was a strong function of Reynolds number. New, higher-Reynolds-number results for the velocity field, which smoothly follow the trends of Mydlarski & Warhaft (1996), are also presented.

The anisotropy of the small scale structure in high Reynolds number (Rλ∼1000) turbulent shear flow

The evolution of temperature variance and heat flux in decaying grid turbulence with a linear cross-stream temperature gradient is studied by producing the temperature gradient by means of two different methods: ( a ) by placing a ‘mandoline’ (Warhaft & Lumley 1978) downstream from the grid but with its wires differentially heated for the present study, and ( b ) by differentially heating ribbons of nichrome (a ‘toaster’) placed in the plenum chamber of the wind tunnel. For the former method the initial thermal/mechanical lengthscale ratio L θ / L was varied by changing the mandoline configuration. For this method it is shown that the gradient causes L θ / L to equilibrate to a value of about 0·9 regardless of its initial value, and that when this value is achieved the temperature variance increases approximately linearly with time. The toaster was used to produce a temperature gradient without the associated initial temperature variance (and initial thermal lengthscale) that is necessarily produced by the mandoline wires; for the toaster the temperature variance was produced solely by the action of turbulence against the temperature gradient. For this experiment too, the thermal variance grew linearly with time, and L θ / L was approximately the same as the equilibrium value for the mandoline experiments. The equilibrium value of the ratio of temperature-variance production to temperature-variance dissipation was approximately 1·5 for all of the experiments. The ratio of the mechanical-dissipation/thermal-dissipation timescales was also found to equilibrate, but there was considerably more scatter in the data for this parameter. The values of the equilibrium length- and timescale ratios were not affected by the magnitude of the temperature gradient, which was varied for both experiments. Good transverse homogeneity in the thermal field was achieved in all cases, in contrast with previous experiments (using heated grids).

The interference of thermal fields from line sources in grid turbulence

The evolution of the scalar probability density function (pdf), the conditional scalar dissipation rate, and other statistics including transport properties are studied for passive temperature fluctuations in decaying grid‐generated turbulence. The effect of filtering and differentiating the time series is also investigated. For a nonzero mean temperature gradient it is shown that the pdf of the temperature fluctuations has pronounced exponential tails for turbulence Reynolds number (Rel) greater than 70 but below this value the pdf is close to Gaussian. The scalar dissipation rate, conditioned on the fluctuations, shows that there is a high expectation of dissipation in the presence of the large, rare fluctuations that produce the exponential tails. Significant positive correlation between the mean square scalar fluctuations and the instantaneous scalar dissipation rate is found when exponential tails occur. The case of temperature fluctuations in the absence of a mean gradient is also studied. Here, the...

The evolution of grid-generated turbulence under conditions of stable thermal stratification

The postulate of local isotropy (PLI) is tested in a wind tunnel uniform shear flow in which the Reynolds number is varied over the range 100⩽Rλ⩽1, 100(6.7×102⩽Rl⩽6.3×104). The high Rλ is achieved by using an active grid [Mydlarski and Warhaft, J. Fluid Mech. 320, 331 (1996)] in conjunction with a shear generator. We focus on the increments of the longitudinal velocity fluctuations in the direction of the mean shear. PLI requires that odd order moments of these quantities approach zero as Rλ→∞. Confirming the lower Reynolds number measurements of Garg and Warhaft [Phys. Fluids 10, 662 (1998)], we show that the skewness of ∂u/∂y decreases as Rλ−0.5 (with a value of 0.2 at Rλ∼1000). Although the decrease is slower than classical scaling arguments suggest, the result is consistent with PLI, indicating a negligible value at high Rλ. However, the normalized fifth moment, 〈(∂u/∂y)5〉/〈(∂u/∂y)2〉5/2, is of order 10, and shows no diminution with Reynolds number, while the normalized seventh moment increases with Rλ...

On passive scalar derivative statistics in grid turbulence

The interference of passive thermal fields produced by two (and more) line sources in decaying grid turbulence is studied by using the inference method described by Warhaft (1981) to determine the cross-correlation coefficient ρ between the temperature fluctuations produced by the sources. The evolution of ρ as a function of downstream distance, for 0.075 d/l d is the wire spacing and l is the integral lengthscale of the turbulence, is determined for a pair of sources located at various distances from the grid. It is found that ρ may be positive or negative (thereby enhancing or diminishing the total temperature variance) depending on the line-source spacing, their location from the grid and the position of measurement. It is also shown that the effects of a mandoline (Warhaft & Lumley 1978) may be idealized as the interference of thermal fields produced by a number of line sources. Thus new light is shed on the rate of decay of scalar-variance dissipation. The thermal field of a single line source is also examined in detail, and these results are compared with other recent measurements.

Passive scalar dispersion and mixing in a turbulent jet

The experiment was carried out in a large, open circuit, low-speed wind tunnel 0.91×0.91 m 2 and 9.14 m in length specially designed for the study of stratified turbulence. The temperature gradient, formed at the entrance to the plenum chamber of the tunnel by means of an array of 72 horizontal, differentially heated elements was varied from 0 to 55°C/m, giving a maximum Brunt-Vaisala frequency N of 1.3 s -1