Decay of Turbulence at High Reynolds Numbers

Abstract

Turbulent motions in a fluid relax at a certain rate once stirring has stopped. The role of the most basic parameter in fluid mechanics, the Reynolds number, in setting the relaxation rate is not generally known. This paper concerns the high-Reynolds-number limit of the process. In a classical grid-turbulence wind-tunnel experiment that both reached higher Reynolds numbers than ever before and covered a wide range of them (10 4 < Re = U M ν < 5 × 10 6 ), we measured the relaxation rate with the unprecedented precision of about 2%. Here U is the mean speed of the flow, M the forcing scale, and ν the kinematic viscosity of the fluid. We observed that the relaxation rate was Reynolds-number independent, which contradicts some models and supports others.PACS numbers: 47.27. Gs, 47.27.Jv Turbulence dissipates kinetic energy, and the easiest way to see this is to let turbulence relax, or freely decay, by removing the agitation that initially set the fluid in motion. In so doing, one observes qualitatively that the fluid comes to rest. The rate at which that happens is the subject at hand, and underlies general turbulence phenomena and modeling. To simplify the problem, the statistics of the turbulent fluctuations are often arranged to be spatially homogeneous and isotropic [1,2]. Even under these conditions, it is not possible quantitatively to predict the decay rate; it is even difficult precisely to measure the decay rate [3][4][5][6]. Yet empirical constraints on the properties of decaying turbulence are what is needed to advance the knowledge of the subject.The physics that control the decay are not fully understood, to the extent that the role of the most basic parameter in fluid mechanics, the Reynolds number, is not clear. What is known is that toward very low Reynolds numbers, the decay rate increases [7,8], as it also does when the spatial structure of the turbulence grows to the size of its container [4,9]. Here, we study the rate of decay in the limit of large Reynolds numbers for turbulence free from boundary effects, a rate which cannot be determined from the previous data.One theoretical framework predicts that the decay rate depends on the large-scale structure of the flow, and not on the Reynolds number once turbulence is fully developed [2, 10-13]. As the Reynolds numbers diverge to infinity, the scale at which energy is dissipated grows arbitrarily small, but these scales continue to dissipate energy as quickly as it is transferred to them by the large scales. This picture is compatible with an emerging consensus that the initial structure of the turbulence sets its decay rate, even when the flow is homogeneous [14][15][16][17].There is also, however, a line of thinking that an elegant and fully self-similar decay emerges in the limit of high Reynolds number [14,15,[18][19][20][21][22][23]. In this description, turbulence tends to become statistically similar to itself, when appropriately rescaled, even as it decays. Observation of a tendency toward slower decay with increasing Reynolds numbers wo...

Effects of multi-scale and regular grid geometries on decaying turbulence

“…While this has been implied or suggested by previous studies, this is the first time it has been explicitly demonstrated for the same Re M for grids with different geometries but the same blockage (including square fractal). This far-field decay rate is not far removed from the measurements of Sinhuber et al (2015) who tested a regular grid at a variety of Re M using a variable density facility, or Krogstad & Davidson (2010 who used both regular and multi-scale cross grids, or the seminal study of Comte-Bellot & Corrsin (1966). Thormann & Meneveau (2014) also found values of −1.16 n −1.19 in the wake of their passive square fractal wing grid which is within 6% of the present measurements.…”

Self-similar decay of high Reynolds number Taylor-Couette turbulence

“…For statistically stationary turbulence, this balance is reflected in the famous picture of the Richardson-Kolmogorov energy cascade [1,2]. While the driving on large scales clearly is non-universal, depending on the flow geometry and stirring mechanism, the energy dissipation mechanism has been hypothesized to be self-similar [3][4][5][6][7][8].How exactly is the energy taken out of the system? A good way to find out is to turn off the driving and follow the then decaying turbulence, as then all scales are probed during the decay process.…”

“…This has been done in various studies over the last decades for homogeneous isotropic turbulence (HIT). Experimentally, the focus of attention was on grid-induced turbulence [8][9][10][11][12][13][14][15], whereas in numerical simulations periodic boundary conditions were used [16][17][18][19]. To what degree the decay of the turbulence depends on the initial conditions [20][21][22] and whether or not it is selfsimilar has controversially been debated [5,11,16,[23][24][25][26][27].…”

“…[8] for decaying homogeneous isotropic turbulence. We suddenly stop the inner cylinder rotation (similarly as in Ref.…”

Validation scheme for small effect of wind tunnel blockage on decaying grid-generated turbulence

“…Although theoretical studies have indicated that the decay exponent can take values of n = 1, 6/5, and 10/7 under high-Reynolds number flow in homogeneous isotropic turbulence, as shown in a histogram of the distribution of the decay exponent in Meldi and Sagaut (2012), rather large variation in the decay exponent has been observed, and most experimental values do not agree with the theoretical predictions. A figure in Sinhuber et al (2015) demonstrates the particularly large variation in the value of the decay exponent at moderate Reynolds numbers. This observation has also been reported in other previous works (e.g., Davidson, 2011).…”