Abstract
From Lie-group (symmetry) analysis of the multi-point correlation equation Oberlack and Günther (Fluid Dyn Res 33:453–476, 2003) found three different solutions for the behavior of shear-free turbulence: (i) a diffusion like solution, in which turbulence diffuses freely into the adjacent calm fluid, (ii) a deceleration wave like solution when there is an upper bound for the integral length scale and (iii) a finite domain solution for the case when rotation is applied to the system. This paper deals with the experimental validation of the theory. We use an oscillating grid to generate turbulence in a water tank and Particle Image Velocimetry (PIV) to determine the two-dimensional velocity and out-of-plane vorticity components. The whole setup is placed on a rotating table. After the forcing is initiated, a turbulent layer develops which is separated from the initially irrotational fluid by a sharp interface, the so-called turbulent/non-turbulent interface (TNTI). The turbulent region grows in time through entrainment of surrounding fluid. We measure the propagation of the TNTI and find quantitative agreement with the predicted spreading laws for case one and two. For case three (system rotation), we observe that there is a sharp transition between a 3D turbulent flow close to the source of energy and a more 2D-like wavy flow further away. We measure that the separation depth becomes constant and in this sense, we confirm the theoretical finite domain solution.
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Notes
Thompson and Turner (1975) proposed that for the case of turbulent diffusion from a planar source, at high Reynolds numbers, the energy is dissipated by “inertial shear” and write du 3/dy = −Bu 3/ℓ, where u and ℓ are the integral velocity and length scales of turbulence and B is a constant of order unity. The LHS of the equation is the divergence of the flux of turbulent energy and the RHS can be interpreted as an eddy viscosity of order uℓ acting on a shear of order u/ℓ, i.e., uℓ(u/ℓ)2 = u 3/ℓ. With ℓ∝ y, case (i), the equation yields a power law for the propagation of the TNTI and with ℓ = const, case (ii), the equation yields a logarithmic law, consistent with the prediction of Oberlack and Günther (2003) and our measurements. At Reynolds number of order unity, uℓ is of order ν, that is, inertial and viscous dissipation are of the same order and the equation given above is no longer valid.
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Acknowledgments
The financial support by the German Research Foundation (DFG) is gratefully acknowledged. Furthermore, the authors would like to thank K. W. Hoyer for his contributions to this work.
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Kinzel, M., Holzner, M., Lüthi, B. et al. Experiments on the spreading of shear-free turbulence under the influence of confinement and rotation. Exp Fluids 47, 801–809 (2009). https://doi.org/10.1007/s00348-009-0724-4
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DOI: https://doi.org/10.1007/s00348-009-0724-4