Papers
The effects of solid boundaries on confined two-dimensional turbulence
- G. J. F. VAN HEIJST, H. J. H. CLERCX, D. MOLENAAR
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- Published online by Cambridge University Press:
- 24 April 2006, pp. 411-431
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This paper addresses the effects of solid boundaries on the evolution of two-dimensional turbulence in a finite square domain, for the cases of both decaying and continuously forced flow. Laboratory experiments and numerical flow simulations have revealed the crucial role of the solid no-slip walls as sources of vorticity filaments, which may significantly affect the flow evolution in the interior. In addition, the walls generally provide normal and tangential stresses that may exert a net torque on the fluid, which can change the total angular momentum of the contained fluid. For the case of decaying turbulence this is observed in so-called ‘spontaneous spin-up’, i.e. a significant increase of the total angular momentum, corresponding to a large domain-filling circulation cell in the organized ‘final’ state. For the case of moderate forcing this phenomenon may still be observed, although the filamentary vortex structures advected away from the walls may cause erosion and possibly a total destruction of the central cell. This disordered stage – characterized by a significantly decreased total angular momentum – is usually followed by a re-organization into a large circulation cell (in either the same or opposite direction) with an increased total angular momentum. The scaling behaviour of vorticity structure functions and the probability distribution function of vorticity increments have been investigated for forced turbulence and indicate a strong anisotropy of the turbulent flow in the range of Reynolds numbers considered.
Statistical mechanics of two-dimensional turbulence
- SUNGHWAN JUNG, P. J. MORRISON, HARRY L. SWINNEY
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- Published online by Cambridge University Press:
- 24 April 2006, pp. 433-456
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The statistical mechanical description of two-dimensional inviscid fluid turbulence is reconsidered. Using this description, we make predictions about turbulent flow in a rapidly rotating laboratory annulus. Measurements on the continuously forced, weakly dissipative flow reveal coherent vortices in a mean zonal flow. Statistical mechanics has two crucial requirements for equilibrium: statistical independence of macro-cells (subsystems) and additivity of invariants of macro-cells. We investigate these requirements in the context of the annulus experiment. The energy invariant, an extensive quantity, should thus be additive, i.e. the interaction energy between a macro-cell and the rest of the system (reservoir) should be small, and this is verified experimentally. Similarly, we use additivity to select the appropriate Casimir invariants from the infinite set available in vortex dynamics, and we do this in such a way that the exchange of micro-cells within a macro-cell does not alter an invariant of a macro-cell. A novel feature of the present study is our choice of macro-cells, which are continuous phase-space curves based on mean values of the streamfunction. Quantities such as energy and enstrophy can be defined on each curve, and these lead to a local canonical distribution that is also defined on each curve. The distribution obtained describes the anisotropic and inhomogeneous properties of a flow. Our approach leads to the prediction that on a mean streamfunction curve there should be a linear relation between the ensemble-averaged potential vorticity and the time-averaged streamfunction, and our laboratory data are in good accord with this prediction. Further, the approach predicts that although the probability distribution function for potential vorticity in the entire system is non-Gaussian, the distribution function of micro-cells should be Gaussian on the macro-cells, i.e. for curves defined by mean values of the streamfunction. This prediction is also supported by the data. While the statistical mechanics approach used was motivated by and applied to experiments on turbulence in a rotating annulus, the approach is quite general and is applicable to a large class of Hamiltonian systems, including drift-wave plasma models, Vlasov–Poisson dynamics, and kinetic theories of stellar dynamics.
The length-scale distribution function of the distance between extremal points in passive scalar turbulence
- LIPO WANG, NORBERT PETERS
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- Published online by Cambridge University Press:
- 24 April 2006, pp. 457-475
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In order to extract small-scale statistical information from passive scalar fields obtained by direct numerical simulation (DNS) a new method of analysis is introduced. It consists of determining local minimum and maximum points of the fluctuating scalar field via gradient trajectories starting from every grid point in the directions of ascending and descending scalar gradients. The ensemble of grid cells from which the same pair of extremal points is reached determines a spatial region which is called a ‘dissipation element’. This region may be highly convoluted but on average it has an elongated shape with, on average, a nearly constant diameter of a few Kolmogorov scales and a variable length that has the mean of a Taylor scale. We parameterize the geometry of these elements by the linear distance between their extremal points and their scalar structure by the absolute value of the scalar difference at these points.
The joint p.d.f. of these two parameters contains most of the information needed to reconstruct the statistics of the scalar field. It is decomposed into a marginal p.d.f. of the linear distance and a conditional p.d.f. of the scalar difference. It is found that the conditional mean of the scalar difference follows the 1/3 inertial-range Kolmogorov scaling over a large range of length-scales even for the relatively small Reynolds number of the present simulations. This surprising result is explained by the additional conditioning on minima and maxima points.
A stochastic evolution equation for the marginal p.d.f. of the linear distance is derived and solved numerically. The stochastic problem that we consider consists of a Poisson process for the cutting of linear elements and a reconnection process due to molecular diffusion. The resulting length-scale distribution compares well with those obtained from the DNS.
Finite-Reynolds-number effects in turbulence using logarithmic expansions
- K. R. SREENIVASAN, A. BERSHADSKII
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- Published online by Cambridge University Press:
- 24 April 2006, pp. 477-498
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Experimental or numerical data in turbulence are invariably obtained at finite Reynolds numbers whereas theories of turbulence correspond to infinitely large Reynolds numbers. A proper merger of the two approaches is possible only if corrections for finite Reynolds numbers can be quantified. This paper heuristically considers examples in two classes of finite-Reynolds-number effects. Expansions in terms of logarithms of appropriate variables are shown to yield results in agreement with experimental and numerical data in the following instances: the third-order structure function in isotropic turbulence, the mixed-order structure function for the passive scalar and the Reynolds shear stress around its maximum point. Results suggestive of expansions in terms of the inverse logarithm of the Reynolds number, also motivated by experimental data, concern the tendency for turbulent structures to cluster along a line of observation and (more speculatively) for the longitudinal velocity derivative to become singular at some finite Reynolds number. We suggest an elementary hydrodynamical process that may provide a physical basis for the expansions considered here, but note that the formal justification remains tantalizingly unclear.
Mechanics of inhomogeneous turbulence and interfacial layers
- J. C. R. HUNT, I. EAMES, J. WESTERWEEL
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- Published online by Cambridge University Press:
- 24 April 2006, pp. 499-519
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The mechanics of inhomogeneous turbulence in and adjacent to interfacial layers bounding turbulent and non-turbulent regions are analysed. Different mechanisms are identified according to the straining by the turbulent eddies in relation to the strength of the mean shear adjacent to, or across, the interfacial layer. How the turbulence is initiated and the topology of the region of turbulence are also significant factors. Specifically the cases of a layer of turbulence bounded on one, or two, sides by a uniform and/or shearing flow, and a circular region of a rotating turbulent vortex are considered and discussed.
The entrainment processes at fluctuating interfaces occur both at the outer edges of turbulent shear layers, with and without free-stream turbulence (e.g. jets, wakes and boundary layers), at internal boundaries such as those at the outside of the non-turbulent core of swirling flows (e.g. the ‘eye-wall’ of a hurricane) or at the top of the viscous sublayer and roughness elements in turbulent boundary layers. Conditionally sampled data enables these concepts to be tested. These concepts lead to physically based estimates for critical modelling parameters such as eddy viscosity near interfaces, entrainment rates, maximum velocity and displacement heights.
Large-scale and very-large-scale motions in turbulent pipe flow
- M. GUALA, S. E. HOMMEMA, R. J. ADRIAN
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- Published online by Cambridge University Press:
- 24 April 2006, pp. 521-542
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In the outer region of fully developed turbulent pipe flow very large-scale motions reach wavelengths more than 8$R$–16$R$ long (where $R$ is the pipe radius), and large-scale motions with wavelengths of $2R$–$3R$ occur throughout the layer. The very-large-scale motions are energetic, typically containing half of the turbulent kinetic energy of the streamwise component, and they are unexpectedly active, typically containing more than half of the Reynolds shear stress. The spectra of the $y$-derivatives of the Reynolds shear stress show that the very-large-scale motions contribute about the same amount to the net Reynolds shear force, d$\overline{-u'v'}/{\rm d}y$, as the combination of all smaller motions, including the large-scale motions and the main turbulent motions. The main turbulent motions, defined as the motions small enough to be in a statistical equilibrium (and hence smaller than the large-scale motions) contribute relatively little to the Reynolds shear stress, but they constitute over half of the net Reynolds shear force.