| dc.description.abstract | Our study contrasts clearly dissipation-reduction phenomena in homogeneous, isotropic turbulence and drag reduction in wall-bounded flows. In both these cases, the polymers increase the overall viscosity of the solution (see, e.g., Fig. 2.4 and Ref. [14]). In wall-bounded flows, the presence of polymers inhibits the flow of the stream-wise component of the momentum into the wall, which, in turn, increases the net throughput of the fluid and thus results in drag reduction-a mechanism that can have no analog in homogeneous, isotropic turbulence.
However, the decrease of ?(f) with increasing c (Fig. 2.2) yields a natural definition of DR [Eq. (2.8)] for this case. Thus, if the term drag reduction must be reserved for wall-bounded flows, then we suggest the expression dissipation reduction for homogeneous, isotropic turbulence. We have shown that DR must be scale-dependent; its counterpart in wall-bounded flows is the position-dependent viscosity of Refs. [5, 8]. Furthermore, as in wall-bounded flows, an increase in c leads to an increase in DR (Fig. 2.2). In channel flows, an increase in We leads to an increase in DR, but we find that DR falls marginally as We increases (Fig. 2.2).
The reduction in the small-scale intermittency (Fig. 2.7) and in the constant-|?| isosurfaces (Fig. 2.8) is in qualitative agreement with channel-flow studies [3], where a decrease in the turbulent volume fraction is seen on the addition of the polymers, and water-jet studies [32], where the addition of the polymers leads to a decrease in small-scale structures. Furthermore, we find that the PDFs of the moduli of the vorticity, the tensor product, and the distribution of the eigenvalues of the rate-of-strain tensor are in qualitative agreement with the experiments of Liberzon et al. [25, 26]. We hope our work will stimulate more experimental studies of dissipation-reduction phenomena in homogeneous, isotropic turbulence.
In the next Chapter, we investigate such phenomena for forced, statistically steady, homogeneous, isotropic turbulence.
In this Chapter, we have carried forward our decaying-turbulence study with polymer additives to the case of statistically steady turbulence. We find that the average viscous dissipation decreases on the addition of polymers. This allows us to extend the definition of dissipation reduction to the regime of statistically steady turbulence. We find that the dissipation reduction increases with an increase in the Weissenberg number We at fixed polymer concentration. The PDFs of the moduli of the vorticity, the tensor product, the distribution of the eigenvalues of the rate-of-strain tensor, and the Q-plot are in qualitative agreement with the experiments of Liberzon et al. [2, 3].
In the deep-dissipation range, the energy spectrum shows a behavior similar to that in our earlier decaying turbulence study [14]. We have carried out a high-resolution study in the dissipation range too. By using the energy spectrum, we have also calculated S?(r) and find trends in qualitative agreement with the experiments.
Some earlier numerical studies of 2D, wall-bounded, statistically steady turbulent flows [22, 23] use forcing functions that are not of the Kolmogorov type; furthermore, they do not include air-drag-induced Ekman friction. Other numerical studies, which include the Ekman friction and Kolmogorov forcing, employ periodic boundary conditions [24, 25, 28]. To the best of our knowledge, our study of 2D turbulent flows is the first one that accounts for the Ekman friction, realistic boundary conditions, and Kolmogorov forcing. We show that, for values of ? that are comparable to those in experiments, the energy dissipation rate because of the Ekman friction is comparable to the energy dissipation rate that arises from the conventional viscosity.
We extract the isotropic part of the structure functions in the forward-cascade regime. We find that velocity structure function exponent ratios show simple scaling, whereas their vorticity counterparts show multiscaling. We also study the topological properties of two-dimensional turbulence by using the Okubo-Weiss criterion, and we find excellent agreement with PDFs that have been obtained experimentally. We hope our results will stimulate experimental studies designed to extract (a) the isotropic parts of structure functions (and thereby to probe the multiscaling of vorticity structure functions [Fig. 4.5(b)]) or (b) the PDF P?(?/????) (Fig. 4.7) near soap-film boundaries.
In Ref. [29], it was argued that, if the Ekman friction is nonzero and in the limit of vanishing viscosity, the third-order velocity structure function shows an anomalous behavior. In our calculations of structure functions of odd orders, we have employed moduli of velocity increments; without these moduli, the error bars are too large in our wall-bounded DNS to obtain good statistics for structure functions of odd order. Thus, we cannot compare our results directly with those of Ref. [29]. The main point of our study is to mimic, as closely as possible, parameters and boundary conditions in soap-film experiments such as those of Ref. [12]. Hence, our viscosity is much higher (and the Reynolds number much lower) than in the DNS of Ref. [27], which was designed to investigate some of the issues raised in Ref. [29]. Therefore, a direct comparison of our structure-function results with those of Ref. [27] is not possible, especially for odd orders because, as mentioned above, we use moduli of velocity increments.
We have, however, checked that our velocity structure functions show simple scaling as in the experiments of Ref. [12]; it would be interesting to explore if these experiments can be extended to confirm the multiscaling of vorticity structure functions that we describe above; such experimental studies might well benefit from the procedures we have used to extract the isotropic parts of structure functions.
We have carried out a detailed numerical study of turbulence-induced melting of a nonequilibrium vortex crystal in a forced, thin fluid film. We use ideas from the density-functional theory of freezing [1, 2, 3, 4], nonlinear dynamics, and turbulence to characterize this. Ideas from liquid-state theory have been used by some recent experiments to analyze the short-range order in the turbulent phase; nonlinear dynamics methods, such as Poincaré-type maps, have been used in the numerical studies of Ref. [10]; experimental studies have used
The curvature of Lagrangian trajectories to identify extrema in vortical and strain-dominated regimes. To the best of our knowledge, there is no study that brings together the methods we do to analyze turbulence-induced melting.
The advantages of our approach are as follows:
(a) It helps us to identify the order parameters for turbulence-induced melting and thus contrast it with conventional melting;
(b) The sequence of transitions can be characterized completely in terms of the Eulerian fields and ? and the total energy E(t) and suitable Fourier transforms of these;
(c) The short-range order in the turbulent phase can be studied conveniently in terms of G.
Equilibrium phase transitions occur strictly only in the thermodynamic limit-that is, roughly speaking, the limit of infinite size. It is interesting to ask how we might take the thermodynamic limit for vortex crystals we have studied here. There seem to be at least two ways to do this:
(a) In the first, the system size should be taken to infinity in such a way that the areal density of the vortical and strain-dominated regimes remains the same in the ordered, crystalline phase;
(b) We can increase the parameter n in the forcing so that more and more unit cells occur in the simulation domain (cf., e.g., Figs. 5.1(a) and (b) for n = 4 and n = 10, respectively).
Such issues have not been addressed in detail by any study, partly because, for large system sizes, it is not possible to obtain the long time series that are required to characterize the temporal evolution of the system (especially in the states we have referred to as spatiotemporal crystals). In particular, it is quite challenging to investigate the system-size dependence of the transitions summarized for n = 4 and n = 10 in Table 5.1.
As we have shown above, the array of transitions that comprise turbulence-induced melting of a vortex crystal is far richer than conventional equilibrium melting. There is another important way in which the former differs from the latter: To maintain the steady states, statistical or otherwise, of our system, we always have a force-thus, in the language of phase transitions, we always have a symmetry-breaking field, both in the ordered and disordered phases. Strictly speaking, therefore, there is no symmetry difference between the “disordered” turbulent state and the vortex crystal, as can be seen directly from the remnants of the dominant peaks in the reciprocal-space spectra E?(k) in Figs. 5.11 and 5.17(c) for n = 4 and n = 10, respectively.
One consequence of this is that the order parameters (?k), with k equal to the forcing wave vectors, do not vanish identically in the disordered, turbulent phase; however, they do assume very small values. Moreover, in the case of turbulence-induced melting, the crystal undergoes a transition from an ordered state to an undulating crystal to a fully turbulent state. Thus, there is noise and hence no fluctuations in the crystalline state; i.e., it is equivalent to a crystal at zero temperature. This has no analogue in the equilibrium melting of a crystal.
In equilibrium, different ensembles are equivalent; we can, e.g., use either the canonical or grand-canonical ensemble to study the statistical mechanics of a system and, in particular, the phase transitions in it. However, this equivalence cannot be taken for granted when we consider nonequilibrium statistical steady states (see, e.g., Ref. [21]). We have seen an example of this in Chapter 4 and in Ref. [14], where certain PDFs show slightly different behaviors depending on whether we keep the Grashof number fixed or whether we keep the Reynolds number fixed.
Turbulence-induced melting offers another example of the inequivalence of dynamical ensembles: The precise sequence of transitions that we encounter in going from the vortex crystal to the turbulent state depends on whether we do so by changing the Grashof number (i.e., the amplitude of the force) as in Ref. [10] or whether we do so by changing ?, as we have done here. We have checked explicitly that we can reproduce the sequence of transitions in Ref. [10] if we tune the Grashof number rather than ?.
Investigations of similar transitions, such as in the route Kolmogorov flow [17], can also benefit by using the combination of methods we have used above. Detailed studies of the effects of confinement and air-drag-induced Ekman friction on turbulence-induced melting, initiated, e.g., in Refs. [11, 22], can also benefit from the use of our methods—but that lies beyond the scope of this Chapter. We hope, too, that our study will encourage experimental groups to analyze turbulence-induced melting by using the set of techniques and ideas that we have described above. | |
