Extension of the "Coherent Structure" Concept: Actual Turbulence

The conditions of chaos formation in typical dynamic systems, formulated in References 169—171, allow us to reveal versatile characteristic features observed at the formation of turbulence (in incipient turbulence). They include [169]: appearance of irregular long-lived spatial structures, whose form (character) is determined by dissipative factors, local instability and fractality of the phase space of such structures, and appearance of the central (at zero frequency) peak in the spectrum.

All these features were observed in the measurements as the spatial structure in the further study was taken the convective Benard cell. Its form depends on viscosity of the medium and geometry (shape) of the space it appears in (i.e., dissipative factors). The cell breaks down into smaller cells (vortices) in a cascade manner. The spectrum of a passive admixture in the cell (temperature) is fractal. Due to nonstationarity of a random process in the cell (bifurcation of stability alternation, see Figure 3.22), the average temperature was not constant. If we do not undertake additional measures to remove nonstationarity (average value of the random function can then be found by the time averaging not over the whole sample length, but only over the length of the characteristic scale of function variation). Then the centered random temperature contains the noncompensated constant component. Just this component lifts the low-frequency part of the Fourier spectrum, in particular, gives the central peak at the zero frequency. This phenomenon is observed for Fourier transforms of both the correlation function and the random nonstationary function itself (see definitions of these functions in Chapter 2).

It is convenient to join all these features by the common term “coherent structure,” if we will extend this, already existing, concept and will include small- scale components into the coherent structure.

Monin and Yaglom (see References 1, 2) defined the coherent structure as a nonrandom nonlinear superposition of large-scale components of turbulence characterized by the high stability. It is likely the simplest definition of all available definitions of the coherent structure (see, for example, References 43, 44, 50—52). The earliest definitions [43,44] deal with the expansion of the instantaneous field of velocities v into the coherent large-scale vcoh and random incoherent turbulent vT components, that is, v = vcoh + vT (double expansion [43]). Later, more complex expansions (e.g., the triple expansion [50,51]) were used. According to these definitions, turbulence consists of nonrandom coherent (large-scale) and purely random (small-scale) motions. Random motions are superimposed on coherent ones and usually extend far beyond the coherent structure. Coherent large-scale vortices are believed to be just main independent energy carriers and usually are not included in the structure of turbulence.

Coherent structures have been actively studied during the recent decades (see, e.g., References 43—79). Near-wall small-scale turbulence, turbulent convection in the near-surface atmospheric layer in the presence of wind shear, “cloud streets” in the atmosphere, and “Langmuir circulation” in seas and lakes are objects of intense studies. In addition, periodic large vortices in jet engine wakes have been investigated. It was shown that the main energy carriers in turbulent flows are large-scale ordered vortices, which influence significantly the formation of all flow characteristics. It was found that large-scale turbulent motions are deterministic, that is, are not random. The studies apply different methods of turbulence visualization (usually, coloring of flow). However, the resolution of the used visualization methods is low. That is why, only large-scale components of turbulent flows can be clearly seen, as a rule. Small-scale inhomogeneities usually remain invisible.

In their research, Sreenivasan et al. [71,72], the assumptions that fractal- ity manifests itself in turbulent flows were confirmed. The fractal dimension of some of them was measured. An assumption was made that turbulence is a set of slightly different fractals or an ensemble of semi-organized motions. The concept of multiplicative processes—multifractals—was used in application to turbulence. The statistical scale invariance of turbulent multifractals was used in the theoretical works by Novikov [79] and Chainais [78]. These works propose the models of generalized infinitely divisible cascades for phenomena of intermittence in hydrodynamic turbulence and introduce the concept of the similarity scale into the theory of infinitely divisible probability distributions, as well as consider the problems of self-similarity and asymptotic behavior of statistical characteristics.

As can be seen from the results presented above, the process of coherent breakdown of the main energy-carrying vortex into smaller ones does not terminate in the region of large-scale (low-frequency) vortices. It continues permanently into the small-scale (high-frequency) region up to the size of small vortices, which still can exist in air (0.6—1.2 mm, see the bifurcation diagram in Table 3.2). The frequencies of small-scale vortices are multiple to the frequency of the main vortex (see Figures 3.27 and 3.28). The oscillation phases in these vortices are rigidly related, and the vortices themselves are coherent (inphase). Therefore, the process of coherent breakdown cannot be limited to the region of large-scale vortices (as was done based on old definitions of the coherent structure).

According to the results presented above, we now can define the coherent structure as a wider concept including small-scale components as well.

The hydrodynamic coherent structure is a compact formation including a long-lived spatial vortex structure (cell) arising as a result of long action of thermodynamic gradients and products of its discrete coherent cascade breakdown.

The breaking-down spatial structure can be called a cell, giving rise to a coherent structure. The parent cell is the main energy-carrying vortex. The breakdown of the parent cell corresponds to breakdown of the main energy-carrying vortex. The frequency of this main vortex can be considered as one of the main attributes of both the parent cell and the coherent structure as a whole. The size of the coherent structure is blurred. Flows, external with respect to the parent cell, can transport its breakdown products to long distances, forming a long turbulent wake. The lifetime of this coherent structure is determined by the time of action of thermodynamic gradients. As a limiting case of the very weak local instability (when the parent structure is locally stable and does not break down), the coherent structure can consist of only one long-lived parent structure. In this case, the parent structure is some configuration of the laminar flow (usually, vortex). We have this situation at the observation of, for example, Benard cells in a thin layer of very viscous fluid.

Thus, the process of turbulence formation (generation) can be attributed to appearance of coherent structures. The compact formation observed in the described above experiments includes a long-lived main energy-carrying vortex and products of its simultaneous cascade discrete breakdown (coherent vortices with multiple frequencies). The frequency of the main vortex is determined by the size of the convective Benard cell. According to our definition, this formation is just the coherent structure.

The issue of formation of coherent structures in fluid, for example, in water seems to be interesting. An example of the smoothed turbulence spectrum in ocean was published in References 1, 2. This spectrum, analogously to the spectrum of coherent structure in air (see Figure 3.26), has frequency intervals with the 5/3-law decrease arranged stepwise. The stepwise character of this spectrum allows us to consider it as a spectrum of coherent structure in water.

Analogous features are also observed as fluid flows around some obstacles. This can be seen from the data of numerous observations [43,44,50—52,62,64,71,172,181] and available numerical solutions of the Navier—Stokes equations [47,48,56— 58,63,65—70,182—184]. On the obstacle surface (near the rear side), a long-lived main vortex is formed (parent structure; there can be several such structures). Behind the obstacle (at some distance from it), well-defined large vortices appear. Their sizes do not exceed the size of the main vortex. The vortex shape is distorted by the external flow (the front surface is concave). With the distance, we observe few smaller vortices and then a continuous turbulent jet consisting of small-scale vortices.

Behind obstacles like a sphere, several parent structures can be formed. Then the presence of even small asymmetry of the obstacle surface distorts the axial symmetry of the flow-around pattern and leads to the competition between parent cells. In this case, the zone behind the obstacle, in which parent structures are formed, extends in length, and the parent cells are usually arranged behind the obstacle as an extended chessboard-like structure. This structure is called a von Karman vortex street. The length of such a vortex street depends on the obstacle

shape, flow velocity, fluid viscosity, and other parameters. The breakdown of parent cells is observed at the end of the von Karman vortex street, where the process of formation of parent vortices terminates and their stability is lost.

As is shown below (see Section 3.3.6), the actual atmospheric turbulence can be considered as a mixture of different coherent structures. The frequencies of the main vortices of these structures are not multiple (not commensurable). The sizes of parent cells can differ significantly from each other. Consequently, the mixture of different coherent structures is not coherent in the general case. Based on our data, if vc str, i is the velocity of motion arising due to existence of the i-th coherent structure, and v is the total velocity, then

The constant Nc determines the number of coherent structures existing in the observation area (usually, Nc ^ 1). Thus, the turbulence can be studied through observation and study of all coherent structures arising in the considered spatial area.

Consider some isolated area, in which the influence of the ambient space is weak or absent. Let it include Nc coherent structures. If the size of the parent cell of one of these structures (assume that with the number i = 1) is far larger than the sizes of parent cells of other structures, then the velocities vc str, i (i = 1, ... , N) can be divided into large-scale and small-scale components

This separation is quite conditional and determined by limiting possibilities of the used visualization methods. Then the total velocity of the medium can be represented as

where

This equation is the double expansion [43]. It is true for the conditions described above and was, in its time, the experimental basis for the earlier definitions of the coherent structure [43,44,51,52]. The considered spatial area can be called the area with decisive influence of one coherent structure or the area with coherent turbulence.

 
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