## 321 Statistical Approach

In this approach, the unsteady processes occurring in turbulent flows are visualized as a combination of some mean process and small-scale fluctuations around it. The typical time variation of fluid velocity at a point in a turbulent flow is shown in Fig. 3.1. In the statistical approach, an instantaneous velocity, U, is visualized as a mean velocity, U (shown by a horizontal line in Fig. 3.1) and fluctuations around it, u. Based on such an approach, the statistical theory of turbulence flows has been developed (see Hinze, 1975 and references cited therein). It has been the basis for most of the engineering modeling of turbulent flow processes. Some of the key concepts of the statistical approach are discussed below.

The large-scale motions (eddies) of turbulence are dominated by inertia effects. For these large-scale motions, viscous effects are negligible. Mean flow stretches these large eddies (in the form of vortices). Angular momentum is conserved during vortex stretching leading to a reduction in cross-section of these vortices. Thus, the process creates motions at smaller length scales (and also at smaller time scales). The stretching work done by the mean flow on large eddies provides the energy which maintains turbulence. The smaller eddies are themselves stretched by somewhat larger FIGURE 3.1 Typical point velocity behavior in turbulent flows.

Dependent on condition of formation

Independent of condition of formation

Dependent on condition of formation I ^ Viscous ^ dissipation

Wavenumber, k

Energy-

Largest eddies of contamrng eddies Universal equilibrium range permanent character ^ ~ *

Inertial subrange

FIGURE 3.2 Energy spectrum for isotropic turbulence (from Hinze, 1975).

eddies and in this way, transfer energy to progressively smaller length scales (larger wave numbers). This process of energy transfer from large scales to small scales is termed as 'energy cascade'. As the length (and time) scales become sufficiently small, viscous effects start becoming important. For these small-scale motions, work is performed against the action of viscous stresses, so that the energy associated with the eddy motions is dissipated. This dissipation results in increased energy losses associated with turbulent flows. For many engineering flows, the smallest scales are of the order of 0.01 to 0.1 mm.

The structure of large eddies is highly flow dependent (and directional) due to their strong interaction with the mean flow. The diffusive action of viscosity tends to smear out directionality at small scales. At high mean flow Reynolds number, the smallest eddies are, therefore, non-directional or isotropic. For large enough Reynolds numbers, there exists an intermediate range of scales ('inertial sub-range') between these anisotropic large eddies and isotropic dissipative eddies. In this inertial sub-range, energy is transferred to smaller scales (higher velocity gradients) with negligible dissipation. As scale decreases below certain length scales (or velocity gradients increase beyond a certain limit), dissipation of kinetic energy by viscous stresses become dominant. Viscous stresses dissipate the kinetic energy and convert it to internal energy. The characteristic length scale at which viscous dissipation becomes important, is called the Kolmogorov scale. Kolmogorov and others (see Hinze, 1975) have described the process of 'energy cascade' in turbulent flows using spectral analysis. A typical energy spectrum of turbulent flows is shown in Fig. 3.2. ke and kd denote wave numbers of energy containing eddies and dissipative eddies, respectively. The -5/3 slope portion of the energy spectrum characterizes the presence of the inertial sub-range. Such spectral analysis has helped to quantify various scales occurring in turbulent flow processes and eventually to the development of computational models, which will be discussed in the following sections. The statistical approach has been used successfully for several applications. However, one of the most important objections to using the statistical approach to describe turbulent flows is that it totally ignores structures occurring within turbulent flows (see for example, Banerjee, 1992). An alternative approach based on turbulence structures is described below.

### 3.2.2. Structural Approach

Practitioners of this approach object to the ubiquitous averaging employed in the statistical approach, which obscures the structures present in the turbulent flows (high vorticity regions, bursts, streaks and so on). The presence of coherent structures in turbulent flows has long been recognized. Banerjee (1992) has given a lucid and very interesting account of the structural approach to studying turbulent flows. In any turbulent flow, such coherent structures exist. For some flows, these structures are more persistent and are not swamped by small-scale fluctuations. For such cases, obviously, the statistical approach may not be appropriate and a structure-based approach may be more fruitful. Banerjee (1992) cited examples where the structural approach can be used to develop quantitative, predictive models. However, in many flow situations, small-scale fluctuations and other factors shadow the coherent structures. In such cases, it is difficult to detect and quantify the characteristics of such structures. Moreover, a consistent theoretical framework to assimilate the random and deterministic elements of structures is lacking (structures are assumed to be randomly distributed in time and space but each occurrence is assumed to be governed by a locally deterministic cause). This approach is, therefore, not frequently used in the engineering modeling of turbulent flows.

### 3.2.3. Deterministic Approach

This is the latest approach to be used to understand turbulence, and is based on recent advances in the theory of non-linear problems and deterministic chaos. Here, attempts are being made to develop quantitative models for transition to turbulence. The tools developed and the results obtained by this approach so far look promising and may throw new light on the mechanism of transition (see Berge et al., 1984 for a more detailed account of the deterministic approach). Efforts to date have been focused on simple systems and the transition to turbulence. The application of these ideas to fully developed turbulent flows has, however, not yet been seriously undertaken. It is not yet evident whether the understanding gained through this approach can be converted into successful predictive models for turbulent flows of practical interest. Reactor engineers, therefore, have to rely on a statistical approach to develop predictive models for most problems of interest.

Before examining approaches to developing predictive models, it will be useful to employ a statistical approach to quantify various relevant length and time scales of turbulent flows, keeping in mind the existence of coherent structures. Estimation of such length and time scales may allow an evaluation of different competing processes such as mixing, heat and mass transfer and chemical reactions. Broadly speaking, turbulent flows are characterized by two length scales: the integral scale, L, where the inertial sub-range begins, and the Kolmogorov length scale, Xk , where the inertial sub-range ends, which are given by (Hinze, 1975):

where E(k) dk is the turbulent kinetic energy contained in the wave number range k to k + dk, v is the kinematic viscosity of the fluid and e is the turbulent energy dissipation rate (which is defined later in this chapter). Each of these length scales is associated with corresponding velocity and time scales (see Hinze, 1975).

Generally, the integral length scale is proportional to the macroscopic length scale of the equipment (for example, about one tenth of the diameter for pipe flows; about half the blade width of an impeller in stirred reactors). The small scales are decided by the fluid viscosity and turbulence energy dissipation rate, e. The small-scale motions exhibit universal characteristics and more or less behave in the same way in all flow processes. The integral scale motions interact with the mean flow field and strongly depend on the boundary conditions of the specific problem under consideration. These motions are sometimes called large eddies; eddy being a hypothetical construct to represent motions covering a small range of length scales. These large eddies are mainly responsible for the transport of momentum, mass and energy, and hence need to be adequately simulated by any turbulence model. Various approaches to modeling turbulent flows are discussed in the following sections.

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