Turbulence wavenumber

We will revisit this topic in Section III when discussing CFD models for mixing-sensitive reactions. Note that while the discussion above applies to RANS turbulence models, the method can be extended to LES by integrating over the SGS wavenumbers (i.e., starting at kc). 

By definition, the turbulent kinetic energy k can be found directly from the turbulent energy spectrum by integrating over wavenumber space ... 

The first term on the right-hand side of (2.61) is the spectral transfer function, and involves two-point correlations between three components of the velocity vector (see McComb (1990) for the exact form). The spectral transfer function is thus unclosed, and models must be formulated in order to proceed in finding solutions to (2.61). However, some useful properties of T (k, t) can be deduced from the spectral transport equation. For example, integrating (2.61) over all wavenumbers yields the transport equation for the turbulent kinetic energy ... 

In words, (2.64) implies that Tu(k, t) is responsible for transferring energy between different wavenumbers without changing the total turbulent kinetic energy. 

From this definition, it can be observed that T,(k. t) is the net rate at which turbulent kinetic energy is transferred from wavenumbers less than k to wavenumbers greater than k. In fully developed turbulent flow, the net flux of turbulent kinetic energy is from large to small scales. Thus, the stationary spectral energy transfer rate Tu(k) will be positive at spectral equilibrium. Moreover, by definition of the inertial range, the net rate of transfer through wavenumbers /cei and kdi will be identical in a fully developed turbulent flow, and thus... 

Thus, E k, t) Ak represents the amount of scalar variance located at wavenumber k. For isotropic turbulence, the scalar integral length scale is related to the scalar energy spectrum by... 

As in Section 2.1 for the turbulent energy spectrum, a model scalar energy spectrum can be developed to describe lop(n). However, one must account for the effect of the Schmidt number. For Sc < 1, the scalar-dissipation wavenumbers, defined by19... 

We have defined two diffusion cut-off wavenumbers in terms of /cdi and in order to be consistent with the model turbulent energy spectrum introduced in Chapter 2. 

For a passive scalar, the turbulent flow will be unaffected by the presence of the scalar. This implies that for wavenumbers above the scalar dissipation range, the characteristic time scale for scalar spectral transport should be equal to that for velocity spectral transport tst defined by (2.67), p. 42. Thus, by equating the scalar and velocity spectral transport time scales, we have23 t)... 

The scalar-dissipation wavenumber /cd is defined in terms of /cdi by /cd = Sc1/2kdi-Like the fraction of the turbulent kinetic energy in the dissipation range kn ((2.139), p. 54), for a fully developed scalar spectrum the fraction of scalar variance in the scalar dissipation range scales with Reynolds number as... 

In homogeneous turbulence, spectral transport can be quantified by the scalar cospectral energy transfer rate Tap(ic, t). We can also define the wavenumber that separates the viscous-convective and the viscous-diffusive sub-ranges nf by introducing the arithmetic-mean molecular diffusivity Tap defined by... 

A detailed description of LES filtering is beyond the scope of this book (see, for example, Meneveau and Katz (2000) or Pope (2000)). However, the basic idea can be understood by considering a so-called sharp-spectral filter in wavenumber space. For this filter, a cut-off frequency kc in the inertial range of the turbulent energy spectrum is chosen (see Fig. 4.1), and a low-pass filter is applied to the Navier-Stokes equation to separate the... 

Figure 4.10. Predictions of the SR model for Re, =90 and Sc = 1 for homogeneous scalar mixing in stationary turbulence. For these initial conditions, all scalar energy is in the first wavenumber band. Curves 1-3 and D correspond to the fraction of scalar energy in each wavenumber band < 2)n/ 4> 2 - Note the relatively long transient period needed for R(t) to approach its asymptotic value of A o = 2.

In an effort to improve the description of the Reynolds stresses in the rapid distortion turbulence (RDT) limit, the velocity PDF description has been extended to include directional information in the form of a random wave vector by Van Slooten and Pope (1997). The added directional information results in a transported PDF model that corresponds to the directional spectrum of the velocity field in wavenumber space. The model thus represents a bridge between Reynolds-stress models and more detailed spectral turbulence models. Due to the exact representation of spatial transport terms in the PDF formulation, the extension to inhomogeneous flows is straightforward (Van Slooten et al. 1998), and maintains the exact solution in the RDT limit. The model has yet to be extensively tested in complex flows (see Van Slooten and Pope 1999) however, it has the potential to improve greatly the turbulence description for high-shear flows. More details on this modeling approach can be found in Pope (2000). 

Note that the right-hand sides of these expressions can be extracted from DNS data for homogeneous turbulence in order to explore the dependence of the rate constants on Rei and Sc. Results from a preliminary investigation (Fox and Yeung 1999) for Rx = 90 have revealed that the backscatter rate constant from the dissipative range has a Schmidt-number dependence like/Son Sc1/2 for Schmidt numbers in the range [1/8, 1], On the other hand, for cut-off wavenumbers in the inertial-convective sub-range, one would expect a 1) and... 

Concept (b) is less useful, except in rare cases where the energy spectrum has been measured. It is common to assume that the turbulence is homogeneous and isotropic and that the eddies in question are in the inertial ( — 5/3 power) subrange. This assumption is unlikely to be valid in an overall sense though it may be reasonable locally (GIO) or for the high wavenumber (small) eddies which are of primary interest. For an example of the application of the theory, see Middleman (Ml3). 

Microturbulence seldom occurs alone it is generally driven by larger-scale motions microturbulence is the high-wavenumber part of the atmospheric spectrum of turbulence. For the stars discussed here the origin of the motion field may be found in pulsations or in convective motions. Such motions have been discovered in a Cyg (Boer et al., 1987 ) they have up- and downward velocities of 14 km s-1 and the... 

It is possible to apply the same consideration to the eddy size distribution in a turbulent flow because eddies in a turbulent flow are produced by the impeller, shear stress, and so on, without artificiality. There is no difficulty in considering the following relationships between the wavenumber k and the diameter of eddy l in a turbulent flow ... 

The recent analysis has been applied to the full three dimensions, and was formulated to include all five variables u, v, w, c, p. In general, a large fraction of the total variance is captured by the first few eigenmodes and only a few wavenumbers, and they adequately capture the structure of the turbulent field. An example of the relative contribution to the total variance by the first five eigenmodes is shown in Figure 4.12 for the 3-dimensional EOF analysis of the u, v, w velocities. The first three wavenumbers of the first eigenmode exceed any wavenumber contribution from the second eigenmode. 

Pao Y-H (1965) Structure of Turbulent Velocity and Scalar Fields at Large Wavenumbers. Phys Fluids 8 (6) 1063-1075... 

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