Turbulence energy spectrum

Fig. 1. The normalized model turbulent energy spectrum for a range of Reynolds numbers.

The source terms on the right-hand sides of Eqs. (25)-(29) are defined as follows. In the momentum balance, g represents gravity and p is the modified pressure. The latter is found by forcing the mean velocity field to be solenoidal (V (U) = 0). In the turbulent-kinetic-energy equation (Eq. 26), Pk is the source term due to mean shear and the final term is dissipation. In the dissipation equation (Eq. 27), the source terms are closures developed on the basis of the form of the turbulent energy spectrum (Pope, 2000). Finally, the source terms... 

The turbulent energy spectrum is defined in terms of the velocity spectrum tensor by integrating out all directional information ... 

For isotropic turbulence, the velocity spectrum tensor is related to the turbulent energy spectrum by... 

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

From the definition of the turbulent energy spectrum, e is related to Eu k, t) by... 

Figure 2.4. Sketch of model turbulent energy spectrum at Rk = 500.

Pope (2000) developed the following model turbulent energy spectrum to describe fully developed homogeneous turbulence 12... 

The model turbulent energy spectrum for Rk = 500 is shown in Fig. 2.4. Note that the turbulent energy spectrum can be divided into roughly three parts ... 

The model turbulent energy spectrum given in (2.53) was introduced to describe fully developed turbulence, i.e., the case where / , (/<. t) does not depend explicitly on t. The case where the turbulent energy spectrum depends explicitly on time can be handled by deriving a transport equation for the velocity spectrum tensor 4> (k, t) starting from the Navier-Stokes equation for homogeneous velocity fields with zero or constant mean velocity (McComb 1990 Lesieur 1997). The resultant expression can be simplified for isotropic turbulence to a transport equation for / ,(/<. t) of the form14... 

In order to understand the role of spectral energy transfer in determining the turbulent energy spectrum at high Reynolds numbers, it is useful to introduce the spectral energy transfer rate Tu(jc,t) defined by... 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

Note that (2.140) will be valid even when the turbulent energy spectrum is not in spectral equilibrium. On the other hand, as shown next, (2.137) is strictly valid only at spectral equilibrium. 

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... 

Note that as Re/, goes to infinity with Sc constant, both the turbulent energy spectrum and the scalar energy spectrum will be dominated by the energy-containing and inertial/inertial-convective sub-ranges. Thus, in this limit, the characteristic time scale for scalar variance dissipation defined by (3.55) becomes... 

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. 

Like the turbulent energy spectrum discussed in Section 2.1, a transport equation can be derived for the scalar energy spectrum lipjn. t) starting from (1.27) and (1.28) for an inert scalar (see McComb (1990) or Lesieur (1997) for details). The resulting equation is21... 

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... 

The maximum wave number resolved with the LES approach is chosen to lie in the inertial sub-range of the turbulence energy spectrum. The governing transport equations are derived either by filtering the Navier-Stokes equation or using volume... 

In this method correlations between various velocity fluctuations are used to determine the turbulent energy spectrum, E k), and several turbulence length scales. The correlations mentioned contain information about how velocities and other flow properties are statistically related in the turbulent flow. Turbulence measured at a fixed point can be described as a fluctuating waveform. If two instantaneous waveforms appear to have a corresponding behavior, they are said to be correlated. Equation (1.311) shows how velocity fluctuations at two points can be statistically correlated if the distance between the two points are small. 

FIGURE 9.9 Turbulence energy spectrum, with as the length scale. 

---