Energy spectrum of turbulence

Fig. 2. Kinetic-energy spectrum of turbulence in shear flow.

For large Reynolds number the kinetic energy spectrum of turbulence has the form E k) oc k-S/3... 

Fluctuations of the velocity field in a turbulent flow vary both in space and time. It is this time-space information that is characterized using the energy spectrum of turbulence. If the variations of one turbulent velocity component u x, f) are known over a length interval [0, L] in a direction Ox at a given time t, the variations of u x, f) may also be expanded into a Fourier series with respect to the space coordinate x Thus, [11.2] is extrapolated into ... 

In other cases the application of this concept has been further extended simulating faster turbulent fluctuations that are within the turbulence spectrum. For such dynamic simulations, using Reynolds averaged models, the Ic-quantity represents the turbulent kinetic energy accumulated on the fraction of the spectrum that is represented by the modeled scales. Therefore, to compare the simulated results obtained with this type of models with experimental data, that is averaged over a sufficient time period to give steady-state data (representing the whole spectrum of turbulence), both the modeled and the resolved scales have to be considered [68]. 

Figure 1.3 Schematic diagram of the kinetic energy spectrum of a turbulent flow.

Since for all dispersion organs in the ranges which are important for the practice the gas dispersion occurs due to turbulence mechanisms it is to be expected that there is a relationship between the turbulence properties and the efficiency of gas dispersion. It has already been pointed out that efficient gas dispersion is only possible, if the microeddy size is smaller than de cind if the percentage of microturbulence is high enough. Therefore the power spectrum of turbulence should influence the efficiency of gas dispersion. Fig. 10 shows one-dimensional power spectra in stirred teink reactors of different sizes according to van der Molen and van Maanen (22). As usual, the energy content of turbulent eddies was plotted as a function of the wave number, K, which is inversely proportional to the diameter of the... 

It is often not sufficient to characterize the magnitude of small-scale fluctuations, but also to understand the scale of the fluctuations or, more precisely, the distribution of the energy of the fluctuations over different frequencies or wave numbers. This is known as the spectrum of turbulence. For example, in the case of shaking of towers or bridges, only relatively high frequencies are important in contrast, high frequencies are relatively inefficient for mixing air masses with different properties. 

Figure 11.2(b) depicts the energy spectrum of a turbulent signal. This curve verifies several essential properties. Function E f ) is always positive, as per its definition (equation [11.4]). It is also found that function E f ) tends toward 0 for n 00. This property results from the Parceval equality ... 

Figure 11.2. Energy spectrum of a signal (a) harmonic signal (b) turbulent signal...

Sirovich, L., Smith, L., Yakhot, V. Energy spectrum of homogeneous and isotropic turbulence in far dissipation range. Physical Review Letters 72(3), 344-347 (1994)... 

Pressure Fluctuation Turbulent pressure fluctuations which develop in the wake of a cylinder or are carried to the cylinder from upstream may provide a potential mechanism for tube vibration. The tubes respond to the portion of the energy spectrum that is close to their natural frequency. 

The next level of complexity looks at the kinetic energy of turbulence. There are several models that are used to study the fluid mechanics, such as the K model. One can also put the velocity measurements through a spectrum analyzer to look at the energy at various wave numbers. 

To estimate the amount of turbulent kinetic energy lost when filtering at a given grid size, it is useful to introduce a normalized model energy spectrum (Pope, 2000) as follows ... 

Note that the final form of the energy spectrum depends only on the local turbulent Reynolds number. As an example, spectra found with different ReL are shown in Fig. 1. 

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

In a RANS simulation of scalar mixing, a model for i ,/, must be supplied to compute (4>a). In fully developed turbulence, t,p can be related to tu by considering the scalar energy spectrum, as first done by Corrsin (1964). 

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 ... 

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.

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). 

As seen in Chapter 2 for turbulent flow, the length-scale information needed to describe a homogeneous scalar field is contained in the scalar energy spectrum E k, t), which we will look at in some detail in Section 3.2. However, in order to gain valuable intuition into the essential physics of scalar mixing, we will look first at the relevant length scales of a turbulent scalar field, and we develop a simple phenomenological model valid for fully developed, statistically stationary turbulent flow. Readers interested in the detailed structure of the scalar fields in turbulent flow should have a look at the remarkable experimental data reported in Dahm et al. (1991), Buch and Dahm (1996) and Buch and Dahm (1998). 

In general, the scalar Taylor microscale will be a function of the Schmidt number. However, for fully developed turbulent flows,18 l.,p L and /, Sc 1/2Xg. Thus, a model for non-equilibrium scalar mixing could be formulated in terms of a dynamic model for Xassociated with working in terms of the scalar spatial correlation function, a simpler approach is to work with the scalar energy spectrum defined next. 

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... 

---