Model velocity spectrum

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

The parameter p0 controls the behavior of the velocity spectrum for small kL . The usual choice is po = 2 leading to EJk) k2 for small k. The alternative choice po = 4 leads to the von Karman spectrum, which has E (k) k4 for small k. For fixed k, e, and v, the model spectrum is determined by setting C = 1.5 and ft = 5.2, and the requirement that Eu k) and 2vk2Eu(k) integrate to k and e, respectively. The turbulence Reynolds number Rei, along with k and Lu, also uniquely determines the model spectrum. 

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  

Because the flow is assumed to be stationary, the time dependence has been dropped. However, the model spectrum could be used to describe a slowly evolving non-stationary spectrum by inserting k(t) and e(t). 

Furthermore, the universal equilibrium range is composed of the inertial range and the dissipation range. As its name indicates, at high Reynolds numbers the universal equilibrium range should have approximately the same form in all turbulent flows. 

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The validity of the stripping model can also be checked by measuring the velocity spectrum of the reactions X+ + HD XH+ + D (and XD+ + H). According to the stripping model, the bands of XH+ and XD+ should appear at the same positions as the bands of these ions, when they are produced by the reactions X+ + H2 and X+ + D2, respectively (the mass of the spectative hydrogen atom should not matter at... 

This relation shows that for homogeneous turbulence, working in terms of the two-point spatial correlation function or in terms of the velocity spectrum tensor is entirely equivalent. In the turbulence literature, models formulated in terms of the velocity spectrum tensor are referred to as spectral models (for further details, see McComb (1990) or Lesieur (1997)). 

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... 

Krt < k scalar spectral transport time scale defined in terms of the velocity spectrum (e.g., rst). 

In many reacting flows, the reactants are introduced into the reactor with an integral scale L that is significantly different from the turbulence integral scale Lu. For example, in a CSTR, Lu is determined primarily by the actions of the impeller. However, is fixed by the feed tube diameter and feed flow rate. Thus, near the feed point the scalar energy spectrum will not be in equilibrium with the velocity spectrum. A relaxation period of duration on the order of xu is required before equilibrium is attained. In a reacting flow, because the relaxation period is relatively long, most of the fast chemical reactions can occur before the equilibrium model, (4.93), is applicable. 

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

The above described statements are confirmed by theory. The authors of works [207,208] established the concept of constant velocity spectrum of elementary chemical reactions, which is due to the different levels of reciprocal arrangement of reactive molecules. This concept is used for the description of chemical processes in liquid condensed systems within the polychronic kinetics model [209,210]. The level of structural organization is characterized by the orientation order parameter a, similar in its physical sense to orientation interaction coefficient 8, which was used by the authors of works referred to in Part 2.6.4. Parameter a can vary from 0 (isotropic system) to 1 (maximum anisotropy, the molecules are parallel). 

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. 

Fig. 1. Model Spectra re-binned to CRIRES Resolution To demonstrate the potential for precise isotopic abundance determination two representative sample absorption spectra, normalized to unity, are shown. They result from a radiative transfer calculation using a hydrostatic MARCS model atmosphere for 3400 K. MARCS stands for Model Atmosphere in a Radiative Convective Scheme the methodology is described in detail e.g. in [1] and references therein. The models are calculated with a spectral bin size corresponding to a Doppler velocity of 1 They are re-binned to the nominal CRIRES resolution (3 p), which even for the slowest rotators is sufficient to resolve absorption lines. The spectral range covers ss of the CRIRES detector-array and has been centered at the band-head of a 29 Si16 O overtone transition at 4029 nm. In both spectra the band-head is clearly visible between the forest of well-separated low- and high-j transitions of the common isotope. The lower spectrum is based on the telluric ratio of the isotopes 28Si/29Si/30Si (92.23 4.67 3.10) whereas the upper spectrum, offset by 0.4 in y-direction, has been calculated for a ratio of 96.00 2.00 2.00.

Figure 3.12. Model scalar energy spectra at Rk = 500 normalized by the integral scales. The velocity energy spectrum is shown as a dotted line for comparison. The Schmidt numbers range from Sc = 10 4 to Sc = 104 in powers of 102.

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). 

Fig. 5. Synthetic spectrum one year after the explosion of model BF7 from Woosley (1987), The ZAMS mass was 20 M and the oxygen core velocity -1200 km/s. 

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