Cut-off wavenumber

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. 

The Smagorinsky constant Cs 0.17 holds at high Reynolds number if the cut-off wavenumber kc falls in the inertial sub-range (Pope 2000). However, the same value is often used for LES at relatively low Reynolds number where no clear inertial sub-range can exist. 

Reynolds numbers, the inertial-convective sub-range contains two stages. The wavenumber bands are defined by the cut-off wavenumbers 26... 

Note that, unlike pap, pD is not bounded by unity. This is because the dissipation-range cut-off wavenumber, (4.101), used to define ( a p) d depends on Scap, which is different for each scalar pair (i.e., Scap / Scaa Scpp). Indeed, for the mean-scalar-gradient case,... 

The SR model can be derived from (A.2) using the cut-off wavenumbers for each wavenumber band. The Kolmogorov wavenumber is defined by... 

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... 

Note that at spectral equilibrium the integral in (A.33) will be constant and proportional to ea (i.e., the scalar spectral energy transfer rate in the inertial-convective sub-range will be constant). The forward rate constants a j will thus depend on the chosen cut-off wavenumbers through their effect on (computationally efficient spectral model possible, the total number of wavenumber bands is minimized subject to the condition that... 

A short time later, Levin s group modified step-scan FT-IR spectrometer to operate with a mid-IR MCT FPA detector. Unlike most MCT detectors used in FT-IR spectrometers, which operate in the photoconductive (PC) mode, the pixels of MCT FPA detectors operate in the photovoltaic (PV) mode. As noted in Section 1.2.2, the cut-off wavenumber of narrow-band PC MCT detectors is about 750 cm" . The PV detector elements used in MCT FPA detectors have the same high sensitivity as narrow-band PC MCT detectors, but the cut-off wavenumber is higher, at about 850 cm" . 

Once the method has been tested for a simple double slit source representing two identical point sources, the next step is to apply this method for different sources. To achieve this, we use again a double slit source but one of the apertures being covered with a low-pass spectral filter (LPF) with cut-off wavenumber at 21 cm, to simulate two different sources in the sky and increase the complexity of the procedure. The new forward modelling method to compute the new source from the single slit data is... 

In this situation the weight coefficient C2 is expected to be substantially lower, as a considerably amount of power is filtered. Applying the previous least squares algorithm for different coefficient values and different cut-off wavenumbers (but... 

Figures.11 (left) shows the reduced distribution of the fit of the real interfero-gram with the simulated interferogram for different cut-off wavenumbers. It can be observed that a minimum peak appears around 20 cm Figure3.11 (right) presents...

Fig. 3.11 Reduced distribution of the fit of the real interferogram with the simulated inter-ferogram for different cut-off wavenumbers (left). Real interferogram (black) and the computed one (W e) when a 21 cm filter is introduced (n gto). The weight coefficients are ci = 1.07 and C2 = 0.45...

Figure4.3 shows an example of a simulated datacube, where 4 sources have been positioned on the sky grid (left) two gaussian sources, and two point sources. The spectra of these sources are different, corresponding to blackbodies of different temperatures which have been multiplied by Alters with different cut-on and cut-off wavenumbers (right). For the loading of more complex sky map data cubes such as science datacubes the simulator will interpolate or decimate to meet the instrument parameters.

Fig. 5.20 Spectra generated with the Sky Simulator for the source consisting of a blackbody at 1800 K with cut-on and cut-off wavenumbers of 5cm and 33cm respectively and attenuated by the atmosphere (left) and the same source with the 21 cm low-pass filter (right)...

Spectrally, the MAL is a blackbody at 1800 K. The cut-on and cut-off wavenumbers are 5 cm and 33 cm respectively, and are defined by the Winston cone and the filters at the bolometer. All such relevant parameters are set in the SkySparams.xlsx file. In this case, as the testbed is set in a laboratory, the atmospheric transmission has also to be applied. 

Three different sources are available point source, gaussian source and elliptical source. The common parameters are x pos and y pos (in arcseconds), the temperature (in Kelvins), the cut-on and cut-off wavenumbers, the emission wavenumber (to place an emission line across the spectrum) and the absorption wavenumber (to place an absorption line across the spectrum). The specific parameters are ... 

On the contrary, for M 0, we observe the existence of a cut-off wavenumber which is function of the three dimensionless parameters W, M and Re and in a such case, each of these parameters can play the role of a bifurcation parameter (in contrast with (8.27), where the bifurcation parameter is only W ). 

Broad-band radiometers, where the spectral range is determined by intrinsic material properties, have generally not been calibrated in orbit with respect to wavenumber. In most cases the cut-off wavelengths of optical materials are predictable functions of temperature, but they are assumed to be relatively stable, otherwise. A major shift in the wavenumber range of such a radiometer is not likely, and the scientific objectives of broad-band measurements rarely depend on the cut-off wavenumbers in a critical manner. 

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