I would like to fit numerical results obtained by NDS of the high order moments of the transverse velocity increment using a generalization of Batchelor parametrization together with a multifractal formalism in order to include the intermittency effect in the energy cascade. A smooth transition has been introduced between the dissipation range behavior (which is scale independent) and the inertial range (which is scale dependent). I used the formula attached below but I do not get a good agreement? Here c denoted a free physical parameter which has to be constant, defining the matching between the two regions. V is the large scale velocity fluctuation Gaussian normal distributed. Is this formula true only for the longitudinal velocity increment or it could also be applied for the transverse component? Does anyone know about an other parametrization used in matching between the two ranges? Any suggestions would be very appreciated! Thanks!
Hello Abdallah,
Where did you find this parametrization ? Could you please add a reference, if any ?
I personnally (see http://dx.doi.org/10.1063/1.4829763) use the extension of Batchelor parametrization as proposed by Aivalis et al (http://dx.doi.org/10.1063/1.1485079) or Kurien and Sreenivasan (http://dx.doi.org/10.1103/PhysRevE.62.2206). These expressions cover the entire distribution of energy from the dissipative up to the largest scales. They are somewhat similar and differ only at large scales. The parameters are the two cross-over length scales (viscous-inertial range and inertial-large scales), the scaling exponent in the inertial range (h in your expression) and the prefactor (V in your expression) which can be easily computed using non-linear least square fit ("nlinfit" function in Matlab). These expressions are (a priori) not restricted to the longitudinal velocity component, but should also hold for transverse structure functions.
Otherwise, if the focus is on the scaling exponent, there is a paper of R.A Antonia, Pearson and Zhou (http://dx.doi.org/10.1063/1.1314339) inspired by a paper of Stolovitzky and Sreenivasan (http://dx.doi.org/10.1103/PhysRevE.48.R3217) in which they use the parametrization of Batchelor to compute the scaling exponent in the (restricted) scaling range. As you may know, this parametrization applies only for the dissipative up to the inertial range.Therefore, you have to fit the data from the smallest separation (\approx \eta) up to a separation r (called r_{up} in Antonia's paper) where the parametrization ceases to apply; i.e somewhere within the inertial range. To assess r_{up}, Antonia et al. used the separation r at which the longitudinal third-order structure function divided by r (D_{LLL]/r) is maximum, i.e. r_{up} \approx \lambda (the Taylor microscale). I used this method, and it worked well, notwithstanding the quite low Reynolds number (see http://doi.org/10.1080/14685248.2013.803556).
I hope it will help you.
G'day.
Dear Fabien.
Thank you for your helpful answer and for the papers (although the links don't work but anyway I could get the paper from their doi).
For the Bachelor parametrization with a MF model, I found it in Physics of Fluids 20, 031703 (2008) of Prof. Biferale (The paper is called: A note on the fluctuation of dissipative scale in turbulence).
I think that the parametrization is correct but the large scale transverse velocity fluctuation has not a Gaussian distribution. I found that for the modulus, V follows a Rayleigh distribution while for the components, it follows a complementary error function distribution. Based on that, a striking agreement was found between the numerical results obtained from DNS and the MF predictions. I have a manuscript written in that direction if we would like to collaborate. Thank you.
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