Turbulence Modeling Using Fractional Derivatives

Abstract

We propose a new two-dimensional turbulence model in this work. The main idea of the model is that the shear stresses are considered to be random variables and we assume that their differences with respect to time are Lévy-type distributions. This is a generalization of the classical Newton’s law of viscosity. We tested the model on the classical Backward Facing Step benchmark problem. The simulation results are in a good accordance with real measurements.

Appendix

Proof

(Theorem 3.1) Let $\alpha \in \mathbb{C}$ be any fixed complex number. Let x be a real or complex number such that | x |  < 1, then

$$\displaystyle{ (1 - x)^{\alpha } =\sum _{ N=0}^{\infty }(-1)^{N}\binom{\alpha }{N}x^{N}. }$$

(3.24)

It is easy to see that

$$\displaystyle{ (1 - x)^{\alpha -1} = \frac{(1 - x)^{\alpha -1}} {1 - x} =\sum _{ N=0}^{\infty }\Bigg(\sum _{ k=0}^{N}(-1)^{k}\binom{\alpha }{k}\Bigg)x^{N}. }$$

(3.25)

On the other hand

$$\displaystyle{ (1 - x)^{\alpha -1} =\sum _{ N=0}^{\infty }(-1)^{N}\binom{\alpha -1}{N}x^{N}, }$$

(3.26)

whence equating the coefficients of x ^N−1, we obtain

$$\displaystyle{ \begin{array}{rl} &\sum _{k=0}^{N-1}(-1)^{k}\binom{\alpha }{k} = (-1)^{N-1}\binom{\alpha -1}{N - 1} \\ & = \binom{N -\alpha -1}{N - 1} = \frac{\varGamma (N-\alpha )} {\varGamma (N)\varGamma (1-\alpha )}. \end{array} }$$

(3.27)

Thus

$$\displaystyle{ \begin{array}{rl} &\lim _{N\rightarrow \infty }N^{\alpha }\sum _{k=0}^{N-1}(-1)^{k}\binom{\alpha }{k} =\lim _{N\rightarrow \infty }\frac{N^{\alpha }\varGamma (N-\alpha )} {\varGamma (N)\varGamma (1-\alpha )} \\ & = \frac{1} {\varGamma (1-\alpha )}\lim _{N\rightarrow \infty }\frac{N^{\alpha }\varGamma (N-\alpha )} {\varGamma (N)} = \frac{1} {\varGamma (1-\alpha )}, \end{array} }$$

(3.28)

where we have used Stirling’s formula (or the known asymptotics for gamma function ratios) in the last step.