Second And Third-Order Structure Functions Of An 'Engineered' Random Field And Emergence Of The Kolmogorov 4/5 And 2/3-Scaling Laws Of Turbulence

Abstract

The 4/5 and 2/3 laws of turbulence can emerge from a theory of 'engineered' random vector fields ${\mathcal{X}}_i(x,t)=X_i(x,t)+\frac{\theta }{\sqrt{d(d+2)}}{X}_i(x,t)\psi (x)$ existing within $\mathbf{D}\subset {\mathbf{R}}^d$. Here, $X_i(x,t)$ is a smooth deterministic vector field obeying a nonlinear PDE for all $(x,t)\in \mathbf{D}\times {\mathbf{R}}^{+}$, and $\theta$ is a small parameter. The field $\psi (x)$ is a regulated and differentiable Gaussian random field with expectation $\mathbb{E}[\psi (x)]=0$, but having an antisymmetric covariance kernel $\mathcal{K}(x,y)=\mathbb{E}[\psi (x)\psi (y)]=f(x,y)K(\Vert x-y\Vert ;\lambda )$ with $f(x,y)=-f(y,x)=1,f(x,x)=f(y,y)=0$ and with $K(\Vert x-y\Vert ;\lambda )$ a standard stationary symmetric kernel. For $0\le \ell \le \lambda <L$ with $X_i(x,t)=X_i=(0,0,X)$ and $\theta =1$ then for $d=3$, the third-order structure function is \begin{align} S_{3}[\ell]=\mathbb{E}\left[|\mathcal{X}_{i}(x+\ell,t)-\mathcal{X}(x,t)|^{3}\right]=-\frac{4}{5}\|X_{i}\|^{3}=-\frac{4}{5}X^{3}\nonumber \end{align} and $S_2[\ell ]=C{X}^2$. The classical 4/5 and 2/3-scaling laws then emerge if one identifies the random field ${\mathcal{X}}_i(x,t)$ with a turbulent fluid flow ${\mathcal{U}}_i(x,t)$ or velocity, with mean flow $\mathbb{E}[{\mathcal{U}}_i(x,t)]=U_i(x,t)=U_i$ being a trivial solution of Burger's equation. Assuming constant dissipation rate $\epsilon$, small constant viscosity $\nu$, corresponding to high Reynolds number, and the standard energy balance law, then for a range $\eta \le \ell \ll \lambda <L$ \begin{align} S_{3}[\ell]=\mathbb{E}\left[|\mathcal{U}_{i}(x+\ell,t)-\mathcal{U}(x,t)|^{3}\right]=-\frac{4}{5}ε\ell\nonumber \end{align} where $\eta =({\nu }^{3/4}\epsilon {)}^{-1/4}$. For the second-order structure function, the 2/3-law emerges as $S_2[\ell ]=C{\epsilon }^{2/3}{\ell }^{2/3}$.