Turbulence in eddy space

The nonlinear integro-differential equation we have derived in the previous section is not very easy to deal with. Since, however, the integral is of convolution type, it transforms to a simple product, through Fourier transform as

$$f_j(\mbox{\boldmath {$x$}},\mbox{\boldmath {$k$}})=(2\pi l... ...ath {$s$}}}q_j(\mbox{\boldmath {$x$}},\mbox{\boldmath {$s$}})$$ (19)

The actual operation of the transform on Eq.(14) gives

$$[-c_r\frac{\partial}{\partial s_r} +\frac{\partial}{\partia... ...rtial x_r}q_r +\partial_r(\mbox{\boldmath {$s$}})(q_jq_r)=0$$ (20)

where

$$\partial_j(\mbox{\boldmath {$s$}})\equiv\partial/\partial x_j+\partial/\partial s_j$$ (21)

Variable $\mbox{\boldmath {$s$}}$ introduced in Eq.(19) as the Fourier variable adjoint to wave number $\mbox{\boldmath {$k$}}$ has dimension of length, which stands for eddy size with direction of the vorticity vector, so may well be called eddy variable. It should be remarked that Eq.(20) is the Navier-Stokes equation in 6D (physical plus eddy) space, describing the motion of turbulent vortices moving with phase velocity $\mbox{\boldmath {$c$}}$ . In fact, this equation governing $q_\lambda(\mbox{\boldmath {$x$}},\mbox{\boldmath {$s$}})$ for prescribed $\mbox{\boldmath {$u$}}(\mbox{\boldmath {$x$}})$ and $p(\mbox{\boldmath {$x$}})$ is alternatively written as

$$[\mbox{\boldmath {$c$}}\cdot\nabla(\mbox{\boldmath {$s$}})]\m... ... NS}(\nabla(\mbox{\boldmath {$x$}}),\mbox{\boldmath {$u$}},p)$$ (22)

with

$$\mbox{\boldmath {$\partial$}}(\mbox{\boldmath {$s$}})\equiv\nabla(\mbox{\boldmath {$x$}})+\nabla(\mbox{\boldmath {$s$}})$$ (23)

where $\mbox{\bf NS}$ has been defined by Eq.(2), and $\nabla$ denote the nabla vector in the respective spaces³. Simplicity in expression for physical quantities is another advantage of working with $\mbox{\boldmath {$s$}}$-space. For example, fluctuation-correlation formula (18) takes remarkably simple form

$$\overline{u_j'\hat{u}_l'}=(2\pi l)^{-3}\int_{-\infty}^{\in... ... {$x$}}},\mbox{\boldmath {$s$}}+\hat{\mbox{\boldmath {$x$}}})$$ (24)

as is easily confirmed by substituting (19) and employing definition of the delta function $\delta(\mbox{\boldmath {$s$}}/l)=(2\pi l)^3 \int e^{i\mbox{\boldmath {$k$}}\cdot\mbox{\boldmath {$s$}}}d\mbox{\boldmath {$k$}}$ . Turbulent dissipation which has been dealt with as an elementary parameter in the classical dimensional analysis can be expressed in　this space by an explicit form: We have, by definition,

$$\begin{array}{rcl} \displaystyle \frac{\epsilon}{2}&= ... ...\mbox{\boldmath {$x$}}}=\mbox{\boldmath {$x$}}} \end{array}$$ (25)

On the other hand, we have from (24)

$$\begin{array}{rcl} \displaystyle \frac{\partial^2 \overli... ...{\mbox{\boldmath {$s$}}}=\mbox{\boldmath {$s$}}} \end{array}$$

Thus we are led to the final expression for dissipation $\epsilon$ as

$$\frac{\epsilon}{2}=\frac{\nu}{(2\pi l)^3}\int_{-\infty}^{\i... ...ldmath {$s$}})q_j) (\partial_j(\mbox{\boldmath {$s$}})q_l)]$$ (26)

This formula will reveal a new facet of the dissipation function hidden in the classical working space.