Dimensional consideration on the existence of pseudo-singularity

In the Kolmogorov regime, dissipation $\epsilon$ was introduced, together with kinematic viscosity $\nu$ , characteristic velocity $v$ and length $l$ , respectively, as an elementary parameter by which to construct the dimensional analysis. They are related to each other through the following relationships

$$\epsilon\sim v^3/l \sim v_{\mbox{\scriptsize K}}^3/l_{\mbox{\scriptsize K}}$$ (27)

where the subscript K denotes the respective quantities in the dissipation region. The following formulae are their immediate consequences

$$\left. \begin{array}{rcl} v_{\mbox{\scriptsize K}}l_{\... ...hspace{10mm} \mbox{(Kolmogorov length)} \end{array}\right\}$$ (28)

where $R$ is the flow Reynolds number

$$R=vl/\nu$$ (29)

Formula (26) we have derived in the newly defined hyperspace sheds some lights on this classical theory that is conducted within the physical space, in the sense that $\epsilon$ is not necessarily an elementary parameter any longer. According to Kolmogorov[1] viscous dissipation occurs only within the scale of Kolmogorov length. Then, the integral region of dissipation function (26) which is proportional to the kinematic viscosity is confined within a small volume of $O(l_{\mbox{\scriptsize K}}^3)$. Dimensional analysis of (26) gives

$$\epsilon\sim\frac{\nu}{l^3}\left(\frac{\mbox{\boldmath {$q... ...{l_{\mbox{\scriptsize K}}}\right)^2l_{\mbox{\scriptsize K}}^3$$ (30)

Thus, from (27) through (30) we are given for the order of magnitude of $\vert\mbox{\boldmath {$q$}}\vert$ :

$$\vert\mbox{\boldmath {$q$}}\vert/v\sim R^{7/8}$$ (31)

It is an indicative of strong in/out mass flow existent (cf. the second of Eq.(28) of the classical theory) in the vicinity of this extremely small `energy black hole' where the flow loses kinetic energy $\epsilon$ converted into heat. Inside this region $\vert\mbox{\boldmath {$s$}}\vert<O(l_{\mbox{\scriptsize K}})$ the turbulence dies off　rapidly towards $q(0)=0$ .There must be, therefore, a drastic variation in the magnitude of $\vert\mbox{\boldmath {$q$}}\vert$ which peaks at the boundary ridgeline between dissipation and inertial ranges　surrounding the origin ( $\mbox{\boldmath {$s$}}$=0), and diminishes quickly inside. In the neighborhood of this `pseudo'-singularity it is obvious for the following conditions to hold:

$$\begin{array}{rcl} \vert\mbox{\boldmath {$q$}}\vert&>> &... ...derbrace{\partial/\partial x_j}_{O(l^{-1})} \\ \end{array}$$ (32)

The latter condition warrants for local homogeneity in the sense of Kolmogorov to hold most strictly in the localized eddy space for any inhomogeneous turbulence. Under these circumstances, the 6D Navier-Stokes equation (20) together with equation of continuity reduce to

$$\partial q_j/\partial s_j=0$$ (33)

$$\left(q_r\frac{\partial}{\partial s_r} -\nu\frac{\partia... ...rtial s_r^2}\right)q_j +\frac{\partial q_4}{\partial s_j}=0$$ (34)

which is nothing but the classical (3D) equations for laminar viscous flows, as disguised through

$$\mbox{\boldmath {$x$}}\rightarrow\mbox{\boldmath {$s$}}\;,... ...oldmath {$u$}},p)\rightarrow(\mbox{\boldmath {$q$}},\rho q_4)$$ (35)

This rule reigns dissipation region as well as the adjacent region of inertial subrange surrounding it, where $\mbox{\boldmath {$q$}}$-function diminishes outward off the pseudo-singularity until flow inhomogeneity starts to make its appearance. The solution as such, to be pursued in this space, is `universal', namely, to be valid even for any shear turbulence. According to Kolmogorov[1], the inertial range is where no viscous effects are operating. So we may claim that the potential flow is prevailing there. This speculation is supported by the assertion: `potential flow is the solution of the Navier-Stokes equation in the region where no solid boundary is existent.' Now is the case with it, because no physical substances are intervening here in this space.