The pseudo-singularity in the inertial subrange

The pseudo-singularity, as viewed from the domain of the inertial range, looks as if a genuine singularity, whose actual form is now to be identified. We start with checking if the local isotropy in the classical sense has physical reality. The isotropy assumption is equivalent to

$$q_j=s_jQ(s)$$ (36)

which is the alternative expression of Robertson's theorem[5]. Substitution of (36) into equation of continuity (33) gives

$$\begin{array}{c} sdQ/ds+3Q=0 \\ .\raisebox{1ex}{.}.\; Q\sim s^{-3} \end{array}$$

Obviously, this is a source/ sink flow, corresponding to the velocity potential

$$\phi^{(0)}=\alpha s^{-1}$$ (37)

If $\alpha$ is positive it represents a sink flow with mass flux $4\pi\alpha$ vanishing at the origin. Reminding that the dissipation range $\vert\mbox{\boldmath {$s$}}\vert<\hspace{-13pt}\raisebox{-1.2ex}{$\sim$}\;l_{\mbox{\scriptsize K}}$ is the `black hole' of turbulent kinetic energy losing the amount by $\epsilon$ converted into heat every second, we see that no mass is supposed to be lost. Thus solution (36) for spherical isotropy is ruled out. Prospective singularity now to take over can be sought within potential flow regime as follows: It is obvious that operator $\partial ^N/\partial s_1^l\partial s_2^m\partial s_3^n (N=l+m+n)$ commutes with Laplacean operator, therefore

$$\phi^{(N)}=\frac{\partial^N \phi^{(0)}}{\partial s_1^l \partial s_2^m\partial s_3^n}$$ (38)

represent a group of potential flows. In particular

$$\phi_j^{(1)}=\frac{\partial \phi^{(0)}}{\partial s_j}$$ (39)

representing a dipole aligned with its axis parallel to $s_j$ direction is seen to meet the purpose. In fact a dipole is made up of a pair of sink/source of equal strength, so no mass flux is lost at this spot. Thus simplest possible candidate for the pseudo-singularity is axially isotropic. The following observation from direct numerical simulation[6] helps us draw a more precise picture of our pseudo-singularity: In the physical space the dissipation occurs only within a confined volume of `elementary particles' of rod shape, randomly dispersed in turbulent medium. They all have a shape like a worm with radius $\sigma_0$ of several Kolmogorov lengths and several ten times of it lengthwise ( $s_0=N\sigma_0$ ), rotating round their own axes. Their lifetime is about $l_{\mbox{\scriptsize K}}/v_{\mbox{\scriptsize K}}=(l/v)R^{-1/2}$. So to say, they are like firefly worms illuminating light for their lifetime as small as of the order of far subseconds. This picture when mapped onto our eddy space is such that two spinning dipoles having opposite sign and rotation are placed at

$$\mbox{\boldmath {$s$}}=(s_1,s_2,s_3)=(\pm s_0,0,0)$$ (40)

respectively, where the axis of symmetry is $s_1$ axis. The direction j = 1 is the direction of motion of turbulence-generating body, representing the only vector prescribing the fluid motion. The flow field induced by the pair of spinning dipoles

$$\left. \begin{array}{rcl} \mbox{\boldmath {$q$}}&= &\m... ... {$q$}}_S^+ +\mbox{\boldmath {$q$}}_S^- \end{array}\right\}$$ (41)

with $\mbox{\boldmath {$q$}}_D^{\pm}$ and $\mbox{\boldmath {$q$}}_S^{\pm}$ standing for velocities induced by dipoles and line vortices, placed at points (40), respectively:

$$\mbox{\boldmath {$q$}}_D^{\pm}=\pm\nabla(\mbox{\boldmath {... ...\vert\mbox{\boldmath {$s$}}\mp\mbox{\boldmath {$i$}}s_0\vert}$$ (42)

$$\mbox{\boldmath {$q$}}_S^{\pm}=\pm\frac{\beta}{4\pi}\delta... ...ert\mbox{\boldmath {$s$}}\mp\mbox{\boldmath {$i$}}s_0\vert^3}$$ (43)

where $\mbox{\boldmath {$i$}}$ is the unit vector designating $s_1$ axis. Expression (42) is direct consequence of (37) and (39), and that for (43) is Biot-Savart's law for a line vortex with infinitesimal length directing $s_1$-axis and with circulation $\beta$. For example, the actual form of $q_1(\mbox{\boldmath {$s$}})$ rewritten in axially isotropic form $q_1(s_1,\sigma)$ with $\sigma^2=s_2^2+s_3^2$ is,

$$\begin{array}{rcl} q_1(s_1,\sigma)&= & \displaystyle ... ...2}{[(s_1+s_0)^2+\sigma^2]^{5/2}} \right\} \\ \end{array}$$ (44)

Note that there is no contribution from $\mbox{\boldmath {$q$}}_S$ to $q_1$. For $\vert s_1\vert>>s_0$ this expression approaches to

$$q_1\sim 6\alpha s_0s_1\left( -\frac{2}{s^5}+\frac{5\sigma^2}{s^7} \right)$$ (45)

which is a quadrupole field, as it should be expected. Streamlines are shown in Fig.1 of the fictitious flow generated by a pair of dipoles given by the potential

$$\left. \begin{array}{rcl} \phi^{(1)}&= &\partial \phi^... ... [(s_1+s_0)^2+\sigma^2]^{-1/2} \right\} \end{array}\right\}$$ (46)

This axi-symmetric flow can also be represented using streamfuntion $\psi$ as

$$\left. \begin{array}{rcl} q_{1}&= &\partial \phi^{(1)}... ...-1} \partial (\sigma \psi)/\partial s_1 \end{array}\right\}$$ (47)

from which we have

$$\psi=-\sigma \partial \phi^{(0)}/\partial \sigma$$ (48)

The flow pattern $\psi$ :const. shows quadrupole-like structure at far field( $\vert\mbox{\boldmath {$s$}}\vert>> s_0$ ) toward which the longitudinal vortices are stretched streamwise and then getting thicker. On the returning path to the dipole core they are chopped off and trim the aspect ratio, getting into the dissipation region. This picture may serve the qualitative description of what is actually observed (Fig.2). Expression (45) gives us estimate for the outer boundary of the locally homogeneous region. That is also the inner boundary of inhomogeneous region where the first condition of inequality (32) ceases to hold;

$$\vert\mbox{\boldmath {$q$}}\vert\sim v$$ (49)

Let this boundary be defined by $s\sim l_0$, then we have $q\sim \alpha s_0l_0^{-4}$ from (45), therefore condition (49) is replaced with a more precise one

$$v_0\sim \alpha s_0 l_0^{-4}$$ (50)

where $v_0$ is the characteristic velocity corresponding to $l_0$. They supplement Kolmogorov formula (27) as

$$\epsilon\sim v_{K}^3/l_K\sim v_0^3/l_0 \sim v^3/l$$ (51)

On the opposite side of the inertial subrange $s-s_0 \sim l_{\mbox{\scriptsize K}}$, $\sigma\sim l_{\mbox{\scriptsize K}}$, $q$ is estimated from (44) as

$$q(s_0)\sim \alpha l_{\mbox{\scriptsize K}}^{-3},$$ (52)

We note that the dimensional analysis developed in Sec.4 still holds by replacing $\vert\mbox{\boldmath {$q$}}\vert$ with $q(s_0)$ of (52), for instance,

$$q(s_0)/v\sim R^{7/8}.$$ (53)

Then, by eliminating $\alpha, q(s_0)$ and $v_0$ from (50) through (53) we have

$$l_0/l=(s_0/l)^{4/13}R^{-11/26}$$ (54)

The relationship between $s_0$ and $l_{\mbox{\scriptsize K}}$ or $l$ is yet to be reconsidered. At present no consensus formula is available for explicit parameter dependence of worm size $s_0$ . A recent observation by numerical experiments[8] is that $s_0$ is of the order of the Taylor microscale ( $\sim l\:\overline{u'}/v,\; \overline{u'}$ ; r.m.s. of the velocity fluctuation ), according to which the outer boundary of the inertial range is

$$l_0/l\sim R^{-11/26}(\overline{u'}/v)^{4/13}$$ (55)

The following expression may serve an easier estimate for the size of the inertial subrange:

$$R^{-1/2}< l_0/l<R^{-1/3}$$