Power spectrum for inertial subrange

The actual form of the pseudo-singularity as predicted in the preceding section enables us to calculate 1D power spectrum for 1D wave number $k_1$ in the inertial subrange. This spectrum function $P_{11}(k_1)$ is written, by definition,

$$\overline{u_1'^2}=\int_{-\infty}^{\infty} P_{11}(k_1)dk_1$$ (56)

The well-known consequence of the classical dimensional analysis is that $P_{11}$ , having the dimension of $v^2l\sim\epsilon^{2/3}l^{5/3}$, reads in the language of wave number space as

$$P_{11} \sim k_1^{-5/3}$$ (57)

An alternative look into this formula on our own basis is the following: The actual form of $P_{11}$ is obtained through comparing formula (18) for $\hat{\mbox{\boldmath {$x$}}}=\mbox{\boldmath {$x$}}$ and $j=l=1$, namely,

$$\overline{u_1'^2}=\mbox{R.P.}l^3 \int_{-\infty}^{\infty}dk_1 \int_{-\infty}^{\infty} dk_2dk_3 f_1f_1^*$$

with (56), which reads

$$P_{11}(k_1)=l^3 \int_{-\infty}^{\infty} dk_2dk_3 f_1f_1^*$$ (58)

This integral is, upon substitution of (19), transformed into the one in $\mbox{\boldmath {$s$}}$-space as

$$P_{11}(k_1)=(2\pi l)^{-3} \int_{-\infty}^{\infty} ds_1 ... ...^{\infty} \sigma d\sigma q_1(s_1,\sigma)q_1(\hat{s}_1,\sigma)$$ (59)

Since the integration spans over the whole $\mbox{\boldmath {$s$}}$-space, we have yet to know solution $q_1$ inside the dissipation range [$O(l_K^3)$] where essentially viscous flow prevails. However, the volume of the dissipation range is by far the smaller than inertial subrange [$O(l_0^3)$], we may dispense with potential flow solution (44) to be integrated over its own region. Thus integral (59) with a small spheroidal regions excepted is to be carried out. (See Fig. 1.) Then, the integral is shown to be converted into a double integral as follows

$$P_{11}(k_1)=\frac{2}{(2\pi l)^3} \int_{0}^{\infty}d(\sigma^2)Q(\sigma^2,k_1)^2$$ (60)

with $Q(\sigma^2,k_1)$ defined by

$$Q(\sigma^2,k_1)=\int_{s_1^\dagger}^{\infty} \sin k_1s_1\frac{\partial^2 \phi^{(0)}}{\partial s_1^2}ds_1$$ (61)

where boundary contour $s_1=s_1^\dagger(\sigma^2)$ is given by.

$$s_1^\dagger(\sigma^2)=s_0(1+\delta)\left[ 1-\frac{\sigma^2}{s_0^2(2\delta+\delta^2)} \right]$$ (62)

The 1D power spectrum calculated is shown in Fig.3. Parameter $\delta$ corresponds to the slenderness ratio of the worm as detected by the direct numerical simulation, which is estimated as the order of $O(10N)^{-1}$ , with $N$ a number of the order of unity to several. For a certain range of this parameter, the spectrum shows $k_1^{-5/3}$ dependence, then with increase in $\delta$ it transits to $k_1^{-2}$ for $\delta >>1$ , where the pair of dipoles are regarded as a quadrupole asymptotically.