2.1 Real space

The generalized velocity $v_{i}(t|\boldsymbol{x}^{\prime },t^{\prime })$ , defined as the velocity measured at time $t$ of a fluid particle that was or will be at the position $\boldsymbol{x}^{\prime }$ at time $t^{\prime }$ , was first introduced by Kraichnan (Reference Kraichnan1965). These two times $t$ and $t^{\prime }$ are called the measuring time and the labelling time, respectively. It should be noted that either of the two times may have occurred first; that is, either $t\leqslant t^{\prime }$ or $t\geqslant t^{\prime }$ may be true. The generalized velocity $v_{i}(t|\boldsymbol{x},t^{\prime })$ yields an Eulerian velocity of

(2.1) $$\begin{eqnarray}u_{i}(\boldsymbol{x},t)=v_{i}(t|\boldsymbol{x},t)\end{eqnarray}$$

for $t^{\prime }=t$ and a Lagrangian velocity at time $t$ for an initial time $t^{\prime }\neq t$ . Because the definition of the generalized velocity $v_{i}(t|\boldsymbol{x}^{\prime },t^{\prime })$ indicates that $v_{i}(t|\boldsymbol{x}^{\prime },t^{\prime })$ is invariant under the transformation of $\boldsymbol{x}^{\prime }$ and $t^{\prime }$ following the motion of the fluid particle, the following is true:

(2.2) $$\begin{eqnarray}\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t^{\prime }}+u_{j}(\boldsymbol{x}^{\prime },t^{\prime })\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}^{\prime }}\right]v_{i}(t|\boldsymbol{x}^{\prime },t^{\prime })=0\quad (t^{\prime }\neq t),\end{eqnarray}$$

where repeated subscripts are summed from 1 to 3.

The Lagrangian position function $\unicode[STIX]{x1D713}(\boldsymbol{x},t|\boldsymbol{x}^{\prime },t^{\prime })$ is defined as (Batchelor Reference Batchelor1949)

(2.3) $$\begin{eqnarray}v_{i}(t|\boldsymbol{x}^{\prime },t^{\prime })=\int _{V_{\infty }}\text{d}\boldsymbol{x}^{\prime \prime }v_{i}(t|\boldsymbol{x}^{\prime \prime },t^{\prime \prime })\unicode[STIX]{x1D713}(\boldsymbol{x}^{\prime \prime },t^{\prime \prime }|\boldsymbol{x}^{\prime },t^{\prime }),\end{eqnarray}$$

where $V_{\infty }$ denotes the entire space. It is important to note that $\int \text{d}\boldsymbol{x}^{\prime \prime }\unicode[STIX]{x1D713}(\boldsymbol{x}^{\prime \prime },t^{\prime \prime }|\boldsymbol{x}^{\prime },t^{\prime })$ is an operator that maps $v_{i}(t|\boldsymbol{x}^{\prime \prime },t^{\prime \prime })$ to $v_{i}(t|\boldsymbol{x}^{\prime },t^{\prime })$ , in other words, you can obtain $v_{i}(t|\boldsymbol{x}^{\prime },t^{\prime })$ at time $t^{\prime }$ by using $\unicode[STIX]{x1D713}(\boldsymbol{x}^{\prime \prime },t^{\prime \prime }|\boldsymbol{x}^{\prime },t^{\prime })$ if $v_{i}(t|\boldsymbol{x}^{\prime \prime },t^{\prime \prime })$ at time $t^{\prime \prime }$ is known in the entire space. Setting $t^{\prime \prime }=t^{\prime }$ in (2.3) yields

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D713}(\boldsymbol{x},t|\boldsymbol{x}^{\prime },t)=\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}^{\prime }),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}(\boldsymbol{x})$ is the Dirac delta function. Furthermore, setting $t^{\prime \prime }=t$ in (2.3) yields the Lagrangian velocity in terms of the Eulerian velocity as

(2.5) $$\begin{eqnarray}v_{i}(t|\boldsymbol{x}^{\prime },t^{\prime })=\int _{V_{\infty }}\text{d}\boldsymbol{x}\,u_{i}(\boldsymbol{x},t)\unicode[STIX]{x1D713}(\boldsymbol{x},t|\boldsymbol{x}^{\prime },t^{\prime }),\end{eqnarray}$$

whereas setting $t^{\prime }=t$ in (2.3) yields the Eulerian velocity in terms of the Lagrangian velocity as

(2.6) $$\begin{eqnarray}u_{i}(\boldsymbol{x},t)=\int _{V_{\infty }}\text{d}\boldsymbol{x}^{\prime }v_{i}(t|\boldsymbol{x}^{\prime },t^{\prime })\unicode[STIX]{x1D713}(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t).\end{eqnarray}$$

Differentiating (2.3) with respect to $t^{\prime \prime }$ , substituting (2.2) into the result, and integrating by parts yields

(2.7) $$\begin{eqnarray}0=\int _{V_{\infty }}\text{d}\boldsymbol{x}^{\prime \prime }v_{i}(t|\boldsymbol{x}^{\prime \prime },t^{\prime \prime })\left\{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}^{\prime \prime }}[u_{j}(\boldsymbol{x}^{\prime \prime },t^{\prime \prime })\unicode[STIX]{x1D713}(\boldsymbol{x}^{\prime \prime },t^{\prime \prime }|\boldsymbol{x}^{\prime },t^{\prime })]+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}(\boldsymbol{x}^{\prime \prime },t^{\prime \prime }|\boldsymbol{x}^{\prime },t^{\prime })}{\unicode[STIX]{x2202}t^{\prime \prime }}\right\}.\end{eqnarray}$$

Using the arbitrariness of $v_{i}(t|\boldsymbol{x}^{\prime \prime },t^{\prime \prime })$ , equation (2.7) yields the evolution equation for the Lagrangian position function as

(2.8) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D713}(\boldsymbol{x},t|\boldsymbol{x}^{\prime },t^{\prime })=-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}}[u_{j}(\boldsymbol{x},t)\unicode[STIX]{x1D713}(\boldsymbol{x},t|\boldsymbol{x}^{\prime },t^{\prime })]\quad (t\neq t^{\prime }),\end{eqnarray}$$

the solution of which is

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D713}(\boldsymbol{x},t|\boldsymbol{x}^{\prime },t^{\prime })=\unicode[STIX]{x1D6FF}\left(\boldsymbol{x}-\boldsymbol{x}^{\prime }-\int _{t^{\prime }}^{t}\text{d}t^{\prime \prime }\boldsymbol{v}(t^{\prime \prime }|\boldsymbol{x}^{\prime },t^{\prime })\right)\end{eqnarray}$$

under the initial condition given by (2.4) and the incompressibility condition

(2.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}u_{i}(\boldsymbol{x},t)=0.\end{eqnarray}$$

Note that Kaneda (Reference Kaneda1981) used the form given in (2.9) instead of that in (2.3) as the definition of the Lagrangian position function.

2.2 Fourier space

In homogeneous turbulence, it is natural to use the Fourier transform. The Fourier transforms of the Eulerian velocity, the Lagrangian velocity and the Lagrangian position function are respectively defined as

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle u_{i}(\boldsymbol{k},t)=\frac{1}{(2\unicode[STIX]{x03C0})^{3}}\int _{V_{\infty }}\text{d}\boldsymbol{x}\,u_{i}(\boldsymbol{x},t)\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}}, & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle v_{i}(t|\boldsymbol{k},t^{\prime })=\frac{1}{(2\unicode[STIX]{x03C0})^{3}}\int _{V_{\infty }}\text{d}\boldsymbol{x}\,v_{i}(t|\boldsymbol{x},t^{\prime })\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}} & \displaystyle\end{eqnarray}$$

and

(2.13) $$\begin{eqnarray}\unicode[STIX]{x1D713}(\boldsymbol{k},t|\boldsymbol{k}^{\prime },t^{\prime })=\frac{1}{(2\unicode[STIX]{x03C0})^{6}}\iint _{V_{\infty }}\text{d}\boldsymbol{x}\,\text{d}\boldsymbol{x}^{\prime }\unicode[STIX]{x1D713}(\boldsymbol{x},t|\boldsymbol{x}^{\prime },t^{\prime })\text{e}^{-\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}+\boldsymbol{k}^{\prime }\boldsymbol{\cdot }\boldsymbol{x}^{\prime })}.\end{eqnarray}$$

Note that we avoid all technical complications by assuming a Fourier transform representation in terms of distributions (Batchelor Reference Batchelor1953; Orszag Reference Orszag1977).

The present investigation focuses on an incompressible viscous flow, the evolution of which is described by the Navier–Stokes equation with forcing

(2.14) $$\begin{eqnarray}\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D708}k^{2}\right]u_{i}(\boldsymbol{k},t)=\unicode[STIX]{x1D614}_{ijm}(\boldsymbol{k})\iint _{V_{\infty }}\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})u_{j}(\,\boldsymbol{p},t)u_{m}(\boldsymbol{q},t)+f_{i}(\boldsymbol{k},t),\end{eqnarray}$$

where

(2.15a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D614}_{ijm}(\boldsymbol{k})=\frac{1}{2\text{i}}[k_{j}\unicode[STIX]{x1D60B}_{im}(\boldsymbol{k})+k_{m}\unicode[STIX]{x1D60B}_{ij}(\boldsymbol{k})],\quad \unicode[STIX]{x1D60B}_{ij}(\boldsymbol{k})=\unicode[STIX]{x1D6FF}_{ij}-\frac{k_{i}k_{j}}{k^{2}}\end{eqnarray}$$

and $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid. From this point forward, $V_{\infty }$ is omitted from integrals with respect to the wavenumber for the sake of brevity. Here, the initial and mean velocities are set to zero:

(2.16) $$\begin{eqnarray}u_{i}(\boldsymbol{k},-\infty )=\langle u_{i}(\boldsymbol{k},t)\rangle =0,\end{eqnarray}$$

where $\langle \cdot \rangle$ denotes an ensemble average. The force $f_{i}(\boldsymbol{k},t)$ is Gaussian white noise with a zero mean:

(2.17) $$\begin{eqnarray}\langle f_{i}(\boldsymbol{k},t)\rangle =0\end{eqnarray}$$

and must satisfy the condition $k_{i}\,f_{i}(\boldsymbol{k},t)=0$ to be compatible with the incompressibility condition

(2.18) $$\begin{eqnarray}k_{i}u_{i}(\boldsymbol{k},t)=0,\end{eqnarray}$$

which is the Fourier transform of (2.10). Furthermore, the force $f_{i}(\boldsymbol{k},t)$ is assumed to satisfy $f_{i}(\boldsymbol{k},t)\neq 0$ only for wavenumbers $|\boldsymbol{k}|\leqslant k_{I}\rightarrow 0$ , which corresponds to the limit of an infinite Reynolds number.

A homogeneous and isotropic force correlation can be written as

(2.19) $$\begin{eqnarray}\langle f_{i}(\boldsymbol{k},t)f_{j}(\boldsymbol{k}^{\prime },t^{\prime })\rangle =\frac{\unicode[STIX]{x1D700}}{4\unicode[STIX]{x03C0}k^{2}}\unicode[STIX]{x1D60B}_{ij}(\boldsymbol{k})\unicode[STIX]{x1D6FF}(k-k_{I})\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })\unicode[STIX]{x1D6FF}(t-t^{\prime }),\end{eqnarray}$$

where $\unicode[STIX]{x1D700}$ is the mean energy dissipation rate per unit mass in stationary turbulence. This force satisfies (see § 4.3.2 in McComb (Reference McComb1990))

(2.20) $$\begin{eqnarray}\langle f_{i}(\boldsymbol{k},t)u_{j}(\boldsymbol{k}^{\prime },t^{\prime })\rangle =\left\{\begin{array}{@{}ll@{}}\displaystyle {\displaystyle \frac{\unicode[STIX]{x1D700}}{8\unicode[STIX]{x03C0}k^{2}}}\unicode[STIX]{x1D60B}_{ij}(\boldsymbol{k})\unicode[STIX]{x1D6FF}(k-k_{I})\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })\quad & (t=t^{\prime })\\ 0\quad & (t>t^{\prime }),\end{array}\right.\end{eqnarray}$$

which indicates that energy is input only for wavenumbers $k\leqslant k_{I}\rightarrow 0$ , and

(2.21) $$\begin{eqnarray}\langle f_{i}(\boldsymbol{k},t)u_{j}(\,\boldsymbol{p},t)u_{m}(\boldsymbol{q},t)\rangle =0,\end{eqnarray}$$

where (2.16) was used. The derivation of (2.20) and (2.21) is simple under (2.19) using the law of causality and the following relation (Novikov Reference Novikov1965):

(2.22) $$\begin{eqnarray}\langle f_{i}(\boldsymbol{k},t)R[\,\boldsymbol{f}]\rangle =\int \text{d}\boldsymbol{k}^{\prime }\int _{-\infty }^{\infty }\text{d}t^{\prime }\langle f_{i}(\boldsymbol{k},t)f_{j}(\boldsymbol{k}^{\prime },t^{\prime })\rangle \left\langle \frac{\unicode[STIX]{x1D6FF}R[\,\boldsymbol{f}]}{\unicode[STIX]{x1D6FF}f_{j}(-\boldsymbol{k}^{\prime },t^{\prime })}\right\rangle ,\end{eqnarray}$$

where $f_{i}(\boldsymbol{k},t)$ is a Gaussian random function, $R[\,\boldsymbol{f}]$ is a functional of $\boldsymbol{f}$ and $\unicode[STIX]{x1D6FF}R[\,\boldsymbol{f}]/\unicode[STIX]{x1D6FF}f_{j}$ denotes a functional derivative.

In Fourier space, equations (2.5) and (2.6) are respectively expressed as

(2.23) $$\begin{eqnarray}v_{i}(t|\boldsymbol{k}^{\prime },t^{\prime })=(2\unicode[STIX]{x03C0})^{3}\int \text{d}\boldsymbol{k}\,u_{i}(\boldsymbol{k},t)\unicode[STIX]{x1D713}(-\boldsymbol{k},t|\boldsymbol{k}^{\prime },t^{\prime })\end{eqnarray}$$

and

(2.24) $$\begin{eqnarray}u_{i}(\boldsymbol{k},t)=(2\unicode[STIX]{x03C0})^{3}\int \text{d}\boldsymbol{k}^{\prime }v_{i}(t|\boldsymbol{k}^{\prime },t^{\prime })\unicode[STIX]{x1D713}(-\boldsymbol{k}^{\prime },t^{\prime }|\boldsymbol{k},t),\end{eqnarray}$$

where $t^{\prime }$ in (2.24) is an arbitrary constant. This arbitrariness is useful in the closure procedure. The Fourier transforms of (2.4) and (2.8) are respectively

(2.25) $$\begin{eqnarray}\unicode[STIX]{x1D713}(\boldsymbol{k},t|\boldsymbol{k}^{\prime },t)=\frac{1}{(2\unicode[STIX]{x03C0})^{3}}\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })\end{eqnarray}$$

and

(2.26) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D713}(\boldsymbol{k},t|\boldsymbol{k}^{\prime },t^{\prime })=-\text{i}k_{j}\iint \text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})u_{j}(\,\boldsymbol{p},t)\unicode[STIX]{x1D713}(\boldsymbol{q},t|\boldsymbol{k}^{\prime },t^{\prime })\quad (t\neq t^{\prime }),\end{eqnarray}$$

the formal solution of which is given by

(2.27) $$\begin{eqnarray}\unicode[STIX]{x1D713}(\boldsymbol{k},t|\boldsymbol{k}^{\prime },t^{\prime })=\frac{\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })}{(2\unicode[STIX]{x03C0})^{3}}-\text{i}k_{j}\int _{t^{\prime }}^{t}\text{d}t^{\prime \prime }\iint \text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})u_{j}(\,\boldsymbol{p},t^{\prime \prime })\unicode[STIX]{x1D713}(\boldsymbol{q},t^{\prime \prime }|\boldsymbol{k}^{\prime },t^{\prime })\end{eqnarray}$$

under the initial condition given in (2.25). Differentiating (2.23) with respect to $t$ and applying (2.14) and (2.26) to the result yields the evolution equation for the Lagrangian velocity:

(2.28) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}v_{i}(t|\boldsymbol{k},t^{\prime }) & = & \displaystyle -(2\unicode[STIX]{x03C0})^{3}\unicode[STIX]{x1D708}\!\int \text{d}\boldsymbol{p}\,p^{2}u_{i}(\,\boldsymbol{p},t)\unicode[STIX]{x1D713}(-\boldsymbol{p},t|\boldsymbol{k},t^{\prime })+(2\unicode[STIX]{x03C0})^{3}\!\int \text{d}\boldsymbol{p}\,f_{i}(\,\boldsymbol{p},t)\unicode[STIX]{x1D713}(-\boldsymbol{p},t|\boldsymbol{k},t^{\prime })\nonumber\\ \displaystyle & & \displaystyle +\,(2\unicode[STIX]{x03C0})^{3}\text{i}\iiint \text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,\text{d}\boldsymbol{r}\,\unicode[STIX]{x1D6FF}(\boldsymbol{r}-\boldsymbol{p}-\boldsymbol{q})\frac{r_{i}r_{j}r_{m}}{r^{2}}u_{j}(\,\boldsymbol{p},t)u_{m}(\boldsymbol{q},t)\unicode[STIX]{x1D713}(-\boldsymbol{r},t|\boldsymbol{k},t^{\prime }).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The Fourier transform of (2.2) is

(2.29) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t^{\prime }}v_{i}(t|\boldsymbol{k},t^{\prime })=-\text{i}k_{j}\iint \text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})u_{j}(\,\boldsymbol{p},t^{\prime })v_{i}(t|\boldsymbol{q},t^{\prime }),\end{eqnarray}$$

where the incompressibility condition (2.18) was used.

2.3 Statistical quantities and their characteristics

Two important velocity correlation functions are

(2.30) $$\begin{eqnarray}\unicode[STIX]{x1D618}_{ij}(\boldsymbol{k},t,t^{\prime })\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })=\langle v_{i}(t|\boldsymbol{k},t^{\prime })u_{j}(\boldsymbol{k}^{\prime },t^{\prime })\rangle \quad (t\geqslant t^{\prime })\end{eqnarray}$$

and

(2.31) $$\begin{eqnarray}\unicode[STIX]{x1D61D}_{ij}(\boldsymbol{k},t,t^{\prime })\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })=\langle v_{i}(t|\boldsymbol{k},t^{\prime })v_{j}(t|\boldsymbol{k}^{\prime },t^{\prime })\rangle \quad (t\leqslant t^{\prime }).\end{eqnarray}$$

The restrictions $t\geqslant t^{\prime }$ in (2.30) and $t\leqslant t^{\prime }$ in (2.31) are important to derive (B 3) and (E 8), respectively. Integrating (2.30) with respect to $\boldsymbol{k}^{\prime }$ over the whole space yields the following alternative form of the correlation function:

(2.32) $$\begin{eqnarray}\unicode[STIX]{x1D618}_{ij}(\boldsymbol{k},t,t^{\prime })=\int \text{d}\boldsymbol{k}^{\prime }\langle v_{i}(t|\boldsymbol{k},t^{\prime })u_{j}(\boldsymbol{k}^{\prime },t^{\prime })\rangle .\end{eqnarray}$$

Combining (2.18) and $\langle v_{i}(t|\boldsymbol{k},t^{\prime })u_{j}(\boldsymbol{k}^{\prime },t^{\prime })\rangle \propto \unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })$ yields

(2.33) $$\begin{eqnarray}k_{j}\unicode[STIX]{x1D618}_{ij}(\boldsymbol{k},t,t^{\prime })=0\end{eqnarray}$$

from (2.32). Using (2.33) and the symmetry of the two indices of the second-order isotropic tensor $\unicode[STIX]{x1D618}_{ij}(\boldsymbol{k},t,t^{\prime })$ (see § 3.3 in Batchelor (Reference Batchelor1953)) yields

(2.34) $$\begin{eqnarray}k_{j}\unicode[STIX]{x1D618}_{ji}(\boldsymbol{k},t,t^{\prime })=k_{j}\unicode[STIX]{x1D618}_{ij}(\boldsymbol{k},t,t^{\prime })=0,\end{eqnarray}$$

which gives the standard form of the correlation function (2.30) in isotropic turbulence

(2.35) $$\begin{eqnarray}\unicode[STIX]{x1D618}_{ij}(\boldsymbol{k},t,t^{\prime })={\textstyle \frac{1}{2}}\unicode[STIX]{x1D60B}_{ij}(\boldsymbol{k})\breve{Q}(k,t,t^{\prime }),\end{eqnarray}$$

where $\breve{Q}(k,t,t^{\prime })$ is the scalar correlation function, and gives a useful relation

(2.36) $$\begin{eqnarray}\unicode[STIX]{x1D618}_{ij}(\boldsymbol{k},t,t^{\prime })=\unicode[STIX]{x1D60B}_{im}(\boldsymbol{k})\unicode[STIX]{x1D618}_{mj}(\boldsymbol{k},t,t^{\prime }).\end{eqnarray}$$

It is important to note that (2.35) is satisfied despite the fact that $k_{i}v_{i}(t|\boldsymbol{k},t^{\prime })\neq 0$ for $t\neq t^{\prime }$ . In stationary turbulence, equation (2.35) is rewritten as

(2.37) $$\begin{eqnarray}\unicode[STIX]{x1D618}_{ij}(\boldsymbol{k},t,t^{\prime })={\textstyle \frac{1}{2}}\unicode[STIX]{x1D60B}_{ij}(\boldsymbol{k})Q(k,t-t^{\prime }),\end{eqnarray}$$

where $Q(k,t)$ is the scalar correlation function, and the energy spectrum $E(k)$ is given by

(2.38) $$\begin{eqnarray}E(k)=2\unicode[STIX]{x03C0}k^{2}Q(k,0).\end{eqnarray}$$

Averaging (2.29) and considering $\langle u_{j}(\,\boldsymbol{p},t^{\prime })v_{i}(t|\boldsymbol{q},t^{\prime })\rangle \propto \unicode[STIX]{x1D6FF}(\,\boldsymbol{p}+\boldsymbol{q})$ yields

(2.39) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t^{\prime }}\langle v_{i}(t|\boldsymbol{k},t^{\prime })\rangle =0.\end{eqnarray}$$

Combining (2.1), (2.16) and (2.39) yields

(2.40) $$\begin{eqnarray}\langle v_{i}(t|\boldsymbol{k},t^{\prime })\rangle =0.\end{eqnarray}$$

3.1 Assumptions and related statistical quantities

This subsection introduces two assumptions to obtain closure equations:

1. (i) Very slow motion is statistically independent of very rapid motion.

2. (ii) The multi-particle joint probability density function of the Lagrangian velocity has a Gaussian distribution.

Assumption (i) can be approximately rewritten as follows. The characteristic time scale of $v_{i}(t|\boldsymbol{x},t^{\prime })$ is much shorter than that of $\int ^{t}\text{d}t^{\prime \prime }v_{i}(t^{\prime \prime }|\boldsymbol{x},t^{\prime })$ because the time structure in $\unicode[STIX]{x2202}v_{i}(t|\boldsymbol{x},t^{\prime })/\unicode[STIX]{x2202}t$ is much finer than that in $v_{i}(t|\boldsymbol{x},t^{\prime })$ under turbulence; in other words, the integral time scale for the acceleration is much shorter than that for the Lagrangian velocity. This is approximately confirmed by direct numerical simulations (Yeung & Pope Reference Yeung and Pope1989; Yeung Reference Yeung1997) and experimental results (Voth, Satyanarayan & Bodenschatz Reference Voth, Satyanarayan and Bodenschatz1998). Equation (2.9) shows that $\unicode[STIX]{x1D713}(\boldsymbol{x},t|\boldsymbol{x}^{\prime },t^{\prime })$ is a function of $\int _{t^{\prime }}^{t}\text{d}t^{\prime \prime }v_{i}(t^{\prime \prime }|\boldsymbol{x}^{\prime },t^{\prime })$ , and hence assumption (i) yields approximately that $\unicode[STIX]{x1D713}(\boldsymbol{x},t|\boldsymbol{x}^{\prime },t^{\prime })$ is statistically independent of $v_{i}(t^{\prime \prime \prime }|\boldsymbol{x}^{\prime \prime },t^{\prime \prime })$ . Thus, assumption (i) can be rewritten as follows:

(i ^′ ) The Lagrangian position function is approximately statistically independent of the Lagrangian velocity.

Note that ‘approximately’ is important not in the closure procedure but in the discussion in § 5.2.

Assumption (i) implies that interactions in the frequency space are local because low-frequency modes are statistically independent of high-frequency modes; this is why this model was named the local interaction in frequency space (LIF) model. Assumption (i) may also be rewritten to state that small-scale motion is statistically independent of large-scale motion, which is fundamental to Kolmogorov’s universal equilibrium theory of the small-scale structure (Tennekes & Lumley Reference Tennekes and Lumley1972), because the characteristic time of small-scale motion is generally shorter than that of large-scale motion.

There is no solid foundation for assumption (ii). Although the one-point, one-time joint probability density function of the Lagrangian velocity is Gaussian, the multi-particle joint probability density function of the Lagrangian velocity is non-Gaussian (Yeung Reference Yeung1994) and, even for one particle, the probability density function of the Lagrangian velocity increment over a time interval is non-Gaussian, especially when the increment is small, which is similar to the case for the Eulerian velocity (Yeung & Pope Reference Yeung and Pope1989).

Let us comment on the Eulerian velocity. The probability density function of the Eulerian velocity is not assumed at all, and thus, in general, it is non-Gaussian. Hence, the energy-transfer function $T(k,p,q)$ , defined in (5.19) and (5.20), is non-zero, although the Lagrangian velocity satisfies the Gaussianity; a detailed discussion is given in § 5.2.

Averaging (2.26) and substituting (2.24) into the result yields

(3.1) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\langle \unicode[STIX]{x1D713}(\boldsymbol{k},t|\boldsymbol{k}^{\prime },t^{\prime })\rangle & = & \displaystyle -(2\unicode[STIX]{x03C0})^{3}\text{i}k_{j}\iiint \text{d}\boldsymbol{k}^{\prime \prime }\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\nonumber\\ \displaystyle & & \displaystyle \times \,\langle v_{j}(t|\boldsymbol{k}^{\prime \prime },t^{\prime \prime })\rangle \langle \unicode[STIX]{x1D713}(-\boldsymbol{k}^{\prime \prime },t^{\prime \prime }|\boldsymbol{p},t^{\prime })\unicode[STIX]{x1D713}(\boldsymbol{q},t|\boldsymbol{k}^{\prime },t^{\prime })\rangle =0,\end{eqnarray}$$

where assumption (i ^′ ) and (2.40) were used. Equations (2.25) and (3.1) yield

(3.2) $$\begin{eqnarray}\langle \unicode[STIX]{x1D713}(\boldsymbol{k},t|\boldsymbol{k}^{\prime },t^{\prime })\rangle =\frac{1}{(2\unicode[STIX]{x03C0})^{3}}\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime }).\end{eqnarray}$$

Furthermore, the Lagrangian position functions are statistically independent of each other, as demonstrated in appendix A.

3.2 Closure procedure

A closed set of three equations is derived from the basic evolution equations (2.14) and (2.28) under assumptions (i ^′ ) and (ii).

First, differentiating (2.32) with respect to $t$ and using (2.36) yields the evolution equation of the correlation function $\unicode[STIX]{x1D618}_{bc}(\boldsymbol{k},t,t^{\prime })$ as

(3.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D618}_{bc}(\boldsymbol{k},t,t^{\prime })=\unicode[STIX]{x1D60B}_{ca}(\boldsymbol{k})\int \text{d}\boldsymbol{k}^{\prime }\left\langle \frac{\unicode[STIX]{x2202}v_{a}(t|\boldsymbol{k},t^{\prime })}{\unicode[STIX]{x2202}t}u_{b}(\boldsymbol{k}^{\prime },t^{\prime })\right\rangle .\end{eqnarray}$$

Considering assumptions (i ^′ ) and (ii) and stationary isotropic turbulence, equation (3.3) yields the first of the three equations in the closed set as

(3.4) $$\begin{eqnarray}\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D708}k^{2}\right]Q(k,t)=-\unicode[STIX]{x03C0}k^{4}Q(k,t)\int _{0}^{\infty }\text{d}p\,I(p/k)\int _{0}^{t}\,\text{d}t^{\prime }Q(p,t^{\prime }),\end{eqnarray}$$

where $I(p)$ is defined in (B 15). The closure equation given by (3.4) is the same as that in the Lagrangian DIA (Kaneda Reference Kaneda1981; Kida & Goto Reference Kida and Goto1997).

Here, a simplified derivation of (3.4) is presented with symbolic quantities; a detailed derivation is given in appendix B. Substituting (2.28) into (3.3) yields the following symbolic expression of (B 1):

(3.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}t}=\left\langle \frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}t}u\right\rangle =-\unicode[STIX]{x1D708}\langle u\unicode[STIX]{x1D713}u\rangle +\langle uu\unicode[STIX]{x1D713}u\rangle +\langle f\unicode[STIX]{x1D713}u\rangle =-\unicode[STIX]{x1D708}\langle u\unicode[STIX]{x1D713}u\rangle +\langle uu\unicode[STIX]{x1D713}u\rangle ,\end{eqnarray}$$

where (2.20) was applied. Substituting the symbolic expression

(3.6) $$\begin{eqnarray}\unicode[STIX]{x1D713}=\unicode[STIX]{x1D6FF}+\int \text{d}t\,u\unicode[STIX]{x1D713}\end{eqnarray}$$

of the formal solution given by (2.27) into the second term on the right-hand side of (3.5) and using the symbolic expression $u=v\unicode[STIX]{x1D713}$ of (2.24) yields a symbolic expression of (3.4) as

(3.7) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}t} & = & \displaystyle -\unicode[STIX]{x1D708}\langle v\unicode[STIX]{x1D713}\unicode[STIX]{x1D713}v\unicode[STIX]{x1D713}\rangle +\int \text{d}t\,\langle v\unicode[STIX]{x1D713}v\unicode[STIX]{x1D713}v\unicode[STIX]{x1D713}\unicode[STIX]{x1D713}v\unicode[STIX]{x1D713}\rangle =-\unicode[STIX]{x1D708}\langle vv\rangle +\int \text{d}t\,\langle vv\rangle \langle vv\rangle \nonumber\\ \displaystyle & = & \displaystyle -\unicode[STIX]{x1D708}Q+\int \text{d}t\,QQ,\end{eqnarray}$$

where assumptions (i ^′ ) and (ii) were applied. Note that $\langle vv\rangle =Q$ was used in the present case (see (B 6)), while $\langle vv\rangle =V$ will be used in the derivation of (3.9) (see (E 8)).

Second, integrating (2.31) with respect to $\boldsymbol{k}^{\prime }$ and differentiating the result with respect to $t$ yields the evolution equation of the correlation function $\unicode[STIX]{x1D61D}_{ab}(\boldsymbol{k},t,t^{\prime })$ as

(3.8) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D61D}_{ab}(\boldsymbol{k},t,t^{\prime })=\int \text{d}\boldsymbol{k}^{\prime }\left[\left\langle \frac{\unicode[STIX]{x2202}v_{a}(t|\boldsymbol{k},t^{\prime })}{\unicode[STIX]{x2202}t}v_{b}(t|\boldsymbol{k}^{\prime },t^{\prime })\right\rangle +\left\langle v_{a}(t|\boldsymbol{k},t^{\prime })\frac{\unicode[STIX]{x2202}v_{b}(t|\boldsymbol{k}^{\prime },t^{\prime })}{\unicode[STIX]{x2202}t}\right\rangle \right].\end{eqnarray}$$

Considering stationary isotropic turbulence and assumptions (i ^′ ) and (ii), in the case of $k>k_{I}$ , equation (3.8) yields the second of the three equations in the closed set as

(3.9) $$\begin{eqnarray}\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+2\unicode[STIX]{x1D708}k^{2}\right]V(k,t)=-2\unicode[STIX]{x03C0}k^{4}V(k,t)\int _{0}^{\infty }\text{d}p\,I(p/k)\int _{0}^{t}\text{d}t^{\prime }Q(p,t^{\prime }),\end{eqnarray}$$

where $V(k,t)$ is defined in (C 12). There is no equation corresponding to (3.9) in the Lagrangian DIA (Kaneda Reference Kaneda1981; Kida & Goto Reference Kida and Goto1997). The derivation of (3.9) is similar to that of (3.4), and hence a simplified derivation is omitted; a detailed derivation is given in appendix C. Combining (3.4) and (3.9) yields

(3.10) $$\begin{eqnarray}V(k,t)=\frac{[Q(k,t)]^{2}}{Q(k,0)}\end{eqnarray}$$

because $V(k,0)=Q(k,0)$ .

Third, setting $t^{\prime }=t$ in (2.32) and differentiating with respect to $t$ yields the evolution equation of $\unicode[STIX]{x1D618}_{ij}(\boldsymbol{k},t,t)$ as

(3.11) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D618}_{ij}(\boldsymbol{k},t,t)=\int \text{d}\boldsymbol{k}^{\prime }\left[\left\langle \frac{\unicode[STIX]{x2202}u_{i}(\boldsymbol{k},t)}{\unicode[STIX]{x2202}t}u_{j}(\boldsymbol{k}^{\prime },t)\right\rangle +\left\langle u_{i}(\boldsymbol{k},t)\frac{\unicode[STIX]{x2202}u_{j}(\boldsymbol{k}^{\prime },t)}{\unicode[STIX]{x2202}t}\right\rangle \right].\end{eqnarray}$$

Considering stationary isotropic turbulence and assumptions (i ^′ ) and (ii), equation (3.11) yields the third of the three equations in the closed set as

(3.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D708}k^{2}Q(k,0) & = & \displaystyle \unicode[STIX]{x03C0}\iint _{I_{k}}\text{d}p\,\text{d}q\,kpq\,b(k,p,q)\int _{0}^{\infty }\text{d}t\,\text{e}^{-\unicode[STIX]{x1D708}(k^{2}+p^{2}+q^{2})t}V(q,t)[V(p,t)-V(k,t)]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D700}}{4\unicode[STIX]{x03C0}k^{2}}\unicode[STIX]{x1D6FF}(k-k_{I}),\end{eqnarray}$$

where $\iint _{I_{k}}\text{d}p\,\text{d}q$ and $b(k,p,q)$ are defined in (B 13) and (E 14), respectively. The closure equation given by (3.12) is different from that in the Lagrangian DIA (Kaneda Reference Kaneda1981; Kida & Goto Reference Kida and Goto1997), but its structure is similar to that in the quasi-normal (QN) model (Millionshtchikov Reference Millionshtchikov1941; Proudman & Reid Reference Proudman and Reid1954; Tatsumi Reference Tatsumi1957).

Here, a simplified derivation of (3.12) is presented; a detailed derivation is given in appendix E. Substituting (2.14) into (3.11) yields the following symbolic expression of (E 1):

(3.13) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\langle uu\rangle =-\unicode[STIX]{x1D708}\langle uu\rangle +\langle fu\rangle +\langle uuu\rangle =-\unicode[STIX]{x1D708}Q+\unicode[STIX]{x1D700}+\langle uuu\rangle .\end{eqnarray}$$

Substituting (2.14) into $\unicode[STIX]{x2202}(uuu)/\unicode[STIX]{x2202}t$ and using the symbolic expression $u=v\unicode[STIX]{x1D713}$ of (2.24) yields

(3.14) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(uuu)=-\unicode[STIX]{x1D708}uuu+v\unicode[STIX]{x1D713}v\unicode[STIX]{x1D713}v\unicode[STIX]{x1D713}v\unicode[STIX]{x1D713}+fuu,\end{eqnarray}$$

the formal solution of which is

(3.15) $$\begin{eqnarray}uuu=\int \text{d}t\,\text{e}^{-\unicode[STIX]{x1D708}t}(v\unicode[STIX]{x1D713}v\unicode[STIX]{x1D713}v\unicode[STIX]{x1D713}v\unicode[STIX]{x1D713}+fuu),\end{eqnarray}$$

which is a symbolic expression of (E 5). Averaging (3.15) yields

(3.16) $$\begin{eqnarray}\langle uuu\rangle =\int \text{d}t\,\text{e}^{-\unicode[STIX]{x1D708}t}\langle vvvv\rangle =\int \text{d}t\,\text{e}^{-\unicode[STIX]{x1D708}t}VV,\end{eqnarray}$$

where (2.21) and assumptions (i ^′ ) and (ii) were applied. For stationary turbulence, substituting (3.16) into (3.13) yields a symbolic expression of (3.12) as

(3.17) $$\begin{eqnarray}0=-\unicode[STIX]{x1D708}Q+\unicode[STIX]{x1D700}+\int \text{d}t\,\text{e}^{-\unicode[STIX]{x1D708}t}VV.\end{eqnarray}$$

In this manner, the closed set of equations consisting of (3.4), (3.9) (or (3.10)) and (3.12) is obtained. It is important to note that the closed set of equations (B 8), (C 10) and (E 11), the correlation functions (2.30) and (2.31), and assumptions (i ^′ ) and (ii) are invariant under a (non-random) Galilean transformation for homogeneous turbulence; a proof is given in appendix F.

4.1 Inertial range

The energy spectrum in the inertial range is considered.

First, integrating (3.12) with respect to $\boldsymbol{k}$ over the whole space yields

(4.1) $$\begin{eqnarray}4\unicode[STIX]{x03C0}\unicode[STIX]{x1D708}\int _{0}^{\infty }\text{d}k\,k^{4}Q(k,0)=\unicode[STIX]{x1D700}.\end{eqnarray}$$

Next, integrating (3.12) with respect to $\boldsymbol{k}$ over a sphere of radius $k^{\prime }$ , interchanging $\boldsymbol{k}$ and $\boldsymbol{k}^{\prime }$ , using (4.1), and taking the limit as $\unicode[STIX]{x1D708}\rightarrow 0$ yields (Kraichnan Reference Kraichnan1966)

(4.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700} & = & \displaystyle 4\unicode[STIX]{x03C0}^{2}\int _{0}^{1}\text{d}p\log \frac{1}{p}\int _{\max (p,1-p)}^{1+p}\text{d}q\,pq\left\{b(1,p,q)\int _{0}^{\infty }\text{d}t\,V(q,t)[V(p,t)-V(1,t)]\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\vphantom{\int _{0}^{\infty }}(p\leftrightarrow q)\right\}.\end{eqnarray}$$

See § 6.4 in Leslie (Reference Leslie1973) for the derivation of (4.2).

The mean energy dissipation rate $\unicode[STIX]{x1D700}$ per unit mass is a unique parameter in the closure equations (3.4), (3.10) and (4.2) in the case of $\unicode[STIX]{x1D708}\rightarrow 0$ ; hence, dimensional analysis yields

(4.3) $$\begin{eqnarray}Q(k,t)=\frac{C_{K}}{2\unicode[STIX]{x03C0}}\unicode[STIX]{x1D700}^{2/3}k^{-11/3}\hat{Q}(\unicode[STIX]{x1D70F}),\quad \unicode[STIX]{x1D70F}=C_{K}^{1/2}\unicode[STIX]{x1D700}^{1/3}k^{2/3}t\end{eqnarray}$$

and

(4.4) $$\begin{eqnarray}V(k,t)=\frac{C_{K}}{2\unicode[STIX]{x03C0}}\unicode[STIX]{x1D700}^{2/3}k^{-11/3}\hat{V}(\unicode[STIX]{x1D70F}),\end{eqnarray}$$

where

(4.5a,b ) $$\begin{eqnarray}\hat{Q}(0)=1,\quad \hat{V}(0)=1\end{eqnarray}$$

and $C_{K}$ is the Kolmogorov constant. Note that (3.10) yields

(4.6) $$\begin{eqnarray}\hat{V}(\unicode[STIX]{x1D70F})=[\hat{Q}(\unicode[STIX]{x1D70F})]^{2}.\end{eqnarray}$$

Substituting (4.3) into (2.38) yields the energy spectrum

(4.7) $$\begin{eqnarray}E(k)=C_{K}\unicode[STIX]{x1D700}^{2/3}k^{-5/3},\end{eqnarray}$$

which is well known as the Kolmogorov $-5/3$ power law.

Substituting (4.3) into (3.4) yields

(4.8) $$\begin{eqnarray}\frac{\text{d}\hat{Q}(\unicode[STIX]{x1D70F})}{\text{d}\unicode[STIX]{x1D70F}}=-\hat{Q}(\unicode[STIX]{x1D70F})\int _{0}^{\infty }\text{d}p\,J(p)\int _{0}^{\unicode[STIX]{x1D70F}}\text{d}t^{\prime }\hat{Q}(pt^{\prime }),\end{eqnarray}$$

where

(4.9) $$\begin{eqnarray}J(p)={\textstyle \frac{3}{4}}p^{-5}I(p^{3/2}).\end{eqnarray}$$

The formal solution of (4.8) with (4.5) is given by

(4.10) $$\begin{eqnarray}\log \hat{Q}(\unicode[STIX]{x1D70F})=-\!\int _{0}^{\infty }\text{d}p\,J(p)\int _{0}^{\unicode[STIX]{x1D70F}}\text{d}t^{\prime }(t-t^{\prime })\hat{Q}(pt^{\prime }),\end{eqnarray}$$

the numerical result of which is shown in figure 1, along with that of $\hat{V}(\unicode[STIX]{x1D70F})$ . This figure indicates that the integral time scale of $\hat{V}(\unicode[STIX]{x1D70F})$ is shorter than that of $\hat{Q}(\unicode[STIX]{x1D70F})$ , which is reasonable because $\hat{V}(\unicode[STIX]{x1D70F})$ includes the effect of the diffusion of two fluid particles.

Figure 1.

Two velocity correlation functions $\hat{Q}(\unicode[STIX]{x1D70F})$ and $\hat{V}(\unicode[STIX]{x1D70F})$ plotted against $\unicode[STIX]{x1D70F}=C_{K}^{1/2}\unicode[STIX]{x1D700}^{1/3}k^{2/3}t$ .

Substituting (3.10) into (4.2) and applying (4.3) yields

(4.11) $$\begin{eqnarray}\displaystyle C_{K}^{-3/2} & = & \displaystyle \int _{0}^{1}\text{d}p\log \frac{1}{p}\int _{\max (p,1-p)}^{1+p}\text{d}q\,(pq)^{-8/3}\nonumber\\ \displaystyle & & \displaystyle \times \,\left\{b(1,p,q)\int _{0}^{\infty }\text{d}t\,[\hat{Q}(q^{2/3}t)]^{2}([\hat{Q}(p^{2/3}t)]^{2}-p^{11/3}[\hat{Q}(t)]^{2})+(p\leftrightarrow q)\right\}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The numerical computation of (4.10) and (4.11) yields

(4.12) $$\begin{eqnarray}C_{K}=1.779\end{eqnarray}$$

in the limit of an infinite Reynolds number. This value is close to both $1.62\pm 0.17$ , which is the mean of numerous experimental and observational data for different turbulent flows (Sreenivasan Reference Sreenivasan1995), and 1.58, which is the mean of direct numerical simulation data for $240\leqslant R_{\unicode[STIX]{x1D706}}\leqslant 1000$ (Donzis & Sreenivasan Reference Donzis and Sreenivasan2010), where $R_{\unicode[STIX]{x1D706}}$ is the Taylor-scale Reynolds number. The other previously developed closure models with no adjustable free parameters for high Reynolds numbers yield the following values for the Kolmogorov constant:

1. (i) $C_{K}=1.77$ (Kraichnan Reference Kraichnan1966) for the abridged Lagrangian DIA,

2. (ii) $C_{K}=2.3$ (McComb & Shanmugasundaram Reference McComb and Shanmugasundaram1984) for the LET model,

3. (iii) $C_{K}=1.72$ (Kaneda Reference Kaneda1986; Kida & Goto Reference Kida and Goto1997) for the Lagrangian DIA,

4. (iv) $C_{K}=3.8$ (Edwards & McComb Reference Edwards and McComb1969) and $C_{K}=1.2$ (Qian Reference Qian1983) for the Fokker–Planck model,

5. (v) $C_{K}=1.73$ (Bos & Bertoglio Reference Bos and Bertoglio2006) for the EDQNM.

4.2 Universal equilibrium range

Let us consider the energy spectrum throughout the universal equilibrium range, which constitutes the inertial and dissipation ranges. In this range, the mean energy dissipation rate $\unicode[STIX]{x1D700}$ per unit mass and the kinematic viscosity $\unicode[STIX]{x1D708}$ are the only parameters in the closure equations (3.4), (3.10) and (3.12); hence, dimensional analysis yields

(4.13) $$\begin{eqnarray}Q(k,t)=\frac{1}{2\unicode[STIX]{x03C0}}C_{K}\unicode[STIX]{x1D700}^{2/3}k^{-11/3}\tilde{Q}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D70F}),\end{eqnarray}$$

where

(4.14a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D705}=C_{K}^{-3/8}\unicode[STIX]{x1D700}^{-1/4}\unicode[STIX]{x1D708}^{3/4}k=C_{K}^{-3/8}k\unicode[STIX]{x1D702},\quad \unicode[STIX]{x1D70F}=C_{K}^{1/2}\unicode[STIX]{x1D700}^{1/3}k^{2/3}t\end{eqnarray}$$

and

(4.15) $$\begin{eqnarray}\tilde{Q}(0,\unicode[STIX]{x1D70F})=\hat{Q}(\unicode[STIX]{x1D70F}).\end{eqnarray}$$

Here, $\unicode[STIX]{x1D702}\;(=\unicode[STIX]{x1D700}^{-1/4}\unicode[STIX]{x1D708}^{3/4})$ denotes the Kolmogorov length scale, and $\hat{Q}(\unicode[STIX]{x1D70F})$ is defined in (4.3).

Substituting (4.13) into (3.4) and (3.12) respectively yields

(4.16) $$\begin{eqnarray}\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\unicode[STIX]{x1D705}^{4/3}\right]\tilde{Q}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D70F})=-\tilde{Q}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D70F})\int _{0}^{\infty }\text{d}p\,J(p)\int _{0}^{\unicode[STIX]{x1D70F}}\text{d}t\,\tilde{Q}(\unicode[STIX]{x1D705}p^{3/2},pt)\end{eqnarray}$$

and

(4.17) $$\begin{eqnarray}\displaystyle \tilde{Q}(\unicode[STIX]{x1D705},0) & = & \displaystyle \frac{1}{2}\iint _{I_{\unicode[STIX]{x1D705}}}\text{d}p\,\text{d}q\,\unicode[STIX]{x1D705}^{-1}(pq)^{-8/3}b(\unicode[STIX]{x1D705},p,q)\int _{0}^{\infty }\text{d}t\exp [-(\unicode[STIX]{x1D705}^{2}+p^{2}+q^{2})t]\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{[\tilde{Q}(q,q^{2/3}t)]^{2}}{\tilde{Q}(q,0)}\left\{\unicode[STIX]{x1D705}^{11/3}\frac{[\tilde{Q}(p,p^{2/3}t)]^{2}}{\tilde{Q}(p,0)}-p^{11/3}\frac{[\tilde{Q}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D705}^{2/3}t)]^{2}}{\tilde{Q}(\unicode[STIX]{x1D705},0)}\right\},\end{eqnarray}$$

where (3.10) was used. The formal solution of (4.16) is given by

(4.18) $$\begin{eqnarray}\log \frac{\tilde{Q}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D70F})}{\tilde{Q}(\unicode[STIX]{x1D705},0)}=-\unicode[STIX]{x1D705}^{4/3}\unicode[STIX]{x1D70F}-\int _{0}^{\infty }\text{d}p\,J(p)\int _{0}^{\unicode[STIX]{x1D70F}}\text{d}t\,(\unicode[STIX]{x1D70F}-t)\tilde{Q}(\unicode[STIX]{x1D705}p^{3/2},pt).\end{eqnarray}$$

Substituting (4.13) into (2.38) yields the compensated energy spectrum

(4.19) $$\begin{eqnarray}\tilde{Q}(\unicode[STIX]{x1D705},0)=\frac{k^{5/3}E(k)}{C_{K}\unicode[STIX]{x1D700}^{2/3}}.\end{eqnarray}$$

The numerical result $\tilde{Q}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D70F})$ of (4.17) and (4.18) at $\unicode[STIX]{x1D70F}=0$ for the LIF model, along with that for the Lagrangian DIA (Kaneda Reference Kaneda1986; Kida & Goto Reference Kida and Goto1997) and the corresponding energy spectrum $E(k)$ for the LIF model are shown in figure 2. Figure 2(a) indicates that the LIF model reproduces the bottleneck effect (a small bump at $\unicode[STIX]{x1D705}\approx 0.1$ ), whereas the Lagrangian DIA does not (see also figure 4). The compensated energy spectrum $\tilde{Q}(\unicode[STIX]{x1D705},0)$ attains a maximum value of 1.02 at $k\unicode[STIX]{x1D702}=C_{K}^{3/8}\unicode[STIX]{x1D705}=0.11$ , which is close to the value of 0.13 obtained from the direct numerical simulation by Donzis & Sreenivasan (Reference Donzis and Sreenivasan2010). They stated that the bottleneck decreases with increasing Reynolds number and would be negligible at $R_{\unicode[STIX]{x1D706}}\sim 2\times 10^{5}$ . This statement is not incompatible with the obtained peak value of $1.02\approx 1$ for the LIF model in the limit of an infinite Reynolds number.

Figure 2.

(a) Compensated energy spectra $\tilde{Q}(\unicode[STIX]{x1D705},0)$ plotted against $\unicode[STIX]{x1D705}=C_{K}^{-3/8}k\unicode[STIX]{x1D702}$ for the LIF model and the Lagrangian DIA. (b) Log–log plot of the energy spectrum $E(k)/(\unicode[STIX]{x1D700}^{1/4}\unicode[STIX]{x1D708}^{5/4})$ plotted against $k\unicode[STIX]{x1D702}$ for the LIF model.

The skewness $S$ of the longitudinal velocity derivative in stationary homogeneous isotropic turbulence is expressed as (see (7.5.15) in Batchelor (Reference Batchelor1953))

(4.20) $$\begin{eqnarray}S=\frac{\displaystyle \left\langle \left({\displaystyle \frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}x_{1}}}\right)^{3}\right\rangle }{\displaystyle \left\langle \left({\displaystyle \frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}x_{1}}}\right)^{2}\right\rangle ^{3/2}}=-\frac{3\sqrt{30}\,\unicode[STIX]{x1D708}}{7}\frac{\displaystyle \int _{0}^{\infty }\text{d}k\,k^{4}E(k)}{\displaystyle \left[\int _{0}^{\infty }\text{d}k\,k^{2}E(k)\right]^{3/2}}=-\frac{3\sqrt{30}}{7}\frac{\displaystyle \int _{0}^{\infty }\text{d}\unicode[STIX]{x1D705}\unicode[STIX]{x1D705}^{7/3}\tilde{Q}(\unicode[STIX]{x1D705},0)}{\displaystyle \left[\displaystyle \int _{0}^{\infty }\text{d}\unicode[STIX]{x1D705}\unicode[STIX]{x1D705}^{1/3}\tilde{Q}(\unicode[STIX]{x1D705},0)\right]^{3/2}},\end{eqnarray}$$

where (4.14a ) and (4.19) were used. Substituting the numerical result $\tilde{Q}(\unicode[STIX]{x1D705},0)$ into (4.20) yields

(4.21) $$\begin{eqnarray}S=-0.49\end{eqnarray}$$

in the limit of an infinite Reynolds number. Antonia et al. (Reference Antonia, Tang, Djenidi and Danaila2015) reexamined numerous experimental and numerical data for different turbulent flows, with the result that the skewness $S$ should approach a constant value of $-0.53$ when the Reynolds number is sufficiently large (see also Sreenivasan & Antonia Reference Sreenivasan and Antonia1997). The other closure models for high Reynolds numbers yield the following values for the skewness:

1. (i) $S=-0.35$ (McComb & Shanmugasundaram Reference McComb and Shanmugasundaram1984) for the LET model,

2. (ii) $S=-0.66$ (Kaneda Reference Kaneda1993; Kida & Goto Reference Kida and Goto1997) for the Lagrangian DIA,

3. (iii) $S=-0.515$ (Qian Reference Qian1994) for the Fokker–Planck model,

4. (iv) $S=-0.4$ (Bos et al. Reference Bos, Chevillard, Scott and Rubinstein2012) for the EDQNM.

Substituting (4.13) into (4.1) yields

(4.22) $$\begin{eqnarray}\frac{1}{2}C_{K}^{-3/2}=\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D705}\unicode[STIX]{x1D705}^{1/3}\tilde{Q}(\unicode[STIX]{x1D705},0),\end{eqnarray}$$

which is useful to check the numerical result $\tilde{Q}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D70F})$ of (4.17) and (4.18). Substituting the numerical result $\tilde{Q}(\unicode[STIX]{x1D705},0)$ into (4.22) yields $C_{K}=1.78$ , which is very close to (4.12).