Energy Partitioning Control in the PITM Hybrid RANS/LES Method for the Simulation of Turbulent Flows

Abstract

The partially integrated transport modeling (PITM) method first introduced in Refs. Schiestel and Dejoan (Theor Comput Fluid Dyn 18:443, 2005) and in Chaouat and Schiestel (Phys Fluids 17:065106, 2005) provides a continuous approach for hybrid RANS-LES simulations. Inspired from the multiple scale approach, the basis of the development of the method is the spectral space in quasi-homogeneous turbulence. The PITM method embodies a partitioning control function that monitors the ratio of subfilter energy to total turbulent energy by reference to the cutoff wavenumber location. How this procedure behaves in inhomogeneous flows is an important question. The present paper demonstrates that the same control function can be used both in homogeneous and in non-homogeneous flows as well. Further on, an analysis of the effect of anisotropic filters generally used for wall flows is conducted for computing the equivalent cutoff wavenumber suitable for determining the subfilter energy of the spectrum and see how it interferes with control function. Illustrations in the turbulent plane channel flow are given that confirm the efficiency of the procedure and DNS data have been used to support and supplement the discussion.

Appendices

Appendix A: Mathematical Properties

Let us consider

$$\begin{aligned} \tau (f,g) = {\overline{fg}} - {{\overline{f}}} {{\overline{g}}} \end{aligned}$$

(129)

and

$$\begin{aligned} \tau (f,g,h)={\overline{fgh}}-{\bar{f}}\tau (g,h)-{\bar{g}}\tau (h,f)-{\bar{h}}\tau (f,g) -{\bar{f}} {\bar{g}} {\bar{h}} \end{aligned}$$

(130)

Eq. (129) can be developed in the following form as

$$\begin{aligned} \tau (f ,g) = \overline{ {\bar{f}} {\bar{g}} } - {\bar{f}} {\bar{g}} + \overline{ {\bar{f}} g^> } + \overline{ {\bar{g}} f^>} + \overline{ f^> g^> } \end{aligned}$$

(131)

It is then possible to get the averaging in statistical sense of (131) as

$$\begin{aligned} \left\langle \tau (f ,g) \right\rangle = \left\langle \overline{ {\bar{f}} {\bar{g}} } \right\rangle - \left\langle {\bar{f}} {\bar{g}} \right\rangle + \left\langle \overline{ {\bar{f}} g^> } \right\rangle + \left\langle \overline{ {\bar{g}} f^>} \right\rangle + \left\langle \overline{ f^> g^> } \right\rangle \end{aligned}$$

(132)

Strictly speaking, the filtering process leads to mathematical non-equality $\overline{ \left\langle \phi \right\rangle } =\left\langle {\bar{\phi }}\right\rangle \ne \left\langle \phi \right\rangle$ for any variable $\phi$. To recover $\left\langle {\bar{\phi }}\right\rangle = \left\langle \phi \right\rangle$, the key concept is to consider the tangent homogeneous anisotropic turbulence field at the physical space location implying that the variation of the mean variable $\phi$ is accounted for by the use of Taylor series expansion in space limited to the linear terms (Chaouat and Schiestel 2007, 2013; Chaouat 2017). In practice, $\left\langle {\bar{\phi }}\right\rangle \approx \left\langle \phi \right\rangle$ only if the variation of the flow velocities over the filter width is not too large. In the framework of the tangent homogeneous space and the spectral cutoff filter, $\overline{ \left\langle \phi \right\rangle } =\left\langle {\bar{\phi }}\right\rangle = \left\langle \phi \right\rangle$ and $\left\langle \phi ^> \phi ^< \right\rangle = 0$, so that Eq. (132) reduces to

$$\begin{aligned} \langle {\tau (f,g)} \rangle = \langle {f^> g^ > } \rangle \end{aligned}$$

(133)

and we get also

$$\begin{aligned} \left\langle \tau (f,g,h) \right\rangle = \left\langle f^> g^> h^>\right\rangle \end{aligned}$$

(134)

In particular, the subfilter scale stress tensor defined in large eddy simulation is then

$$\begin{aligned} \tau _{ij}^{(s)} =\tau (u_i, u_j)= \overline{u_i u_j} - \bar{ u}_i {\bar{u}}_j \end{aligned}$$

(135)

while the resolved scale stress tensor is

$$\begin{aligned} {\tau _{ij} ^{(r)} }={\bar{u}}_i {\bar{u}}_j- \left\langle u_i \right\rangle \left\langle u_j \right\rangle \end{aligned}$$

(136)

so that

$$\begin{aligned} \tau _{ij} ^{(s+r)} = \tau _{ij}^{(s)} + \tau _{ij}^{(r)} = \overline{u_i u_j} - \left\langle u_i \right\rangle \left\langle u_j \right\rangle \end{aligned}$$

(137)

In RANS modeling, the Reynolds stress tensor accounting for the total fluctuating velocities is defined as

$$\begin{aligned} R_{ij} = \left\langle u_i u_j \right\rangle - \left\langle u_i \right\rangle \left\langle u_j \right\rangle \end{aligned}$$

(138)

Using the decomposition $\phi =\left\langle \phi \right\rangle +\phi ^<+ \phi ^>$, the Reynolds stress tensor $R_{ij}$ can be rewritten as

$$\begin{aligned} R_{ij}= \left\langle u^<_i u^<_j \right\rangle + \left\langle u^>_i u^>_j\right\rangle + \left\langle u^<_i u^>_j\right\rangle + \left\langle u^>_i u^<_j\right\rangle \end{aligned}$$

(139)

So that, assuming that the correlation between the small scale and large scale $\left\langle u^<_i u^>_j\right\rangle$ are small compared to the other correlations, $R_{ij}$ can be computed in a first approximation as the sum of the statistical average of subfilter and resolved stresses

$$\begin{aligned} R_{ij} \approx \left\langle \tau _{ij} ^{(s+r)} \right\rangle = \left\langle \tau _{ij} ^{(s)} \right\rangle + \left\langle \tau _{ij} ^{(r)} \right\rangle \end{aligned}$$

(140)

and the contraction of the tensors appearing in Eq. (140) leads to

$$\begin{aligned} k \approx \left\langle k^{(s+r)} \right\rangle = \left\langle k^{(s)} \right\rangle + \left\langle k^{(r)} \right\rangle \end{aligned}$$

(141)

where $k^{(s)}$ and $k^{(r)}$ are the filtered subfilter and resolved turbulence energy, respectively. This relation is strictly true in the case of the spectral cutoff filter, and only approximate in the case of a sharp filter.

Appendix B: Formal Derivative Operators

For sake of clarity, we consider in this section the use of a uniform spectral cutoff filter (Chaouat and Schiestel 2007, 2013; Chaouat 2017). As recalled in “Appendix A”, each variable $\phi$ can be decomposed into a statistical mean $\left\langle \phi \right\rangle$, a macro-scale fluctuation $\phi ^<$ and a micro-scale fluctuation $\phi ^>$. Using this decomposition for the velocity, it is a simple matter to see that the mass conservation implies

$$\begin{aligned} \frac{\partial u^<_j }{\partial x_j} + \frac{\partial u^>_j}{\partial x_j}=0 \end{aligned}$$

(142)

But the spectral cutoff filter also yields

$$\begin{aligned} \frac{\partial u^<_j }{\partial x_j} = \frac{\partial u^>_j }{\partial x_j} =0 \end{aligned}$$

(143)

The particle derivative in the instantaneous flow is given by

$$\begin{aligned} \frac{d\phi }{dt} = \frac{\partial \phi }{\partial t} + u_j \frac{\partial \phi }{\partial x_j } \end{aligned}$$

(144)

The material derivative accounting for the statistical velocity is

$$\begin{aligned} \frac{ D \phi }{Dt} = \frac{\partial \phi }{\partial t} + \left\langle u_j \right\rangle \frac{\partial \phi }{\partial x_j } \end{aligned}$$

(145)

and in particular

$$\begin{aligned} \left\langle \frac{ D \phi }{Dt} \right\rangle = \frac{ D \left\langle \phi \right\rangle }{Dt} \end{aligned}$$

(146)

The material derivative accounting for the filtered velocity is

$$\begin{aligned} \frac{{\mathcal {D}} \phi }{{\mathcal {D}}t} = \frac{\partial \phi }{\partial t} + {\bar{u}} _j \frac{\partial \phi }{\partial x_j } = \frac{\partial \phi }{\partial t} + \left\langle u_j \right\rangle \frac{\partial \phi }{\partial x_j } + u^<_j \frac{\partial \phi }{\partial x_j } \end{aligned}$$

(147)

or equivalently

$$\begin{aligned} \frac{{\mathcal {D}} \phi }{{\mathcal {D}}t} = \frac{{D} \phi }{{D}t} + u^<_j \frac{\partial \phi }{\partial x_j } \end{aligned}$$

(148)

so that the averaging in the statistical sense of Eq. (147) is

$$\begin{aligned} \left\langle \frac{{\mathcal {D}} \phi }{{\mathcal {D}}t} \right\rangle = \frac{\partial \left\langle \phi \right\rangle }{\partial t} + \left\langle u_j \right\rangle \frac{\partial \left\langle \phi \right\rangle }{\partial x_j } + \frac{\partial }{\partial x_j } \left\langle u^<_j \phi \right\rangle + u^<_j \frac{\partial \phi }{\partial x_j } \end{aligned}$$

(149)

On the other hand, applying a filtering process on Eq. (147) gives

$$\begin{aligned} \overline{\frac{{\mathcal {D}} \phi }{{\mathcal {D}} t} } = \frac{{\mathcal {D}} \overline{\phi } }{{\mathcal {D}} t} \end{aligned}$$

(150)

Using (143), Eq. (149) can be rewritten as

$$\begin{aligned} \left\langle \frac{{\mathcal {D}} \phi }{{\mathcal {D}}t} \right\rangle = \frac{\partial \left\langle \phi \right\rangle }{\partial t} + \left\langle u_j \right\rangle \frac{\partial \left\langle \phi \right\rangle }{\partial x_j } + \frac{\partial }{\partial x_j } \left\langle u^<_j \phi \right\rangle \end{aligned}$$

(151)

leading to

$$\begin{aligned} \left\langle \frac{{\mathcal {D}} \phi }{{\mathcal {D}}t} \right\rangle = \frac{ D \left\langle \phi \right\rangle }{Dt} + \frac{\partial }{\partial x_j } \left\langle u^<_j \phi \right\rangle \end{aligned}$$

(152)

More generally, from the definition

$$\begin{aligned} \frac{d \phi }{dt} = \frac{\partial \phi }{\partial t} + \left\langle u_j \right\rangle \frac{\partial \phi }{\partial x_j } + u^<_j \frac{\partial \phi }{\partial x_j } + u^>_j \frac{\partial \phi }{\partial x_j } \end{aligned}$$

(153)

and one can also easily derive some useful relations such as

$$\begin{aligned}{}&\overline{\frac{d \phi }{dt} } = \frac{{\mathcal {D}} \overline{\phi } }{{\mathcal {D}}t} + \frac{\partial \overline{u^>_j \phi }}{\partial x_j } \end{aligned}$$

(154)

$$\begin{aligned}{}&\left\langle \frac{d \phi }{dt} \right\rangle = \frac{ D \left\langle \phi \right\rangle }{Dt} + \frac{\partial }{\partial x_j }\left\langle u^<_j \phi \right\rangle + \frac{\partial }{\partial x_j }\left\langle u^>_j \phi \right\rangle \end{aligned}$$

(155)

Finally, note that

$$\begin{aligned} \frac{\partial }{\partial x_j} \left\langle u^<_j \overline{\phi } \right\rangle = \frac{\partial }{\partial x_j} \left\langle u^<_j \phi \right\rangle \end{aligned}$$

(156)

because $\left\langle u^<_i \phi ^>\right\rangle$ reduces to zero.

Appendix C: Remark on Turbulent Flows Near Energetic Equilibrium

In the framework of first-moment closure referring to eddy-viscosity turbulence models, the production term, without prejudice to the reasoning, can be expressed as

$$\begin{aligned} P^{(s)} = 2C_\mu \frac{{{k^{(s)} }^2 }}{{\varepsilon ^{(s)} }}\left( {{{\overline{S}}} _{ij} {{\overline{S}}} _{ij} } \right) \end{aligned}$$

(157)

with

$$\begin{aligned} S_{ij} = \frac{1}{2}\left( {\frac{{\partial u_i }}{{\partial x_j }} + \frac{{\partial u_j }}{{\partial x_i }}} \right) \end{aligned}$$

(158)

and

$$\begin{aligned} \nu _t ^{(s)} = C_\mu \frac{{{k^{(s)} }^2 }}{{\varepsilon ^{(s)} }} \end{aligned}$$

(159)

so that

$$\begin{aligned}{}&\left\langle {P^{(s)} } \right\rangle \approx 2\left\langle {\nu _t ^{(s)} } \right\rangle \left\langle {S_{ij} S_{ij} } \right\rangle \approx 2\left\langle {\nu _t ^{(s)} } \right\rangle \left\langle {S_{ij} } \right\rangle \left\langle {S_{ij} } \right\rangle + 2\left\langle {\nu _t ^{(s)} } \right\rangle \left\langle {S_{ij} ^< S_{ij} ^< } \right\rangle \nonumber \\&\quad \approx 2\left\langle {\nu _t ^{(s)} } \right\rangle \left\langle {S_{ij} } \right\rangle \left\langle {S_{ij} } \right\rangle + \frac{1}{2}\left\langle {\nu _t ^{(s)} } \right\rangle \left\langle {\left( {\frac{{\partial u_i ^< }}{{\partial x_j }} + \frac{{\partial u_j ^< }}{{\partial x_i }}} \right) \left( {\frac{{\partial u_i ^< }}{{\partial x_j }} + \frac{{\partial u_j ^ < }}{{\partial x_i }}} \right) } \right\rangle \end{aligned}$$

(160)

where the last term is the mean spectral flux $F^{(1)}$. As it is clear from Eq. (160), the averaging in a statistical sense of the subfilter production (including the mean flow production and the production from the large resolved scales giving rise to $F^{(1)}$ approaches the RANS production, i.e., $\left\langle P^{(s)} \right\rangle \approx P$ and the same result prevails for the subfilter dissipation-rate $\left\langle \varepsilon ^{(s)} \right\rangle \approx \epsilon$ for flows close to energetic equilibrium so that

$$\begin{aligned} P \approx \left\langle {P^{(s)} } \right\rangle \approx 2\nu _{t} \left\langle {S_{ij} } \right\rangle \left\langle {S_{ij} } \right\rangle \end{aligned}$$

(161)

where $\nu _{t}$ denotes the turbulent eddy viscosity in RANS modeling given by

$$\begin{aligned} \nu _{t} = C_\mu \frac{{k^2 }}{\varepsilon } \end{aligned}$$

(162)

involving the total kinetic energy. All these relations can be also rewritten

$$\begin{aligned}{}&P \approx 2\left\langle {\nu _t ^{(s)} } \right\rangle \left\langle {S_{ij} } \right\rangle \left\langle {S_{ij} } \right\rangle + 2\left\langle {\nu _t ^{(s)} } \right\rangle \left\langle {S_{ij} ^< S_{ij} ^ < } \right\rangle \nonumber \\&\quad \approx 2\left\langle {\nu _t ^{(s)} } \right\rangle \left\langle {{{\overline{S}}} _{ij} {{\overline{S}}} _{ij} } \right\rangle \approx 2\nu _{t} \left\langle {S_{ij} } \right\rangle \left\langle {S_{ij} } \right\rangle \end{aligned}$$

(163)

These relations are recalling the Heisenberg hypothesis is spectral space (Hinze 1975) assuming that the effect of the small eddies with wavenumbers larger than $\kappa$ by which energy is withdrawn from larger eddies with wavenumbers smaller than $\kappa$ is described through a small scales eddy viscosity. Then the spectral flux is represented as the product of two integrals in the domains $\left[ {0,\kappa } \right]$ and $\left[ {\kappa ,\infty } \right]$ as in Eq. (163).

Appendix D: Order of Magnitude of the Resolved Part of the Turbulent Diffusion

We consider the resolved part of the turbulent advection denoted $J^{(r)}$ of any arbitrary variable $\psi$ due to the macro-scale fluctuation velocity $u^<_i$ as

$$\begin{aligned} J^{(r)}_\psi (\psi ) = - \frac{\partial }{{\partial x_j }} ( {u^< _j \psi ^ < } ) \end{aligned}$$

(164)

and which becomes a turbulent diffusion term once statistically averaged.

$$\begin{aligned} \left\langle J^{(r)}_\psi (\psi )\right\rangle = - \frac{\partial }{{\partial x_j}} \left\langle {u^< _j \psi ^ < } \right\rangle \end{aligned}$$

(165)

The large scale fluctuation variable $\psi ^< = {\bar{\psi }} -\left\langle \psi \right\rangle$ is fully resolved in the numerical simulation but for convenience an analytical approximation can be easily obtained considering the convection process by means of the large scale fluctuating velocity $u^<_i$ and using the time scale of the resolved eddies $\tau ^{(r)}$ as

$$\begin{aligned} \psi ^< \approx { \left\langle \psi \right\rangle } \left( x_i + u^<_i \tau ^{(r)} \right) - { \left\langle \psi \right\rangle } (x_i ) \end{aligned}$$

(166)

where

$$\begin{aligned} \tau ^{(r)} \approx \frac{ k^{(r)} }{\epsilon ^{(s)} } \end{aligned}$$

(167)

corresponds to the displacement time of an eddy before loosing its individuality. The approximation of the fluctuation corresponding to a wavelength range $[\kappa _1,\kappa _2]$ is approached by characteristic the time scale of this range of eddies multiplied by the gradient of the filtered field with a cutoff at $\kappa _1$. If $\kappa _1=0$, then the use of the gradient of the mean value is natural. Then, Taylor series expansion of Eq. (166) limited to the first order yields

$$\begin{aligned} \psi ^< \approx \tau ^{(r)} u^< _m \frac{\partial {\left\langle \psi \right\rangle } }{\partial x_m } \end{aligned}$$

(168)

So that

$$\begin{aligned} J^{(r)}_\psi = \frac{\partial }{\partial x_j} \left[ {u^<_j u^<_m} \tau ^{(r)} \frac{\partial {\left\langle \psi \right\rangle }}{\partial x_m } \right] \end{aligned}$$

(169)

or in a first approximation by neglecting the cross-correlations

$$\begin{aligned} J^{(r)}_\psi (\psi )= - \frac{\partial }{{\partial x_j }} {u^< _j \psi ^< } \approx \frac{\partial }{\partial x_j} \left[ {\frac{2}{3}u^<_m u^<_m} \tau ^{(r)} \frac{\partial {\left\langle \psi \right\rangle }}{\partial x_j } \right] \approx C_\psi \frac{\partial }{\partial x_j} \left[ \frac{ (k^{(r)} )^2}{\epsilon ^{(s)}} \frac{\partial {\left\langle \psi \right\rangle }}{\partial x_j } \right] \end{aligned}$$

(170)

where $C_\psi$ is a numerical coefficient introduced to describe the diffusion process, leading to

$$\begin{aligned} \left\langle J^{(r)}_\psi \right\rangle \approx C_\psi \frac{\partial }{\partial x_j} \left[ \left\langle \frac{ \left( k^{(r)} \right) ^2}{\epsilon ^{(s)}}\right\rangle \frac{\partial {\left\langle \psi \right\rangle }}{\partial x_j } \right] \end{aligned}$$

(171)

Moreover, for the instantaneous value ${\bar{\psi }}= \left\langle \psi \right\rangle + \psi ^<$, a fluctuating part remains

$$\begin{aligned} J^{(r)}_\psi (\psi )= \left\langle J^{(r)}_\psi \right\rangle +{J^{(r)}_\psi }^<=C_\psi \frac{\partial }{\partial x_j} \left[ \left\langle \frac{ \left( k^{(r)} \right) ^2}{\epsilon ^{(s)}}\right\rangle \frac{\partial \left\langle \psi \right\rangle }{\partial x_j } \right] +{J^{(r)}_\psi }^< \end{aligned}$$

(172)

Similarly, the diffusion process caused by the micro-scale fluctuation $\psi ^>$ is then can be computed as

$$\begin{aligned} \psi ^> \approx { {\bar{\psi }} } \left( x_i + u^>_i \tau ^{(s)} \right) - { {\bar{\psi }} } (x_i ) \end{aligned}$$

(173)

with

$$\begin{aligned} \tau ^{(s)} \approx \frac{ k^{(s)} }{\epsilon ^{(s)} } \end{aligned}$$

(174)

leading to

$$\begin{aligned} \psi ^> \approx \tau ^{(s)} u^> _m \frac{\partial {{\bar{\psi }}} }{\partial x_m } \end{aligned}$$

(175)

so that

$$\begin{aligned} J^{(s)}_\psi (\psi ) = - \frac{\partial }{{\partial x_j }} (\overline{u^> _j \psi ^ > } ) = C_\psi \frac{\partial }{\partial x_j} \left[ \frac{ \left( k^{(s)} \right) ^2}{\epsilon ^{(s)}} \frac{\partial {\bar{\psi }}}{\partial x_j } \right] = C_\psi \frac{\partial }{\partial x_j} \left[ \frac{ \left( k^{(s)} \right) ^2}{\epsilon ^{(s)}} \frac{\partial \left\langle {\psi }\right\rangle }{\partial x_j } \right] +C_\psi \frac{\partial }{\partial x_j} \left[ \frac{ \left( k^{(s)} \right) ^2}{\epsilon ^{(s)}} \frac{\partial {\psi ^<} }{\partial x_j } \right] \end{aligned}$$

(176)

The total mean diffusion involving the micro-scale and the macro-scale is therefore

$$\begin{aligned} \left\langle J^{(s+r)}_\psi \right\rangle = \left\langle J^{(s)}_\psi \right\rangle + \left\langle J^{(r)}_\psi \right\rangle = C_\psi \frac{\partial }{\partial x_j} \left[ \left\langle \frac{ \left( k^{(s)} \right) ^2 + \left( k^{(r)} \right) ^2 }{\epsilon ^{(s)}} \right\rangle \frac{\partial \left\langle {\psi }\right\rangle }{\partial x_j } \right] \approx C_\psi \frac{\partial }{\partial x_j} \left[ \left\langle \frac{\left( k^{(s)}+ k^{(r)} \right) ^2 }{\epsilon ^{(s)}}\right\rangle \frac{\partial \left\langle {\psi }\right\rangle }{\partial x_j } \right] \end{aligned}$$

(177)

or in a more compact form

$$\begin{aligned} \left\langle J^{(s+r)}_\psi \right\rangle \approx C_\psi \frac{\partial }{\partial x_j} \left[ \left\langle \frac{\left( k^{(s+r)}\right) ^2 }{\epsilon ^{(s)}}\right\rangle \frac{\partial \left\langle {\psi }\right\rangle }{\partial x_j } \right] \end{aligned}$$

(178)

that is to say

$$\begin{aligned} \left\langle J^{(s+r)}_\psi \right\rangle \approx C_\psi \frac{\partial }{\partial x_j} \left[ \frac{k^2 }{\epsilon } \frac{\partial \left\langle {\bar{\psi }} \right\rangle }{\partial x_j } \right] \end{aligned}$$

(179)

Note that the mean diffusion originating from the resolved scales is non zero only in the case of triple correlations. Indeed, as already remarked, double correlations between large scale and small scale fluctuations are always zero $\left\langle u_i^{<}u_i^{>}\right\rangle =0$. In contrast, for triple correlations like $\left\langle u_j^{<}k^{(s)'}\right\rangle$ with $k^{(s)'}=k^{(s)<}+k^{(s)>}$, $k^{(s)<}=u_i^{<}u_i^{<}-\left\langle u_i^{<}u_i^{<}\right\rangle$ and $k^{(s)>}=u_i^{>}u_i^{>}-\left\langle u_i^{>}u_i^{>}\right\rangle$, it appears $\left\langle u_j^{<} k^{(s)'}\right\rangle =\left\langle u_j^{<} k^{(s)<}\right\rangle$. In practice, an approximation of $\left\langle J^{(s)}_\psi \right\rangle$ and $\left<J^{(r)}_\psi \right>$ for practical use are thus

$$\begin{aligned} \left\langle J^{(s)}_\psi \right\rangle = C_\psi \frac{\partial }{\partial x_j} \left[ \frac{ \left\langle k^{(s)} \right\rangle ^2}{\left\langle \epsilon ^{(s)}\right\rangle } \frac{\partial \left\langle {\bar{\psi }}\right\rangle }{\partial x_j } \right] \end{aligned}$$

(180)

and

$$\begin{aligned} \left\langle J^{(r)}_\psi \right\rangle = C_\psi \frac{\partial }{\partial x_j} \left[ \frac{ \left\langle k^{(r)} \right\rangle ^2}{\left\langle \epsilon ^{(s)}\right\rangle } \frac{\partial \left\langle {\bar{\psi }}\right\rangle }{\partial x_j } \right] \end{aligned}$$

(181)

We have thus approximated both the large scale fluctuating diffusion term and the small scale averaged one in a consistent way.

Appendix E: Variational Calculus of the Diffusion Terms $J^{(s+r)}_{\psi} (\psi )$

The general expressions of the diffusion terms $J^{(r)}_\psi (\psi )$ and $J^{(s)}_\psi (\psi )$ are given by Eqs. (172) and (176) in which $k^{(r)}$, $k^{(s)}$ and $\epsilon ^{(s)}$ are all functions of $(x_i,t)$. In a first approximation, the functional variations of $\psi (x_i,t)$ can be computed as $\delta \psi (x_i,t) = \zeta (x_i , t)\delta \alpha$, where $\zeta (x_i ,t)$ is a function in space and time while $\delta \alpha$ is a small scalar increment, it clearly emphasizes the fact that $\delta$ is a functional variation of $\psi$ whose magnitude is controlled by $\delta \alpha$ while keeping the same shape $\zeta (x_i ,t)$. If $\partial /\partial x_i$ is the usual partial differential operator in $x_i$ (or t ) corresponding to the usual first partial derivative, then one can write

$$\begin{aligned} \delta \left( \frac{\partial \psi }{\partial x_i} \right) (x_i ,t) = \frac{\partial (\delta \psi )}{\partial x_i} (x_i ,t)= \frac{\partial }{\partial x_i} \left[ \zeta (x_i , t) \right] \delta \alpha \end{aligned}$$

(182)

regardless the particular shape of the $\psi (x_i , t)$ curve. Then, assuming that the function $\zeta (x_i ,t)$ roughly keeps the same shape as the original function $\psi (x_i , t)$, that is to say

$$\begin{aligned} \zeta (x_i ,t) \propto \psi (x_i , t) \end{aligned}$$

(183)

we get the following estimate

$$\begin{aligned} \frac{\delta (\frac{\partial \psi }{\partial x_i})}{\frac{\partial \psi }{\partial x_i}}(x_i ,t)= \frac{\delta \psi }{\psi }(x_i ,t) = \delta \alpha \end{aligned}$$

(184)

Applied to Eq. (178), one gets

$$\begin{aligned} \frac{\delta \left\langle J^{(s+r)}_\psi \right\rangle }{\left\langle J^{(s+r)}_\psi \right\rangle }(x_i ,t) \approx \frac{\delta \left[ \left\langle \frac{\left( k^{(s+r)}\right) ^2 }{\epsilon ^{(s)}}\right\rangle \frac{\partial \left\langle {\psi }\right\rangle }{\partial x_j }\right] }{ \left[ \left\langle \frac{\left( k^{(s+r)}\right) ^2 }{\epsilon ^{(s)}}\right\rangle \frac{\partial \left\langle {\psi }\right\rangle }{\partial x_j } \right] }(x_i ,t) \end{aligned}$$

(185)

and thus

$$\begin{aligned} \frac{\delta \left\langle J^{(s+r)}_\psi \right\rangle }{ \left\langle J^{(s+r)}_\psi \right\rangle }(x_i ,t) \approx 2 \frac{\delta \left\langle k^{(s+r)}\right\rangle }{ \left\langle k^{(s+r) }\right\rangle }(x_i ,t) -\frac{\delta \left\langle \epsilon ^{(s)}\right\rangle }{\left\langle \epsilon ^{(s)}\right\rangle }(x_i ,t) +\frac{\delta \left\langle \psi \right\rangle }{\left\langle \psi \right\rangle }(x_i ,t) \end{aligned}$$

(186)

for ${\psi } = k^{(s)}(x_i ,t)$ and ${\psi }=\epsilon ^{(s)}(x_i ,t)$, respectively, considering that $k^{(s+r)}=k^{(s)}+k^{(r)}$ remains approximatively constant when the cutoff is varied, Eq. (186) reduces to

$$\begin{aligned} \frac{\delta \left\langle J^{(s+r)}_\psi \right\rangle }{\left\langle J^{(s+r)}_\psi \right\rangle }(x_i ,t) = - \frac{\delta \left\langle \epsilon ^{(s)}\right\rangle }{\left\langle \epsilon ^{(s)}\right\rangle }(x_i ,t) + \frac{\delta \left\langle \psi \right\rangle }{\left\langle \psi \right\rangle }(x_i ,t) \end{aligned}$$

(187)

Due to the implicit j summation in Eq. (178), one may reasonably wonder if Eq. (185) is valid. This equation is still verified because for each value of j, (j=1,2 and 3), the same approximation given by Eq. (186) is obtained so that

$$\begin{aligned} \frac{\delta \left\langle J^{(s+r)}_\psi \right\rangle }{ \left\langle J^{(s+r)}_\psi \right\rangle } = \frac{\delta \left\langle J^{(s+r)}_{\psi ,j=1 }\right\rangle }{ \left\langle J^{(s+r)}_{\psi ,j=1 }\right\rangle } =\frac{\delta \left\langle J^{(s+r)}_{\psi ,j=2 }\right\rangle }{ \left\langle J^{(s+r)}_{\psi ,j=2 } \right\rangle } =\frac{\delta \left\langle J^{(s+r)}_{\psi ,j=3 }\right\rangle }{\left\langle J^{(s+r)}_{\psi ,j=3 } \right\rangle } = \frac{\delta \left\langle J^{(s+r)}_{\psi ,j=1,3} \right\rangle }{ \left\langle J^{(s+r)}_{\psi ,j=1,3 } \right\rangle } \end{aligned}$$

(188)

Note that the relations (182) and (184) are also valid for instantaneous quantities but in this case $\delta \alpha$ becomes fluctuating.