Kármán-Howarth formalism revisited as governing inhomogeneous turbulence

In 1938, von Kármán and Howarth proposed an equation governing homogeneous isotropic turbulence whose original form is written as[3]

$$\overline{u_j'(\widehat{\mbox{NS}})_l + \hat{u}_l'(\mbox{NS})_j}=0$$ (1)

In this equation $u_j'=u_j'(\mbox{\boldmath {$x$}},t)$ and $\hat{u}_l'=u_l'(\widehat{\mbox{\boldmath {$x$}}},t)$ are instantaneous velocity fluctuation at $\mbox{\boldmath {$x$}}$ and $\widehat{\mbox{\boldmath {$x$}}}$ , respectively, overbar ( $\hspace{.1zw}\rule[.7zh]{1zw}{.03zh}\hspace{.1zw}$) denotes the conventional (ensemble) average and

$$\mbox{\bf NS} \equiv \mbox{\bf NS}(\nabla , \underline{\mbox{\boldmath {$u$}}}, \underline{p})$$

$$= (\frac{\partial}{\partial t}+\underline{\mbox{\boldmath {$u$}}}\c... ...nu\nabla^2)\underline{\mbox{\boldmath {$u$}}} + \rho^{-1}\nabla \underline{p}=0$$ (2)

denotes the Navier-Stokes equation written in terms of the instantaneous fluid quantities

$$\begin{array}{rcl} \underline{\mbox{\boldmath {$u$}}}&=&\... ...mbox{\boldmath {$u$}}'\\ \underline{p}&=&p+p' \end{array}$$ (3)

where $\mbox{\boldmath {$u$}}$ and $p$ are the average velocity and pressure, respectively, $\nu$ is the kinematic viscosity, and $\nabla$ denotes nabla vector. Likewise $\widehat{\mbox{\bf NS}}$ is defined by Eq.(2) in which ( $\nabla,\underline{\mbox{\boldmath {$u$}}},\underline{p}$ ) is replaced with ( $\hat{\nabla},\hat{\underline{\mbox{\boldmath {$u$}}}}, \hat{\underline{p}}$). Eq.(1) is an equation in 6-D physical space and time, namely, has seven independent variables ( $\mbox{\boldmath {$x$}}, \hat{\mbox{\boldmath {$x$}}}, t$). The classical theory[3] chose to discuss homogeneous and isotropic turbulence to reduce independent variables to ( $\vert\mbox{\boldmath {$x$}}-\hat{\mbox{\boldmath {$x$}}}\vert,t$). We can show, however, that the equation can be renovated to be available for inhomogeneous turbulence as well by introducing separation of variables, together with a closure condition which seems plausible intuitively. The actual derivation proceeds as follows: Eq.(1) in which $\mbox{\bf NS}(\nabla, \underline{\mbox{\boldmath {$u$}}}, \underline{p})$ is decomposed into average and fluctuating parts using (3),reads

$$\overline{u_j'(\widehat{\mbox{NS}})_l' + \hat{u}_l'(\mbox{NS})_j'} = 0$$ (4)

where

$$(\mbox{NS})_j'\equiv(\frac{\partial}{\partial t} +u_r\fr... ... p'}{\partial x_j} +\frac{\partial}{\partial x_r}(u_j'u_r')$$ (5)

together with $(\widehat{\mbox{NS}})_l'$ defined as in Eq.(1). Upon substituting Eq.(5) into Eq.(4) we see that the equation consists of terms of double and triple fluctuation correlations, which we will decompose as

$$\overline{u_j'\hat{u}_\lambda'} = \mbox{R.P.}l^6 \int_{-\infty}^{\infty} g_j(\mbox{\boldmath {$x$}}... ...{$k$}}}) d\mbox{\boldmath {$k$}}d\hat{\mbox{\boldmath {$k$}}}\;,\;\lambda=(l,4)$$ (6)

$$\overline{u_j'\hat{u}_l'\tilde{u}_r'} = \mbox{R.P.}l^9 \int_{-\infty}^{\infty} g_j(\mbox{\boldmath {$x$}}... ...ox{\boldmath {$k$}}d\hat{\mbox{\boldmath {$k$}}}d\tilde{\mbox{\boldmath {$k$}}}$$ (7)

where $\mbox{\boldmath {$k$}}$ is the wave number, $l$ is the characteristic length of flow geometry, R.P. denotes taking real part, and subscript $\lambda$ runs 1 through 4. A supplementary definition

$$u_4'=\rho^{-1}p'$$ (8)

stands for the pressure fluctuation. It is more convenient for later use to employ the following alternative expressions:

$$\overline{u_j'\hat{u}_\lambda'}=\mbox{R.P.}l^3 \int_{-\i... ...)d\mbox{\boldmath {$k$}}\;,\;\lambda=(l,4) \hspace{35mm}(6')$$

$$\overline{u_j'\hat{u}_l'\tilde{u}_r'}=\mbox{R.P.}l^6 \int... ...mbox{or}\; \hat{\mbox{\boldmath {$x$}}}) \hspace{4.5mm}(7')$$

It is to be noted that there is no a priori reason why the double and triple correlations are to be expressed in terms of the same $g_\lambda$. At this point it is simply invoked as the closure condition to truncate the chain of equations at the level of Eq.(4). This closure can be justified only through the quality of outcome as to whether it fits with physical reality of turbulence. With these preliminaries, Eq.(4) with assumption (6',7') substituted into, leads to the form typical of separation of variables, here into ( $\mbox{\boldmath {$x$}}$,t) and ( $\hat{\mbox{\boldmath {$x$}}}$ ,t), respectively;

$$\int_{-\infty}^{\infty} d\mbox{\boldmath {$k$}}g_j\hat{g}_... ...(\widehat{\mbox{ns}})_l^{(0)}}_{\displaystyle =\;-i\omega}]=0$$ (9)

where $\omega$ is the separation parameter having the dimension of the frequency, and $\hat{g}_l$ stands for $g_l(\hat{\mbox{\boldmath {$x$}}},\mbox{\boldmath {$k$}})$ , and $(\mbox{ns})^{(0)}$ is defined by

$$(\mbox{ns})_j^{(0)}\equiv [\mbox{ns}(\mbox{\boldmath {$g$}}... ... g_4}{\partial x_j} +\frac{\partial}{\partial x_r}I(g_jg_r)$$ (10)

with

$$I(g_jg_r)\equiv l^3\int_{-\infty}^{\infty} g_j(\mbox{\bo... ..._r(\hat{\mbox{\boldmath {$k$}}})d\hat{\mbox{\boldmath {$k$}}}$$

Imaginary factor $i$ as appearing in Eq.(9) reflects the statistical symmetry of the tensor $\overline{u_j'\hat{u}_l'}$ , which is met under the following additional condition

$$g_l(\hat{\mbox{\boldmath {$x$}}},-\mbox{\boldmath {$k$}})=[g_l(\hat{\mbox{\boldmath {$x$}}},\mbox{\boldmath {$k$}})]^*$$ (11)

In fact, then, the two terms inside[ ] of integral (9) are commutable to each other through taking complex conjugate(*), if $i\omega$ is purely imaginary. Thus we are led to the assertion that the only equation that need to be solved is

$$i\omega g_j=(\mbox{ns})_j^{(0)}$$ (12)

To go further from this point on, we need to have relationship between frequency $\omega$ introduced as the separation parameter and wave number $\mbox{\boldmath {$k$}}$ by which the solution is constructed in the form of bilinear integral. Physically they are related to each other by the dispersion relation $\omega(\mbox{\boldmath {$k$}})$ . Or, instead, we may introduce phase velocity $\mbox{\boldmath {$c$}}$ by

$$\omega=\mbox{\boldmath {$c$}}\cdot\mbox{\boldmath {$k$}}$$ (13)

with no loss of generality. In fact, its constancy does not mean the turbulent eddies being assumed as nondispersive. For the unsteady term in the equation $(\partial/\partial t \neq 0)$ is responsible for the dispersive part, if any. Owing to the convolution form of integral (10) periodic factor drops off from Eq.(12) despite its nonlinear structure by putting

$$g_\lambda(\mbox{\boldmath {$x$}},\mbox{\boldmath {$k$}})=e... ...ox{\boldmath {$x$}},\mbox{\boldmath {$k$}})\;,\;\lambda=(l,4)$$ (14)

thereby we have the following equation governing its amplitude $f_\lambda$,

$$i\omega f_j=(\mbox{ns})_j$$ (15)

where

$$(\mbox{ns})_j \equiv [\mbox{ns}(\mbox{\boldmath {$f$}},f_4)]_j$$

$$= [\frac{\partial}{\partial t} +u_r\partial_r(\mbox{\boldmath {$k$}... ...rtial_j(\mbox{\boldmath {$k$}})f_4 +\partial_r(\mbox{\boldmath {$k$}})I(f_jf_r)$$ (16)

with

$$\mbox{\boldmath {$\partial$}}(\mbox{\boldmath {$k$}})\equiv\nabla(\mbox{\boldmath {$x$}})+i\mbox{\boldmath {$k$}}$$ (17)

It is readily checked that expressions defined by (15) and (10) are related to each other by

$$(\mbox{ns})_j^0 =[(\mbox{ns})_j]_{\footnotesize {\mbox{\... ...\boldmath {$k$}})\rightarrow \nabla(\mbox{\boldmath {$x$}})}}$$

Several remarks are in order with regards to physical implications of variables having appeared in this section: Amplitude function $\mbox{\boldmath {$f$}}$ as obeying Eq.(14) is essentially complex and is not an observable. It is related to the observable quantity of fluctuation correlation through

$$\overline{u_j'\hat{u}_l'}=\mbox{R.P.}l^3\int_{-\infty}^{\i... ...ldmath {$x$}}},\mbox{\boldmath {$k$}})d\mbox{\boldmath {$k$}}$$ (18)

as is confirmed by (6'),(11)and (13). Its complex variable structure has its origin in Eq.(9) where the imaginary unit $i$ is introduced from the symmetry postulate of statistical mechanics[2] for correlation tensor (18). We may note some coincidental parallelism to Schroedinger's wave equation where the imaginary factor secures the corresponding tensor to be Hermitean as it should. The most crucial on which this paper rests is the soundness of the basis of Eq.(1). Originally it was a product of intuition by Theodore von Kármán without any `first principle' ground shown in the paper[3]. In fact, then, any linear combinations of $(\mbox{\bf NS})'$ would be claimed as equally qualified. Firm basis is provided by nonequilibrium statistical mechanics[4] to justify that the linear combination only of the form (1) is consistent with Liouville's equation, namely, the equation of continuity in the phase space of Hamiltonian mechanics. .