Batchelor–Chandrasekhar equation

The theory is based on the principle that the statistical properties are invariant for rotations about a particular direction $\backslash boldsymbol\{\backslash lambda\}$ (say), and reflections in planes containing $\backslash boldsymbol\{\backslash lambda\}$ and perpendicular to $\backslash boldsymbol\{\backslash lambda\}$. This type of axisymmetry is sometimes referred to as strong axisymmetry or axisymmetry in the strong sense, opposed to weak axisymmetry, where reflections in planes perpendicular to $\backslash boldsymbol\{\backslash lambda\}$ or planes containing $\backslash boldsymbol\{\backslash lambda\}$ are not allowed.{{cite journal

| last1=Lindborg | first1=Erik

| date=1995

| title=Kinematics of homogeneous axisymmetric tubulence

| journal=Journal of Fluid Mechanics

| volume=302

| pages=179-201

| doi=10.1017/S002211209500406X}}

Let the two-point correlation for homogeneous turbulence be

:$R\_\{ij\}(\backslash mathbf\{r\},t)\; =\; \backslash overline\{u\_i(\backslash mathbf\{x\},t)u\_j(\backslash mathbf\{x\}+\backslash mathbf\{r\},t)\}.$

A single scalar describes this correlation tensor in isotropic turbulence, whereas, it turns out for axisymmetric turbulence, two scalar functions are enough to uniquely specify the correlation tensor. In fact, Batchelor was unable to express the correlation tensor in terms of two scalar functions, but ended up with four scalar functions, nevertheless, Chandrasekhar showed that it could be expressed with only two scalar functions by expressing the solenoidal axisymmetric tensor as the curl of a general axisymmetric skew tensor (reflectionally non-invariant tensor).

Let $\backslash boldsymbol\{\backslash lambda\}$ be the unit vector which defines the axis of symmetry of the flow, then we have two scalar variables, $\backslash mathbf\{r\}\backslash cdot\backslash mathbf\{r\}=r^2$ and $\backslash mathbf\{r\}\backslash cdot\backslash boldsymbol\{\backslash lambda\}=r\backslash mu$. Since $|\backslash boldsymbol\{\backslash lambda\}|=1$, it is clear that $\backslash mu$ represents the cosine of the angle between $\backslash boldsymbol\{\backslash lambda\}$ and $\backslash mathbf\{r\}$. Let $Q\_1(r,\backslash mu,t)$ and $Q\_2(r,\backslash mu,t)$ be the two scalar functions that describes the correlation function, then the most general axisymmetric tensor which is solenoidal (incompressible) is given by,

:$R\_\{ij\}\; =\; Ar\_ir\_j\; +\; B\backslash delta\_\{ij\}\; +\; C\backslash lambda\_i\backslash lambda\_j\; +\; D\; \backslash left\; (\backslash lambda\_i\; r\_j\; +\; r\_i\; \backslash lambda\_j\; \backslash right\; )$

where

: $\backslash begin\{align\}$

A &= \left (D_r-D_{\mu\mu} \right )Q_1+ D_r Q_2, \\

B &= \left [- \left (r^2D_r+r\mu D_\mu+2 \right )+r^2 \left (1-\mu^2 \right )D_{\mu\mu}-r\mu D_\mu \right ]Q_1 - \left [r^2 \left (1-\mu^2 \right )D_r+1 \right ]Q_2, \\

C &= -r^2 D_{\mu\mu}Q_1 + \left (r^2 D_r+1 \right )Q_2, \\

D &= \left (r\mu D_\mu +1 \right )D_\mu Q_1 - r\mu D_r Q_2.

\end{align}

The differential operators appearing in the above expressions are defined as

:$\backslash begin\{align\}$

D_r &= \frac{1}{r}\frac{\partial }{\partial r} - \frac{\mu}{r^2} \frac{\partial }{\partial \mu}, \\

D_\mu &= \frac{1}{r} \frac{\partial }{\partial \mu}, \\

D_{\mu\mu} &= D_\mu D_\mu = \frac{1}{r^2} \frac{\partial^2 }{\partial \mu^2}.

\end{align}

Then the evolution equations (equivalent form of Kármán–Howarth equation) for the two scalar functions are given by

:$\backslash begin\{align\}$

\frac{\partial Q_1}{\partial t} &= 2\nu\Delta Q_1 + S_1, \\

\frac{\partial Q_2}{\partial t} &= 2\nu \left (\Delta Q_2 + 2 D_{\mu\mu} Q_1 \right ) + S_2

\end{align}

where $\backslash nu$ is the kinematic viscosity and

:$\backslash Delta\; =\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; r^2\}\; +\; \backslash frac\{4\}\{r\}\backslash frac\{\backslash partial\; \}\{\backslash partial\; r\}\; +\; \backslash frac\{1-\backslash mu^2\}\{r^2\}\backslash frac\{\backslash partial^2\; \}\{\backslash partial\; \backslash mu^2\}\; -\; \backslash frac\{4\backslash mu\}\{r^2\}\backslash frac\{\backslash partial\; \}\{\backslash partial\; \backslash mu\}.$

The scalar functions $S\_1(r,\backslash mu,t)$ and $S\_2(r,\backslash mu,t)$ are related to triply correlated tensor $S\_\{ij\}$, exactly the same way $Q\_1(r,\backslash mu,t)$ and $Q\_2(r,\backslash mu,t)$ are related to the two point correlated tensor $R\_\{ij\}$. The triply correlated tensor is

: $S\_\{ij\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; r\_k\}\; \backslash left(\; \backslash overline\{u\_i(\backslash mathbf\{x\},t)\; u\_k(\backslash mathbf\{x\},t)u\_j(\backslash mathbf\{x\}+\backslash mathbf\{r\},t)\}-\backslash overline\{u\_i(\backslash mathbf\{x\},t)\; u\_k(\backslash mathbf\{x\}+\backslash mathbf\{r\},t)u\_j(\backslash mathbf\{x\}+\backslash mathbf\{r\},t)\}\backslash right)\; +\; \backslash frac\{1\}\{\backslash rho\}\; \backslash left(\backslash frac\{\backslash overline\{\backslash partial\; p(\backslash mathbf\{x\},t)\; u\_j(\backslash mathbf\{x\}\; +\; \backslash mathbf\{r\},t)\}\}\{\backslash partial\; r\_i\}\; -\; \backslash frac\{\backslash overline\{\backslash partial\; p(\backslash mathbf\{x\}\; +\; \backslash mathbf\{r\},t)\; u\_i(\backslash mathbf\{x\},t)\}\}\{\backslash partial\; r\_j\}\; \backslash right).$

Here $\backslash rho$ is the density of the fluid.

• The trace of the correlation tensor reduces to

::$R\_\{ii\}\; =r^2\; \backslash left\; (1-\backslash mu^2\; \backslash right\; )\; \backslash left\; (D\_\{\backslash mu\backslash mu\}Q\_1-D\_rQ\_2\; \backslash right\; )-2Q\_2-2\; \backslash left\; (r^2D\_r+2r\backslash mu\; D\_\backslash mu\; +3\; \backslash right\; )Q\_1.$

• The homogeneity condition $R\_\{ij\}(-\backslash mathbf\{r\})=R\_\{ji\}(\backslash mathbf\{r\})$ implies that both $Q\_1$ and $Q\_2$ are even functions of $r$ and $r\backslash mu$.

During decay, if we neglect the triple correlation scalars, then the equations reduce to axially symmetric five-dimensional heat equations,

:$\backslash begin\{align\}$

\frac{\partial Q_1}{\partial t} &= 2\nu\Delta Q_1, \\

\frac{\partial Q_2}{\partial t} &= 2\nu \left ( \Delta Q_2 + 2 D_{\mu\mu} Q_1 \right )

\end{align}

Solutions to these five-dimensional heat equation was solved by Chandrasekhar. The initial conditions can be expressed in terms of Gegenbauer polynomials (without loss of generality),

:$\backslash begin\{align\}$

Q_1(r,\mu,0) &= \sum_{n=0}^\infty q_{2n}^{(1)}(r)C_{2n}^{\frac{3}{2}}(\mu), \\

Q_2(r,\mu,0) &= \sum_{n=0}^\infty q_{2n}^{(2)}(r)C_{2n}^{\frac{3}{2}}(\mu),

\end{align}

where $C\_\{2n\}^\{\backslash frac\{3\}\{2\}\}(\backslash mu)$ are Gegenbauer polynomials. The required solutions are

:$\backslash begin\{align\}$

Q_1(r,\mu,t) &= \frac{e^{-\frac{r^2}{8\nu t}}}{32(\nu t)^{\frac{5}{2}}} \sum_{n=0}^\infty C_{2n}^{\frac{3}{2}}(\mu) \int_0^\infty e^{-\frac{r'^2}{8\nu t}}r'^4 q_{2n}^{(1)}(r')\frac{I_{2n+\frac{3}{2}} \left (\frac{rr'}{4\nu t} \right )}{\left (\frac{rr'}{4\nu t} \right )^{\frac{3}{2}}}\ dr', \\[8pt]

Q_2(r,\mu,t) &= \frac{e^{-\frac{r^2}{8\nu t}}}{32(\nu t)^{\frac{5}{2}}}\sum_{n=0}^\infty C_{2n}^{\frac{3}{2}}(\mu) \int_0^\infty e^{-\frac{r'^2}{8\nu t}}r'^4 q_{2n}^{(2)}(r')\frac{I_{2n+\frac{3}{2}}\left (\frac{rr'}{4\nu t} \right )}{\left (\frac{rr'}{4\nu t} \right )^{\frac{3}{2}}}\ dr' +4\nu\int_0^t\frac{dt'}{[8\pi\nu(t-t')]^{\frac{5}{2}}} \int\cdots\int\left(\frac{1}{r^2}\frac{\partial^2 Q_1}{\partial \mu^2}\right)_{r',\mu',t'} e^{-\frac{|r-r'|^2}{8\nu(t-t')}}\ dx_1'\cdots dx_5',

\end{align}

where $I\_\{2n+\backslash frac\{3\}\{2\}\}$ is the Bessel function of the first kind.

As $t\backslash to\backslash infty,$ the solutions become independent of $\backslash mu$

:$\backslash begin\{align\}$

Q_1(r,\mu,t) &\to -\frac{\Lambda_1 e^{-\frac{r^2}{8\nu t}}}{48 \sqrt{2\pi}(\nu t)^{\frac{5}{2}}}, \\

Q_2(r,\mu,t) &\to -\frac{\Lambda_2 e^{-\frac{r^2}{8\nu t}}}{48 \sqrt{2\pi}(\nu t)^{\frac{5}{2}}},

\end{align}

where

:$\backslash begin\{align\}$

\Lambda_1 &=-\int_0^\infty q_{2n}^{(1)}(r)\ dr \\

\Lambda_2 &=-\int_0^\infty q_{2n}^{(2)}(r)\ dr

\end{align}

• Kármán–Howarth equation
• Kármán–Howarth–Monin equation

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Category:Equations of fluid dynamics

Category:Fluid dynamics

Category:Turbulence