Kármán–Howarth equation

Consider a two-point velocity correlation tensor for homogeneous turbulence

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

For isotropic turbulence, this correlation tensor can be expressed in terms of two scalar functions, using the invariant theory of full rotation group, first derived by Howard P. Robertson in 1940,Robertson, H. P. (1940, April). The invariant theory of isotropic turbulence. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 36, No. 2, pp. 209–223). Cambridge University Press.

:$R\_\{ij\}(\backslash mathbf\{r\},t)\; =\; u\text{'}^2\; \backslash left\backslash \{\; [f(r,t)-g(r,t)]\backslash frac\{r\_ir\_j\}\{r^2\}\; +\; g(r,t)\; \backslash delta\_\{ij\}\backslash right\backslash \},\; \backslash quad\; f(r,t)\; =\; \backslash frac\{R\_\{11\}\}\{u\text{'}^2\},\; \backslash quad\; g(r,t)\; =\; \backslash frac\{R\_\{22\}\}\{u\text{'}^2\}$

where $u\text{'}$ is the root mean square turbulent velocity and $u\_1,\backslash \; u\_2,\; \backslash \; u\_3$ are turbulent velocity in all three directions. Here, $f(r)$ is the longitudinal correlation and $g(r)$ is the lateral correlation of velocity at two different points. From continuity equation, we have

:$\backslash frac\{\backslash partial\; R\_\{ij\}\}\{\backslash partial\; r\_j\}=0\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; g(r,t)\; =\; f(r,t)\; +\; \backslash frac\{r\}\{2\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; r\}f(r,t)$

Thus $f(r,t)$ uniquely determines the two-point correlation function. Theodore von Kármán and Leslie Howarth derived the evolution equation for $f(r,t)$ from Navier–Stokes equation as

:$\backslash frac\; \backslash partial\; \{\backslash partial\; t\}\; (u\text{'}^2\; f)\; -\; \backslash frac\{u\text{'}^3\}\{r^4\}\; \backslash frac\; \backslash partial\; \{\backslash partial\; r\}\; (r^4\; h)\; =\; \backslash frac\{2\backslash nu\; u\text{'}^2\}\{r^4\}\; \backslash frac\backslash partial\; \{\backslash partial\; r\}\; \backslash left(r^4\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; r\}\backslash right)$

where $h(r,t)$ uniquely determines the triple correlation tensor

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

L.G. Loitsianskii derived an integral invariant for the decay of the turbulence by taking the fourth moment of the Kármán–Howarth equation in 1939,Loitsianskii, L. G. (1939) Einige Grundgesetze einer isotropen turbulenten Strömung. Arbeiten d. Zentr. Aero-Hydrdyn. Inst., 440.Landau, L. D., & Lifshitz, E. M. (1959). Fluid Mechanics Pergamon. New York, 61. i.e.,

:$\backslash frac\; \backslash partial\; \{\backslash partial\; t\}\; \backslash left(u\text{'}^2\; \backslash int\_0^\backslash infty\; r^4\; f\backslash \; dr\backslash right)\; =\; \backslash left[2\backslash nu\; u\text{'}^2\; r^4\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; r\}\; +\; u\text{'}^3\; r^4\; h\backslash right]\_0^\backslash infty.$

If $f(r)$ decays faster than $r^\{-3\}$ as $r\backslash rightarrow\backslash infty$ and also in this limit, if we assume that $r^4\; h$ vanishes, we have the quantity,

:$\backslash Lambda\; =\; u\text{'}^2\; \backslash int\_0^\backslash infty\; r^4\; f\backslash \; dr\; =\; \backslash mathrm\{constant\}$

which is invariant. Lev Landau and Evgeny Lifshitz showed that this invariant is equivalent to conservation of angular momentum.Landau, L. D., & Lifshitz, E. M. (1987). Fluid mechanics. 1987. Course of Theoretical Physics. However, Ian Proudman and W.H. Reid showed that this invariant does not hold always since $\backslash lim\_\{r\backslash rightarrow\backslash infty\}\; (r^4\; h)$ is not in general zero, at least, in the initial period of the decay.Proudman, I., & Reid, W. H. (1954). On the decay of a normally distributed and homogeneous turbulent velocity field. Phil. Trans. R. Soc. Lond. A, 247(926), 163-189.Batchelor, G. K., & Proudman, I. (1956) The large-scale structure of homogeneous turbulence. Phil. Trans. R. Soc. Lond. A, 248(949), 369-405. In 1967, Philip Saffman showed that this integral depends on the initial conditions and the integral can diverge under certain conditions.Saffman, P. G. (1967). The large-scale structure of homogeneous turbulence. Journal of Fluid Mechanics, 27(3), 581-593.

For the viscosity dominated flows, during the decay of turbulence, the Kármán–Howarth equation reduces to a heat equation once the triple correlation tensor is neglected, i.e.,

:$\backslash frac\; \backslash partial\; \{\backslash partial\; t\}\; (u\text{'}^2\; f)\; =\; \backslash frac\{2\backslash nu\; u\text{'}^2\}\{r^4\}\; \backslash frac\backslash partial\; \{\backslash partial\; r\}\; \backslash left(r^4\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; r\}\backslash right).$

With suitable boundary conditions, the solution to above equation is given bySpiegel, E. A. (Ed.). (2010). The Theory of Turbulence: Subrahmanyan Chandrasekhar's 1954 Lectures (Vol. 810). Springer.

:$f(r,t)\; =\; e^\{-r^2/8\backslash nu\; t\},\; \backslash quad\; u\text{'}^2\; =\; \backslash mathrm\{const.\}\backslash times\; (\backslash nu\; t)^\{-5/2\}$

so that,

:$R\_\{ij\}(r,t)\; \backslash sim\; (\backslash nu\; t)^\{-5/2\}\; e^\{-r^2/8\backslash nu\; t\}.$

• Kármán–Howarth–Monin equation (Andrei Monin's anisotropic generalization of the Kármán–Howarth relation)
• Batchelor–Chandrasekhar equation (homogeneous axisymmetric turbulence)
• Corrsin equation (Kármán–Howarth relation for scalar transport equation)
• Chandrasekhar invariant (density fluctuation invariant in isotropic homogeneous turbulence)

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