Kármán–Howarth equation

In isotropic turbulence the Kármán-Howarth equation (after Theodore von Kármán and Leslie Howarth 1938), which is derived from the Navier-Stokes equations, is used to describe the evolution of non-dimensional longitudinal .

Consider a two-point correlation for homogeneous turbulence ${\displaystyle R_{ij}(\mathbf {r} ,t)={\overline {u_{i}(\mathbf {x} +\mathbf {r} ,t)u_{j}(\mathbf {x} ,t)}}}$. For isotroppic turbulence the correlation function can be expressed in terms of two scalar function

where ${\displaystyle u'=\left[{\frac {1}{3}}({\bar {u_{1}^{2}}}+{\bar {u_{2}^{2}}}+{\bar {u_{3}^{2}}})\right]^{1/2}}$ is the root mean square turbulent velocity and ${\displaystyle u_{1},\ u_{2},\ u_{3}}$ are turbulent fluctuating velocity in all three directions. From continuity equation we have ${\displaystyle {\frac {\partial R_{ij}}{\partial r_{j}}}=0}$, so

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