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 autocorrelation.

Mathematical description

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

$R_{ij}(\mathbf {r} ,t)=u'^{2}\left\{g(r,t)\delta _{ij}+[f(r,t)-g(r,t)]{\frac {r_{i}r_{j}}{r^{2}}}\right\},\quad f(r,t)={\frac {R_{11}}{u'^{2}}},\quad g(r,t)={\frac {R_{22}}{u'^{2}}}$

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

$g(r,t)=f(r,t)+{\frac {r}{2}}{\frac {\partial }{\partial r}}f(r,t)$

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

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

where $h(r,t)$ uniquely determines the triple correlation function $S_{ijk}={\overline {u_{i}(\mathbf {x} ,t)u_{j}(\mathbf {x} ,t)u_{k}(\mathbf {x} +\mathbf {r} ,t)}}$ i.e., $h(r,t)={\frac {S_{111}}{u'^{3}}}$.

See also

• Kármán–Howarth–Monin equation (Andrei Monin's anisotropic generalization of the Kármán–Howarth relation)