Prandtl 1926: the Austauschansatz

Prandtl proposed an ansatz (Austauschansatz) $$ T' \approx - l' \, \overline{T}_x \Rightarrow u'T'_x \;\approx\; u' l' \overline{T}_{xx} $$

By introducing the turbulent diffusivity $k_t$, which depends on size and strength of the waves or $$ k_t \;\equiv\; u' l' $$

Approximate $$ \overline{u'T'_x} \;\approx\; k_t \, \overline{T}_{xx} $$

$$ \overline{T}_t = - \overline{u} \, \overline{T}_x + (k + k_t)\, \overline{T}_{xx} + \overline{H} $$ Or in vector form $$ \overline{T}_t = - \overline{\mathbf{u}} \cdot \nabla \overline{T} + k_{\text{turb}} \nabla^2 \overline{T} + \overline{H} $$

For waves of $1 \, \text{m}$ amplitude and $1 \, \text{mm/s}$ speed of water, $$ k_t \approx 10^{-3} \, \text{m}^2 \, \text{s}^{-1} $$
For comparison, the molecular diffusion $$ k_{\text{Molecular}} = 10^{-7} \, \text{m}^2 \, \text{s}^{-1} $$

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