Stability of explicit models

A physically reasonable condition for an explicit second order model expressed the fact that increasing (non-dimensional) shear $\overline{S}$ should lead to increasing vertical shear-anisotropies of turbulence, $b_{13}$ and $b_{23}$. It has been shown by Burchard and Deleersnijder (2001) that a violation of this condition may lead to numerical instabilities of the models.

Mathematically, the shear-condition is expressed by

$$\dfrac{\partial {(b_{13}^2 + b_{23}^2)^\frac{1}{2}}}{\partial {\o... ...\partial {\tilde{c}_\mu \overline{S}}}{\partial {\overline{S}}}\ge 0 \; , \quad$$ (82)

where (74) has been used. Using the equilibrium form of the stability function described in section 4.7.39, this condition leads to a cubic equation in $\alpha_M = \overline{S}^2$. A simpler condition can be obtained, when this equation is solved after terms multiplied by $d_5$ and $n_2$, which usually are very small, are neglected.

The resulting approximate condition is

$$\alpha_M \le \dfrac{d_0 n_0 + (d_0 n_1 + d_1 n_0) \alpha_N + (d_1... ...pha_N^3}{ d_2 n_0 + (d_2 n_1 + d_3 n_0 ) \alpha_N + d_3 n_1 \alpha_N^2} \quad .$$ (83)

Burchard and Deleersnijder (2001) showed that using (83) the most well-known models yield numerically stable results. However, for some models like those of Mellor and Yamada (1982) and Kantha and Clayson (1994), the limiter (83) is almost always `active', and hence replaces the actual turbulence model in a questionable way.