Analyse the turbulence models


    subroutine analyse_model

This routine analyses all models in GOTM for their physical properties implied by chosen model parameters. These results can be displayed by calling the internal routine report_model(), also defined in the turbulence module (see section 4.7.5).

In most cases, the relations connecting model parameters and physical properties have already been derived in section 4.7.3: the von Kármán constant, $ \kappa$, follows from (102), the decay rate in homogeneous turbulence , $ d$, from (104), and the steady-state Richardson-number from (114). These relations have been obtained in `generic' form (see section 4.7.29), but relations for specific models, like the $ k$-$ \epsilon$ model or the $ k$-$ \omega$ model, can be derived by simply adopting the parameters compiled in table 8 and table 9 in section 4.7.29.

The decay rates $ \alpha$ and $ L$ in shear-free turbulence follow from the physically meaningful roots of (110) and (111), which are

\begin{displaymath}\begin{array}{rcl} \alpha &=& - \dfrac{4 n {(\sigma_k^\psi)}^...
...frac{1}{2} }{12 n^2} \right)^\frac{1}{2} \; , \quad \end{array}\end{displaymath} (115)

where it should be recalled that $ R=c_\mu^0/c_\mu$. For the standard models (without ASM), $ R=1$ may be assumed. Then, with the values from table 8 and table 9, solutions for the $ k$-$ \epsilon$ model of Rodi (1987), and the $ k$-$ \omega$ model of Umlauf et al. (2003) can be directly recovered as special cases of this equation.

Due to its wall-functions, the model of Mellor and Yamada (1982) described in section 4.7.27 requires a slightly more complicated analysis. For this model, the von Kármán constant is computed according to

$\displaystyle \kappa = \sqrt{\dfrac{E_2 - E_1 + 1}{S_l B_1}} \quad .$ (116)

The decay rates in shear-free turbulence can be shown to be

\begin{displaymath}\begin{array}{rcl} \alpha &=& \dfrac{5 \kappa B_1^{\frac{1}{2...
..._1 \kappa^2 S_l )^2} \right)^\frac{1}{2} \; , \quad \end{array}\end{displaymath} (117)

where we introduced the abbreviation

\begin{displaymath}\begin{array}{rcl} {\cal N} &=& 6 E_2 \left( 2 S_l - S_q \rig...^2 S_l (S_l + 12 S_q )\right)^\frac{1}{2} \quad . \end{array}\end{displaymath} (118)

These equations replace (115) for the model of Mellor and Yamada (1982). Decay-rates for this model do not at all depend on the stability functions. However, they depend on the parameter $ E_2$ of the wall-functions. This parameter, however, has been derived for wall-bounded shear flows, and it is not very plausible to find it in an expression for shear-free flows.

The routine analyse_model() works also for one-equation models, where the length-scale, $ l$, is prescribed by an analytical expression (see section 4.7.32). However, some attention has to be paid in interpreting the results. First, it is clear that these models cannot predict homogeneous turbulence, simply because all formulations rely on some type of modified boundary layer expressions for the length-scale. This impies that a well-defined decay rate, $ d$, and a steady-state Richardson-number, $ Ri_{st}$, cannot be computed. Second, the von Kármán constant, $ \kappa$, does not follow from (102) or (116), because $ \kappa$ now relates directly to the prescribed slope of the length-scale close to the bottom or the surface. Third, in shear-free flows, (115)$ _1$ or (117)$ _1$ remain valid, provided the planar source of the spatially decaying turbulence is located at $ z=0$. Then, the slope of the length-scale, $ L$, defined in (108) can be identified with the prescribed slope, $ \kappa$, and (115)$ _1$ or (117)$ _1$ are identical to the solutions suggested by Craig and Banner (1994).

In this context, it should be pointed out that the shear-free solutions also have a direct relation to an important oceanic situation. If the planar source of turbulence is assumed to be located at $ z=0$, and if the injected turbulence is identified with turbulence caused by breaking surface-waves, then it can be shown that (115) or (117) are valid in a thin boundary layer adjacent to the suface. Further below, to classical law of the wall determines the flow, see Craig and Banner (1994) and citeUmlaufetal2003.


    Original author(s): Lars Umlauf

Karsten Bolding 2012-01-24