The model of Canuto et al. (2001)

Canuto et al. (2001) and Cheng et al. (2002) use a model that is virtually identical to the traditional model of Launder et al. (1975) and Gibson and Launder (1978). The values of their model parameters and their notation, however, are somewhat different.

Looking for conversion relations, it should be noted that the anisotropy tensor $b^$CCHD$_{ij}$ used by Canuto et al. (2001) is twice the tensor defined in (52), $b^$CCHD$_{ij} = 2 k b_{ij}$. Also the dissipative time scale $\tau^$CCHD of Canuto et al. (2001) is twice the time scal defined in (56), $\tau^$CCHD$= 2 \tau$. If one further notes that the turbulent heat flux $h_i= {\langle u'_i \theta' \rangle}$ is related to the buoyancy flux according to ${\langle u'_i b' \rangle} = \alpha g h_i$, relations between the model parameters can be found.

With these relations, equation (10a) of Canuto et al. (2001) can be re-written as

$$b_{ij} = - \lambda_1 \overline{S}_{ij} - 2 \lambda_2 \overline{\S... ...{ij} - 2 \lambda_3 \overline{Z}_{ij} - \lambda_4 \overline{\Gamma}_{ij} \quad .$$ (89)

The return-to-isotropy part of the pressure-strain model of Canuto et al. (2001) reads

$$\Phi_{ij} = - \dfrac{2}{\lambda} \epsilon b_{ij} \; , \quad$$ (90)

from which, by comparing with (51), it follows that $c_1 = 2/\lambda$ and $c^*_1=0$, and hence from (65) ${\cal N} = 1/\lambda$. Thus, adopting the relations $a_1 = \lambda_1/\lambda$, $a_2 = 2 \lambda_2/\lambda$, $a_3 = 2 \lambda_3/\lambda$, $a_4=0$, and $a_5 = \lambda_4/\lambda$, (89) corresponds exactly to (59),

Similarly, equation (10a) of Cheng et al. (2002) can be re-expressed in the form

$$\dfrac{\lambda_5}{2}\gamma_{i} = - \lambda_6 \overline{S}_{ij} \g... ...j} + \dfrac{2}{3} \overline{N}_{i} - \lambda_0 \overline{T} \delta_{i3} \quad .$$ (91)

The somewhat simpler model of Canuto et al. (2001) adopts the equilibrium assumption (72), and replaces the last term in (91) by $-\lambda_0 r \gamma_3 \overline{N}^2 \delta_{i3}$ and, assuming constant $r$, identifies $\lambda_0 r = \lambda_8$. The time scale ratio $r$ is computed in equation (20a) of Canuto et al. (2001).

The return-to-isotropy part of this model (see equation (6c) of Cheng et al. (2002)) reads

$$\Phi^b_i = - \dfrac{\lambda_5}{2} \dfrac{\epsilon}{k} {\langle u'_i b' \rangle} \; , \quad$$ (92)

from which follows, by comparison with (57) and (65), that ${\cal N}_b = c_{b1} = \lambda_5/2$. Comparing (91) with (63) one finds, by inspection, the relations $a_{b1} = \lambda_6$, $a_{b2} = \lambda_7$, $a_{b3} = 2$, $a_{b4} = 2 \lambda_0$, and $a_{b5} = 2 \lambda_8$. Some parameter sets for this model are compiled in table 3.

Table 3: Some parameter sets for the model of Canuto et al. (2001)

	$\lambda$	$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$	$\lambda_4$	$\lambda_5$	$\lambda_6$	$\lambda_7$	$\lambda_8$
CHCD01A	$0.4$	$2/3$	$0.107$	$0.0032$	$0.0864$	$0.12$	$11.9$	$0.4$	0	$0.48$
CHCD01B	$0.4$	$2/3$	$0.127$	$0.00336$	$0.0906$	$0.101$	$11.2$	$0.4$	0	$0.318$
CCH02	$0.4$	$2/3$	$0.107$	$0.0032$	$0.0864$	$0.1$	$11.04$	$0.786$	$0.643$	$0.547$