The model of Gibson and Launder (1978)

The pressure-strain model of this important class of engineering models has been originally suggested by Launder et al. (1975). It can be written as

$$\Phi_{ij} = - 2 \tilde{c}_1 \epsilon b_{ij} - \tilde{c}_2 k S_{ij... ..._{ij} - \dfrac{2}{3} P \delta_{ij} \right) + \tilde{c}_6 \Gamma_{ij} \; , \quad$$ (84)

where that last term has been added by Gibson and Launder (1978) to account for the effects of gravity in stratified fluids. This term is identical to the last term in (51). The new production-of-anisotropy tensor $D_{ij}$ is defined as

$$D_{ij} = - {\langle u'_i u'_m \rangle} \dfrac{\partial {U_m}}{\pa... ...}} - {\langle u'_j u'_m \rangle} \dfrac{\partial {U_m}}{\partial {x_i}} \quad .$$ (85)

Using the tensor relations

$$\begin{array}{rcl} P_{ij} = - 2 k \Sigma_{ij} - 2 k Z_{ij} + ... ...3} P \delta_{ij} - \dfrac{4}{3} k S_{ij} \; , \quad \end{array}$$ (86)

(84) can be re-written in the form

$$\Phi_{ij} = - 2 \tilde{c}_1 \epsilon b_{ij} + \left( \dfrac{4}{3}... ...t( \tilde{c}_3 - \tilde{c}_4 \right) k Z_{ij} + \tilde{c}_6 \Gamma_{ij} \quad .$$ (87)

Comparting with (51), the following relations can be estabilished: $c_1 = 2 \tilde{c}_1$, $c_2 = 4/3 ( \tilde{c}_3 + \tilde{c}_4 )- \tilde{c}_2$, $c_3 = 2 ( \tilde{c}_3 + \tilde{c}_4 )$, $c_3 = 2 ( \tilde{c}_3 - \tilde{c}_4 )$, $c_5 = 0$, and $c_6 = \tilde{c}_6$.

Gibson and Launder (1978) use a slightly different notation for the pressure-scambling model (57). Their model is somewhat simplified form of the model of Jin et al. (2003), which can be written as

$$\begin{array}{rcl} \Phi^b_i &=& - \tilde{c}_{b 1} \dfrac{\eps... ... {x_j}} - 2 \tilde{c}_{b 5} k_b \delta_{i3} \quad . \end{array}$$ (88)

Using the decomposition of the velocity gradient in its symmetric and anti- symmetric part, (54), the following parameter relation are evident: $c_{b1} = \tilde{c}_{b1}$, $c_{b2} = \tilde{c}_{b2} + \tilde{c}_{b3}$, $c_{b3} = \tilde{c}_{b2} - \tilde{c}_{b3}$, $c_{b4} = \tilde{c}_{b4}$, $c_{b5} = \tilde{c}_{b5}$.

Parameter values for this model are compiled in table 3. `GLNEW' denotes the revised parameter set for the pressure-strain model given in Wilcox (1998) and for the pressure-buoyancy gradient model in Zhao et al. (2001).

Table 2: Some parameter sets for the model of Gibson and Launder (1978)

	$\tilde{c}_1$	$\tilde{c}_2$	$\tilde{c}_3$	$\tilde{c}_4$	$\tilde{c}_6$	$\tilde{c}_{b1}$	$\tilde{c}_{b2}$	$\tilde{c}_{b3}$	$\tilde{c}_{b4}$	$\tilde{c}_{b5}$	$r$
GL78	$1.8$	0	$0.6$	0	$0.5$	$3$	$0.33$	0	0	$0.33$	$0.8$
GLNEW	$1.8$	0	$0.78$	$0.2545$	$0.3$	$3.28$	$0.4$	0	0	$0.4$	$0.8$