Compute special values of stability functions


INTERFACE:

   subroutine compute_cm0(turb_method,stab_method,scnd_method)
DESCRIPTION:

Computes the values of the stability function $ c_\mu$ defined in (46) in the logarithmic boundary-layer, $ c_\mu^0$, and in shear-free, spatially decaying turbulence, $ c_\mu^$sf (see section 4.7.4).

$ c_\mu^0$ is the value of $ c_\mu$ in unstratified equilibrium flows, i.e. in the logarithmic wall region. It can be obtained from the relation $ P=\epsilon$, according to (80) written in the form

$\displaystyle \dfrac{P}{\epsilon} = \hat{c}_\mu \alpha_M = 1 \quad .$ (122)

In unstratified flows, $ \hat{c}_\mu$ only depends on $ \alpha_M$ (see sections 4.24-4.26), and (122) is a polynomial equation for the value of $ \alpha_M$ in equilibrium. Its solution is

$\displaystyle \alpha_M = \dfrac{3 {\cal N}^2}{a_2^2 - 3 a_3^2 + 3 a_1 {\cal N}} \; , \quad$ (123)

where, according to (67) in equilibrium $ {\cal N} = (c_1 + c^*_1)/2$. The value of the stability function in equilibrium follows directly from (122),

$\displaystyle \hat{c}_\mu^0 = \dfrac{a_2^2 - 3 a_3^2 + 3 a_1 {\cal N}}{3 {\cal N}^2} \quad .$ (124)

Note that $ \hat{c}_\mu^0 = (c_\mu^0)^4$ according to (78).

Algebraic Stress Models exhibit an interesting behaviour in unstratified, shear-free turbulence. Clearly, in the absence of shear, these models predict isotropic turbulence, $ b_{ij} = 0$, according to (61). This is a direct consequence of the assumption (60), implying an infinitely small return-to-isotropy time scale. Formally, however, the limit of the stability function $ \hat{c}_\mu$ for $ \alpha_M \rightarrow 0$ follows from (76) and the definitions given in sections 4.24-4.26. The limiting value is

$\displaystyle \lim_{\alpha_M \rightarrow 0} \hat{c}_\mu = \hat{c}_\mu^$sf$\displaystyle = \dfrac{a_1}{{\cal N}} \; , \quad$ (125)

where, according to (67), one has either $ {\cal N} = c_1/2-1$ or $ {\cal N} = (c_1 + c^*_1)/2$, see section 4.24 and section 4.26, respectively. The above limit corresponds to nearly isotropic turbulence supporting a very small momentum flux caused by a very small shear.

Note that $ \hat{c}_\mu^$sf$ = (c_\mu^0)^3 c_\mu^$sf according to (78).


USES:

   IMPLICIT NONE
INPUT PARAMETERS:
   integer, intent(in)                 :: turb_method
   integer, intent(in)                 :: stab_method
   integer, intent(in)                 :: scnd_method
REVISION HISTORY:
   Original author(s): Lars Umlauf

Karsten Bolding 2012-12-28