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 (44) 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 (78) written in the form

$$\dfrac{P}{\epsilon} = \hat{c}_\mu \alpha_M = 1 \quad .$$ (120)

In unstratified flows, $\hat{c}_\mu$ only depends on $\alpha_M$ (see sections 4.7.37-4.7.39), and (120) is a polynomial equation for the value of $\alpha_M$ in equilibrium. Its solution is

$$\alpha_M = \dfrac{3 {\cal N}^2}{a_2^2 - 3 a_3^2 + 3 a_1 {\cal N}} \; , \quad$$ (121)

where, according to (65) in equilibrium ${\cal N} = (c_1 + c^*_1)/2$. The value of the stability function in equilibrium follows directly from (120),

$$\hat{c}_\mu^0 = \dfrac{a_2^2 - 3 a_3^2 + 3 a_1 {\cal N}}{3 {\cal N}^2} \quad .$$ (122)

Note that $\hat{c}_\mu^0 = (c_\mu^0)^4$ according to (76).

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 (59). This is a direct consequence of the assumption (58), 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 (74) and the definitions given in sections 4.7.37-4.7.39. The limiting value is

$$\lim_{\alpha_M \rightarrow 0} \hat{c}_\mu = \hat{c}_\mu^ = \dfrac{a_1}{{\cal N}} \; , \quad$$ (123)

where, according to (65), one has either ${\cal N} = c_1/2-1$ or ${\cal N} = (c_1 + c^*_1)/2$, see section 4.7.37 and section 4.7.39, 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 (76).

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