Boundary conditons for the epsilon-equation

INTERFACE:

   REALTYPE function  epsilon_bc(bc,type,zi,ki,z0,u_tau)

DESCRIPTION:

Computes prescribed and flux boundary conditions for the transport equation (163). The formal parameter bc determines whether Dirchlet or Neumann-type boundary conditions are computed. Depending on the physical properties of the boundary-layer, the parameter type relates either to a visous, a logarithmic, or an injection-type boundary-layer. In the latter case, the flux of TKE caused by breaking surface waves has to be specified. Presently, there is only one possibility to do so implemented in GOTM. It is described in section 4.34. All parameters that determine the boundary layer have to be set in gotmturb.nml.

Note that in this section, for brevity, $z$ denotes the distance from the wall (or the surface), and not the standard coordinate of the same name used in GOTM.

Viscous boundary-layers

This type is not implemented yet in GOTM.

Logarithmic boundary-layers

The Dirichlet (prescribed) boundary condition follows from (153) as

$$\epsilon= \dfrac{(c_\mu^0)^3 k^\frac{3}{2}}{\kappa (z+z_0)} \; , \quad$$ (134)

where we used the law-of-the-wall relation $l=\kappa(z+z_0)$.

The Neumann (flux) boundary condition can be expressed as

$$F_\epsilon = - \dfrac{\nu_t}{\sigma_\epsilon} \dfrac{\partial {\e... ...rtial {z}} = \dfrac{(c_\mu^0)^4}{\sigma_\epsilon} \dfrac{k^2}{z+z_0} \; , \quad$$ (135)

by inserting $l=\kappa(z+z_0)$ into the expression for the diffusivity in (44). Note, that in (134) and (135), we use ki, the value of $k$ at the current time step, to compute the boundary conditions. By means of (101), it would have been also possible to express the boundary conditions in terms of the friction velocity, $u_*$. This, however, causes numerical difficulties in case of a stress-free surface boundary-layer as for example in the pressure-driven open channel flow.

Shear-free boundary-layers with injection of TKE

The Dirichlet (prescribed) boundary condition follows simply from the power-law (108) inserted in (153). This yields

$$\epsilon= (c_\mu^0)^3 K^\frac{3}{2} L^{-1} (z+z_0)^{\frac{3}{2}\alpha - 1} \quad .$$ (136)

The Neumann (flux) boundary condition is

$$F_\epsilon = - \dfrac{\nu_t}{\sigma_\epsilon} \dfrac{\partial {\e... ...silon} K^2 \left( \dfrac{3}{2} \alpha -1 \right) (z+z_0)^{2\alpha-1} \; , \quad$$ (137)

which follows from (108) and (44). The parameter $K$ is computed as described in the context of (128).

USES:

   IMPLICIT NONE

INPUT PARAMETERS:

   integer, intent(in)                 :: bc,type
   REALTYPE, intent(in)                :: zi,ki,z0,u_tau

REVISION HISTORY:

   Original author(s): Lars Umlauf