Viscous Flux Discretization

Interior Faces

To evaluate the stress tensor , the velocity gradient tensor at the face must be written in terms of the cell velocities for purposes of linearization.

Using Eqn. 354, the velocity gradient tensor at a face may be written:

(71)

where:

(72)

(73)

where and are the explicitly computed velocity gradient tensors in the cells

Boundary Faces

For slip walls, the viscous shear force at the wall boundary face, , simply set to zero.

For no-slip walls in laminar flow, Eqn. 356 is used to represent the velocity gradient tensor at the face:

(74)

where:

(75)

and is the specified wall tangential motion.

The stress tensor at the face is then given by:

(76)

For no-slip walls in turbulent flow, it is assumed that only the component of velocity parallel to the wall is of interest. A linear relationship between the wall shear force and the wall-parallel component of the difference in velocity between the wall and the cell is hypothesized:

(77)

Using the definitions of wall shear stress magnitude () and reference velocity (), the coefficient of proportionality is:

(78)

The reference velocity is computed according to the specific turbulence model. The value of is obtained (as a function of ) from the appropriate wall law.