IV.4 Boundary Conditions

We need to specify additional boundary conditions for our new transport equations. Unfortunately k and e are not always measured quantities, so we may have to estimate them.

IV.4.1 Inlet boundary

For laminar (incompressible) flow, we specify u- at the inlet. For turbulence modelled with the k - e model we specify u-, k, e at the inlet. However k,e values are not always known. We have to estimate the values to use as b.c. and it is always worth checking to see that the calculated result is not sensitive to the inlet conditions. Conventionally : k ~ 5 - 10% of the inlet kinetic energy. We can also guess the turbulent length scale to be

l ~ 0.07L
for some length scale L (eg. diameter of a pipe). With this, an estimate of the inlet e would be
e = ---------

IV.4.2 Outlet boundary

For laminar flow, we usually specify p. For turbulent flow, specify

               @k    @e
p,     and     ---=  --- = 0
               @n    @n

IV.4.3 Solid walls

Treatment of solid walls is quite complex. Near-wall turbulent flow is not like that in free stream - instead have a boundary layer consisting of laminar sublayer, transition region and log-law region. We could make the mesh fine enough to resolve all the detail in this region, but this would be very expensive. Hence we need to treat this region of the flow differently. Luckly near-wall flow is universal and we can introduce a separate wall model to describe the flow in the layer of cells close to the wall.

Wall functions of the form

       2             3
k =   V~ u*-,    e = -u*
       Cm          ky
are often used. Note the introduction of wall parameters such as the friction velocity u* and the von Karman constant k. The velocity in this cell given by usual log-law
 +    u--   1-    +
u  =  u* =  k log y  + B
and similar formulae can be derived to take account of things like heat transfer from surfaces (strongly affected by the b.l. flow, so the modelling has to be good).

However such functions have been derived under the assumption that we do indeed have turbulent flow in the near-wall cell, i.e. the cell centre needs to be within the transition region. This provides a restriction on the y+ value for the first cell yp+ :

y+p > 11.63
which represents a restriction on the mesh in this region. If we cannot satisfy this relationship we would need to use alternative (low-Re) turbulence models.