The UMIST-N Near-Wall Treatment Applied to Periodic Channel Flow

CHAPTER 5. RESULTS 69
The solution procedure employed involved initialising flow variables to zero,
applying the driving pressure gradient, and iteratively updating the solution
until convergence was achieved. Gradients in time ( ∂
∂t
terms) were included
in the solution, so that unconverged results may have been regarded con-
ceptually as the transient response of the flow to a step change in pressure.
However, transient results are not shown. ∂
∂t
terms do not affect the con-
verged result in steady flow and were included primarily for debugging.
In Figure 5.1, the k-ε solutions overpredict peak velocity and bulk flow, while
the k-ω model underpredicts. Figure 5.3 shows that the 〈U〉 + results nearer
to the wall are similar for each model and match the DNS result closely.
Equation 3.17 can be evaluated at y = 0 to obtain
d 〈U〉 y=0
τw = ρν
dy
d〈P 〉
dx by
Recalling Equation 3.19, τw is related to
d 〈P 〉
τw = −δ
dx
(5.1)
(5.2)
Therefore, close agreement on 〈U〉 + near the wall is to be expected from the
prescription of
d〈P 〉
dx . The deviation in 〈U〉+ occurs farther from the wall and
is likely due to the prediction of νt.
The subgrid solution produces results that are very similar to those of the
low-Reynolds-number approach. This is to be expected, since the internal
subgrid calculation is identical to a standard low-Reynolds-number calcu-
lation in a parabolic solution scheme. The unique aspect of the subgrid
approach as compared to a low-Reynolds-number parabolic solution is in the
transferral of information between the subgrid and the main grid via mutu-
ally dependent boundary conditions. However, this complexity is unlikely to