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

CHAPTER 5. RESULTS 73
Integrating with respect to y,
δ
∂
1 ∂ 〈P 〉
〈U〉 dy = −
y|
∂t
ρ ∂x
y=δ
τ
y=0 +
ρ
0
τ=0
τ=τw
Dividing by δ and noting that 1
δ
〈U〉 dy = U, Equation 5.7 becomes
δ 0
dU
dt
+ τw
ρδ
∂ 〈P 〉
= −1
ρ ∂x
(5.7)
(5.8)
Thus, in unsteady channel flow, the driving pressure gradient applies energy
to both overcome a wall shear stress and to accelerate the flow. The fact that
peaks and troughs in U are underpredicted in Figure 5.5 indicates that dU
dt is
underestimated by the turbulence models. τw also displays a lower amplitude
of variation than expected, as seen in Figure 5.6. This is consistent with
(Equation 5.1). Together, these
Figure 5.5, since τw is a function of ∂〈U〉
∂y
results suggest that the energy is added to and removed from the flow at a
less than realistic rate.
The equality in Equation 5.8 appears to be compromised when using these
turbulence models in periodic flow. In terms of the x-momentum equation
(Equation 3.5), this suggests that the models are not as well tuned to predict
〈uv〉 in periodic channel flow as in steady channel flow. Errors in predicting
〈uv〉 produce deviations in 〈U〉, particularly farther from the wall, as seen in
Figures 5.11 & 5.12. This effects the dU
dt
y=0
term in Equation 5.8.
Figures 5.7, 5.8, 5.9 & 5.10 show flow variables plotted as a function of phase
angle at various locations throughout the channel (y/δ = 0.1, 0.2, 0.5 & 0.9,
respectively).
Results for the k-ε log law do not appear in Figure 5.7, because y/δ = 0.1
is outside the calculated flow field when the log law is used. In Figure 5.8,