Three-Dimensional Wall Effects
In a freestream, recall that a lifting body can e modeled by a horseshoe vortex:
V∞
Γ
Consider a rectangular cross-section tunnel:
Flow is into page
Γ
Γ
wing
The image system for this looks like:
y
y
y
images
y y y
y
y
images
y y y
y
y
y
images
images
y
y
y
y
images
y
y
y
y
y
y
y
actual tunnel
images
y y y
y
y
y
y
Γ
y
y y y
images
y
y
y y y
images
y
y
y
y
y
y
y
y
y
y
y y y
Three-Dimensional Wall Effects
The effect of these images is:
For fixed lift, such that Γ is constant,
∗ an upwash exists due to images ⇒ α is effectively larger
α∞
α tunnel
≅
effective
freestream
AOA
AOA of
model in
tunnel
+
∆α i
correction due to
upwash induced by
images
∗ Similarly, this creates decrease in induced drag relative to freestream flight:
Recall,
C Di ∝ C Lα i
⇒ ∆C Di = C L ∆α i
⇒ C Di = C Di
∞
tunnel
+ ∆C Di
Or, since we are interested in the total drag:
C D∞ = C Dtunnel + ∆C Di
Specific formulas derived in detailed analysis give that:
S
∆α i = δ ( )C L
C
where S = reference area
C = tunnel cross - sectional area
δ = factor which depends on tunnel & model geometry
Wright Brothers is an elliptic cross-section with dimensions 10 ft wide by 7 ft high.
b
h
B
16.100 2002
2
Three-Dimensional Wall Effects
Define:
be
h
B
b
k≡ e
B
be ≡ effective span ≈ 0.9b
λ≡
horseshoe
model
b
caused because wing tip trailing
vortices are inboard of tips a
short distance downstream
actual trailing vortex
Boundary Correction Factor, δ
Values of δ for a wing with uniform loading in a closed elliptical jet
K = Effective span/Jet width
16.100 2002
3