Project Assignment #1 Aerodynamic Analysis of the BWB Wind Tunnel Model

Project Assignment #1
Aerodynamic Analysis of the BWB Wind Tunnel Model
Sample Solution
Prof. David L. Darmofal
Department of Aeronautics and Astronautics
Massachusetts Institute of Technology
September 29, 2005
1
Data
Table 1 contains geometric data for the full-scale BWB. Table 2 contains data for the wind tunnel.
For now, assume that the air is at standard sea level conditions (i.e. the temperature, density,
pressure, and viscosity) in the wind tunnel test section.
Symbol
Sref
Swet
b
cmac
Parameter
Reference Area (trapezoidal wing)
Wetted Area
Wing span
Mean aerodynamic chord
Value
7,840 ft 2
31,324 ft2
280 ft
30.75 ft
Table 1: Geometric data for full-scale BWB
Symbol
U�
��
��
a�
q�
Parameter
Model scale
freestream velocity
freestream density
freestream kinemativ viscosity
freestream speed of sound
freestream dynamic pressure
Value
1/47
50 and 100 mph
0.00237 sl/ft3
1.58e-04 ft 2 /s
1116.4 ft/s
25.56 lb/ft 2
Table 2: Wind tunnel data
2
Aerodynamic Model
To estimate the aerodynamic forces and moment on the BWB in the tunnel, a vortex lattice
calculation is combined with a semi-empirical estimate of the skin friction and pressure drag (C D f
and CDp , respectively). The vortex lattice calculation provides estimates for the lift coefficient
(CL ), the induced drag coefficient (CDi ), and the pitching moment coefficient about the aircraft
nose (CM 0 ). The total drag is the sum of the three drag estimates, Note, since the wind tunnel
test is at subsonic conditions, wave drag is not expected to occur.
1
3
Required Tasks
3.1
Mach and Reynolds Number for Wind Tunnel Tests
The Reynolds number reported here is based the mean aerodynamic chord, i.e.,
U� cmac
Rec =
.
��
Note that the value of cmac is 1/47 of the value listed in Table 1 because of the scale of the wind
tunnel model.
For the 50 and 100 mph tunel speeds, the Mach and Reynolds number are given in Table 3.
U�
50 mph
100 mph
M�
0.065
0.130
Rec
3.03e5
6.06e5
Table 3: Freestream Mach and Reynolds number for tunnel tests.
3.2
Description of Friction and Pressure Drag Estimation
The skin friction and pressure drag are estimated using the approach described in the Lift & Drag
Primer. Specifically, for the friction drag,
Swet
.
CDf � cf
Sref
The estimate of the average skin-friction coefficient will be based on turbulent flat plate data at
a representative Reynolds number. For this purpose, we will use the flat plate data in Figure 1
in the Lift & Drag Primer. As specified in the assignment, we assume that the boundary layer is
turbulent and use the analytic equation given in Figure 1 to determine the skin friction coefficient,
0.37
cf =
.
(1)
(log10 Re)2.6
We will assume that the average skin friction c f can be estimated from Equation (1) using the
Reynolds number based on the mean aerodynamic chord, Re c , i.e.
0.37
cf =
.
(log 10 Rec )2.6
At the two tunnel conditions, this gives,
At 50 mph, CD f = 0.0177,
At 100 mph, CD f = 0.0154.
Following the Lift & Drag primer, the pressure drag coefficient can be estimated as,
�
CDp = (CDp )min + CDf + CDp
(CDp )min = 60
�
tmax
c
�4
CDf .
min
CL 2 ,
However, since the BWB is relatively thin, the minimum pressure drag is expected to be small
compared the skin friction. For example, if the maximum thickness-to-chord ratio were 20%, then
(CDp )min � 0.1CD f . Thus, the minimum pressure drag is about 10% of the skin friction drag and
therefore will be neglected. As a result, the pressure drag estimate reduces to,
CDp � CDf CL 2 .
2
3.3
Plots
The vortex lattice code, AVL, was run at a series of angles for both Mach numbers. The aero­
dynamic data from the AVL simulations is shown in Table 4 and 5. This data was then used to
produce the required plots. Note, the neutral point which is reported by AVL is the distance in
feet from the nose of aircraft at fullscale. This location is normalized by the fullscale c mac = 30.75
feet and reported in the tables. As can be seen from the tables, there is very little difference in the
inviscid predictions from AVL due to the change in Mach number.
�
-3
-2
-1
0
1
2
3
6
9
12
15
CL
-0.152
-0.033
0.087
0.206
0.326
0.445
0.565
0.921
1.276
1.626
1.973
CDi
0.0017
0.0013
0.0019
0.0034
0.0058
0.0092
0.0136
0.0323
0.0591
0.0940
0.1364
CM 0
0.335
0.013
-0.310
-0.634
-0.958
-1.281
-1.604
-2.564
-3.505
-4.415
-5.284
xac /cmac
2.69
2.70
2.70
2.71
2.71
2.71
2.71
2.70
2.68
2.64
2.59
Table 4: AVL results for 50 mph (M� = 0.065).
�
-3
-2
-1
0
1
2
3
6
9
12
15
CL
-0.153
-0.033
0.087
0.207
0.327
0.447
0.567
0.925
1.280
1.632
1.980
CD i
0.0017
0.0013
0.0019
0.0034
0.0059
0.0093
0.0137
0.0324
0.0595
0.0945
0.1372
CM 0
0.336
0.012
-0.312
-0.637
-0.962
-1.287
-1.611
-2.575
-3.519
-4.432
-5.305
xac /cmac
2.69
2.70
2.70
2.71
2.71
2.71
2.71
2.70
2.68
2.64
2.59
Table 5: AVL results for 100 mph (M� = 0.130).
The required plots are shown in Figure 1-4; both tunnel speeds are included in each plot.
However, for everything except the drag polar (Figure 2), the curves overlap.
3.4
Questions
1. How did the change in velocity impact the force and moment coefficients? Answer: the
coefficients were almost unchanged by the velocity change except for a small dependence of
the CD . The dependence of CD on the velocity arises from the change in the friction drag
estimate which is a function of the Reynolds number.
3
2
1.5
C
L
1
0.5
0
−0.5
−4
−2
0
2
4
6
�
8
10
12
14
16
Figure 1: CL versus � for 50 (blue) and 100 (red) mph.
2
1.5
C
L
1
0.5
0
−0.5
0
0.05
0.1
C
0.15
0.2
0.25
D
Figure 2: CL versus CD for 50 (blue) and 100 (red) mph.
4
0.2
0.15
0.1
Mac
0.05
C
0
−0.05
−0.1
−0.15
−0.2
−0.5
0
0.5
C
1
1.5
2
L
Figure 3: CM ac versus CL for 50 (blue) and 100 (red) mph.
4
3.5
x /c
cp mac
, x /c
ac mac
3
2.5
2
1.5
1
0.5
0
−0.5
0
0.5
C
1
1.5
2
L
Figure 4: xcp (solid) and xac (dashed) versus CL for 50 (blue) and 100 (red) mph.
5
2. Given the response to the previous question, what will the impact of the changing the velocity
by a factor of two have on the force and moments (i.e. not the force and moment coefficients)?
Answer: Given that the force and moment coefficients are relatively unchanged by the
velocity in the tunnel, the force and moments are expected to increase by a factor of 4 for a
factor 2 increase in the velocity. For example, the lift is related to the lift coefficient by,
L = q� Sref CL .
2 and C was insensitive to the velocity change, then L V 2 which results in
Since q� V�
L
�
the lift having a squared dependence on the velocity.
6