Aerodynamic Analysis of Drag Reduction Devices on the Underbody for SAAB
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8/12/2019 Aerodynamic Analysis of Drag Reduction Devices on the Underbody for SAAB
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y+
C d
C l
C p
C p
V
V
V
C m
C d
C l
C p
V
V
V
C m
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V
τ ω
C p
C p
C p
10◦
Af
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Af
m2
C d
C l
C m
C p
c p [J/kg · K ]
ρ
kg/m3
F A [N ]
F D [N ]
F G [N ]
F L [N ]
F R [N ]
g
m/s2
k
[W/K · m]
p [P a]
T [K ]
U [m/s]
V [m/s]
µ [kg/m · s]
Φ
y+
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Ma = 0.3 ≈ 100 [m/s]
Continuity equation : ∂u
∂x +
∂ v
∂y +
∂ w
∂z = 0
2nd
ρgx − ∂p∂x + µ
∂ 2u∂x2 + ∂ 2u
∂y2 + ∂ 2u∂z2
= ρ du
dt
ρgy − ∂p∂y + µ ∂ 2v
∂x2 + ∂ 2v∂y2 + ∂ 2v
∂z2 = ρ dvdt
ρgz − ∂p∂z + µ
∂ 2w∂x2 + ∂ 2w
∂y2 + ∂ 2w∂z2
= ρ dw
dt
1st
ρc pdT
dt = k∇2T + Φ
ρ = densityt = time
p = pressureµ = static viscosity
g = gravity constant
c p = specific heat capacityT = temperaturek = coefficient of thermal conductivity
Φ = viscous dissipation f unction
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F X = F A + F R + F G + F D = ma + f R mg cosα + mg sinα + 12 ρC DAf U 2
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0 20 40 60 80 100 120 140 1600
100
200
300
400
500
600
700
800
Speed [km/h]
R e s i s t a n c e [ N ]
Aerodynamic drag vs. Rolling resistance
Aerodynamic dragRolling resistance
(F D)
p1 + ρV 21
2 = p2 +
ρV 222
p = static pressure
ρV 2
2 = dynamic pressure
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C d = F
D1
2ρU 2A
ρ = air density
U = f reestream velocity
F D = drag force
A = frontal area
(F L)
C l = F L1
2ρU 2A
C p
C p
C p
C p C p
C p = p − p∞1
2ρV 2
= p − p∞
dynamic pressure
p∞ p
Re = ρU l
µ
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dpdx
< 0
Continuity : ∂u
∂x +
∂ v
∂y = 0
Momentum along wall : ∂u
∂x + v
∂u
∂y ≈ U
dU
dx +
1
ρ
∂τ
∂y
u = v = 0
∂τ
∂y
wall
= µ∂ 2u
∂y2
wall
= −ρU dU
dx =
dp
dx
xs
τ w = 0
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φ
30◦
γ,
γ,
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h2
h1
N
θ
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C diffuserD = tanθ
−C diffuser
L
C pd
C pd = p2 − p1
1
2ρU 2
1
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ρgx − dpdx + µ d2u
dx2 + d2udy2 + d2u
dz2 = ρ dudt
ρgy − dpdy + µ
d2vdx2 + d2v
dy2 + d2vdz2
= ρ dv
dt
ρgz − dpdz + µ
d2wdx2 + d2w
dy2 + d2wdz2
= ρ dw
dt
∂ρ
∂t +
∂ (ρu)
∂x +
∂ (ρv)
∂y +
∂ (ρw)
∂z = 0
∂u
∂x
+ ∂ v
∂y
+ ∂ w
∂z
= 0
u, v, w & p
u = 1
T
T ˆ
0
u dt
p = ¯ p + p
u = u + u
v = v + v
w = w + w
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∂ u
∂x +
∂ v
∂y +
∂ w
∂z = 0
ρgx − ∂ p
∂x +
∂
∂x
µ
∂ u
∂x − ρu2
+
∂
∂y
µ
∂ u
∂y − ρuv
+
∂
∂z
µ
∂ u
∂z − ρuw
= ρ
du
dt
ρuv
k −
k −
k −
µt
k k
k =u2 + v2 + w2
1/2
k −
k−
k −
k−
µt = Cµρk2
ρuv k −
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− ρu
iu
j = µt
∂ui∂xj
+ ∂ uj∂xi
− 23
ρk + µt ∂ui
∂xi
δ ij
y+
y+ = yu∗
ν
u∗ = 0.05U
u∗
ν
y
y+ = 1
y+
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y+
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1
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Af
Af
m2
m
m
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S 1,√
S 2,√
S 3,√
S 4,√
S 5,√
S 6,√
S 7,√
S 8,√
S 9,√
S 10,√
S 11,√
S 12,√
S 13,√
S 14,√
W 1,√
W 2,√
W 3,√
W 4,√
W 5√
W 6,√
W 7,√
W 8,√
W 9,√
W 10,√
W 11,√
W 12,√
W 13,√
W 14,√
W 15,√
W 16,√
W 17,√
k − ε
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C d
C l
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0 5 10 15 20 25 30 35 400
0.05
0.1
0.15
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Velocitymagnitude [m/s]
Z − c o o r d i n a t e [ m ]
Boundary Layer on diffuser
5 degreesdiffuser xy−cord: (5.0;0)
0 5 10 15 20 25 30 35 401.455
1.46
1.465
1.47
1.475
1.48
Velocity magnitude [m/s]
Z − c o o r d i n a t e [ m ]
Boundary Layer on roof
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V
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8◦ 5◦
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±
±
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τ ω
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[%]
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k − ω
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