Aerodynamics
Masters of Mechanical Engineering
• Steady-flow,
• Flow independent of z direction,(two-dimensional)
• Fully developed flow,
h
U
x
0=∂
∂
t
0=∂
∂
z
0=∂
∂
x
vr
y
Incompressible, Laminar Couette Flow
Aerodynamics
Masters of Mechanical Engineering
y
h
U
• Boundary conditions
- Impermeability of the walls:
- No-slip condition:
x
000 =⇒==⇒= vhyvy
Uuhyuy ˆ00 =⇒==⇒=
Incompressible, Laminar Couette Flow
Aerodynamics
Masters of Mechanical Engineering
• Continuity equation
• Boundary condition
0=v
.0 constvy
v=⇔=
∂
∂
Incompressible, Laminar Couette Flow
Aerodynamics
Masters of Mechanical Engineering
• Momentum balance,
• Momentum balance,
• Pressure is only a function of
must be indepedent of
y
2
21
0y
u
x
p
∂
∂+
∂
∂−= ν
ρ
y
p
∂
∂−=
ρ
10
x
x
=
∂
∂0
x
vx
r
dx
dp
Incompressible, Laminar Couette Flow
Aerodynamics
Masters of Mechanical Engineering
• Momentum balance,
• Boundary conditions
y
u
ydx
dp
y
u
yx
yx
∂
∂=
∂
∂==
∂
∂
µτ
τ
ρρν
112
2
x
Uuhy
uy
ˆ
00
=⇒=
=⇒=
Incompressible, Laminar Couette Flow
Aerodynamics
Masters of Mechanical Engineering
• Solution
• Reference values for length and velocity
( )
−+=
−−=
2
ˆ
2
1ˆ
hy
dx
dp
h
U
yhydx
dpU
h
yu
yx µτ
µ
UU
hL
ref
ref
ˆ=
=
Incompressible, Laminar Couette Flow
Aerodynamics
Masters of Mechanical Engineering
• Solution in dimensionless variables
• Non-dimensional numbers
−Λ−=
−Λ−=
h
y
U
h
y
h
y
U
u
yx
2
121
Re
2
ˆ21
11ˆ
2ρ
τ
dx
dp
U
h
hURe
µ
ν
2
ˆ
2
=Λ
= Reynolds number
Pressure gradient parameter
Incompressible, Laminar Couette Flow
Aerodynamics
Masters of Mechanical Engineering
• Non-dimensional numbers
Reynolds number
Pressure gradient parameter
2
2
2
ˆ
ˆ
ˆ
h
U
dx
dp
h
U
h
U
Re
µ
µ
ρ
=Λ
∝
Incompressible, Laminar Couette Flow
Aerodynamics
Masters of Mechanical Engineering
-0.25 0 0.25 0.5 0.75 1 1.250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Λ=-2
Λ=-1
Λ=0
Λ=1
Λ=2
-3 -2 -1 0 1 2 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Λ=-2
Λ=-1
Λ=0
Λ=1
Λ=2
h
y
h
y
UU ˆ 2UR yxe ρτ
Incompressible, Laminar Couette Flow
Aerodynamics
Masters of Mechanical Engineering
∂
∂
∂
∂+
∂
∂+
∂
∂
∂
∂+
∂
∂−=
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂−=
∂
∂+
∂
∂
=∂
∂+
∂
∂
y
v
yx
v
y
u
xy
p
y
vv
x
vu
x
v
y
u
yx
u
xx
p
y
uv
x
uu
y
v
x
u
ννρ
ννρ
21
21
0
Two-dimensional, IncompressibleSteady Flow
Aerodynamics
Masters of Mechanical Engineering
Constant viscosity, ν=constant
∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂
∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂
=∂
∂+
∂
∂
2
2
2
2
2
2
2
2
1
1
0
y
v
x
v
y
p
y
vv
x
vu
y
u
x
u
x
p
y
uv
x
uu
y
v
x
u
νρ
νρ
Two-dimensional, IncompressibleSteady Flow
Aerodynamics
Masters of Mechanical Engineering
Two-dimensional, IncompressibleSteady Flow
Making the equations dimensionless
Reference values
Velocity
LengthPressure *22
**
**
,
,
pUpU
LyyLxxL
vUvuUuU
ee
eee
ρρ =→
==→
==→
Aerodynamics
Masters of Mechanical Engineering
convectiondiffusionO[ ]
Two-dimensional, IncompressibleSteady Flow
( ) ( )
( ) ( )
====
∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂
∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂
=∂
∂+
∂
∂
2
2*
*2
2*
*2
*
*
*
*
*
*
*
*
2*
*2
2*
*2
*
*
*
*
*
*
*
*
*
*
*
*
1
1
0
L
UL
UU
LULUR
y
v
x
v
Ry
p
y
vv
x
vu
y
u
x
u
Rx
p
y
uv
x
uu
y
v
x
u
e
ee
eee
e
e
µ
ρ
νµ
ρ
Aerodynamics
Masters of Mechanical Engineering
y
u
∂
∂= µτ (uni-dimensional shear-stress)
Ar µ � 1,8×10-5kgm-1s-1 ν �1,1×10-5m2s-1
Água µ � 1,0×10-3kgm-1s-1 ν �1,0×10-6m2s-1
Two-dimensional, IncompressibleSteady Flow
• Practical applications are usually flows athigh Reynolds numbers, 5
10>eR
Aerodynamics
Masters of Mechanical Engineering
Two-dimensional, IncompressibleSteady Flow
• Effects of shear-stresses are restricted to small regions that exhibit large velocity variations in small distances
• Thin shear layers- Thickness of the shear layer, δ, is much smaller than the reference length L, δ/L≪ 1
Aerodynamics
Masters of Mechanical Engineering
Boundary-layer Wake
Mixing layerJet
Two-dimensional, IncompressibleSteady Flow
Aerodynamics
Masters of Mechanical Engineering
Thick shear layers (Bluff bodies)
Two-dimensional, IncompressibleSteady Flow
Aerodynamics
Masters of Mechanical Engineering
Boundary-Layer Approximations
Prandtl simplifications (1904)
Analysis of the order of magnitude of the terms included in the continuity and momentum balance equations
Starting hypothesis: Re≫1. (δ/L≪1)ν
xUR e
e =
Aerodynamics
Masters of Mechanical Engineering
Boundary-Layer Approximations
Prandtl Simplifications (1904)
Order of magnitude of variable ξ, O[ξ], is given bythe upper limit of the ξ variation
Known orders of magnitude
O[x]→ L
O[y] → δ O[u]→ Ue
Aerodynamics
Masters of Mechanical Engineering
[ ]
[ ]L
Uv
v
L
U
y
v
x
u
e
e
δ
δ
=
=+
=∂
∂+
∂
∂
0
0
O
O
Boundary-Layer Approximations
Continuity equation
Aerodynamics
Masters of Mechanical Engineering
L
U
L
p
dx
dp
dx
dUU
dx
dp
constUp
ee
ee
e
e
2
2
11
0
.2
1
==
=+
=+
ρρ
ρ
ρ
O
Boundary-Layer Approximations
Bernoulli’s equation applied to the outer flow(ideal fluid)
Aerodynamics
Masters of Mechanical Engineering
++=+
++=+
++=+
∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂
2
22222
22
222
2
2
2
2
11
111
1
1
δ
δ
ν
δν
νρ
L
R
L
LUL
U
L
U
L
U
L
U
U
L
U
L
U
L
U
L
U
y
u
x
u
x
p
y
uv
x
uu
e
e
eeee
eeeee
Boundary-Layer Approximations
Momentum balance in the x direction
Aerodynamics
Masters of Mechanical Engineering
+
2
11
δ
L
Re
111
01
2
2
2
2
2
ee
e
RL
L
Rx
u
Rx
u
=⇒
=
∂
∂
≅=
∂
∂
δ
δν
ν
O
O
≪
Boundary-Layer Approximations
Momentum balance in the x direction
Analysis of diffusion
Aerodynamics
Masters of Mechanical Engineering
O
O
++
∂
∂−=+
++
∂
∂−=+
∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂
2
2
2
2
2
2
2
2
2
32
2
2
2
2
2
2
2
1
1
1
δ
δδν
ρ
δδ
δ
δν
ρ
δδ
νρ
L
L
U
L
U
LUy
p
L
U
L
U
L
U
L
U
y
p
L
U
L
U
y
v
x
v
y
p
y
vv
x
vu
ee
e
ee
eeee
Boundary-Layer Approximations
Momentum balance in the y direction
Aerodynamics
Masters of Mechanical Engineering
2
2
2
2
1
111
11
L
U
y
p
Ry
p
U
L
e
ee
δ
ρ
ρδ
=
∂
∂−
++
∂
∂−=+
eRL
=
2
δ
O
O
Boundary-Layer Approximations
Momentum balance in the y direction
Using we obtain
Aerodynamics
Masters of Mechanical Engineering
0
2
11 222
20
≅∂
∂
=
=
∂
∂∫
y
p
UUR
UL
dyy
pee
e
e ρρρδδ
O ≪
Boundary-Layer Approximations
Momentum balance in the y direction
Across the boundary-layer
Therefore,
Aerodynamics
Masters of Mechanical Engineering
∂
∂+−=
∂
∂+
∂
∂
=∂
∂+
∂
∂
2
21
0
y
u
dx
dp
y
uv
x
uu
y
v
x
u
νρ
• The selected coordinate system must respect thefollowing conditions:
1. The x coordinate must be aligned with the outer flow
2. The y coordinate is normal to the surface
Boundary-Layer Approximations
Aerodynamics
Masters of Mechanical Engineering
∂
∂+−=
∂
∂+
∂
∂
=∂
∂+
∂
∂
2
21
0
y
u
dx
dp
y
uv
x
uu
y
v
x
u
νρ
• Static pressure is independent of the coordinate y.Pressure change with x (dp/dx) may be obtainedfrom the outer flow, p(x)≃pe(x). Therefore, the pressure does not belong to the unknowns.
The pressure is part of the input
of a boundary-layer problem
Boundary-Layer Approximations
Aerodynamics
Masters of Mechanical Engineering
• The equations are no longer elliptic in the xdirection. For a given value of x, the solution dependsonly on the upstream conditions. Therefore, it is possible to solve the problem using a marchingprocedure in the x direction (initial value problem).
∂
∂+−=
∂
∂+
∂
∂
=∂
∂+
∂
∂
2
21
0
y
u
dx
dp
y
uv
x
uu
y
v
x
u
νρ
Boundary-Layer Approximations
Aerodynamics
Masters of Mechanical Engineering
0≅∂
∂
y
p
02
2
≅
∂
∂
x
uν
Simplified Forms of the Navier-Stokes Equations
• Boundary layer, thin shear layer equations
― Pressure determined by the outer flow,
― Diffusion in the main direction of the flowneglected,
Aerodynamics
Masters of Mechanical Engineering
x
p
x
p e
∂
∂≅
∂
∂
02
2
≅
∂
∂
x
uν
Simplified Forms of the Navier-Stokes Equations
• Parabolized Navier-Stokes equations
― Pressure derivative in the main direction of theflow determined by the outer flow,
― Diffusion in the main direction of the flowneglected,
Aerodynamics
Masters of Mechanical Engineering
02
2
≅
∂
∂
x
uν
Simplified Forms of the Navier-Stokes Equations
• Reduced Navier-Stokes equations
― Diffusion in the main direction of the flowneglected,
― Pressure determination makes the problem elliptic