Student Handout 21 2014
22
External incompressible viscous H:B CHEE 3363 Spring 2014 Handout 21 Reading: Fox 9.1--9.3 1
Transcript of Student Handout 21 2014
External incompressible viscous
CHEE 3363Spring 2014Handout 21
�Reading: Fox 9.1--9.3
�1
Learning objectives for lecture
1. Use dimensional analysis to obtain the scaling form of the
submerged object.�
2. Apply the momentum integral equation to calculate the
�2
boundary layer
�3
�- Airfoils�- Automobiles�- Airplanes�
�- �- �- �- �
�- �
increasing pressure�
�4
Boundary layer introduction
Classical theoretical hydrodynamics based on Euler’s equation�
- �
-�
�
-important�
-�
�5
�- �
�- Displacement thickness, δ �
- Disturbance thickness, δ �
- Momentum thickness, θ
�6
y ≥ δ�7
�
δ, δ*, θ are all functions of x �8
θ/δδ∗/δ
�9
1Given �Determine
u/U = y/δ
�
2
Assumptions for analyzing boundary layers
�1.�
2.�
3.�
4.
�11
Scaling solution to boundary layerGivenpressure gradient.Determine
Boundary conditions �1. �2.
(assumes dimensionless velocity x; δ natural choice
�12
∂u
∂x+
∂v
∂y= 0
u
@u
@x
+ v
@u
@y
= ⌫
@
2u
@y
2
Approximate solution for boundary-layer 1Given
Determineδ as a
function of x.
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∂
∂t
∫
CV
ρdV +
∫
CS
ρv · dA = 0
∫
CS
ρv · dA = 0 mab + mbc + mcd = 0
mbc = 0
Approximate solution for boundary-layer 2x + dx
(mfs xthrough surface s
�14
mbc = −mab − mcd = −w
[∂
∂x
∫ δ
0
ρu dy
]dx
FSx+ FBx
=
∂
∂t
∫
CV
uρ dV +
∫
CS
uρv · dA
FSx= mfab + mfbc + mfcd
mfab = −
∫ δ
0
uρuw dy
mfx+dx = mfx +
∂mf
∂x
]
x
dx
mfcd = −
∫ δ
0
uρuw dy +
∂
∂x
[∫ δ
0
uρuw dy
]dx
mx+dx = ms +
∂m
∂x
]
x
dx
x is p (neglecting changes in y
Approximate solution for boundary-layer 3x is U
�15
mfbc = Umbc = −U∂
∂x
[∫ δ
0
ρuw dy
]dx
∫
CS
uρv · dA = −
∫ b
0
uρuw dy+
∫ δ
0
uρuw dy+
∂
∂x
[∫ δ
0
uρuw dy
]dx−U
∂
∂x
[∫ δ
0
ρuw dy
]dx
Fab = pwδ
px+dx = p +
dp
dx
]
x
dx Fcd = −
(p +
dp
dx
]
x
dx
)w(δ + dδ)
p +
1
2
dp
dx
]
x
dx Fbc =
(p +
1
2
dp
dx
]
x
dx
)wdδ
Approximate solution for boundary-layer 4
x
�16
Fad = −
(τw +
1
2
dτw
)wdx
FSx= −
[dp
dxδ dx −
1
2
dp
dxdx dδ − τwdx −
1
2
dτw dx
]w
−
[dp
dxδ dx − τwdx
]w =
[∂
∂x
(∫ δ
0
uρu dy
)dx − U
∂
∂x
(∫ δ
0
ρu dy
)dx
]w
Approximate solution for boundary-layer 5
�17
δ =
∫ δ
0
dy
Approximate solution for boundary-layer 6
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momentum integral� equation
parallel to surface
Substitute and divide by U2
�1.
�2. �3. Derive an expression for τw
Givenpressure gradient.Pressure p is constant, U(x U = constantAssume velocity distribution u/U is similar for all values of x, and is a function of y/δ
�19
Assume velocity distribution of form��
1. �
2. �
Change variables to η = y / δ
�
u vanishing at y a
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4
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