Momentum Integral Equation

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Momentum Integral Equation
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Von Karman and Poulhausen derived momentum
integral equation (approximation) which can be
used for both laminar and turbulent flow (with
and without pressure gradient)
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Von Karman and Polhausen devised a simplified
method by satisfying only the boundary conditions
of the boundary layer flow rather than
satisfying Prandtls differential equations for
each and every particle within the boundary layer.
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Want to solve for ? in Laminar Flow

• assume velocity profile,
• u/U f(y/??), similar profiles
• u ? Ue at y ?
• ?u/?y ? 0 at y ?
• u 0 at y 0
• ?w ??u/?y ?U/?d(u/U)/d?

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LAMINAR FLOW
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(plate is 2 thick, RexL 10,000 air bubbles
in water)
For flat plate with dp/dx 0, dU/dx 0
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• Realize (like Blasius) that u/U
• similar for all x when plotted
• as a function of y/?.
• Substitutions ? y/? so dy ?d?
• 0 when y0 ?1 when y ?

Not f(x)
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Strategy assume velocity profile u/Uo f(?),
obtain an expression for ?w as a function of ?,
and solve for ?
f (?)
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Laminar Flow Over a Flat Plate, dp/dx 0
Want to know ?w(x)
Assume velocity profile u a by cy2 B.C. at
y 0 u 0 so a 0 at y
? u U so U b? c?2 at y ?
?u/?y 0 b 2c? so b -2c? U
-2c?2 c?2 -c?2 so c -U/?2 b 2U/?
u a by cy2 0 2Uy/? Uy2/?2
u/U 2? -?2
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Laminar Flow Over a Flat Plate, dp/dx 0
Want to know ?w(x)
Assume velocity profile u a by cy2 u a
by cy2 0 2Uy/? Uy2/?2 u/U 2(y/?)
(y/?)2 Let y/? ? u/U 2? -?2
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Laminar Flow Over a Flat Plate, dp/dx 0
u/U 2? -?2
Strategy obtain an expression for ?w as a
function of ?, and solve for ?(x)
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?w 2?U/? u/U 2? -?2
2? - 4?2 2?3 - ?2 2?3 - ?4
Strategy obtain an expression for ?w as a
function of ?, and solve for ?(x)
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2?U/(??U2) (d?/dx) (?2 (5/3)?3 ?4
(1/5)?5)01 2?U/(??U2) (d?/dx) (1 5/3 1
1/5) (d?/dx) (2/15) 15?dx ??U(d?)
Assuming ? 0 at x 0, then c 0
?2/2 15?x/(?U)
Strategy obtain an expression for ?w as a
function of ?, and solve for ?(x)
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?2/2 15?x/(?U) ?2/x2 30?/(?Ux) 30
Rex ?/x 5.48 (Rex)-1/2 Exact Solution ?/x
5(Rex)-1/2
Can also calculate drag on plate by integrating
over ?w since know ?w 2?U/? Since know ? and
u(x,y) can also calculate ?.
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TURBULENT FLOW
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BREATH
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Want to solve for ? in Laminar Flow

• u/Uo (y/?)1/n (from pipe)
• u/Uo ?1/n similar profiles
• 2. ?w ??u/?y BLOWS UP at y 0
• ?w 0.0332? (V)2?/(RV)1/4

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Calculating drag on a flat plate, zero pressure
gradient turbulent flow u/Uo (y/?)1/8

Cant use ?wall ?du/dy? y0
?u/?y blows up at y 0
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Calculating drag on a flat plate, zero pressure
gradient turbulent flow
u/Uo (y/?)1/8 Uo Uc/l R ?
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?w 0.0332 ? V2 ?/(RV)1/4
PIPE
TO USE FOR FLAT PLATE need to Uavg to Uc/l
and R to ?
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?w 0.0332 ? V2 ?/(RV)1/4 V 0.837 Uc/l R
? ?w 0.0243 ? Uc/l2 ?/(Uc/l ?)1/4
u/Uo (y/?)1/8 ?1/8
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u/Uo (y/?)1/8 ?1/8
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?w 0.0243 ? Uo2 ?/(Uo ?)1/4 ?w (8/90) ?
Uo2d?/dx ?1/4d? 0.274(?/U)1/4dx (4/5)?5/4
0.274 (?/U)1/4x c
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Turbulent Flow
Assume tripped at leading edge so turbulent flow
everywhere on plate
(4/5)?5/4 0.274 (?/U)1/4x c
Assume ? 0 at x 0, so c 0

• (5/4) 0.274 (?/U)1/4x4/5
• 0.424(?/U)1/5x4/5

? 0.424 Rex-1/5x
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? 0.424 Rex-1/5x
?w 0.0243 ? U2 (?/(U?))1/4 ?w 0.0243 ? U2
(?/(U 0.424 Rex-1/5x ))1/4
?w 0.0301 ? U2 Rex-1/5
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Re increases, n increases, wall shear stress
increases, boundary layer increases, viscous
sublayer decrease
u/U
u/U (y / ?)1/6 u/U (y / ?)1/7 u/U (y / ?)1/8
y/?
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LAMINAR BOUNDARY LAYER AT SEPARATION
Given u/U a b? c?2 d ?3 What are
boundary conditions?
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Given u/U a b? c?2 d ?3
? y/?
Separating ?u/?y 0

• 0 u 0 a 0
• 0 ?u/?y 0 b 0
• ? u U 1 c d
• ? ?u/?y 0 2c 3d 0
• 2(1-d) 3d 2 d 0
• so d -2 and c 3

u/U 3 ?2 -2 ?3
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Separating Flow u/U 3 ?2 -2 ?3
dp/dx gt 0
dp/dx 0
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QUESTIONS
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Which way is flow moving?
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• ? In laminar flow along a plate, ?(x), ?(x),
  ?(x) and?w(x)
• Continually decreases
• Continually increases
• Stays the same
• ? In turbulent flow along a plate, ?(x), ?(x),
  ?(x) and?w(x)
• Continually decreases
• Continually increases
• Stays the same
• ?At transition from laminar to turbulent flow,
  ?(x), ?(x), ?(x)
• and?w(x)
• Abruptly decreases
• Abruptly increases
• Stays the same

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61 ellipsoid
?wall
Turbulent Laminar
forced
natural
Turbulent Laminar
?
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What is wrong with this figure?
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What is wrong with this figure?
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80 of fresh water found in world Antarctic ice
Are Antarctic Icebergs Towable Arctic News Record
Summer 1984 36
Cf 0.074/Re1/5 (FoxCf 0.0594/Re1/5) Area
1 km long x 0.5 km wide ?sea water 1030 kg/m3
?sea water 1.5 x 10-6 m2/sec Power available
10 kW Maximum speed ?
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P UD 104 U Cf½ ? U2A U 0.074
?1/5/(U1/5L1/5)( ½ ? U2A) 104 U14/5
0.074(1.5x10-6)1/5/10001/5 x ½
(1030)(1000)(500) U14/5 0.03054 U 0.288
m/s Rex 3x108