a model of a river boat is to be tested at 1:13.5 scale. the boat is designed to travel at 9mph in...
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A model of a river boat is to be tested at 1:13.5 scale. The boat is designed to travel at 9mph in fresh water at 10o C. (1)Estimate the distance from the bow on the full-scaleship where transition occurs. (2) Where should transition be stimulated on model?
What equations are relevant to this problem?
What approach should we take?
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A model of a river boat is to be tested at 1:13.5 scale. The boat is designed to travel at 9mph in fresh water at 10o C.Estimate the distance from the bow where transition occurs.Where should transition be stimulated on model?
Things we know: Re = UL/ ReL transitions ~ 5 x 105 (experimentally determined for flat plate and dP/dx = 0)
Want same fraction of L to be turbulent for both cases: [xtr/L]model = [xtr/L]boat so ration of xtr = ratio of boat lengths
= 13.5
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?
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Three unknowns, A, B, and C- will need three boundary conditions. What are they?
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(*/ = 0.344)
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0.344
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y /
u / U
Sinusoidal, parabolic, cubic look similar to Blasius solution.
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u/Ue = 2(y/) – (y/)2
u/Umax = (y/)1/7
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R
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R
(interesting)
u = (-R2/4)(dp/dx)(1-[r/R])2 eq. 8.12
wall = (R/2)(dp/dx) Umax = (-R2/4)(dp/dx)
du/dr = -2Umax/Rdu/dy = 2Umax/R
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*
*
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1 m
U = 2.7 m/s
cd
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(5) Two-Dimensional(6) Incompressible assumed assumptions
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cd
u(x,y)/U = y/ = (y)
1
0
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cd
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FDrag
cd
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d c
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U1 = Uo = 80 ft/sec1= 0.8 in.
U2 = ?P2 – P1 = ?2= 1.2 in.
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Continuity U1A1eff = U2A2eff
A1eff = (h -2 *1 ) (h -2 *1 ) A2eff = (h -2 *2 ) (h -2 *2 )
*1
*1
* = 0.8 in *1
u/U = y/ =
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Continuity U1A1eff = U2A2eff
A1eff = (h -2 *1 ) (h -2 *1 ) A2eff = (h -2 *2 ) (h -2 *2 )
*2
*2
*2 = 1.2 in *2
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First a bit of mathematic legerdemain before attempting Ex. 9.3 ~
= ?
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L = 1.8 m
U = 3.2 m/s
= ?; * = ?; w = ?
b = 0.9m
Laminar and assume u/U = sin(/2); = y/
*/ = 1o (1-u/U) d
/ = 1o u/U(1-u/U) d
w = U2 d/dx = u/y0
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w = u/y0 = (U/)d(u/U)/d(y/)0
w = (U/)dsin(/2)/d0
w = (U/)(/2)cos(/2)0
w = (U/)(/2) = U/(2)
u/U = sin(/2); = y/
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*/ = 1o (1-u/U) d
u/U = sin(/2); = y/
*/ = 1o (1 – sin(/2) )d
*/ = + (2/)cos (/2)from 1 to 0
*/ = 1 - 2/ = 0.363* = 0.363
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/ = 1o u/U(1-u/U) d
= 0.137
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Finding */ = 1o (1-u/U) d
/ = 1o u/U(1-u/U) d
w = U2 d/dx
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w = U2 d/dxw = u/y0
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…..
u/U = sin(/2); = y/
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u/U = sin(/2); = y/
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Fdrag = 0L wbdx = 0
L U2(d/dx)bdx = 0
L U2bd = U2b0L
Fdrag = U2bL
(whew!)
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Calculating drag on a flat plate, zero pressure gradient – turbulent flow
*
*u/y blows up at y = 0
Can’t usewall = du/dy y=0*
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Turbulent Flow
Tripped at leading edge so turbulent flow everywhere on plate
= 0.424 Rex-1/5x
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w = 0.0243 U2 (/(U))1/4
w = 0.0243 U2 (/(U 0.424 Rex-1/5x ))1/4
= 0.424 Rex-1/5x
w = 0.0301 U2 Rex-1/5
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0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
u/U = (y / )1/6
u/U = (y / )1/7
u/U = (y / )1/8
u/U
y/
Re increases, n increases, wall shear stress increases, boundary layer increases, viscous sublayer decrease
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LAMINAR BOUNDARY LAYER AT SEPARATION
Given: u/U = a + b + c2 + d 3
What are boundary conditions?
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Given: u/U = a + b + c2 + d 3
= y/
= 0; u = 0; a = 0= 0; u/y = 0; b = 0= ; u =U; 1 = c + d= ; u/y = 0; 2c + 3d = 02(1-d) + 3d = 2 + d = 0 so d = -2 and c = 3
Separatingu/y = 0
u/U = 3 2 -2 3
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dp/dx = 0
Separating Flowu/U = 3 2 -2 3
dp/dx > 0
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U
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If Re transition occurs at 500,000 evaluate ReL, xtr, and power to overcome skin friction drag on fin
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ReL = UL/ = 1.54 x 107
Calculating transition location, x:Assume xtr occurs at 500,000
Rex = Ux/ = 500,000Rex/ReL = xtr/L = 500,000 / 1.54 x 107
xtr = 0.0325 L = 0.0325 x 1.65 = 0.0536 m
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FLAT PLATE dp/dx = 0
LAMINAR:From Theory
CD = 1.33 / ReL1/2
Eq. 9.33
TURBULENT:From Experiment
5x105< ReL<107
xtransition = 0CD = 0.0742/ReL
1/5
ReL<109
xtransition = 0CD = 0.455/(logReL)2.58
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For Re transition of 500,000 and 5x105 < ReL < 109
CD = 0.455/(log ReL)2.58 – 1610/ReL (Eq. 9.37b) CD = 0.00281 – 0.000105 = 0.0027
~ 4% less
FD = CD2LH(1/2)U2 = 91.6 N2 sides so 2 x Area
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Smooth flat plate, dp/dx = 0; flow parallel to plate
CD ~ 0.0028
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Reh = Uh/, don’t know UAssume: turbulent flow, transition at leading edge,
doesn’t flutter, 5 x 105 < ReL < 107
Then CD = 0.0742/ReL1/5
Plastic plate falling in water, find terminal velocity -
Sum of forces = 0Drag Force = f (U)
DRAGBOUYANCY
WEIGHT
+ z
- z
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Net Force = 0 = FB + FD - W
FB – W = (plastic - water) g (hLt)FD = CD ½ U2A = [0.0742/ReL
1/5] ½ water U2AReh = Uh/
Solving for U get 11.0 ft /sec
Check Re number: 1.86 x 106
5 x 105 < ReL < 107
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mg = 120 kgU = 6 m/s
FD = CD ½ U2A
Sum of forces = 0,pick A so U = 6 m/s
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mg = 120 kgU = 6 m/s
FD = CD ½ U2A
Sum of forces = 0,pick A so U = 6 m/s
CD = 1.42 for open hemisphereFD = CD ½ U2A = W = mg
A = d2/4d2 = [mg]/[CD U2 /8]
d = [(8/)(120kg)(9.8 m/s2) (1/1.42)(1/1.23 kg/m3)(1/6m/s)2
d = 6.9 m
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FD = CD ½ U2A
FD = ma = m(dU/dt) = m(dU/dx)(dx/dt)
CD ½ U2A = m (dU/dx) U
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CD ½ U2A = m (dU/dx) UdU/U = CD A dx /(2m)
Integrating from x=0
to x=x gives:ln Uf – lnUi = CD Ax /(2m)CD = 2m ln {Uf/Ui} /[ Ax]
CD = 0.299
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Find POWER = F U for this conditionAnd then see if that is enough to win bet.
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air
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8.33 m/s
2.78 m/s
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FD = CD ½ U2AA = 23.4 ft2
CD = 0.5FR = 0.015 x 4500 lbf
FD = FR to find U where aerodynamic force = frictional force
Total Power = FDU + FRU
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FD [CD ½ U2A] = FR[.015xW](0.5)½ 0.00238slug/ft3U2(23.4ft2)
= 0.015 x 4500 lbf
U2 = 67.5/0.0139
U = 69.7 ft/s = 47.5 mph
Where does aerodynamic force = frictional force ?
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Average of 44 sport cars is 0 43, not much better than the 0.47 of ’37 the Lincoln Zephyr
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(245 km/hr)