ENGN 311 Fluid Mechanics Winter 2018 Final Exam Equation...
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ENGN 311 Fluid Mechanics Winter 2018 Final Exam Equation Sheet Fluid Statics Pressure Fluid Statics Equation: - - → ∇ p + ρg =0 Incompressible Fluid, Gravity Opposing z-Axis: p - p o = -ρg (z - z o ) Submerged Surfaces Resultant Force: F R = γh c A Line of Action: y P = I xx,c y c A + y c Buoyancy: F B = γ f V i
Transcript of ENGN 311 Fluid Mechanics Winter 2018 Final Exam Equation...
ENGN 311
Fluid Mechanics
Winter 2018
Final Exam Equation Sheet
Fluid Statics
Pressure
Fluid Statics Equation: −−→∇p+ ρ~g = 0
Incompressible Fluid, Gravity Opposing z-Axis: p− po = −ρg(z − zo)
Submerged Surfaces
Resultant Force: FR = γhcA
Line of Action: yP =Ixx,c
ycA+ yc
Buoyancy: FB = γfV
i
Fluid Dynamics
Fluid Kinematics
Material Derivate:D
Dt=
∂
∂t+ (
−→V · −→∇)
Reynolds Transport Theorem:DB
Dt=
∂
∂t
∫
cvρb dV +
∫
csρb(−→V · n̂
)dA
Fluid Deformation and Rotation
Volumetric Dilatation Rate:1
V
dV
dt=
−→∇ · −→V
Vorticity: ~ζ =−→∇ ×−→
V
Linear Strain Rates
εxx =∂u
∂x
εyy =∂v
∂y
εzz =∂w
∂z
Shear Strain Rates
εxy =1
2
(∂u
∂y+∂v
∂x
)
εyz =1
2
(∂v
∂z+∂w
∂y
)
εzx =1
2
(∂w
∂x+∂u
∂z
)
Along a Streamline
Bernoulli’s Equation:p
γ+V 2
2g+ z = constant
Finite Control Volume
Continuity: 0 =∂
∂t
∫
cvρ dV +
∫
csρ(−→V · n̂
)dA
Linear Momentum:∑
~F =∂
∂t
∫
cvρ−→V dV +
∫
csρ−→V(−→V · n̂
)dA
Energy Equation:Pin
γ+ αin
V 2in
2g+ zin + hpump =
Pout
γ+ αout
V 2out
2g+ zout + hturbine + hL
ii
General Pipe Flow
Major Losses: hL = fL
D
V 2
2g
Minor Losses: hL = KL
V 2
2g
Hydraulic Diameter: Dh =4Ac
p
Laminar, Fully-Developed, Steady, Incompressible Pipe Flow
Volumetric Flow Rate: V̇ =∆PπD4
128µL
Elevation Change: ∆P is replaced by ∆P − ρgL sin θ
Drag and Lift
Drag Coefficient: CD =FD
12ρV 2A
Lift Coefficient: CL =FL
12ρV 2A
Differential Analysis
Continuity:∂ρ
∂t+−→∇ · (ρ−→V ) = 0
Stream Function: u =∂ψ
∂y, v = −∂ψ
∂x, ur =
1
r
∂ψ
∂θ, uθ = −∂ψ
∂r
Incompressible Navier-Stokes: ρD−→V
Dt= ρ~g −−→∇P + µ∇2−→V
Shear Stress for xy/yx: τxy = τyx = µ
(∂u
∂y+∂v
∂x
)
Shear Stress for yz/zy: τyz = τzy = µ
(∂v
∂z+∂w
∂y
)
Shear Stress for zx/xz: τzx = τxz = µ
(∂w
∂x+∂u
∂z
)
Boundary Layer Analysis
Displacement Thickness: δ∗ =∫ ∞
0
(1 − u
U
)dy
Momentum Thickness: θ =∫ ∞
0
u
U
(1 − u
U
)dy
Skin Friction Coefficient: Cf,x =τw
12ρU2
iii
103
104
105
106
107
108
10−2
10−1
8
9
1.2
1.4
1.6
1.8
2
2.5
3
3.5
4
4.5
5
5.5
6
7
8
9
6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8
1e−005
2e−005
5e−005
0.0001
0.0002
0.0004
0.0006
0.00080.001
0.0015
0.002
0.003
0.004
0.006
0.008
0.01
0.0125
0.015
0.01750.02
0.025
0.03
0.035
0.040.0450.05
0.06
0.07
Laminarflow
Criticalzone
Transition zone Complete turbulence, rough pipes, R > 3500/r, 1/√f = 1.14 − 2 log r→ →← ←
Dar
cy−
Wei
sbac
h fr
ictio
n fa
ctor
fÍ ÄÄÄÄ
2hD
g
LV 2
Moody Diagram
r = 5e−006
r = 1e−006
Smooth pipes, r = 01/√f = 2 log(R √f ) − 0.8
MaterialÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
Riveted steelConcreteWood staveCast ironGalvanized ironAsphalted cast ironCommercial steelDrawn tubing
Reynolds number R Í ÄÄ (V in fps, D in ft, ν in ft 2/s)VDν
Rel
ativ
e ro
ughn
ess
rÍ Ä
(ε
in ft
, D in
ft)
ε D
Hagen−Poisseuille equationR ≤ 2300, f = 64/R
Colebrook equation, R ≥ 23001/√f = −2 log(r /3.7 + 2.51/(R √f ))
Acceleration at sea levellatitude 45°, g = 32.1740 ft/s2
VD for water at 60°F (V in fps, D in inches)0.1|
0.2|
0.4|
0.6|
0.8|
1|
2|
4|
6|
8|
10|
20|
40|
60| |
100|
200|
400|
600| |
1000|
2000|
4000|
6000| |
10000|____________________________________________________________________________________________________________________________________________________________________________________________
VD for atmospheric air at 60°F2|
4|
6|
8|
10|
20|
40|
60| |
100|
200|
400|
600| |
1000|
2000|
4000|
6000| |
10000|
20000|
40000|
60000| |
100000|
ε (ft)ÄÄÄÄÄÄÄÄÄÄ
0.003−0.030.001−0.010.0006−0.0030.000850.00050.00040.000150.000005
Fluid at 60°FÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
WaterAir (14.70 psia)
ν (ft2/s)ÄÄÄÄÄÄÄÄÄÄ
1.217e−0050.0001583
iv
v
vi
