Fluid Mechanics Wrap Up

1
Fluid Mechanics Wrap Up

• CEE 331
• April 19, 2014

2
Review

• Fluid Properties
• Fluid Statics
• Control Volume Equations
• Navier Stokes
• Dimensional Analysis and Similitude
• Viscous Flow Pipes
• External Flows
• Open Channel Flow

3
Shear Stress
dimension of
Tangential force per unit area
Rate of angular deformation
change in velocity with respect to distance
rate of shear
4
Pressure Variation When the Specific Weight is
Constant
Piezometric head
5
Center of Pressure yp
Sum of the moments
y 0 where p datum pressure
Transfer equation
6
Inclined Surface Findings

• The horizontal center of pressure and the
  horizontal centroid ________ when the surface has
  either a horizontal or vertical axis of symmetry
• The center of pressure is always _______ the
  centroid
• The vertical distance between the centroid and
  the center of pressure _________ as the surface
  is lowered deeper into the liquid
• What do you do if there isnt a free surface?

coincide
below
decreases
7
Forces on Curved Surfaces Horizontal Component

• The horizontal component of pressure force on a
  curved surface is equal to the pressure force
  exerted on a horizontal ________ of the curved
  surface
• The horizontal component of pressure force on a
  closed body is always _____
• The center of pressure is located on the
  projected area using the moment of inertia

projection
zero
8
Forces on Curved Surfaces Vertical Component

• The vertical component of pressure force on a
  curved surface is equal to the weight of liquid
  vertically above the curved surface and extending
  up to the (virtual or real) free surface
• 
  Streeter, et. al

9
Cylindrical Surface Force Check
89.7kN
0.948 m

• All pressure forces pass through point C.
• The pressure force applies no moment about point
  C.
• The resultant must pass through point C.

C
1.083 m
78.5kN
(78.5kN)(1.083m) - (89.7kN)(0.948m) ___
0
10
Uniform Acceleration

• How can we apply our equations to a frame of
  reference that is accelerating at a constant
  rate? _______________________________
  _______________________

Use total acceleration including acceleration due
to gravity.
Free surface is always normal to total
acceleration
11
Conservation of Mass
N Total amount of ____ in the system h ____
per unit mass __
mass
1
mass
cv equation
But dm/dt 0!
mass leaving - mass entering - rate of increase
of mass in cv
12
EGL (or TEL) and HGL

• The energy grade line may never be horizontal or
  slope upward (in direction of flow) unless energy
  is added (______)
• The decrease in total energy represents the head
  loss or energy dissipation per unit weight
• EGL and HGL are ____________and lie at the free
  surface for water at rest (reservoir)
• Whenever the HGL falls below the point in the
  system for which it is plotted, the local
  pressures are lower than the __________________

pump
coincident
reference pressure
13
Losses and Efficiencies

• Electrical power
• Shaft power
• Impeller power
• Fluid power

Motor losses
IE
bearing losses
Tw
pump losses
Tw
gQHp
14
Linear Momentum Equation
Fp2
M2
Fssx
The momentum vectors have the same direction as
the velocity vectors
M1
Fssy
Fp1
W
15
Vector Addition
q2
cs2
cs1
cs3
q1
q3
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Summary

• Control volumes should be drawn so that the
  surfaces are either tangent (no flow) or normal
  (flow) to streamlines.
• In order to solve a problem the flow surfaces
  need to be at locations where all but 1 or 2 of
  the energy terms are known
• The control volume can not change shape over time
• When possible choose a frame of reference so the
  flows are steady

17
Summary

• Control volume equation Required to make the
  switch from a closed to an open system
• Any conservative property can be evaluated using
  the control volume equation
• mass, energy, momentum, concentrations of species
• Many problems require the use of several
  conservation laws to obtain a solution

18
Navier-Stokes Equations
Navier-Stokes Equation
h is vertical (positive up)
Inertial forces N/m3
Pressure gradient (not due to change in elevation)
Shear stress gradient
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Summary

• Navier-Stokes Equations and the Continuity
  Equation describe complex flow including
  turbulence, but are difficult to solve
• The Navier-Stokes Equations can be solved
  analytically for several simple flows

20
Dimensionless parameters

• Reynolds Number
• Froude Number
• Weber Number
• Mach Number
• Pressure Coefficient
• (the dependent variable that we measure
  experimentally)

21
Froude similarity

• Froude number the same in model and prototype
• ________________________
• define length ratio (usually larger than 1)
• velocity ratio
• time ratio
• discharge ratio
• force ratio

difficult to change g
22
Laminar Flow through Circular Tubes
Laminar flow
Shear at the wall
True for Laminar or Turbulent flow
23
Pipe Flow Energy Losses
Dimensional Analysis
Darcy-Weisbach equation
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Laminar Flow Friction Factor
Hagen-Poiseuille
Darcy-Weisbach
-1
Slope of ___ on log-log plot
25
Moody Diagram
0.10
0.08
0.05
0.04
0.06
0.03
0.05
0.02
0.015
0.04
0.01
0.008
friction factor
0.006
0.03
0.004
laminar
0.002
0.02
0.001
0.0008
0.0004
0.0002
0.0001
0.00005
0.01
smooth
1E03
1E04
1E05
1E06
1E07
1E08
R
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Solution Techniques

• find head loss given (D, type of pipe, Q)

• find flow rate given (head, D, L, type of pipe)

• find pipe size given (head, type of pipe,L, Q)

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Minor Losses

• We previously obtained losses through an
  expansion using conservation of energy, momentum,
  and mass
• Most minor losses can not be obtained
  analytically, so they must be measured
• Minor losses are often expressed as a loss
  coefficient, K, times the velocity head.

High Re
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Swamee Jain Iterative Technique for D and Q
(given hl)

• Assume all head loss is major head loss.
• Calculate D or Q using Swamee-Jain equations
• Calculate minor losses
• Find new major losses by subtracting minor losses
  from total head loss

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Darcy Weisbach/Moody Iterative Technique Q (given
hl)

• Assume a value for the friction factor.
• Calculate Q using head loss equations
• Find new friction factor

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Open ConduitsDimensional Analysis

• Geometric parameters
• ___________________
• ___________________
• ___________________
• Write the functional relationship

Hydraulic radius (Rh)
Channel length (l)
Roughness (e)
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Open Channel Flow Formulas
Chezy formula
Manning formula (MKS units!)
T /L1/3
Dimensions of n?
NO!
Is n only a function of roughness?
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Boundary Layer Thickness

• Water flows over a flat plate at 1 m/s. Plot the
  thickness of the boundary layer. How long is the
  laminar region?

x 0.5 m
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Flat PlateStreamlines
3
2
4
0
Point v Cp p 1 2 3 4
1
gtp0
0
1
ltU
gtp0
gt0
ltp0
gtU
lt0
ltp0
Points outside boundary layer!
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Flat Plate Drag Coefficients
35
Drag Coefficient on a Sphere
1000
100
Stokes Law
Drag Coefficient
10
1
0.1
0.1
1
10
102
103
104
105
106
107
Re500000
Reynolds Number
Turbulent Boundary Layer
36
More Fluids?

• Hydraulic Engineering (CEE 332 in 2003)
• Hydrology
• Measurement Techniques
• Model Pipe Networks (computer software)
• Open Channel Flow (computer software)
• Pumps and Turbines
• Design Project
• Pollutant Transport and Transformation (CEE 655)