Fluid Dynamics
 The study of the physical processes governing fluid
motion is called fluid dynamics.
 Some hydrologic problems requiring knowledge in F.D:
 Flow of water in streams
 Flow of water in subsurface aquifers
 Infiltration of precipitation into soils
 Design of flood control measures
 Transport of groundwater contaminants
Fluid Dynamics
 The basic principle used in F.D. is Newton’s second law of
motion:
F = m ·a
 Forces that affect movement of fluids are
 Gravity
 Pressure difference
 Surface stress
 Fluid: “Substance that continuously deforms when subjected to
a shear stress”. Gas & liquids…
 Shear Stress: tangential force per unit area acting on a surface
Fluid Dynamics
 Viscosity: Property of fluid describing its resistance to motion
(The opposite is called fluidity but not commonly used)
 Shear stress: t = F / A
du
t m
dz
 M 
 LT 2 
 m : dynamic viscosity
(varies with 0T)
 u : kinematic viscosity
=m/r
 m for water at 20 0C = 1x10-3 N·sec/m2 (Pa·sec)
 u = 1x10-6 m2/s
Fluid Dynamics
Relative Values of Properties of Pure Liquid Water as f (T0)
Fluid Statics
 When fluid is not moving  No shear stress
 Hydrostatic equation:
dP
  rg  g
dz
r : density of the fluid
g : gravity
g : specific weight
(+)
z
dP
g
dz
(+)
z
 Integrating with depth gives: P2  P1  g z 2  z1   P  g z
 Note that we assumed incompressibility, i.e. the density does
not change with depth : dr /dz=0
 Even in the deep oceans at 4 km depth, where pressures are 40
MPa, there is only a 1.8% decrease in volume
Horizontal Force on a Wall
P( z )  g z
w
P ( z  h)  g h
z
h
h
F   P( z ) wdz
0
h
F  w (g z )dz 
0
gwh2
2
Manometer
 Manometer is a device used in measuring fluid pressures
 Working principle is balance of hydrostatic pressures
PB  PA  g Hg h
Air
PC  PB  PA  g Hg h
D
A
H 2O
l
h
C
B
Hg
PC  PD  g H 2O l
PD  PC  g H 2O l
PD  PA  g Hg h  g H 2O l
Fluid Dynamics
 Again the basic principle is F = m∙a
F: Sum of all forces causing fluid motion
a: Rate of change of fluid velocity with time
 Fluid does not move as a body with a single velocity. It is a
collection of many small particles
 The velocity can vary in space and time
 Local acceleration: Change in velocity with time at a single
point
 Convective acceleration: Change in velocity from point to
point along the flow path at a single time
 Total acceleration: Sum of local and convective acceleration
Example
 Suppose you leave Chicago at 7 am on a summer day and head
south. Suppose temperature increases at a rate of 2 0C/hr during
morning. There is also warming trend in southerly direction at
a rate of 0.8 0C per 100 km. If your speed is 75 km/hr what is
the total rate of change in temperature with time?
 Total rate of change in T:
= 2 0C/hr + 0.8 0C/100km*75 km/hr
= 2.6 0C/hr
 Local acceleration:
aL  T t
 Convective acceleration:
ac  uT y 
 Total acceleration:
aT  T t  uT y 
More on Acceleration
 In the previous example T was varying with location in the N-S
direction (y) and with time (t), i.e. T = f (y,t)
 Local acceleration:
 Convective acceleration:
 Total acceleration:
aL  T t
ac  uT y 
aT  T t  uT y 
 For flow: ax  u t  uu x 
where u: velocity in –x direction
 u t  0 : Steady flow - velocity does not change with time
(opposite is unsteady flow)
 u x  0 :Uniform flow - velocity does not vary from point to
point (opposite is non-uniform flow)
Bernoulli Equation:
Newton’s 3rd law: When one body exerts a force on a second body, the second body simultaneously
exerts a force equal in magnitude and opposite in direction to that of the first body.
F  m a
P1 A  P2 A  Fgs  r  A  ds  a
P2  P1 
P
ds
s
 dz 
Fgs  Fg  sin   rgAds  
 ds 
P 
dz

ds  A  rgAds  rAds  a
 P1  P1 
s 
ds

Bernoulli Equation

P
dz
ds  A  rgAds  rAds  a
s
ds
a
du du ds
du

u
dt ds dt
ds
 Dividing both sides by (rAds) and assuming P=f(s)
u
du
dz 1 dP
g 
0
ds
ds r ds
 
1 d u2
du
 Recall that
u
2 ds
ds
d u
P 

  0
z
ds  2 g
rg 
2
 H = constant, called total head
2
u
P
z H
2g
g
Bernoulli Equation
2
u
P
z H
2g
g
Velocity head:
u2 / 2g
[L]
Elevation head:
z
[L]
Pressure head:
P/g
[L]
 Note that all the terms have the units of length
Conservation of Energy
 Multiplying Bernoulli Equation with weight (mg) gives
mu 2
 mgz  PV  cst.
2
First term: Kinetic Energy
Second term: Potential Energy
Third term: Flow work (work due to pressure)
 Therefore, Bernoulli Equation is a form of conservation of
energy
 NOTE: Only valid along path of integration (-s axis), a line
everywhere parallel to flow field “streamline”
Example
 At (1) and (2)
u12
P1 u22
P2
 z1  
 z2 
2g
g 2g
g
 P1 = P2 = Patm
u22
u22
z1 
 z2 
 z1  z2  h
2g
2g
u  2 gh
(z1 kept constant)
 If we put a plug at point (2) to stop flow: (u2=0 and P2 ≠ Patm)
z1 
P1
g
 z2 
P2
g
 P2  P1  gh  P2  P1  gh (hydrostatic eqn.)
Continuity Equation
 For steady flow amount of water in any segment does not
change with time, i.e. inflow = outflow
Q1  Q2  u1 A1  u2 A2
u1

4
D12  u2

4
2
D22
 D1 
 u1
 u 2  
 D2 
 Assume D2 = 2D1, then
u2=u1/4
 Bernoulli Eqn:
u12
P1 u22
P2
 z1  
 z2 
2g
g 2g
g
P2  P1
g
u12  u22
15 2

 P2  P1  ru1
2g
32
Energy Loss
 Height of each fountain is a
measure of internal pressure
 h1 > h2 > h3
u12
P
H
 z1  1  hL1 
2g
g
u22
P2
 z 2   hL 2 
2g
g
 Loss in the total head: Head loss, hL
u32
P3
 z3   hL 3
2g
g
 In reality there is always a friction loss between the fluid and the wall
causing head loss
Friction Loss
 Friction loss is given by Darcy-Weisbach equation
L u2
hL  f
D 2g
where,
f : Friction factor
L : Length
D : Diameter
 Solving for u gives
2g
u
DS f
f
Sf : friction slope (hL/L)
 f is function of Reynolds Number (Re) and boundary roughness
Re 
uD

(Remember  = m / r)
Laminar-Turbulent Flow
 Laminar flow is characterized by layers, or laminas, of fluid moving
at the same speed and in the same direction. No fluid is exchanged
between the laminas
Re < 2000
 In turbulent flow, the streamlines or flow patterns are disorganized
and there is an exchange of fluid between these areas. Momentum is
also exchanged such that slow moving fluid particles speed up and
fast moving particles give up their momentum to the slower moving
particles and slow down themselves
Re > 4000
 Transition zone 2000 < Re < 4000
 Functional form of friction factor differs for turbulent/laminar flows
Moody Diagram
Open Channel Hydraulics
 Any unpressurized flow is
considered open channel
flow.
h
 E.g. Surface water flow in
datum
z0
u2
H  z0  h 
2g
rivers, streams, and canals
u2
E  h
2g
(E= Specific Energy)
 Assume rectangular cross-section
 Specific Discharge: q  Q w  u  h

w: width of channel
q2
E  h
2gh 2
Open Channel Hydraulics
 What is the minimum specific energy if q is fixed?
1
q  3
 hc   
 g 
uc2
 Using q = uh
 hc 
 uc  ghc
g
 When mean velocity is equal to ghc specific energy is min.
2
2
dE
q
 1 3  0
dh
ghc
 Froude Number: Fr  u
Fr = 1
Fr < 1
Fr > 1
gh
In natural streams:
h ~ Area/(top width), i.e. A/T
: critical flow
: sub-critical flow
: super-critical flow
Open Channel Hydraulics
Fr  u
gh
c  gh
c : celerity
(wave speed)
 Critical flows do not
exist naturally (they are
unstable)
 If a flow passes through
Fr = 1, a hydraulic jump
will occur
Discharge Measurements
using Control Structures
 Artificial obstructions like a step or dam over which all the water
in a channel must flow is called “weir”
 Two common types: Sharp crested & broad crested
Broad Crested Weir
 upstream flow is subcritical,
downstream is supercritical
 At some point over the crest
flow must be critical
u02
u2
hweir 
 h0 
2g
2g
 Assuming u is negligible we obtain hweir
Q  u0 A  u0 wh0 
1/ 2
 8 
3
2
 Q  
 hweir
w
 27 g 
u2 3
u0  gh0  hweir 
 h0
2g 2
3
 h0 and u0  2 ghweir
2
3
 Other weir types have similar relationships. Discharge can be computed
easily by measuring hweir (i.e., h1-z0)
Steady-Uniform Flow
u12 2 g
hf
u22 2 g
h1
h2
datum
z1
u12
u22
z1  h1 
 z2  h2 
 hf
2g
2g
z2
 h1 = h2 and u1 = u2  hf = z1 - z2
z1  z2
 Sf 

L
L
hf
 Bed slope: S0  tan   sin  (for small  < 100)
z1  z 2
 S0 
L
 S f  S0
Wall Shear Stress
p : wetted perimeter
L

A : cross sectional area
 Assume prismatic channel, i.e. cross
A
p
W
section does not change in direction
of flow
W = g·A·L
A
W sin   t 0 pL  0  gALS0  t 0 pL  t 0  g S0
p
 t 0  g RS 0
t0: Wall shear stress
R: Hydraulic Radius (A/p)
Channel Flow Equations

L u2
(Darcy-Weisbach) for pipes
hL  f
D 2g
A D 2 / 4 D
R 

 D  4R
p
D
4
hL
f u2
8g
 Sf 
 u
RS f
L
4R 2 g
f
 Chezy’s Eq: u  c RS f
 Manning’s Eq: u 
a
n
with c  8g / f
R2/3 S f
(C=R1/6 /n in Chezy’s eq.)
a = 1 if SI units are used, = 1.49 if English units are used
n: Manning’s roughness (TABLE & FHWA report)
Measuring Flow in Natural Channels
Measuring Flow in Natural Channels
 Discharge is not directly measured, instead flow velocities at
subsections are measured
 First step is selecting cross sections across the total width of stream
 Select a straight reach where the streambed is uniform and relatively free
of boulders and aquatic growth
 flow should be uniform and free of eddies, dead water near banks, and
excessive turbulence
 Determine the width of the stream by stringing a measuring tape from
bank to bank at right angles to the direction of flow
 Next, determine the spacing or width of the verticals. Space the verticals
so that no subsection has more than 10 percent of the total discharge
 If the stream width is less than 5 ft, use vertical spacing widths of 0.5 ft
 If the stream width is greater than 5 ft, the minimum number of verticals is
10 to 25. The preferred number of verticals is 20 to 30
Measuring Flow in Natural Channels
 Depths < 2.5 Feet – Six-tenths method: Only one measurement is
required at each measurement section. Velocity is measured at 60% of
the depth from surface
 Depths >2.5 Feet – Two point method: If the depth is greater than
2.5 feet, two measurements should be taken at 20% and 80% of the
total depth. Mean velocity is the average of two measurements
 Three point method: When velocities in the vertical are abnormally
distributed three velocity should be observed at 0.2, 0.6 and 0.8 of the
depth. The mean velocity is computed by averaging 0.2 and 0.8 depth
observations and then averaging the result with the 0.6 depth
observation
1m
 hz 
 h=depth, m~7
 There is a mathematical basis: u ( z )  u0 
 h  z0 