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Introduction
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Introduction to hydraulics
2
Overview
• Introduction • Pressure • Viscosity • Visualising fluid flow • Real and ideal fluids • Laminar and turbulent flow • Boundary layers • Flow classification • Re-cap
3
Introduction definition of a fluid
• Distinction between solid and fluid? ◦ solid - can resist an applied shear (may deform) - deformation disappears once force removed
(assuming elastic limit not reached) ◦ fluid - deforms continuously under applied shear - deformation permanent
4
• A fluid is a substance in gaseous or liquid form
• Gas ◦ expands until it encounters container walls ◦ cannot form free surface ◦ readily compressible
• Liquid ◦ takes shape of container ◦ forms a free surface in
the presence of gravity ◦ difficult to compress
liquid gas
Introduction definition of a fluid
5
• We are not really interested in gases in this module
• When we talk about fluids, you can take it to mean that we are talking about liquids
• For the vast majority of civil engineering problems, the liquid we deal with is water
• Note that velocity is represented by the symbols u, U, v, V ◦ lower case usually indicates local velocity ◦ upper case usually indicates mean velocity
Introduction definition of a fluid
6
• Mass ◦ amount of matter in a body (kg)
• Weight ◦ force of gravity on a mass (kgm/s2 or N)
• Density ◦ ratio of mass to volume (kg/m3)
• Specific weight ◦ ratio of weight to volume (kg/m2s2 or N/m3)
• Relative density ◦ ratio of fluid density to density of water
Introduction common properties
7
Pressure definition
• Pressure is defined as force per unit area ◦ what is the pressure exerted by a square box
of dimensions 0.5m2 and mass 100kg? ◦ remember that: force = mass x acceleration - on earth, acceleration is due to gravity which
is 9.81m/s2
� � 2N/m196250100819
AF
u
.
.P
8
Pressure hydrostatic pressure
• Pressure in a stationary fluid (hydrostatic pressure) equal in any direction at a given depth
• Hydrostatic pressure acts perpendicular to any surface and is equal in all directions ◦ otherwise shear forces would exist � water would move
• Hydrostatic pressure varies linearly with depth
ghP U
h
9
Pressure force on a plane horizontal surface
• The force acting on a plane horizontal surface due to hydrostatic pressure is:
i.e. pressure multiplied by
area pressure acts upon
• This force acts at the centre of pressure ◦ on a horizontal surface,
this coincides with the centroid of the surface
ghAPAFP U � AF
h
10
Pressure force on a plane vertical surface
• The force acting on a plane vertical surface due to hydrostatic pressure is:
i.e. mean pressure multiplied
by area over which pressure acts
◦ on a vertical surface, the mean pressure equates to the pressure at centroid of the surface
AhhgAPF mean ¸¹·
¨©§ �
2
21U
h1
h2
11
• The force again acts at the centre of pressure ◦ but on a vertical surface this does not coincide
with the centroid of the surface ◦ centre of pressure
is the centroid of the “pressure intensity” diagram
• Text books give centroid data for commonly occurring geometries
Pressure force on a plane vertical surface
12
Pressure force on a plane surface: centroids
D
2Dhc
D
2Dhc
R
S34Rhc
D
3Dhc
13
Pressure force on a plane surface general orientation
• Generally, the hydrostatic force on a plane surface of any orientation is given by: pressure at centroid u area of surface
• Force acts at centre of pressure (centroid of “pressure intensity” diagram)
• Centre of pressure always below centroid of surface
14
Pressure units
• Variety of different units: ◦ atmospheric pressure:
1.013 bar 101.3 kN/m2
101.3 kPa 0.76 mHg 10.2 mH2O
15
Pressure example 1
• The lock below is installed in a section of canal. If the lock gate is 3m wide, determine: a. hydrostatic force on each side of gate b. where forces act c. magnitude and point of action of resultant
hydrostatic force on gate
3.5m
2.0m
16
Pressure solution 1
a. hydrostatic force given by:
3.5m
2.0m
� �
� � kN5930.220.281.91000
kN18035.325.381.91000
2
1
u¸¹·
¨©§uu �
u¸¹·
¨©§uu �
F
F
AghF cU
F2 F1
17
Pressure solution 1
b. as both pressure intensity diagrams are triangular:
m67.030.22
m17.135.3
1
y
y
59kN 180kN
y1 y2
18
Pressure solution 1
c. magnitude of resultant force given by:
taking moments about O gives
kN1215918021 � � FFFr
� � � �m41.1
67.0517.11801210 �
u�u u� ¦r
rO
yyM
59kN 180kN
1.17m 0.67m
Fr yr
O
19
Viscosity definition
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress
Viscosity is "thickness" or "internal friction"
Water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity
The less viscous the fluid is, the greater its ease of movement (fluidity)
20
Viscosity definition
• A property that represents the internal resistance of a fluid to motion ◦ n viscosity � p deformation under stress
• Arises due to the variations in velocity between different layers of a fluid ◦ these variations generate shear stresses
• The force a flowing fluid exerts on a body in the flow direction is called the drag force ◦ magnitude of drag force depends partly on
viscosity
21
Viscosity definition
• Consider a fluid between two horizontal plates
• If lower plate is fixed and upper plate can move horizontally ◦ shear stress acting on the fluid in contact with
upper plate is the force divided by the area
a b
c d F AF
W
22
• The fluid in contact with the upper plate will travel at the same velocity as the plate
• The fluid in contact with the stationary lower plate will be stationary
� the fluid will deform (abcd o abc*d*)
• Also means that there is a variation in fluid velocity with depth
Viscosity definition
a b
c d c* d* V F
23
F
Viscosity definition
• The force (F) required to move the upper plate is related to the plate area (A), plate velocity (V) and distance between plates (d) thus:
• This implies that: nV � nF pd � nF
a b
c d c* d* V
d
dVAF v
24
Viscosity definition
• We could rewrite this expression by introducing a coefficient of viscosity (P) to represent this proportional relationship:
• Recalling that:
dV
AF
dVAF
dVAF PP � �v
AF
W
dV
dV
AF PWP � �
25
F
• Taking an infinitely thin filament of fluid at height y above the lower plate
� �� �
dydu
dV
dV
dydu
dydu
ydyyuduu
PW
PW
�
����
:As
filamentacrossgradientvelocity
Viscosity definition
c d c* d* V
u + du u d
y
dy
26
Viscosity definition
• If we draw a graph of W against du/dy, the gradient of the line is the coefficient of viscosity
• Newtonian fluid ◦ varies with temperature ◦ constant with deformation ◦ straight line
• Non-Newtonian fluid ◦ varies with temperature
and deformation ◦ curved line
W
dydu
water, air
polymers, toothpaste, mayo
27
Viscosity definition
• P is the coefficient of dynamic viscosity ◦ called coefficient of absolute viscosity ◦ units of Ns/m2 or kg/ms (1N = 1kgm/s2)
• Another measure is coefficient of kinematic viscosity (X), defined as:
◦ units of m2/s (kg/ms y kg/m3)
UPX
28
Viscosity typical values
Put these fluids in order of increasing viscosity
air fresh water
blood
peanut butter
motor oil
29
Viscosity typical values
Fluid P (kg/ms) air 0.0000017
fresh water 0.001
blood 0.01
motor oil 0.05-2.0
peanut butter 150-250
Note: 1kg/ms = 1Pas = 1Ns/m2 = 10poise
30
Viscosity example 2
• Determine the force needed to maintain a relative velocity of 2m/s between two plates 1.2m2 separated by a 1.5mm thick film of lubricant with a dynamic viscosity of 1.55x10-3kg/ms
31
Viscosity example 2
� � N48.20015.021055.12.1 3 uu �
� �
�F
dVAF
dV
AF
dV PPPW
32
• A 26mm wide vertical gap between two plates is filled with a liquid of viscosity 1.49kg/ms.
A 2mm thick plate (0.75mx1.2m) is pulled vertically through the gap at a speed of 0.15m/s.
Determine the force required to overcome the viscous resistance provided the plate is in the centre of the gap.
Viscosity example 3
26mm
2mm
33
• We can isolate each side of the system and treat separately:
The total force required is thus 33.5N (i.e. 2 u 16.75N)
Viscosity solution 3
� � N75.16
012.015.049.12.175.0 uuu �
� �
F
dVAF
dV
AF
dV PPPW
12mm 12mm 2mm
34
• A cylinder 100mm diameter and 750mm long is contained within a vertical tube 103mm internal diameter.
The space between the cylinder and the tube is filled with a lubricant with a kinematic viscosity of 4.5 x 10-4m2/s and relative density of 0.92.
If the cylinder has a mass of 3.06kg, determine its terminal velocity when it slides down the tube ignoring all forces except gravity and viscous friction.
Viscosity example 4
35
• To determine the terminal velocity under the action of gravity:
Viscosity solution 4
� �
� �
m/s461041402360105130
kg/ms141010541000920m236075010 area surface wetted
m10512
100010302
D thickness film
N30819063
3
4
2
3tube
...
.V
......DLA
...Dd
..maFAFdV
dV
AF
dV
cylinder
uuu
�
uuu
uu
u �
�
u
� �
�
�
�
UQP
SS
PPPW
36
Recap
• Fluids: we are concerned with water
• Pressure ◦ hydrostatic force on plane surface is pressure at centroid u area of surface ◦ force acts at centroid of “pressure intensity”
diagram
• Viscosity ◦ represents internal resistance of a fluid to motion ◦ kinematic viscosity and dynamic viscosity used
to represent the viscosity of a fluid
37
Today
• Visualising fluid flow • Real and ideal fluids • Laminar and turbulent flow • Boundary layers • Flow classification • Re-cap
38
• Two important concepts ◦ pathlines - represent the “paths” followed by individual
fluid particles ◦ streamlines - represent the “paths” that fluid particles starting
from the same point will travel - area bounded by number of
streamlines called a streamtube (no flow across boundary)
Visualising fluid flow
39
• A streamline is one that drawn is tangential to the velocity vector at every point in the flow at a given instant and forms a powerful tool in understanding flows.
Visualising fluid flow
40
Visualising fluid flow
• Laminar flow ◦ pathlines ( ) { streamlines ( ) ◦ streamlines only truly valid
for laminar flows
• Turbulent flow ◦ pathlines { streamlines ◦ meaningless to draw
pathlines for all particles ◦ streamlines used to represent
general flow patterns
laminar
turbulent
X
41
Visualising fluid flow
• Streamlines ◦ cannot cross (cannot have 2 velocity vectors) ◦ one at free surface and solid boundary ◦ parallel { constant v, constant p ◦ converging { n v, p p (static to kinetic energy) ◦ diverging { p v, n p (kinetic to static energy)
e.g. flow through a bridge
42
Real and ideal fluids
• Ideal fluids ◦ inviscid ◦ incompressible ◦ no surface tension effects
• Real fluids ◦ viscous ◦ compressible ◦ surface tension effects
• Ideal fluids do not actually exist, but are sometime used to simplify complex problems
43
Laminar and turbulent flow
• At low velocities, in straight “smooth” pipes, flow can be: ◦ highly ordered ◦ have smooth streamlines
• As flow velocities increase, the effect of small disturbances (pipe wall roughness, vibration, etc) can lead to less uniform flow
• Reynolds (1884) injected dye into a flow of water and observed 3 different flow paths at different velocities
44
Laminar and turbulent flow Reynolds experiment
• Laminar (low vel.) ◦ smooth dye flow
• Transitional (medium vel.) ◦ wavy dye flow
• Turbulent (high vel.) ◦ random dye flow ◦ dye mixes with water
45
Laminar and turbulent flow flow classification
• Laminar flow ◦ fluid flows in discrete layers ◦ no mixing
• Transitional ◦ “bursts” of turbulence over laminar flow
• Turbulent ◦ random fluid motion (velocity and direction)
• Most flows we are concerned with are turbulent
46
Laminar and turbulent flow Reynolds number
• Onset of turbulence found to be related to: ◦ velocity ◦ viscosity ◦ some representative dimension (l)
• Leads to expression for Reynolds Number
• In a pipe, representative dimension is ??
diameter (D)
XPU VlVlRe
PUVD
�Re
47
Laminar and turbulent flow Reynolds number
• Reynolds number represents ratio of: inertia force to viscous force “get going” forces to “stopping” forces
• Reynolds number used to compare flow types
• For typical pipeline: ◦ laminar flow: Re < 2000 ◦ transitional flow: 2000 < Re < 4000 ◦ turbulent flow: Re > 4000
48
Laminar and turbulent flow Reynolds number
• Reynolds number is dimensionless
� � � �
cancel all which
33
¸¹·
¨©§
¸¹·
¨©§
¸¹·
¨©§
¸¹·
¨©§
¸¹·
¨©§
¸¹·
¨©§
LTM
TLL
LM
mskg
smm
mkg
DVRe PU
49
Laminar and turbulent flow turbulence models
• Turbulence is a very complex phenomenon
• Individual fluid particles vary in direction and velocity randomly � impossible to accurately model (at fine scale)
with generally applicable numerical model � normally use empirical data
50
Boundary layers
• Imagine a flat plate in a uniform flow (velocity U)
• Friction between fluid and plate � velocity of 1st fluid layer (surface) is zero
• Velocity of 2nd layer should be U except for the shearing action between 1st & 2nd layer � p velocity of 2nd layer � shearing action between 2nd and 3rd layer….
U
51
Boundary layers
• This mechanism continues until shearing forces become negligible � original uniform velocity (U)
• Hence, velocity varies from zero (at fluid/plate boundary) to U (some distance from plate)
• Zone over which velocity variation occurs termed Boundary Layer (BL)
U
u = U u = U u = U u = 0.75U u = 0.5 U
52
Boundary layers
• As the fluid passes along plate, more of the flow is affected by the shearing forces setup at the fluid/plate boundary � BL thickness increases
• Variation in layer thickness is not constant
• 3 regions occur
U
53
U
Boundary layers
◦ laminar region - fluid motion maintained by viscous shearing
action between layers - smooth velocity distribution � mathematical function can describe
velocity distribution reasonable accurately
laminar
54
U
Boundary layers
◦ turbulent region - eddies form due to faster moving flow
passing over slower moving laminar region - eddies cause some particles to move
between fast moving flow and laminar region � momentum transfer maintains motion
(highly turbulent process)
laminar
turbulent
55
U
Boundary layers
◦ transitional region - balance between viscous shear and
momentum transfer changing
• Laminar sub-layer (LSL) ◦ there is always a very thin sublayer below the
turbulent region, as fluid velocity is always zero at plate boundary
laminar transitional
turbulent
laminar sublayer
56
Boundary layers bounded flows
• Most civil engineering flows are completely or partially “bounded” e.g. pipes, open channels
• In pipe flow the wall BLs converge at some distance downstream of the flow entry point ◦ entry length is the distance to convergence ◦ fully developed flow downstream of entry length
U fully
developed flow
entry length
57
Boundary layers bounded flows
• Entry lengths typically short e.g. 50 – 100 diameters � normally safe to assume that civil engineering
flows are fully developed
• As velocity varies in the BL � velocity varies across whole pipe diameter
58
Boundary layers relative roughness
• The effect of pipe roughness depends on the physical roughness of the pipe walls relative to the depth of the laminar sub-layer (LSL)
k GL k GL
k GL
Smooth turbulent • roughness
protrusions lie within LSL
• fluid “trapped” in-between protrusions
• smooth flow
Transitional • roughness
protrusions just penetrate LSL
• transitional turbulent flow
Turbulent • roughness
protrusions fully penetrate LSL
• rough, turbulent flow
59
Boundary layers relative roughness
• Hydraulically smooth pipe ◦ exhibits smooth turbulent flow
• Hydraulically rough pipe ◦ exhibits rough turbulent flow
• Hence concept of relative roughness (pipe roughness relative to flow conditions)
k GL k GL
k GL
Smooth turbulent
Transitional Turbulent
60
• BL concept can be used to explain formation of turbulent wakes downstream of an object in a real fluid flow
• Imagine a circular object in a fluid flow
• The streamlines will converge as they pass the object � flow acceleration � BL will also form
Boundary layers flow separation
61
• Velocity will vary within the BL
• As the fluid passes the centreline of the object � streamlines re-converge (flow deceleration)
• As BL velocity is lower than flow � forms wake � energy loss � p pressure
Boundary layers flow separation
high pressure
low pressure
62
Boundary layers drag
• Total drag (profile drag) on an object in a flowing fluid consists of two components ◦ pressure drag (form drag) - due to pressure difference front-back
◦ skin friction drag (viscous drag) - due to object roughness
• Contribution to total drag depends on ◦ object shape ◦ object orientation
high skin friction
drag high pressure
drag
63
Boundary layers drag
• Total drag determined using:
A = cross-sectional area of object presented to flow
V = fluid velocity
U = fluid density
Cdr = drag coefficient (empirically determined) = f(shape, roughness, Re) | 0.5 for sphere, 0.1 for streamlined body
2force Drag
2AVCdrU
64
Boundary layers velocity distribution
• Velocity distribution depends on type of flow
• Based on empirical data
◦ Laminar:
◦ Turbulent: Uf = 0.99U (velocity asymptotic to U) u = velocity at depth y G = BL thickness A, B, n = coefficients
»¼
º«¬
ª¸¹·
¨©§�
f
2
GGyByA
Uu
nyUu
1
¸¹·
¨©§
f G
65
Flow classification temporal and spatial
• Temporal variation ◦ steady flow: conditions do not vary with
time ◦ unsteady flow: conditions do vary with time
• Spatial variation ◦ uniform flow: conditions do not vary with
distance along channel ◦ non-uniform flow: conditions do vary with
distance along channel (gradual or rapid)
66
Flow classification temporal and spatial
• Steady, uniform flow e.g. constant flow through pipe of constant
cross-section
• Steady, non-uniform flow e.g. constant flow through tapering pipe
• Unsteady, uniform flow e.g. “instantaneous” pressure surge in pipe of
constant diameter (impossible!)
• Unsteady, non-uniform flow e.g. flood wave in natural river channel
67
Flow classification dimensions
• Most flows are 3-D (+ 1!) ◦ parameters can vary in three directions (x, y, z) ◦ parameters can vary with time
• Analysis of such flows is very complex, even with today’s computing power
• Normally appropriate to consider flows in 1-D i.e. consider variations in 1 physical dimension
(general direction of flow) and with time ◦ velocity and pressure variations across section
are accounted for elsewhere
68
Recap
• Fluid visualisation ◦ pathlines represent the “paths” followed by
individual fluid particles ◦ streamlines represent the “paths” that fluid
particles starting from the same point will travel
• Ideal fluids ◦ do not actually exist, but are sometimes used to
simplify complex problems
69
Recap
• Flow classification ◦ type (Reynolds number) - laminar flow - transitional - turbulent
◦ temporal/spatial - steady or unsteady - uniform or non-uniform
• Boundary layers ◦ lead to velocity variations and turbulence