Presentation - Hydraulics 1 - Introduction to Hydraulics

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

Ronak Kandachia

Ronak Kandachia

Ronak Kandachia

Ronak Kandachia

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


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


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)

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• 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


• 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


a b

c d c* d* V F

23

F


• 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


• 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


c d c* d* V

u + du u d

y

dy

26


• 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

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• 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

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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

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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.


40


• 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


• 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


• 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


• 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

Documents

Presentation - Hydraulics 1 - Introduction to Hydraulics