Course overviewReview of fluid mechanics course

Spring, 2019

DM23862: Applied fluid mechanics

June Kee Min

1/57

Course overview

2/57Syllabus

Course description:

This course aims to establish some basic methodologies to solve practical fluid flow problems including

both internal and external flows using mathematical and experimental ways. Detail applications of those

methodologies to various engineering fluid flows will be covered for inviscid flows, pipe flows, boundary

layers, flow separation, drag and lift forces exerting on immersed bodies in fluid flows.

Instructor:

Prof. June Kee Min

Office: M914, Tel: 2598, e-mail: jkmin@pusan.ac.kr, Office hour: Friday 10:00-12:00

TA:

Mr. Jun Seok Lee, (Office: 3301, Tel: 7842, e-mail: junseok.lee@pusan.ac.kr)

Lecture hours: Mon/Wed 15:00 -16:15

Lecture room: 3214

Textbook:

Munson, Young and Okiishi’s Fundamentals of Fluid Mechanics, 8-th Edition, Wiley. (유체역학, 윤순현 외,

퍼스트북)

References:

F.M. White, Fluid Mechanics, 7th Edition, McGraw Hill.

G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 2000.

Course materials:

Downloadable through http://cmfd.pusan.ac.kr Lecture-Graduate-Fluid Mechanics

Grade policy:

Homework (20%), Midterm (30%), Final (40%), Attendance (10%)

3/57Schedule

Week Date(MM/DD) Topics

1 03/04, 03/06 Introduction, Review of (Ch.1-5, Ch.6.1-3, 8-9 )

2 03/11, 03/13 Ch.6.4-7 Potential flow

3 03/18, 03/20 Ch.7.1-4 Dimensional analysis HW#1

4 03/25, 03/27 Ch.7.5-8 Modeling and similitude

5 04/01, 04/03 Ch.7.9-11 Dimensionless governing equations

6 04/08, 04/10 Ch.8.1-2 General characteristics of pipe flow HW#2

7 04/15, 04/17 Mid-term exam

8 04/22, 04/24 Ch.8.3-4 Turbulent pipe flow Ch. 6-7+

9 04/29, 05/01 Ch.8.5-6 Minor loss, Pipe flow examples

10 (05/06), 05/08 Ch.9.1General characteristics of external flow HW#3

11 05/13, (05/15) Ch.9.2 Boundary layer

12 05/20, 05/22 Ch.9.3 Drag

13 05/27, 05/29 Ch.9.4 Lift

14 06/03, 06/05Brief introduction to open channel flow, compressible flow,

and turbo machinery (Ch. 10, 11 and 12)HW#4

15 06/10, 06/12 Buffer

16 06/17, 06/19 Final exam Ch. 8-9+

4/57Governing equations of fluid flow (Ch.6)

• Approaches to analyze the fluid motion

Seeking an estimate of gross effects (mass flow, induced force, energy change) over finite

region or control volume control volume approach.

Seeking a point-by-point details of flow pattern by analyzing an infinitesimal region of the flow

differential approach

The differential approach yields the basic differential equations (governing equations) of fluid

motion. analytic solutions for idealized flow, computational fluid dynamics (CFD)

2 2 2

2 2 2

2 2 2

2 2 2

2 2

2

0

x

y

z

u v w

x y z

u u u u p u u uu v w g

t x y z x x y z

v v v v p v v vu v w g

t x y z y x y z

w w w w p wu v w g

t x y z z x

2

2 2

w w

y z

Mass conservation Momentum conservation

Governing differential equations

Potential flow Viscous flow (CFD)

5/57Dimensional analysis (Ch.7)

Similitude and model scaling

• Dimensional analysis and similitude

Experimental and/or analysis data can be best presented in dimensionless form. Various forms

of data can be reduced to a single set of curve when suitably nondimensionalized.

dimensional analysis.

Engineering test models are usually smaller than the final design. Scale models allow testing of

a design prior to building, and in many cases are a critical step in the development process. A

model is said to have similitude with the real application if the two share geometric similarity,

kinematic similarity and dynamic similarity

Dimensionless group in fluid mechanics

6/57Internal and external flows (Ch.8 and 9)

• Internal flow

• External flow

All the boundaries of the flow are surrounded by solid walls except for the inlet and outlet.

Viscous boundary layer grows from the wall and merge in the core of the pipe

ex) Flow in pipe and/or duct

The flow is partially confined by the solid wall

The viscous boundary layer is free to grow

ex) Flow around airplane/marine and land vehicle, open channel flow

<External flow><Internal flow>

7/57Laminar and turbulent flows in a pipe (Ch.8)

• Reynolds number

• Laminar and turbulent flow

Dimensionless number representing the ratio between the viscous and inertial forces

The most important parameter in fluid mechanics

R ea vg

D

V D

R e 2 3 0 0D

Laminar regime: Perturbation decays out, smooth streamline (stable state)

Turbulent regime: Perturbation grows, random flow characteristic (unstable state)

R e 2 3 0 0D

R e 2 3 0 0D

<da Vinci’s schetch of turbulent flow>

Transition Reynolds number

<Moody diagram – head loss in pipes>

8/57Topics in external flows (Ch.9)

• Boundary layer on a flat plate

• Drag • Lift

9/57

Review of Fluid-Mechanics course(Ch.1-5)

10/57

Ch1. Introduction

11/57Difference between solid and fluid

• Solid• Can resist an applied shear by deforming.• Stress is proportional to strain.

• Fluid• Deforms continuously under applied shear.• Stress is proportional to strain rate.

(a) Solid (b) Fluid

F

A

F V

A h

12/57Liquid and gas: in detail

• Liquid• A state of matter in which the molecules are relatively free to change their

positions with respect to each other but restricted by cohesive forces so as to maintain a relatively fixed volume.

• Gas• A state of matter in which the molecules are practically unrestricted by cohesive

forces. A gas has neither definite shape nor volume.

retain a definite volume in a container form a free surface in gravitational field strong intermolecular cohesive force

volume of gas=volume of container Gas cannot form a free surface, and thus gas flows rarely concerned with

gravitational effects other than buoyancy. negligible cohesive forces A gas is free to expand until it encounters confining walls. A gas has no definite volume and left to itself without confinement

(cf) PlasmaHeating a gas may ionize its molecules or atoms, thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions. (The fourth state of matter)

13/57Continuum

• Real nature of matter• All substances are composed of an extremely large number of molecules.• Molecules interact with each other via collision and intermolecular forces. • Spacing between molecules (at normal pressure and temperature)

- (Liquid) ~10-7 mm, (Gas) ~10-6

It is not yet practical to study the behavior of individual molecules when trying to describe the behavior of fluids.

• Continuum• Characterize the behavior of fluid by considering the average, or macroscopic, value

of the quantity of interest (density, pressure, velocity etc.), where the average is evaluated over a small volume containing a large number of molecules.

Continuouslyvarying density

distribution

Calculated density vs. elemental volume

14/57Density and related properties

• Density•Mass per unit volume

•The primary property used to determine if the continuum assumption is appropriate.•At standard atmospheric conditions (sea-level, patm = 101.3 kPa, 15℃)

3

0

lim [ k g /m ]V

m

V

3 31 .2 2 5 / , 9 9 8 .9 5 /

a ir a irk g m k g m

• Specific volume•Reciprocal of the density

31[ m /k g ]v

• Specific weight•Weight per unit volume

3[ N /m ]g

• Specific Gravity

• the ratio of the density of a substance to that of water

@ 4 C

[ ]

ow a te r

S G

1 3 .5 5H g

S G

3 3

1 3 .5 5 1 0 0 0

1 3 .6 1 0 k g /m

H g

31 0 0 0 .0 k g /m

15/57Ideal gas law

• Compressible gas•Gases are highly compressible in comparison to liquids, with changes in gas density directly

related to changes in pressure and temperature.

• Ideal gas• Fluid whose molecules have a mutual effect arising solely from perfectly elastic collision (i.e.,

neglecting intermolecular forces).

• Ideal gas law (equation of state)

p

R T

wherep : absolute pressure (cf: gauge pressure) [Pa=N/m2]T : absolute temperatureR : gas constant

Atmosphericpressure

VacuumuR

RM

Universal gas constant(=8.314 kJ/kg-mol K)

Molar mass

• Close approximation of the behavior of real gases under normal conditions when the gases are not approaching liquefaction.

16/57Viscosity

• The “fluidity” of the fluidcf) Density: the “heaviness” of the fluid

Fluid

Fluid “sticks” to the solid boundary NO-SLIP condition

d u

d y

u(y)

A

/d u d y

: Shear stress [Pa] or [N/m2]

: (absolute or dynamic) viscosity [Pa∙s] or [N∙s/m2]

: rate of shearing strain (vel. gradient) [1/s]

0

lim

ta n

t t

U t

b

U d u d u

b d y d y

Ex) For fluid between parallel plates

17/57No-slip condition

• A fluid in direct contact with a solid ”sticks” to the surface due to viscous effects

• Responsible for• Wall shear stress τw• Surface drag D= ∫ τw dA

• The development of the boundary layer

f lu id so lid

V V

water

turbulent

laminar

Boundary layer

Boundary condition for the fluid flow

18/57Newtonian and Non-Newtonian fluids

• Newtonian fluid: Shear stress is directly proportional to the velocity gradient: Most common fluids (ex: water, air, oil…)

• Non-Newtonian fluid: Viscosity is dependent on the shear rate (not constant): Shear thinning, shear thickening and Bingham plastic

d u

d y c o n s ta n t

( / )f d u d y d u

d y

Newtonian fluid Non-Newtonian fluid

Apparent viscosity

Yield stress

(ex: latex paint)

(ex: water-corn starch mixture)

(ex: tooth paste)

Bingham plastic

19/57Surface tension

• Definition

• A force developed at the interface between a liquid and a second liquid or gas.

• Results from attractive force between molecules.

• Surface tension coefficient, σ [N/m]

22 R p R

• Force balance of a liquid drop

2p

R

• Force balance of a bubble

22 ( 2 )R p R

4p

R

2d L

2d L

1d L

1d L

• In general

1 2

1 1p

R R

20/57Contact angle

• Appears when liquid interface intersects with a solid surface.

Non-wetting

(hydrophobic)

Wetting

(hydrophilic)

• The wettability of a solid surface by a liquid.

21/57Capillary rise

• Effect of capillary action in small tubes

22 c o sg R h R

• Force balance in the tube

2 c o sh

g R

22/57

Ch2. Fluid statics

23/57Pascal’s law

•Pascal’s law• Pressure acts uniformly in all directions on a small volume (point) of a fluid.• In a fluid confined by solid boundaries, pressure acts perpendicular to the boundary

– it is a normal force.

Furnace duct Pipe or tube

Coaxial pipe

Dam

Pressure is a Normal Force(acts perpendicular to surfaces)It is also called a Surface Force.

24/57Pressure variation in a fluid at rest

• For a fluid at rest, a = 0

thenˆ

ˆ ˆ ˆ ˆ

p g

p p pg

x x x

k

i j k k

0 0p p p

gx y z

• p is function of z only!

( )d p

gd z

For liquids or gases at rest, the pressure gradient in the vertical directionat any point in a fluid depends only on the specific weight of the fluid at that point.

25/57Pressure variation : incompressible fluid (1)

2 2

1 1

p z

p z

d p g d z

• For incompressible fluid, ρ = constant, then the integration yields

2 1 2 1( )p p g z z

2 1 1 2( )p p g z z

1 2p g h p

: Hydrostatic pressure distribution

1 2p p

hg

: Pressure head [L] = [m](pressure difference in terms of fluid height)

1 2

3

1 2

6 9 k P a

6 9 1 07 .0 4 m

1 0 0 0 9 .8 1

p p

p ph

g

26/57Measurement of pressure (1)

• Absolute pressure: The pressure of a fluid is expressed relative to that of vacuum (=0)

• Gage pressure: Pressure expressed as the difference between the pressure of the fluid and

that of the surrounding atmosphere.

a b s a tm g a g ep p p

27/57Measurement of pressure (2)

• Mercury barometer

5

7 6 0 m m H g

=

(1 3 6 0 0 )(9 .8 1)(0 .7 6 0 )

1 .0 1 3 1 0 P a = 1 0 1 .3 k P a

a tm

H g

p

g h

Torricelli’s experiment (1644)

• Atmospheric pressure (barometric pressure)

28/57Manometry

•U-Tube manometer

2 3

1 1 2 2

2 2 1 1

A a tm

A

p p

p g h p g h

p g h g h

0 (gage pressure)

•For gas

2 2 1 1 2( 1 )

A Ap g h g h a n d p p

•Pressure difference between A and B

2 3

1 1 3 3 2 2

3 3 2 2 1 1

A B

A B

p p

p g h p g h g h

p p g h g h g h

Fluid A can be different.(gas or liquid)

•Pressure at A

1 1g

3 3g

2 2g

2 2g

29/57Magnitude of the hydrostatic force

•Suppose a submerged plane surface is inclined at an angle θ to the free surface of a liquid.

0s inp p g y

s iny

C a v ep p

R CF p A

s inR

A A A

F d F g h d A g y d A

CA

y d A y A

•Using the definition of centroid, yC

•Resultant force, FR

( s in )R C C

F g y A g h A

The resultant force on one side of any plane submergedsurface in a uniform fluid is- equal to the pressure at the centroid of the surface

times the area of the surface,- independent of the shape of the plane or the angle θ

30/57Location of center of pressure force

•Taking the moment of FR about O

2

( s in ) ( s in )

( )

R RA

C RA

C RA

F y y d F

g y A y y g y d A

y A y y d A

•Using the moment of inertia

2 2

x O x C CA

y d A I I A y

2

2

( )

s in

C R x C C

x C x C

R C R C

C C

y A y I A y

I Iy y o r h h

y A h A

•Finally

The depth of the center of pressure depends on- the shape of the surface and the angle of inclination- and is always below the depth of the centroid.

•Similarly x

R

C

y C

C

Ix x

Ay

(Parallel axis theorem)

31/57Buoyancy

• Buoyant force• A body immersed in a fluid experiences a vertical buoyant force equal to the weight

of the fluid it displaces. (Archimedes’ principle)• A floating body displaces its own weight in the fluid in which it floats.

1 2V a irF W W W

W e ig h t o f th e f lu id c o lu m n

a b o v e th e s u rfa c e

VF

( 2 ) (1)

(f lu id w e ig h t a b o v e 2 ) - ( f lu id w e ig h t a b o v e 1 )

= w e ig h t o f f lu id e q u iv a le n t to b o d y v o lu m e

B V VF F F

2 1

2 1

( )

( )

= ( b o d y v o lu m e )

B H

b o d y

H

b o d y

F p p d A

g z z d A

g

Upward force

32/57Archimedes’ principle

• An immersed body is buoyed up by a force equal to the weight of the fluid it displaces.

• If the buoyant force on an object is greater than the force of gravity acting on the object, the object will float.

Apparent weight = Weight of object - Weight of displaced fluid

Density of object

Density of fluid

Weight of object

Weight of displaced fluid=

Weight of object

Weight of object – Apparent weight =

Archimedes (287 BC – 212 BC)

33/57Linear motion (1)

• Rigid-body uniform motion

• An open container of liquid translating along a straight path with constant acceleration a (and ax = 0).

( )y z

p pa g a

y z

• The change in pressure between (x, y, z) and (x, y+dy, z+dz)

( )y z

p pd p d y d z a d y g a d z

y z

• Along a line of constant pressure, dp = 0.

y

z

ad z

d y g a

: The free surface will be inclined. : All lines of constant pressure will be parallel to the free surface.

0y

a • If

34/57Linear motion (2)

• The free surface will be horizontal• But the pressure distribution will be governed by

0 , 0y z

a a • If

( )z

d pg a

d z

35/57Rigid-body rotation (1)

• Liquid in a tank rotating with constant angular velocity ω

ˆz

p g e a

• In Cylindrical polar coordinate

1ˆ ˆ ˆ

r z

p p pp

r r z

e e e

2ˆ ˆ ˆ0 0

r zr

a e e e

20

p p pr g

r z

• Differential pressure

2p pd p d r d z r d r g d z

r z

Centripetal acceleration

36/57Rigid-body rotation (2)

• On a horizontal plane (dz = 0)

20

d pr

d r

: The pressure increases in the radial direction.

• Along the constant-pressure surface (p = constant, dp = 0)2

2

0r d r g d z

d z r

d r g

2 2

c o n s ta n t2

rz

g

• Integrating

2

2 2

c o n s ta n t2

d p r d r g d z

rp g z

: at fixed r, the pressure varies hydrostatically.

37/57

Ch3. The Bernoulli equation

38/57Summary

21

2p V g z c o n s t

• Assumptions

along a streamline (Bernoulli equation)

• Newton’s second law (F = ma)

2V

p d n g z c o n s tR

across the streamline

• Inviscid fluid• Incompressible fluid• Steady state• Gravity and pressure only

39/57Static, dynamic, and total pressure

Static pressure

21c o n s t (a lo n g a s tre a m lin e )

2T

p V g z p

Hydrostatic pressure

Dynamic pressure Total pressure

• Static pressure

• The actual thermodynamic pressure of the fluid as it flows.

• Hydrostatic pressure

• The change in pressure possible due to potential energy variations of the fluid as a result of elevation changes.

• Dynamic pressure

• The conversion of kinetic energy in a flowing fluid into a "pressure rise" as the fluid is brought to rest.

• Total pressure

• The Bernoulli equation: c o n s t (a lo n g a s tre a m lin e )T

p

40/57Various forms of Bernoulli equation

• Energy per unit volume

Pressure head

Velocity head

Elevation head

2 2

1 1 1 2 2 2

1 1

2 2p V g z p V g z Dimension: [N·m/m3= N/m2 = Pa]

• Energy per unit mass

2 2

1 1 2 2

1 2

2 2

p V p Vg z g z

Dimension: [N·m/(N·s2/m) = m2/s2]

• Energy per unit weight (head)

2 2

1 1 2 2

1 2

2 2

p V p Vz z

g g g g Dimension: [N·m/N = m]

41/57Pitot tube

• Calculation of flow speed

• Dynamic pressure = Stagnation pressure - Static pressure

• Principle of Pitot static tube

2

2 3

1

2p p p V

• Stagnation pressure at (2) and (3)

21

2s ta g n a tio n s ta tic

V p p

• Static pressure at (1) and (4)

Concentrictubes

Pressure gage

Pressure gage

0z

1 4p p p

Assuming negligible elevation change

• Combining

measured

measured

2

3 4

1

2p p V

• Finally

3 42 ( ) /V p p

“Measures velocity using pressure difference”

42/57

Ch4. Fluid kinematics

43/57Lagrangian & Eulerian descriptions (1)

• Lagrangian method (particle concept)

• Following individual fluid particles as they move.• The fluid particles are tagged or identified.• Determining how the fluid properties associated with these particles change

as a function of time.

• Eulerian method (field concept)

• The fluid motion is given by completely prescribing the necessary properties as a functions of space and time.

• Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past those points.

• Preferred in fluid mechanics

( )( )

A

A A

d tt

d t

rV V

ˆ ˆ ˆ( , , , ) ( , , , ) ( , , , ) ( , , , )x y z t u x y z t v x y z t w x y z t V V i j k

44/57Lagrangian & Eulerian descriptions (2)

Eulerian methodLagrangian method

45/57Streamline (1)

• Streamline

• A line to which, at each instant, the velocity vectors are tangent

• Consider an arc length

ˆ ˆ ˆd d x d y d z r i j k

• dr must be parallel to the local velocity vector

ˆ ˆ ˆu v w V i j k

• Therefore,

d r d x d y d z

V u v w (equation for streamline)

46/57Acceleration field

• Derivation of acceleration (Eulerian description)

ˆ ˆ ˆ( , , , ) ( ( ) , ( ) , ( ) , )x y z t x t y t z t t u v w V V V i j k• Velocity field

• Acceleration field d d x d y d zu v w

d t x d t y d t x d t t t x y x

V V V V V V V V Va

( )d

d t t

V Va V Vor

in scalar form

x

y

z

u u u ua u v w

t x y

v v v va u v w

t x

z

zy

w w w wa u v w

t x y z

Local (unsteady) acceleration

Convective (advective) acceleration

47/57Material derivative

• Material derivativeD

D t(= Total derivative, substantial derivative, Eulerian derivative)

• Transformation between Lagrangian and Eulerian frames

( )

D d

D t d t t

V

local (unsteady) term convective term

• Advective acceleration is nonlinear: source of many phenomenon and primary challenge in solving fluid flow problems.

( )d

d t t

V Va V V

48/57Control volume and system representations

• System

• A collection of matter of fixed identity (always the same atoms or fluid particles), which may move, flow, and interact with its surroundings.

• In mechanics courses.• Dealing with an easily identifiable rigid body.

• Control volume

• A volume in space (geometric entity, independent of mass) through which fluid may flow.

• In fluid mechanics course.• Difficult to focus attention on a fixed identifiable quantity of mass.• Dealing with the flow of fluids.

49/57Reynolds Transport Theorem (RTT)

ˆs y s

c v c s

D Bb d V b d A

D t t

V n

• Time rate of change of the property B of the system(B: mass, momentum, energy or angular momentum etc)

• The time rate of change of B within the control volume

• The net flux of B out of the control volume through the control surface

50/57

Ch5. Finite control volume analysis

51/57Conservation of mass

• Continuity equation

ˆs y s

c v c s

D Bb d V b d A

D t t

V n• RTT

• Let , 1B M b

• Thenˆ 0

s y s

c v c s

D Md V d A

D t t

V n

ˆ 0c v c s

d V d At

V n

or

• Finally, the continuity equation becomes

0o u t in

c v

d V m mt

52/57Fixed, non-deforming control volumes

• For steady flow 0o u t in

c v

d V m mt

0o u t in

m m

• For steady and incompressible 0o u t in

Q Q

• For unsteady0

o u t inc v

d V m mt

“+” : the mass inside the control volume is increasing“-” : the mass inside the control volume is decreasing.

1 2m m

1 1 1 2 2 2V A V A

53/57Linear momentum equation (1)

• Newton’s second law

Time rate of change of thelinear momentum of the system

Sum of external forcesacting on the system

=

s y s S B F F F

( ) ( )M s y s te m V s y s te m

D D Dd m d V

D t D t D t

PV V

( )

ˆV s y s te m C V C S

Dd V d V d A

D t t

V V V V n

• Reynolds transport theorem

Surface force(ex. Pressure,Shear stress)

Body force(ex. Gravity)

Net rate of flow of linear momentumthrough the control surface

Time rate of change of the linear momentum of the content of thecoincident control volume

( )B a n d b P V

( )d d mm m

d t d t

V VF a

54/57Linear momentum equation (2)

• When a control volume is coincident with a system at an instant of time

c o n te n ts o f th e c o in c id e n t c o n tro l v o lu m es y s F F

• For a fixed and non-deforming control volume

C o n te n ts o f th e c o in c id e n t

c o n tro l v o lu m e

ˆC V C S

d V d At

V V V n F

Linear momentum equation(Newton’s second law)

55/57Application of Linear momentum equation (1)

• Steady flow with uniform velocity distribution

C Vˆ

C V C S

d V d At

V V V n F

1

ˆ( )

N

i i i i i

i

A

F V V n

Where N = number of flow exit/entrance areas

• At entrance

• At exit

1 1 1ˆ V V n

2 2 2ˆ V V n

• Continuity1 1 1 2 2 2V A V A m

1 1 1 1 2 2 2 2

1 1 1 2 1

2 1

( )

( )

A V A V

A V

m

F V V

V V

V V

56/57Convenient forms of energy equation

• Energy per unit weight (Head) [m]

2 2

1 1 2 2

1 2

2 2P T L

p V p Vz h z h h

g g g g

: p u m p h e a d , : tu rb in e h e a d , : h e a d lo s s

S u b s c r ip ts 1 : in le t , 2 : o u tle t

P T Lh h h

• Energy per unit volume [N/m2 = Pa]

2 2

1 1 1 2 2 2

1 1( )

2 2P T

p V g z w p V g z w lo s s

where

where[J / k g ] : p u m p w o rk p e r u n it m a s s

[J / k g ] : tu rb in e w o rk p e r u n it m a s s

P

T

w

w

• Energy per unit mass [m2/s2 = J/kg]

2 2

1 1 2 2

1 2

2 2P T

p V p Vg z w g z w lo s s

57/57

Thank you!