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Fluid Mechanics Review

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Fundamental Concepts

Physic Properties: , , , , , ,

Forces on Fluid:

Internal Forces

External ForcesSurface

Volume

Ideal Fluid/Real Fluid

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Fluid Statics

Basic Differential Equation:

),,(

0

zyxpp

pgradF

=

=

Potential Force:

0:. =+

=

pgradgradEqDiff

gradF

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Fluid Statics

Gravity Force:

gzgF ==

constzp =+

Hydrostatics:

=2

1

12

z

z

gdzpp Aerostatics:

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Fluid Statics

Archimedes Buoyancy:

WgPz =

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Fluid Kinematics

Motion classification:

Viscous Friction Ideal Fluid

Viscous Fluid Laminar

Turbulence Time Stable

Unstable

Dimension 1D, 2D, 3D

Steady/Unsteady

Incompressible/Compressible

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Fluid Kinematics

Two methods for the description of fluid motion

Lagrangian Method

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Eulerian Method

Fluid Kinematics

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Fluid Kinematics

)( ugradut

u

dt

uda

+

==

Acceleration of a Fluid Element

z

z

y

z

x

zzz

z

z

y

y

y

x

yyy

y

zx

yx

xxxx

x

uz

u

uy

u

ux

u

t

u

dt

du

a

uz

uu

y

uu

x

u

t

u

dt

dua

uz

uu

y

uu

x

u

t

u

dt

dua

+

+

+

==

+

+

+

==

+

+

+

==

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Fluid Kinematics

Relation of System Derivatives to theControl Volume Formulation

+

=SCVsystem

dAnut

X

dt

dX

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Fluid Kinematics

0)( =+

udiv

t

01

)(1

0

=

+

+

=

+

+

z

uu

rru

rr

z

u

y

u

x

u

zr

zyx

Continuity Equation

Incompressible ( = const):

Irrotational/Rotational Flow urot

2

1=

1D, steady flow: Q1 = Q2

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Fluid Dynamics

Cu

t

gradu

=+++

=

2

2

Cu

=++2

2

Irrotational, potential flow

Steady flow (integration along a streamline )

Steady flow (integration along a vortex)

Cu

=++2

2

Steady flow (integration along a normal to the streamline)

( )R

u

n

2

=+

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Fluid Dynamics

Cup

gzt

=+++

2

2

Cup

gz =++2

2

Irrotational, potential flow

Steady flow (integration along a streamline )

Steady flow (integration along a vortex)

Steady flow (integration along a normal to the streamline)

R

upgz

n

2

=

+

Cup

gz =++2

2

(Bernoulli Equation)

Gravity, ideal, incompressible fluid:

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Fluid Dynamics

dtududivgradupgradF

=++

)(3

1 2

0=udiv

Equations of Motion for Viscous Fluid(Navier-Stokes Equations)

Incompressible Flows:

2

2

2

2

2

22

zyx

+

+

=

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Fluid Dynamics

dt

QdPP

dt

dE

sm

~

++=

Energy Equation

Int. Formulation

Diff. Formulation

+=

+ dAnqdAndwuFdwu

edt

d

WW

2

2

Gravity, incompressible, steady flow:

fhg

Vpz

g

Vpz +++=++

22

2

22

22

2

11

11

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Fluid Dynamics

Momentum Equation

( )

[ ] =

+

=

outoutoutoutoutout

W

VQVQF

dAnuudwt

u

F

)(

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Dimensional Analysis

[ ] tml

TMLa =

)...,,,(121

=

n

aaafa

( ) 1,...,,, 121 == nsf s

Dimensional/Non-dimensional Quantities

Fundamental Dimensions/ Derivative Dimensions

Buckingham Theorem

k fundamental dimensions

Dimensionless Parameterk

k

iki

aaa

a

...21 21

+=

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Dimensional Analysis

VLVL

ForceViscous

ForceInertia===Re

2

2

1V

ppCa v

=

gL

V

ForceGravity

ForceInertiaFr ==

Significant Dimensionless Group in Fluid Dynamics

Reynolds number:

Cavitation number:

Froude number:

Weber number:

LV

ForceTensionSurface

ForceInertiaWe

2

==

Mach number:c

V

ilityCompressibtodueForce

ForceInertiaM ==

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Similarity

Geometric Similarity

Kinematically Similarity

Dynamically Similarity

Each dimensionless group has the same value

in the model and in the prototype