MECH 221 FLUID MECHANIC
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Transcript MECH 221 FLUID MECHANIC
MECH 221 FLUID MECHANICS
(Fall 06/07)
Chapter 1: INTRODUCTION
Instructor: Professor C. T. HSU
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MECH 221 – Chapter 1
1.1. Historic Background
Until the turn of the century, there were two
main disciplines studying fluids:
• Hydraulics -
engineers utilizing empirical
formulas from experiments for
practical applications.
• Mathematics - Scientists utilizing analytical
methods to solve simple
problems.
(Aero/Hydrodynamics)
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MECH 221 – Chapter 1
1.1. Historic Background
Fluid Mechanics is the modern
science developed mainly by
Prandtl and von Karman to study
fluid motion by matching
experimental data with
theoretical models. Thus,
combining Aero/Hydrodynamics
with Hydraulics.
Prandtl
Indeed, modern research
facilities employ mathematicians,
physicists, engineers and
technicians, who working in
teams to bring together both
view points: experiment and
theory.
Von Karman
(1875-1953)
(1881-1963)
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MECH 221 – Chapter 1
1.1. Historic Background
Some examples of fluid flow phenomena:-
Aerodynamics design : the engagement of
a wing from static state using a suitable
angle of attack will produce a start vortex.
The strength of it is very important for the
airplane to obtain high upwards lift force,
especially in aircraft takeoff on carrier. This
photo shows a model wing suddenly starts
its motion in a wind tunnel.
Waves motion : Much of the propulsive
force of a ship is wasted on the wave action
around it. The distinctive wave patterns
around a ships is the source of this wave
drag. The study of these waves, therefore,
is of practical importance for the efficient
design of ship.
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MECH 221 – Chapter 1
1.1. Historic Background
Hydraulic Jump
A circular hydraulic jump in the kitchen
sink. Hydraulic jump is a fluid
phenomenon important to fluid
engineers. This is one type of
supercritical flow, which is a rapid
change of flow depth due to the
difference in strength of inertial and
gravitational forces
Structure-Fluid interaction
Vortices generated due to motion in
fluid is of great important in structural
design. The relation of a structure’s
natural frequency with the shedding
spectrum affect many fields of
engineering, e.g. building of bridges
and piers. Photo shows the vortex
resembling the wake after a teaspoon
handle when stirring a cup of tea.
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MECH 221 – Chapter 1
1.1. Historic Background
Tidal Bore
Tidal bore is a kind of hydraulic jump,
and can be regarded as a kind of
shockwave in fluid. The knowledge of
its propagation is critical in some river
engineering projects and ship
scheduling. The photo shows the
famous tidal bore in Qiantang River,
China.
Droplets dynamics
Fluid dynamics sometimes is useful in
microelectronic applications. Droplets
dynamics is crucial to the bubblejet
printing and active cooling technology.
Photo shows a drop of water just
hitting a rigid surface, recorded by
high speed photography.
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MECH 221 – Chapter 1
1.2. Fundamental Concepts
The Continuum Assumption
Thermodynamical Properties
Physical Properties
Force & Acceleration (Newton’s Law)
Viscosity
Equation of State
Surface Tension
Vapour Pressure
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MECH 221 – Chapter 1
1.2.1. The Continuum Assumption
Fluids are composed of many finite-size
molecules with finite distance between them.
These molecules are in constant random
motion and collisions
This motion is described by statistical
mechanics (Kinetic Theory)
This approach is acceptable, for the time
being, in almost all practical flows
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MECH 221 – Chapter 1
1.2.1. The Continuum Assumption
Within the continuum assumption there are
no molecules. The fluid is continuous.
Fluid properties as density, velocity etc. are
continuous and differentiable in space & time.
A fluid particle is a volume large enough to
contain a sufficient number of molecules of
the fluid to give an average value for any
property that is continuous in space,
independent of the number of molecules.
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MECH 221 – Chapter 1
1.2.1. The Continuum Assumption
Characteristic scales for standard atmosphere:
- atomic diameter ~ 10-10 m
- distance between molecules ~ 10-8 m
- mean free path, (sea level) ~ 10-7 m
const. 100,000m ; = .000006 m
250,000m ; = 0.0012 m
Knudsen number: Kn = / L
- mean free path
L - characteristic length
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MECH 221 – Chapter 1
1.2.1. The Continuum Assumption
For continuum assumption: Kn << 1
•
Kn < 0.001
• 0.001< Kn < 0.1
- Non-slip fluid flow
- B.C.s: no velocity slip
- No temp. jump
- Classical fluid mechanics
- Slip fluid flow
- Continuum with slip B.C.s
• 0.1< Kn< 10
- Transition flow
• 10<Kn
- Free molecular flow
- No continuum, kinetic gas
- Molecular dynamics
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MECH 221 – Chapter 1
1.2.2. Thermodynamical Properties
Thermodynamics - static situation of equilibrium
n - mean free time
a – speed of molecular motion (~ speed of sound: c)
n = /a –microscopic time scale to equilibrium
Liquid
Gas
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MECH 221 – Chapter 1
1.2.2. Thermodynamical Properties
Convection time scale s = L / U
- L : characteristic length
- U : fluid velocity (macroscopic scale)
Local thermodynamic equilibrium
assumption: n«s
- /a « L/U (/L).(U/a) « 1 Kn.M « 1
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MECH 221 – Chapter 1
1.2.2. Thermodynamical Properties
Mach number: M = U / a
- Incompressible flow: M0, U«a
- Compressible flow:
- Gas dynamics
-
M<1 : subsonic
M~1 : transonic
M>1 : supersonic (1<M<5)
M»1 : hypersonic (5<M<40)
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MECH 221 – Chapter 1
1.2.3. Physical Properties
Example: density at point P
• = density, mass/volume [kg/m3]
• = specific weight [N/m3]
=g
• average density in
a small volume V
= m / V
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MECH 221 – Chapter 1
1.2.3. Physical Properties
P ≠ lim(m/V) as V 0
P = lim(m/V) as V V*
V*~=R.E.V. (representative
elementary volume)
Fluid particle with volume:
V*~=(1 m)3 ~109 particles
Specific gravity, S.G.:
the ratio of a liquid's density to
that of pure water at 4oC (39.2oF)
H2O @ 4oC
= 1000 kg/m3
= 1 g/cm3
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MECH 221 – Chapter 1
1.2.3. Physical Properties
Similarly, other macroscopic physical
properties or physical quantities can be
defined from this microscopic viewpoint
Momentum M,
Velocity u
Acceleration a
Temperature T
Pressure, viscosity, etc…
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MECH 221 – Chapter 1
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MECH 221 – Chapter 1
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MECH 221 – Chapter 1
1.2.4. Force & Acceleration (Newton’s Law)
The force on a body is proportional to the
resulting acceleration
F = ma ; unit: 1N = 1kg . 1m/s2
The force of attraction between two bodies is
proportional to the masses of the bodies
m1m2
F G
r2
m1 : Body1
r = Distance
m2 : Body 2
G = Gravitational Constant
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MECH 221 – Chapter 1
1.2.4. Force & Acceleration (Newton’s Law)
Various kinds of forces
Static pressure
Dynamic pressure
Shear force
Body force (weight)
Surface tension
Coriolis force
Lorentz force, etc…
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MECH 221 – Chapter 1
1.2.4. Force & Acceleration (Newton’s Law)
Newton’s law is a conservation law. It describes the
conservation of linear momentum in a system.
Different kinds of conservation Laws, e.g.
Conservation of mass
Conservation of linear momentum
Conservation of energy, etc…
Continuity equation
Navier-Stokes equations
Energy equation, etc…
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MECH 221 – Chapter 1
1.2.5. Viscosity
The shear stress on an interface tangent to
the direction of flow is proportional to the
strain rate (velocity gradient normal to the
interface)
= µu/y
µ is the (dynamic) viscosity [kg/(m.s)]
Kinematic viscosity: = µ/ [m2/s]
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MECH 221 – Chapter 1
1.2.5. Viscosity
Power law:
= k ( u/ y)m
Newtonian fluid: k = µ, m=1
Non-Newtonian fluid: m1
Bingham plastic fluid:
= 0 +µu/y
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MECH 221 – Chapter 1
1.2.5. Viscosity
No-slip condition
From observation of real fluid, it
is found that it always ‘stick’ to
the solid boundaries containing
them, i.e. the fluid there will not
slip pass the solid surface.
This effect is the result of fluid
viscosity in real fluid, however
small its viscosity may be.
A useful boundary condition for
fluid problem.
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MECH 221 – Chapter 1
1.2.6. Equation of State (Perfect Gas)
Equation of state is a constitutive equation
describing the state of matter
Ideal gas: the molecules of the fluid
have perfectly elastic collisions
Ideal gas law: p = R T
R is universal gas constant
Speed of sound: c=(dp/d)1/2
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MECH 221 – Chapter 1
1.2.7. Surface Tension
At the interface of a liquid and a gas
the molecular attraction between like
molecules (cohesion) exceed the
molecular attraction between unlike
molecules (adhesion). This results in a
tensile force distributed along the
surface, which is the surface tension.
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MECH 221 – Chapter 1
1.2.7. Surface Tension
For a liquid droplet in gas in
equilibrium:
-(∆p)R2 + (2R) = 0
∆p is the inside pressure in
the droplet above that of the
atmosphere
∆p=pi- pe = 2 / R
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MECH 221 – Chapter 1
1.2.7. Surface Tension
For liquids in contact with gas and solid, if the
adhesion of the liquid to the solid exceeds the
cohesion in the liquids, then the liquid will rise
curving upward toward the solid. If the adhesion
to the solid is less than the cohesion in the liquid,
then the liquid will be depressed curving
downward. These effects are called capillary
effects.
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MECH 221 – Chapter 1
1.2.7. Surface Tension
The capillary distance, h, depends for a given liquid and
solid on the curvature measured by the contact angle ,
which in turn depends on the internal diameter.
(2R) cos - g(R2)h = 0
→ h=2 cos/gR
The pressure jump across an interface in general is
p = (1/R1 + 1/R2)
For a free surface described by z=x3=η(x1,x2),
1/Ri= ( 2η/ xi2)/[1+( η/ xi)2]3/2
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MECH 221 – Chapter 1
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MECH 221 – Chapter 1
1.2.8. Vapour Pressure
When the pressure of a liquid falls below the vapor pressure it
evaporates, i.e., changes to a gas. If the pressure drop is due
to temperature effects alone, the process is called boiling. If
the pressure drop is due to fluid velocity, the process is called
cavitation. Cavitation is common in regions of high velocity,
i.e., low p such as on turbine blades and marine propellers.
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MECH 221 – Chapter 1
1.2.8. Vapour Pressure
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