Title: MECH 3021 – Viscous Flow
1- MECH 3021 Viscous Flow
- (1) Basic concept Shear stress, velocity
gradient, coefficient of viscosity (friction in
fluid) - (2) Shear stress
- (coefficient of viscosity) X (velocity gradient)
- (3) Examples Water less viscous Honey more
viscous
2Compressible Flows
- (1) Fluid Mechanics of Flight Burn fuel to
overcome the drag force, or roughly the shear
stress at the wall / airplane wing. Hence the
dynamics of boundary layer is crucial, i.e. we
need to calculate the velocity profile and its
derivative tw is the shear stress divided by
viscosity
3Compressible Flows
- (2) The nature of the flow, i.e. laminar (slow,
orderly) versus turbulent (fast, chaotic), is
critical to this stress, and hence the study of
stability is important.
4Aerospace
- (1) Conventional commercial aircrafts typically
fly at a Mach number of 0.4 to 0.7. At such
speeds, the fluid starts to become compressible. - (2) The stability of compressible boundary
layers thus deserves serious studies. Given our
limitations, we embark on an elementary
introduction of compressible flows.
5Speed of Sound
- (1) Signals, information and to some extent
energy are transmitted by small disturbances in
fluids. Small disturbances in air are known as
sound (waves). - (2) Speed of sound can be computed from a dynamic
/ thermodynamic consideration.
Animation courtesy of Dr. Dan Russell, Kettering
University
6Reviews of Thermodynamics
- (1) Adiabatic processes (no
heat exchange, temperature will vary) - versus
- Isothermal processes
(temperature constant, permit heat exchange). - (2) First Law dQ dW
dE -
Heat intothe system
work onthe system
increase ininternal energy
7The Momentum Theorems
- (1) Linear and angular momentum theorems
- (2) Control surfaces conceptual (imaginary)
surfaces fixed in space
Credit Cimbala, Cengel (2008), Essentials of
Fluid Mechanics Fundamentals and applications
8- Volume flow rate (area)(velocity)
- Mass flow rate (density)(area)(velocity)
- Momentum flow rate (density)(area)(velocity)2
9Linear Momentum theorem The total external
force rate of change of momentum inside the
control surface the net outflux of momentum
from the surface (outflux influx)
10A Heuristic Explanation
A typical control volume in a funnel-shaped pipe
Fluid contained at t t0
The fluid at t t0 dt
Credit http//www.eng.fsu.edu/dommelen/courses/f
lm/rey_tran/index.html
11A Heuristic Explanation (Contd)
To calculate the change of momentum of the fluid
considered (Fdt, caused by the external force),
the inflow and outflow of the control volume have
to be accounted for
12Supersonic, subsonic and transonic flows
- (1) Flows where the fluid speed exceeds the
LOCAL sound speed are known as SUPERSONIC flows.
- (2) Flows where the fluid speed remains less
than the LOCAL sound speed are known as SUBSONIC
flows. - (3) Flows with both supersonic and subsonic
portions are known as TRANSONIC flows.
13Basic Gas Laws / Processes
- (1) Isothermal compression / expansion
- (2) Adiabatic compression / expansion
-
14Sir Isaac Newton
- (1) Newton invented calculus when he was 23. An
epidemic broke out in the UK about three or four
hundred years ago. He returned to his rural home
and invented calculus in about 18 months. - (2) Inspired by an apple falling from a tree,
Newton discovered or invented gravitational force.
15Sir Isaac Newton (contd)
- (3) Consider the equations of motion in polar
coordinates. If the tangential component of the
force is zero, while the radial component is an
inverse square law in the radius. The solution is
an ellipse, and hence he solved the dynamics of
the solar system The Earth moves around the Sun
in an elliptical path.
16Sir Isaac Newton (contd)
- (4) Newton solved the problem of brachistochrone
minimum time of travel for a falling particle
between two points not on the same vertical
straight line. - (5) BUT, Newton made a mistake in calculating the
speed of sound. He used an isothermal, but not
the adiabatic, condition. The adiabatic condition
is the correct one.
17Speed of Sound
- (1) Speed of sound change of pressure with
respect to density at ADIABATIC conditions. -
- (2)
-
- (3) Sound travels FASTER in water than air (and
still faster in solids). Contrast this feature
with light.
18The Energy Equation
- (1) We first have a reservoir of gas at rest
(or gas in stagnant condition). - (2) The gas is then allowed to flow out from this
reservoir through an attached tube / pipe say by
a pressure differential (higher pressure in the
tank / reservoir). - (3) The velocity / kinetic energy acquired must
come at the expense of the internal energy, and
hence the temperature of the gas MUST DROP as it
flows.
Reservoir
V, ?
P0, T0V0
19Thermodynamic consideration
- (1) Energy of the gas is measured by the
enthalpy, or roughly, - internal energy (pressure)(volume).
- (roughly CVT P V) CV T RT (CV R)T
- (specific heat capacity at constant pressure)
- Multiplied by (temperature) CpT
- (2) Energy conservation
- Enthalpy at reservoir enthalpy in flow
kinetic energy
20Flows through a duct of varying area
- (1) When a fluid flowing in a pipe / tube
encounters a reduction in cross sectional area,
our intuition and quantitative studies in
incompressible flow earlier tell us that the
fluid will speed up. - (2) While this remains broadly true for subsonic
flows, the statement in point (1) may NOT hold
for supersonic flows.
21Compressible flows in ducts / pipes / tubes with
varying cross sectional areas
- (1) The law of conservation of mass still holds,
which means that the product of (density)(cross
sectional area)(velocity) must remain constant.
For incompressible fluids, the density is
constant and hence our intuition is correct (i.e.
area goes down, velocity goes up). - (2) For supersonic flows, the density changes
drastically with velocity and hence the flow may
SLOW down on approaching a reduction in area.
22Analysis of flows through a duct of varying area
- (1) ? V A constant and hence
-
- (2) but from the one dimensional (1D)
equations of steady (!!) motion -
- (3) Speed of sound
23Analysis (continued)
- (1)
- (2) M lt 1, dA lt 0, dV gt 0,
- (3) M gt 1, dA lt 0, dV lt 0 (Counterintuituve!!).
24Convergent Nozzles
Boeing 757 Nozzle
Airbus A330 Nozzle
Extracted from http//www.astechmfg.com/
Extracted fromhttp//en.wikipedia.org/wiki/Image
Turbofan_operation.png
25Converging-Diverging Nozzles
- The nozzle accelerates the flow of a gas from a
subsonic to supersonic speed. - In the initial stage, decreasing flow area
results in subsonic (Mlt1) acceleration of the
gas. - The area decreases until the throat area is
reached, where M1. - Increasing flow area accelerates the flow
supersonically (Mgt1) thereafter.
Cimbala and Cengel (2006), Fluid Mechanics
Fundamentals and Applications
26Converging-Diverging Nozzles
- It is used in rocket engines or other supersonic
applications.
Photos of a NERVA rocket nozzle on display at the
Michigan Space and Science Center (These photos
were taken by Richard Kruse in 2002)http//histor
icspacecraft.com
Space Shuttle Main Engine nozzle
http//www.k-makris.gr/
27Shock Waves
- (1) Generally even the qualitative features of
supersonic flows are completely different from
those of subsonic flows. - (2) Another further distinction is that shock
waves, or surfaces of discontinuities, can occur
in supersonic but NOT subsonic flows.
28Shock Waves (contd)
- (1) Thought experiment A gas in a very long
cylinder is at rest. A piston is pushed into the
gas and the piston is allowed to accelerate. - (2) As the gas is compressed, a sequence of wave
fronts is generated. However, the wave fronts
generated more recently have a higher velocity as
the piston is accelerating.
29Shock Waves (contd)
- (3) As these younger (i.e. generated more
recently) wave fronts travel faster than the
older wave fronts, we eventually have a piling up
of wave fronts, and they form a relatively sharp
discontinuity which is known as a shock wave. - (4) The thickness of a shock wave is of a few
mean free paths, but in practice the shock is
taken as having zero thickness.
30Supersonic Flights
M6
M3.5
Free-flight models of the X-15 being fired into a
wind Tunnel vividly detail the shock-wave
patterns for airflow.
Credit NASA History Divisionhttp//history.nasa.
gov/SP-60/ch-5.html
31The Normal Shock Waves
- (1) A discontinuity of gas properties (in
practice, a thin region of rapid changes)
perpendicular (normal) to the flow direction. - (2) In calculations of textbooks / notes, we go
to a frame where the airplane is at rest, and the
shock is then stationary ahead of the
(supersonic) airplane. In practice, the
(supersonic) airplane is flying and the shock is
rapidly advancing in a region of much slower
moving air (hence the term wave).
slow moving air
supersonic air
advancingsupersonic airplane
at rest
advancing shock
stationary shock
In calculation
In practice
32The Normal Shock Waves
- (3) Principles Conservation of mass, momentum,
energy, AND the equation of state make up FOUR
equations in four unknowns, velocity, pressure,
density and temperature. - (4) Conceptually simple but BEWARE of algebra.
33Analysis of shock waves
34Shock waves (continued)
- (1) Eliminate temperature in the energy equation
by the equation of state - (2) Use this as the equation for p2 and
substitute into the momentum equation. - (3) Eliminate ?2 by the continuity equation to
obtain a relation between u2 and u1.
35Prandtls relation
- u1 u2 (a)2
- The product of upstream and downstream velocities
(relative to the shock) is equal to the square of
the sound speed at the place where the flow is
sonic.
36Rankine Hugoniot Relation
-
- This is the relation between the pressure and
density ratios across the shock.
37Qualitative features of subsonic flows
- (1) It is possible to hear the sound generated
by the aircraft / disturbance from all directions
in a sufficiently small neighborhood of the
aircraft. - (2) A higher frequency is heard in front of the
source and a lower frequency is heard behind the
source. (Doppler Effect)
M 0.7
Animation courtesy of Dr. Dan Russell, Kettering
University
38Qualitative features of a sonic flow
- An observer in front of the source will detect
nothing until the source arrives.
M 1
Animation courtesy of Dr. Dan Russell, Kettering
University
39Qualitative features of supersonic flows
- Sound waves generated will be confined to within
a cone (3D) or triangular region (2D) of half
angle sin1(1/M) where M is the Mach number.
M 1.4
Animation courtesy of Dr. Dan Russell, Kettering
University
40Waves and Stabilities in Fluid Flows
- (1) Stabilities of flows have traditionally been
studied by imposing wavy disturbances. - (2) For sufficiently small amplitude, the
equations of motion are linearized (i.e. linear
stability). - (3) Physically, interactions among waves are
ignored.
41Stability (continued)
- (1) Wave grows instability Wave decays
stability - General disturbance superposition, Fourier
analysis. - (2) Temporal stability for a fixed wave number
or wave length, find the (complex) frequency. - Real frequency neutral or propagating waves.
- Complex frequency growing or decaying waves.
42Stability (continued)
- (1) Finite amplitude effects difficult to
impossible to analyze. Numerical or computational
approaches. - (2) Quasi parallel flows e.g. boundary layer,
treated as a parallel flow in the leading order
approximation.
43Dispersive Waves
- If the (phase) velocity of the wave
- depends on the wave number (or wave
- length), the wave is termed dispersive.
- Implications a group of waves will disperse
- or disintegrates into components in the far
- field.
44Dispersive Waves Animation
- Initial profile is a Gaussian pulse
- Black Non-dispersive wave (maintain a constant
shape) - Blue dispersive wave (disperse or
broaden)
Animation courtesy of Dr. Dan Russell, Kettering
University
45Dispersive Waves
- (1) The speed / velocity of a wave depends on the
wavelength or frequency Dispersive waves. - (2) Examples of NON-dispersive waves
electromagnetic waves in vacuum. - (3) Examples of dispersive waves almost all
wave motions in fluids and solids (except sound
waves).
46Basic terminology on wave motion
- (1) Meaning of wavelength, frequency, period,
velocity (frequency X wavelength) taken as well
known. - (2) Wave number number of waves per 2 pi length
(or k 2 pi/wavelength). - (3) Angular frequency 2 pi times frequency.
- (4) Velocity (angular frequency)/k
47Simple linear partial differential equations as
examples
- (1) Certain simple linear, partial differential
equations can be taken as examples for dispersive
waves. - (2) Relation between angular frequency and wave
number Dispersion Relation.
48Dispersion of light in prism
- (1) Waves with different wavelengths (colors)
travel at different speeds. - (2) The difference in speed results in different
refraction angles (Refractive index ratio of
speeds). - (3) Thus, splitting of white light into a
rainbow.
(Extracted from http//electron9.phys.utk.edu/phys
136d/modules/m10/geometrical.htm)
49Vorticity
- (1) Theoretical definition Curl of the
velocity field . - (2) Practical Significance Twice the local
angular velocities.
Movie by National Committee for Fluid Mechanics
Films (Prof. Ascher Shapiro) Vorticity Meter
From 230 to 318 Vorticity in straight channel
330 to 352 Free vortex 455 to 535
50Two dimensional ideal incompressible fluid
- Continuity equation (linear)
- ux vy 0
- Equations of motion (nonlinear)
- ut uux vuy px /?
- vt uvx vvy py /?
51Irrotational Flows
- (1) Flows with no vorticity Irrotational flows.
- (2) A velocity potential exists such that the
gradient of the potential gives the velocity
field, and this satisfies the INVISCID equation
of motions identically provided a suitable
pressure is chosen
52Stream function
- (1) A mathematical device to satisfy the
continuity equation identically. - (2) Does NOT impose any constraint on the
dynamics (i.e. say NOTHING about the nature of
the flow). - (3) Difference of adjacent streamlines mass (or
volume) flow rate between streamlines.
53Irrotational Free Surface Waves
- (1) Velocity potential exists.
- (2) Mass conservation implies the velocity
satisfies the Laplaces equation.
54Integration of the equations of Motion
Bernoullis equation
- The equations of motion can be integrated
under the irrotational flow assumption to produce
the Bernoullis equation, i.e. sum of pressure,
kinetic energy and gravity (potential energy)
terms being constant or at worst a function of
time.
55Basic governing equations
- (1) Conservation of mass Continuity equation
-
- (2) Newtons second law (rate of change of
momentum) Equations of Motion
56Irrotational Free Surface Waves (contd)
- (1) Within the fluid (or in the fluid field)
Laplace equation for the velocity potential
satisfy the equation of motion identically
(irrotational flow) and mass conservation. - (2) Flat rigid bottom no penetration (or
vertical velocity being zero), but can still slip
(no slip for a viscous fluid).
57Free Surface Boundary Conditions
- (1) Kinematic boundary condition A fluid
particle initially at the free surface will
remain in the free surface. - (2) Dynamic boundary condition Fluid pressure at
the free surface being atmospheric (if surface
tension is absent).
58Analytical Formulation
- Kinematic boundary condition Material derivative
of the free surface equation being zero -
59Analytical Formulation (contd)
- Dynamic boundary condition From the Bernoullis
equation
60Full formulation for nonlinear surface waves
- (1) Irrotational flow velocity potential
exists, satisfies the momentum equations
identically. - (2) Inside the fluid / field Mass conservation
implies Laplace equation for the velocity
potential. - (3) Bottom boundary condition normal velocity
of fluid same as that of the boundary ( 0, if
boundary is at rest.)
61Full formulation for nonlinear surface waves
(contd)
- (4) Kinematic boundary condition particles
remain on the free surface. - (5) Dynamic boundary condition pressure at the
free surface remains constant (atmospheric). - (6) Still too difficult to solve!! Look for small
amplitude (or linear) waves.
62Linearized Boundary Conditions
- (1) Solve the Laplace equation in terms of
elementary functions, then use the - (2) Linearized boundary conditions to obtain the
dispersion relation.
63Dispersion relation
- (1) Dispersion relation for small amplitude
(linear) water waves - ?2 g k tanh (k H)
- where H is the water depth.
- (2) Graph of the dispersion relation.
64Dispersion relation (contd)
- (1) Long waves k tends to zero tanh(z) z for
small z, hence - ?2 gk kH k2 gH
- c2 gH
- Long wave phase speed is (gH)1/2.
65Dispersion relation (contd)
- (2) Short waves k becomes large tanh(z) 1
for large z, hence - ?2 gk
- ?/k (g/k)1/2
- For large k this is less than (gH)1/2. Hence
short waves (or deep water waves) are DISPERSIVE
and move SLOWER than shallow water waves.
66Group velocity and phase velocity
- For a wave packet, each local oscillation (crest
or trough) moves with the phase velocity but the
packet (or envelope) moves with the group
velocity ( ??/?k). - Animations
- (1)http//www.phys.virginia.edu/classes/109N/more
_stuff/Applets/sines/GroupVelocity.html - (2)http//ftp.ccp14.ac.uk/ccp/web-
mirrors/isotropy/stokesh/vgroup_flash.html
67Particle paths
- Individual fluid particles undergo oscillations
in closed curves and do NOT advance with the wave
form (for small amplitude waves only). - Fluids of finite depth a fluid particle moves
in an ellipse. The semi-minor axis decreases
steadily with depth and the path reduces to a
simple harmonic motion in a straight line at the
bottom. - Fluids of an infinite depth a fluid particle
moves in a circle. The size of the circle
decreases with depth.
Extracted from http//piru.alexandria.ucsb.edu/col
lections/geography3b/mike/Coastal_Pix/wave_animati
on1.gif
68Energy of surface waves
- (1) The kinetic and potential energy of the wave
per wavelength can be calculated from first
principles e.g. - Mass ? (dxdy) (velocity)2/2 and then integrate
over the depth and one wavelength for KE - (density) g (amplitude)2 wavelength/2
- (Same result for PE).
69Surface tension (capillary waves)
- (1) Surface tension force per unit length on
the surface of the fluid. - (2) Surface tension only affects short waves
(wavelength 5cm or less). - (3) Changes the dynamic boundary condition
(Excess pressure atmospheric pressure fluid
pressure) surface tension/curvature.
70Surface tension (contd)
- (4) Effects change the dispersion curves for
large wave numbers. Presence of inflexion point
(triad resonance is now possible).
71Other features
- (1) Standing waves two identical waves
propagating in opposite directions. Presence of
nodes. - (2) Instability of interface 2layer fluid.
Solve Laplace equation in each fluid. Match
boundary conditions across the boundary. Waves
grow instability.
72Exact solutions of the viscous Navier Stokes
equations
- Rectangular Geometry
- (1) Couette Flow, 0 lt y lt 1,
- U(y) y
- Physically Flow between two parallel, rigid
- plates with one or both plate(s) in motion NO
- externally applied pressure gradient. Note how
the - equations of motion, boundary conditions and
- continuity equation are satisfied.
Extracted from I. G. Currie (2003), Fundamental
Fluid Mechanics
73Exact solutions of the Navier Stokes equations
- (2) Plane Channel 1 lt y lt 1,
- U(y) 1 y2
- Physically Flow between two parallel, rigid
- plates with both plates fixed but with an
- externally applied pressure gradient. Note how
- the equations of motion, boundary conditions
- and continuity equation are satisfied
Extracted from I. G. Currie (2003), Fundamental
Fluid Mechanics
74Exact solutions of the Navier Stokes equations
- (3) Mixed Couette and Channel Flows
- 0 lt y lt 1,
- U(y) 1 y2 ay
- Physically flow with one or two plate(s) in
- motion, plus an externally applied pressure
- gradient.
Extracted from I. G. Currie (2003), Fundamental
Fluid Mechanics
75Exact solutions of the Navier Stokes equations
- Cylindrical Geometry
- (4) Pipe Flow 0 lt r lt a
- U(r) a2 r2
Extracted from I. G. Currie (2003), Fundamental
Fluid Mechanics
76Stability of a viscous fluid
- The Orr Sommerfeld equation (theory)
- (1) We start with an exact solution of the Navier
Stokes equations. - (2) Impose a wavy disturbance.
- (3) Sinusoidal (normal) modes in the streamwise
(x) direction - (4) Vertical structure and boundary conditions
eigenvalue problem.
77Stability of a viscous fluid (contd)
- The Orr Sommerfeld equation (physics)
- (1) We start with a parallel flow or very slowly
diverging flow (boundary layer). - (2) Impose a wavy disturbance.
- (3) Wave propagating in the streamwise (x)
direction - (4) No slip, no penetrating boundary conditions
vertical structure of the flow deduced.
78History
- (1) First derived by Orr and Sommerfeld around
1906/1907. - Awarded 50 pounds for solving the problem of
turbulence. - (2) Analytically impossible for any smooth
profile. Asymptotic analysis possible but
involved (1910s 1970s). - (3) Early numerical works 1950s only rough
estimate of critical Reynolds number.
79History (contd)
- (4) Accurate numerical solution (by spectral
method) of the Orr Sommerfeld (OS) equation for
channel flow S. Orszag - (1971) about 60 years after the first
derivation of OS.
80Basic Properties
- (1) For Couette and cylindrical pipe flows NO
linear instability. - How can the flow go turbulent?
- (Finite amplitude effects?)
81Basic Properties
- (1) For channel flow the lowest critical
Reynolds number is about 5772.2 for a wave
number of about 1. - Problems
- (A) 5772.2 is way too high, transition can occur
from Re 1000 to 3000. - (B) Growth rate is on a viscous time scale, too
slow (travel about 300 wavelengths before
doubling in amplitude).
82Flow over bluff bodies
- (1) Flow over a bluff (i.e. of a significant
cross sectional area) versus a streamlined (i.e.
long and thin) body. - (2) Stagnation point in front of the body
deceleration from free stream velocity to zero
velocity on the body (no penetration and no slip
conditions)
83Flow regimes around an immersed body
84Flow over a bluff body (contd)
- (3) Potential flow region For high
Reynolds number flow, a regime most frequently
encountered in practice, viscous effects are only
important in a thin region around the body, i.e.
in the boundary layer only. - (a) High Reynolds number implies
- (inertial force)/(viscous force) gtgt 1 Viscous
force NOT large enough to drive the flow. - (b) Examples of low Re flows fluid moving
slowly or small scale (e.g. micro) fluid flows
85Flow over a bluff body (contd)
- (4) Boundary layer and separation
- (a) The boundary layer will grow as viscous
effects accumulate (typically like the square
root of the streamwise coordinate). - (b) Velocity profile determined by momentum
balance. - (5) Separation occurs when there is an adverse
pressure gradient, or abrupt change in geometry.
86Bluff bodies (contd)
- (6) Wake portion(s) of fluid behind the bluff
body which has experienced viscous effects.
Typically characterized by two or more eddies and
thus high energy dissipation rate, implying
pressure loss. - (7) Stagnation in front of the body velocity
small in that region, and hence high pressure in
front of the body. - (8) Differential in pressure Form drag
87Momentum and Force in Fluids
- From the Bernoullis equation, a fluid moving at
a HIGH SPEED will have a LOWER PRESSURE. - Conversely, a fluid moving at a low speed will
have a higher pressure. - DO NOT MIX UP this with the common day experience
that stopping a high speed will cause say pain to
your hand. That force / pain is associated with
the momentum theorem.
88Formation of vortex in a wake
89Bluff bodies (contd)
- The other component of drag Skin friction drag.
- Velocity must go from zero at the wall (no slip
condition) to the free stream value far away from
the body. Hence the velocity profile typically
must have a positive, non-zero slope at the wall.
This will result in a shear stress as a result of
the concept shear stress (coefficient of
viscosity) X (velocity gradient)
90Bluff bodies (contd)
- Drag Coefficient
-
- Drag force divided by a nondimensional inertial
force (pressure times area) - Important Area Projection of the bluff body
perpendicular to the direction of the flow.
91Flow over a cylinder
- Various regimes
- (1) Very low Reynolds number (Re) no boundary
layer separation. Appearance very similar to
inviscid flows, but note that the boundary
conditions are completely different!!
92Flow over a cylinder
- (2) Slightly higher Re Two symmetric eddies,
rotating in opposite directions, separated but
NOT carried downstream.
93Cylinder (contd)
- (3) Still higher Re Eddies start to vibrate and
eventually become detached from the cylinder.
They are convected downstream. - (4) Further increase of Re process becomes more
violent alternate rows of vortices, the vortex
street. - This process produces circulation around the
cylinder and thus will produce a lift.
Credit Cesareo de la Rosa Siqueira at the
University of Sao Paulo, Brazil
94Flow behind a cylinder (contd)
- This lift force will be periodic, as the shedding
of vortices is periodic as well. Thus the
cylinder is subject to a periodic vibration. - General comment The existence of a circulation
around a bluff body will generally generate a
force perpendicular to the direction of the flow
(a lift force). This can be rigorously proven for
inviscid flows using complex variables.
95Cylinder (contd)
Extracted from http//www.youtube.com/watch?vPdaC
hF24Jj8
96Flow behind a bluff body
- http//ptonline.aip.org/journals/doc/PHTOAD-ft/vol
_63/iss_9/68_1s.shtmlvideo
97Flow behind a sphere
- (1) Qualitatively similar to the flow behind a
cylinder, but - (2) Instead of vortex streets, vortex rings are
formed. - (3) Formed around Re 10, and convected
downstream around 200 lt Re lt 2000. - (4) Not periodic, and thus sphere not subject to
periodic vibration.
98The Stokes equations
- (1) Ignore or neglect the quadratic, convective
acceleration terms and this thus results in a
LINEAR system of equations. - (2) (Contrast Navier Stokes equations are
nonlinear, even though the continuity equation is
always linear.)
99Stokes law or formula
- (1) A formula for the TOTAL viscous drag force
(i.e. form drag skin friction drag) for a
SPHERE falling in an otherwise INFINITE fluid in
LOW Reynolds number. - F 6 p µ a V
- a radius, µ coefficient of viscosity
- V terminal velocity
100Stokes drag law for a sphere
- Assumptions
- (1) The fluid is of an infinite extent. In
practice, as long as the sphere is far away from
the wall, this assumption is taken as satisfied. - (2) Re must be small. (Re lt 0.1, safe, 0.1 lt Re
lt 1, reasonable). - (3) Steady state, i.e. terminal velocity, is
reached.
101A sphere falling in a very viscous fluid
- Analytically we can show that
-
- (projection of the sphere perpendicular to the
flow)
102A falling sphere
- (1) Consider a sphere released from rest in an
otherwise infinite fluid, and Re is small. - (2) Initially, acceleration due to gravity,
moderated somewhat by buoyancy. - (3) Eventually dynamic equilibrium attained among
gravity, buoyancy and viscous drag Sphere falls
with a TERMINAL VELOCITY.
103Experiment A falling sphere
Movie by National Committee for Fluid Mechanics
Films (Prof. Ascher Shapiro) Falling sphere
experiment 1900 to 2000
104Dynamic equilibrium of a falling sphere
- Relevant forces
- (1) gravity (weight of sphere)
- (2) buoyancy (floating upthrust of the fluid)
- (3) viscous drag (Stokes law)
- After some calculations
- (viscosity)(terminal velocity)
- constant (radius)2(difference in density)
105Falling sphere (contd)
- (1) Densities, viscosity fixed the smaller the
sphere, the smaller the terminal velocity (makes
sense as the Reynolds number is smaller for
smaller a). - (2) Densities, radius fixed the larger the
viscosity, the smaller the terminal velocity. - (makes sense as V is smaller for a more viscous
fluid).
106History and Fluid Mechanics
- (1) The leading place for aerospace research
around 1900 1920 is probably Germany, where
Ludwig Prandtl and his school were performing
frontier works. - His student von Karman eventually settled in
California, USA, and the legendary H. S. Tsien
from China worked with von Karman until he was
sent back to China in the early 1950s.
107History and Fluid Mechanics
- (2a) The pressure exerted by a column of fluid
with density ?, height h is ?gh, where g is
acceleration due to gravity. If a column of solid
displaces the fluid, and the fluid depth at the
upper and lower ends are h1 and h2 respectively,
then a difference in pressure of ?g(h2 h1) will
arise, resulting in a force of ?g(h2 h1)X(cross
sectional area) weight of fluid the solid
displaces buoyancy force.
108History and Fluid Mechanics
- (2b) Empress Dowager of the Ching (Qing) Dynasty
did NOT believe that a ship made of iron (or
metals) can float, and she therefore diverted the
money intended for the Beiyang fleet to build one
of the imperial gardens. As a result, the Chinese
navy had to use wooden ships to fight the
Japanese navy in the first Sino Japanese war
around 1890.
109History and Fluid Mechaics
- (3a) The Millikan oil drop experiment The first
experiment to measure the charge of the electron
around 1910, and he was awarded the Nobel prize
in 1924, in part or mainly because of this. - (3b) Hong Kong This was first put in the
Physics Advanced Level Examinations the year I
took the A Levels. This topic was later withdrawn.
110- History
- (1) Ludwig Prantdl Germany 1900s to 1940s
Boundary layer, rocket propulsion, aerospace
engineering - (2) Th. Von Karman California, USA
- (3) H. S. Tsien California, then deported
back to China in the 1950s Rocket enterprise in
China.