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

2
Compressible 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

3
Compressible 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.

4
Aerospace

• (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.

5
Speed 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
6
Reviews 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
7
The 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

9
Linear 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)
10
A 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
11
A 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
12
Supersonic, 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.

13
Basic Gas Laws / Processes

• (1) Isothermal compression / expansion
• (2) Adiabatic compression / expansion

14
Sir 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.

15
Sir 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.

16
Sir 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.

17
Speed 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.

18
The 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
19
Thermodynamic 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

20
Flows 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.

21
Compressible 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.

22
Analysis 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

23
Analysis (continued)

• (1)
• (2) M lt 1, dA lt 0, dV gt 0,
• (3) M gt 1, dA lt 0, dV lt 0 (Counterintuituve!!).
  

24
Convergent Nozzles

• Subsonic applications

Boeing 757 Nozzle
Airbus A330 Nozzle
Extracted from http//www.astechmfg.com/
Extracted fromhttp//en.wikipedia.org/wiki/Image
Turbofan_operation.png
25
Converging-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
26
Converging-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/
27
Shock 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.

28
Shock 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.

29
Shock 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.

30
Supersonic 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
31
The 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
32
The 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.

33
Analysis of shock waves
34
Shock 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.

35
Prandtls 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.

36
Rankine Hugoniot Relation

• This is the relation between the pressure and
  density ratios across the shock.

37
Qualitative 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
38
Qualitative 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
39
Qualitative 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
40
Waves 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.

41
Stability (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.

42
Stability (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.

43
Dispersive 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.

44
Dispersive 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
45
Dispersive 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).

46
Basic 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

47
Simple 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.

48
Dispersion 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)
49
Vorticity

• (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
50
Two dimensional ideal incompressible fluid

• Continuity equation (linear)
• ux vy 0
• Equations of motion (nonlinear)
• ut uux vuy px /?
• vt uvx vvy py /?

51
Irrotational 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

52
Stream 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.

53
Irrotational Free Surface Waves

• (1) Velocity potential exists.
• (2) Mass conservation implies the velocity
  satisfies the Laplaces equation.

54
Integration 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.

55
Basic governing equations

• (1) Conservation of mass Continuity equation
• (2) Newtons second law (rate of change of
  momentum) Equations of Motion

56
Irrotational 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).

57
Free 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).

58
Analytical Formulation

• Kinematic boundary condition Material derivative
  of the free surface equation being zero

59
Analytical Formulation (contd)

• Dynamic boundary condition From the Bernoullis
  equation

60
Full 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.)

61
Full 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.

62
Linearized Boundary Conditions

• (1) Solve the Laplace equation in terms of
  elementary functions, then use the
• (2) Linearized boundary conditions to obtain the
  dispersion relation.

63
Dispersion 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.

64
Dispersion 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.

65
Dispersion 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.

66
Group 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

67
Particle 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
68
Energy 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).

69
Surface 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.

70
Surface tension (contd)

• (4) Effects change the dispersion curves for
  large wave numbers. Presence of inflexion point
  (triad resonance is now possible).

71
Other 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.

72
Exact 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
73
Exact 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
74
Exact 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
75
Exact 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
76
Stability 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.

77
Stability 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.

78
History

• (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.

79
History (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.

80
Basic Properties

• (1) For Couette and cylindrical pipe flows NO
  linear instability.
• How can the flow go turbulent?
• (Finite amplitude effects?)

81
Basic 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).

82
Flow 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)

83
Flow regimes around an immersed body
84
Flow 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

85
Flow 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.

86
Bluff 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

87
Momentum 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.

88
Formation of vortex in a wake
89
Bluff 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)

90
Bluff 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.

91
Flow 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!!

92
Flow over a cylinder

• (2) Slightly higher Re Two symmetric eddies,
  rotating in opposite directions, separated but
  NOT carried downstream.

93
Cylinder (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
94
Flow 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.

95
Cylinder (contd)
Extracted from http//www.youtube.com/watch?vPdaC
hF24Jj8
96
Flow behind a bluff body

• http//ptonline.aip.org/journals/doc/PHTOAD-ft/vol
  _63/iss_9/68_1s.shtmlvideo

97
Flow 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.

98
The 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.)

99
Stokes 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

100
Stokes 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.

101
A sphere falling in a very viscous fluid

• Analytically we can show that
• (projection of the sphere perpendicular to the
  flow)

102
A 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.

103
Experiment A falling sphere
Movie by National Committee for Fluid Mechanics
Films (Prof. Ascher Shapiro) Falling sphere
experiment 1900 to 2000
104
Dynamic 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)

105
Falling 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).

106
History 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.

107
History 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.

108
History 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.

109
History 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.