Embed Size (px)
Citation preview
FLUID MECHANICS: AMME2261 summary DONβT FORGET TUTE QUESTIONS AVAILABLE THURSDAY FOR THE NEXT TUTE!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Week 1
Contents Module 1: Introduction to fluid properties ............................................................................................ 5
Definition of fluids: ............................................................................................................................. 5
Methods of analysing fluids: ............................................................................................................... 5
Methods of analysis: ........................................................................................................................... 5
Control mass: .................................................................................................................................. 5
Control volume: .................................................................................................................................. 5
Way to answer questions: ...................................................................................................................... 6
Idealised 1 dimensional fluid flow .......................................................................................................... 6
Continuity equation: ........................................................................................................................... 6
Reference frames: Lagrangian ................................................................................................................ 8
Eulerian: .................................................................................................................................................. 8
Pressure, density and continuum ........................................................................................................... 8
Viscosity: ................................................................................................................................................. 9
Dynamics viscosity .................................................................................................................................. 9
Kinematic viscosity: ......................................................................................................................... 9
Surface tension: .................................................................................................................................... 11
gases: .................................................................................................................................................... 12
Standard atmosphere: .......................................................................................................................... 12
Mamometers: ................................................................................................................................... 13
Hydrostatic force on a submerged surface ........................................................................................... 14
Submerged plane surface ................................................................................................................. 14
Centre of pressure ................................................................................................................................ 15
Moment areas of standard shapes ....................................................................................................... 16
Hydrostatic forces on submerged surfaces .......................................................................................... 18
Example ......................................................................................................................................... 18
Buoyancy: .............................................................................................................................................. 19
Example ............................................................................................................................................. 19
Trapezoidal rule: ............................................................................................................................... 20
Stability analysis .................................................................................................................................... 21
Unstable: ........................................................................................................................................... 21
Unconditionally/neutral stable ......................................................................................................... 21
G above B: ......................................................................................................................................... 22
Example ......................................................................................................................................... 23
Fundamentals of fluid dynamics ........................................................................................................... 24
Integral form of fluid dynamics: ........................................................................................................ 24
Conservation of mass ............................................................................................................................ 24
Special forms of flow: ....................................................................................................................... 24
Volume flow rate: ............................................................................................................................. 25
Mass flow rate: ................................................................................................................................. 25
Example ......................................................................................................................................... 25
Conservation of linear momentum ....................................................................................................... 25
Special forms of flow: ....................................................................................................................... 26
Example ......................................................................................................................................... 26
Conservation of angular momentum .................................................................................................... 27
Differential forms of flow ...................................................................................................................... 28
Conservation of mass/ continuity equation ..................................................................................... 28
Cylindrical coordinates: ................................................................................................................. 28
Conservation of linear momentum ................................................................................................... 28
Shear stress in 3D: ......................................................................................................................... 29
Naviar stokes equation: ........................................................................................................................ 29
Fully developed flow: ........................................................................................................................ 29
Euler equation, inviscid fluids: (easier to solve) ................................................................................... 29
Dimensional analysis ............................................................................................................................. 31
Buckingham pi theorem: ....................................................................................................................... 31
To determine pi groups:.................................................................................................................... 32
Significant Ξ groups: ......................................................................................................................... 32
Reynolds number: ................................................................................................................................. 32
Euler number: ....................................................................................................................................... 33
Froude number: .................................................................................................................................... 33
Weber number: ..................................................................................................................................... 33
Flow similarity: ...................................................................................................................................... 33
Incomplete similarlity: .......................................................................................................................... 33
Inviscid flow .......................................................................................................................................... 35
Euler equation ................................................................................................................................... 35
Inviscid flow over wings: ............................................................................................................... 35
Euler equation along a streamline: ................................................................................................... 35
Streamline: .................................................................................................................................... 35
Euler equation along streamline steady flow: .............................................................................. 36
Bernoulli equation ............................................................................................................................ 37
Example ......................................................................................................................................... 37
Static, stagnation and dynamic pressure .......................................................................................... 38
Static pressure ...................................................................................................................................... 38
Stagnation pressure: ......................................................................................................................... 38
Measuring velocity: ........................................................................................................................... 38
Module 5: potential flow theory ........................................................................................................... 39
Stream function: ............................................................................................................................... 39
Definition of stream function ........................................................................................................... 39
Stream function in polar coordinates: .......................................................................................... 40
Potential function: ................................................................................................................................ 42
Definition of Potential function ππ₯, π¦, π‘: .......................................................................................... 42
Potential function polar coordinates ............................................................................................ 42
Laplaveβs equation: ............................................................................................................................... 42
Laplace equation: .............................................................................................................................. 42
Example: ........................................................................................................................................ 43
Elementary plane flow: ......................................................................................................................... 43
Uniform flow: .................................................................................................................................... 43
Source flow: ...................................................................................................................................... 43
Sink: ................................................................................................................................................... 44
Irrotational vortex: ............................................................................................................................ 45
Doublet ............................................................................................................................................. 45
Superposition of elementary plane flows: ............................................................................................ 45
Direct method (simple approach) ..................................................................................................... 46
Flow past a bluff body: ...................................................................................................................... 46
Rankine body (source, sink, uniform) ............................................................................................... 47
Example: flow over a cylinder ....................................................................................................... 47
Turbomachinary of inviscid fluids ......................................................................................................... 48
Pumps/fans/ blowers/compressors: ............................................................................................. 48
Positive displacement pumps: .......................................................................................................... 48
Dynamic pumps: ............................................................................................................................... 49
Comparison of pump types ........................................................................................................... 50
Euler turbomachine equations: ........................................................................................................ 50
Torque: .......................................................................................................................................... 50
Power: ........................................................................................................................................... 51
Head rise/drop: ............................................................................................................................. 51
Radial flow turbomachiary: ............................................................................................................... 51
Viscous flow: ......................................................................................................................................... 52
Internal flow development: .............................................................................................................. 52
For laminar flow: ........................................................................................................................... 53
Turbulent flow: ............................................................................................................................. 53
Transition to turbulence: .............................................................................................................. 53
Fully developed laminar flow in a pipe: ............................................................................................ 53
Laminar pipe flow equations: ........................................................................................................... 54
Reduced naviar stokes: ................................................................................................................. 54
Velocity distribution: ..................................................................................................................... 54
Shear stress: .................................................................................................................................. 54
Volumetric flow rate: .................................................................................................................... 54
Pressure gradient: ......................................................................................................................... 54
Mean velocity: ............................................................................................................................... 54
Max velocity: ................................................................................................................................. 54
Introduction to external viscous flow: .............................................................................................. 56
Pitch, roll, yaw/side,lift,drag ......................................................................................................... 57
Coefficient of lift and drag ................................................................................................................ 57
Drag: .................................................................................................................................................. 57
Drag force ...................................................................................................................................... 58
Drag on a sphere: .......................................................................................................................... 58
Steamlining: ...................................................................................................................................... 59
Common drag coefficients: ............................................................................................................... 60
Lifting bodies: Wing .......................................................................................................................... 62
Lift and drag as a function of angle of attack.................................................................................... 63
Flaps .............................................................................................................................................. 63
Module 1: Introduction to fluid properties
Definition of fluids: - A solid can resist shear stresses, and will undergo static deflection (up to a point) if a stress is
applied
- A fluid cannot resist shear stress, and will translate (move) if stress applied
o Liquids: are incompressible and will retain their volume
o Gases: can be compressed and will take the volume of their container
o Fluids are not elastic, but have viscosity
Methods of analysing fluids: 1. Analytical analysis
o Uses equations such as:
Conservation of mass
Newtonβs equations
Conservation of angular momentum
1st and 2nd laws of thermodynamics
To analysis and give exact answers to fluid analysic
2. Computational/numerical (CFD)
3. Experimental
o Partial image velocimetry
o Streak/smoke lines
Methods of analysis: 1st step: define the system involved (boundaries, forces, ect)
Usually either control mass or control volme:
Control mass: - Fixed mass of fluid, fluid mass does not cross boundaries, mass doesnβt change, volume can
- Eg: piston is control mass
Control volume: - Fixed volume, mass can change but volume constant; mass flux across boundaries
- Eg: pipe junction
-
Way to answer questions: 1. Diagram, labelling control surface, control volume/mass, inlets/outlets, forces ect
2. Write assumptions: eg constant densit
3. Start with fundamental equations
4. Simplify, finally add numbers
Idealised 1 dimensional fluid flow
Continuity equation: If flow is a βcontinuumβ (the difference between the fluid little volumes is βsmoothβ, can be called a
continuum
π1π΄1π1 = π2π΄2π2
Eg 1.1:
ANSWER:
π΄π π π’πππ‘ππππ : ππππ π‘πππ‘ ππππ ππ‘π¦ (π€ππ‘ππ ππ π’π π’ππππ¦ π ππππ ππ π π’πππ‘πππ πππ π‘βππ )
οΏ½ΜοΏ½1 = οΏ½ΜοΏ½2 (ππππ π‘πππ‘ πππ π ππππ€ πππ‘π πππ‘π€πππ πππππ‘ 1 πππ 2)
πΆπππππ’ππ‘π¦ πππ’ππ‘πππ:
π1π΄1π£1 = π2π΄2π2
β΄ π2 =π΄1π1
π΄2 (π1 = π2 (ππ π π’πππ‘πππ))
=π (
π·12
4 ) π1
π (π·2
2
4 )
(ππππ ππ ππππππ)
= (π·1
π·2)
2
π2
= (50
30)
2
(2.5)π
π
= 6.9 ππ β1
Average speed: is the average speed of all the particles along the inlet/outlet (speed of 0 in contact
of pipe, higher speed in centre)
2 1
Control system
Control volume π£2 π£1
Reference frames: Lagrangian
- Considers elemental globs of fluid (control masses), and forces are solved for each glob
- - Moving reference frame
-
- This is very time costly though, solving βοΏ½βοΏ½ = ποΏ½βοΏ½; π‘π πππ‘ οΏ½ββοΏ½(π‘) πππ π(π‘) for every particle.
- Can be used to analyse discrete phases (eg- a water spray)
Eulerian: - Fixed reference frame
- Make a grid, and monitor the flow through each section of grid
οΏ½ββοΏ½ = π(π, π‘ )
Pressure, density and continuum Fluids are aggregations of molecules, and the distance between molecules can be very large
compared to molecular diameter
- Density on a small scale does not have much meaning, due to microscopic uncertainty.
- Microscopic uncertainty diminishes when you increase the volume and is large compared to
molecular spacing β 10β9π (most problems will be above microscopic uncertainty)
- For very large observations, there can be smooth variations in density too, called
macroscopic uncertainty (eg: density difference in a room, slightly higher on the floor than
the ceiling)
- This fluid is called a continuum, where the variation in fluid property is smooth enough to
perform calculus on. (at very low pressure (eg- atmosphere renty), the molecular spacing
can become too large, as the spacing is comparable to the system size, and so molecular
theory of rarefied gas flow must be used)
Viscosity:
Dynamics viscosity - Measure of a fluidβs resistance to shear stress
- If a stress of π is applied, in Newtonian fluids, there is a linear relationship between shear
and resulting strain rate. The top surface moved ππ’, and the bottom surface is static
π βππ
ππ‘ =
πππ
ππ‘
β΄
β΄ π = ππ π½
π π= π
π π
π π
(π’ = π£ππππππ‘π¦ ππ π π‘πππ π ππππππ‘πππ; π¦ ππ πβππππ ππ βπππβπ‘) π€βπππ π ππ ππ¦ππππππ π£ππ πππ ππ‘π¦
π ππ ππ: ππ. πβ1. π β1 ππ ππ. π
Kinematic viscosity: Ratio of dynamics viscosity to density
π =π
π, ππ ππ
π2
π
Example 1.2
π΄π π π’πππ‘ππππ : πππ’ππ‘π‘π ππππ€ (ππ π· β« β, ππ‘ πππ ππ ππππππππ ππ π‘ππππ πππ‘πππ βππππ§πππ‘ππππ¦)
π = πππ’
ππ¦= π
π’
β
ππ πΆππ’ππ‘π‘π ππππ€: π’ππππ ππππ‘π πππ£πππ, πππ‘π‘ππ ππ π π‘πππ‘πππππ¦ πππππ‘ππ£π π‘π ππ‘
π’ = π (π·
2) = (
2ππ
60) (
π·
2) =
πππ·
60
β΄ π =π (
πππ·60 )
β
π =πΉ
π΄=
(ππ·2
)
π΄
π = ππ΄ (π·
2) = π(ππ·πΏ) (
π·
2) =
1
2πππ·2πΏ
π·
2
β
ππ₯
ππ₯
=1
2(
π (πππ·
60 )
β ) ππ·2πΏ
=π2πππ·3πΏ
120
Non newtonima fluids do not have linear relationships between π and deformation rate
π = πππ’
ππ¦; π = ππππππππ‘ π£ππ πππ ππ‘π¦
Surface tension: Fluid behaves like elastic membraine in tension: between a fluid and another fluid or solid
πΉππππ ππ π π’πππππ π‘πππ πππ: πΉπ‘ = π(π)
π = πππππ πππ π’πππ‘ πππππ‘β; π = πππππ‘β ππ ππππ‘πππ‘ ππππ πππ‘π€πππ ππππππππ
fluid statics:
dP
dz= βΟg = βΞ³ (specific weight)
P β P0 = βΟg h
gases:
π = π0πβπβπ π
Standard atmosphere:
Pressure is function of depth, and does not depend on geometry
Hydrolic jack:
πΉ1
πΉ2=
π΄1
π΄2
Mamometers:
π3 = πππ‘π + ππβ
π4 = πππ‘π + ππ»20πβ β πππππβ2
π = π1π΄ = ππβ
β΄ πΉππππ πππππππ = π2π΄ β π = ππ(β2 β β1)
Hydrostatic force on a submerged surface
Submerged plane surface
πΉπ = β« ππΉπ΄
= β« πππ΄π΄
= β« (π0 + ππβ)ππ΄ = β« (π0 + πππ ππππ¦)ππ΄
= π0π΄ + ππ π πππ β« π¦π΄
ππ΄
(β« π¦π΄
ππ΄ = π¦ππ΄ = 1π π‘ ππππππ‘ ππ ππππ)
πΉπ = (π0 + πππ ππππ¦π)π΄ = πππ΄ (ππ = ππππ π π’ππ ππ‘ ππππ‘ππππ)
Note: if air was on both sides of the surface (eg, at a gate): ππ =
πππ ππππ¦π
Eg:
πΉπ = β«πππ΄π΄
= β«ππ(π· + π sin 30)π€πππ΄
(πππ‘π: πππ‘ππππππππ ππ ππ πππ‘β π ππππ ππ πππ‘π)
= β« ππ(π· + π sin 30)π€πππΏ
0
= ππ (π·π +π2
4)
0
πΏ
Centre of pressure Even though πΉπ is calculated with
π¦π (ππππ‘ππππ ππ ππππ), πΉπ πππ‘π π‘βπππ’πβ π¦β² (ππππ‘ππ ππ ππππ π π’ππ)
To find centre of pressure:
βπ0 = 0
β΄
π¦β²πΉπ = β« π¦πππ΄π΄
=
= β« π¦(π0 + ππβ)ππ΄π΄
= β« π¦(π0 + πππ¦π πππ)π€ππΏπ΄
= π0 β« π¦ππ΄π΄
+ πππ πππ β« π¦2ππ΄π΄
(1π π‘ ππππππ‘ ππ ππππ) + (2ππ ππππππ‘ ππ ππππ ππππ’π‘ π₯: πΌπ₯π₯)
π€π πππ πβππππ πΌπ₯π₯πππ‘π π ππππππ‘ ππππ’π‘ π‘βπ ππππ‘ππππ ππ ππππ πππ π‘πππ ππ ππππ’π‘ π
β΄ πΌπ₯π₯ = πΌοΏ½ΜοΏ½οΏ½ΜοΏ½ + π΄π¦π2
β¦
β΄ π¦β² = π¦π +πππ ππππΌπ₯π₯
πΉπ
Moment areas of standard shapes
Hydrostatic forces on submerged surfaces ππ = βπππ¨
ππΉ = β β« πππ¨ππ¨π
β β« πππ¨ππ¨π
β β« πππ¨ππ¨π
ππΉ = πΉπ₯π + πΉπ¦π + πΉπ§π
the resultant:
πΉπ = πΉπ£ + πΉπ»
Example
β» +βΆ βπ0 = 0 (ππ πππ‘π ππ ππππ ππ)
β΄ πΉππ β πΉπ»π¦β² β πΉππ₯β² = 0
πΉπ» = πππ΄ = πππ·
2Γ π·π€ = 396 ππ
π¦β² = π¦πβ +
πΌπ₯οΏ½ΜοΏ½
π΄π¦πβ
=π·
2+
π€π·3
12
π·π€π·2
= 2.67π
β΄ π¦β² = 4 β π¦β² = 1.33
πΉπ£ = π€πππβπ‘ ππ πππ’ππ = ππ β« (π β π¦)π€ππ₯
π·2
4
0
= πππ€
π₯β² = 1.2
Buoyancy:
Buoyancy is the net vertical pressure acting on the object
ππΉπ = ππ(β2 β β1)ππ΄ = πππβππ΄
ππΉπ = ππππ
πΉπ = πππ (π€βπππ π ππ π‘βπ πππ ππππππ π£πππ’ππ)
Example A hot air balloon is to lift a basket and payload weighing 270 kg. The balloon may be approximated
as a sphere of diameter of 16 m. To what temperature must the air be heated in order to achieve
lift-off?
πΉπ
ππππ ππππ πππ‘
If in equilibrium:
πΉπ = ππ + ππ
β΄ πβππ‘ πππππ = ππππ πππ‘π + ππππ π = π(ππ + πππ‘ππ)
πβ = πππ‘π +ππ
π
As ideal gas:
π1π1
π1=
π2π2
π2
β΄ π2 =π2π1
π1
In ship design and operation, buoyancy is a critical parameter. Will the ship float? How much more
load can be added? Shipsβ hulls are rarely nice simple shapes. There is a tradition of designing ships
with βfair linesβ for hydrodynamic (e.g. drag) and aesthetic reasons. Example lines plan on the next
page, and example table of offsets is on the next after that. It is generally not possible to analytically
integrate the buoyancy force over the irregular shaped volume that is floating. Numerical integration
(quadrature) is required. Simple quadrature methods are Simpsonβs Rule and the Trapezoidal Rule.
Trapezoidal rule:
πΌππ‘πππππ β β (π(π₯0)
2+ π(π₯1) + β― + π(π₯πβ1) +
π(π₯π)
2)
