Download Fluid Mechanics, Lecture Notes - Engineering - 6 and more Study notes Mechanical Engineering in PDF only on Docsity!
CIVE1400: An Introduction to Fluid Mechanics
Dr P A Sleigh [email protected]
Dr CJ Noakes [email protected]
January 2008 Module web site: www.efm.leeds.ac.uk/CIVE/FluidsLevel
Unit 1: Fluid Mechanics Basics 3 lectures Flow Pressure Properties of Fluids Fluids vs. Solids Viscosity Unit 2: Statics 3 lectures Hydrostatic pressure Manometry/Pressure measurement Hydrostatic forces on submerged surfaces Unit 3: Dynamics 7 lectures The continuity equation. The Bernoulli Equation. Application of Bernoulli equation. The momentum equation. Application of momentum equation. Unit 4: Effect of the boundary on flow 4 lectures Laminar and turbulent flow Boundary layer theory An Intro to Dimensional analysis Similarity
Real fluids
Flowing real fluids exhibit __________ effects, they:
- “______” to solid surfaces
- have _________ within their body.
From earlier we saw this relationship between shear stress and velocity gradient:
τ ∝
The shear stress, τ, in a fluid
is proportional to the velocity gradient
- the rate of change of velocity across the flow.
For a “__________” fluid we can write:
τ =
where μ, is coefficient of “dynamic viscosity”
(or simply “viscosity”). Here we look at the influence of forces due to ___________ in a moving fluid.
__________and _________flow
__ ______ would happen - but for different flow rates.
Top:____________ Middle: ______________ Bottom: ______________
Top: _________ flow Middle: _________ flow Bottom: _________ flow
________ flow:
Motion of the fluid particles is ____ ______all particles moving in straight lines parallel to the pipe walls.
Turbulent flow:
Motion is, locally, _________ ________ but the overall direction of flow is one way.
But what is fast or slow? At what speed does the flow pattern change? And why might we want to know this?
The was first investigated in the 1880s by Osbourne Reynolds in a classic experiment in fluid mechanics.
A tank arranged as below:
What are the units of Reynolds number?
We can fill in the equation with SI units:
ρ μ
ρ μ
Re
u d
ud
It has ______ units
A quantity with _____ units is known as a _______________ (or ______________ ) quantity.
(We will see more of these in the section on dimensional analysis.)
The Reynolds number, Re, is a ___________________ number.
At what speed does the flow pattern change?
We use the Reynolds number in an example:
A pipe and the fluid flowing have the following properties:
water density pipe diameter (dynamic) viscosity,
What is the _________ velocity when flow is laminar i.e. Re = ________
Re = =
ρ μ
ud
u
u
What does this abstract number mean?
We can give the Re number a physical meaning.
This may help to understand some of the reasons for the changes from laminar to turbulent flow.
Re =
ρ μ
ud
When _________ forces dominate (when the fluid is flowing _____ and Re is larger) the flow is ________
When the _________ forces are dominant (slow flow, _____ Re) they keep the fluid particles in line, the flow is _________.
__________flow
- Re <
- ‘ ’ velocity
- Dye does not mix with water
- Fluid particles move in lines
- Simple mathematical analysis possible
- Rare in practice in water systems.
____________flow
Re <
- ‘ ’ velocity
- Dye stream wavers - mixes slightly.
__________flow
- Re >
- ‘ ’ velocity
- Dye mixes rapidly and completely
- Particle paths
- Average motion is in flow direction
- Cannot be seen by the naked eye
- Changes/fluctuations are very difficult to detect. Must use laser.
- Mathematical analysis very difficult - so experimental measures are used
- Most common type of flow.
Attaching a manometer gives _______ (______) loss due to the energy lost by the fluid overcoming the ______ ______.
L
Δ p
The pressure at 1 (upstream) is _______ than the pressure at 2.
How can we quantify this pressure loss in terms of the forces acting on the fluid?
Consider a cylindrical element of incompressible fluid flowing in the pipe,
area A
τw
τw
τ w is the mean shear stress on the boundary Upstream pressure is p , Downstream pressure falls by Δ p to ( p - Δ p )
The driving force due to __________
driving force
pA (^) ( p p A ) p A p
d
π 2
The ____________ is due to the shear stress
= ×
×
shear stress area over which it acts
= area of pipe wall
w w
τ τ π dL
What is the variation of shear stress in the flow?
τw
τw
r
R
At the wall
τ w
R p
L
At a radius r
τ
τ τ
r p
L
r
w R
A linear variation in shear stress.
This is valid for:
- _______flow
- ________flow
- __________flow
Shear stress and hence pressure loss varies with _________ of flow and hence with _____.
Many experiments have been done with various fluids measuring the pressure loss at various Reynolds numbers.
A graph of pressure loss against Re look like:
This graph shows that the relationship between pressure loss and Re can be expressed as
r
δ r r R
The fluid is in equilibrium, shearing forces equal the ___________ forces. τ π π
τ
r L p A p r^2
p
L
r
Newton’s law of viscosity says τ = μ
du
dy
We are measuring from the pipe centre, so
τ = − μ
du
dr
Giving:
p
L
r du
dr
du
dr
p
L
r
μ
μ
In an integral form this gives an expression for velocity,
u
p
L
= − ∫ r dr
The value of velocity at a point distance r from the centre
u
p
L
r
r = −^ + C
At r = 0 , (the centre of the pipe), u = umax , at r = R (the pipe wall) u = 0 ;
C
p
L
R
At a point r from the pipe centre when the flow is laminar:
u ( )
p
L
r =^ R^ − r
2 2 μ This is a _______ profile (of the form y = ) so the velocity profile in the pipe looks similar to
v
