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CIVE1400: An Introduction to Fluid Mechanics
Unit 3: Fluid Dynamics
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
Fluid Dynamics
Objectives
1.Identify differences between:
- steady/unsteady
- uniform/non-uniform
- compressible/incompressible flow
2.Demonstrate streamlines and stream tubes
3.Introduce the Continuity principle
4.Derive the Bernoulli (energy) equation
5.Use the continuity equations to predict pressure and velocity in flowing fluids
6.Introduce the momentum equation for a fluid
7.Demonstrate use of the momentum equation to predict forces induced by flowing fluids
Flow Classification
Fluid flow may be classified under the following headings
_______________:
Flow conditions (velocity, pressure, cross-section or depth) are the same at every point in the fluid. _________________:_ Flow conditions are not the same at every point.
________________:
Flow conditions may differ from point to point but DO NOT change with time.
________________:
Flow conditions change with time at any point.
Fluid flowing under normal circumstances
- a river for example - conditions vary from point to point we have non-uniform flow.
If the conditions at one point vary as time passes then we have unsteady flow.
Combining these four gives.
________________________. Conditions do not change with position in the stream or with time. E.g. flow of water in a pipe of constant diameter at constant velocity.
Conditions change from point to point in the stream but do not change with time. E.g. Flow in a tapering pipe with constant velocity at the inlet.
At a given instant in time the conditions at every point are the same, but will change with time. E.g. A pipe of constant diameter connected to a pump pumping at a constant rate which is then switched off.
Every condition of the flow may change from point to point and with time at every point. E.g. Waves in a channel.
This course is restricted to Steady uniform flow
- the most simple of the four.
One dimensional flow:
Conditions vary only _______________________ not across the cross-section.
The flow may be unsteady with the parameters varying in time but not across the cross-section. E.g. Flow in a pipe.
But:
Since flow must be zero at the pipe wall
- yet non-zero in the centre - there is a difference of parameters across the cross-section.
Pipe Ideal flow Real flow
Should this be treated as two-dimensional flow? Possibly - but it is only necessary if very high accuracy is required.
Two-dimensional flow
Conditions vary in the direction of flow and in
___________________ at right angles to this.
Flow patterns in two-dimensional flow can be shown by curved lines on a plane.
Below shows flow pattern over a weir.
In this course we will be considering:
Some points about streamlines:
- Close to a solid boundary, streamlines are _______________ to that boundary_
- The direction of the streamline is the ________ of the fluid velocity
- Fluid can not _______ a streamline
- Streamlines can not cross ______________
- Any particles starting on one streamline will stay on that same streamline
- In __________ flow streamlines can change position with time
- In _______ flow, the position of streamlines does not change.
Streamtubes
A circle of points in a flowing fluid each has a streamline passing through it.
These streamlines make a tube-like shape known as a streamtube
In a two-dimensional flow the streamtube is flat (in the plane of the paper):
Flow rate
Mass flow rate
m
dm
dt
mass
time taken to accumulate this mass
Volume flow rate - Discharge.
More commonly we use volume flow rate Also know as discharge.
The symbol normally used for discharge is Q.
discharge, Q
volume of fluid
time
Discharge and mean velocity
Cross sectional area of a pipe is A Mean velocity is um.
Q = Au m
We usually drop the “m” and imply mean velocity.
Continuity
Mass entering = Mass leaving + Increase
per unit time per unit time of mass in
control vol per unit time
For steady flow there is no increase in the mass within the control volume, so
For steady flow
Mass entering = Mass leaving per unit time per unit time
Q 1 = Q 2 = A 1 u 1 = A 2 u (^2)
Mass flow in Cvolumeontrol
Mass flow out
In a real pipe (or any other vessel) we use the mean velocity and write
ρ 1 A 1 u m 1 = = =
For incompressible, fluid ρ 1 = ρ 2 = ρ
(dropping the m subscript)
This is the continuity equation most often used.
This equation is a very powerful tool. It will be used repeatedly throughout the rest of this course.
Some example applications of Continuity
- What is the outflow?
- What is the inflow?
Lecture 9: The Bernoulli Equation
Unit 3: Fluid Dynamics
The Bernoulli equation is a statement of the principle of conservation of energy along a streamline
It can be written:
p
g
u
g
1 1 z
2 1
+ + = H = Constant
These terms represent:
Pressure
energy per
unit weight
Kinetic
energy per
unit weight
Potential
energy per
unit weight
Total
energy per
unit weight
These term all have units of length,
they are often referred to as the following:
pressure head = velocity head =
potential head = total head =
Restrictions in application of Bernoulli’s equation:
- Flow is _________
- Density is __________ (incompressible)
- ____________ losses are __________
- It relates the states at two points along a single streamline, (not conditions on two different streamlines)
All these conditions are impossible to satisfy at any instant in time!
Fortunately, for many real situations where the conditions are approximately satisfied, the equation gives very good results.
