fluid dynamics

CAMS

in the School of Computing, Engineering and Physical Sciences

Introductory fluid dynamics

by Dr J. Whitty

3 2

1 m m

Lessons structure

• The lessons will in general be subdivided in to eight number of parts, viz.:

1) Statement of learning objectives 2) Points of orders

3) Introductory material (Types of flow)

4) Concept introduction (The conservation of mass)

5) Development of related principles (flow continuity)

6) Concrete principle examples via –

reinforcement examination type exercises 7) Summary and feedback

Learning Objectives

– State and use the basic thermodynamic laws

– Derive the conservation of mass

– Describe the differences between flow regimes

– Calculate simple fluid flow mechanisms – Evaluate volumetric flow rates in fluid

simple systems

Recap: Laws of thermodynamics

• These are quite simply the 4 axioms (self evident truths) of all modern Physics, they are known as the four Laws of Thermodynamics and relate to the

quantities of

– Zeroth: Temperature – First: Energy

– Second: Disorder (Entropy) – Third: Balance of them all

Consequences of the first law:

Flow Processes

• If we consider the

first law based on

some fluid passing

through a control

volume above a

datum (at sea-level)

for consentience.

Application of the

first law, with the

following

assumption:

1. The mass flow is

constant and equal to the outlet mass flow 2. The cross-section

properties of the inlet and outlet are constant

Conservation of Mass

Conservation of mass

Both Heat and fluid flow must adhere to the principal of the flow of mass and energy. Here we can consider a

system (sometimes referred to as a control volume) with fluid flow (or heat) in and out of the system

The unit of mass flow the kg per second (kg/s). Because speed has magnitude and direction, it vector quantity. 2 1 m m m_in     _m ₃  _m _out Consequence?? out in

m

m







i.e.

The Consequence of to

Conservation

of

Mass

1. The mass (and sometimes volume) flow

rate of a in-viscid, incompressible fluid

(like water or oil) is constant.

2. This principle is one of probably the

fundamental assumption in the field of

Fluid Mechanics, this will now be

explored!

Class Examples Time:

Fluids in motion

As an example of this principle we will investigate the concept of a fluid (say water) in motion. There is still a little terminology that is

required before we proceed, these being:

1. Assumptions regarding the fluid in motion, namely: a) Viscid

b) In-viscid

2. Assumptions regarding the type of flow regime’ a) Laminar

b) Transition c) Turbulent

3. Assumptions regarding Compressibility: 1. Compressible, or

1. Viscosity

• The viscosity of a fluid is the internal resistance to a change in the shape. Typically viscous fluids are treacle like:

glycerine and thick oils. All fluids have some type of viscosity, however some fluids have such small viscosities have (e.g. water, air) can be considered in-viscid i.e. the viscosity of the fluid can be ignored! It is these type of fluids we considered here.

• Hence we have:

1. Viscid fluids (includes fluid viscosity effects) 2. In-viscid fluids (neglects fluid viscosity effects)

Since the math is considerably reduced when in-viscid fluids are concerned it is these types we consider!

2. Flow regime’

• Laminar

• Turbulent

• Transition flow

Class Exercise:

3.

Compressibility

• Incompressible fluid: Where the density of the fluid remains constant! (This course)

• Incompressible fluid: Where the density of the fluid

changes during the flow process! (Not this course)

• When the Compressibility (Bulk) Modulus is?

Class Question:

What? z y x v p p K                   z y x v p p K    

Continuity of flow

• For the system shown, given that the

flow is laminar, in-viscid and

incompressible, find the flow rate at the

outlet.

A₁

v₂ m/s

v₁ m/s

Continuity of flow; Solution:

• Here we could just apply the

conservation of mass, as we know it is a

consequence of the first law of

thermodynamics, thus:

which implies

t x t x t x

_A

_A

A

₁



₁ _



₂



₂ _



₃



₃ _



A x



_t



A x



_t



A x



t 3 3 1 2 2 1 1 1 1         3 2 1

m

m

m











and gives:

_{As density and the volume} of then control volume are constant!

The Continuity Equation:

• We have now we’ve proved the

continuity equitation (I wonder why I

have spent so many slides on it?)



 

 



3 3 2 2 1 1 3 2 1 v A v A v A A A A x_t t x t x      _    

Using the fact that. The flow is in-compressible:

3 3 2 2 1 1

v

A

v

A

v

A





Example #2

• Evaluate the velocity of the fluid exiting

the barrel of beer:

20mm DIA

1 m/s

6 m/s

Example #2; solution:

• Apply the continuity equation, thus:

     

3 2 2 2 3 2 3 2 2 2 1 2 1 20 1 30 6 20 4 4 4 v v D v D v D                       _{  } Hence:

   

₂ -1 2 2 3 8.25ms 20 30 20 6    v

Class

Problems

3. A system has two inlet rates of 3m3_{/s &}

2m3/s what is the approximate output

velocity [2]; and what assumptions did you make [3]?

4. For the system shown, determine the

volumetric flow rate and velocity at the out-let. Given the large diameter pipe is 1.25 that of the smaller.

3.2m/s

1.6m/s

18

Class

problem; solution #4:

• Here were are given the volumetric flow

rate, hence by continuity we have:

• There are three assumptions in place

here:

– The flow regime is laminarB1

– The fluid is incompressibleB1

– The fluid is in-viscidB1

1 -3 3 2 1 3 3 2 2 1 1 s m 5 2 3       Q Q Q v A v A v A M1A1

Class

problem; solution #2:

• Apply the continuity equation taking D and 1.25D along as parameter, thus:

The required velocity can be found from the flow rate thus:





2 3 3 2 2 3 2 2 537 . 5 6 . 1 5 . 1 2 . 3 4 6 . 1 ) 5 . 1 ( 4 2 . 3 4 D Q Q D Q D D             _M1 M2 A1 1 3 3 2 2 3 2 3 3 3 ms 05 . 7 537 . 5 4 4 537 . 5 4          v v D D v D v A Q M1 M1 A1

Examination

type questions

1. Explain, using cogent examples: three laws of thermodynamics [6].

a) Use formulae to describe three mechanisms of heat transfer [6].

b) Find the total heat lost an asbestos (thermal conductivity 0.15W/mK) reinforced steel wall (thermal conductivity 50W/mK), given that the concrete is twice the thickness of the steel. [8]

Examination

type questions

2. State three states of matter. [3]

a) Explain the meaning of incompressible flow [2].

b) Given that the large pipe is 1.4 times the diameter of the small pipe evaluate the velocity at the output [12],

c) Clearly state the assumptions of the modelling process [3].

3.4m/s

Summary

• Have we met our learning objectives:

specifically

, are you now able to do:

– State and use the basic thermodynamic laws

– Derive the conservation of mass

– Describe the differences between flow regimes

– Calculate simple fluid flow mechanisms – Evaluate volumetric flow rates in fluid