Lecture 01-11-2010

8/8/2019 Lecture 01-11-2010

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Mathematical Description

of

Physical Phenomenon

8/8/2019 Lecture 01-11-2010


•

expressed in

mathematical

form

(usually

• A equations possess a common orm

– Identification of the common form

– Help to

construct

a

general

solution

procedure

8/8/2019 Lecture 01-11-2010


8/8/2019 Lecture 01-11-2010


Physical Phenomenon & Differential

Equation

•

– Frequently used

as

a

dependent

variable

but

not

a

– Arises from more basic equations employing

s ecific internal ener or s ecific enthal

8/8/2019 Lecture 01-11-2010



Equation

•

– ‘J’ any

flux

dependent

variable

– Balance over

a control

volume

8/8/2019 Lecture 01-11-2010



Equation

•

– A term

expressed

on

unit

volume

basis

‘ ’ –

– ‘rho’ is the density

– e pro uc r o n o represen s e amoun o

extensive property contained in CV

–

relevant property per unit volume

8/8/2019 Lecture 01-11-2010



Equation

• . .

• Each term

represent

an

influence

on

a

unit

•

All the

terms

collectively

imply

a

balance

or

conservation

8/8/2019 Lecture 01-11-2010


a e o c ange o mass o c em ca spec e per un vo ume

Rate of generation of specie per unit volume (may be –ve, +ve or zero

Convection flux carried by flow field ‘rho into u’

Diffusion

flux caused

by

gradients

of

mass

fraction

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’

Diffusion Coefficient

8/8/2019 Lecture 01-11-2010


For steady low‐velocity with negligible viscous dissipation,Specific enthalpy

Thermal

conductivity

Temperature

Rate of change of enthalpyInfluent of conduction heat transfer within fluid

Volumetric rate of heat eneration ‐ve, +ve or zero

8/8/2019 Lecture 01-11-2010


• Substitutin and sim lification

• Setting velocity

zero,

gives

steady

state

conduction

equation

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Viscous term in addition to the first term

Momentum flux

Rate of chan e of momentum x‐direction

Viscous term

Body force per unit volume

8/8/2019 Lecture 01-11-2010


The Turbulence‐Kinetic‐Energy

Equation• From ‘two‐equation model’ of turbulence, the

equation of kinetic energy ‘k’ of the fluctuating motion

Diffusion coefficient for k Rate of generation of turbulence energy

Kinematic rate of dissipation

Net source term

• Similar differential equation governs ‘epsilon’

8/8/2019 Lecture 01-11-2010


8/8/2019 Lecture 01-11-2010


•

quantities (mass

fraction,

enthalpy,

temperature,

etc.

• A ro riate meanin will be iven to diffusion

coefficient and source term

• Density may

be

related

to

mass

fraction,

temperature, etc. via an equation (like equation of

state)

• Flow field

should

satisfy

continuity

equation

8/8/2019 Lecture 01-11-2010


• . .

can be

represented

in

Cartesian

‐tensor

form

• Subscript ‘j’

can

take

the

values

1,

2,

3,

denotin the three s ace coordinates

8/8/2019 Lecture 01-11-2010


• ,

summation of

three

terms

is

implied

• One‐dimensional form can alwa s be obtained

by dropping the subscript ‘j’

8/8/2019 Lecture 01-11-2010


• . . ,

convection and

diffusion

terms

conform

to

• More convenient to work with dimensionless

8/8/2019 Lecture 01-11-2010


• G.D.E ‐ An extremel im ortant conclusion‐‐‐‐‐‐

Recognize this!!!

• A time saving step

• Concern only with numerical solution of G.D.E.

• Even with

com uter

ro ram,

sufficient

to

write

a

general sequence of instructions for solving

• General purpose program can be developed

• Variables, initial

&

boundary

conditions

can

be

adjusted for making the specific case

8/8/2019 Lecture 01-11-2010


•

a stationary

frame

of

reference,

but

steady

in

airplane

• Axisymmetric flow in a circular pipe appears

to

e‐

n a

cartes an

coor nate

system

ut

is 2‐D in cylindrical polar coordinates

8/8/2019 Lecture 01-11-2010


‐ ‐

• ‐

influenced by

changes

in

conditions

on

either

side of location

• One‐Way: conditions at a given location are

influenced by

changes

in

conditions

on

only

one side of location

• Time – always one‐way coordinate

• Space – mostly

two

‐way,

but

may

nearly

become one‐way in some conditions

8/8/2019 Lecture 01-11-2010


Computer Implications of One‐Way

Two‐Way

Coordinates

• ‐

given situation,

computer

storage

and

• Example

8/8/2019 Lecture 01-11-2010


•

to obey

a

generalized

conservation

principle

• . . .

equations can be addressed later on and different

cases can be accommodated then after.

• Things become simpler if we make right selection

‐

coordinates into

one

‐way

coordinates

Lecture 01-11-2010

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