chapter_8 turbulence and its modeling

7/30/2019 Chapter_8 Turbulence and Its Modeling

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Chapter 8: Finite Volume Method forUnsteady Flows

ra m ezaDepartment of Mechanical Engineering

Eastern Mediterranean University

Spring 2008-2009

8.1 Introduction

The conservation law for the transport of a scalar in an

unsteady flow has the general form

div div rad S

+ = +u (8.1)

by replacing the volume integrals of the convective and

diffusive terms with surface integrals as before (see section

2.5) and changing the order of integration in the rate of

change term we obtain:

t

I. Sezai Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 2

( ) ( )

( )

t t t t

CV t t At t t t

t A t CV

dt dV n dA dt t

n grad dA dt S dVdt

+ +

+ +

+

= +

u

(8.2)


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Introduction

Unsteady one-dimensional heat conduction is governed by the equation

In addition to usual variables we have c, the specific heat of material

T Tc k St x x

= + (8.3)

(J/kg/K).

Consider the one-dimensional control volume in Figure 8.1. Integration of

equation (8.3) over the control volume and over a time interval from tto

t+tgives


This may be written ast CV t CV t CV

T Tc dVdt k dVdt sdVdt

t x x

+ + + = +

(8.4)

e t t t t t t

e ww t t t

T T Tc dt dV kA kA dt S Vdt

t x x

+ + + = +

(8.5)

The left hand side can be written as

( )0t t

P P

CV t

Tc dt dV c T T V

t

+ =

(8.6)

n equa on . superscr p o re ers o empera ures a me t.

Temperatures at time level t+tare not superscripted

Eqn(8.6) could also be obtained by substituting0

P PT TT

t t

=


So, first order (backward) differencing scheme has been used. If weapply central differencing to rhs of eqn (8.5),

(8.7)

( )0t t t t

P WE PP P e w

PE WPt t

T TT Tc T T V k A k A dt S Vdt

x x

+ + = +


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To calculate the integrals we have to make an assumption

about the variation ofTP, TEand TWwith time, we could usetemperaturesat time t, or

at time t+ t

or combination of both.

Integral of temperature TPwith respect to time can be writtenas;

0(1 )t t

T P P P

t

I T dt T T t +

= = + (8.8)


= a weighting parameter between zero and one.

( )0 00 1/ 2 1

12

T P P P P I T t T T t T t

+

Using formula (8.8) forTWand TE in equation (8.7), and

dividing byAtthroughout, we have

( ) ( )0 e E P w P W P P k T T k T T T Tc x

=

which may be re-arranged to give

(8.9)

( )( ) ( )0 0 0 0

1

PE WP

e E P w P W

PE WP

t x x

k T T k T T S x

x x

+ +

k k k k x


(8.10)

( ) ( ) 0

1 1

1 1

e w e wP E E W W

PE WP PE WP

e wP

PE WP

c T T T T T t x x x x

k kxc T S x

t x x

+ + = + + +

+ +


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Now we identify the coefficients ofTWand TEas aWand aEand write equation(8.10) in familiar standard form:

( ) ( )( ) ( )

0 0

0 0

1 1

1 1

p P W W W E E E

P W E P

a T a T T a T T

a a a T b

= + + + + +

(8.11)

where

and

with

( ) 0P W E Pa a a a= + +

0

P

xa c

t

=

W Ea a b


For = 0 explicit scheme

0 < < 1 implicit scheme, for= 0.5 Crank-Nicolson scheme

= 1 Fully implicit scheme

w e

WP PE

S xx x

8.2.1 Explicit scheme

In the explicit scheme the source term is linearized as b=Su+SpTp0.Now the substitution of= 0 into (8.11) gives the explicitdiscretisation of the unsteady conductive heat transfer equation:

(8.12)( )0 0 0 0P P W W E E P W E P P ua T a T a T a a a S T S = + + + + where

and

0

P Pa a=

0

P

xa c

t

=

W Ea a


The right hand side of eqn (8.12) only contains values at the old timestep so the left hand side can be calculated by forward marching intime. The scheme is based on backward differencing, and is of firstorder accurate.

w e

WP PE x x


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All coefficients should be positive aP0- aW- aE>0

or ifk = const. and xPE= xWP=x this condition can

be written as

Or

This ine ualit sets a strin ent maximum limit to the

2x kc

t x

>

(8.13a)

(8.13b)( )

2

2t c

k

= = >

The analytical solution is given in Ozisik (1985) as

( ) ( )1

2

1

( , ) 4 ( 1)exp cos

200 2 1

n

n n

n

T x tt x

n

+

=

=

(8.18)

The numerical solution with the explicit method is generated

by dividing the domain width L into five equal control

volumes with x = 0.004m. The resulting one-dimensional

(2 1)/

2n

nWhere and k c

L

= =


grid is shown in Figure 8.2.


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The time step for the explicit method is subject to the condition that2

c x


( )262

10 10 0.004

2 10

8

tk

t

t s

chapter_8 turbulence and its modeling

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