chapter_8 turbulence and its modeling
TRANSCRIPT
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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
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( ) ( )
( )
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
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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
=
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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)
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= 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
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(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
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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
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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
= =
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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
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( )262
10 10 0.004
2 10
8
tk
t
t s