Eastern Mediterranean University - Chapter 8: Finite Volume Method...

Chapter 8: Finite Volume Method for

Unsteady Flows

Ibrahim Sezai Department of Mechanical Engineering

Eastern Mediterranean University

Spring 2013-2014

I. Sezai – Eastern Mediterranean University ME555 : Computational Fluid Dynamics 2

8.1 Introduction

The conservation law for the transport of a scalar in an

unsteady flow has the general form

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:

( ) ( ) ( )div div grad St

u (8.1)

( ) ( )

( )

t t t t

CV t t At t t t

t A t CV

dt dV n dA dtt

n grad dA dt S dVdt

u

(8.2)


Introduction

Unsteady one-dimensional heat conduction is governed by the equation

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

(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 t to

t+Δt gives

This may be written as

T Tc k S

t x x

(8.3)

t t t t t t

t 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

In equation(8.6) superscript ‘o’ refers to temperatures at time t.

Temperatures at time level t+Δt are not superscripted.

Eqn(8.6) could also be obtained by substituting

So, first order (backward) differencing scheme has been used. If we

apply central differencing to rhs of eqn (8.5),

0

t t

P P

CV t

Tc dt dV c T T V

t

(8.6)

0

P PT TT

t t

(8.7)

0

t 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


To calculate the integrals we have to make an assumption about the variation of TP, TE and TW with time, we could use temperatures

at time t, or

at time t+Δt

or combination of both.

Integral of temperature TP with respect to time can be written as;

Hence

θ = a weighting parameter between zero and one.

0(1 )

t t

T P P P

t

I T dt T T t

(8.8)

0 0

0 1/ 2 1

1

2T P P P PI T t T T t T t


Using formula (8.8) for TW and TE in equation (8.7), and

dividing by AΔt throughout, we have

which may be re-arranged to give

(8.9)

(8.10)

0

0 0 0 0

1

e E P w P WP P

PE WP

e E P w P W

PE WP

k T T k T TT Tc x

t x x

k T T k T TS x

x x

0 0

0

1 1

1 1

e w e wP E E W W

PE WP PE WP

e wP

PE WP

k k k kxc T T T T T

t x x x x

k kxc T S x

t x x


The source term is linearized as b=Su+SPTP. Now we identify the coefficients of TW and TE as aW and aE and write equation (8.10) in familiar standard form:

where

and

with

For θ = 0 explicit scheme

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

θ = 1 Fully implicit scheme

(8.11)

0

P W E P Pa a a a S

0

P

xa c

t

0(1 )

W E

w eu P P

WP PE

a a b

k kS S T

x x

0 0

0 0

(1 ) (1 )

(1 ) (1 )p P W W E E W W E E

P W E P

a T a T a T a T a T

a a a T b


8.2.1 Explicit scheme

In the explicit scheme the source term is linearized as b=Su+SpTp0.

Now the substitution of θ = 0 into (8.11) gives the explicit discretisation of the unsteady conductive heat transfer equation:

where

and

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

(8.12) 0 0 0 0

P P W W E E P W E P P ua T a T a T a a a S T S

0

P Pa a

0

P

xa c

t

W E

w e

WP PE

a a

k k

x x


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

or if k = const. and δxPE= δxWP=Δx, this condition can

be written as

Or

This inequality sets a stringent maximum limit to the

time step size and represents a serious limitation for

the explicit scheme.

Not recommended for the explicit scheme problems.

2x kc

t x

(8.13a)

(8.13b)

2

2

xt c

k


8.2.2 Crank – Nicolson scheme

The Crank – Nicolson method results from setting θ = ½ in eqn.

(8.11). Now the discretised unsteady heat conduction equation is

where

and

01 1

2 2P W E P Pa a a a S

0

P

xa c

t

01

2

W E

w eu P P

WP PE

a a b

k kS S T

x x

(8.14) 0 0 0 01 1 1 1

2 2 2 2 2 2

W Ep P W W E E W W E E P P

a aa T a T a T a T a T a T b


Schemes with ½ ≤ θ ≤ 1 are unconditionally stable for all values of time step Δt.

However, for physically realistic results all coefficients should be positive. Then, coefficients of TP

0 should be positive, or

which leads to

This time step limitation is only slightly less restrictive than (8.13) associated with the explicit method

The method is based on central differencing Second order accurate in time

Is more accurate than explicit method

2xt c

k

0

2

W EP

a aa

(8.15)


The fully implicit scheme When θ =1 we obtain the fully implicit scheme. The source term is linearized as b=Su+SPTP. The discretised equation is

where

and

with

Both sides of the equation contains temperatures at the new time step, and a system of algebraic equations must be solved at each time level

is unconditionally stable for any Δt

is only first order accurate in time

small time steps are needed to ensure accuracy of results

0 0

p P W W E E P P ua T a T a T a T S

0

P P W E Pa a a a S

0

P

xa c

t

W E

w e

WP PE

a a

k k

x x

(8.16)


8.3 Illustrative examples Example 8.1

A thin plate is initially at a uniform temperature of 200 ºC. At a certain time t=0 the temperature of the east side of the plate is suddenly reduced to 0ºC. The other surface is insulated. Use the explicit finite volume method in conjunction with a suitable time step size to calculate the transient temperature distribution of the slab and compare it with the analytical solution at time (i) t = 40s, (ii) t = 80s and (iii) t = 120s. Recalculate the numerical solution using a time step size equal to the limit given by (8.13) for t = 40s and compare the results with the analytical solution. The data are: plate thickness L=2cm, thermal conductivity k = 10 W/m/K and ρc = 10 × 106 J/m3/K.

Solution: the one–dimensional transient heat conduction eqn is

The initial and boundary conditions:

T Tc k S

t x x

(8.17)

200 at 0

0 at 0, 0

0 at , 0

T t

Tx t

t

T x L t


The analytical solution is given in Ozisik (1985) as

The numerical solution with the explicit method is generated

by dividing the domain width L into five equal control

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

grid is shown in Figure 8.2.

1

2

1

( , ) 4 ( 1)exp cos

200 2 1

(2 1)where and /

2

n

n n

n

n

T x tt x

n

nk c

L

(8.18)


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

2

26

2

10 10 0.004

2 10

8

c xt

k

t

t s


Example 8.2 solve the problem of Example 8.1

again, using the fully implicit method and compare

the explicit and implicit method solutions for a time

step of 8 s.


8.4 Implicit Method for Two-and Three- dimensional Problems

Transient diffusion equation in three-dimensions is governed by

A three dimensional control volume is considered for the

discretisation. The resulting equation is

where

S = (Su + SPP) is the linearized source

c k k k St x x y y z z

(8.27)

0 0

P P W W E E S S N N B B T T

P P u

a a a a a a a

a S

(8.28)

0

P W E S N B T P Pa a a a a a a a S

0

P

Va c

t


A summary of the relevant neighbour coefficients is given below

The following values for the volume and cell face areas apply in three

cases:

1

2

3

W E S N B T

w w e e

WP PE

w w e e s s n n

WP PE SP PN

w w e e s s n n b b t t

WP PE SP PN BP PT

a a a a a a

A AD

x x

A A A AD

x x y y

A A A A A AD

x x y y z z

1 2 3

1w e

n s

b t

D D D

V x x y x y z

A A y y z

A A x x z

A A x y


8.5 Discretisation of transient convection-diffusion equation

The unsteady transport of a property is given by

Here, we quote the implicit/hybrid difference form of the

transient convection-diffusion equations.

Transient three-dimensional convection-diffusion of a general

property in a velocity field u is governed by

( ) ( ) ( )div div grad St

u (8.29)

u v w

t x y z

Sx x y y z z

(8.30)


The discretised equation at location P with degree of implicitness θ is

0 0 0 0

0

(1 ) (1 ) (1 ) (1 )

(1 ) (1 ) (1 ) (1 )

p P W W E E S S N N W W E E S S N N

W E S N P

a a a a a a a a a

a a a a S

0

0

(1 ) ,

, (transient terms), = + (body forces per unit volume in the differential equation)

max , 0 ,

body trans dc pres body

C P P P

trans o o o body body bodyP PP P P C P P

eE e

e

S s V S S S s

VS a a s s s

t

ya F

x

max , 0 , max , 0 , max , 0

( ) ,

,

[( ) / ] ( ) for x-momentum eq

w n sW w N n S s

w n s

bodyP PP E W N S P

e w n s

e w e w

pres

y x xa F a F a F

x y y

Va a a a a s x y F

t

F F F F F

pV p p x V p p y

xS

uation

[( ) / ] ( ) for y-momentum equation

max , 0 max , 0

n s n s

dc

e e P e e E

pV p p y V p p x

y

S F F

max , 0 max , 0

max , 0 max , 0

max , 0 max , 0

, , , , ( , , , are found by MIM method)

, , ,

w w P w w W

n n P n n N

s s P s s S

e w n s e w n se w n s

e w n

F F

F F

F F

F u y F u y F v x F v x u u v v

= face values found from a high order (higher than 1st order) convection scheme such as QUICK or CDs


8.6 Worked example of transient convection-diffusion using QUICK differencing

Example 8.3 consider convection and diffusion in the one-dimensional domain sketched in Figure 8.7. Calculate the transient temperature field if the initial temperature is zero everywhere and the boundary conditions are =0 at x=0 and ∂ /∂x=0 at x=L. the data are L=1.5m, u=2m/s, ρ=1.0kg/m3 and Γ=0.03kg/m/s.

the source distribution defined by Figure 8.8 applies at times t>0 with a=-200, b=100, x1= 0.6m, x2= 0.2m. Write a computer program to calculate the transient temperature distribution until it reaches a steady state using the implicit method for time integration and the Hayase et al variant of the QUICK scheme for the convective and diffusive terms and compare this result with the analytical steady state solution


Solution convection-diffusion of a property subjected to a distributed source term is governed by

We use a 45 point grid to subdivide the domain and perform all calculations with a computer program. It is convenient to use the Hayase et al formulation of QUICK (see section 5.9.3)since it gives a tri-diagonal system of equations which can be solved iteratively with the TDMA (see section 7.2).

uS

t x x x

(8.32)


The velocity is u=2.0m/s and the cell width is Δx=0.0333 so F=ρu=2.0 and D=Γ/δx=0.9 everywhere. The Hayase et al formulation gives φ at cell faces by means of the following formulae:

13 2

81

3 28

e P E P W

w W P W WW

(8.33)

(8.34)

(8.35)


A time step Δt = 0.01 is selected, which is well within stability limit for explicit schemes

Start with an initial field of P0 =0 at all nodes.

Solve iteratively for values until a converged solution is obtained

Set P0 P and proceed to the next time level

To monitor whether steady state reached:

if P – P0 < ε steady state . (ε may be 10-9)


The Analytical Solution

Under the given boundary conditions the solution to the problem is as

follows:

with

and

and

(8.41) 01 2 2

2

2

1

( ) 1

sin cos

Px

n

n

ax C C e Px

P

L n x n n x na P P

L L L Ln

2

202 2

1

; cosn

PLPLn

au a nP C n P

e LP e

2

201 2 2

1

n

n

a nC C a P

LP

1 2 1 1

0

1 2 1 21 1

2 2

2 2

2

2cos cosn

x x ax b bxa

L

a x x b n x xn x ax bLa a

n x L x L


The analytical and numerical steady state solutions are compared

in Figure 8.9. as can be seen the use of the QUICK scheme and a

fine grid for spatial discretisation ensure near-perfect agreement


8.7 Solution procedures for unsteady flow calculations

8.7.1 Transient SIMPLE

The continuity equation in a transient two-dimensional flow is given by

The integrated form of this eqn over a two-dimensional scalar CV is

The equivalent of pressure correction equation (5.32) for a two-dimensional transient flow will take the form

(8.42)

(8.43)

(8.44)

0

u v

t x y

0

0P P

e w n sV uA uA uA uA

t

P P W W E E S S N Na p a p a p a p a p b

* * * *

( ) ( ) ( ) ( )

( )

E e W w N n S s

P w e s n

o

P P

w e s n

a Ad a Ad a Ad a Ad

a a a a a

Vb u A u A v A v A

t


A higher order differencing scheme may also be used for time

derivative

A second order accurate scheme with respect to time is

Tn and Tn-1 are known from previous time steps they are

treated as source terms and are placed on the rhs of the

equation.

1 113 4

2

n n nTT T T

t t

(8.45)


8.8 Steady state calculations using the pseudo-transient

approach

It was mentioned in chapter 6 that under-relaxation is

necessary to stabilize the iterative process of obtaining

steady state solutions. The under-relaxed form of the two-

dimensional u-momentum equation, for example, takes the

form

Compare this with the transient (implicit) u-momentum

equation

(8.46)

(8.47)

, , ( 1)

, 1, , , , ,1i J i J n

i J nb nb I J I J i J i J u i J

u u

a au a u p p A b u

0 0

, , 0

, , 1, , , , ,

i J i J

i J i J nb nb I J I J i J i J i J

V Va u a u p p A b u

t t


In equation (8.46) the superscript n-1 indicates the previous

iteration and in equation(8.47) superscript 0 represents the

previous time level. We immediately note a clear analogy

between transient calculations and under-relaxation in

steady state calculations. It can be easily deduced that

This formula shows that it is possible to achieve the effects

of under-relaxed iterative steady state calculations from a

given initial field by means of a pseudo-transient

computation starting from the same initial field by taking a

step size that satisfies (8.48).

(8.48) 0

, ,1

i J i J

u

u

a V

t

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