Lid Driven Cavity Flow OK Final Eldwin

1

COMPUTATIONAL

HEAT TRANSFER (ME 673)

FINAL PROJECT

LID DRIVEN CAVITY FLOW

Name: Eldwin Djajadiwinata

Student ID: 434107763

Lecturer: Dr. Shereef Sadek

2

Numerical Heat Transfer ME673

Final-Project

Second Semester 1435

Due date Wednesday June 4th

Consider the two-dimensional steady state flow of air in a closed lid-driven square cavity as

shown in Figure 1, Lid-driven cavity flow Figure 1,

defined over the domain , subject to the shown boundary conditions. Requirements: 1. Using vorticity-stream function approach write down the non-dimensional governing

equations using as reference values 2. Discretize the governing PDEs using finite volume discretization 3. Solve the flow problem described above using ADI scheme for the following parameters:

4. Compare results from part 3 with results in the literature 5. Plot stream-function and vorticity contours, estimate the position and magnitude of

maximum stream-function and vorticity values

6. Compute the temperature field

7. Calculate and plot number along the horizontal walls defined as

Where (*) denotes non-dimensional variables.

Note: All submitted work should be entirely yours. Include source code in your report.

Figure 1, Lid-driven cavity flow

3

Question 1

Transforming the governing equations into non-dimensionalized forms.

The governing equations are as follows.

(1)

(2)

(3)

Let us first define the following non-dimensional variables.

(4)

a) Transforming the continuity/stream function equation

(5)

Substitute the non-dimensional variables in Eq. (4) into Eq. (5)

4

Finally, we will get the non-dimensional continuity/function equation

(6)

b) Transforming the vorticity equation

If we open the Del operator and multiply the dot matrix we get:

(

)

(7)

Again, substitute the non-dimensional variable into the equation above.

[

]

(

)

5

(

)

Finally we obtain the non-dimensional vorticity equation as shown in Eq. (8)

(

) (8)

c) Transforming the energy equation

If we expand the Del term, the energy equation will be as follows.

(

) (9)

Substitute the non-dimensional variables in Eq. (4) into the Eq. (9)

(

)

(

)

(10)

(

)

(

)

6

Multiplying the R.H.S with , then we will get the final form of the non-dimensional energy equation.

(

) (11)

Where

Question 2

The non-dimensional governing equations are as follows. The star superscript is intentionally not included for simplicity.

(12)

(13)

(14)

Discretizing the governing equations

The index in this discretization is based on the figure below.

7

Figure 2: The control volumes used in the discretization

a) The stream (continuity) equation

(15)

From the previous homework on finite volume, we have developed in detail the L.H.S term of

Eq. (15). Thus, here we will directly show the final form which was obtained from that

homework.

(

) (16)

And

8

(17)

Thus the discretized equation is

(

) (18)

b) Discretize the vorticity equation

(

)

(19)

Discretizing the first term on the L.H.S, (

)

:

(

)

(

)

(20)

Using central difference at time level we obtain Eq. (21). We intentionally expand the

equation at this time level in order to be used in ADI scheme which method is by splitting a

single time step into two of half size time step.

(

)

(21)

9

Discretizing the second term on L.H.S,

:

[ ]

(22)

Since our problem is an incompressible flow then the divergence of velocity will be zero. Thus

the equation becomes:

[ ]

(23)

Using the divergence theorem we will convert the volume integral on the R.H.S to become a

surface integral.

[ ]

(24)

[ ]

(25)

Evaluate the R.H.S of Eq. (25) one by one.

Evaluate

:

(26)

10

(27)

{[ ]

[ ]

}

[ ]

(28)

Evaluate the

:

(29)

11

(30)

{[ ]

[ ]

}

[ ]

(31)

Substituting Eq. (28) and Eq. (31) into Eq. (25) we obtain:

[ ]

[ ]

[ ]

(32)

Evaluate

Using the same method as that to find the previous volume integral of Laplacian we know that:

(

)

(

) (33)

12

Substitute Eq. (21), Eq. (32), and Eq. (33) into Eq. (19) and divide it by we will get the

discretized equation for vorticity equation.

[ ]

[ ]

(

)

(34)

We notice on the R.H.S that the space discretization has not been indexed with the time level.

We will put the index based on the ADI method that we are going to apply when solving.

Step one for ADI:

[

]

[

]

(

)

(35)

13

Step two for ADI:

[

]

[

]

(

)

(36)

c) Discretize the energy equation

The differential form energy equation has exactly the same form as the vorticity equation. Thus,

the discretized equation will also be the same except that the and are substituted by and

, respectively. Please note, as mentioned before, all of the parameters here have been non-

dimensionalized. The discretized energy equation used for ADI scheme can be seen in Eq. (37)

and Eq. (38). If we solve the vorticity and stream function equation decoupled from the energy

equation, i.e., the temperature gradient does not affect the velocity gradient, and if our target is

the steady state condition, then the transient terms in Eq. (37) and Eq. (38) will vanish.

The first step for ADI:

[

]

[

]

(

)

(37)

14

The second step for ADI:

[

]

[

]

(

)

(38)

Question 3

In this question we are asked to solve the flow problem presented numerically using ADI

scheme with finite volume approach. However, after several attempts on solving the flow

problem using ADI scheme with finite volume were not successful, another method was

searched for.

Explicit upwind scheme with finite difference approach was finally chosen to solve the

set of equations. The form of the discretized governing equations remains the same except for the

boundary.

The boundary condition for the control volume (finite volume) approach is as follows.

Using the Ghost cell method together with the Taylor Series expansion we can relate the

information on the boundary to the information of the cell nodes. Also, since the given boundary

conditions are not explicitly written in terms of vorticity and stream function (instead, it is in

terms of velocities), we need to convert/relate the velocity terms into the vorticity and stream

function terms.

For the finite difference approach, in this particular problem, creating Ghost nodes is

not required. Only Taylor series expansion is used as well as the relation between the related

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three parameters, i.e., velocity, stream function, and vorticity as it is also done in finite volume

method.

Boundary condition treatment used

At the bottom wall

To find the boundary condition for stream function we will use its definition as follows.

(39)

The u will be zero because of no slip condition while the v will be zero due to the impermeable

wall. Thus, and it follows that will be a constant that can be chosen to be zero.

Boundary condition for vorticity is obtained by the following method.

It is also known by definition that

(40)

To evaluate

, we will use Taylor series expansion at the bottom wall.

16

(

)

(

)

Rearranging the equation above we get

(

) [ (

) ]

(41)

Based on the definition of stream function, we also know that

(42)

Substituting Eq. (42) and Eq. (41) into Eq. (40) and also knowing that for bottom wall, the

stream function and u velocity is zero, we will obtain the boundary condition for vorticity at wall

bottom as follows.

(43)

Using the grid notation (instead of distance, x) shown at the beginning of this report, we can

write the B.C. for bottom wall as

(44)

Using the same procedure, boundary conditions on the other walls can be obtained. These

boundary conditions can be seen inside the code that will be presented in this section.

17

The upwind explicit scheme is intentionally used to reduce the stability constraint on the

time step which arises when central differencing scheme is used for the convective term.

Moreover, upwind scheme also takes into account the effect of flow direction since information

is travelling in the same direction with the flow. Hence, a more realistic result is expected to be

obtained. To do this, upwind scheme is using backward differencing when the velocity is

positive and forward differencing when the velocity is negative. Therefore, in the FORTRAN

code, if-conditions are used to switch from one to another. The FORTRAN code used is

presented at the end of this report.

Question 4

In this section we will compare our results with those from Ghia et al. As can be seen in

Figure 3 and Figure 4 that the the current results match very closely to the published-Ghia et al.s

results. Moreover, it can also be seen that the grid independence has been reached since the

results at 128x128 are very close to that at 64x64. However, when higher Re is attempted, i.e.

400, it is found that the discrepancy is quite large. This might be because the different scheme

used or due to the current simulation has not reached steady state yet as will be shown next.

From Figure 5 it can be seen that, also for Re = 400, the higher the grid and the longer the

flow time, the closer the current solution to that of Ghias et al. This result explains that higher

Re requires finer mesh and longer simulation time to reach steady state. The current result also

indicates that upwind explicit scheme can also predict the flow as good as the scheme used by

Ghia et al.

18

Figure 3: Comparison of u velocity between the current results and the Ghia et al.s results at Re=100 at the vertical bisection of the cavity

Figure 4: Comparison of v velocity between the current results and the Ghia et al.s results at Re=100 at

the horizontal bisection of the cavity

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

-0.30 -0.10 0.10 0.30 0.50 0.70 0.90 1.10

Dim

ensi

on

less

y-d

irec

tio

n, y

*

Dimensionless x-velocity, u*

Ghia et al., 128x128, Re=100

eldwin, 128x128, 6.5s, Re=100

eldwin, 64x64, 6s, Re=100

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Dim

ensi

on

less

y-v

elo

city

, v*

Dimensionless x-direction

Ghia et al., 128x128, Re=100


eldwin, 128x128, 6.5s, Re=100

19

Figure 5: Comparison of u velocity between the current results and the Ghia et al.s results at

Re=400 at the vertical bisection of the cavity

Question 5 & 6

Plots of various parameters are presented in this section. In Figure 6, it can be seen that the

maximum vorticity happens on the top-corner of the domain for Re=100. However, it is confined

on a small region.

0

20

40

60

80

100

120

140

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

Dim

enso

inle

ss y

-dir

ecti

on

, y*

Dimensionless x-velocity, u*

Ghia et al., 128x128, Re=400

eldwin, 64x64, 15.6s, Re=400


eldwin, 128x128, 7.8s, Re=400

eldwin, 128x128, 20.8s, Re=400

20

Figure 6: Vorticity contour plot for grid size 64x64, Re=100, and 6 sec. flow time.

For the stream function (Figure 7), the maximum value occurs between the center of the

cavity and the moving lid and it is rather shifted to the right (the direction of the lid movement).

The maximum (absolute) value is predicted around -0.1. The velocities contour and temperature

contour are presented in Figure 8 Figure 10 below.

21

Figure 7: Stream function contour plot for grid size 64x64, Re=100, and 6 sec. flow time.

Figure 8: velocity at u-dir. contour plot for grid size 64x64, Re=100, and 6 sec. flow time.

22

Figure 9: velocity at v-dir. contour plot for grid size 64x64, Re=100, and 6 sec. flow time.

Figure 10: Temperature contour plot for grid size 64x64, Re=100, and 6 sec. flow time.

23

Question 7

The Nusselt number was calculated using Excel program based on the simulation data.

Figure 11 shows the plot of Nusselt number along the horizontal walls.

Figure 11: Plot of Nusselt number along the top and the bottom boundary for grid size 128x128,

Re=400, and 7.8 sec. flow time.

24

PROGRAM lid_driven_cavity_flow

!SOLVING THE LID DRIVEN CAVITY FLOW

!USING EXPLICIT UPWIND METHOD

IMPLICIT NONE

INTEGER :: i,j,m,n,k,nts

!m is number of segments in x direction

!n is number of segments in y direction

!nts is the number of time step

REAL, PARAMETER :: Pr=0.7, Re = 400, Gr=0

DOUBLE PRECISION :: xsize, ysize, as, bs, cs, a, b, p, omega

!xsize is the nondimensional size of the domain in x dir (0

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!T is the dimensionless temperature

!It is designated to be

!allocatable, i.e., 2D matrix with adjustable size.

WRITE(*,10,ADVANCE='no')

READ(*,*) xsize


READ(*,*) ysize


READ(*,*) m


READ(*,*) n

WRITE(*,*) 'Number of time steps ='

READ(*,*) nts

WRITE(*,*) 'Relaxation factor for temperature calculation ='

READ(*,*) omega

WRITE(*,*) 'The values that you have entered are:'

WRITE(*,*)

WRITE(*,*) 'x =', xsize

WRITE(*,*) 'y =', ysize

WRITE(*,*) 'm =', m

WRITE(*,*) 'n =', n

WRITE(*,*) 'Number of time steps =', nts

!Calculate dx, dy, dt

dx = dble(xsize)/dble(m)

dy = dble(ysize)/dble(n)

dt = dble(dx)/3.d0

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OPEN (unit = 1 , file = "result_cavity_flow_explicit_upwind")

WRITE(1,*) 'The values that you use are :'

WRITE(1,*)

WRITE(1,*) 'x =', xsize

WRITE(1,*) 'y =', ysize

WRITE(1,*) 'm =', m

WRITE(1,*) 'n =', n

WRITE(1,*) 'time step size =', dt

WRITE(1,*) 'the actual flow time =', nts*dt

WRITE(1,*)

ALLOCATE(z(1:m+1,1:n+1))

ALLOCATE(s(1:m+1,1:n+1))

ALLOCATE(u(1:m+1,1:n+1))

ALLOCATE(v(1:m+1,1:n+1))

!Initial condition

z(:,:) = 1.d-32

s(:,:) = 1.d-32

u(:,:) = 1.d-32

v(:,:) = 1.d-32

!Boundary condition

s(1,:)=0 ; s(:,1)=0;

s(m+1,:)=0;s(:,n+1)=0;

u(1,:)=0 ; u(:,1)=0;

v(1,:)=0 ; v(:,1)=0;

27

v(m+1,:)=0;v(:,n+1)=0;

u(m+1,:)=0; u(:,n+1)=1

!Calculate the vorticity distribution

DO k=1,nts

DO j=2,n

DO i=2,m

z(1,:)= -(2/dx/dx)*(s(2,:)) !left wall

z(m+1,:)= -(2/dx/dx)*(s(m,:)) !right wall

z(:,1)= -(2/dy/dy)*(s(:,2))!bottom wall

z(:,n+1)= -(2/dy/dy)*(1.d0*dy + s(:,n)) !top wall

IF (u(i,j)>=0) THEN

a = (z(i,j)-z(i-1,j))/dx

ELSE

a = (z(i+1,j)-z(i,j))/dx

ENDIF

IF (v(i,j)>=0) THEN

b = (z(i,j)-z(i,j-1))/dy

ELSE

b = (z(i,j+1)-z(i,j))/dy

ENDIF

z(i,j)=dt*(-u(i,j)*(a)-v(i,j)*(b)&

& + 1/Re*((z(i-1,j)-2*z(i,j)+z(i+1,j))/dx/dx&

& +(z(i,j-1)-2*z(i,j)+z(i,j+1))/dy/dy))+z(i,j)

28

ENDDO

ENDDO

!CALCULATE THE STREAM FUNCTION

as = 1.d0/(dy*dy)

bs = 1.d0/(dx*dx)

cs = -2.d0*(as + bs)

!-----------------------------------------------------------------------------------

DO j=2,n

DO i=2,m

s(i,j)=1/cs*(-as*s(i,j-1) - bs*s(i-1,j) - bs*s(i+1,j)&

&- as*s(i,j+1) - z(i,j))

ENDDO

ENDDO

!------------------------------------------------------------------------------

!FINDING THE VELOCITY U AND V

DO j=2,n

DO i=2,m

u(i,j)= (s(i,j+1)-s(i,j-1))/2/dy

v(i,j) = -(s(i+1,j)-s(i-1,j))/2/dx

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ENDDO

ENDDO

ENDDO !ENDDO for the time step

!-----------------------------------------------------------

!CALCULATE THE TEMPERATURE DISTRIBUTION

ALLOCATE(T(1:m+1,1:n+1))

ALLOCATE(T_old(1:m+1,1:n+1))

!INITIALIZATION TO START THE ITERATION

T(:,:) = 0 !choose any random value

!BOUNDARY CONDITION

T(:,1) = 1 !Temp at the bottom

T(:,n+1)= 0 !Temp at the top

!T(1,:) = T(2,:) !USING FORWARD DIFFERENCE

!T(m+1,:)= T(m,:) !USING FINTE DIFFERENCE

p = 1 !first guess for p (any number > 1e-5)

DO WHILE (p >= 1e-5)

T_old(:,:) = T(:,:)

DO j=2,n

DO i=2,m

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T(1,:) = T(2,:) !USING FORWARD DIFFERENCE

T(m+1,:)= T(m,:) !USING FINTE DIFFERENCE

T(i,j) = T(i,j) + omega*(1/cs*(-as*T(i,j-1) - bs*T(i-1,j) -

bs*T_old(i+1,j)&

&- as*T_old(i,j+1) + Re*Pr*(u(i,j)*(T_old(i+1,j)-T(i-1,j))/2/dx +&

& v(i,j)*(T_old(i,j+1)-T(i,j-1))/2/dy))-T(i,j))

T(1,:) = T(2,:) !USING FORWARD DIFFERENCE

T(m+1,:)= T(m,:) !USING FINTE DIFFERENCE

ENDDO

ENDDO

p = ABS(MAXVAL(T(2:n,2:m)-T_old(2:n,2:m))/MAXVAL((T(2:n,2:m))))

ENDDO

!-------------------------------------------------------------------

!PRINTING RESULTS TO DATA FILE NO.1

WRITE(1,*)'The vorticity distribution:'

WRITE(1,*)

WRITE(1,*)' i=1 i=2 i=3 ...(so on if available)'

DO j=1,n+1

DO i=1,m+1

WRITE(1,21,ADVANCE='no') z(i,j)

ENDDO

WRITE(1,*)

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ENDDO

WRITE(1,*)

WRITE(1,*)

!-------------------------------------------------------------------------------

WRITE(1,*)'The stream function distribution:'

WRITE(1,*)


DO j=1,n+1

DO i=1,m+1

WRITE(1,21,ADVANCE='no') s(i,j)

ENDDO

WRITE(1,*)

ENDDO

WRITE(1,*)

!--------------------------------------------------------------------------------------------

WRITE(1,*)'The u distribution:'

WRITE(1,*)


DO j=1,n+1

DO i=1,m+1

WRITE(1,21,ADVANCE='no') u(i,j)

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ENDDO

WRITE(1,*)

ENDDO

WRITE(1,*)

!--------------------------------------------------------------------------------------

WRITE(1,*)'The v distribution:'

WRITE(1,*)


DO j=1,n+1

DO i=1,m+1

WRITE(1,21,ADVANCE='no') v(i,j)

ENDDO

WRITE(1,*)

ENDDO

WRITE(1,*)

!-------------------------------------------------------------------

WRITE(1,*)'The temperature distribution:'

WRITE(1,*)


DO j=1,n+1

DO i=1,m+1

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WRITE(1,21,ADVANCE='no') T(i,j)

ENDDO

WRITE(1,*)

ENDDO

WRITE(1,*)

!-----------------------------------------------------------------------

10 FORMAT('The size of the domain in x direction, x = ')

11 FORMAT('The size of the domain in y direction, y = ')

12 FORMAT('The number of segments in x direction, m = ')

13 FORMAT('The number of segments in y direction, n = ')

21 FORMAT(F13.5)

!FORMAT no.21 is to let the value using this format to be Real decimal form (for 'F') and

!the last digit is written at the 6th space (for '13') and consist of 5 decimal number after point (for

.5)

!22 FORMAT(A3,I4)_not used

END PROGRAM lid_driven_cavity_flow

Lid Driven Cavity Flow OK Final Eldwin

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