 Open Access
 Total Downloads : 0
 Authors : Banamali Dalai
 Paper ID : IJERTCONV6IS16005
 Volume & Issue : RDME – 2018 (Volume 06 – Issue 16)
 Published (First Online): 05012019
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Solution of Incompressible Viscous Flow in a LidDriven Cavity using CrankNicholson Scheme
Solution of Incompressible Viscous Flow in a LidDriven Cavity using CrankNicholson Scheme
Banamali Dalai
Centre for Advanced PostGraduate Studies, Mechanical Engineering, Biju Patanaik University of Technology,
Odisha, Rourkela, India,
AbstractThe stream functionvorticity form of the unsteady NavierStokes equation is solved using second order accurate in space and first order accurate in time CrankNicholson scheme in a finite difference mesh. The scheme shows implicit in nature. The geometry of the problem is a liddriven cavity. The solution is obtained up to maximum Reynolds number 22,500 in the grid size 129×129 and 257×257. The flow pattern is studied in the form of stream function & vorticity contours, central velocity profiles and location of the eddies.
Keywords – Unsteady, Incompressible, CrankNicholson
In this paper, the NavierStokes equation is solved in a liddriven cavity using CrankNicholson scheme. The Crank Nicholson scheme has the flexibility of taking the average value at n+1 and n time steps. This scheme is implicit in nature.

FORMULATION
The governing equation in stream functionvorticity form is expressed as:
scheme, Finite difference scheme.
1
2
2

INTRODUCTION

u v

t x y
Re x2
2
y
(1)
The liddriven cavity is a square cavity in which all the walls except the top lid are fixed. The top lid is allowed to
The stream function equation is expressed as:
2 2
move towards right with nondimensional velocity unity. Initially, the cavity is filled with fluid which is at rest. When the lid starts moving towards right the fluid flow in the cavity
x2
y 2
(2)
is set up. The fluid is assumed to be viscous and incompressible.
In 1966, Burggraf[1] studied the viscous flow in two dimensional liddriven cavity. He solved the stream function and vorticity form of the NavierStokes equation up to
Where , and Re represent the stream function,
vorticity and Reynolds number respectively. u and v represent the components of velocities along x and ydirections respectively.
The boundary conditions for the liddriven cavity are:
Reynolds number 400 in the grid size 40×40 and 60×60. He has proved analytically that the vorticity value at the centre of the primary vortex will not exceed 1.8859. Later Erturk et
At y 0, 0 x 1, u v 0; At y 1, 0 x 1, u 1, v 0;
(3)
al.[2] solved the stream functionvorticity form of unsteady
NavierStokes equation up to maximum Reynolds number
At x 0, 1; 0
y 1, u v 0
21000 in the grid sizes 401×401, 501×501 and 601×601. Their solution for vorticity value at the centre of the primary
Eq.(1) & (2) are discretised using CrankNicholson
scheme in finite difference uniform mesh.
vortex crosses the theoretical limit of 1.8859 within
n1 n
un n1 n1
n n
Reynolds number 21000. Erturk & Gockol[3] solved the
ij ij ij ij 1 ij 1 ij 1 ij 1
stream functionvorticity form of the NavierStokes equation using fourth order accurate alternating direction implicit
t 2
2x
2x
vn n1 n1
n n
method upto Reynolds number 20,000 in the grid sizes
ij i 1 j i 1 j
i 1 j i 1 j
2
401×401, 501×501 and 601×601. Here Erturk and
Gockols[3] solution for vorticity value at the centre of the
primary vortex does not cross the theoretical limit of 1.8859. Erturk[4] solved the stream functionvorticity form of the NavierStokes equation using GaussSeidel iteration techniques in a grid size 1025×1025. He observed that the vorticity value at the centre of the primary eddy does not cross the theoretical limit of 1.8859 within Reynolds number 20000.
2y
2y
n1
n1 n1 n
n n
various Reynolds numbers are shown in Table.1. There are
1
ij 1
2.0 ij
2
ij 1
ij 1 2.0
ij
2
ij 1
small eddies at the corners along the lower wall of the cavity. As the Reynolds number increases, the stream function
Re
n
2x
2.0 n1 n1
2x
n 2.0 n n
(4)
contours in the primary eddy become circular in nature and its centre moves towards the geometric centre of the cavity.
i 1 j ij i 1 j i 1 j ij i 1 j
As the Reynolds number increases more and more, the stream
2y 2
2y 2
functions in the primary eddy become more circular in nature
and the centre of the primary eddy moves towards the
The discretised equation (3) is first order accurate in time and second order accurate in space.
Similarly, the stream function equation (2) is discretised using CrankNicholson scheme as:
1 n1 2.0 n1 n1
geometric centre of the cavity as shown in Fig.1. The number of secondary eddies increase in the lower corner of the cavity. The number of secondary eddies become more than the right corner as the Reynolds number increases. The number of secondary eddies also increase in the left vertical wall of the cavity after Reynolds number 2500 as shown in Fig.1. The
n1
ij 1 ij ij 1
2
ij
n 2.0 n n
x 2
n1 2.0 n1 n1
vorticity contours also show large variation near to the walls
of the cavity. The centre of the primary eddy becomes constant vorticity core with increase of Reynolds number as
ij 1 ij ij 1 i 1 j ij i 1 j
shown in Fig.2. The radius of the constant vorticity core
x 2
n n n
y 2
increases with increase of Reynolds number.
i1 j 2.0 ij
y 2
i1 j
(5)
TABLE I: Location and strength of the centre of the primary eddy
Re
Grid
x
y
Ref.
100
129×129
0.617
0.734
0.100
3.046
257×257
0.617
0.738
0.102
3.121
400
129×129
0.563
0.609
0.107
2.156
257×257
0.555
0.606
0.111
2.231
1000
129×129
0.531
0.563
0.107
1.855
257×257
0.531
0.566
0.113
1.966
601×601
0.530
0.565
0.119
2.067
Ref[2]
2500
129×129
0.523
0.539
0.101
1.632
257×257
0.523
0.543
0.112
1.814
601×601
0.520
0.543
0.121
1.974
Ref[2]
5000
129×129
0.524
0.531
0.101
1.632
257×257
0.511
0.535
0.108
1.706
601×601
0.515
0.536
0.122
1.936
Ref[2]
7500
129×129
0.523
0.531
0.084
1.312
257×257
0.516
0.531
0.104
1.633
601×601
0.513
0.532
0.122
1.919
Ref[2]
10000
129×129
0.523
0.531
0.078
1.211
257×257
0.516
0.527
0.101
1.573
601×601
0.512
0.530
0.122
1.908
Ref[2]
12500
129×129
0.523
0.523
0.073
1.129
257×257
0.516
0.527
0.098
1.522
601×601
0.512
0.528
0.121
1.899
Ref[2]
15000
129×129
0.523
0.523
0.069
1.060
257×257
0.516
0.527
0.095
1.477
601×601
0.510
0.528
0.121
1.893
Ref[2]
17500
129×129
0.523
0.523
0.065
1.001
257×257
0.516
0.527
0.093
1.436
601×601
0.510
0.527
0.121
1.887
Ref[2]
20000
129×129
0.523
0.523
0.062
0.095
257×257
0.516
0.523
0.090
1.399
601×601
0.510
0.527
0.121
1.881
Ref[2]
22500
129×129
0.523
0.523
0.059
0.906
The equation (4) and (5) are second order accurate in space. Equation (4) and (5) produces the vorticity and stream function values respectively in new time steps (n+1). Applying Eq.(3) in Thoms formula[1], the first order accurate boundary values for vorticity can be obtained. The discretised boundary conditions are represented as:
ij
2
p
ij 1
ij
O(h)
for j 0 and all i;
ij
2
h 2 ij
ij 1
O(h)
for j N and all i;
ij
2
p
i1 j
ij
O(h)
for i 0 and all j &
ij
(6)
2
h 2
ij 1
ij
O(h)
for i N and all j
Where N is the total number of grid points along i and j directions respectively. Here the boundary conditions are first order accurate in space. Since the discretised equations are implicit in nature, the change of time step does not have any effect to the convergence. The solution of the discretised equation (4) and (5) are obtained using GaussSeidel iteration techniques with relaxation parameter. The convergence of the solution is assumed to be the residue of discretised equation
(1) and (2) approaches towards 1010. The grid meshes used for the solution are 129×129, 257×257.


RESULTS & DISCUSSION
The solution for the incompressible viscous flow in the liddriven cavity is obtained up to Reynolds number 22,500 in both grid sizes 129×129 and 257×257. It is observed that at lower Reynolds number 100, the stream function contour forms closed structures and its centre is towards right near the lid. This is named as primary eddy of the cavity. The location and strength of the primary eddy for
The u and vvelocity profiles along the centre of the cavity at different Reynolds numbers are shown in Fig.3 and 4 respectively. Both the velocity profiles match very well up to Reynolds number 5000. At Reynolds number 10,000 and 20,000 the computed velocity profiles do not match perfectly with Erturk et al[2]. This happens due to large grid size difference with Erturk et al[2]. The computed velocity profiles are at grid size 257×257 whereas Erturk et als.[2] grid size is 401×401. Here the important point is to be
observed that the vorticity contour plot at Reynolds number 20,000 show zigzag nature near the top right corner of the cavity in the grid size 257×257 which shows the insufficient grid resolution at that Reynolds number because the u & v velocity profiles do not match with Erturk et al.[2]. So the CrankNicholson scheme seems to produce better result than Erturk et als[2] result but accuracy is the main point of concern.
Figure: 1 Stream function contour plots at various Reynolds numbers
Figure: 2 Vorticity contour plots at various Reynolds numbers
Figure: 3 uvelocity profiles along the centerline of the cavity at different Reynolds numbers
Figure: 4 vvelocity profiles along the centerline of the cavity at different Re
From Fig.5 it is seen that the vorticity value in the grid size 257×257 crosses the theoretical limit value of – 1.8859 near to Re=2000 which is much earlier than Erturk et al[2]. So, it seen that though the high Reynolds number
solution is obtained at lower grid size but the accuracy is very poor. From Fig.6 comparing the magnitude of the velocity values along the centre of the cavity, it seen that as the Reynolds number increases the length of the central portion of linear profile increases. The linear nature of the velocity profile shows the constant vorticity region which increases with increase of Reynolds number. The velocity profiles become closer to the wall with increase of Reynolds number which indicates the presence of boundary layers on the walls. This type of flow feature can also be verified from the vorticity contour plot in Fig.2.
Figure: 5 Vorticity value at the centre of the primary eddy with Re
Figure: 6 u and vvelocity profiles along the centreline of the cavity with Re
IV CONCLUSION
The stream function vorticity form of the Navier Stokes equation is solved using CrankNicholson scheme. The solution is obtained up to maximum Reynolds number 22,500 using the grid size 129×129 and 257×257. The Crank Nicholson scheme produces better result at lower grid size than that of Erturk et al but the accuracy of the former scheme is very poor.
REFERENCES
[1] R Burggraf, Analytical and numerical studies of the structure of steady separated flows, Journal of Fluid Mechanics 24 pp.113151 (1966). [2] E. Erturk, T. C. Corke and C Gockol, Numerical solutions of 2D steady incompressible driven cavity flow at highe Reynolds numbers, Int. Journal of Numerical Methods in Fluids 48 pp. 747 774 (2005). [3] E Erturk and C. Gokcol, Fourthorder compact formulation of NavierStokes equations and driven cavity flow at high Reynolds numbers, Int. J. for Num. Methods in Fluids. 50 pp. 421436 (2006). [4] E Erturk, Discussions on driven cavity flow. Int. J. for Num.Methods in Fluids. 60 pp. 275294 (2009).