Simulation of Moffat Vortices in Two-Dimensional Lid-Driven

Cavity Flows

ME 412 Fluent Project 1

Lihan Xu, Jingwei Zhu

February 14, 2014

Instructor: Surya Pratap Vanka

1 Problem Description

Moffat vortices are often observed in cavity flows when one of the boundaries is inflicted with shear

stress. In this project, Moffat vortices will be examined in two-dimensional lid-driven cavity flow.

The cavity flow chosen here is square shape. We will assume the top wall of the quadrilateral is

moving while the other three remain quiescent. Reynolds number is set to be 1000 which indicates

laminar flow condition. In addition, the grids (number of control volumes) used in the simulation

will be varied from 20×20, 40×40, 80×80. The main objectives of this project can be summarized

as:

• Simulation of Moffat Vortices in a square shape cavity flow.

• Test the sensitivity of simulation results by varying the grids.

• Improve understanding of numerical computation of fluid dynamics.

• Examination of flow physics.

2 Details of the Code

Fluent 14.5 was used for the simulation.

Regarding the solver, we chose pressure-based type, absolute velocity formulation, steady in time.

The flow of interest is planar.

We set the side length of the square to be 1 meter. The maximum face size was set to be 0.05m,

0.025m and 0.0125m to achieve 20 × 20, 40 × 40 and 80 × 80 grid respectively. Since the problem

is two-dimensional and Reynolds number is 1000, we assumed the flow to be laminar.

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Regarding the fluid we used in our calculations, we chose air which has density of 1kg/m3 and

viscosity of 0.001kg/m · s.

The boundary on the top was set to be a moving wall whose speed is 1m/s to the right. The rest

of the boundary was set to be stationary.

Regarding the solution methods, SIMPLE scheme was used for pressure-velocity coupling. As for

spatial discretization, we selected least squares cell based gradient, standard pressure and second

order upwind momentum.

Initialization method was chosen to be standard and we computed from the top since this part of

boundary is moving while the rest is stationary. Initial values were all set to zero. Convergence

criteria were all set to 1 × 10−3. We set the number of iterations to 1000.

3 Numerical Parameters

Air density and dynamic viscosity were assumed to be 1kg/m3 and 0.001kg/m · s respectively. The

size of the square cavity was set to be 1m× 1m. Thus for Reynolds number of 1000, velocity of the

shearing boundary was calculated to be 1m/s using the equation below:

Re =ρ · V · L

µ(1)

The parameters used in this numerical analysis are listed in the table below.

Flow regime Reynolds number Density [kg/m3]

Laminar 1000 1

Viscosity [kg/m · s] Square Length [m] Boundary Velocity [m/s]

0.001 1 1

Table 1: List of numerical parameters

4 Computational Times

Iterations and CPU time required to complete computation for each grid are shown in the Table

2.

5 Observations of Numerical Behavior

Numerical solutions are calculated based on each mesh elements. Finer mesh usually gives rise to

more accurate results. On the other hand, finer mesh requires longer computation time and con-

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Grid Iterations Computational time (s)

20 × 20 159 3.9

40 × 40 323 8.3

80 × 80 693 23.8

Table 2: Iterations and CPU time to complete computation

sequently more cost. Therefore there is a trade-off between accuracy and cost. Figure 1 compares

the mesh behavior of the 2-D square cavity for different grid sizes. As we can see, the larger the

grid size, the finer the mesh is.

Figure 2 shows the residual convergence process for the three grid sizes. For 20 × 20 case, mass

residuals reach 1× 10−3, the criterion for convergence we set, after 159 iterations; for 40× 40 case,

mass residuals are 1 × 10−3 after 323 iterations while for 80 × 80 case, 693 iterations is required to

reach mass residuals of 1×10−3. As we can see, larger grid size requires more iterations to complete

computation and thus longer calculation time. The number of iterations required to reach the same

level of convergence will double when grid density increases fourfold.

6 Discussion of the Flow Physics

Streamlines of the cavity flow for the three grid sizes are shown in Figure 3. They are colored by

stream function.

Line plots of y-velocity along x direction as well as x-velocity along y direction for the three grid

sizes are shown in Figure 4. The comparison of y-velocity in different locations y=0.1, 0.3, 0.5,

0.7, 0.9 and comparison of x-velocity in locations x=0.1, 0.3, 0.5, 0.7, 0.9 are shown in two figures

respectively for each grid size (a) (b) (c), represented by the five colored lines.

As we can see, for each grid size, the centerline seems to have the highest velocity magnitude

compared to other locations. The velocity deceases in the direction of pointing away from the

centerline. Also, for each line, the magnitude of velocity goes through a process that it decreases

to zero first and then increases in inverse direction. This is mainly caused by the vortices in which

the center velocity is zero while velocity away from the center gets higher. The velocities at edges

are bounded by boundary conditions.

Comparison between different grid sizes is plotted in Figure 5. Centerline velocity profile was used

for both x and y directions. We can see that the magnitude gets bigger as the grid size increases.

Contours of pressure for the three grid sizes are shown in Figure 6.

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(a) 20 × 20

(b) 40 × 40

(c) 80 × 80

Figure 1: (a)(b)(c): Mesh behavior for three grid sizes

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(a) 20 × 20

(b) 40 × 40

(c) 80 × 80

Figure 2: (a)(b)(c): Residual convergence for three grid sizes

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(a) 20 × 20

(b) 40 × 40

(c) 80 × 80

Figure 3: (a)(b)(c): Streamlines of the cavity flow for the three grid sizes

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(a) 20 × 20

(b) 40 × 40

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(c) 80 × 80

Figure 4: (a)(b)(c): Line plots of velocity along y and x direction for the three grid sizes

Figure 5: Comparison of velocity at the centerline of the square for different grid sizes

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(a) 20 × 20

(b) 40 × 40

(c) 80 × 80

Figure 6: (a)(b)(c): Contours of pressure for the three grid sizes

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According to Figure 3, we observed that there are 3 vortices in the cavity flow. The major one is

at the center of the square cavity while the other two are at the two bottom corner of the cavity.

The major vortex is clockwise and the two vortices at the corner are counterclockwise.

From Figure 6, we can see that the highest pressure in the flow occurs at the upper right corner of

the cavity. The flow driven to the right by the moving wall gets blocked by the wall on the right at

the upper right corner and thus it is being compressed to very high pressure. The lowest pressure

occurs at the center of the major vortex, which is colored by blue. This is because the pressure

drops sharply as approaching the vortex.

References

[1] Surya. P. Vanka, “ME 412 Fluent project manual,” Spring 2014.

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