Alex Wright

100963406

1

MECH 5001: THEORY OF VISCOUS FLOW

ASSIGNMENT 1

Alex Wright

100963406

ABSTRACT

A cross flow over a cylinder in a quasi-3-dimensional domain

was simulated using FINE/Hexa. A mesh independence study

was conducted to discover a mesh with greater than 300,000

cells was necessary to accurately resolve physical flow

structures. The simulation convergence was adversely affected

by the use of a steady state, time invariant solver for a naturally

dynamic physical flow. The drag force was used to conduct the

mesh independence study. The drag coefficient was determined

to be approximately 0.8 based on the simulation. This is in

contrast to the empirically determined value of approximately 1

for this flow with Reynolds number on the order of 103. A

combination of factors can be attributed to the discrepancy

between observed and expected results. They include the domain

geometry and the computational strategy. This exercise can be

considered a success despite the poor quality of results obtained

via numerical simulation.

INTRODUCTION

The purpose of this assignment is to introduce the students

to the practice of computational fluid dynamics by investigating

a cross flow over an infinite cylinder. This is a simulation that

has been investigated in great detail experimentally and

numerically. This experiment is popular for its geometric

simplicity while providing a variety of flow structures in the

wake of the cylinder at various flow conditions.

Experimental investigations have developed an empirical

relationship between the Reynolds number and the drag

coefficient of the flow. The relationship is shown in Figure 1.

Figure 1: Reynolds number versus Drag Coefficient for an infinite

cylinder. (Weisstein)

Depending on the Reynolds number, the relationship

between the pressure driven drag forces and the total observed

drag force can change. For very low Reynolds numbers the drag

is dominated by skin friction and 1

. At moderate

Reynolds numbers, the total drag is roughly equal to the pressure

driven drag and = 1. At very high Reynolds numbers, the flow becomes turbulent modifying the fluid properties in the

boundary layer. At this point, there is a brief drop as separation

is delayed. The drag coefficient then continues to increase with

Reynolds number.

This study will use the drag force as a method of mesh

independence validation. The observed drag will be compared to

the expected value determined by the equation

= 1

22

Where , the drag coefficient, is determined from Figure 1, is the fluid density, is the free stream velocity and is the interfacial area perpendicular to the flow.

METHODS

The domain for this study was modeled in Pro|Engineer

Wildfire 4.0, a solid modeling package. The domain is shown in

fig 2, which includes the dimensions

Figure 2: The domain geometry. DCylinder=10mm (Matida)

The solid model was triangulated and imported into the

domain meshing software Hexpress 2.10-4. This software

develops fully hexahedral unstructured meshes for complex

geometries. The software allows for mesh refinement around

desired geometry features as well as automatic insertion of a

viscous boundary layer of cells to ensure a non-dimensional wall

distance of y+=1.0 for accurate modeling of the viscous layer.

The flow solver used in this study was FINE/Hexa 2.10-4.

It is an unstructured, density based, finite volume solver which

solves the Reynolds Averaged Navier Stokes equations.

Convective fluxes were discretized via Roes second order upwind scheme. Diffusive fluxes were discretized via the central

difference scheme. The general Navier-Stokes equation solved

by FINE/Hexa is

Alex Wright

100963406

2 Copyright 2014 by The Crown in Right of Canada

+

S

S

=

Where is the control volume, is the control surface, is

the set of conservative variables, is the advective fluxes, is the diffusive fluxes and contains the source terms. These are further defined in the Theory Manual for FINE/Hexa.

(Numeca International) The relevant chapter is included in

Appendix A for convenience.

The boundary conditions are outlined in Figure 4. The inlet

velocity was 6 m/s, orthogonal to the inlet plane. The outlet

pressure was set as 101300 Pa, or 0 Pa gauge. The domain walls

were set as inviscid in order to approximate a 2-dimensional flow

using this quasi 3-dimensional domain. The cylinder wall was a

standard wall with a no-slip condition applied. These conditions

yield a Reynolds number of approximately 4x103, where 1 according to Figure 1.

Figure 4: Domain Boundary Conditions. Inlet - Green, Outlet - Red,

Domain Walls - Blue, Cylinder Wall - Black.

MESH INDEPENDENCE

This study was initiated with a very coarse mesh of only

8256 cells. The mesh was successively refined in 5 steps up to a

maximum of 518896 cells. The meshes are shown in Figure 3.

The intermediate meshes are listed in Table 1. Table 1 relates the

number of cells to the drag force on the cylinder calculated by

the flow solver. The empirical expectation of the drag force from

= 1

22 is 0.000425 N.

Table 1: Mesh Independence Data

Simulation Cells FD (x10-4)N

1 8256 4.61

2 31764 3.14

3 127808 4.60

4 326696 3.23

5 518896 3.78

6 264299 3.26

The poor convergence of the drag force can be attributed to

the unsteady nature of the simulation. The coarse meshes

converged well because they could not properly resolve the

dynamic vortex shedding in the wake of the cylinder. As the cell

count passed 100,000, the mesh became fine enough to resolve

the vortices. The vortices are physical and are known as von

Karman vortices. The vortex shedding made steady state

convergence difficult to attain. The drag force calculated by the

solver would oscillate as the vortices developed. While the mesh

independence criterion did not converge to the expected value,

they did converge to a value near 0.00032 N. Therefore, one may

assume that the simulations are mesh independent around

300,000 cells and could capture physical phenomena. The

deviation from the expected value for drag force may have been

due to the choice to use a quasi-3-dimensional model rather than

a truly 3-dimensional or 2-dimensional model for an infinite

cylinder. The numerically calculated drag force yield a drag

coefficient of around 0.8.

Figure 3: Computational Meshes. From top to bottom: Coarsest Mesh,

Finest Mesh, and Finest Mesh near Cylinder

Alex Wright

100963406

3 Copyright 2014 by The Crown in Right of Canada

DISCUSSIONS AND RESULTS

The following figures are from simulation 5, which had

approximately 500,000 cells.

Figure 5: Velocity Vector Plots. Top: Whole Domain. Bottom: Zoomed

in near Cylinder Wake

Figure 5 shows the velocity vector plot at the central x-y

plane of the domain. The unsteadiness of the wake can be clearly

observed in this plot.

Figure 6: Magnitude of Velocity with Velocity Streamlines

Figure 6 shows the magnitude of velocity contour plot with

velocity vector streamlines. This plot clarifies the instability in

the wake of the cylinder. The streamlines indicate the direction

of flow and can clearly be observed forming discrete vortex

cores. The streamlines illustrate the complexity of the flow in the

wake as well as the vortex cores developing therein.

Figure 7: Total Pressure Contour Plot

The total pressure contour plot in Figure 7 displays the low

pressure zone within the wake of the cylinder. The lower total

pressure indicates a loss of energy in the flow as it passed the

cylinder. The loss is a combination of skin friction and viscous

mixing in the wake.

Figure 8: Static Pressure Contour Plot and with Velocity Streamlines

The static pressure contour plot in Figure 8 shows regions

of the wake with lower static pressure. Fluid is drawn to regions

of low static pressure from regions of higher static pressure.

These low pressure regions are clearly indicated by the velocity

streamlines as vortex cores. As this simulation approaches a

dynamics steady state, the vortex shedding would become more

regular in size and location. These simulations had not reached

the dynamic steady state.

Alex Wright

100963406

4 Copyright 2014 by The Crown in Right of Canada

CONCLUSIONS

The simulation of cross flow over a cylinder using a quasi-

3-dimensional domain was an interesting exercise with a myriad

of challenges and fascinating results. The dynamic nature of the

flow did not lend itself to a simple simulation. The simulation

had a Reynolds number around 4x103 which is in the laminar

flow regime wherein vortices are periodically shed in the wake

of the cylinder. The vortex shedding resulted in fluctuations of

convergence criteria for the steady state solver. Regardless of the

convergence difficulties, the dynamic flow structures were

resolved when the mesh was fine enough (n>300,000). The

vortex cores have been shown to be regions of low static pressure

separating from the aft end of the cylinder. In regard to the drag

force and the discrepancy observed between the computed and

expected value could be due to the geometry of the domain as

well as the dynamic nature of the flow. A quasi-3-dimensional

domain does not allow for true 3- or 2-dimensional observation

of the phenomenon. The cylinder therefore was not truly

representative of an infinitely long cylinder. A better strategy for

simulating this flow would be to use mirror or periodic boundary

conditions on the domain side walls and use an unsteady flow

solver to capture the dynamic vortex shedding.

REFERENCES

Matida, Edgar. "Assignment 1." Ottawa, January 2014. Numeca International. "Theoretical Manual FINE/Hexa."

Brussels: Numeca International, February 2010.

Weisstein, Eric W. Cylinder Drag. 4 February 2014.

.