Computational Fluid Dynamics Across a Cylindrical Sample

Brendan Welch

UNC Charlotte, Department of Mechanical Engineering, MEGR 3114

Fluid Mechanics

December 7, 2020

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Table of Contents

Table of Contents .......................................................................................................................... 2

Introduction ................................................................................................................................... 3

Objectives....................................................................................................................................... 3

Results and Discussion .................................................................................................................. 3

Reynold’s Number of 104 .......................................................................................................... 3

Reynold’s Number of 106 .......................................................................................................... 8

References .................................................................................................................................... 13

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Introduction

Computational fluid dynamics, or CFD, is an alternate method of gathering fluid mechanical data

to pure experimentation with exhaustive set up and procedure. According to J. Anderson in the

book Computational Fluid Dynamics: And Introduction, CFD takes the three principles on which

Fluid Mechanics is built, conservation of mass, Newton’s second law, and conservation of

momentum, and applies them to problems complex enough to require the use of some computing

software1.

CFD is a preferable approach to several problems in terms of time, labor, and cost. The amount

of resources required to set up wind tunnel testing with sensors at every location of interest and

fluids with interchangeable viscosities and pressures far exceeds the simple steps of generating a

mesh over a CAD model in a CFD software.

Objectives

This CFD project was completed to generate plots and contours of the velocity, pressure,

vorticity, and shear stress across a cylinder as well as calculating various physical parameters.

The geometry constructed in this project is a cylinder with a 100 mm diameter positioned within

a 2 m by 3 m rectangular space. The leftmost edge of this rectangle is sanctioned as the input.

The height of the cylinder was assumed to be 1 m for simplicity in calculations involving the

drag coefficient (exposed area length times diameter of 0.1 m2). This geometry was generated in

Space Claim and further analysis involving fluid flow was conducted using Fluent.

Air was selected as the input fluid with a density of 1.225 kg/m3 and a dynamic viscosity of

1.7894x10-5 Pa s. In order to achieve a laminar flow with a Reynold’s number of 104 a flow

velocity of 1.5 m/s was chosen. At this Reynold’s number, the stagnation pressure, separation

angle, and drag coefficient were all calculated and each of the previously mentioned7

contours/plots were generated.

The same geometric setup, but with a new input velocity of 150 m/s was evaluated to produce

and compare all of the same data. This velocity yields a new Reynold’s number of 106, implying

a turbulent flow.

Results and Discussion

Reynold’s Number of 104

The velocity and pressure contours at a Reynold’s number of 104 shown below (See Figures 1 &

2).

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Figure 1: Velocity contour across cylinder. Scale of velocity's ranging from 0 m/s to 2 m/s shown at left.

Figure 2: Pressure contour across cylinder. Scale of pressures ranging from 0 Pa to 1.5 Pa shown at left.

In terms of color scale, these two figures are nearly inverted images of each other. The main

high-pressure zone shown just to the left of the specimen, or at the stagnation point, in Figure 2

directly correlates with the point of lowest velocity at the stagnation point shown on Figure 1.

Based on these contours, points of high velocity correlate to points of low pressure, and vice

versa. This observation stands to reason with the example of a nozzle, where the pressure

decreases as the velocity increases.

The plot of pressure versus position is shown below (See Figure 3).

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Figure 3: Pressure versus horizontal position along mesh.

The horizontal axis of this graph corresponds to the position in the ‘x’ direction along the entire

mesh. The cylinder occupies horizontal position values from 0.95 m to 1.05 m. Based on the plot

in the figure above, the static pressure at the stagnation point is equal to 1.5 Pa.

The stagnation pressure was calculated separately using the (1), the Bernoulli Equation where

‘Px’ is the pressure, ‘vx’ is the velocity, ‘zx’ is the height of the fluid, ‘ρ’ is the fluid density, and

‘g’ is the acceleration due to gravity.

𝑃1

𝜌+

𝑣12

2+ 𝑔𝑧1 =

𝑃2

𝜌+

𝑣22

2+ 𝑔𝑧2 (1)

(1) was simplified to (2) using the following assumptions: the gauge pressure at the inlet (P1) is

zero, the velocity at the stagnation point (v2) is zero, and the heights do not change from point

one to point two (z1 = z2).

𝑣1

2

2=

𝑃2

𝜌 ∴ 𝑃2 =

𝜌𝑣12

2 (2)

Velocity v1 = 1.5m/s and fluid density ρ = 1.225 kg/m3 were plugged in to yield a stagnation

pressure of roughly 1.38 Pa. Between this Bernoulli calculated value and the value measured by

the Fluent software, the percent error is only 8%.

The zoomed in vorticity contour across the cylinder as generated by default is shown below (See

Figure 4).

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Figure 4: Vorticity contour with default range across cylinder. Default scale of roughly 0 rotations per second to

2700 rotations per second shown at left.

This vorticity contour operated off a massive range that did not accurately depict lower spin rates

that extend past the cylinder into the wake. A new range of 0 rotations per second to 100

rotations per second was chosen and a new vorticity profile was generated (See Figure 5).

Figure 5: Vorticity contour across cylinder with newly selected range of 0 rotations per second to 100 rotations per

second. Scale with this range shown at left.

With a more refined vorticity scale, fluid rotation can be seen much further downstream.

Plots of shear stress and static pressure along the arc length of the cylinder are shown below (See

Figures 6 & 7).

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Figure 6: Shear stress along arc length of cylinder. Stagnation point marked by red line. Separation point marked by

blue line.

Figure 7: Pressure along arc length of cylinder. Stagnation point marked by red line.

Since the stagnation pressure has been found to be 1.5 Pa, the stagnation point on the cylinder

was identified at roughly 0.178 m on the pressure plot above (See Figure 7). Since the shear

stress of an external flow is equal to zero at the separation point, the point of separation can be

identified to be at roughly 0.252 m on the shear stress plot above (See Figure 6). Thus, the

measured arc length between the stagnation point and separation point is equal to the difference

in the two, or 0.074 m. Using (3), the relationship between angle and arc length, where ‘s’ is the

arc length in meters, ‘r’ is the radius in meters, and ‘ϴ’ us the angle in radians, the angle of

separation was calculated.

𝑠 = 𝑟𝛳 ∴ 𝛳 =𝑠

𝑟 (3)

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Given that r = 50 mm = 0.05 m and s = 0.074 m, separation angle ϴ was found to be 1.48 radians

or 84.80°.

The total drag force exerted on the cylinder was reported by the software to be 0.11245975 N.

(4), the expression for the drag coefficient, where ‘CD’ is the drag coefficient, ‘FD’ is the drag

force, ‘ρ’ is the fluid density, ‘v’ is the fluid velocity, and ‘A’ is the surface area, was used to

calculate the drag coefficient for this Reynolds’ number.

𝐶𝐷 =𝐹𝐷

1

2𝜌𝑣2𝐴

(4)

Using (5) and the values for each physical parameter previously mentioned, a drag coefficient of

0.81633 per unit length was calculated.

Reynold’s Number of 106

The velocity and pressure contours at a Reynold’s number of 104 shown below (See Figures 8 &

9).

Figure 8: Velocity contour across cylinder. Scale of velocity's ranging from 0 m/s to 270 m/s shown at left.

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Figure 9: Pressure contour across cylinder. Scale of pressures from 0 Pa to 14 kPa shown at left.

These contours keep to the trend of high-pressure regions associating with low velocity regions

and vice versa. The stagnation pressure reported by the software at this Reynold’s number is

equal to 15 kPa. Using the Bernoulli equation solved for stagnation pressure, (2), the stagnation

pressure was calculated to be 13.78 kPa with the same 8% error.

The default vorticity contour across the cylinder and the refined vorticity contour with a range of

0 rotations per second to 500 rotations per second are shown below (See Figures 10 & 11).

Figure 10: Vorticity contour with default range across cylinder. Default scale of roughly 0 rotations per second to

181,000 rotations per second shown at left.

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Figure 11: Vorticity contour with range 0 rotations per second to 500 rotations per second. Scale reflecting this

range shown at left.

The vorticity continues much more strongly and for a much longer distance with this increased

velocity. Because of this, the range was increased further with a new maximum of 2000 rotations

per second (See Figure 12).

Figure 12: Vorticity contour with range 0 rotations per second to 2000 rotations per second. Scale reflecting this

range shown at left.

Plots of shear stress and static pressure along the circumference of the cylinder are shown below

(See Figures 13 & 14).

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Figure 13: Shear stress along the arc length of the cylinder.

Figure 14: Pressure along the arc length of the cylinder.

With the stagnation pressure equaling 15 kPa, the stagnation point can be found on the pressure

plot at roughly 0.235 m (See Figure 14). Given that shear stress is equal to zero at the separation

point, the separation point of this cylinder can be found on the shear stress plot at roughly 0.125

m (See Figure 13). Using these values to calculate the arc length, equal to 0.155 m, and (3) the

separation angle ‘ϴ’ was calculated to be 2.2 rad or 126.05°.

The net drag force on the cylinder was reported by the software to be 340.32716 N. Drag

coefficient ‘CD’ was calculated again using (4) with new velocity and drag force inputs and the

same surface area input. The resulting drag coefficient was calculated to be 0.2469 per unit

length. The drag coefficient decreased by nearly 70% by increasing the Reynold’s number from

104 to 106. Despite the substantial increase in drag force, the drag coefficient for turbulent flow is

lower because of the steeper velocity gradient at the interface of the cylinder wall.

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Additional Physical Quantities

The turbulent kinetic energy, or the kinetic energy generated by turbulent motion per unit mass,

is also attainable using the Fluent software2. The contour for turbulent kinetic energy in the flow

with a Reynold’s number of 104 is shown below (See Figure 15).

Figure 15: Turbulent kinetic energy across the cylinder. Energy scale of roughly 0 m2/s2 to 0.22 m2/s2 shown at left.

The highest points of turbulent kinetic energy appear behind the cylinder in the wake of the fluid.

Oddly enough, this area of high turbulent kinetic energy dissipation occurs within an area of

extremely low vorticity (Compare to Figure 5). This implies that the turbulent motion dissipating

kinetic energy is not necessarily the spinning of each parcel of air, but rather some other form of

fluctuation or collision.

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References

1. Anderson, Degrez, Degroote, et al. Computational Fluid Mechanics: An Introduction.

(Waterloo, Belgium, 2009). p. 1-25.

2. “Turbulence Kinetic Energy.” CFD Online. https://www.cfd-

online.com/Wiki/Turbulence_kinetic_energy