1

Computational Fluid Dynamics

Assignment B

1 INTRODUCTION

A common CFD application is the simulation of external aerodynamics. In the automotive industry, it is

important to make sure the drag of the vehicle is kept as low as possible, as it can lead to savings in fuel

consumption and shape design. The Ahmed [1] model is often used in experiments to represent car shapes

due to its simple geometry and the ease of varying parameters.

The aim of this simulation is to achieve the following goals:

1. Create a representative 3-dimensional model geometry of an Ahmed car, within a computational fluid domain with key design parameters

2. Create and perform quality CFD simulations of the 3D Ahmed [1] car model and to extract meaningful data

3. Gain an understanding on model requirements for turbulent flows and the importance of y+ values to satisfactorily capture the boundary layer in these conditions

4. Understand the usage of boundary layers and size functions 5. To learn how to compare and discuss results with published experimental results

1.1 Problem description and report outline

This report will conduct a number of different fluid flow simulations for a single 3D Ahmed car model

[1] and acquire the necessary data. Figure 1 below illustrates a typical Ahmed car body design.

Figure 1: Side view (left) and front view (right) of the Ahmed car model with slant angle ()

In order to conduct a comprehensive analysis of the above problem; the Ahmed car model will be

subjected to four different flow simulations throughout this report. Initially, the model will be tested with

a 30 slant angle under two different turbulence models; k-Epsilon and k-Omega. Furthermore, the

Ahmed car model will be tested with a reduced slant angle under the k-epsilon turbulence model. The

final simulation will analyse the flow characteristics of the Ahmed car model with the addition of a car-

spoiler under the k-epsilon turbulence model.

2

2 MODEL DESCRIPTION AND CREATION

2.1 Model Geometry and Dimensions

In order achieve an accurate simulation of airflow over the car body; a 3D computational domain must be

constructed with the car body enclosed within. Figure 2 and table 1 illustrate the geometry and

dimensions of the computational domain. These domain coordinates will provide an adequate

computational space to resolve and capture the required airflow data prior to reaching the outlet of the

domain.

Figure 2: 2D geometry of the computational domain

Table 1: Computational domain dimensions

In addition to the 2D geometry shown above, a 3D extrusion of 4000 mm depth is used to complete the

computational domain.

Figure 3 illustrates the geometric design and associated dimensions of the Ahmed car body within the

computational domain. Initially, the car body will include a slant angle of 30 degrees; however the slant

angle is a variable subject to change depending on the simulation model. The 2D geometry shown below

is extruded to a depth of 194.5 mm to achieve the desired 3D geometry.

Figure 3: Car body geometry and dimensions

V1 (mm) L2 (mm) L3 (mm) L4 (mm)

3000 50 16000 6300

3

2.2 Boundary Conditions

The overall geometry in question is composed of seven named section boundaries based on the

computational domain and car body design. The seven boundaries include the inlet, outlet, sky, ground,

symm-plane, side-plane, and car-body. Figure 4 illustrates the overall geometry of the Ahmed car model

simulation and the corresponding boundaries.

Figure 4: 3D Ahmed car model geometry

Density: = 1.185 [kg m^-3]; Dynamic Viscosity: = 1.831E-05 [kg m^-1 s^-1].

Reynolds number of 2.3 106. The flow turbulence intensity is set to 1.8%.

2.3 Mesh Details

Figures 5 & 6 illustrate the details of the boundary layer mesh (Inflation) and size functions settings

(sizing).

Figure 5: Boundary layer mesh (Inflation)

Figure 6: Sizing function settings (Sizing)

4

In order to save time while meshing, it is very useful to estimate the distance between the wall and the

first grid node (y1). To estimate the y1 value based on a desired y+ value, ANSYS CFX recommends the

use of the following equation:

1 = + 74

13 14 Where:

L is the flow length scale.

ReL is the Reynolds number based on the model length scale. Car length in this case.

In the case of the k-epsilon turbulence model; ANSYS CFX requires that:

(+) < 300

In this simulation, a conservative value of y+=150 will be used as the desired y+ value.

Therefore, based on the above formula and considerations; the estimated first inflation layer height (y1) is:

1 = 1.6 []

Figure 7: Mesh inflation

5

2.4 Mesh Quality

In order to achieve an accurate mesh of high quality one must focus on three fundamental measurements

of quality: Skewness, Smoothness, and Aspect Ratio [5].

The skewness of a grid is an appropriate indicator of the mesh quality and suitability [5]. Large mesh

skewness will effectively compromise the accuracy of the interpolated regions [5]. While there are several

methods for determining the skewness of a grid; this report will focus specifically on the method of

Equiangular skewness[5].

Equiangular skewness is defined as:

[ 180

,

]

Where:

max = largest angle in the face or cell. min = smallest angle in the face or cell. e = angle for an equiangular face/cell (e.g., 60 for a triangle, 90 for a square).

Equiangular skewness varies within a range from zero to one (0 1), with zero being the best possible scenario, while a skewness of one is almost never desirable. Table 2 1 illustrates a more detailed

breakdown of a typical skewness range.

Table 2: Typical skewness range

Value of Skewness Cell Quality

1 Degenerate

0.9 - 1 Bad

0.75 - 0.9 Poor

0.5 - 0.75 Fair

0.25 - 0.5 Good

0 - 0.25 Excellent

0 Equilateral

Smoothness refers to the change in size of the cell. For an accurate high quality mesh, it is important to

minimize any sudden jumps in cell size. A large sudden change in size will result in incorrect results at

nearby nodes [5].

Figure 9: Basic smoothness example

min

max

Figure 8: Basic skewness example

6

Aspect ratio is a ratio of the longest to the shortest side of a cell. An ideal aspect ratio is equal to one.

Maintaining an aspect ratio close to one is extremely important as it minimizes interpolation error and

achieves accurate multidimensional flow [5].

Figure 10: Basic Aspect ratio example

Figure 11& 12 illustrate a 2D and 3D visual overview of the mesh surrounding the car and the

computational domain. It is clear from the below images that the mesh in question maintains a relative

consistency in the change in size between cells. The mesh gradually reduces its cell size as it approaches

the more complex geometry surrounding the car body; this is necessary as the mesh needs to be refined

into smaller cells to capture more complicated flow patterns around the car body.

Based on a visual observation; the mesh does not achieve a perfect aspect ratio equal to one. However,

the majority of mesh cells appear to have an aspect ratio very close to what would be considered ideal.

The mesh also appears to maintain its cell aspect ratio over the gradual change in cell size.

Figure 11: 3D body mesh over computational domain

Figure 12: Mesh around car body

7

The mesh skewness can be observed analytically. Figure 13 below illustrates a set of numerical data

extracted from the Ahmed car simulation; this data is crucial to equiangular mesh skewness.

Figure 13: Mesh details

Based on the data presented in figure 13 and the skewness range demonstrated in table 2 it can be

observed that the average equiangular skewness of [0.2253] with a deviation of [0.1221] lies with the

excellent quality category while sometimes deviating into the good quality category.

3 AHMED CAR SIMULATION (30 SLANT ANGLE)

This section of the report will present and discuss all relevant observations and results generated by the

ANSYS CFX simulation of the Ahmed car body with a 30 slant angle. The model will be subjected to

both k-epsilon and k-omega turbulence models.

The main objectives are as follows:

- Understand and discuss the importance of y+ (dimensionless wall distance) values in turbulent

flows.

- Compare and discuss the simulated drag coefficient CD and lift coefficient CL of the CFD Ahmed car model against the experimental data of Watkins and Vino [3].

3.1 K-Epsilon vs. K-Omega Turbulence Models

Turbulence modeling is the construction and use of a model to predict the effects of turbulence. The K-

Epsilon (k-) and K-Omega (k-) turbulence models are amongst the most commonly used turbulence models in CFD; due to their ease of use, and low cost computation. Both are considered two equation models meaning that they include two extra transport equations to represent the turbulent properties of the flow [2].

The k- turbulence model has been shown to be useful or free shear layer flow applications with small pressure gradients. Accuracy has a tendency to reduce for flows containing large pressure gradients. K- models have been proven inadequate for flows with strong curvature, strong buoyancy effects, and strong

swirls [2].

8

Figures 14 & 15 demonstrate the y+ contour effects on the car body for the two turbulence models. While

both images appear almost identical; keen observation suggests that the k- model achieves a slightly higher y+ range in comparison to the k- model. However, it appears that the k- model maintains a higher range and more uniform contour as the flow moves of the slant angle.

Figures 16 & 17 depict pressure contours for the two turbulence models. Visually, it is difficult to

determine any differences between the two models. However, it appears that the k- model achieves a miniscule maximum pressure range; while the k- model has a slightly lower minimum pressure range.

Y+, the non-dimensional wall distance for a wall-bounded flow states that the average velocity of a

turbulent flow at a particular point is directly proportional to the logarithm of the distance from that point

to the wall or the boundary of the fluid region [6]. The following formulae explain the relationship

between Y+ and the dimensionless velocity of the flow:

+ =1

+ + +, where + =

Figure 15: Y+ contour plot (k-) Figure 14: Y+ contour plot (k-)

Figure 17: Pressure contour plot (k-) Figure 16: Pressure contour plot (k-)

9

Similar to the previous plots; the velocity vector plots depicted in figures 18 & 19 have almost identical

visual flow characteristics. Velocity ranges are near identical.

Figures 20 & 21 illustrate the streamline plots of the two turbulence models. Keen observation suggests

that the k- turbulence model has developed a slightly denser turbulent swirl in the wake of the car body and a slight increase in velocity.

Figure 19: Velocity vector plot (k-) Figure 18: Velocity vector plot (k-)

Figure 21: Streamline plot (k-) Figure 20: Streamline plot (k-)

10

The turbulent kinetic energy (TKE) plots illustrated in figures 22 & 23 demonstrate major differences

between the two turbulence models. Visual observation suggests that the k- model experiences a greater build-up of turbulent kinetic energy as the flow passes of the front of the car body; resulting in a much

larger distribution of energy. The k- model on the other hand experiences almost no distribution of energy over the car body; instead achieving a greater energy distribution in the wake of the car body. Also,

the k- model achieves a maximum TKE almost twice that of its comparator.

Figures 24 & 25 represent turbulent kinetic energy plots in the wake of the car body. While both models

have identical contour ranges; the k- appears to have less energy development in the wake of the car body than the k- model; the energy appears to disperse. However, similar to the previous TKE plot illustrated in figures 22 & 23 the k- model achieves a larger distribution in the wake, just not as intense as the comparator model.

Figure 23: TKE plot symm-plane (k-) Figure 22: TKE plot symm-plane (k-)

Figure 25: TKE plot car body (k-) Figure 24: TKE plot car body (k-)

11

3.2 Lift (CL) and Drag (CD) Coefficients

In the automotive industry, the coefficients of lift and drag are defined as:

=

1

22

=

1

22

Where A is the reference area perpendicular to the drag force; which in this case of an automobile is the

frontal area. However, this report will utilize ANSYS CFX to solve the lift and drag coefficients

numerically using the following CFX expression definitions:

Figure 26: CFX expression definitions

According to the expression definitions illustrated in figure 26 the coefficients of lift & drag for the

Ahmed car body with a 30 slant angle is as follows:

Table 3: Lift & Drag Coefficients

However, experimental data provided by Watkins and Vino; [3] which underwent testing similar to the

simulation of the Ahmed car body presented above suggest that the Drag and Lift coefficients should be

in the range of 0.32 and 0.5 respectively. Why is this so? Why is there such a large difference between

the experimental and simulation results?

After thorough observation of the literature provided by Watkins and Vino, [3] the reasoning behind the

variation in results is clear. The experiment undertaken by Watkins and Vino involved a setup to test the

drafting effects on multiple car models. The addition of a second car model and the consequential drafting

effect generated from this setup resulted in a considerable decrease in drag and an increase in lift. It is

also apparent that the experiment conducted by Watkins and Vino utilized a test speed of 35 m/s; slightly

higher than the test speed of 34.04 m/s used in the Ahmed car simulation demonstrated in this report.

4 AHMED CAR SIMULATION (12.5 SLANT ANGLE)

This section of the report will present and discuss all relevant observations and results generated by the

ANSYS CFX simulation of the Ahmed car body with a 12.5 slant angle. The model will be subjected to

the k-epsilon turbulence model only.

The main objectives are as follows:

- Obtain the drag coefficient CD for the Ahmed car model with reduced slant angle using the k-Epsilon

turbulence model only.

Drag Coefficient (CD) 0.4879

Lift Coefficient (CL) 0.2994

12

- Analyse and discuss the differences of the drag coefficient CD between the experimental fluid dynamics (EFD) data [1] and the Ahmed car simulation.

4.1 30 vs. 12.5 slant angle

Figures 15 & 28 depict Y+ contour comparison between the 30 and 12.5 Ahmed car models. The main

visual difference between the two contours is the presence of a more evenly distributed contour over the

12.5 slant. This is most likely dude to a more accurate simulation as the k- turbulence model performs better over smaller curvatures.

The most apparent difference between the pressure contours illustrated in figure 17 & 27 is the lack of

pressure build-up at the top slant edge for the 12.5. The two contours appear near identical otherwise.

The 12.5 model also experiences less stagnation over the slant and achieves less flow separation and less

pressure change.

The streamline plots representing the two slant angles seen in figures 21 & 30 have a very apparent

difference. Due to the reduced slant; the 12.5 model has much larger and more aggressive turbulent

vortices in the wake of the car body. This also results in a slightly reduce flow velocity that is less

concentrated as the flow enters the wake as seen in the velocity vector comparison in figures 19 & 31.

Figure 28: Y+ contour Figure 27: Pressure contour

Figure 30: Streamline plot Figure 29: Velocity vector plot

13

Figures 23, 25, 31 and 32 demonstrate the turbulent kinetic energy plot comparison between the 30 and

12.5 slant angles. The 12.5 model has a significant increase in TKE by approximately 30 [m2 s-2]. The

12.5 model also appears to have a much more aggressive energy distribution in the wake of the car body.

4.2 Lift and drag coefficients

Using that same method demonstrated in section 3.2 the Lift and Drag coefficients for the Ahmed car

model with a 12.5 slant angle are as follows:

Table 4: Lift and Drag Coefficients

According to the results gathered from the SAE paper and experimental fluid dynamics data (EFD) [1] of

the Ahmed car; a CD value of 0.23 is more appropriate for the Ahmed configuration demonstrated in this

report. Why is there such a significant difference between the experimental and numerical results?

While the Ahmed configuration may be similar to that in the simulation and have an identical slant angle;

there is one considerable difference that accounts for the drastic variation in CD values. The test section in

the SAE paper utilized a Reynolds number of 4.29 million; almost double the Reynolds number of 2.6

million used for the Ahmed simulation in this report [1]. In order to achieve a Reynolds number of this

magnitude, the test was performed at a wind speed of 60 m/s. Increasing the wind speed and Reynolds

number by almost double that used in the simulation results in a drastic increase in drag.

5 CAR-SPOILER MODEL

This section of the report will present and discuss all relevant observations and results generated by the

ANSYS CFX simulation of the Ahmed car body with a 30 slant angle and the addition of a car-spoiler.

The model will be subjected to only the k-epsilon turbulence model.

The main objectives are as follows:

- Calculate lift and drag coefficients for the car-spoiler model

- Compare and discuss the difference of the CD and CL data between the single Ahmed car with a 30 slant angle and the car spoiler model.

Drag Coefficient (CD) 0.4663

Lift Coefficient (CL) 0.0461

Figure 32: TKE symm-plane Figure 31: TKE car body

14

The above figures suggest that the addition of a spoiler increases flow rate over the slant as well as

increased pressure and energy dissipation in the wake.

Using that same method demonstrated in section 3.2 the Lift and Drag coefficients for the Ahmed car

model with a 30 slant angle and mounted car-spoiler are as follows:

Table 5: Lift & Drag Coefficients

Observation of the above contour plots and Lift & Drag coefficients; in comparison with the material

presented in section three; suggests that the car-spoiler simulation results in an increase in drag and a

slight decrease in lift. This increase in and drag and decrease in lift results in flow down force on the car

body; forcing the car body into the ground.

Drag Coefficient (CD) 0.6055

Lift Coefficient (CL) 0.2644

Figure 33: Y+ contour Figure 34: Pressure contour

Figure 36: TKE symm-plane Figure 35: TKE car body

15

6 CONCLUSIONS

The CFD simulations conducted in this report demonstrate the effects of external aerodynamics on an

Ahmed car body with varying geometry. It is apparent that the meshing method used in these simulations

is effective as it resulted in an accurate mesh with little deviation in consistency within its three main

quality assurance characteristics; skewness, smoothness, and aspect ratio. Observations of the 30 slant

model under different turbulence conditions suggest that both the k- and k- experience similar results and in some cases converge; however, the TKE plots depict a clear dissipation effect in the wake of the

car body for the k- model. The comparison of the 30 and 12.5 models suggest that the 30 slant achieves a greater flow separation and pressure difference to pressure stagnation over the slant; resulting

in increased lift. Observation of the car-spoiler comparison leads to the conclusion that the addition of a

spoiler results in higher flow speeds over the slant and more energy dissipation in the wake of the car. It is

also apparent that the main reasons for the variations in lift and drag coefficients between the CFD

simulations and the experimental procedures seen in Watkins & Vino and the SAE paper lies within the

setup of the test section. The experiment conducted by Watkins and Vino utilized a drafting setup and a

slightly increased flow velocity; while the experiment conducted in the SAE paper used a flow velocity

and Reynolds number almost double that of what was used in the simulation.

7 REFERENCES

[1] S. R. Ahmed, G. Ramm, Some salient features of the time-averaged ground vehicle wake, SAE-Paper

840300, 1984.

[2] K-epsilon models. (2011, june 18). Retrieved from CFD Online: http://www.cfd-online.com/Wiki/K-

epsilon_models

k-omega models. (2011, 10 12). Retrieved from CFD Online: http://www.cfd-online.com/Wiki/K-

omega_models

Bakker, A. (2006). Applied Computational Fluid Dynamics - Meshing. Retrieved from bakker:

http://www.bakker.org/dartmouth06/engs150/07-mesh.pdf

J.Y. Tu, G. Y. (n.d.). Computational Fluid Dynamics - A Practical Approach. UK: Elsevier Science

Limited.