Drag Force acting on a cyclist using MATLAB

AIM: To calculate the Drag force against a cyclist and plot the characteristics using MATLAB.

GOVERNING EQUATION: The equation to calculate the Drag force is as follows:

`F_d = frac{1}{2} .rho .A .v^2. C_d`

where;

`F_d` = Drag force (N)

`rho` = Density of air (kg/m³)

`A` = Frontal area (m²)

`v` = Velocity (m/s)

`C_d` = Drag coefficient (unitless)

OBJECTIVE:

To calculate the Drag force and plot the characteristics of:

1. Velocity vs Drag force and interpret the results.

2. Drag coefficient vs Drag force for different bodies and interpret the results.

THEORY: In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. This can exist between two fluid layers (or surfaces) or a fluid and a solid surface. Unlike other resistive forces, such as dry friction, which are nearly independent of velocity, drag forces depend on velocity. Drag force is proportional to the velocity for a laminar flow and the squared velocity for a turbulent flow. Even though the ultimate cause of a drag is viscous friction, the turbulent drag is independent of viscosity. Drag forces always decrease fluid velocity relative to the solid object in the fluid's path.

SOLUTION:

Programming Language: MATLAB

Code #1: The MATLAB program to plot Velocity vs Drag force is shown below.

% Program to plot Velocity vs Drag force acting on a cyclist.

% Inputs:

% Density of air (kg/m^3)
rho = 1.225;

% Frontal area (m^2)
A = 1;

% Velocity (m/s)
v = (1:30)

% Drag coefficient (unitless)
C_d = 0.47;


% Solution:

% Drag force (N)
F_d = 0.5 * rho * A * v.^2 * C_d


% Plot:

plot(v,F_d)
xlabel('Velocity (m/s)')
ylabel('Drag force (N)')
title('Velocity vs Drag force')
grid on

Output #1: Different values of Drag force (F_d) are computed for a range of Velocities (v) by keeping a constant frontal area as shown below.

Plot #1: Drag force (F_d) varies in a parabolic fashion with respect to the Velocity (v) as shown below.

Code #2: The MATLAB program to plot Drag coefficient vs Drag force is shown below.

% Program to plot Drag coefficient vs Drag force acting on a cyclist.

% Inputs:

% Density of air (kg/m^3)
rho = 1.225;

% Frontal area (m^2)
A = 1;

% Velocity (m/s)
v = 15;

% Drag coefficient (unitless)
C_d = [0.47 0.42 0.50 1.05 0.8 0.82 1.15 0.04 0.09]
label = {'Sphere','Half-sphere','Cone','Cube','Angled cube','Long cylinder','Short cylinder','Streamlined body','Streamlined half-body'}


% Solution:

% Drag force (N)
F_d = 0.5 * rho * A * v.^2 * C_d


% Plot:

plot(C_d,F_d,'marker','s','markerfacecolor','r')
text(C_d,F_d,label)
xlabel('Drag coefficient (unitless)')
ylabel('Drag force (N)')
title('Drag coefficient vs Drag force')
grid on

The Drag coefficients for different shapes are shown below.

Output #2: Different values of Drag force (F_d) are computed for various Drag coefficients (C_d) according to their respective shapes by keeping a constant frontal area as shown below.

Plot #2: Drag force (F_d) varies in a linear fashion with respect to various Drag coefficients (C_d) as shown below. Streamlined bodies should be preferred as they tend to contribute less to the Drag force as compared to the short cylinder and cube.

CONCLUSION: From the above results, we can infer that the Drag force is:

1. Directly proportional to the Velocity square.

2. Directly proportional to the Drag coefficient.

Hence, the Drag force should be as low as possible as it tends to lower the fuel economy and efficiency of the moving object.

REFERENCES: Image Source