Fluid Flow over a Bicycle

Aim of the Project:- Analyze the fluid flow over a bicycle and visualize the relationship between different parameters involved in the case study in matlab.

Background Knowledge:- For fluid flow over any body, there are mainly two forces acting on the body; one is the force that is parallel to the fluid flow direction which occurs due to body moving in a viscous fluid medium and other one is the lift force which is caused due to the pressure difference between the upper and lower part of the body that causes the lift above the surface.

The forces acting on a body in a fluid flow can be seen in the figure,

Other forces are thrust force which is created by the propelling effect by different mechanisms and the weight force which is a body force.

Our analysis of this project is concerned on the drag force and its relationship with different parameters.

Drag force, also know as air resistance (in case of air in fluid flow) is the force that acts parallel to the body over which the fluid flow is taking place. Drag force is of two types, one is Form or Pressure drag and the other one is Skin Friction drag. The form drag is caused due to the shape of the body and skin friction drag is caused by the viscous effect of fluid flow over a surface.

The fluid flow over a bicycle case we are looking on has form drag dominating over skin friction drag because of the configuration of its shape and it can be seen in the figure below,

Drag force in general can be calculated as the change in pressure times the projected area,

where, P1 = Upstream pressure, P2 = Downstream pressure, A = Projected area

In general, Drag force (`F_D`) is calculated as `F_D` = (P2 - P1) * A

Governing Equation:-

The equation of the drag force `F_D` can be found out for a body having drag co-efficient `C_D`, shear stress ocuring at the surface as `τ_w`, body moving with velocity 'V' having frontal area 'A' and the fluid flowing having a density `ρ` as,

Drag co - efficient, `C_D= (Shear Stress At The Surface Of The Body)/(Dynamic Pressure)`

or `C_D= ( τ_w)/(1/2 ρV^2 )` , where `τ_w= F_D/A`

or `C_D= ( F_D)/(1/2 ρAV^2 )`

or Drag Force, `F_D= 1/2 ρAV^2*C_D`

Objectives of the project:-

• Calculate the drag force against a cyclist
• Plot Velocity vs Drag Force graph
• Plot Drag Co-efficient vs Drag Force graph

Case Study:-

Tools Used:- Matlab

In this program, the parameters of drag force over a bicycle are:-

1. Denisity of the fluid medium, `ρ`. Fluid flowing past the bicycle is air.
2. Velocity of the body relative to the fluid medium, `V`.
3. Drag Co-efficient, `C_D`.
4. Projected area of the body against fluid fow, `A`.

Part 1:- Calculation of drag force and plotting Velocity vs Drag Force graph, to observe the dependence of varying velocity of the body with drag force.

The shape of the cyclist in the direction of fluid flow is an angular cube, that can be seen as below,

The fluid flowing across the body can be seen as below,

The Drag Co-efficient of the shape of the body i.e angular cube is 0.8, can be seen below,

The code snippet of the program can be written as,

% Aim of the program is to calculate the drag force for fluid flow(i.e air) past a moving bicycle and plot Velocity vs Drag force

close all
clear all
clc

%Inputs 
%Drag co-efficient, dimentionless term
C_d = 0.8;

%Projected area of the body, m^2
A = 1;  %Assuming each side as 1 metre 

%Density of the fluid, Kg/m^3
rho = 1.27;

%Velocity of the fluid flow, m/sec
V = [1:50];

% The drag force F_d (N) can be calculated as following
F_d = 0.5*rho*V.^2*A*C_d

%Plotting Velocity Vs Drag Force graph
plot(V, F_d)
xlabel('Velocity(m/sec)')
ylabel('Drag Force(N)')
grid on 

The program for this case can be seen in matlab as,

The output of the program is different values of drag force by varrying velocity from 1 m/sec to 50 m/sec,

And the plot between Velocity and Drag Force as output is shown as below,

Inference:- As the drag force is a squaring function of the velocity, hence with increase in velocity (say 2 times), the drag force increases (4 times).

Part 2:- Calculation of drag force and plotting Drag co-efficient vs Drag Force graph, to observe the dependence of varying Drag co-efficient of the body with drag force.

The values of drag co-efficient for different bodies varies between 0 to 1; closer to 0 is for the streamlined body and closer to 1 is for the bluff body.

The value of speed of the bicycle assumed for this case is 15 mph or approx. 6.7 m/sec,

The code snippet for the program can be written as,

% Aim of the program is to calculate the drag force for fluid flow(i.e air) past a moving bicycle and plot Drag Co-efficient vs Drag force

close all
clear all
clc

%Inputs 
%Drag co-efficient, dimentionless term
C_d = linspace(0,1,50);

%Projected area of the body, m^2
A = 1;   %Assuming each side as 1 metre

%Density of the fluid, Kg/m^3
rho = 1.27;

%Velocity of the fluid flow, m/sec
V = 6.7;  %Average reasonably experience cyclist speed is 15 mph or ~6.7 m/sec

% The drag force F_d (N) can be calculated as following
F_d = 0.5*rho*V^2*A.*C_d

%Plotting Drag Co-efficient Vs Drag Force graph
plot(C_d, F_d)
xlabel('Drag co-efficient')
ylabel('Drag Force(N)')
grid on

The program for this case can be seen in matlab as below,

The output of the program is different values of drag froce by varrying drag co-efficient from 0 to 1 as below,

And the plot between Drag Co-efficient and Drag Force as output is shown as below,

Inference:- As the drag force is a linear function of the drag co-efficient, hence with increase in drag co-efficient, the drag force increases linearly.

Conclusion:-

For better performance of the cyclist, the drag force can be reduced by making the body streamlined to reduce the drag co-efficient. An illustrative example of

the same can be seen as below,