Drag force calculation for flow over a Bicycle

Introduction

Drag force is one of the single most important force in field of aerodynamics. It is the resistance for the object to move through the fluid at a velocity. Higher speed, higher the drag and it increases quadratically with changes in velocity. It can be classified into pressure drag and friction drag. It is the function of frontal area, velocity, density of fluid and Coefficient of discharge which in turn is a function of shape of the body, inclination, fluid viscosity and compressibility.

Drag Force = 0.5*coefficient of discharge * rho*Area*(velocity)^2

Octave Code

The Octave code is written to visualise the effects of velocity and coefficient of discharge on the drag force. Density and frontal area is assumed and is kept constant throughout the code. Velocity values are swept from 1ms-1 to 50ms-1 and various values of Cd for different shapes are fed into the 1D array.

In first part of the code, effect of velocity is seen on drag force while changing the Cd value in the for loop. This lets us see that the behaviour of drag force with respect to velocity is same for different shapes of body.

In second part of code, velocity is assumed constant and effect of coefficient of drag on Drag force is noted and results are plotted.

%Program is used to analyse Drag force with variaton in velocity and coefficient of discharge
%Drag effect is an important aspect from the aerodynamics perspective
%Coefficient of discharge changes for different shapes and same frontal area
%Cd values for different shapes are as follows
% Flat plate = 1.28
% Prism = 1.14
% Bullet = 0.295
% Airfoil = 0.045
% Sphere = 0.5
%Source: Nasa (grc.nasa.gov/WWW/K-12/airplane/shaped.html
% Drag force is given by the formula shown below
%drag_force = 0.5*rho*Area*v.^2*c_d;
clear all
close all
clc

%Assumptions
rho = 1.2;
Area = 0.1;

%sweeping velocity
v = [1:50];

%Cd curve array for different shapes
c_d = [1.28 1.14 0.295 0.045 0.5];


% To solve for Drag force for various values of Cd and Velocity
ct = 1;
for i = 1:length(c_d)
  drag_force = 0.5.*rho.*Area.*v.^2.*c_d(i);
  figure(ct) % comment this line if all the plots are required in same plot window
  plot(v,drag_force,'color',[rand rand rand],'linewidth',2)
  %hold on %use this function and comment figure(ct) when all curves are required
           %in the same plot
  xlabel ('velocity[ms-1]')
  ylabel ('Drag Force[N]')
  grid on
  ct = ct+1;
  printf('figure(%d) is for Cd value of %f \n',i,c_d(i)) %For user ref in command window
end

% To find the effect of Cd on drag force, assuming constant velcoity and frontal area
%Assumptions 
v1 = 100;
for j = 1:length(c_d)
  drag_force1(j) = 0.5*rho*Area*v1^2*c_d(j);
end
%plotting Coefficient of discharge against Drag force for different shapes
figure(ct)
plot(c_d,drag_force1,'color','r','marker','o','linewidth',3)
xlabel ('Coefficient of discharge')
ylabel ('Drag Force[N]')
title('Coefficient of discharge vs Drag force')
grid on
printf('minimum drag force is %f for Cd value of %f\n', min(drag_force1), min(c_d))
printf('Minimum drag force is experienced by a streamlined body and maximum by Plate\n')

Results

Sample plot for the code is attached here.

Discussion on results

1) Drag force varies quadratically with changes in velocity

2) Shape of body has a significant impact on the drag force

3) Lower the coefficient of discharge, lower is the Drag force for same frontal area, velocity and density

4) Streamlined bodies have lesser drag coefficient and hence can attain higher velocities for same amount of drag

5) Above result is one of the reason for design of cars and aeroplane as a streamlined body

6) Higher drag results in increased fuel consumption and hence not economical

7) However, drag is not all bad and sometimes used for advantage like in race cars with spoliers