FLOW OVER BICYCLE USING MATLAB
AIM: To calculate the Drag force acting against a cyclist and to determine its behavior by varying the coefficient of drag and velocity using MATLAB
GOVERNING EQUATIONS:
The Drag equation:
Drag depends on the density of the air`(ρ)`, the square of the velocity `( u^2),`, the air's viscosity and compressibility, the size and shape of the body, and the body's inclination to the flow. In general, the dependence on body shape, inclination, air viscosity, and compressibility is very complex.
One way to deal with complex dependencies is to characterize the dependence by a single variable. For drag, this variable is called the drag coefficient, designated "`C_d`" This allows us to collect all the effects, simple and complex, into a single equation. The drag equation states that drag `F_d` is equal to the drag coefficient `C_d` times the density `ρ` times half of the velocity `u` squared times the reference area `A`.
`F_d=1/2 ρu^2 C_d A`
`F_d=`Drag force, force component in the direction of the velocity.
`ρ=` Mass density of the fluid, in this case, air =1.225 `kg/m^3`
`u=` flow speed of the object relative to the air.
`A=` frontal area or reference area.
`C_d=`Drag co-efficient
For given air conditions, shape, and inclination of the object, we must determine a value for `C_d` to determine drag.
Aerodynamicists use a number to model all of the complex items that affect drag, including the shape of an object, the size of an object, and the angle at which the object meets the air. The number scientists use is called the drag coefficient (`C_d`).
This slide shows typical values of the drag coefficient for a variety of shapes.
This slide shows the typical values of the drag coefficient for a variety of shapes. The values shown here were determined experimentally by placing models in a wind tunnel and measuring the amount of drag and the tunnel conditions of velocity and density. The drag equation was then used to produce the coefficient. The projected frontal area (the maximum cross-sectional area) of each object was used as the reference area (A). A flat plate has `C_d`= 1.28, a wedge-shaped prism with the wedge facing downstream has `C_d`= 1.14, a sphere has a `C_d` that varies from .07 to .5, a bullet has a `C_d` = .295, and a typical airfoil has a `C_d`= .045.
We can study the effect of shape on drag by comparing the values of the drag coefficient for any two objects as long as the same reference area is used and other variables are matched. All of the drag coefficients on this slide were produced in low speed (subsonic) wind tunnels. A quick comparison shows that a flat plate gives the highest drag, and a streamlined symmetric airfoil gives the lowest drag--by a factor of almost 30! Therefore, we can conclude that:
Shape has a very large effect on the amount of drag produced.
OBJECTIVES:
- To calculate the Drag force acting against a cyclist
- To plot the Velocity of the object relative to air v/s Drag force
- To plot the Drag coefficient `C_d`v/s Drag force `F_d` for different cyclist positions with the same frontal area.
CODE 1- To plot the Velocity of the object relative to air v/s Drag force :
Here the Reference area depends on what type of drag coefficient is being measured. For automobiles, bicycles, and many other objects, the reference area is the projected frontal area of the object or a vehicle. Here the frontal area `A` is taken as 0.3405 `m^2` from the research article, is calculated by taking the average of the 8 different cyclist positions (average is taken because the area does not change much and lies between the range of 0.331 to 0.35).
`C_d` is not a constant but varies as a function of flow speed, flow direction, object position, object size, fluid density, and fluid viscosity. Speed, kinematic viscosity, and a characteristic length scale of the object.
%%%%%%%%%%%%%%%%%%%%% Drag force against a Cyclist %%%%%%%%%%%%%%%%%%%%%%%
%The main factors involved are the Velocity, Frontal Area, Density of air
% Write a program to plot Velocity vs Drag force.
clear all
close all
clc
rho=1.225; %desnisty of air in [kg/m^3]
%By taking the average frontal area of 8 different 8 cyclist positions
%data is taken from reasearch article
A=0.3405; %frontal area matrix [m^2]
V=1:1:25; %velocity matrix [m/s]
%By taking the average coefficent of drag of 8 different cyclist positions
cd=0.62; %drag coefficient [dimesnionless quantity]
Drag_force=0.5*rho*A*(V.^2)*cd %Drag force on bicycle [N]or[kg*m/s^2]
%plot for velocity vs Drag froce
figure(1)
plot(V,Drag_force,'b-*','linewidth',1)
xlabel('Velocity [m/s]')
ylabel('Drag force [N]')
title('Velocity V/s Drag force')
grid on
Maximum_drag_Force=max(Drag_force)
Minimum_drag_Force=min(Drag_force)EXPLANATION FOR CODE 1:
1) clc, clear all, and close all is used to start the code as it clears the command window, clears the variables from the workspace, and closes all the plot respectively.
2) rho, A, V, cd are the input user values taken from the article.
3) plot(V,Drag_force,'b-*','linewidth',1) is used to plot the graph between velocity and drag force and 'b-*' is used to get blue solidline with * at different points, 'linewidth' is taken as 1 and can be taken any value using this.
4) xlabel and ylabel functions are used to label the x-axis and y-axis
5) title function is used to give a title to the plot
6)grid on switches on the grids on the plot.
7)max function is used to call the maximum and minimum value of drag force from the drag force coefficient.
8)% sign is used to write comments.
RESULT FOR CODE 1:
CODE 2: To plot the Drag coefficient `C_d` v/s Drag force `F_d` for different cyclist positions with same frontal area.
The figure below shows the 8 cyclist positions and the corresponding Drag coefficients
The data is taken from a research article mentioned in the references
|
Cyclist Position |
`C_d` |
|
1. Time trial top tube |
0.568 |
|
2. Top tube 4 |
0.604 |
|
3. Back down 2 |
0.588 |
|
4. Back down 1 |
0.618 |
|
5. Pantani |
0.641 |
|
6. Froome |
0.611 |
|
7. Top tube 3 |
0.655 |
|
8. Top tube 2 |
0.677 |
%%%%%%%%%%%%%%%%%%%%% Drag force against a Cyclist %%%%%%%%%%%%%%%%%%%%%%%
% Write a program to plot Drag Co-efficient vs Drag force.
clear all
close all
clc
rho=1.225; %desnisty of air in [kg/m^3]
%By taking the average frontal area of 8 different 8 cyclist positions
%data is taken from reasearch article
A=[0.3405]; %frontal area matrix [m^2]Time trial top tube
V=20; %velocity matrix [m/s]
%By taking the coefficent of drag of 8 different cyclist positions
cd=[0.568 0.604 0.588 0.618 0.641 0.611 0.655 0.677];
%drag coefficient [dimesnionless quantity]
Drag_force=0.5*rho*A*(V^2).*cd %Drag force on bicycle [N]or[kg*m/s^2]
%plot for Coefficent of drag vs Drag force
figure(1)
plot(cd,Drag_force,'r-*','linewidth',1)
xlabel('Coefficent of drag')
ylabel('Drag force [N]')
title('Coefficent of drag V/s Drag force')
grid on
Maximum_drag_Force=Drag_force(1,8)
Drag_force_at_Top_tube_2_postion=Maximum_drag_Force
Minimum_drag_Force=Drag_force(1,1)
Drag_force_at_Time_trial_top_tube_postion=Minimum_drag_Force
EXPLANATION FOR CODE 2:
1) clc, clear all, and close all is used to start the code as it clears the command window, clears the variables from the workspace, and closes all the plot respectively.
2) rho, A, V, are the input user values taken from the article.
3)Cd values are taken from 8 cyclist positions from the article.
3) plot(cd,Drag_force,'r-*','linewidth',1) is used to plot the graph between Coefficient of drag and drag force and 'r-*' is used to get red solidline with * at different points, 'linewidth' is taken as 1 and can be taken any value using this.
4) xlabel and ylabel functions are used to label the x-axis and y-axis
5) title function is used to give a title to the plot
6)grid on switches on the grids on the plot.
7)max function is used to call the maximum and minimum value of drag force from the drag force coefficient.
8)% sign is used to write comments.
RESULT FOR CODE 2:
CONCLUSION:
- As seen from the Results and Plot of the CODE 1- Velocity of the object vs Drag force, it can be seen that the Drag force varies parabolically with the `(F_d∝ u^2)` and if the velocity increases by a factor of 2 then the Drag force increases by a factor of 4 due to this more power will be required by the cyclist at higher velocities to overcome Drag.
- From the Results and Plot of the CODE 2- Co-efficient of drag vs Drag force, it can be seen that the drag force varies linearly with Co-efficient of drag`(F_d∝ C_d)`.
- From the above, we can see that the maximum drag force occurs in (1,8) cell of the Drag force matrix that happens when the cyclist position is Top tube 2
- The minimum drag force occurs in (1,1) cell of the Drag force matrix that happens when the cyclist position is time trial top tube. Hence Shape of the object even when the frontal area is the same has a very large effect on the amount of drag coefficient and hence the amount of Drag force.
REFERENCES: