# Finding an expression for the minimum drag coefficient

• ryan88
In summary, the minimum drag coefficient, C_D, of an aircraft during straight and level flight can be found by maximizing the performance (lift-to-drag ratio) of the aircraft, which is the same as finding the minimum of the function (C_D/C_L) = (C_{D_0}/C_L) + kC_L. This occurs when the second derivative of this function with respect to C_D is less than zero.

## Homework Statement

I need to find an expression for the minimum drag coefficient, $$C_D$$, of an aircraft during straight and level flight.

## Homework Equations

$$C_D=C_{D_0}+kC_L^2$$

Where:
$$C_{D_0}$$ is the profile drag coefficent
$$C_L$$ is the lift coefficient
$$k$$ is just a constant

## The Attempt at a Solution

I started off by plotting the equation for $$C_D$$ shown above and got the following:
http://g.imagehost.org/view/0218/drag [Broken]

So I thought that the minimum drag would be when:
$$\frac{dC_D}{dC_L}=0$$

However, I remember my lecturer saying something about I would have to do:
$$\frac{d}{dC_L}\left(\frac{C_D}{C_L}\right)=0$$

But I am unsure why, I was wondering if someone could shed some light on this please?

Thanks,

Ryan

Last edited by a moderator:
What do your variables stand for?

Sorry, I have updated my first post.

Ryan

Maybe I'm still not following you. It seems as if you have two independent variables, which may or may not be related. Are the two drag coefficients (lift and profile) functions of anything or what?

Otherwise we simply have two positive numbers adding each other.

The profile drag coefficient is constant and the lift coefficient is dependant upon the angle of attack of the aircraft. Therefore as the angle of attack changes, so does the lift coefficient and also the drag coefficient.

Ryan

...OK now we're getting somewhere. How do the coefficients change as a function of angle of attack?

Ok, I have just seen my lecturer and cleared this up.

Differentiating $$C_D$$ with respect to $$C_L$$ does find the minimum drag, however this happens when $$C_L = 0$$. Since the question states that the aircraft is in straight and level flight, it is obvious that $$C_L \neq 0$$. The minimum drag during flight would be when the performance of the aircraft (i.e. the lift-to-drag ratio) is at its maximum, i.e:

$$\frac{d^2C_L}{dC_D^2} < 0$$

Drag is defined as:

$$C_D=C_{D_0}+kC_L^2$$

Dividing through by $$C_L$$ gives:

$$\frac{C_D}{C_L}=\frac{C_{D_0}}{C_L}+kC_L$$

Since this is the reciprocal of the performance (lift-to-drag ratio), finding the minimum of this function will be the same as finding the maximum of the performance, which will give the minimum drag of the aircraft.

Hence to answer this question you need to check for when:

$$\frac{d\frac{C_D}{C_L}}{dC_L}=0$$

Not:

$$\frac{dC_D}{dC_L}=0$$

Thanks minger for your help on this.

Ryan

Oh, I thought we were acutally finding the expression, which is why I wanted to know the equations.

Either way, glad you got it figured out.

## 1. What is the minimum drag coefficient?

The minimum drag coefficient is the smallest value of the drag coefficient that can be achieved for a particular object or system. It represents the amount of resistance or drag that the object experiences as it moves through a fluid, such as air or water.

## 2. Why is finding an expression for the minimum drag coefficient important?

Finding an expression for the minimum drag coefficient is important because it allows scientists and engineers to optimize the design of objects and systems for efficient movement through fluids. This can lead to improvements in areas such as transportation, energy efficiency, and sports performance.

## 3. How is the minimum drag coefficient calculated?

The minimum drag coefficient is typically calculated using mathematical equations and models that take into account factors such as the shape and size of the object, the properties of the fluid, and the velocity at which the object is moving. These calculations can be complex and may require advanced computational methods.

## 4. What are some real-world applications of the minimum drag coefficient?

The minimum drag coefficient has numerous real-world applications, including designing efficient aircraft, ships, and cars, as well as improving the performance of athletes and sports equipment. It is also important in understanding and predicting the behavior of natural phenomena such as weather patterns and ocean currents.

## 5. Can the minimum drag coefficient be reduced to zero?

No, the minimum drag coefficient cannot be reduced to zero. Even for the most streamlined and aerodynamic shapes, there will always be some level of drag due to the physical properties of fluids. However, by finding the minimum drag coefficient, scientists and engineers can strive to minimize the resistance and improve overall efficiency.