Fluid Mechanics: Acceleration Field

[Seraph042]

Homework Statement

A picture of the problem is attached

Assume that the streamlines for the wingtip vortices of an airplane can be approximated by circles of radius r, and that the speed is V = K/r where K is a constant

Determine the streamline acceleration a_s and the normal acceleration a_n for this flow.

Homework Equations

$$\vec{a}$$ = V dV/dS $$\hat{s}$$ + V²/R $$\hat{n}$$

The Attempt at a Solution

The solution has it that a_s = 0, which I can understand because dV/dS is 0 for this problem, but it says that a_n = K/r³, which doesn't make sense unless R = K*r because V² = K² / r²

Where did I go wrong?

 

[KataKoniK]

it is important to carefully analyze the given problem and equations before attempting to solve it. In this case, it seems that the given solution may have made a mistake in the derivation of the normal acceleration an. Let's take a closer look at the problem and equations to see if we can find the correct solution.

First, we are given that the streamlines for the wingtip vortices can be approximated by circles of radius r and that the speed is V = K/r. This means that the velocity is inversely proportional to the radius, which is consistent with the formula V = rω for circular motion.

Next, we are given the equations for the streamline acceleration as and the normal acceleration an. The streamline acceleration is given by:

\vec{a} = V \frac{dV}{dS} \hat{s} + \frac{V^2}{R} \hat{n}

where V is the velocity, dV/dS is the rate of change of velocity along the streamline, and R is the radius of curvature of the streamline.

Now, let's consider the normal acceleration an. This is given by:

an = \frac{V^2}{R}

Substituting the given equation for V, we get:

an = \frac{K^2}{r^2 R}

This is where the mistake in the given solution may have occurred. The correct formula for an in terms of K and r is:

an = \frac{K^2}{r^3}

This can be derived by substituting V = K/r into the equation for an and simplifying. Therefore, the correct solution for an is an = K^2/r^3, not K/r^3 as given in the solution.

In conclusion, as a scientist, it is important to carefully check and verify the equations and solutions before accepting them as correct. In this case, it seems that there was a mistake in the given solution, and the correct solution for the normal acceleration an is an = K^2/r^3.

 

[Mene]

I would first check the assumptions made in the problem. Is it appropriate to approximate the streamlines as circles? Is the speed equation accurate for this particular scenario? If the assumptions are valid, then I would proceed with the solution as follows:

The streamline acceleration can be calculated using the equation given in the problem statement:

\vec{a} = V \frac{dV}{dS} \hat{s} + \frac{V^2}{R} \hat{n}

Since dV/dS is 0 for this problem, the streamline acceleration is also 0.

To calculate the normal acceleration, we first need to find the radius of curvature, R. Since we are assuming that the streamlines are circles, the radius of curvature is equal to the radius of the circle, r. Therefore, R = r.

Substituting this into the equation for normal acceleration, we get:

an = \frac{V^2}{R} = \frac{K^2}{r^2}

This is the correct equation for normal acceleration in this scenario. The error in the given solution is that they have used V2 = K2 / r2, which is not correct. The correct equation is V2 = K/r, as stated in the problem statement.

In conclusion, there was a mistake in the given solution and the correct equation for normal acceleration is an = K2/r2. It is important to carefully check all assumptions and equations when solving a problem in fluid mechanics to ensure accurate results.