Aerodynamics and hydrodynamics

[niko2000]

Hi,
Let's tie a sphere, drop it into a water and trawl it with a boat. How do we calculate the angle between a vertical line and rope?
Actually I don't know much about aerodynamics and hydrodynamics so I don't know how drag force is calculated from shape, mass and speed of an object. Recently I have started using Simulink and Matlab so it would really help me to get a some formula so I could try to do some model.
Thank you,
Regards,
Niko

 

[da_willem]

I don't know much about aerodynamics and hydrodynamics

Well you need quite a bit of knowledge of these fields to properly do the calculations. For instance engineers usually model the drag force by using the empirical relation:

$$F_D=C_D(Re)A\frac{1}{2}\rho v^2$$

With A the frontal surface of your sphere. rho the density of water and v your velocity. The drag coefficient ([itex]C_D[/tex]) depends on the Reynolds number so imlicitly on the velocity. If you want to find the drag force for a certain fixed velocity of the boat you could calculate the Reynolds number from its definition:

$$Re=\frac{\rho v D}{\mu}$$

With D the characteristic length scale, in your case the diameter of the sphere and [itex]\mu[/tex] the dynamic viscosity of water. Next you could look up the drag coefficient in a graph or something, only for certain Reynolds numbers (<1) this can be analytically found:

$$C_D= \frac{24}{Re}$$

But for Re<1 your boat would have to move very! slowly. For 1E3<Re<2E5 the drag coëfficiënt is approximately constant (~0,4). See: http://www.uh.edu/engines/spheredrag.jpg for an example of how to find the drag coefficient from the Reynolds number.

With the drag coëfficient you can calculate the drag with the first formula. But again, I think a good calculation involves a lot of knowledge of the fields you mention and will be quite labourous, so you might still want to change your mind...

 

[da_willem]

Well, giving it another thought. If you discard the effect of the boat on the water, so you can evaluate the flow as homogeneous, a first approximation is not too difficult.