What is the Geometric Balance of an Object in a Fluid Flow?

[Raparicio]

Dear Friends,

I have a problem on fluid mechanics that I can't solve.

The question is one boat that has great velocity. In front of it, it's created a wave because of the boat. Now, imagine that this boat has greater and greater velocity. The angle of the wave in the front, at great velocity, will be more orthogonal to the trajectory of the boat.

Now imagine that one circle of this wave arrounds the boat. The lines of flux that enter this circle, will be the same they go out (conservation of mass), well, how can be calculated this?

There's one formula that says shows the flux that enter and go out one surface is the same that enters and go out the volume. It could be calculate with the navier-stokes formula and the conservation of the mass.

The question is about an object that is into a flow, that is arrounded by this flow, and finally one geometric balance.

best reggards.

 

[Clausius2]

Dear Friends,

I have a problem on fluid mechanics that I can't solve.

The question is one boat that has great velocity. In front of it, it's created a wave because of the boat. Now, imagine that this boat has greater and greater velocity. The angle of the wave in the front, at great velocity, will be more orthogonal to the trajectory of the boat.

Now imagine that one circle of this wave arrounds the boat. The lines of flux that enter this circle, will be the same they go out (conservation of mass), well, how can be calculated this?

There's one formula that says shows the flux that enter and go out one surface is the same that enters and go out the volume. It could be calculate with the navier-stokes formula and the conservation of the mass.

The question is about an object that is into a flow, that is arrounded by this flow, and finally one geometric balance.

best reggards.

I am not too sure about what you want to calculate. I don't know what circle are you referring to. Maybe some picture would help a bit. Anyway, no matter how is the control surface you have defined, the integral mass conservation law is:

$$\oint_S \overline{v}\cdot \overline{dS}=0$$

The problem you are describing is a typical free surface flow. IF and only IF the boat velocity is less than the propagation speed of the superficial waves, there will be such circles around the boat (i.e. the Froude number is less than unity). As you may have seen on TV or in some pictures, usually there is a weak around the boat which has the shape of an opened triangle such a shock wave. This means the flow is supercritical in upstream the boat and transforms into a subcritical one just downstream of an small hydraulic jump. See my figure attached:

 

[Raparicio]

Dear Clausius:

The question is similar that first picture. It's like an airplane, flying at the same velocity of the sound: it generates arround it, a wave that is increasing in the time, always more radius, but never pass it. The doppler efect it's very strange in this example.

There are 2 questions that I want to understand:

1) how to calculate the doppler efect exactly at the velocity of sound
2) what's the effect of a sphere, going at a very great velocity, that makes arround it a great wave that is bigger than the sphere, and is composed of water.
3) If it exists any calculations that we can establish one balance between this mass that envolves sphere and what is arrounding it.

I don't know how to calculate it.

 

[Clausius2]

Dear Clausius:

The question is similar that first picture. It's like an airplane, flying at the same velocity of the sound: it generates arround it, a wave that is increasing in the time, always more radius, but never pass it.

You're wrong. The first picture is a subcritical flow, which is analogous to subsonic compressible flow. The wave is reaching zones of upstreaming fluid. When the flow is critical (incompressible and free surface flow) or sonic (compressible flow) the wave speed is the same than the free stream velocity, and so one should see an steady front of waves just in front of the body nose. As you may check the Froude / Mach angle in this cases is 90º.
As a cautionary note, I am comparing free surface flows with compressible flow because both behaviors are similar as far as changes in flow phenomena made by the two most important numbers ($$Fr=U/\sqrt{gH}$$ and $$M=U/c$$).

There are 2 questions that I want to understand:

1) how to calculate the doppler efect exactly at the velocity of sound

Just at sonic flow, same particle just upstream the body nose would sense a wave length 0, because no pressure information can reach upstream zones.
Don't be surprised the equations for frequency will give you a singularity. The critical /sonic flow is singular and unsteady effects are very important in these cases.

2) what's the effect of a sphere, going at a very great velocity, that makes arround it a great wave that is bigger than the sphere, and is composed of water.

I haven't understood nothing. Escríbelo en español a ver si entiendo lo que quieres decir.

 

[Raparicio]

Imagine one object at the same velocity of sound (exactly mach 1), and now translate this waves to another "fluid". It's the most approached I can explain.