Formation of shock waves and gas dynamics of flow in a nozzle.

[Urmi Roy]

Hi I have a few questions regarding gas dynamics...

1. when M=1, dA=0...what does this mean? Does it imply that if we have a converging (subsonic) nozzle with its exit mach no,say 0.8, and then direct the flow to a constant area tube, then we have to attain M=1?

2. When we know the length of a duct, and the entering mach no, can we predict from only this information whether there will be a shock or not? (i.e what is the minimum info we require to predict whether there will be a shock or not?).

3. Is it true that while a subsonic flow can be continuously accelerated to supersonic flow, the converse (i.e continuous deceleration of a supersonic flow to subsonic speed) is not true and has to take place via a shock wave?(but I thought this was possible in a diverging diffusor...

Thanks a lot.

 

[Mech_Engineer]

I can't answer your shock questions, but for the first one it seems to me when you're considering a nozzle and you are given the conditions M=1 and dA=0, this means in my mind it is a boundary condition where the speed at the nozzle's throat (where the rate of change of the nozzle's cross-sectional area approaches zero) is Mach 1. In other words, the nozzle is a subsonic nozzle accelerating the flow to Mach 1, a.k.a. choked flow.

Whether the speed MUST be M=1 is dependent on geometry, but a tube is not a nozzle so it seems to me it will not have any further acceleration of the flow.

 

[Urmi Roy]

Thanks for your answer, Mech_Engineer :-)

The inference in the first question comes from the equation in red on
http://exploration.grc.nasa.gov/education/rocket/nozzle.html

To me, it seems that for subsonic flow with increasing area causes flow velocity to decrease whereas for supersonic flow with increasing area, the flow velocity decreases whereas a kind of singular/detached phenomenon is when M=1, and from this, we simply gather that dA=0...it doesn't seem to have any connection with the other cases.

dA could be zero even if we had a maximum area at a crossection, and the passage were diverging-converging...

 

[boneh3ad]

1. when M=1, dA=0...what does this mean? Does it imply that if we have a converging (subsonic) nozzle with its exit mach no,say 0.8, and then direct the flow to a constant area tube, then we have to attain M=1?

Is there a context for this? If this is in the context of a nozzle, then Mech_Engineer hit it on the head. If you have a subsonic converging nozzle that ends such that the exit Mach number is 0.8, it will never reach sonic conditions without converging further. So in the case you mention, $M=1$ where $dA=0$, the upstream stagnation pressure is such that at the exit of the converging portion of the nozzle, the Mach number is unity. This would be the throat of the nozzle in the case of a converging-diverging nozzle.

2. When we know the length of a duct, and the entering mach no, can we predict from only this information whether there will be a shock or not? (i.e what is the minimum info we require to predict whether there will be a shock or not?).

No. Imagine you have adiabatic flow through a constant-area duct and you have the conditions at the entrance (station 1) and the exit (station 2). For now, don't worry about if they are the same or not. Starting from conservation of mass, momentum, energy and the equation of state, we have between the two stations:
Mass:
$$\rho_1 u_1 = \rho_2 u_2$$
Momentum:
$$\rho_1 u_1^2 + p_1 = \rho_2 u_2^2 + p_2$$
Energy:
$$c_p T_1 + \frac{1}{2}u_1^2 = c_p T_2 + \frac{1}{2}u_2^2$$
State:
$$\frac{p_1}{\rho_1 T_1} = \frac{p_2}{\rho_2 T_2}$$

There is obviously the trivial solution where $u_1 = u_2$, etc., but we are more interested in shocks, which represent a discontinuity here. So let's assume then that $u_1 \neq u_2$. In compressible flow, we generally like to work in the variables $M$, $p$, and $T$, so we can change the previous four equations into three equations in terms of these variables and solve for $M_1^2/M_2^2$:
$$\frac{M_1^2}{M_2^2} = \frac{\left(1+\gamma M_1^2\right)^2\left(1+\frac{\gamma-1}{2}M_2^2\right)}{\left(1+\gamma M_2^2\right)^2\left(1+\frac{\gamma-1}{2}M_1^2\right)}$$

Which is a quadratic equation with two solutions. One is simply what I mentioned earlier, or $M_1 = M_2$. The other is a discontinuity, or the shock. In other words, if you know the area of a duct and the incoming Mach number, you still have two solutions: one where the exit mach number is the same and one where it is different. To know whether there is a shock, you need to know that you have a discontinuity there. Most often, you know because you have a stagnation pressure difference between the inlet and the outlet. The important thing, though, is that you just need something to indicate the need for a flow discontinuity.

3. Is it true that while a subsonic flow can be continuously accelerated to supersonic flow, the converse (i.e continuous deceleration of a supersonic flow to subsonic speed) is not true and has to take place via a shock wave?(but I thought this was possible in a diverging diffusor...

In general, you can't continuously decelerate a supersonic flow. In theory this is possible, but practically it isn't the case. If you had an established supersonic flow, you could have your supersonic diffuser designed such that you move back to Mach 1 at the second throat and then either have an infinitesimally weak shock bring it subsonic speeds and slow down through the diverging section. Problem number one is that you can and often do have shocks in the converging section of the diffuser, which can alter your flow.

The bigger problem, however, is that in a real flow in a supersonic wind tunnel, you have to start the tunnel first. When you do this, the starting shock (itself a normal shock) makes its way into the test section, and downstream of this the flow is subsonic. That means that it actually speeds up in the converging section, so in order to pass the mass flow of the nozzle through the diffuser as well, the diffuser throat has to be sized so as to go at most sonic at the Mach number downstream of the starting shock. This means the diffuser throat is necessarily much larger than that needed to bring the steady flow down to sonic conditions. What you end up with is a supersonic diffuser that slows the flow down and reduces the strength of the normal shock at its exit, but can never practically slow it down without a shock.

 

[Urmi Roy]

Thanks for your help Boneh3ad :-)

Is there a context for this?

Yes, please see pg 84 in the pdf http://ebooksgo.org/engineering-technology/fluidmechanics.pdf ...it is within the chapter variable area ducts...

In any case, I think I understand Mech_Engineer's point.

To know whether there is a shock, you need to know that you have a discontinuity there. Most often, you know because you have a stagnation pressure difference between the inlet and the outlet. The important thing, though, is that you just need something to indicate the need for a flow discontinuity.

So unless we look for things that are hard to find, like stagnation pressure loss etc, we can't know if there would be a shock or not?
Given a particular numerical problem in gas dynamics, even if its in real life, we couldn't just see the setup and tell if its design or off -design?

 

[boneh3ad]

So unless we look for things that are hard to find, like stagnation pressure loss etc, we can't know if there would be a shock or not?
Given a particular numerical problem in gas dynamics, even if its in real life, we couldn't just see the setup and tell if its design or off -design?

Those aren't even hard to measure. At any rate, you usually won't even need to other than to confirm your design is functioning properly. For example, in a supersonic wind tunnel, you know both stagnation pressures because you have to know what pressure you are putting in and what you are exhausting to (typically either atmosphere or near-vacuum) in order to design the tunnel. You can get the upstream total pressure from a simple Pitot probe or from a pressure transducer in the air supply line and you can get the stagnation pressure of your exhaust reservoir through another pressure transducer. Simple.

In a more general sense, you need to know the conditions. It isn't enough just to say "you have air moving at Mach 2 through a duct, is there a shock?" If there doesn't need to be a shock, there won't be, so you need some sort of indication that nature will require a shock. Just looking at a constant-area duct and knowing a Mach number gives you none of that insight.

 

[Urmi Roy]

In a more general sense, you need to know the conditions.

Right...so that includes stagnation pressure, temperature, perhaps the static values of these also...we just need the values of these quantities at certain cross-sections (usually inlet and test sections)...

Whether the speed MUST be M=1 is dependent on geometry, but a tube is not a nozzle so it seems to me it will not have any further acceleration of the flow.

We have a problem in our book which says :
"Air at p0= 10 bar, T0= 400K is supplied to a 50mm diameter pipe (constant area). The friction factor for the pipe (Fanno flow) for the pipe surface is 0.002...if the mach no changes from 3 at the entry to 1 at the exit, determine...etc etc..."

So, as Mech_Engineer said, for a constant area tube, you'd expect no change in mach no. However, in this example, clearly the mach number changes...is this just because there is friction?
Also, since friction causes increase in entropy, wouldn't we expect the mach no. to increase along the pipe, instead of dropping from 3 to 1?


Similarly, is it possible to intuitively explain why heat addition in a supersonic flow results in decrease of velocity?

 

[Urmi Roy]

I think what I'm looking for is an intuitive explanation of these two lines:

" In other words, subsonic flow through a pipe with friction will accelerate, and supersonic flow will decelerate."

"Adding heat to a fluid flowing at subsonic velocities in a pipe will cause the flow to accelerate, and adding heat to supersonic flow in a pipe will cause the flow to decelerate"