Urmi Roy said:
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.
Urmi Roy said:
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.
Urmi Roy said:
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.