Symmetric Converging-Diverging Nozzle Areas?

[nakas12]

Hey guys I was wondering if a symmetric C-D nozzle has the same area at the inlet and exit, can the mach numbers at both points be the same?

This situation would involve a normal shock which would be near the exit as well. I would assume the Mach number at the throat would be 1.

With that said in terms of area ratios between area 1 and 2 at the inlet and exit and the throat, that the ratios would be equal to each other where:

A1/A* = A2/A*

A* = Throat area

Looking at the isentropic flow table I would assume both mach number equal to each other.

 

[jambaugh]

The Mach number typically will not be the same. In fact given a shock is the transition from sub to super-sonic flow the Mach number on one side will be M>1 while on the other M<1. The normal shock breaks the symmetry of the problem and changes the stagnation point of the flow. So even with a second shock reverting the flow to subsonic you will get via change in stagnation point, a different Mach number.

Recall that for subsonic flow increasing area decreases flow velocity but for supersonic increasing area implies increasing flow velocity.

In the absence of a shock you would have in the ideal case equal Mach numbers.

 

[nakas12]

Hmm ok. Well the problem I have been looking at involves a subsonic inflow and subsonic outflow. At the throat I am sure the Mach number will be 1 and will increase until the shock. After the shock the flow will be subsonic.The problem I am having is that I've been given the outflow and inflow pressure ratio of P2/P1 = .90 and a polynomial equation of A(x) = 1 - 0.8x + 0.8x^2; where the cross sectional area varies with the equation given. The length of the nozzle is 1 meter.

I solved for the Area of the throat to be .8. Once I plugged in X = 0.5 since its in the middle of the nozzle

If I solved for it at the inlet and exit locations I would get the same area ratios which is confusing me alot.

 

[boneh3ad]

Assuming inviscid flow, without a shock the Mach numbers will be the same but since you are assuming there is a shock there then you can't make that assumption. It is no longer isentropic in that case since, obviously, there is a shock there.

 

[nakas12]

When I saw this problem though it said to assume isentropic flow except across the normal shock. Really the only problem I'm having is understand how to find the mach number at the inlet

 

[jambaugh]

What bon3head said,
You need to use isentropic flow between the shock and each end independently then determine the ratios at the shock. It has been a while since I took compressible flow but I recall one needs to express things in terms of stagnation pressure/temp, etc and then determine the ratio of stagnation values on each side of the shock front where they change due to the non-isentropic nature of the shock.

The fact that you are given (for a symmetric nozzle) a ratio of pressures which is not 1 should be significant to you.

 

[nakas12]

Ok I think I understand now. I just have one last question if you guys don't mind answering.

I'm going to assume M = 1 at the throat. The area ratio for that is 1. A/A* =1. Would the ratio be referring to the area at the inlet over the throat area? Just want to make sure I'm doing this correctly

 

[jambaugh]

Ok I think I understand now. I just have one last question if you guys don't mind answering.

I'm going to assume M = 1 at the throat. The area ratio for that is 1. A/A* =1. Would the ratio be referring to the area at the inlet over the throat area? Just want to make sure I'm doing this correctly

A* is defined as the area at which M=1 so that is what you are assuming, that A=A*. As to the validity of that assumption, check your notes and text but I believe that is correct. As the subsonic speed must increase as the area decreases but the supersonic speed must decrease, it can't stay supersonic if it decreases from its M=1 value. Thus for a stable shock to appear with speed increasing approaching and leaving the shock, the area must decrease up to and then increase past the shock.

 

[nakas12]

But if you see the polynomial equation of A(x) = 1 - 0.8x + 0.8x^2; where the cross sectional area varies with the equation given. I already figured out A/A* which gives me an entirely different value from the ratio at the throat. The length of the nozzle is 1 meter and the throat area from this equation would give 0.8 and throat inlet would be 1 meter.

Basically I am getting 2 different A/A* values. One from the Mach number at the throat and one from the polynomial equation

I've discussed this with some others at class and none of them have any idea about how to solve this.

 

[boneh3ad]

Well if you know that there is a shock somewhere in the nozzle, then the nozzle must be choked and assuming sonic flow a the throat is valid. As for A/A*, that value will vary depending on where you are in x in the nozzle. There isn't a single value that describes the nozzle.