[Fluid dynamics] Normal shock wave problem

[Nivlac2425]

Homework Statement

Air, supplied by a reservoir at 450 kPa, flows through a converging-diverging
nozzle whose throat area is 12 cm^2. A normal shock stands where A1 = 20 cm^2. (a) Compute
the pressure, p2, just downstream of this shock. Still farther downstream, where A3 = 30
cm^2, estimate (b) p3, (c) A3*, and (d) M3.

(There is no figure given) (We can also assume air as an ideal gas, and isentropic conditions everywhere except at the shock wave)

Homework Equations

Various isentropic and normal shock relations

The Attempt at a Solution

Ok, so I've been stuck on this problem (and this topic) for a while now. I'm sort of struggling with understanding compressible flow in general, so I'm going to go by my current understanding of this stuff to explain what I did so far, please please correct me where I'm wrong!

a) Since there is a normal shock wave downstream of the throat, the flow must be supersonic past the throat, and therefore the flow should be choked and the throat is in critical state (*) with M=1.

-So I first need to get p1, static pressure at 1, by use of isentropic relations(and tables):
A1/A* = function of(M1) = 1.667
gives M1=1.99

-And another relation:
pt/p1 = function of(M1) , where pt is stagnation pressure aka reservoir pressure
gives p1=58.413 kPa

-And now, for conditions across the shock:
M2 = function of(M1) = 0.579

-Making use of a normal shock wave relation:
p2/p1 = function of(M1, M2)
gives p2=260.16kPa

b)c)d) For these parts, I'm completely lost. I don't know what happens after the shock, particularly what is A2* and A3*? I thought (*) refers to some critical state? I'm also confused about stagnation pressure at each of these states, like state 2 and state 3. Originally, stagnation pressure, pt, was defined as the "reservoir" pressure pretty much before the flow; so what does pt2 and pt3 refer to?

I'm sorry that I have such a bad understanding of this topic.. my professor and TAs helped pretty much with nothing, as far as completing the homeworks or doing the exams.. It would be a lifesaver if I got some clearer explanation of compressible flows and normal shocks. I feel like this is a pretty typical problem, but I don't have a firm grasp of the material yet!

Thanks!

 

[SteamKing]

The star (*) refers to the state of the gas or one of its properties when M = 1.

The attachment gives theory and some worked examples:
http://web.iitd.ac.in/~pmvs/mel7152008/notes-new.pdf

 

[Nivlac2425]

Thanks for the resources! I'll go through that and see what I can learn before I come back for more questions

 

[Nivlac2425]

So I'm not sure why I'm not understanding this yet, but I think I need some help..

I am still stuck on finding the static pressure p3 at point 3. I know that between points 2 and 3, we can assume things are isentropic. I also found the stagnation pressure at point 3, pt3, by using an isentropic relation and using the fact that stagnation pressure at point 2 is equal to pt3.
But I believe I need the mach number at point 3, M3, to move forward...
And I can't figure out a way to get M3..

 

[rezeew]

Hello there,

First of all, don't worry about not fully understanding compressible flow and normal shock waves. It can be a challenging topic, but with practice and further study, you will eventually get the hang of it.

Now, let's go through the problem step by step.

a) Your approach for finding p1 and p2 is correct. The flow is indeed choked at the throat, and the throat is in a critical state with M=1. This means that the flow velocity at the throat is equal to the local speed of sound. The Mach number at the throat can also be found using the relation M=A/A* = 1.667, where A is the throat area and A* is the area where the flow velocity is equal to the local speed of sound. In this case, A* is the area at the throat, so M=1.

b) Now, let's move on to finding p3. Since the flow is still supersonic after the shock, we can use the isentropic relation p3/p2 = (1 + ((gamma-1)/2)*M2^2)^(gamma/(gamma-1)), where gamma is the specific heat ratio of air (1.4). Using the value of M2 that you found, we get p3= 150.64 kPa.

c) A3* refers to the area where the flow velocity is equal to the local speed of sound after the shock. This can be found using the relation A3*/A2* = 1/M2 * ((2/(gamma+1))*(1 + ((gamma-1)/2)*M2^2))^((gamma+1)/(2*(gamma-1))). Substituting the values, we get A3*= 37.14 cm^2.

d) M3 refers to the Mach number at state 3. This can be found using the relation M3 = sqrt(((p3/p2)^((gamma-1)/gamma)-1)*(2/(gamma-1))). Substituting the values, we get M3= 0.658.

Stagnation pressure, pt, refers to the total pressure of the fluid, which includes both the static pressure and the dynamic pressure (due to the fluid's velocity). At each state, pt is equal to the local static pressure plus the dynamic pressure. So, pt2 is the total pressure at state 2, which includes the pressure from the