Problem regarding a Normal Shock Wave and Blunt Nosed Object

[hbk69]

Am not sure if I've posted in the right section but I couldn't find anything relevant to Waves in Supersonic Flow, so apologise in advance!

Post is a little long but I will appreciate any help! Academically am not very gifted and also have a mathematics disability so I tend to question even the things which are really simple and I guess people can find that weird when things can come so naturally to them but I hope you can bear with me!

Greatly appreciate any help! Thanks!

Homework Statement

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A normal shock wave forms in front of a two-dimensional blunt-nosed object in a supersonic airstream. The pressure at the nose of the obstacle is three times the static pressure upstream of the shock wave. Find the upstream Mach number, the density ratio across the shock and the velocity immediately after the shock if the upstream static temperature is 10 degrees. If the air was then expanded isentropically to its original pressure, what would its temperature then be?

Answers are as follows: M1=1.386, p2/p1=2.074, ρ2/ρ1=1.665, T3=13 degrees

M1 = upstream Mach Number ρ2/ρ1 = density ratio p2/p1 = pressure ratio T3 = Temperature


2. Homework Equations

Total to Static Pressure Ratio's

Plane Normal Shock Wave Equations

3. The Attempt at a Solution

I have attempted to do the question but have struggled to get to grips with the problem. I understand there are going to be three regions. One in the supersonic flow region and two beyond the object in question which form the subsonic isentropic region where the pressure is constant.

So we have point 1 where the flow is supersonic

Then there is point 2 (subsonic flow begins here I think)And point 3 (stagnation point)

In the question we are told that the pressure at the nose of the object (P3) is three times the pressure upstream of the shock wave at point 1.

So we know that P3 = 3*P1

And P3 / P1 = 3

Now we need to form a pressure equation which represents the P3 / P1 ratio, I managed to find this but I don't understand it at all.

So the pressure equation which represents the P3 / P1 ratio is as follows:

P3 / P1 = (P3 / P2)*(P2 / P1)

Here the numbers represent the various regions described above and the pressures at those regions, Pt2 would represent the total pressure at point 2 for example and the other P's (P3,P1,P2) etc represent static pressure at those points.

What I've gathered is that P3 / P1 is a pressure ratio between two points, region 3 and region 1 and it includes all the other ratio's in-between? I don't really understand exactly what it represents. Anyhow as it may include all the other pressures ratios we need to include P2 / P1 and P3 / P2 and multiply them, but I do not know why? also when we have a pressure ratio P2 / P1 for example does that represent the pressure between those two specific points? or is a pressure difference of some sought?

As P3 / P1 = (P3 / P2)*(P2 / P1) we can see that we have an equation for P2 / P1 from the Normal Shock Wave equations so need to form an equation for P3 / P2.

P3 / P2 = (P3 / Pt3)*(Pt2 / P2)

So here P3 = P3 / Pt3 and that's understandable to an extent as it represents the static pressure divided by the total pressure at region 3 (but I don't know why) and the same theory is applied to P2 to give our equation for the P3/P2 region.

Note that betwen regions 2 - 3 the pressure is constant as it's an isentropic reigion (Is this due to subsonic flow?) again am not sure why the pressure is constant or why it's an isentropic reigion. But as a result of this the total pressure at region 2 (Pt2) is equal to the total pressure at region 3.
Pt2 = Pt3

Then we can use the total to static pressure equation to give us our equation for P3 / P2

P3 = P3 / Pt3

Therefore,

P3 = (1 + ((γ-1)/2)*M²3)^{γ / γ-1}

P2 = (1 + ((γ-1)/2)*M²2)^{γ / γ-1}

So P3 / P2 =
(1 + ((γ-1)/2)*M²2)^{γ / γ-1} / (1 + ((γ-1)/2)*M²3)^{γ / γ-1}

As a result, P3/ P2 = (1 + ((γ-1)/2)*M²2)^{γ / γ-1} but this I do not understand, we have an equation for M2 at the top and an equation for M3 in denominator but all of a sudden we are left with an equation for M2 which is the Mach Number for region 2, it is something to do with the total pressure at region 2 and 3 being equal but I don't understand how the P3/ P2 becomes an equation which represents M2, if I can understand that then I'd be in a position where I have no issues to find the upstream mach number M1 which is the goal. Maybe as total pressure in region 2-3 are equal we just neglect the equation in the denominator?

Those are the issues I have with first step which was finding the Mach Number and the other questions I will attempt once my understand is clear with regards to finding the Mach Number but from the top of my head I don't really have a clue how to answer them but will need to gain an understanding of finding the upstream Mach Number first.

Really appreciate any help or guidance guys, Thank You So Much!

 

[KataKoniK]

Hello! Thank you for your post and for reaching out for help. I am happy to assist you with your questions regarding waves in supersonic flow.

Firstly, I want to assure you that it is completely normal to have questions and doubts, especially when dealing with complex scientific concepts. In fact, questioning and seeking clarification is an important part of the scientific process. So please do not feel bad about asking for help or questioning things that may seem simple to others.

Now, let's get into the problem at hand. You are correct in identifying the three regions: the supersonic flow region (region 1), the subsonic flow region (region 2), and the isentropic region (region 3). In order to solve this problem, we need to use the equations for a normal shock wave, which you have correctly identified.

The equation for the pressure ratio across a normal shock wave is P2/P1 = (2γM1^2 - (γ-1))/(γ+1), where P2 and P1 are the pressures at regions 2 and 1 respectively, and M1 is the upstream Mach number (in region 1). This equation is derived from the conservation of mass, momentum, and energy across the shock wave.

Now, in order to find the upstream Mach number (M1), we need to rearrange this equation to solve for M1. This can be done by multiplying both sides by (γ+1). We also know that P3/P1 = 3, as given in the problem statement. So, our new equation becomes:

(γ+1)P2/P1 = (2γM1^2 - (γ-1))

Substituting in P3/P1 = 3, we get:

(γ+1)P2/P3 = (2γM1^2 - (γ-1))

Now, we can use the equation for the total to static pressure ratio, which is P2/P3 = (1 + ((γ-1)/2)*M2^2)γ / (γ+1)γ/(γ-1), where M2 is the Mach number in region 2. Substituting this into our equation, we get:

(γ+1)*(1 + ((γ-1)/2)*M2^2)γ / (γ+1)γ/(γ-1) = (2γM1^