Explained: Normal Shock Relations
ฝัง
- เผยแพร่เมื่อ 5 ก.ย. 2024
- In this video we will go through the full derivation of the normal shock relations. We will solve for downstream Mach number (M2), velocity ratio (u2/u1), density ratio (rho2/rho1), pressure ratio (P2/P1), and temperature ratio (T2/T1). We will assume the gas is calorically perfect (i.e. the specific heats are constant), which allows us to solve for all these values as only a function of specific heat ratio (gamma) and the upstream Mach number (M1).
===== ERRORS =====
► For the board from 9:09 - 10:33, the terms in brackets on the top line are correct, but the terms in brackets on the next two lines are incorrect (I switched a negative sign with a positive sign). When I bring the expression over to the next board at 10:34, the bracketed term is back to being correct, with a negative sign instead of the positive sign. [Thank you Pratheesh Prabhakar]
► From 21:44 to the end of the video, the second bracketed term in the final T2/T1 equation should be inverted (i.e, I have written it there as rho2/rho1, where it should really be rho1/rho2 as mentioned on the line above). [Thank you Pratheesh Prabhakar and Osama Hamdy]
===== RELEVANT VIDEOS =====
→ 1D Mass Eqn
goo.gl/0vesye
→ 1D Momentum Eqn
goo.gl/FHFUi4
→ 1D Energy Eqn
goo.gl/RSXVyc
→ Thermally Perfect Gas
goo.gl/maElnm
→ Specific Heats
goo.gl/kdf1B7
→ Isentropic Relations
goo.gl/Q8Rv9O
===== REFERENCES =====
► Notes by Matt MacLean
► Modern Compressible Flow, Anderson
► Elements of Gasdynamics, Liepmann and Roshko
► Gas Dynamics, Zucrow and Hoffman
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Please see the "Errors" section in the video description for two errors that were found during the video.
another error btw @16:33... Msub2 squared is the function of gamma and Msub1 .. you dint take the square root and by NOT doing so you didnt complete the relation of the function as you desired ... you asked for Msub2 as a function of gamma and Msub1 ... for that to be true the function needs the square root in the final derivation ... the grammar of Math gotta hate it
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and Please stop using that red thing ... all some of us can see is a darker white where you used red ... it DOESNT SHOW UP WELL AT ALL and 200% zoom is hard on the eyes just trying to read the stuff in red
You explain better than any of my current professors. I wish you taught the class
Hah, thanks!
Thanks for constantly uploading the videos, really appreciate your love towards aeronautics and engineering..
You're welcome! I enjoy doing it, and I'm glad they can help others that have an interest in these topics too.
You have no idea how much help you've been. You're going straight to heaven sir!
Thanks! Always nice to know I've been helping.
You teach better than my teacher.
Thank you
No problem!
Thank you
We'll pass tomorrow because of you
Thanks
I really needed this to learn these relations as fast as pussible.
My pleasure!
bro u have just explained it very nicely.
Thanks!
14:40 after a quick look, this equation results in gamma^2 + 2*gamma + 1 - gamma^2 + 2*gamma - 1 = 48
Josh, I realize you are busy ?working on your phd?, but you have some good stuff here. my suggestion is retitling numbering these lectures by numbers and subject. Each video refers to another for "the derivation" but I cannot find any of them. Note also that time resolution here after one year is in years, so there's no easy way to know the which is older of a 5 year+1 month video vs. 5 year+2 month-old video. Re-titling might be a good solution to better access.
Thanks for the feedback! My videos aren't made for a typical classroom-style curriculum layout. They're pretty much just whatever I felt like making a video on at the time. I agree it would be helpful to give an overview, but all the relevant videos should be included in the video description, with their names and links. If there's something you can't find in the description, just let me know and I'll be sure to add it. I've been trying to use cards in my newer videos, but I just don't have the time or energy to go back to my older videos to update them. TH-cam keeps changing the way they do things anyway, so in the future, cards might not even be a thing. That's what happened to the older text overlay feature, which is why some of my older videos may have references to something on-screen that isn't there anymore. Maybe I'll figure out a better organization strategy in the future.
Is a typical control valve considered a Converge Diverge nozzle?
And also i have doubt in the equation for finding the temperature ratio while substituting density ratio
Yes, you're correct. Thanks for pointing that out!
Sir, @9:27 i am stuck, becoz of 10:35 equation..is there any mistake? Help me please!
You're right, it looks like I put a '+' into a couple equations instead of the correct '-'. The top equation at 9:27 is correct, while the next two lines have the incorrect '+' sign. Then the equation at 10:35 is back to being correct with the '-' sign.
You forgot to flip the equation of (rho2/rho1) at the last equation of (T2/T1)
Yep, you're correct. I'm adding that to my errors in the video description. Thanks!
Can upstream and downstream values be related without shock? How can I obtain the exit velocity of the fluid after normal shock has occurred?
Without a shock present, there won't be any changes in the flow properties. When there is a shock, the flow properties change across the shock, and they can be calculated based on the expressions derived in this video. If you want to get the Mach number after the shock, you can use the Mach number after the shock (M2). The velocity is equal to the Mach number times the speed of sound. The speed of sound can be expressed as a function of the temperature (T2). So to get the velocity after the shock, you can compute the Mach number after the shock, the static temperature after the shock, and then use those in the equation: u2 = a2*M2 = sqrt(gamma*R*T2)*M2. In this equation, gamma is the ratio of specific heats and R is the specific gas constant.
So the mach number after shock is constant? In other words, if the shock occurs at 75% of the length of the nozzle, the shock number at 75% of the length is the same as the mach number at the exit of the nozzle? However the velocity is different because the speed of sound is dependent on temperature?
Thank you for this video too and your reply, dude. It's so hard to find good information outside of my texbook on this stuff. Not to mention my textbooks equations have errors in them half the time!!!
So we need to be careful about terminology here. You mention a nozzle, which means that the area is changing, and the flow is either 2D or something called quasi-1D (ignoring 3D for simplicity). The normal shock relations were derived from the 1D conservation equations. These relations from the video relate properties just upstream of the shock to properties just downstream of the shock. If the flow is truly 1D (in this straight duct), and we are neglecting viscous effects and heat addition (as we did in our conservation equations), then the values upstream of the shock will always be the upstream values, and the values downstream of the shock will always be the downstream values. That is, the Mach number right after the shock is the same as the Mach number 10 or 100 meters farther downstream of the shock.
But if we are talking about a nozzle now, then the area is changing. Instead of going full-blown 2D flow, we can make an intermediate step to quasi-1D flow. In 1D flow, all the properties of the flow were independent of the area, but in quasi-1D flow, the area is also allowed to change, and is now a function of the X-direction.
I'm going to assume that you are talking about the diverging section of a converging diverging nozzle. For certain conditions (reservoir-to-back pressure ratios), a normal shock will form in the diverging section of the nozzle. Across this normal shock, we can use the relations derived in this video. We need to know the properties just upstream of the shock, and we can calculate the properties just downstream of the shock. So let's say that the shock is at 75% of the nozzle length. We need to know the properties (u1, M1, T1, P1, rho1, etc.) at 75% of the nozzle length, and we can calculate the properties right after the shock (u2, M2, T2, P2, rho2, etc.). Now, these values will not stay the same up until the exit of the nozzle, because the area is still increasing. The conservation equations are different. They are not the same ones we used for the normal shock relations. I have a couple videos on the area-Mach number relation, and the resulting equation shows how the Mach number changes for varying areas. Note that the area-Mach number relation works for both the supersonic flow upstream of the shock and for subsonic flow downstream of the shock. So the properties are changing up until the normal shock in the nozzle, and then after they pass through the shock, they continue to change (but now they are subsonic) until the exit of the nozzle.
I'm actually currently making a video about converging-diverging nozzles, so your question came at a good time. My future videos will go into more detail about the different types of states of the nozzles, and will go into detail how to solve each state. One of those states is the normal shock in the nozzle. I'm not sure when I'll post it, but I am working on them now. In the meantime, you can check out my converging-diverging nozzle MATLAB code on my GitHub: github.com/jte0419/Converging_Diverging_Nozzle
I hope that sorta answered your question!
Not a problem. I know how confusing these topics can be, and I made these videos to try to help out.
you don't need to edit out every time you take a breath