What areas of maths and physics do I need to understand explosion physics?

[Ax_xiom]

I was watching Jorge S. Diaz's Seminar on blast waves where he derived a formula for the position of a blast wave in respect to time. Despite learning multivariable and vector calculus, I'm still confused about some of the steps he takes to derive the final expression. So what other areas of mathematics do I need to know?

Is it also fine for me to ask questions about specific steps he took in the video?

 

[berkeman]

What is your interest in studying blast physics? Are you wanting to better understand blast injuries? Or are you wanting to understand something else about explosions and the resulting blast wave?

https://www.sciencedirect.com/science/article/pii/S0196064416310319

 

[Ax_xiom]

What is your interest in studying blast physics? Are you wanting to better understand blast injuries? Or are you wanting to understand something else about explosions and the resulting blast wave?

https://www.sciencedirect.com/science/article/pii/S0196064416310319

I more want to understand how air speed and overpressure scale with the size of the explosion and the distance from the epicentre (and maybe use fluid dynamics to create an equation for both). I wanted to look at the video to see what I need to know and I don't understand everything yet

 

[berkeman]

I more want to understand how air speed and overpressure scale with the size of the explosion and the distance from the epicentre

What is your interest in this topic?

 

[Ax_xiom]

What is your interest in this topic?

Just a general curiosity, why do you ask?

 

[berkeman]

In addition to being an EE, I'm also a Medic. My interests in explosion physics are from the blast injury angle. When others express interests in explosives, my other trainings kick in and I ask questions. I don't believe you just have a general curiosity about blast physics, to be honest. Do you want links to the fluid dynamics involved, or do you just want simple equations to do some blast injury calculations?

What is your physics and math training level so far in case you want the former?

 

[Ax_xiom]

In addition to being an EE, I'm also a Medic. My interests in explosion physics are from the blast injury angle. When others express interests in explosives, my other trainings kick in and I ask questions. I don't believe you just have a general curiosity about blast physics, to be honest. Do you want links to the fluid dynamics involved, or do you just want simple equations to do some blast injury calculations?

What is your physics and math training level so far in case you want the former?

I want more links to the fluid dynamics involved. I know some vector and multivariable calculus although I could just learn whatever extra maths is needed later

 

[berkeman]

Then follow the links at the Wikipedia article: https://en.wikipedia.org/wiki/Blast_wave

I'll also ping our local blast physics expert @Dr. Courtney in case he is still monitoring PF.

 

[Ax_xiom]

Then follow the links at the Wikipedia article: https://en.wikipedia.org/wiki/Blast_wave

I'll also ping our local blast physics expert @Dr. Courtney in case he is still monitoring PF.

Ok, Is it alos fine if I ask question about specific steps Jorge takes in the video?

 

[berkeman]

Ok, Is it alos fine if I ask question about specific steps Jorge takes in the video?

Probably, but don't make us watch the video to help you. Take screenshots of the equations or reproduce them here using LaTeX (see the LaTeX Guide link below the Edit window).

 

[Frabjous]

Try
explosive shocks in air by Kinney
Blast waves by needham
Introduction to simple shock waves in air by Prunty
Gas Dynamics of Explosion by Lee

It’s technically compressible fluid dynamics, but scaling laws and approximations can take you a long way

 

[Ax_xiom]

Probably, but don't make us watch the video to help you. Take screenshots of the equations or reproduce them here using LaTeX (see the LaTeX Guide link below the Edit window).

Alright.

The first step that confuses me is this:

I fully understand the left hand side, as he just found the value of the integral, but what confuses me is the right hand side and specifically the $\frac{\gamma}{2}\psi\phi^{2}$ term. I'm pretty sure that what he did is made some substitutions and moved the constants $\frac{4 \pi R^{3}P_{0}}{\lambda}$ out of the integral, but when I do that I can't get those constants to come out of the integral.

 

[Frabjous]

What is a₀ (it is some sort of velocity)? If M_s is the mach number, I am unsure why one can pull it outside the integral.

 

[Ax_xiom]

What is a₀?

I don't think he explains in the video but I think it's the speed of sound in the medium based on what he says about the final equation. He did mention that he would use these equations:

And the Rankine-Hugonoit boundary conditions to solve the equations so maybe that is what he did there.

 

[Frabjous]

I don't think he explains in the video but I think it's the speed of sound in the medium based on what he says about the final equation. He did mention that he would use these equations:
View attachment 363424
And the Rankine-Hugonoit boundary conditions to solve the equations so maybe that is what he did there.

If the video does not define everything, trying to fill in the steps is going to be really difficult. I would suggest finding another reference.

 

[Frabjous]

Diaz has a paper on archive.
https://arxiv.org/pdf/2110.09488
It assumes that you already know a lot.

 

[Ax_xiom]

Diaz has a paper on archive.
https://arxiv.org/pdf/2110.09488
It assumes that you already know a lot.

Ok thanks, I'll see what I can do here

 

[Ax_xiom]

I wonder if it's possible to use these equations to calculate the air velocity and pressure of a blast wave at a given distance with a given energy. $M_{s} = a_{0}^{-1}(\frac{dR}{dt})$ and we already have an expression for $\frac{dz}{d\tau}$ which is the scaled distance over the scaled time.

 

[Frabjous]

You can calculate them at the shock front.

 

[renormalize]

Diaz has a paper on archive.
https://arxiv.org/pdf/2110.09488

Thanks for citing the Diaz & Rigby paper. We had a poster some time back who vehemently insisted that a blast-wave shock-front could propagate at the speed-of-sound $c$ and cited data from the 2020 Beirut chemical explosion to back his claim. But the analysis in this paper clearly demonstrates that the Beirut shock front traveled supersonically and only approached $c$ ("acoustic wave") as $t\rightarrow\infty$:

 

[Ax_xiom]

You can calculate them at the shock front.

I did it with overpressure and I'm getting some odd results:
dR/dt transforms with the chain rule like this:
$$\frac{dR}{dt} = \frac{dR}{dz}\frac{dz}{d\tau}\frac{d\tau}{dt}$$
And our variables are these:
$$
\begin{align*}
R_{0}z &= R \\
\tau &= \frac{a_{0}t}{R_{0}}
\end{align*}
$$
So their final derivatives and the final value of dR/dt is as follows:
$$
\begin{align*}
\frac{dR}{dz} &= R_{0}\\
\frac{d\tau}{dt} &= \frac{a_{0}}{R_{0}}\\
\frac{dR}{dz}\frac{dz}{d\tau}\frac{d\tau}{dt} &= R_{0}\frac{dz}{d\tau}\frac{a_{0}}{R_{0}} = a_{0}\frac{dz}{d\tau}
\end{align*}
$$
We know what $\frac{dz}{d\tau}$ is as that is the final result of the paper and earlier in the paper we get this expression $$P(R) = (\frac{2 \gamma M_{s}^{2}-(\gamma-1)}{\gamma+1})P_{0}$$ for the pressure at the shock front where $$M_{s} = a_{0}^{-1}\frac{dR}{dt}$$ this means $$M_{s} = \frac{dz}{d\tau}$$ The rest of what I did is just agebraic manipulation to get it into units that aren't dimensionless.
$$
\begin{align*}
\frac{dz}{d\tau} &= (1+\frac{1}{K_{0}z^{3}})^{\frac{1}{2}}\\
z &= \frac{RP_{0}^{\frac{1}{3}}}{E_{0}^{\frac{1}{3}}} \\
\frac{dz}{d\tau} &= (1+\frac{E_{0}}{K_{0}R^{3}P_{0}})^{\frac{1}{2}} = M_s\\
P(R) &= (\frac{2 \gamma(1+\frac{E_{0}}{K_{0}R^{3}P_{0}})-(\gamma-1)}{\gamma+1})P_{0}\\
&= (\frac{\frac{2 \gamma E_0}{K_{0}R^{3}P_{0}}+\gamma+1}{\gamma+1})P_{0}\\
&= (\frac{2 \gamma E_0}{K_{0}R^{3}P_{0}(\gamma+1)}+1)P_0\\
&= \frac{2 \gamma E_0}{K_{0}R^{3}(\gamma+1)}+P_0
\end{align*}
$$
Overpressure is just $P-P_0$ so the expression for overpressure is $$\frac{2 \gamma E_0}{K_{0}R^{3}(\gamma+1)}$$ which implies that overpressure increases linearly with explosion size, which looking at nukemap doesn't seem to be true

 

[Frabjous]

What nukemap? Overpressure going as E/R³ is common result.

 

[Ax_xiom]

What nukemap? Overpressure going as E/R³ is common result.

This one by Alex Wellerstein. The dimensional analysis makes sense but plotting yield and overpressure on a graph shows that it isn't linear:

Taking the logarithm of both sides and checking the gradient of the line makes the non-linear trend of the line even more clear:

It looks more like a square root relationship rather than a linear one based on the line

Edit: The sample points were taken 4.78km away from the blast site, if that information is relevant

 

[Frabjous]

You have nukemap and you have an equation. Have you tried plotting nukemap using the equation?

The distance matters.

Does nukemap have mach reflection?

 

[Ax_xiom]

You have nukemap and you have an equation. Have you tried plotting nukemap using the equation?

Yeah, I've done the same for distance and the equation also disagrees:

The overpressure seems to be decreasing similarly to the inverse square law but my model predicts a cubic decrease.

Does nukemap have mach reflection?

It does, that's why it can also model airbursts aswell

 

[Frabjous]

In the language of the paper, is nukemap using the STvN solution or something more complicated? One also needs to think about the error bars of the synthetic data.

To confirm, mach reflection is turned off in the nukemap numbers?

The hugoniot equation is a thermodynamic equation, so one would expect that a good mach number will result in a good pressure. On the other hand, the paper does not make pressure comparisons at all and ultimately was not published, so there might be some loose ends that the authors still need to address. For example: On page two, there is the description of the auxillary function K. In figure 1, the particle velocity does not have a negative region at late times, which is not adequately discussed.

 

[Ax_xiom]

In the language of the paper, is nukemap using the STvN solution or something more complicated? One also needs to think about the error bars of the synthetic data.

What Nukemap does is that it takes Glasstone and Dolan's data on nuclear weapons and interpolates between them to allow Nukemap to simulate airbursts.

More detail about nukemap can be found here.

Also isn't the STvN solution for the position of a shockwave? How would you use it for the overpressure?

To confirm, mach reflection is turned off in the nukemap numbers?

Again, since nukemap can simulate airbursts and the data it's built from has mach reflections I'd assume its turned on.

For example: On page two, there is the description of the auxillary function K.

How does that change things? Should I find what $z$ is with that in mind?

 

[Frabjous]

Again, since nukemap can simulate airbursts and the data it's built from has mach reflections I'd assume its turned on.

The paper is describing spherical explosions. So you are calculating two different things.

Do you have access to the book by Kinney? The first edition has a table for a one ton explosion as a point source and as a spherical charge. The second edition has a table for a 1 kg and a 1 kt tnt explosion. Try comparing the code and the paper to either of these data sets

 

[Ax_xiom]

The paper is describing spherical explosions. So you are calculating two different things.

Do mach reflections affect the overpressure that much? Even when the explosion is on the ground? I reasoned that the effects would only be some sort of scaling factor as the solution that Diaz came up with was able to model surface bursts by dividing the calculated yield by 1.8 to account for reflections

Do you have access to the book by Kinney? The first edition has a table for a one ton explosion as a point source and as a spherical charge. The second edition has a table for a 1 kg and a 1 kt tnt explosion. Try comparing the code and the paper to either of these data sets

Unfortunately not, could you link the table here by any chance?

 

[Frabjous]

Unfortunately not, could you link the table here by any chance?

Give me a day or two.