[Physics] How to model/simulate pressures and flows in a network of pipes

classical-mechanicsfluid dynamicspressuresimulations

I'm having a hard time finding information on how to model/simulate this. I attached a couple files, both of which show an example tank & pump network. It's just nonsense that I made up for this post, but it shows what kind of system I'm trying to simulate.

(I just put it on the web instead:
pdf link,
vsd link)

I've been looking on the web to find information on how to do this, but I keep running into formulas that are way more complex than what I need to do. I just want to know what the pressure and flow is at any node in the system, with 80% or so accuracy. There's no need to consider fluid compression, and we can just use a constant for the resistance experienced through valves or narrowing pipes. We don't need to consider friction of fluid against piping. We can ignore heat. It's important to note, again, that I don't need complete accuracy, it's much more the behavior of the simulated system that is important.

I've tried this going in both directions, that is following a flow path down from the tanks/sources and propogating pressure changes through the system, and I've also tried it directly from the node in question, and considering that each node has a number of source nodes in terms of pressures and flows. Both of these simulated SW systems become highly erratic the further downstream from the pumps that you get, and the more on/off valves that I put into the system.

Even if you just told me where to look or what book to read, that would be a huge help. If there's example source code out there somewhere that does this, I will buy you a 60 oz beer of your choice.

Thanks in advance.

(post edit 4/12)

Thanks for the answers so far, I've ended up spending quite a bit of time looking around for a solution that already exists, because I was just convinced that many have run into this problem before, and knew that it was solvable in a general sense. Part of the reason I was so interested in a general solution is that I can rip through 12 more of these simulations that I have to do quickly and accurately if that's the case.

I've found another piece of software called EPAnet, which seems like a simpler solution than what was listed below, and I'm looking through their SDK at the moment and playing around with it. I'll keep you posted, the few that are interested … this might not be a big deal to most, but to the few that it matters to it will be huge, so I'll write up what I find.

(edit 4/13)

I've spent the last two days building a solid wrapper around the EPAnet programmer's toolkit, and making a scratch program that does some of the things I want to do. (Incidentally, I've broken a new record for the # of times in a day I can make LabVIEW crash quickly and silently.)

It's really quite complex underneath, and I'm delighted by how much mathematical power I can stand on. I don't think they've fleshed out all the issues, so I've been working around some of them. But it's looking like it's 80% chance so far that it's going to work for what I want.

Best Answer

My advice is to look into the Lattice Boltzmann Method, which is robust and simple for athermal non-compressible flows. Try out this open source GPL3 software for example:

http://www.lbmethod.org/palabos/

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[Physics] How to model water flow with and without friction against other objects

Modelling flow of fluids in general is a big research (and industrial) field. This field is called Computational Fluid Dynamics (CFD)

All CFD methods are based on the so-called Navier-Stokes equations and the continuity equation, which is nothing more than conservation of momentum and mass.

There are different flavours in the methods of discretizing these equations:

Finite Volume
Finite Element
Finite Difference

Which all have pro's and cons.

In all these methods, friction is implicitly introduced by the boundary conditions. A common assumption for boundaries, as solid walls, is the so-called no-slip condition. This basically means that the local velocity at the wall is zero. This is basically nothing more than a sink of momentum via the wall shear stress. Making the model neglect friction, you should assume a zero-shear boundary, which means that the wall-normal gradient of the velocity component is zero.

As you're talking about water flows and objects, you know that these flows become turbulent. The non-linear term in the Navier-Stokes equation will be responsible for flow phenomena at lots of scales. Therefor, it is common practice to model this turbulence. Also here, we have different approaches:

DNS: Direct Numerical Simulation: No modelling at all, but the smallest grid size is smaller than the smallest turbulent scales
LES: Large Eddy Simulation: The Navier-Stokes equation is filtered for different scales. The largest scales are solved for, the smallest sub-filter scales are models
RANS: Reynolds Average Navier-Stokes: The Navier-Stokes equations are averaged (over time or realizations), and the non-linear term in the velocity fluctuations (also called Reynolds stress), is modelled, often via some a so-called turbulent viscosity (and by solving additional transport equations for turbulent properties).

Advised reading may be the book by Ferziger and Peric : http://www.amazon.com/Computational-Methods-Fluid-Dynamics-Ferziger/dp/3540420746

Newtonian Mechanics – How to Simulate Rotational Instability

$\vec\omega = I^{-1} \vec L$, and $\vec L$ is constant in the absence of external forces. The bit that I think you're missing is that $I$ rotates with the rigid body, so it is not constant in general and neither is $\vec\omega$.

I played with your online example, and the angular velocity does seem to always remain constant when I'm not poking the block, which is consistent with an inertia tensor that's a scalar multiple of the identity. I've never used Unity, but it seems to support arbitrary tensors of inertia via Rigidbody.inertiaTensor and Rigidbody.inertiaTensorRotation, so maybe you just need to set those properly.

If you end up having to roll your own physics, here's some naive, untested, and potentially wrong pseudocode:

const double time_step = ...;
const Quaternion L = {0, Lx, Ly, Lz};  // angular momentum
const double K1 = 1/I1, K2 = 1/I2, K3 = 1/I3;  // reciprocals of principal-axis moments of inertia; no idea if there's a standard letter for this
Quaternion orientation = {1, 0, 0, 0};
while (1) {
    Quaternion transformed_L = conjugate(orientation) * L * orientation;
    Quaternion transformed_omega = {0, K1 * transformed_L.i, K2 * transformed_L.j, K3 * transformed_L.k};
    Quaternion omega = orientation * transformed_omega * conjugate(orientation);
    orientation = quaternion_exp(time_step * omega) * orientation;
    // ... or orientation = normalize((1 + time_step * omega) * orientation);
    // ... or orientation = normalize((1 + time_step * omega + 0.5 * (time_step * omega)**2) * orientation);
    output(orientation);
}

Best Answer

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[Physics] How to model water flow with and without friction against other objects

Newtonian Mechanics – How to Simulate Rotational Instability

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