Understanding the Navier-Stokes Equations for Smooth Particle Hydrodynamics

In summary, the student is trying to understand the Navier-Stokes equations and figure out how to break them down into variables and functions. He is looking for papers and resources that can help him with this task.
  • #1
Bucky
82
0
Hi, hope this is the right area. Also please excuse me completley ignoring the template, I don't think it's applicable for the problem.

I'm an honours year student in a comp sci course and I've decided to do an implimentation of Smooth Particle Hydrodynamics in a 3d application as my topic. Currently I'm trying to understand the Navier-Stokes equations and figure out how to break them down into variables and functions.

My two comparisons of the equation at the moment are a paper on the subject:
http://graphics.ethz.ch/Downloads/Publications/Papers/2003/mue03b/p_Mue03b.pdf

and the wikipedia article on navier stokes:
http://en.wikipedia.org/wiki/Navier-Stokes_equations


that muller 03 paper lists the navier-stokes equation as:

ρ (∂v / ∂t + v·∇v) = −∇p+ρg+μ∇2v,

(am I right in saying that ∂v / ∂t is just acceleration?)

whereas wikipedia gives it as:

ρ (∂v / ∂t + v·∇v) = −∇p + f +μ∇2v,

wheeere...
ρ = density
v = velocity
t = time
p = pressure
μ = viscosity

the notable change is f (representing external forces) to ρg. where, rho represents density, and g represents a density field(?).

firstly, I don't get these 'field' things. are they just the weighted average of certain variables of particles surrounding the particle we're currently looking at? is this what smoothing kernels are used for?

if i "solve" this equation, what do i actually end up with?

also the upside down triangle (grad?), represents the gradient of the element? how do you even find that?
 
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  • #2
Bucky said:
also the upside down triangle (grad?), represents the gradient of the element? how do you even find that?

I believe that is the del operator...http://en.wikipedia.org/wiki/Del
 
  • #3
ok i looked at that wiki article, and del is used to mean a lot of things. what's it supposed to be representing here? I noticed it has some relation to fields too :/


if i was planning to use this in an application, say to find the acceleration each frame. could i just rearrange the equation to get the acceleration term on it's own (or with velocity, since that'd be a known variable)?
 
  • #4
if you didn't found anything
I recommend you the fallowing papers:
http://liu.diva-portal.org/smash/get/diva2:324983/FULLTEXT01
Here they are showing some kernel functions

also a paper with a different (hack) approach that is easier to implement:
http://www.iro.umontreal.ca/labs/infographie/papers/Clavet-2005-PVFS/pvfs.pdf
 
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Related to Understanding the Navier-Stokes Equations for Smooth Particle Hydrodynamics

1. What is Smooth Particle Hydrodynamics (SPH)?

Smooth Particle Hydrodynamics (SPH) is a computational method used in fluid dynamics to simulate the motion and behavior of fluids. It is a meshless method that uses particles to represent the fluid instead of a traditional grid. This allows for a more accurate and efficient simulation of complex fluid motion.

2. How does SPH work?

In SPH, the fluid is represented by a set of particles, each with its own properties such as position, velocity, and density. These particles interact with each other through a smoothing kernel function, which calculates the contribution of each particle to the properties of its neighboring particles. By solving the equations of motion for each particle, the behavior of the fluid can be simulated over time.

3. What are the advantages of using SPH over other fluid simulation methods?

One of the main advantages of SPH is its ability to handle large deformations and complex boundary conditions, making it suitable for simulating a wide range of fluid phenomena. It is also a meshless method, which means it can handle irregular geometries without the need for complex mesh generation. Additionally, SPH is a Lagrangian method, meaning it can easily track individual particles and their properties over time.

4. What are some common applications of SPH?

SPH is commonly used in fields such as astrophysics, oceanography, and engineering to simulate fluid dynamics in a variety of scenarios. Specific applications include simulating the behavior of ocean waves, studying the dynamics of galaxy formation, and modeling the impact of a fluid on a structure.

5. What are the limitations of SPH?

While SPH has many advantages, it also has some limitations. One of the main challenges is accurately simulating fluids with large density variations, such as water and air. Another limitation is the computational cost, as SPH requires a large number of particles to accurately represent the fluid, which can be time-consuming and resource-intensive.

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