Navier Stokes Equations - Helmholtz-Hodge decomposition and pressure

In summary, there is a relationship between the helmholtz-hodge decomposition and the pressure equation in terms of resolving the problem of keeping the flow divergence-free. However, they are not directly related and the pressure equation is only applicable for certain inviscid flows. In general, the continuity equation is necessary for solving Navier-Stokes problems. The Leray Projection uses Hodge orthogonal decomposition to project the Sobolev space onto the space of divergence-free functions, eliminating the pressure term in the weak statement of the problem.
  • #1
Bucky
82
0
Hi, I've been doing some work with the NS equations. I've read a few papers by fellow undergrads that imply a relationship between the helmholtz-hodge decomposition and the pressure equation.

As far as I can see, they're both separate ways of resolving the problem of keeping the flow divergence free. Am I wrong in thinking this?
 
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  • #2
What "pressure equation" are you talking about?
Euler's pressure equation? That only holds for certain inviscid flows, most notably for irrotational flow, but an analogue can be made when a Clebsch decomposition* can be made (i.e, when the helicity in closed vortex tubes is zero, rather than just constant (the latter being the general case for inviscid flows).

Anyhow, none of this is applicable for general NS problems; for those, the continuity equation (rather than the condition of a solenoidal field) is an indispensable fourth equation along with the three momentum equations.


Possibly, I've misunderstood what you are referring to; it's been awhile since I was slightly acquainted with all of this.


* The Clebsch decomposition is not a general decomposition like the Helmholtz decomposition, and looks like this:
[tex]\vec{v}=\nabla\phi+\alpha\nabla\beta[/tex]
where alpha and beta are concerned with, and in their form represents a limitation on, the vorticity in the fluid field.
 
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  • #3
sorry I guess my post was rather vague. Let's try again

The Navier Stokes equations for incompressible fluid flow are the following:

[tex] \frac{\partial u} {\partial t} = -(u. \nabla )u - \frac {1} {\rho} \nabla p + v \nabla ^2 u + f [/tex]

[tex] \nabla .u = 0[/tex]

The second equation (the incompressibility equation) is the one I'm curious about.

I've read that this can be resolved through the pressure equation, or through the helmholtz-hodge decomposition.

One paper suggested this was done through substituting a pressure update formula into the divergence formula.

the pressure update formula is
[tex] u^{n+1} _{i+1/2,j} = u_{i+1/2,j} - \delta t \frac{1}{\rho} \frac{p_{i+1,j} - pi,j}{\delta x}[/tex]
[tex] u^{n+1} _{i,j+1/2} = u_{i,j+1/2} - \delta t \frac{1}{\rho} \frac{p_{i,j+1} - pi,j}{\delta x}[/tex]

substituting into the divergence formula gives

[tex] \frac{\delta t}{\rho} (\frac{4p_{i,j}-p_{i,j+1}-p_{i+1,j}-p_{i-1,j}-p_{i,j-1}}{\Delta x^2}) = -\frac{u_{i+1/2,j}-u_{i-1/2,j}}{\Delta x} + \frac{v_{i,j+1/2} - v_{i,j-1/2}}{\Delta x} [/tex]

The helmholtz-hodge decomposition is

[tex]

\xi = \nabla u + \nabla .v +h
[/tex]

where u is a scalar potential field (note that [tex]\nabla * (\nabla u) = 0[/tex]
where v is a vector potential field (note that [tex]\nabla .(\nabla * v) = 0[/tex]
where h is the harmonic vector field (note that [tex]\nabla .h = 0[/tex]

One of the papers I've read states that it uses a Poisson equation to derive a height field, which is subtracted from [tex]\xi[/tex] to yield a divergence free flow (it should be noted that this paper altogether ignores the harmonic vector field). The paper can be found here:
http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/GDC03.pdf


in a few papers I've read the authors have implied a relationship between the pressure and helmholtz-hodge solutions. However I don't see how they're related; in fact in that paper I just mentioned the author doesn't even acknowledge the pressure equation in his statement of the NS equations as pertaining to his solver, so I'm finding difficulty in tieing these things together.

I hope this has made my question more clear, and look forward to any answers.
 
  • #4
hi. I am interested in the HH decomposition too. Unfortunately, I am stuck on that. Please be in touch if you have found something interesting.
P.S. Are you from CG?
 
  • #5
From my understanding, the way the second equation is manifests itself onto the pressure field is through the Leray Projection. The Leray Projection, using Hodge orthogonal decomposition, projects the Sobolev space onto the space of divergence free functions (satisfying the second equation). If looking for a weak solution to the problem, then we can integrate by parts and the pressure term becomes the L2 inner product between (p,div(v)). Since we projected v onto the space of divergence free functions this term is zero and the weak statement no longer involves the pressure field. Is that what you are asking about?
 

1. What are the Navier-Stokes equations?

The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluids. They take into account factors such as pressure, viscosity, and acceleration to model the behavior of fluids in various scenarios.

2. What is the Helmholtz-Hodge decomposition in relation to the Navier-Stokes equations?

The Helmholtz-Hodge decomposition is a mathematical technique used to break down a vector field into its components. In the context of the Navier-Stokes equations, it is used to separate the velocity field into a solenoidal (divergence-free) component and an irrotational (curl-free) component.

3. How does the Helmholtz-Hodge decomposition help in solving the Navier-Stokes equations?

By separating the velocity field into its solenoidal and irrotational components, the Helmholtz-Hodge decomposition simplifies the Navier-Stokes equations and makes them easier to solve. This technique is commonly used in computational fluid dynamics to improve the accuracy and efficiency of simulations.

4. Why is the pressure term important in the Navier-Stokes equations?

The pressure term in the Navier-Stokes equations is essential for modeling the behavior of fluids. It represents the force per unit area exerted by the fluid on its surroundings and is a crucial factor in determining the velocity and acceleration of the fluid.

5. Can the Navier-Stokes equations be used to model all types of fluids?

No, the Navier-Stokes equations are only applicable to Newtonian fluids, which are fluids that exhibit a linear relationship between stress and strain rate. They are not suitable for non-Newtonian fluids such as blood or non-homogeneous fluids like sand-water mixtures.

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