Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New features!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

123 submissions , 104 unreviewed
3,547 questions , 1,198 unanswered
4,552 answers , 19,366 comments
1,470 users with positive rep
411 active unimported users
More ...

The Euler equations as a RNG fixed point

+ 5 like - 0 dislike
26 views

In this paper at the at the beginning of the last paragraph on p.2 it is said, that the Euler equations, which are an infinite Reynolds number limit of the Navier-Stokes equations, arise as an RNG fixed point. This fixed point is said to be non-unique for a system witn N mixing species, because it can be choosen from an N+1 dimensional parameter space spanned by N-1 dimensionless mass diffusivities, the dimensionless thermal diffusivity (or Prantl Number) and the dimensionless ratio of the anisotropic and isotropic viscosity.

About this issue I have several closely related questions:

  1. What kind of fixed point is this when considering a classification into Gaussian/interacting, trivial/nontrivial, etc fixed points

  2. What is the expected behaviour of the RG flow around this fixed point?

  3. Is the non-uniqueness of this fixed point the same kind of non-uniquenes as the one described on p67 if this paper, which explains that the presence of redundant marginal operators can lead to a whole line of physically equivalent fixed points? If this way of analyzing a fixed point can be applied in my example, would the Euler equations fixed point then correspond to some kind of an N+1 dimensional surface of fixed points where the mass diffusivities, the prantl number, and the ratio of the anisotropic and isotropic viscosity play the role of such redundant marginal operators?

asked Jul 23, 2013 in Theoretical Physics by Dilaton (4,175 points) [ revision history ]
recategorized Apr 11, 2014 by dimension10

1 Answer

+ 1 like - 0 dislike

Ok, here is how I see these issues now after taking a bit more time to think about them:

In the first paper it is said that the Euler equations emerge as the infinite Reynolds number limit of the Navier Stokes equation which means that according to the definition of the Reynolds number

$$ RE = \frac{V L}{\nu} $$

the molecular diffusion can be neglected in this case. As is known that to fully developped turbulence all scales are expected to conribute (there is no characteristic scale present) and molecular diffusion can be neglected at larg scales, the fixed point corresponding to the Euler equations from a RG point of view is a critical IR fixed point. However as mentioned here when looking at LES dynamic subgrid scale parameterizations from a RG point of view, talking of a (scale invariant) fixed point is not exactly justified because the rescaling is missing in the renormalization step, and the IR limit the system approaches when repeating this modified renormalization step is more exactly called a limit point.

Looking at the non-uniqueness of this fixed (or limit) point mentioned in the first paper from the point of view explained around p. 67 of the second paper, this non-uniqueness does not mean that the N+1 parameters span some kind of a higher dimensional generalization of a line of fixed points, as the corresponding operators are relevant instead of redundant and marginal. What is meant instead, is when doing a linear analysis of the RG flow around the fixed (limit) point such that the nearby action is given by

$$ S_t(\phi) = S_{*}(\phi) + \sum\limits_i \alpha_i e^{\lambda_i t}O_i(\phi) $$

where $S_{*}(\phi)$ is the action at the fixed (relevant) point and $\alpha_i$ are integration constants, there are N+1 relevant operators $O_i(\phi)$ for a system with N mixing species with Eigenvalues $\lambda_i > 0$ which are determined from the Eigenvalue equation

$$ M O_i(\phi) = \lambda_i O_i(\phi) $$

The non-uniqueness alluded to in the first paper corresponds to the fact that the integration constants $\alpha_i$ are not determined by the renormalization procedure itself but as explained in the second paper have to be determined by the bare action or perfect action which lies on a renormalized trajectory.

In the context of Large Eddy Simulations (LES) that make use of dynamic subgrid scale parameterizations for turbulent diffusion for example, it is possible to dispense with the non-uniqueness by calculating the corresponding integration constant directly from the resolved scale by making use of the Germano identity Eq. (4.2) in the second paper and application of the Smagorinsky scheme to calculate a dynamic mixing length.

answered Jul 26, 2013 by Dilaton (4,175 points) [ revision history ]
Pointing out errors, mistakes, confusions, etc I would highly appreciate :-)

This post imported from StackExchange Physics at 2014-03-09 16:11 (UCT), posted by SE-user Dilaton

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar\varnothing$sicsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...