CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > General Forums > Main CFD Forum

Analytic solution for 2D steady Euler equations

Register Blogs Members List Search Today's Posts Mark Forums Read

Reply
 
LinkBack Thread Tools Search this Thread Display Modes
Old   October 15, 2012, 12:05
Default Analytic solution for 2D steady Euler equations
  #1
New Member
 
Join Date: Oct 2012
Posts: 1
Rep Power: 0
jojo81 is on a distinguished road
Hello,

I'm trying to verifiy my CFD code for low Mach number flows with some analytical or manufactured solutions.

The convection and/or diffusion equation for a scalar is fine and now I focus on the Navier Stokes equations:

\nabla \cdot \underline{u} = 0
\frac{\partial \underline{u}}{\partial t} + \left( \underline{u} \cdot \nabla \right) \underline{u} - \nu \Delta \underline{u} = - \frac{1}{\rho_0} \nabla p

First, I want to check the non linear convection term, so I suppress the viscosity considering a perfect fluid and starting from the Green Taylor vorticies solution for Navier Stokes:

u(x,y,t) = - \cos (2 \pi x) \sin (2 \pi y) e^{-8 \pi^2 \nu t}
v(x,y,t) = + \sin (2 \pi x) \cos (2 \pi y) e^{-8 \pi^2 \nu t}
p(x,y,t) = -\frac{1}{4}\left[\cos(4 \pi x) + \cos (4 \pi y)\right] e^{- 16 \pi^2 \nu t}

I get this solution:

u(x,y,t) = - \cos (2 \pi x) \sin (2 \pi y)
v(x,y,t) = + \sin (2 \pi x) \cos (2 \pi y)
p(x,y,t) = -\frac{1}{4}\left[\cos(4 \pi x) + \cos (4 \pi y)\right]

which satisfies the steady Euler equations:

\nabla \cdot \underline{u} = 0
\left( \underline{u} \cdot \nabla \right) \underline{u} = - \frac{1}{\rho_0} \nabla p

I tried this solution with my CFD code setting boundary and initial conditions with the solution, the pressure is also imposed, not computed, with the solution.

At each iteration, I solve the velocity components and then update the convective flux, solve velocity, update the convective flux, etc. I use finite volume, upwind or centered convective scheme.

My problem: after some oscillations, the flow diverges and I don't understand why. Is the solution unstable ? Can't a solver for Navier Stokes solve an Euler problem ?

Did or is anybody try this kind of problem ?

Thanks.
jojo81 is offline   Reply With Quote

Reply

Thread Tools Search this Thread
Search this Thread:

Advanced Search
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are Off
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
Delta form of Heat, Euler and NS equations RameshK Main CFD Forum 3 May 30, 2012 10:41
initialize flow field with steady state solution holg FLUENT 0 July 13, 2009 17:10
Euler equations Brian Main CFD Forum 0 September 8, 2008 06:19
Euler equations? Jan Ramboer Main CFD Forum 2 August 19, 1999 01:58
2d analytic Euler solutions? niles pierce Main CFD Forum 1 July 14, 1998 12:22


All times are GMT -4. The time now is 00:21.