Free convection

ebrattr · August 17, 2012, 11:29

I have had a big problem trying to simulate free convection over a vertical plate. One of my problems is i want to control the pressure gradient, I mean one of the unknown variables is the pressure p. But, in free convection or at least, in my model I don't take into account the x-momentum equation. In fact, my equations are the following

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$
$u \frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}=\zeta \beta g(T-T_{\infty})+\nu \frac{\partial^2 u}{\partial y^2}$
$u\frac{\partial T}{\partial x}+ v\frac{\partial T}{\partial y} = \alpha \frac{\partial^2 T}{\partial y^2}$

With this boundary conditions

$u(x,\infty) = 0$

$T(x,\infty) = T_{\infty}$

This is the problem,

Link: http://s7.postimage.org/xobd7x255/image.png

I take $F_x = -\zeta\rho g$ and $\frac{\partial p}{\partial x} = -\zeta \rho_{\infty}g$

And I compute $\rho$ as, $\rho = \rho_{\infty}+\left(\frac{\partial \rho}{\partial T} \right)_P (T-T_{\infty})$ . Replacing, $\beta=-\frac{1}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_P$ .

I will initially define:
$T_w(x), \, u_w(x)$ .

Finally,
$\zeta \in \{-1,0,1\}$ .

In fact, is it possible to compute this in a COMSOL? Mi unknown are only u,v and T not P. Because I assume the pressure gradient as I explained before.

leflix · August 17, 2012, 14:30

Quote:

Originally Posted by ebrattr

I have had a big problem trying to simulate free convection over a vertical plate. One of my problems is i want to control the pressure gradient, I mean one of the unknown variables is the pressure p. But, in free convection or at least, in my model I don't take into account the x-momentum equation.

Hi ebrattr,

No but in the model you have presented here you have taken the x-momentum equation into account.
I'm afraid your model is completely rubbish.
Free convection is a kind of flow which is vertically dominated as it is induced by gravity.
So taking x momentum equation does not make sense. If you assume that your flow is unidirectional then choose rather the y momentum because the buoyancy force which drives such flow is oriented along y axis.
I even do not undertand how you can have your buoyancy term in the x momentum or I have missed something.

f= rho*g with vector g=-9.81y where here f g and y are vectors.

Quote:

In fact, is it possible to compute this in a COMSOL? Mi unknown are only u,v and T not P. Because I assume the pressure gradient as I explained before.

If COMSOL is available for you, why do you want to use a simplified model and solve it with COMSOL ? Solve the full 2D Navier-Stokes equations as COMSOL is capable to do it.
Then you could verify a posteriory if your assumptions were right or not especially concerning the pressure gradient, but I guess it won't

ebrattr · August 17, 2012, 14:38

I turned over the axes, look at my axes. I will follow you advice. But I want to compute also a solution for those equations because I want to learn about it.

What is the simplest way to compute it ? With a good solution (exact) it doesnt matter if the convergence takes to much for the algorithm jaja

leflix · August 17, 2012, 20:33

Quote:

Originally Posted by ebrattr

I turned over the axes, look at my axes.

Don't do such stuff when you submit a problem for which you look for a solution ....fluid mechanic is already arduous and strenuous enough that changing axes brings more difficulties than help. You confused us (at least me) for no reasons.
Better keep the usual notations well mastered by every one here

Quote:

What is the simplest way to compute it ? With a good solution (exact) it doesnt matter if the convergence takes to much for the algorithm jaja

I would say try finite difference or finite volume method
It should be easy if you can extract the v componnent from integrating the continuity equation.

ebrattr · August 18, 2012, 01:00

Quote:

Originally Posted by leflix

I would say try finite difference or finite volume method
It should be easy if you can extract the v componnent from integrating the continuity equation.

Dont you have any reference guide ?

leflix · August 18, 2012, 04:28

Quote:

Originally Posted by ebrattr

Dont you have any reference guide ?

You have to learn howto discetize a partial derivative equation based on finite difference, finite volume or finite element. Just choose your method.

If you want to use finite volume books of Versteek or Peric are great
http://www.amazon.com/An-Introductio.../dp/0131274988
http://www.amazon.com/Computational-...keywords=peric

for finite differences or finite element I don't have particular books in mind
but you can find many lecture notes on internet.

once you discretized your equations, in one time step solve:
your momentum equation, then use continuity equation to obtain the second velocity component, then solve heat equation finaly advance time step..

ebrattr · August 20, 2012, 16:05

I solved the problem in MATLAB.

Now I'm going to compare my results with COMSOL results. But I see this weird behaviour and I don't know why ?

Imagen: http://s10.postimage.org/5q2xjuz2v/Temperature.png

Imagen: http://s12.postimage.org/mmjk7giq3/velocity.png

Here is my COMSOL file: http://www.mediafire.com/?fsdkoqa1uumm3cc

Thanks !

August 17, 2012, 11:29	Free convection	#1
ebrattr New Member Join Date: Jul 2012 Posts: 4 Rep Power: 13	I have had a big problem trying to simulate free convection over a vertical plate. One of my problems is i want to control the pressure gradient, I mean one of the unknown variables is the pressure p. But, in free convection or at least, in my model I don't take into account the x-momentum equation. In fact, my equations are the following $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ $u \frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}=\zeta \beta g(T-T_{\infty})+\nu \frac{\partial^2 u}{\partial y^2}$ $u\frac{\partial T}{\partial x}+ v\frac{\partial T}{\partial y} = \alpha \frac{\partial^2 T}{\partial y^2}$ With this boundary conditions $u(x,\infty) = 0$ $T(x,\infty) = T_{\infty}$ This is the problem, Link: http://s7.postimage.org/xobd7x255/image.png I take $F_x = -\zeta\rho g$ and $\frac{\partial p}{\partial x} = -\zeta \rho_{\infty}g$ And I compute $\rho$ as, $\rho = \rho_{\infty}+\left(\frac{\partial \rho}{\partial T} \right)_P (T-T_{\infty})$ . Replacing, $\beta=-\frac{1}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_P$ . I will initially define: $T_w(x), \, u_w(x)$ . Finally, $\zeta \in \{-1,0,1\}$ . In fact, is it possible to compute this in a COMSOL? Mi unknown are only u,v and T not P. Because I assume the pressure gradient as I explained before. Last edited by ebrattr; August 17, 2012 at 14:39.

August 20, 2012, 16:05	Efficiency	#7
ebrattr New Member Join Date: Jul 2012 Posts: 4 Rep Power: 13	I solved the problem in MATLAB. Now I'm going to compare my results with COMSOL results. But I see this weird behaviour and I don't know why ? Imagen: http://s10.postimage.org/5q2xjuz2v/Temperature.png Imagen: http://s12.postimage.org/mmjk7giq3/velocity.png Here is my COMSOL file: http://www.mediafire.com/?fsdkoqa1uumm3cc Thanks ! Last edited by ebrattr; August 20, 2012 at 20:27.

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August 17, 2012, 14:38		#3
ebrattr New Member Join Date: Jul 2012 Posts: 4 Rep Power: 13	I turned over the axes, look at my axes. I will follow you advice. But I want to compute also a solution for those equations because I want to learn about it. What is the simplest way to compute it ? With a good solution (exact) it doesnt matter if the convergence takes to much for the algorithm jaja