solving for denstiy and velocity implicitly

Hooman · June 1, 2012, 06:49

Hi,

I have been trying to solve the following systems of equations implicitly:

$\frac{\partial\rho}{\partial t}=f\left(\rho,t\right)$

$\frac{\partial \rho v}{\partial t}=g\left(\rho,v,t\right)$

I don't want to change the second equation to:
$\frac{\partial(\rho v)}{\partial t}=\rho\frac{\partial(v)}{\partial t}+v\frac{\partial(\rho)}{\partial t}$

I have the initial variables and $\rho$ and v are vectors. I have solved the equations explicitly and I have solved one system of equations implicitly before but not sure how to approach this problem with two systems of equations implicitly. Do I need to turn them all into one system of equations?

Thanks!

FMDenaro · June 1, 2012, 07:14

Quote:

Originally Posted by Hooman

Hi,

I have been trying to solve the following systems of equations implicitly:

$\frac{\partial\rho}{\partial t}=f\left(\rho,t\right)$

$\frac{\partial \rho v}{\partial t}=g\left(\rho,v,t\right)$

I don't want to change the second equation to:
$\frac{\partial(\rho v)}{\partial t}=\rho\frac{\partial(v)}{\partial t}+v\frac{\partial(\rho)}{\partial t}$

I have the initial variables and $\rho$ and v are vectors. I have solved the equations explicitly and I have solved one system of equations implicitly before but not sure how to approach this problem with two systems of equations implicitly. Do I need to turn them all into one system of equations?

Thanks!

but I think that your system requires the energy equation, too... for the compressible flow model the pressure is thermodinamic and you need to supply the state equation ...
However, your system is intrinsicly coupled, you can use several methods to solve it

Hooman · June 1, 2012, 07:30

I know about the energy equation to solve for pressure.
Right now I just need to know how to approach solving these two equations in the form that they are and with only rho and v the variable.

Can you name some of these methods please?

Thanks!

FMDenaro · June 1, 2012, 08:04

Quote:

Originally Posted by Hooman

I know about the energy equation to solve for pressure.
Right now I just need to know how to approach solving these two equations in the form that they are and with only rho and v the variable.

Can you name some of these methods please?

Thanks!

you can see: Chung T.J. Computational fluid dynamics (CUP, 2002)
and look at the PISO method

Hooman · June 4, 2012, 10:30

Thanks for this. I looked it up.

Can I just ask if I had a nonlinear term in the right hand side of the second equation, for instance $\rho v^2$ , would I need to linearize it somehow or can I just leave it as it is?

Thanks again!

Hooman · June 7, 2012, 05:46

Hi again,

I just had a look at some books and articles and from what I understood correction methods such as PISO and SIMPLE are used when the pressure-based approach is used in solving compressible flows. When using density-based approaches are used there is no need to use these equations we have the continuity equation. I was wondering if you could correct me if I'm mistaken about this and also how I can go on about solving these equations using when using the density-based approach?

I hope u see this message.

Thanks again.

June 1, 2012, 06:49	solving for denstiy and velocity implicitly	#1
Hooman Senior Member Join Date: Jun 2010 Posts: 111 Rep Power: 17	Hi, I have been trying to solve the following systems of equations implicitly: $\frac{\partial\rho}{\partial t}=f\left(\rho,t\right)$ $\frac{\partial \rho v}{\partial t}=g\left(\rho,v,t\right)$ I don't want to change the second equation to: $\frac{\partial(\rho v)}{\partial t}=\rho\frac{\partial(v)}{\partial t}+v\frac{\partial(\rho)}{\partial t}$ I have the initial variables and $\rho$ and v are vectors. I have solved the equations explicitly and I have solved one system of equations implicitly before but not sure how to approach this problem with two systems of equations implicitly. Do I need to turn them all into one system of equations? Thanks!

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June 1, 2012, 07:30		#3
Hooman Senior Member Join Date: Jun 2010 Posts: 111 Rep Power: 17	I know about the energy equation to solve for pressure. Right now I just need to know how to approach solving these two equations in the form that they are and with only rho and v the variable. Can you name some of these methods please? Thanks!

June 4, 2012, 10:30		#5
Hooman Senior Member Join Date: Jun 2010 Posts: 111 Rep Power: 17	Thanks for this. I looked it up. Can I just ask if I had a nonlinear term in the right hand side of the second equation, for instance $\rho v^2$ , would I need to linearize it somehow or can I just leave it as it is? Thanks again!

June 7, 2012, 05:46		#6
Hooman Senior Member Join Date: Jun 2010 Posts: 111 Rep Power: 17	Hi again, I just had a look at some books and articles and from what I understood correction methods such as PISO and SIMPLE are used when the pressure-based approach is used in solving compressible flows. When using density-based approaches are used there is no need to use these equations we have the continuity equation. I was wondering if you could correct me if I'm mistaken about this and also how I can go on about solving these equations using when using the density-based approach? I hope u see this message. Thanks again.