[Neelay Doshi]

Hello everyone!

I am trying to solve 1-dimension, inviscid and compressible Navier-Stokes equation using the ode45 function in MATLAB. I have derived the necessary governing equation and have attached them in the post.

I understand that normally these are solved using FDM, FVM etc, but I wish to solve them using ode45 as it would require much lesser computational time.

As you can see, the equations are coupled and every equation contains a derivative term of another variable which may not be calculated yet.

For instance, as per my understanding, if I were create an ode function containing the 4 system of equations as per the order given in the attachment, then ode45 would first try to solve the energy equation (dT/dx) but it wouldn't know the value of the derivative of velocity (du/dx) as it has not been calculated yet, and so the solver would use du/dx=0, which would be incorrect.

Is their anyother way to solve these coupled odes? Kindly correct me if I am mistaken.

[FMDenaro]

Note sure you addressed correctly the question.
The inviscid flow model is the Euler system of equations not the Navier-Stokes.
Then you are assuming a steady condition that means this set of equations



rho*u=G=constant (mass)



G*du/dx +dp/dx=0 (momentum)


G*d/dx (E+p/rho) =0 (total energy)


Are you really want to solve numerically this system??

[Neelay Doshi]

Yes I meant Euler system, but the governing equations you have written and the ones i have attached are essentially the same, i have simply rearranged the terms.

Yes I would like to try to solve it numerically (using ode45). Do you suggest that it is not possible?

[FMDenaro]

Originally Posted by Neelay Doshi

Yes I meant Euler system, but the governing equations you have written and the ones i have attached are essentially the same, i have simply rearranged the terms.

Yes I would like to try to solve it numerically (using ode45). Do you suggest that it is not possible?

As you see, the energy equation is simply integrated as E+p/rho=constant, along with the mass constraint G=constant you get from the momentum G*u+p= constant.


There is nothing to solve numerically

[LuckyTran]

Originally Posted by Neelay Doshi

then ode45 would first try to solve the energy equation (dT/dx) but it wouldn't know the value of the derivative of velocity (du/dx) as it has not been calculated yet, and so the solver would use du/dx=0, which would be incorrect.
.

but why would the solver choose du/dx=0 and not -12345, hmm? Think about it...


What you are missing are the initial conditions and boundary conditions for Euler equations. Your problem definition is not complete you state the governing equations, the initial conditions, and boundary conditions. You should know the values of rho,p,T,U as well as drho/dx,dp/dx,dT/dx,dU/dx on the boundaries. You are choosing to approach this with ode45, making it akin to an intitial value problem instead of a boundary value problem. Still, you should know their values at least at x=0.

[Neelay Doshi]

Originally Posted by LuckyTran

but why would the solver choose du/dx=0 and not -12345, hmm? Think about it...

I have attached a snip of the ode45 function I have written, as you can see, I need to initialize the derivative column matrix to zero. If there is another method then kindly guide me.

Originally Posted by LuckyTran

Your problem definition is not complete you state the governing equations, the initial conditions, and boundary conditions. You should know the values of rho,p,T,U as well as drho/dx,dp/dx,dT/dx,dU/dx on the boundaries.

I do have the initial conditions (at x=0). I am feeding the ode45 function with the initial values of T, P, rho and u as the input "y". As the equations are 1st order odes, these conditions should suffice as per my understanding and I should not require the derivative values (drho/dx,dp/dx,dT/dx,dU/dx) and infact these values are supposed to be calculated by ode45.

Kindly tell me where have I gone wrong in writing the function code.

[LuckyTran]

Originally Posted by Neelay Doshi

I have attached a snip of the ode45 function I have written, as you can see, I need to initialize the derivative column matrix to zero. If there is another method then kindly guide me.

You don't NEED to. You could just as easily initialize it with ones(). Actually you can even initialize it with malloc(). Allocating space for a variable and initializing the problem are different things.

What you need to do is put in x=0 (i.e. put the initial conditions in) into the system and solve for all the missing derivatives (not using ode45 because it is not a coupled system of ode's you are solving but an algebraic linear system).