Eulerian Position and Velocity Updates - Compressible Fluid Dynamics

Hello Everyone,

It is my first time writing in this forum, so please be patient with me :)

I am writing a code following Smoothed Particle Computation (SPC) method for solving a Compressible Fluid Dynamics Problem.

In my code I got to a point where I need to do the following:

Basically, say at time = 0, I have a certain velocity u, a position x and acceleration a.

Now, I want to update u and x for an amount of time steps, using the following fomulae:

u (t+1) = u(t) + acceleration*dt

x(t+1) = x(t) + u(t+1)*dt

t = original time/time index

t+1 = new time

dt = 1 (for example), acceleration = same for every time step. In my case I would like to have 10 time steps.

How do I do it?

Thanks, Fed

Here is my code so far, just for fashion :

clc, clear
N=61; sigma = 5; m = 1;  a=1;
j = 1:61; 
x_j = j - 1 + 0.5*sign((N+1)/2 - j).*(1-exp(-0.2.*( j-(N+1)./2).^2) )

    for x_i = [1:61]
    ro = m/(sqrt(pi)*sigma).*exp((-(x_i-x_j).^2)./sigma^2)
    del_ro = (2*m)/(sqrt(pi)*sigma^3).*(x_i-x_j).^2.*exp((-(x_i-x_j).^2)./sigma^2)
    u_j = (-(a^2.*log(ro)))./2
    acc = (- a^2./ro).*del_ro
    end

for k = 1:10
u_j_new(k+1) = u_j(k) +acc(k)
     x_j_new(k+1) = [x_j(k).*u_j_new(k)]
     ro_new(k+1) = m/(sqrt(pi)*sigma).*exp((-(x_i-x_j_new).^2)./sigma^2)
end

Answer 1

[dwight nwaigwe]

Are u_j x_j acc vectors corresponding to 61 different points? Do you want to treat each point individually? In other words, do you want to find the new position,velocity, and ro of each index?

In that case you would do

for k = 1:10  

u_j = u_j +acc

x_j = x_j(k)+u_j_new

ro = m/(sqrt(pi)*sigma).*exp((-(x_i-x_j).^2)./sigma^2)

end

Answer 2

[dwight nwaigwe]

whoops, the second line should be

x_j = x_j+u_j_new

Answer 3

[Federico]

Thank you very much. I think what you wrote is right. I just cannot put it together with other loops.

I'll keep this posted as I go further with my code.

Anyway, in this code I will be using the Smoothed Particle Hydrodynamics method, so yes I am treating the flow into 61 finite smoothed particle.

Are you into this field (CFD etc etc? Because I feel like I will need some more help and am completely doing this by myself.

cheers

Answer 4

[dwight nwaigwe]

Hi Federico, Well, the loop over k is responsible for the evolution of your points in time. Is your first loop for initialization? Where does del_ro come into play?

I'm not into CFD but I am familiar with fluid mechanics and numerical methods (although I may be rusty).

Are you new to programming? If so, don't be afraid because it's not that hard. You just have to learn the "rules" of programming.