Determine particle position knowing the velocity field

[Zoli]

Dear all,

I solved the Navier-Stokes equations in Eulerian description. I would like to illustrate it as follows:
I thought to place particles in the domain which will characterize the fluid flow. However I must know the particle position in the Lagrangian specification. As I place the particles at specified positions at t=0, I know the initial state. But how to step further?

Thanks,
Zoli

 

[olivermsun]

The idea is to integrate the velocity of the particle u(x0) for a time step. Then the particle has reached a new position (x1) where it has a new velocity u(x1), and so you integrate that velocity, and so on…

Many fancier schemes abound for doing this estimation more accurately, but that's the basic idea.

 

[vanhees71]

This is given by the transformation between Euler and Lagrange coordinates:
$$\vec{r}=\vec{r}(t,\vec{R}).$$
It describes the motion of a fluid element that was at the initial time, $t=0$ at position $\vec{R}$. This mapping between $\vec{r}$ and $\vec{R}$ is invertible as long as there are no singularities like shock waves etc. Then you can express every quantity in both the Lagrange coordinates $\vec{R}$ and Euler coordinates $\vec{r}$. E.g., the velocity of a fluid element is given by
$$\vec{v}(t,\vec{R})=\partial_t \vec{r}(t,\vec{R}).$$
You can as well take the velocity field as function of the Euler coordinates. Then the relation is
$$\vec{v}(t,\vec{R})=\vec{v}[t,\vec{r}(t,\vec{R})] \equiv \vec{v}(t,\vec{r}).$$
Then you have to distinguish between partial time derivatives (at fixed Euler coordinates $\vec{r}$) and "material time derivatives (time derivatives at fixed Lagrange coordinates $\vec{R}$. E.g., for the equations of motion (Euler perfect fluid or viscous Navier Stokes etc) you need the acceleration of a given fluid element, i.e., the material time derivative
$$\mathrm{D}_t \vec{v}=\frac{\partial}{\partial t} \vec{v}(t,\vec{r})+\frac{\partial \vec{r}(t,\vec{R})}{\partial t} \cdot \vec{\nabla}_{\vec{r}} \vec{v}(t,\vec{r})=\frac{\partial}{\partial t} \vec{v}(t,\vec{r})+(\vec{v} \cdot \vec{\nabla}_r) \vec{v}(t,\vec{r}).$$

If you have the velocity field in Euler coordinates, $\vec{v}(t,\vec{r})$, you can thus simply calculate the trajectories of the fluid elements by solving the set of coupled differential equations as an initial-value problem:
$$\frac{\mathrm{d} \vec{r}}{\mathrm{d} t}=\vec{v}(t,\vec{r}), \quad \vec{r}(t=0)=\vec{R}$$
This gives you the Euler coordinates in terms of the Lagrange coordinates, $\vec{r}=\vec{r}(t,\vec{R}).$

 

[Zoli]

So if I am not mistaken I should do the following for each particle:
1. Solve the initial-value problem to gain $\vec{r}(t,\vec{R})$ at different time steps. For this I must solve an IVP consisting of 2 equations because of the 2D-problem. But I have $\vec{v}$ at discrete points at each time.
2. How to continue?

Could you please specify algorithmically how to solve it? An example would help me a lot. E.g. let us regard I have the analytical solution $\vec{v} = e^{-t}\cos(x)\sin(y)$ on $[-1,1]\times[-1,1]$ in Euler coordinates given at $(x_1,y_1),\ldots, (x_N,y_N)$ at $t_0,t_1,\ldots,t_M$. How could I determine the position of the particle (that is $\vec{R}(t)$) starting at $\vec{R}(t=0)=(0,0)$?

 

[Zoli]

This is given by the transformation between Euler and Lagrange coordinates:
$$\vec{r}=\vec{r}(t,\vec{R}).$$
It describes the motion of a fluid element that was at the initial time, $t=0$ at position $\vec{R}$. This mapping between $\vec{r}$ and $\vec{R}$ is invertible as long as there are no singularities like shock waves etc. Then you can express every quantity in both the Lagrange coordinates $\vec{R}$ and Euler coordinates $\vec{r}$. E.g., the velocity of a fluid element is given by
$$\vec{v}(t,\vec{R})=\partial_t \vec{r}(t,\vec{R}).$$
You can as well take the velocity field as function of the Euler coordinates. Then the relation is
$$\vec{v}(t,\vec{R})=\vec{v}[t,\vec{r}(t,\vec{R})] \equiv \vec{v}(t,\vec{r}).$$
Then you have to distinguish between partial time derivatives (at fixed Euler coordinates $\vec{r}$) and "material time derivatives (time derivatives at fixed Lagrange coordinates $\vec{R}$. E.g., for the equations of motion (Euler perfect fluid or viscous Navier Stokes etc) you need the acceleration of a given fluid element, i.e., the material time derivative
$$\mathrm{D}_t \vec{v}=\frac{\partial}{\partial t} \vec{v}(t,\vec{r})+\frac{\partial \vec{r}(t,\vec{R})}{\partial t} \cdot \vec{\nabla}_{\vec{r}} \vec{v}(t,\vec{r})=\frac{\partial}{\partial t} \vec{v}(t,\vec{r})+(\vec{v} \cdot \vec{\nabla}_r) \vec{v}(t,\vec{r}).$$

If you have the velocity field in Euler coordinates, $\vec{v}(t,\vec{r})$, you can thus simply calculate the trajectories of the fluid elements by solving the set of coupled differential equations as an initial-value problem:
$$\frac{\mathrm{d} \vec{r}}{\mathrm{d} t}=\vec{v}(t,\vec{r}), \quad \vec{r}(t=0)=\vec{R}$$
This gives you the Euler coordinates in terms of the Lagrange coordinates, $\vec{r}=\vec{r}(t,\vec{R}).$

Using this idea, I managed to perform what I wanted. Thank you!