1
$\begingroup$

My input is a height map, i.e. the ground of the 3D world, represented by a function defined on a $N$-sized grid $h:\{0,N\}^2 \rightarrow \mathbb{R}$. The height of the ground over $(x,y,0)$ is $h(x,y)$.

I would like to model an approximate fluid flow over this height map. As a first attempt, I assume the fluid is perfect, incompressible, and constrained to the ground. A solution to the problem can then be another function $f:\{0,N\}^2 \rightarrow \mathbb{R}$ where $f(x,y)$ is the height of fluid over the ground point $(x,y,h(x,y))$. The fluid top surface is free.

With the perfect and incompressible fluid hypotheses, Navier-Stokes equations are as follow, with $\vec{V}$ the speed vector, $P$ the pressure, and $f$ all other mass forces:

$$\nabla\cdot \vec{V} = 0$$

$$\frac{\partial\vec{V}}{\partial t} + \vec{V}\cdot\nabla\vec{V}= f - \frac{1}{\rho} \nabla P~~(Eq 1)$$

But I think I can simplify even more these equations due to the last hypothesis, yet I do not know how.


EDIT:

Maybe I could assume $$\frac {dV_z}{dt} <<< 1$$ and use the projection of the previous equation over $z$ axis to derive, for any $x,y,z$ such as the point $(x,y,z) $ is in the fluid : $$ P (x,y,z) = P_0 + \rho g (z_{fluid surface} - z) $$

I can then integrate this pressure over the height of the fluid column for a given $x,y $ and find the mean pressure of this column, then use this value in the $x $ and $y $ axes projections of the equation 1 in order to find the speeds.

Would this be a correct reasoning?

$\endgroup$
  • $\begingroup$ Is this a 3D flow or a 2D flow? What happened to the convective terms in the differential momentum balance? $\endgroup$ – Chet Miller Mar 7 '18 at 20:48
  • $\begingroup$ The world is 3D but the fluid is constrained to the ground. If the height map is modelled as a function $h(x,y)=z$, then the fluid can be fully described by its top surface $f(x,y)=z' \ge h (x,y) $ $\endgroup$ – Silverspur Mar 8 '18 at 0:16
  • $\begingroup$ As for the convective terms, unless I'm mistaken (which might very well be the case), they refer to viscosity effects and disappear in the case of a perfect fluid. $\endgroup$ – Silverspur Mar 8 '18 at 0:18
  • $\begingroup$ Yes, you are mistaken. The convective terms are not zero for a perfect fluid if the height of the channel is a function of x and y. $\endgroup$ – Chet Miller Mar 8 '18 at 0:26
  • $\begingroup$ My bad, I've mistaken them for the diffusive ones. I've corrected the post to something I hope is correct. $\endgroup$ – Silverspur Mar 8 '18 at 9:54
1
$\begingroup$

I assume you want to solve the shallow water equations, which are the depth-averaged versions of the Euler equations. They can be written as

$\frac{\partial h}{\partial t}+\frac{\partial uh}{\partial x}+\frac{\partial vh}{\partial y}=0$

$\frac{\partial v}{\partial t}+\frac{\partial}{\partial y}\big(v^2+\frac{1}{2}gh^2\big)+\frac{\partial}{\partial x}(uvh)=0$

$\frac{\partial u}{\partial t}+\frac{\partial}{\partial x}\big(u^2+\frac{1}{2}gh^2\big)+\frac{\partial}{\partial y}(uvh)=0$

where $h=h(x,y)$ is the water depth field, $u$ and $v$ are the velocity component fields in $x$ and $y$ directions.

These equations can be solved with many numerical schemes for instance finite difference methods (e.g. using the Lax-Wendroff scheme).

However, your concept of using particles seems to require a Lagrangian approach. From this point of view, the smoothed particle hydrodynamics method (SPH) is a possible choice. Despite of certain difficulties, fortunately, the shallow water equations above are already solved with SPH successfully. I would suggest to read the papers below (if you have access to them) for the technical details of the solution.

[*] Giulia Rossi, Michael Dumbser, Aronne Armanini, A well-balanced path conservative SPH scheme for nonconservative hyperbolic systems with applications to shallow water and multi-phase flows, Computers & Fluids, Volume 154, 2017, Pages 102-122,

[**] Xilin Xia, Qiuhua Liang, Manuel Pastor, Weilie Zou, Yan-Feng Zhuang, Balancing the source terms in a SPH model for solving the shallow water equations, Advances in Water Resources, Volume 59, 2013, Pages 25-38

Edit: Although the implementation of SWE-SPH is easy compared to finite element or finite volume solvers, I think it is better to use an existing solver like SWE-SPHysics, which is an open source tool and a little brother of DualSPHysics. As long as you desire a little more programming and higher configurability, there are further options like Aboria or Nauticle (my own project), being general purpose particle simulation tools.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.