# How to simply model fluid flow over heigtmap?

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?

• Is this a 3D flow or a 2D flow? What happened to the convective terms in the differential momentum balance? – Chet Miller Mar 7 '18 at 20:48
• 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 $f(x,y)=z' \ge h (x,y)$ – Silverspur Mar 8 '18 at 0:16
• 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. – Silverspur Mar 8 '18 at 0:18
• 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. – Chet Miller Mar 8 '18 at 0:26
• My bad, I've mistaken them for the diffusive ones. I've corrected the post to something I hope is correct. – Silverspur Mar 8 '18 at 9:54

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.