Fluid Dynamics

Important: Please read the installation page for details about how to install the toolboxes. $\newcommand{\dotp}[2]{\langle #1, #2 \rangle}$ $\newcommand{\enscond}[2]{\lbrace #1, #2 \rbrace}$ $\newcommand{\pd}[2]{ \frac{ \partial #1}{\partial #2} }$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\umax}[1]{\underset{#1}{\max}\;}$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\uargmin}[1]{\underset{#1}{argmin}\;}$ $\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\abs}[1]{\left|#1\right|}$ $\newcommand{\choice}[1]{ \left\{ \begin{array}{l} #1 \end{array} \right. }$ $\newcommand{\pa}[1]{\left(#1\right)}$ $\newcommand{\diag}[1]{{diag}\left( #1 \right)}$ $\newcommand{\qandq}{\quad\text{and}\quad}$ $\newcommand{\qwhereq}{\quad\text{where}\quad}$ $\newcommand{\qifq}{ \quad \text{if} \quad }$ $\newcommand{\qarrq}{ \quad \Longrightarrow \quad }$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\EE}{\mathbb{E}}$ $\newcommand{\Zz}{\mathcal{Z}}$ $\newcommand{\Ww}{\mathcal{W}}$ $\newcommand{\Vv}{\mathcal{V}}$ $\newcommand{\Nn}{\mathcal{N}}$ $\newcommand{\NN}{\mathcal{N}}$ $\newcommand{\Hh}{\mathcal{H}}$ $\newcommand{\Bb}{\mathcal{B}}$ $\newcommand{\Ee}{\mathcal{E}}$ $\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Gg}{\mathcal{G}}$ $\newcommand{\Ss}{\mathcal{S}}$ $\newcommand{\Pp}{\mathcal{P}}$ $\newcommand{\Ff}{\mathcal{F}}$ $\newcommand{\Xx}{\mathcal{X}}$ $\newcommand{\Mm}{\mathcal{M}}$ $\newcommand{\Ii}{\mathcal{I}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Ll}{\mathcal{L}}$ $\newcommand{\Tt}{\mathcal{T}}$ $\newcommand{\si}{\sigma}$ $\newcommand{\al}{\alpha}$ $\newcommand{\la}{\lambda}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\La}{\Lambda}$ $\newcommand{\si}{\sigma}$ $\newcommand{\Si}{\Sigma}$ $\newcommand{\be}{\beta}$ $\newcommand{\de}{\delta}$ $\newcommand{\De}{\Delta}$ $\newcommand{\phi}{\varphi}$ $\newcommand{\th}{\theta}$ $\newcommand{\om}{\omega}$ $\newcommand{\Om}{\Omega}$

This numerical tour explores fluid dynamics for image generation.

In [2]:

Velocity Flow Field

A velocity flow is simply a 2-D vector field $V = (V_i)_{i=1}^N \in \RR^{n \times n \times 2}$ where $V_i \in \RR^2$ is one of the $N=n \times n$ vectors at a position indexed by $i$.

It can be generated as a realization of Gaussian process. The blurring creates correlations in the flow.

In [3]:
n = 128;
options.bound = 'per';
V = perform_blurring(randn(n,n,2), 40, options);

Subsampling display operator.

In [4]:
myplot = @(V)plot_vf( V(1:6:n,1:6:n, :) );

We can display the vector field using arrow.

In [5]:

We can renormalize the flow, which enhance the singularities. It defines $\tilde V$ as $\tilde V_i = V_i/\norm{V_i}$.

In [6]:
normalize = @(V)V ./ repmat( max(1e-9,sqrt(sum3(V.^2, 3))) , [1 1 2]);


In [7]:

Incompressible Flows

An incompressible flow as vanishing divergence. The set of vector incompressible flow defines a sub-space of $\RR^{n \times n \times 2}$ $$ \Ii = \enscond{V}{ \text{div}(V)=0 }$ \qwhereq \text{div}(V) = \pd{V}{x_1} + \pd{V}{x_2} \in \RR^{n \times n}. $$ Here $\pd{}{x_s}$ for $s=1,2$ are finite differences approximation of the horizontal and vertical derivative operators (we suppose here periodic boundary conditions).

The orthogonal projection $U = \text{Proj}_{\Ii}(V)$ on $\Ii$ is computed by solving a Poisson equation $$ U = V-\nabla A \qwhereq \Delta A = \text{div}(V). $$

This is especially simple for periodic boundary conditions since $A$ can be compute over the Fourier domain as $$ \forall \om \neq 0, \quad \hat A(\om) = \frac{\hat Y(\om)}{\mu(\om)}$ \qwhereq Y = \text{div}(V) \qandq \mu(\om_1,\om_2) = -4 \sin(\om_1 \pi / n)^2 -4 \sin(\om_2 \pi / n)^2 $$ and $\hat A(0)=0$.

Compute the kernel $\mu(\om)$.

In [8]:
[Y,X] = meshgrid(0:n-1,0:n-1);
mu = sin(X*pi()/n).^2; mu = -4*( mu+mu' );
mu(1) = 1;

Computation of $A$.

In [9]:
A = @(V)real( ifft2( fft2( div(V, options) ) ./ mu ) );

Projection on incompressible flows.

In [10]:
ProjI = @(V)V - grad(A(V), options);

Display $U=\text{Proj}_{\Ii}(V)$.

In [11]:
U = ProjI(V);

Display $W=U-V$ the irrotational component of $V$.

In [12]:

Note that the decomposition $V=U+W$ is called the Hoge decomposition of the vector field.

Image Advection Along the Flow

A flow defines a warping operator that transport the content of an image along the streaming of the flow.

We load an image $f$.

In [13]:
name = 'lena';
f = crop( load_image(name, 2*n), n);

Given some vector field $U$, the warping operator $f_1 = \Ww_U(f)$ along the flow is define $$ f_1(x) = f(x+U(x)) $$ i.e. it advects the values of $f$ by the vector field $U$ to obtain the values of $f_1$.

We define $U$ as a scaled normalized incompressible flow.

In [14]:
U = normalize(ProjI(V));

Helper function: enforce periodicity.

In [15]:
periodic = @(P)cat(3, mod(P(:,:,1)-1,n)+1, mod(P(:,:,2)-1,n)+1 );

Helper function: extend an image by 1 pixel to avoid boundary problems.

In [16]:
extend1 = @(f)[f f(:,1)];
extend = @(f)extend1(extend1(f)')';

Helper function: bilinear interpolation on a grid.

In [17]:
myinterp = @(P1,f1,Pi)interp2( P1(:,:,2), P1(:,:,1),f1, Pi(:,:,2), Pi(:,:,1) );

First we compute the initial and warped grids.

In [18]:
[Y,X] = meshgrid(1:n,1:n); 
P = cat(3, X,Y);
[Y1,X1] = meshgrid(1:n+1,1:n+1);
P1 = cat(3, X1,Y1);

Defines the warping operator $\Ww_U$.

In [19]:
W = @(f,U)myinterp( P1, extend(f), periodic( P - U ) );

Display a warped image $\Ww_{\rho U}(f)$ for some scaling $\rho$.

In [20]:
rho = 2;

Exercise 1

Display $\Ww_{\rho U}(f)$ for various values of $\rho$.

In [21]:
In [22]:
%% Insert your code here.

Exercise 2

Define an iterative scheme via: $$ f^{(\ell+1)} = \Ww_{\rho U}(f^{(\ell)}). $$ Display the result $f^{(\ell)}$, which corresponds approximately to solving an advection equation at time $t=\ell \rho$.

In [23]:
In [24]:
%% Insert your code here.

Fluid Dynamics

Fluid dynamics solves the incompressible Navier-Stokes equations to evolve in time the vector field.

We discribe here a simple algorithm introduced in:

J. Stam, Stable Fluids, SIGGRAPH'99, 1999, p. 121-128.

It proposes a semi-implicit scheme for the resolution of the Navier Stockes equations for the movement of incompressible fluids $$ \pd{V}{t} = \text{Proj}_{\Ii}\pa{ -(V \cdot \nabla) V + \nu \Delta V + W }. $$ Here $\nu \geq 0$ is the viscosity of the fluid, $W$ is a source term, $\Delta$ is the Laplacian, and $-(V \cdot \nabla) V$ is the non-linear self-advection, where we have used the short-hand notation $V \cdot \nabla$ for the derivative operator along a flow $V$: $$ (V \cdot \nabla)U = ( V_1 \pd{U_1}{x_1} + V_2 \pd{U_1}{x_2}, V_1 \pd{U_2}{x_1} + V_2 \pd{U_2}{x_2} ).$$

In order to visualize the flow, we also advect and diffuse along the flow a density $g$ of particules, which is a scalar field. Once $V$ has been computed, it follows a linear PDE $$ \pd{g}{t} = -(V \cdot \nabla) g + \mu \Delta g + h $$ with some initial condition at time $t=0$, where $h$ is a source for the density.

In practice, we solve this PDE in parallel to the PDE for $V$.

In the following, we use $W=0$ and $h=0$ (no sources).

Set the viscosity $\nu$ for the velocity field.

In [25]:
nu = 1/10;

We use a larger viscosity $\mu$ for the evolution of the density of particules.

In [26]:
mu = 2*nu;

Extend the warping operator $\Ww_U$ to work with vector fields as input. This will apply $\Ww_U$ on each channel of the vector field (X and Y coordinates).

In [27]:
Wt = @(V,U)cat(3, W(V(:,:,1),U), W(V(:,:,2),U) );

We discretize the PDE's using some time step $\tau$.

In [28]:
tau = .5;

The algorithm computes $V^{(\ell)}$ at iteration $\ell$ which is an approximation of the PDE solution at time $\ell \tau$. It is computed itertatively as $$ \tilde V^{(\ell)} = \Ww_{\tau V^{(\ell)}}( V^{(\ell)} ) \qandq V^{(\ell+1)} = \text{Proj}_{\Ii}\pa{ \tilde V^{(\ell)} + \tau\nu\Delta \tilde V^{(\ell)} + \tau W }$ $$

It computes in parallel the evolution of the density as $$ \tilde g^{(\ell)} = \Ww_{\tau V^{(\ell)}}( g^{(\ell)} ) \qandq g^{(\ell+1)} = \tilde g^{(\ell)} + \tau\nu\Delta \tilde g^{(\ell)} + \tau h $$

Set the initial field $V=V^{(0)}$ at time $t=0$.

In [29]:
V = normalize(ProjI(V));

Set the initial density $g=g^{(0)}$ at time $t=0$.

In [30]:
g = f;

The first step is to advect the vector field $V$ and $g$ along the flow $V$ itself. This corresponds to an implict discretization of the term $-(V \cdot \nabla) V$.

In [31]:
g = W (g,tau*U);
V = Wt(V,tau*U);

We implement the Laplacian using finite difference.

In [32]:
s1 = [2:n 1]; s2 = [n 1:n-1];
Delta = @(g)1/4 *( g(s1,:,:) + g(s2,:,:) + g(:,s1,:) + g(:,s2,:) ) - g;

The second step is to diffuse the vector field and the density.

In [33]:
V = V + tau*nu*Delta(V);
g = g + tau*mu*Delta(g);

The last step is to ensure incompressibility of $V$ by projecting on $\Ii$.

In [34]:
V = ProjI(V);

Exercise 3

Compute the fluid dynamic by iterating these steps. nitialization

In [35]:
In [36]:
%% Insert your code here.