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# Fun with compute shaders and fluid dynamics
## First for some theory…
What we want (Navier-Stokes):
$$\begin{aligned} \partial_t \rho + \nabla \cdot (\rho u) &= 0 \\ \partial_t u + (u \cdot \nabla) u &= -\frac{1}{\rho} \nabla p + 2\nu\nabla \cdot (\mathrm{S})\end{aligned}$$
Pressure $p = c_s^2 \rho$
Kinetic viscosity: $\nu = c_s^2 \tau$
Tensor: $\mathrm{S} = \frac{1}{2} (\nabla u + (\nabla u)^\top)$
What we use (Boltzmann equilibrium):
$$\left( \partial_t + \xi \cdot \partial_x + \frac{F}{\rho} \cdot \partial_\xi \right) f = \Omega(f) \left( = \partial_x f \cdot \frac{dx}{dt} + \partial_\xi f \cdot \frac{d\xi}{dt} + \partial_t f \right)$$
How we get there (BGK LBM):
$$\Omega(f) := -\frac{f-f^\text{eq}}{\tau} \Delta t$$
$$(\partial_t + \xi \cdot \nabla_x) f = -\frac{1}{\tau} (f(x,\xi,t) - f^\text{eq}(x,\xi,t))$$
$$\newcommand{\V}[2]{\begin{pmatrix}#1\\#2\end{pmatrix}} \{\xi_i\}_{i=0}^8 = \left\{ \V{0}{0}, \V{-1}{\phantom{-}1}, \V{-1}{\phantom{-}0}, \V{-1}{-1}, \V{\phantom{-}0}{-1}, \V{\phantom{-}1}{-1}, \V{1}{0}, \V{1}{1}, \V{0}{1} \right\}$$
$$(\partial_t + \xi_i \cdot \nabla_x) f_i(x,t) = -\frac{1}{\tau} (f_i(x,t) - f_i^\text{eq}(x,t))$$
$$f_i^\text{eq} = w_i \rho \left( 1 + \frac{u \cdot \xi_i}{c_s^2} + \frac{(u \cdot \xi_i)^2}{2c_s^4} - \frac{u \cdot u}{2c_s^2} \right)$$
$$\rho(x,t) = \sum_{i=0}^{q-1} f_i(x,t)$$
$$\rho u(x,t) = \sum_{i=0}^{q-1} \xi_i f_i(x,t)$$
$$w_0 = \frac{4}{9}, \ w_{2,4,6,8} = \frac{1}{9}, \ w_{1,3,5,7} = \frac{1}{36}$$
$$\overline{f_i} = f_i + \frac{1}{2\tau}(f_i - f_i^\text{eq})$$
$$\overline\tau = \tau + \frac{1}{2}$$
$$\overline{f_i}(x+\xi_i,t+1) = \overline{f_i}(x,t) - \frac{1}{\overline\tau} (\overline{f_i}(x,t) - f_i^\text{eq}(x,t))$$
$$f_i^\text{out}(x,t) = f_i(x,t) - \frac{1}{\tau}(f_i(x,t) - f_i^\text{eq}(x,t))$$
$$f_i(x+\xi_i,t+1) = f_i^\text{out}(x,t)$$
## …translated into GLSL compute shaders
```cpp
layout (local_size_x = 1, local_size_y = 1) in;
layout (std430, binding=1) buffer bufferCollide{ float collideCells[]; };
layout (std430, binding=2) buffer bufferStream{ float streamCells[]; };
layout (std430, binding=3) buffer bufferFluid{ float fluidCells[]; };
uniform uint nX;
uniform uint nY;
```
```cpp
const uint q = 9;
const float weight[q] = float[](
1./36., 1./9., 1./36.,
1./9. , 4./9., 1./9. ,
1./36 , 1./9., 1./36.
);
```
```cpp
uint indexOfDirection(int i, int j) {
return 3*(j+1) + (i+1);
}
uint indexOfLatticeCell(uint x, uint y) {
return q*nX*y + q*x;
}
/* [...] */
float get(uint x, uint y, int i, int j) {
return collideCells[indexOfLatticeCell(x,y) + indexOfDirection(i,j)];
}
```
```cpp
float equilibrium(float d, vec2 v, int i, int j) {
return w(i,j) * d * (1 + 3*comp(i,j,v) + 4.5*sq(comp(i,j,v)) - 1.5*sq(norm(v)));
}
```
```cpp
void main() {
const uint x = gl_GlobalInvocationID.x;
const uint y = gl_GlobalInvocationID.y;
const float d = density(x,y);
const vec2 v = velocity(x,y,d);
setFluid(x,y,v,d);
for ( int i = -1; i <= 1; ++i ) {
for ( int j = -1; j <= 1; ++j ) {
set(x,y,i,j, get(x,y,i,j) + omega * (equilibrium(d,v,i,j) - get(x,y,i,j)));
}
}
}
```
```cpp
void main() {
const uint x = gl_GlobalInvocationID.x;
const uint y = gl_GlobalInvocationID.y;
if ( x != 0 && x != nX-1 && y != 0 && y != nY-1 ) {
for ( int i = -1; i <= 1; ++i ) {
for ( int j = -1; j <= 1; ++j ) {
set(x+i,y+j,i,j, get(x,y,i,j));
}
}
} else {
// rudimentary bounce back boundary handling
[...]
}
}
```
## Visuals
## Reaching down from the heavens
|
