Lattice Boltzmann Equation: D2Q9 Model Simulation

Topic 6 Complex Systems Lecture 8
Lattice Boltzmann Equation: D2Q9 Model Simulation
The following program lbe.cpp simulates two-dimensional flow in a channel with a linear obstacle using the Lattice
Boltzmann Equation in the Bhatnagar-Gross-Krook approximation. The program is based on lbe.f by Dr. Sauro Succi.
lbe.cpp
// Lattice Boltzmann Equation: D2Q9 Model Simulation 1
// Flow in a channel with a linear obstacle 2
#include 4
#include 5
#include 6
using namespace std; 8
#include 10
int Nx = 64; // number of cells in x direction 12
int Ny = 32; // number of cells in y direction 13
int Nc = 9; // number of discrete velocities 15
enum direction { 16
East = 1, North, West, South, 17
NorthEast, NorthWest, SouthWest, SouthEast 18
}; 19
double ***n; // number densities 21
double ***neq; // equilibrium number densities 22
double ***u; // fluid velocity 23
PHY 411-506 Computational Physics II 1 Friday, April 16, 2004

Topic 6 Complex Systems Lecture 8
void allocate() { 25
static int oldNx = 0, oldNy = 0; 26
if (Nx != oldNx) { 27
if (n != 0) { 28
for (int x = 0; x < oldNx+2; x++) { 29
for (int y = 0; y < oldNy+2; y++) { 30
delete [] n[x][y]; delete [] neq[x][y]; delete [] u[x][y]; 31
} 32
delete [] n[x]; delete [] neq[x]; delete [] u[x]; 33
} 34
} 35
delete [] n; delete [] neq; delete [] u; 36
oldNx = Nx; 37
oldNy = Ny; 38
n = new double** [Nx+2]; 39
neq = new double** [Nx+2]; 40
u = new double** [Nx+2]; 41
for (int x = 0; x < Nx+2; x++) { 42
n[x] = new double* [Ny+2]; 43
neq[x] = new double* [Ny+2]; 44
u[x] = new double* [Ny+2]; 45
for (int y = 0; y < Ny+2; y++) { 46
n[x][y] = new double [Nc]; 47
neq[x][y] = new double [Nc]; 48
u[x][y] = new double [2]; 49
} 50
} 51
} 52
} 53
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Topic 6 Complex Systems Lecture 8
double rho; // fluid density 55
double u0[2]; // initial fluid velocity 56
double uf[2]; // final fluid velocity 57
double tau; // relaxation time 58
int nObstacle; // extent of obstacle in y direction 59
void getInput() { 61
cout > u0[0]; u0[1] = 0; 64
cout > rho; 66
cout > uf[0]; uf[1] = 0; 68
cout > tau; 70
cout > nObstacle; 72
} 73
const double w0 = 16 / 36.0, 75
w1 = 4 / 36.0, 76
w2 = 1 / 36.0; 77
const double w[9] = { // directional weights 78
w0, // zero 79
w1, w1, w1, w1, // east, north, west, south 80
w2, w2, w2, w2 // north-east, north-west, south-west, south-east 81
}; 82
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Topic 6 Complex Systems Lecture 8
const double c[9][2] = { // particle velocities 83
{0, 0}, // zero 84
{1, 0}, {0, 1}, // east, north 85
{-1, 0}, {0, -1}, // west, south 86
{1, 1}, {-1, 1}, // north-east, north-west 87
{-1, -1}, {1, -1} // south-west, south-east 88
}; 89
const double csSqd = 1 / 3.0; // speed of sound squared 90
void initialize_u() { 92
for (int x = 0; x < Nx; x++) 93
for (int y = 0; y < Ny; y++) { 94
for (int d = 0; d < 2; d++) 95
u[x][y][d] = u0[d]; 96
} 97
} 98
void compute_neq() { 100
for (int x = 0; x < Nx+2; x++) 101
for (int y = 0; y < Ny+2; y++) { 102
double uSqd = u[x][y][0] * u[x][y][0] + u[x][y][1] * u[x][y][1]; 103
uSqd /= 2 * csSqd; 104
for (int i = 0; i < Nc; i++) { 105
double uci = u[x][y][0] * c[i][0] + u[x][y][1] * c[i][1]; 106
uci /= csSqd; 107
neq[x][y][i] = rho * w[i] * (1 + uci * (1 + uci / 2) - uSqd); 108
} 109
} 110
} 111
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Topic 6 Complex Systems Lecture 8
void initialize_n() { 113
for (int x = 0; x < Nx + 2; x++) 114
for (int y = 0; y < Ny + 2; y++) 115
for (int i = 0; i < Nc; i++) 116
n[x][y][i] = neq[x][y][i]; 117
} 118
void initialize() { 120
getInput(); 121
allocate(); 122
initialize_u(); 123
compute_neq(); 124
initialize_n(); 125
} 126
void boundaryConditions() { 128
// periodic boundary conditions for east moving particles at east/west ends 130
for (int y = 1; y

Topic 6 Complex Systems Lecture 8
n[Nx + 1][y][SouthWest] = n[1][y][SouthWest]; 141
} 142
// no-slip boundary conditions for north moving particles at the top end 144
for (int x = 1; x

Topic 6 Complex Systems Lecture 8
for (int y = Ny; y >= 1; y--) { 170
n[x][y][East] = n[x - 1][y][East]; 171
n[x][y][NorthEast] = n[x - 1][y - 1][NorthEast]; 172
} 173
// start at SE corner and stream south and south-east moving particles 175
for (int x = Nx; x >= 1; x--) 176
for (int y = 1; y

Topic 6 Complex Systems Lecture 8
} 199
} 200
void collide() { 202
for (int x = 1; x

Topic 6 Complex Systems Lecture 8
int x = Nx / 4; 229
int y1 = Ny / 2 - nObstacle / 2; 230
int y2 = Ny / 2 + nObstacle / 2; 231
for (int y = y1 + 1; y

Topic 6 Complex Systems Lecture 8
obstacle(); 257
glutPostRedisplay(); 259
} 260
const double toDegrees = 180.0 / (4 * atan(1.0)); 262
void display() { 264
glClear(GL_COLOR_BUFFER_BIT); 265
glColor3ub(0, 0, 255); 266
glRectd(0, - 0.05 * (Ny + 2), Nx + 2, 0); 267
glRectd(0, Ny + 1, Nx + 2, (Ny + 1) * 1.05); 268
if (nObstacle > 1) { 269
glColor3ub(0, 255, 0); 270
int x = Nx / 4, y1 = Ny / 2 - nObstacle / 2, 271
y2 = Ny / 2 + nObstacle / 2; 272
glRectd(x, y1, x + 1, y2); 273
} 274
double uMax = 0; 276
for (int x = 0; x < Nx + 2; x++) 277
for (int y = 0; y < Ny + 2; y++) { 278
double uMag = sqrt(u[x][y][0] * u[x][y][0] + u[x][y][1] * u[x][y][1]); 279
if (uMag > uMax) 280
uMax = uMag; 281
} 282
if (uMax == 0) uMax = 1; 283
glColor3ub(255, 0, 0); 285
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Topic 6 Complex Systems Lecture 8
for (int x = 0; x < Nx + 2; x++) 286
for (int y = 0; y < Ny + 2; y++) { 287
glPushMatrix(); 288
double uMag = sqrt(u[x][y][0] * u[x][y][0] + 289
u[x][y][1] * u[x][y][1]); 290
double scale = uMag / uMax; 291
double angle = atan2(u[x][y][1], u[x][y][0]) * toDegrees; 292
glTranslated(x, y, 0); 293
glScaled(scale, scale, 1); 294
glRotated(angle, 0, 0, 1); 295
glBegin(GL_LINES); 296
glVertex2d(0, 0); glVertex2d(1, 0); 297
glVertex2d(1, 0); glVertex2d(0.8, 0.2); 298
glVertex2d(1, 0); glVertex2d(0.8, -0.2); 299
glEnd(); 300
glPopMatrix(); 301
} 302
glutSwapBuffers(); 304
} 305
void reshape(int w, int h) { 307
int x0 = 0, y0 = 0, dx = w, dy = h; 308
double aspect = (w * (Ny + 2) * 1.1) / (h * (Nx + 2)); 309
if (aspect > 1) { 310
dx = int(dx / aspect); 311
x0 = (w - dx) / 2; 312
} else { 313
dy = int(dy * aspect); 314
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Topic 6 Complex Systems Lecture 8
y0 = (h - dy) / 2; 315
} 316
glViewport(x0, y0, dx, dy); 317
glMatrixMode(GL_PROJECTION); 318
glLoadIdentity(); 319
gluOrtho2D(0, Nx + 2, - 0.05 * (Ny + 2), (Ny + 2) * 1.05); 320
glMatrixMode(GL_MODELVIEW); 321
glLoadIdentity(); 322
} 324
void mouse(int button, int state, int x, int y) { 326
switch (button) { 327
case GLUT_LEFT_BUTTON: 328
if (state == GLUT_DOWN) 329
; 330
break; 331
default: 332
break; 333
} 334
} 335
int main(int argc, char *argv[]) { 337
glutInit(&argc, argv); 338
if (argc > 1) { 339
Nx = atoi(argv[1]); 340
Ny = atoi(argv[2]); 341
} 342
initialize(); 343
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Topic 6 Complex Systems Lecture 8
glutInitDisplayMode(GLUT_DOUBLE | GLUT_RGB); 344
glutInitWindowSize(600, 330); 345
glutInitWindowPosition(100, 100); 346
glutCreateWindow("Lattice Boltzmann Equation: D2Q9 Model"); 347
glClearColor(1.0, 1.0, 1.0, 0.0); 348
glShadeModel(GL_FLAT); 349
glPixelStorei(GL_UNPACK_ALIGNMENT, 1); 350
glutDisplayFunc(display); 351
glutReshapeFunc(reshape); 352
glutIdleFunc(takeStep); 353
glutMouseFunc(mouse); 354
glutMainLoop(); 355
} 356
PHY 411-506 Computational Physics II 13 Friday, April 16, 2004