import matplotlib.pyplot as plt import numpy as np from FluidSim.FluidSimParameters import * """ Create Your Own Lattice Boltzmann Simulation (With Python) Philip Mocz (2020) Princeton Univeristy, @PMocz Simulate flow past cylinder for an isothermal fluid """ def main(): """ Finite Volume simulation """ # Simulation parameters epsilon = 0.000000001 Nx = 400 # resolution x-dir Ny = 100 # resolution y-dir rho0 = 1 # average density tau = 0.6 # collision timescale Nt = 80000 # number of timesteps plotRealTime = True # switch on for plotting as the simulation goes along params = FluidSimParameter(Ny) # params = WaterParameter(Ny) # params = MagmaParameter(Ny) # Lattice speeds / weights NL = 9 idxs = np.arange(NL) cxs = np.array([0, 0, 1, 1, 1, 0, -1, -1, -1]) cys = np.array([0, 1, 1, 0, -1, -1, -1, 0, 1]) weights = np.array([4 / 9, 1 / 9, 1 / 36, 1 / 9, 1 / 36, 1 / 9, 1 / 36, 1 / 9, 1 / 36]) # sums to 1 xx, yy = np.meshgrid(range(Nx), range(Ny)) # Initial Conditions N = np.ones((Ny, Nx, NL)) # * rho0 / NL temperature = np.ones((Ny, Nx, NL), np.float) # * rho0 / NL has_fluid = np.ones((Ny, Nx), dtype=np.bool) has_fluid[int(Ny/2):, :] = False np.random.seed(42) N += 0.01 * np.random.randn(Ny, Nx, NL) X, Y = np.meshgrid(range(Nx), range(Ny)) N[:, :, 3] += 2 * (1 + 0.2 * np.cos(2 * np.pi * X / Nx * 4)) # N[:, :, 5] += 1 rho = np.sum(N, 2) temperature_rho = np.sum(temperature, 2) for i in idxs: N[:, :, i] *= rho0 / rho temperature[:, :, i] *= 1 / temperature_rho # N[50:, :] = 0 temperature[:, :] = 0 # temperature += 0.01 * np.random.randn(Ny, Nx, NL) # Cylinder boundary X, Y = np.meshgrid(range(Nx), range(Ny)) cylinder = (X - Nx / 4) ** 2 + (Y - Ny / 2) ** 2 < (Ny / 4) ** 2 inner_cylinder = (X - Nx / 4) ** 2 + (Y - Ny / 2) ** 2 < (Ny / 4 - 2) ** 2 N[cylinder] = 0 N[0, :] = 0 N[Ny - 1, :] = 0 temperature[cylinder] = 0 # N[int(Ny/2):, :] = 0 has_fluid[cylinder] = False has_fluid[0, :] = False has_fluid[Ny - 1, :] = False # for i in idxs: # N[:, :, i] *= has_fluid # Prep figure fig = plt.figure(figsize=(4, 2), dpi=80) reflection_mapping = [0, 5, 6, 7, 8, 1, 2, 3, 4] # Simulation Main Loop for it in range(Nt): print(it) # Drift new_has_fluid = np.zeros((Ny, Nx)) F_sum = np.sum(N, 2) for i, cx, cy in zip(idxs, cxs, cys): F_part = N[:, :, i] / F_sum F_part[F_sum == 0] = 0 to_move = F_part * (has_fluid * 1.0) to_move = (np.roll(to_move, cx, axis=1)) to_move = (np.roll(to_move, cy, axis=0)) new_has_fluid += to_move N[:, :, i] = np.roll(N[:, :, i], cx, axis=1) N[:, :, i] = np.roll(N[:, :, i], cy, axis=0) temperature[:, :, i] = np.roll(temperature[:, :, i], cx, axis=1) temperature[:, :, i] = np.roll(temperature[:, :, i], cy, axis=0) # has_fluid = new_has_fluid > 0.5 # new_has_fluid[F_sum == 0] += has_fluid[F_sum == 0] * 1.0 # new_has_fluid[(np.abs(F_sum) < 0.000000001)] = 0 fluid_sum = np.sum(has_fluid * 1.0) has_fluid = (new_has_fluid / np.sum(new_has_fluid * 1.0)) * fluid_sum print('fluid_cells: %d' % np.sum(has_fluid * 1)) # for i in idxs: # N[:, :, i] *= has_fluid bndry = np.zeros((Ny, Nx), dtype=np.bool) bndry[0, :] = True bndry[Ny - 1, :] = True # bndry[:, 0] = True # bndry[:, Nx - 1] = True bndry = np.logical_or(bndry, cylinder) # bndry = np.logical_or(bndry, has_fluid < 0.5) # Set reflective boundaries bndryN = N[bndry, :] bndryN = bndryN[:, reflection_mapping] bndryTemp = temperature[bndry, :] bndryTemp = bndryTemp[:, reflection_mapping] sum_f = np.sum(N) print('Sum of Particles: %f' % sum_f) print('Sum of Temperature: %f' % np.sum(temperature)) # sum_f_cyl = np.sum(N[cylinder]) # print('Sum of Forces in cylinder: %f' % sum_f_cyl) # sum_f_inner_cyl = np.sum(N[inner_cylinder]) # print('Sum of Forces in inner cylinder: %f' % sum_f_inner_cyl) # if sum_f > 4000000.000000: # test = 1 # N[Ny - 1, :, 5] += 0.1 # N[0, :, 1] -= 0.1 # N[0, :, 5] += 0.1 # N[Ny - 1, :, 1] -= 0.1 # Calculate fluid variables rho = np.sum(N, 2) temp_rho = np.sum(temperature, 2) ux = np.sum(N * cxs, 2) / rho uy = np.sum(N * cys, 2) / rho ux[(np.abs(rho) < epsilon)] = 0 uy[(np.abs(rho) < epsilon)] = 0 g = -params.g * (temp_rho - yy / Ny) # uy[np.abs(rho) >= epsilon] += g[np.abs(rho) >= epsilon] / 2.0 uy += g / 2.0 # u_length = np.maximum(np.abs(ux), np.abs(uy)) u_length = np.sqrt(np.square(ux) + np.square(uy)) u_max_length = np.max(u_length) if u_max_length > np.sqrt(2): ux = (ux / u_max_length) * np.sqrt(2) uy = (uy / u_max_length) * np.sqrt(2) print('max vector part: %f' % u_max_length) # ux /= u_max_length # uy /= u_max_length # scale = abs(np.max(np.maximum(np.abs(ux), np.abs(uy))) - 1.0) < epsilon # if scale: # g = 0.01 * (temp_rho - yy / Ny) # # # F = np.zeros((Ny, Nx), dtype=np.bool) # # F = -0.1 * rho # # # uy[np.abs(rho) >= epsilon] += tau * F[np.abs(rho) >= epsilon] / rho[np.abs(rho) >= epsilon] # uy[np.abs(rho) >= epsilon] += g[np.abs(rho) >= epsilon] / 2.0 # # u_length = np.maximum(np.abs(ux), np.abs(uy)) # u_max_length = np.max(u_length) # # print('max vector part: %f' % u_max_length) # # ux /= u_max_length # uy /= u_max_length # print('minimum rho: %f' % np.min(np.abs(rho))) # print('Maximum N: %f' % np.max(N)) # print('Minimum N: %f' % np.min(N)) # Apply Collision temperature_eq = np.zeros(temperature.shape) Neq = np.zeros(N.shape) for i, cx, cy, w in zip(idxs, cxs, cys, weights): Neq[:, :, i] = rho * w * ( 1 + 3 * (cx * ux + cy * uy) + 9 * (cx * ux + cy * uy) ** 2 / 2 - 3 * (ux ** 2 + uy ** 2) / 2) temperature_eq[:, :, i] = temp_rho * w * ( 1 + 3 * (cx * ux + cy * uy) + 9 * (cx * ux + cy * uy) ** 2 / 2 - 3 * (ux ** 2 + uy ** 2) / 2) # test1 = np.sum(Neq) test2 = np.sum(N-Neq) if abs(test2) > 0.0001: test = '' print('Overall change: %f' % test2) n_pre_sum = np.sum(N[np.logical_not(bndry)]) temperature_pre_sum = np.sum(temperature[np.logical_not(bndry)]) N += -(1.0 / params.t1) * (N - Neq) temperature += -(1.0 / params.t2) * (temperature - temperature_eq) # Apply boundary N[bndry, :] = bndryN temperature[bndry, :] = bndryTemp # temperature[0, :, 0] = 0 # temperature[1, :, 0] = 0 temperature[0, :, 0] /= 2 temperature[1, :, 0] /= 2 temperature[Ny - 1, :, 0] = 1 temperature[Ny - 2, :, 0] = 1 # n_sum = np.sum(N, 2) # n_sum_min = np.min(n_sum) # if n_sum_min < 0: # N[np.logical_not(bndry)] += abs(n_sum_min) # N[np.logical_not(bndry)] /= np.sum(N[np.logical_not(bndry)]) # N[np.logical_not(bndry)] *= n_pre_sum # print('Sum of Forces: %f' % np.sum(N)) # temperature_sum = np.sum(temperature, 2) # temperature_sum_min = np.min(temperature_sum) # if temperature_sum_min < 0: # temperature[np.logical_not(bndry)] += abs(temperature_sum_min) # temperature[np.logical_not(bndry)] /= np.sum(temperature[np.logical_not(bndry)]) # temperature[np.logical_not(bndry)] *= temperature_pre_sum # print('Sum of Temperature: %f' % np.sum(temperature)) no_cylinder_mask = np.sum(N, 2) != 0 print('min N: %f' % np.min(np.sum(N, 2)[no_cylinder_mask])) print('max N: %f' % np.max(np.sum(N, 2))) print('min Temp: %f' % np.min(np.sum(temperature, 2)[no_cylinder_mask])) print('max Temp: %f' % np.max(np.sum(temperature, 2))) # plot in real time - color 1/2 particles blue, other half red if (plotRealTime and (it % 10) == 0) or (it == Nt - 1): fig.clear() plt.cla() ux[cylinder] = 0 uy[cylinder] = 0 vorticity = (np.roll(ux, -1, axis=0) - np.roll(ux, 1, axis=0)) - ( np.roll(uy, -1, axis=1) - np.roll(uy, 1, axis=1)) vorticity[cylinder] = np.nan # vorticity *= has_fluid cmap = plt.cm.bwr cmap.set_bad('black') plt.subplot(2, 2, 1) plt.imshow(vorticity, cmap='bwr') plt.clim(-.1, .1) # plt.imshow(has_fluid * 2.0 - 1.0, cmap='bwr') # plt.imshow(bndry * 2.0 - 1.0, cmap='bwr') # plt.imshow(np.sum(N, 2) * 2.0 - 1.0, cmap='bwr') # plt.imshow((np.sum(temperature, 2) / np.max(np.sum(temperature, 2))) * 2.0 - 1.0, cmap='bwr') plt.subplot(2, 2, 2) max_temp = np.max(np.sum(temperature, 2)) # plt.imshow(np.sum(temperature, 2) / max_temp * 2.0 - 1.0, cmap='bwr') plt.imshow(np.sum(temperature, 2) * 2.0 - 1.0, cmap='bwr') plt.clim(-.1, .1) plt.subplot(2, 2, 3) max_N = np.max(np.sum(N, 2)) plt.imshow(np.sum(N, 2) / max_N * 2.0 - 1.0, cmap='bwr') plt.clim(-.1, .1) # ax = plt.gca() # ax.invert_yaxis() # ax.get_xaxis().set_visible(False) # ax.get_yaxis().set_visible(False) # ax.set_aspect('equal') plt.pause(0.001) # Save figure # plt.savefig('latticeboltzmann.png', dpi=240) plt.show() return 0 if __name__ == "__main__": main()