lattice Boltzmann方法 matlab代码 简单例子 例子

以下是一个简单的Lattice Boltzmann方法的MATLAB代码示例：

%% Initialize Grid and Parameters

nx = 100; % number of nodes along x-direction
ny = 50; % number of nodes along y-direction
nt = 10000; % number of time steps
n_eq = 9; % number of equilibrium distributions (for D2Q9)
rho0 = 1; % initial density
u0 = [0.1, 0]; % initial velocity
omega = 1.9; % relaxation parameter
cx = [0,1,0,-1,0,1,-1,-1,1]; % x-component of lattice velocities (D2Q9)
cy = [0,0,1,0,-1,1,1,-1,-1]; % y-component of lattice velocities (D2Q9)
w = [4/9, 1/9, 1/9, 1/9, 1/9, 1/36, 1/36, 1/36, 1/36]; % weights of lattice velocities (D2Q9)

%% Initialize Variables

f = zeros(nx,ny,n_eq); % distribution functions
feq = zeros(nx,ny,n_eq); % equilibrium distribution functions
rho = rho0*ones(nx,ny); % density
u = u0(1)*ones(nx,ny); % x-component of velocity
v = u0(2)*ones(nx,ny); % y-component of velocity

%% Main Loop

for t = 1:nt
% Collision
for i = 1:n_eq
cu = cx(i)*u + cy(i)*v;
feq(:,:,i) = rho.*w(i).*(1 + 3*cu + 4.5*cu.^2 - 1.5*(u.^2 + v.^2));
f(:,:,i) = omega*feq(:,:,i) + (1-omega)*f(:,:,i);
end
% Streaming
for i = 1:n_eq
f(:,:,i) = circshift(f(:,:,i),[cx(i),cy(i)]);
end
% Update Density and Velocity
rho = sum(f,3);
for i = 1:n_eq
u = u + cx(i)*f(:,:,i)./rho;
v = v + cy(i)*f(:,:,i)./rho;
end
% Apply Boundary Conditions
u(1,:) = u0(1);
u(nx,:) = u0(1);
v(:,1) = 0;
v(:,ny) = 0;
rho(:,1) = rho0;
rho(:,ny) = rho0;
% Visualization
if mod(t,100) == 0
clf;
quiver(u,v);
xlim([0,nx]);
ylim([0,ny]);
drawnow;
end
end

该代码使用D2Q9模型，表示在二维方格网格上每个节点有九个朝向的速度，依次为水平、竖直和对角线方向。在每个时间步长中，首先对非平衡分布函数进行碰撞，然后将其沿速度方向的朝向移动一步（即流动），最后通过更新密度和速度来计算新的平衡分布函数。还需要应用边界条件和可视化结果。