Lid driven cavity
\[\renewcommand{\DdQq}[2]{{\mathrm D}_{#1}{\mathrm Q}_{#2}}
\renewcommand{\drondt}{\partial_t}
\renewcommand{\drondx}{\partial_x}
\renewcommand{\drondy}{\partial_y}
\renewcommand{\drondtt}{\partial_{tt}}
\renewcommand{\drondxx}{\partial_{xx}}
\renewcommand{\drondyy}{\partial_{yy}}
\renewcommand{\dx}{\Delta x}
\renewcommand{\dt}{\Delta t}
\renewcommand{\grandO}{{\mathcal O}}
\renewcommand{\density}[2]{\,f_{#1}^{#2}}
\renewcommand{\fk}[1]{\density{#1}{\vphantom{\star}}}
\renewcommand{\fks}[1]{\density{#1}{\star}}
\renewcommand{\moment}[2]{\,m_{#1}^{#2}}
\renewcommand{\mk}[1]{\moment{#1}{\vphantom{\star}}}
\renewcommand{\mke}[1]{\moment{#1}{e}}
\renewcommand{\mks}[1]{\moment{#1}{\star}}\]
In this tutorial, we consider the classical \(\DdQq{2}{9}\) and
\(\DdQq{3}{15}\) to simulate a lid driven acvity modeling by the
Navier-Stokes equations. The \(\DdQq{2}{9}\) is used in dimension
\(2\) and the \(\DdQq{3}{15}\) in dimension \(3\).
The \(\DdQq{2}{9}\) for Navier-Stokes
The \(\DdQq{2}{9}\) is defined by:
- a space step \(\dx\) and a time step \(\dt\) related to the
scheme velocity \(\lambda\) by the relation
\(\lambda=\dx/\dt\),
- nine velocities \(\{(0,0), (\pm1,0), (0,\pm1), (\pm1, \pm1)\}\),
identified in pyLBM by the numbers \(0\) to \(8\),
- nine polynomials used to build the moments
\[\{1, \lambda X, \lambda Y, 3E-4, (9E^2-21E+8)/2, 3XE-5X, 3YE-5Y,X^2-Y^2, XY\},\]
where \(E = X^2+Y^2\).
- three conserved moments \(\rho\), \(q_x\), and \(q_y\),
- nine relaxation parameters (three are \(0\) corresponding to
conserved moments):
\(\{0,0,0,s_\mu,s_\mu,s_\eta,s_\eta,s_\eta,s_\eta\}\), where
\(s_\mu\) and \(s_\eta\) are in \((0,2)\),
- equilibrium value of the non conserved moments
\[\begin{split}\begin{aligned}\mke{3} &= -2\rho + 3(q_x^2+q_y^2)/(\rho_0\lambda^2), \\ \mke{4} &= \rho-3(q_x^2+q_y^2)/(\rho_0\lambda^2), \\ \mke{5} &= -q_x/\lambda, \\ \mke{6} &= -q_y/\lambda, \\ \mke{7} &= (q_x^2-q_y^2)/(\rho_0\lambda^2), \\ \mke{8} &= q_xq_y/(\rho_0\lambda^2),\end{aligned}\end{split}\]
where \(\rho_0\) is a given scalar.
This scheme is consistant at second order with the following equations
(taken \(\rho_0=1\))
\[\begin{split}\begin{gathered} \drondt\rho + \drondx q_x + \drondy q_y = 0,\\ \drondt q_x + \drondx (q_x^2+p) + \drondy (q_xq_y) = \mu \drondx (\drondx q_x + \drondy q_y ) + \eta (\drondxx+\drondyy)q_x, \\ \drondt q_y + \drondx (q_xq_y) + \drondy (q_y^2+p) = \mu \drondy (\drondx q_x + \drondy q_y ) + \eta (\drondxx+\drondyy)q_y,\end{gathered}\end{split}\]
with \(p=\rho\lambda^2/3\).
We write the dictionary for a simulation of the Navier-Stokes equations
on \((0,1)^2\).
In order to impose the boundary conditions, we use the bounce-back
conditions to fix \(q_x=q_y=0\) at south, east, and west and
\(q_x=\rho u\), \(q_y=0\) at north. The driven velocity
\(u\) could be \(u=\lambda/10\).
The solution is governed by the Reynolds number
\(Re = \rho_0u / \eta\). We fix the relaxation parameters to have
\(Re=1000\). The relaxation parameters related to the bulk viscosity
\(\mu\) should be large enough to ensure the stability (for instance
\(\mu=10^{-3}\)).
We compute the stationary solution of the problem obtained for large
enough final time. We plot the solution with the function quiver of
matplotlib.
Reynolds number: 1.000e+03
Bulk viscosity : 1.000e-04
Shear viscosity: 2.000e-04
relaxation parameters: [0.0, 0.0, 0.0, 1.8573551263001487, 1.8573551263001487, 1.7337031900138697, 1.7337031900138697, 1.7337031900138697, 1.7337031900138697]
The \(\DdQq{3}{15}\) for Navier-Stokes
The \(\DdQq{3}{15}\) is defined by:
- a space step \(\dx\) and a time step \(\dt\) related to the
scheme velocity \(\lambda\) by the relation
\(\lambda=\dx/\dt\),
- fifteen velocities
\(\{(0,0,0), (\pm1,0,0), (0,\pm1,0), (0,0,\pm1), (\pm1, \pm1,\pm1)\}\),
identified in pyLBM by the numbers
\(\{0,\ldots,6,19,\ldots,26\}\),
- fifteen polynomials used to build the moments
\[\{1, E-2, (15E^2-55E+32)/2, X, X(5E-13)/2, Y, Y(5E-13)/2, Z, Z(5E-13)/2, 3X^2-E, Y^2-Z^2, XY, YZ, ZX, XYZ \},\]
where \(E = X^2+Y^2+Z^2\).
- four conserved moments \(\rho\), \(q_x\), \(q_y\), and
\(q_z\),
- fifteen relaxation parameters (four are \(0\) corresponding to
conserved moments):
\(\{0, s_1, s_2, 0, s_4, 0, s_4, 0, s_4, s_9, s_9, s_{11}, s_{11}, s_{11}, s_{14}\}\),
- equilibrium value of the non conserved moments
\[\begin{split}\begin{aligned} \mke{1} &= -\rho + q_x^2 + q_y^2 + q_z^2,\\ \mke{2} &= -\rho,\\ \mke{4} &= -7q_x/3, \\ \mke{6} &= -7q_y/3, \\ \mke{8} &= -7q_z/3, \\ \mke{9} &= (2q_x^2-(q_y^2+q_z^2))/3, \\ \mke{10} &= q_y^2-q_z^2, \\ \mke{11} &= q_xq_y, \\ \mke{12} &= q_yq_z, \\ \mke{13} &= q_zq_x, \\ \mke{14} &= 0. \end{aligned}\end{split}\]
This scheme is consistant at second order with the Navier-Stokes
equations with the shear viscosity \(\eta\) and the relaxation
parameter \(s_9\) linked by the relation
\[s_9 = \frac{2}{1 + 6\eta /\dx}.\]
We write a dictionary for a simulation of the Navier-Stokes equations on
\((0,1)^3\).
In order to impose the boundary conditions, we use the bounce-back
conditions to fix \(q_x=q_y=q_z=0\) at south, north, east, west, and
bottom and \(q_x=\rho u\), \(q_y=q_z=0\) at top. The driven
velocity \(u\) could be \(u=\lambda/10\).
We compute the stationary solution of the problem obtained for large
enough final time. We plot the solution with the function quiver of
matplotlib.
Reynolds number: 2.000e+03
Shear viscosity: 5.000e-05
