Poiseuille flow is the flow through a pipe or (in our case) a slit under a homogeneous force density, e.g. gravity. In the limit of small Reynolds numbers, the flow can be described with the Stokes equation. We assume the slit being infinitely extended in $y$ and $z$ direction and a force density $f_y$ on the fluid in $y$ direction. No slip-boundary conditions (i.e. $\vec{u}=0$) are located at $x = \pm h/2$. Assuming invariance in $y$ and $z$ direction and a steady state, the Stokes equation is simplified to:
\begin{equation} \mu \partial_x^2 u_y = f_y \end{equation}where $f_y$ denotes the force density and $\mu$ the dynamic viscosity. This can be integrated twice and the integration constants are chosen so that $u_y=0$ at $x = \pm h/2$ to obtain the solution to the planar Poiseuille flow [8]:
\begin{equation} u_y(x) = \frac{f_y}{2\mu} \left(h^2/4-x^2\right) \end{equation}We will simulate a planar Poiseuille flow using a square box, two walls with normal vectors $\left(\pm 1, 0, 0 \right)$, and an external force density applied to every node.
Use the data to fit a parabolic function. Can you confirm the analytic solution?
import logging
import sys
import numpy as np
import espressomd
import espressomd.lb
import espressomd.lbboundaries
import espressomd.shapes
logging.basicConfig(level=logging.INFO, stream=sys.stdout)
espressomd.assert_features(['LB_BOUNDARIES_GPU'])
# System constants
BOX_L = 16.0
AGRID = 0.5
VISCOSITY = 2.0
FORCE_DENSITY = [0.0, 0.001, 0.0]
DENSITY = 1.5
TIME_STEP = 0.01
system = espressomd.System(box_l=[BOX_L]*3)
system.time_step = TIME_STEP
system.cell_system.skin = 0.4
lbf = espressomd.lb.LBFluidGPU(agrid=AGRID, dens=DENSITY, visc=VISCOSITY, tau=TIME_STEP,
ext_force_density=FORCE_DENSITY)
system.actors.add(lbf)
logging.info("Setup LB boundaries.")
WALL_OFFSET = 2.0 * AGRID
top_wall = espressomd.lbboundaries.LBBoundary(shape=espressomd.shapes.Wall(normal=[1, 0, 0], dist=WALL_OFFSET))
bottom_wall = espressomd.lbboundaries.LBBoundary(shape=espressomd.shapes.Wall(normal=[-1, 0, 0], dist=-(BOX_L - WALL_OFFSET)))
system.lbboundaries.add(top_wall)
system.lbboundaries.add(bottom_wall)
logging.info("Iterate until the flow profile converges (5000 LB updates).")
system.integrator.run(5000)
logging.info("Extract fluid velocity along the x-axis")
fluid_velocities = np.zeros(lbf.shape[0])
for x in range(lbf.shape[0]):
# Average over the nodes in y direction
v_tmp = np.zeros(lbf.shape[1])
for y in range(lbf.shape[1]):
v_tmp[y] = lbf[x, y, 0].velocity[1]
fluid_velocities[x] = np.average(v_tmp)
INFO:root:Setup LB boundaries. INFO:root:Iterate until the flow profile converges (5000 LB updates). INFO:root:Extract fluid velocity along the x-axis
The solution is available at /doc/tutorials/04-lattice_boltzmann/scripts/04-lattice_boltzmann_part4_solution.py
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