Von Karman vortex street¶

\[\renewcommand{\DdQq}[2]{{\mathrm D}_{#1}{\mathrm Q}_{#2}} \renewcommand{\drondt}{\partial_t} \renewcommand{\drondx}{\partial_x} \renewcommand{\drondy}{\partial_y} \renewcommand{\drondtt}{\partial_{tt}} \renewcommand{\drondxx}{\partial_{xx}} \renewcommand{\drondyy}{\partial_{yy}} \renewcommand{\dx}{\Delta x} \renewcommand{\dt}{\Delta t} \renewcommand{\grandO}{{\mathcal O}} \renewcommand{\density}[2]{\,f_{#1}^{#2}} \renewcommand{\fk}[1]{\density{#1}{\vphantom{\star}}} \renewcommand{\fks}[1]{\density{#1}{\star}} \renewcommand{\moment}[2]{\,m_{#1}^{#2}} \renewcommand{\mk}[1]{\moment{#1}{\vphantom{\star}}} \renewcommand{\mke}[1]{\moment{#1}{e}} \renewcommand{\mks}[1]{\moment{#1}{\star}}\]

In this tutorial, we consider the classical \(\DdQq{2}{9}\) to simulate the Von Karman vortex street modeling by the Navier-Stokes equations.

In fluid dynamics, a Von Karman vortex street is a repeating pattern of swirling vortices caused by the unsteady separation of flow of a fluid around blunt bodies. It is named after the engineer and fluid dynamicist Theodore von Karman. For the simulation, we propose to simulate the Navier-Stokes equation into a rectangular domain with a circular hole of diameter \(d\).

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 a 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 \(q_x=q_y=\rho v_0\) at south, east, and north where the velocity \(v_0\) could be \(v_0=\lambda/20\). At west, we impose the simple output condition of Neumann by repeating the second to last cells into the last cells.

The solution is governed by the Reynolds number \(Re = \rho_0v_0d / \eta\), where \(d\) is the diameter of the circle. Fix the relaxation parameters to have \(Re=500\). 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 vorticity of the solution with the function imshow of matplotlib.

[1]:

%matplotlib inline

[2]:

import numpy as np
import sympy as sp
import pylbm

X, Y, LA = sp.symbols('X, Y, LA')
rho, qx, qy = sp.symbols('rho, qx, qy')

def bc_in(f, m, x, y):
    m[qx] = rhoo * v0

def vorticity(sol):
    ux = sol.m[qx] / sol.m[rho]
    uy = sol.m[qy] / sol.m[rho]
    V = np.abs(uy[2:,1:-1] - uy[0:-2,1:-1] - ux[1:-1,2:] + ux[1:-1,0:-2])/(2*sol.domain.dx)
    return -V

# parameters
rayon = 0.05
Re = 500
dx = 1./64   # spatial step
la = 1.      # velocity of the scheme
Tf = 75      # final time of the simulation
v0 = la/20   # maximal velocity obtained in the middle of the channel
rhoo = 1.    # mean value of the density
mu = 1.e-3   # bulk viscosity
eta = rhoo*v0*2*rayon/Re  # shear viscosity
# initialization
xmin, xmax, ymin, ymax = 0., 3., 0., 1.
dummy = 3.0/(la*rhoo*dx)
s_mu = 1.0/(0.5+mu*dummy)
s_eta = 1.0/(0.5+eta*dummy)
s_q = s_eta
s_es = s_mu
s  = [0.,0.,0.,s_mu,s_es,s_q,s_q,s_eta,s_eta]
dummy = 1./(LA**2*rhoo)
qx2 = dummy*qx**2
qy2 = dummy*qy**2
q2  = qx2+qy2
qxy = dummy*qx*qy

print("Reynolds number: {0:10.3e}".format(Re))
print("Bulk viscosity : {0:10.3e}".format(mu))
print("Shear viscosity: {0:10.3e}".format(eta))
print("relaxation parameters: {0}".format(s))

dico = {
    'box': {'x': [xmin, xmax],
            'y': [ymin, ymax],
            'label': [0, 2, 0, 0]
           },
    'elements': [pylbm.Circle([.3, 0.5*(ymin+ymax)+dx], rayon, label=1)],
    'space_step': dx,
    'scheme_velocity': la,
    'parameters': {LA: la},
    'schemes': [
        {
            'velocities': list(range(9)),
            'conserved_moments': [rho, qx, qy],
            'polynomials': [
                1, LA*X, LA*Y,
                3*(X**2+Y**2)-4,
                (9*(X**2+Y**2)**2-21*(X**2+Y**2)+8)/2,
                3*X*(X**2+Y**2)-5*X, 3*Y*(X**2+Y**2)-5*Y,
                X**2-Y**2, X*Y
            ],
            'relaxation_parameters': s,
            'equilibrium': [
                rho, qx, qy,
                -2*rho + 3*q2,
                rho-3*q2,
                -qx/LA, -qy/LA,
                qx2-qy2, qxy
            ],
        },
    ],
    'init': {rho:rhoo,
             qx:0.,
             qy:0.
    },
    'boundary_conditions': {
        0: {'method': {0: pylbm.bc.BouzidiBounceBack}, 'value': bc_in},
        1: {'method': {0: pylbm.bc.BouzidiBounceBack}},
        2: {'method': {0: pylbm.bc.NeumannX}},
    },
    'generator': 'cython',
}

sol = pylbm.Simulation(dico)
while sol.t < Tf:
    sol.one_time_step()

viewer = pylbm.viewer.matplotlib_viewer
fig = viewer.Fig()
ax = fig[0]
im = ax.image(vorticity(sol).transpose(), clim = [-3., 0])
ax.ellipse([.3/dx, 0.5*(ymin+ymax)/dx], [rayon/dx,rayon/dx], 'r')
ax.title = 'Von Karman vortex street at t = {0:f}'.format(sol.t)
fig.show()

Reynolds number:  5.000e+02
Bulk viscosity :  1.000e-03
Shear viscosity:  1.000e-05
relaxation parameters: [0.0, 0.0, 0.0, 1.4450867052023122, 1.4450867052023122, 1.9923493783869939, 1.9923493783869939, 1.9923493783869939, 1.9923493783869939]

../_images/notebooks_07_Von_Karman_vortex_street_2_1.png

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