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Non-uniqueness of Falkner-Skan solutions with adverse pressure gradient
import scipy as sp
from scipy.integrate import ode
from scipy.optimize import fsolve
from matplotlib import rc
rc('text', usetex=True)
tick_size = 'large'
label_size = 'xx-large'
rc('xtick', labelsize=8)
rc('ytick', labelsize=8)
rc('legend', fontsize=12)
rc('axes', labelsize=12)
fig_width_pt = 469.75502
inches_per_pt = 1.0/72.27
golden_mean = (sp.sqrt(5.0)-1.0)/2.0
fig_width = fig_width_pt *inches_per_pt
fig_height = fig_width * golden_mean
fig_size = [fig_width, fig_height]
rc('figure', figsize=fig_size)
from matplotlib import pylab as plt
from fs import *
def int_fs(beta_0, beta, alpha, eta_inf):
args = sp.array([beta_0, beta, eta_inf])
y0 = sp.array([0.0, 0.0, alpha])
r = ode(fs.rates, fs.jac).set_integrator('zvode', rtol=1e-14, atol=1e-14, method='adams', with_jacobian=True, order=12)
t0 = 0.0
t1 = 1.0
nt = 200
dt = t1 / float(nt)
r.set_initial_value(y0, t0).set_f_params(args).set_jac_params(args)
for i in xrange(nt):
r.integrate(r.t+dt)
return(r)
def int_tanh_fs(beta_0, beta, alpha, eta_inf):
args = sp.array([beta_0, beta, eta_inf])
y0 = sp.array([0.0, 0.0, alpha])
r = ode(fs.rates, fs.jac).set_integrator('zvode', rtol=1e-14, atol=1e-14, method='adams', with_jacobian=True, order=12)
t0 = 0.0
t1 = 1.0
# instead of using a uniform point distribution, use a tanh
# mapping, which clusters the points near the wall where the
# variation is largest
nt = 64
t = sp.arctanh(sp.linspace(t0,t1,nt,endpoint=False))
t = t/t.max()
dt = sp.diff(t)
r.set_initial_value(y0, t0).set_f_params(args).set_jac_params(args)
for i in xrange(nt-1):
r.integrate(r.t+dt[i])
return(r)
def objective_func(x, beta_0, beta):
alpha = x[0]
eta_inf = x[1]
r = int_fs(beta_0, beta, alpha, eta_inf)
return(sp.array([(r.y[1]-1.0).real, (r.y[2]).real]))
def objective_func2(x, beta_0, beta):
alpha = x[0]
eta_inf = x[1]
r = int_tanh_fs(beta_0, beta, alpha, eta_inf)
return(sp.array([(r.y[1]-1.0).real, (r.y[2]).real]))
def extrap(y):
return(2*y[1]-y[0])
def extrap2(y):
return(-(4*y[1]-y[0]-5*y[2])/2.0)
eta_inf = 5.0
alpha = 1e-2
beta_0 = 1.0
x = sp.array([alpha, eta_inf])
nbeta = 71
beta = sp.linspace(-0.199010445, 0.199010445, nbeta)
foo = sp.zeros((nbeta,2))
for i in xrange(nbeta):
args = (beta_0, beta[i])
foo[i] = fsolve(objective_func2, x, args=args, xtol=1e-12)
if(i==0):
x[0] = alpha # foo[i,0]
elif(i==1):
x[0] = extrap(foo[i-1:i+1,0])
else:
x[0] = extrap2(foo[i-2:i+1,0])
x[1] = eta_inf
goo = sp.zeros((nbeta,2))
alpha2 = -3e-1
eta_inf2 = 6.0
for i in xrange(nbeta):
args = (beta_0, beta[i])
goo[i] = fsolve(objective_func2, x, args=args, xtol=1e-12)
if(i==0):
x[0] = alpha2 # foo[i,0]
x[1] = eta_inf2
elif(i==1):
x[0] = extrap(goo[i-1:i+1,0])
x[1] = eta_inf2
else:
x[0] = extrap2(goo[i-2:i+1,0])
x[1] = goo[i,1]
# x[1] = eta_inf2
sp.savetxt("foo.txt", foo)
sp.savetxt("beta.txt", beta)
r = ode(fs.rates, fs.jac).set_integrator('zvode', rtol=1e-14, atol=1e-14, method='adams', with_jacobian=True, order=12)
nt = 64
u1 = sp.zeros(nt)
u1[0] = 0.0
y0 = sp.array([0.0,0.0,foo[0,0]])
t0 = 0.0
t1 = 1.0
t = sp.arctanh(sp.linspace(t0,t1,nt,endpoint=False))
dt = sp.diff(t)
args = sp.array([beta_0, beta[0], foo[0,1]])
r.set_initial_value(y0, 0.0).set_f_params(args).set_jac_params(args)
for i in xrange(nt-1):
r.integrate(r.t+dt[i])
u1[i+1] = r.y[1].real
u2 = sp.zeros(nt)
u2[0] = 0.0
y0 = sp.array([0.0,0.0,foo[int(nbeta/2),0]])
args = sp.array([beta_0, beta[int(nbeta/2)], foo[int(nbeta/2),1]])
r.set_initial_value(y0, 0.0).set_f_params(args).set_jac_params(args)
for i in xrange(nt-1):
r.integrate(r.t+dt[i])
u2[i+1] = r.y[1].real
u3 = sp.zeros(nt)
u3[0] = 0.0
y0 = sp.array([0.0,0.0,foo[nbeta-1,0]])
args = sp.array([beta_0, beta[nbeta-1], foo[nbeta-1,1]])
r.set_initial_value(y0, 0.0).set_f_params(args).set_jac_params(args)
for i in xrange(nt-1):
r.integrate(r.t+dt[i])
u3[i+1] = r.y[1].real
ni = 14
u4 = sp.zeros(nt)
u4[0] = 0.0
y0 = sp.array([0.0,0.0,goo[ni,0]])
args = sp.array([beta_0, beta[ni], goo[ni,1]])
r.set_initial_value(y0, 0.0).set_f_params(args).set_jac_params(args)
for i in xrange(nt-1):
r.integrate(r.t+dt[i])
u4[i+1] = r.y[1].real
u5 = sp.zeros(nt)
u5[0] = 0.0
y0 = sp.array([0.0,0.0,foo[ni,0]])
args = sp.array([beta_0, beta[ni], foo[ni,1]])
r.set_initial_value(y0, 0.0).set_f_params(args).set_jac_params(args)
for i in xrange(nt-1):
r.integrate(r.t+dt[i])
u5[i+1] = r.y[1].real
plt.figure()
plt.plot(beta, foo[:,1])
plt.xlabel(r'$\beta$')
plt.ylabel(r'$\eta_{\infty}$')
plt.savefig("eta_inf.png")
plt.figure()
plt.plot(beta, foo[:,0])
plt.plot(beta, goo[:,0])
plt.xlabel(r'$\beta$')
plt.ylabel(r'$\alpha$')
plt.savefig("alpha.png")
plt.figure()
plt.plot(u1, foo[0,1]*t, label=r'$\beta=%g$'%beta[0])
plt.plot(u2, foo[int(nbeta/2),1]*t, label=r'$\beta=%g$'%beta[int(nbeta/2)])
plt.plot(u3, foo[nbeta-1,1]*t, label=r'$\beta=%g$'%beta[nbeta-1])
plt.plot(u4, goo[10,1]*t, label=r'$\beta=%g$'%beta[10])
plt.plot(u5, foo[10,1]*t, label=r'$\beta=%g$'%beta[10])
plt.legend(loc=0)
plt.xlabel(r"$f'$")
plt.ylabel(r'$\eta$')
xmin,xmax,ymin,ymax = plt.axis()
plt.axis([xmin,xmax,0.0,8.0])
plt.savefig("fs_profiles.png")
plt.show()
@jstults

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jstults commented Feb 5, 2012

non-unique alpha vs beta

Laminar separation (negative wall shear) with adverse pressure gradient:
non-unique velocity profiles

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