### Tutorials/demos on multi-phase or free-surface Navier Stokes flows?

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Hi all,

Does anyone know of or have access to any fenics tutorials or examples for multiphase and/or free surface navier stokes flows? After searching for quite some time, the material I've found is limited. Specifically, I found some libraries, https://bitbucket.org/trlandet/ocellaris

and an older (incomplete?) tutorial using level set method for interface tracking: http://www.karlin.mff.cuni.cz/~hron/fenics-tutorial/multiphase/doc.html

Apologies for the general question, but I would think there is enough interest in these topics to warrant an "official" fenics tutorial.

Thank you!

Does anyone know of or have access to any fenics tutorials or examples for multiphase and/or free surface navier stokes flows? After searching for quite some time, the material I've found is limited. Specifically, I found some libraries, https://bitbucket.org/trlandet/ocellaris

and an older (incomplete?) tutorial using level set method for interface tracking: http://www.karlin.mff.cuni.cz/~hron/fenics-tutorial/multiphase/doc.html

Apologies for the general question, but I would think there is enough interest in these topics to warrant an "official" fenics tutorial.

Thank you!

Community: FEniCS Project

### 1 Answer

4

Hi,

This is something that I had spent same time working with. I hope this can help you.

This algorithm simulates a bubble problem with superficial tension.

```
from fenics import *
import time
import mshr
import ufl
import os
from math import floor, ceil
get_ipython().magic('matplotlib inline')
# In[2]:
dt = 0.005 # Time Step
k = Constant(dt) # Time Step
t_end = 3.0 # Total simulation time
theta = Constant(0.5) # Interpolation schema
g = Constant((0.0,-0.98)) # Gravity
rho1 = 1000.0 # Surround density
rho2 = 100.0 # Bubble density
nu1 = 10.0 # Surround viscosity
nu2 = 1.0 # Bubble viscosity
sigma = 24.5
# In[3]:
Dx = 100
Dy = 2*Dx
mesh = RectangleMesh(Point(0.0,0.0), Point(1.0,2.0), Dx, Dy,'crossed')
# In[4]:
beta=0.4
epsilon = (beta*((1.0/Dx)**(0.9)))
# In[5]:
center = Point(0.5,0.5)
radius = 0.25
phi = Expression('1.0/(1.0+exp((sqrt((x[0]-A)*(x[0]-A) + (x[1]-B)*(x[1]-B))-r)/(eps)))',degree=2, eps=epsilon, A=center[0], B=center[1],r=radius)
# In[6]:
V = VectorElement("Lagrange", mesh.ufl_cell(), 2) # Velocity vector field
P = FiniteElement("Lagrange", mesh.ufl_cell(), 1) # Pressure field
L = FiniteElement("Lagrange", mesh.ufl_cell(), 2) # Levelset field
N = VectorFunctionSpace(mesh, "CG", 1, dim=2) # Normal vector field
VP = MixedElement([V,P])
W = FunctionSpace(mesh,VP)
# In[7]:
I = Identity(V.cell().geometric_dimension()) # Identity tensor
n = FacetNormal(mesh)
h = CellSize(mesh) # Mesh size
# In[8]:
bcs = list()
bcs.append(DirichletBC(W.sub(0),Constant((0.0,0.0)),"near(x[1],0.0)||near(x[1],2.0)"))
bcs.append(DirichletBC(W.sub(0).sub(0),Constant(0.0),"near(x[0],0.0)||near(x[0],1.0)"))
# In[9]:
w = Function(W); w0 = Function(W)
v,p = split(w); v0,p0 = split(w0)
# In[10]:
SL = FunctionSpace(mesh,L);
l = Function(SL); l0 = Function(SL)
l.assign(interpolate(phi,SL)); l0.assign(interpolate(phi,SL))
# In[11]:
def delta(l):
grad_phi = project(grad(l),N)
return(sqrt(dot(grad_phi,grad_phi)))
def rho(l):
return(rho1+(rho2-rho1)*l)
def nu(l):
return(nu1+(nu2-nu1)*l)
# In[12]:
def NS(v,p,l,v_):
grad_phi = project(grad(l),N)
nls = grad_phi/sqrt(dot(grad_phi,grad_phi))
Ts = delta(l)*sigma*(I-outer(nls,nls))
T = -p*I + nu(l)*(grad(v)+grad(v).T)
return(inner(T,grad(v_))*dx + rho(l)*inner(grad(v)*v,v_)*dx - rho(l)*inner(g,v_)*dx + inner(Ts,grad(v_))*dx)
#def NS(v,p,l,v_):
# grad_phi = project(grad(l),N)
# mgrad=sqrt(dot(grad_phi,grad_phi))
# nls = grad_phi/mgrad
# ft = sigma*div(nls)*grad_phi
# Ts = delta(l)*sigma*(I-outer(nls,nls))
# T = -p*I + nu(l)*(grad(v)+grad(v).T)
# return(inner(T,grad(v_))*dx + rho(l)*inner(grad(v)*v,v_)*dx - rho(l)*inner(g,v_)*dx + inner(ft,v_)*dx)
# In[13]:
def navier_stokes():
v,p = split(w); v0,p0 = split(w0)
v_,p_ = TestFunctions(W)
F = inner((rho(l)*v-rho(l0)*v0)/k,v_)*dx + theta*NS(v,p,l,v_) + (1.0-theta)*NS(v0,p,l0,v_) + div(v)*p_*dx
begin("navier_stokes")
J = derivative(F,w)
problem=NonlinearVariationalProblem(F,w,bcs,J)
solver=NonlinearVariationalSolver(problem)
solver.solve()
end()
return(w)
# In[14]:
alpha=Constant(1.0)
def IP(l,l_):
h_avg = (h('+') + h('-'))/2.0
r = alpha('+')*h_avg*h_avg*inner(jump(grad(l),n), jump(grad(l_),n))*dS
return (r)
def LS(l,v,l_):
return(inner(v,grad(l))*l_*dx)
# In[15]:
def level_set():
v,p = split(w); v0,p0 = split(w0)
l_ = TestFunction(SL)
F = inner((l-l0)/k,l_)*dx + theta*LS(l,v,l_) + (1.0-theta)*LS(l0,v0,l_) + IP(l,l_)
bc = []
begin("level_set")
J = derivative(F,l)
problem=NonlinearVariationalProblem(F,l,bc,J)
solver=NonlinearVariationalSolver(problem)
solver.solve()
end()
return(l)
# In[16]:
def reinit(l,epsilon,beta,Dx,mesh):
# time-step
dtau = Constant(1.0/(beta*((1.0/Dx)**(1.1))))
# space definition
V = VectorFunctionSpace(mesh, "CG", 1, dim=2)
FE = FunctionSpace(mesh, "CG", 2)
# functions setup
phi = Function(FE); phi0 = Function(FE)
w = TestFunction(FE)
# intial value
phi0.assign(interpolate(l,FE))
# Unit normal vector (does not change during this process)
grad_n = project(grad(l),V)
n = grad_n/(sqrt(dot(grad_n,grad_n)))
# FEM linearization
F = dtau*(phi-phi0)*w*dx - 0.5*(phi+phi0)*dot(grad(w),n)*dx + phi*phi0*dot(grad(w),n)*dx + epsilon*0.5*dot(grad(phi+phi0),n)*dot(grad(w),n)*dx
# Newton-Raphson parameters
bc = []
E = 1e10; E_old = 1e10
cont = 0; num_steps = 10
for n in range(num_steps):
begin("Reinitialization")
solve(F == 0, phi, bc)
end()
error = (((phi - phi0)*dtau)**2)*dx
E = sqrt(abs(assemble(error)))
fail = 0
if (E_old < E ):
fail = 1
print('fail',"at:", cont)
break
phi0.assign(phi)
cont +=1
E_old = E
#print("Error:", E, "nincre", cont)
return phi
# In[17]:
vfile = File("P1_IP_S1_1/velocity.pvd")
pfile = File("P1_IP_S1_1/pressure.pvd")
lfile = File("P1_IP_S1_1/level.pvd")
# In[18]:
Vc = VectorFunctionSpace(mesh,"CG",2)
R = VectorFunctionSpace(mesh,"R",0,dim=2)
position = Function(Vc)
position.assign(Expression(["x[0]","x[1]"], element=Vc.ufl_element()))
c = TestFunction(R)
# In[19]:
out_dt = 0.1; count = 0
t = dt
if __name__ == "__main__":
print("P1_IP_S1_1")
print("IP", "alpha",alpha, "inR",10, "beta", beta)
while t < t_end:
for problem in [navier_stokes, level_set]:
if (problem == navier_stokes):
w = problem()
else:
l1 = problem()
l2 = reinit(l1,epsilon,beta,Dx,mesh)
l.assign(interpolate(l2,SL))
# Extract solutions
w0.assign(w); l0.assign(l)
v,p = w.split()
# Mass
V=assemble(conditional(gt(l,0.5),1.0,0.0)*dx)
# Center of mass
volume = assemble(conditional(ge(l,0.5),Constant(1.0),0.0)*dx)
centroid = assemble(conditional(ge(l,0.5),dot(c,position),0.0)*dx)
centroid /= volume
xc = centroid[0][0]; yc = centroid[1][0]
# Velocity
u_c = assemble(conditional(ge(l,0.5),dot(c,v),0.0)*dx)
u_c /= volume
vxc = u_c[0][0]; vyc = u_c[1][0]
#myfile3.write('%e %e' '\n' % (vyc, t))
print("%e %e %e %e" %(V, yc, vyc, t))
# Save solution
if (t >= float(count)*out_dt):
count+=1
v.rename("velocity", "velocity")
p.rename("pressure", "pressure")
l.rename("level", "level")
vfile << v
pfile << p
lfile << l
t += dt
v.rename("velocity", "velocity")
p.rename("pressure", "pressure")
l.rename("level", "level_set")
vfile << v
pfile << p
lfile << l
```

Hi, thanks so much for sharing this code. I tried running it, and it seems that the NS eqns blow up as the bubble starts rising, but it does give a good starting point. Also, could you by any chance provide some references for how you're solving the level set equation/handling the reinitialization? Thanks again!

written
10 days ago by
Alexander Niewiarowski

File attached: Paulo___Levelset_Implementation.pdf (263.44 KB)

File attached: Paulo___Bubble_Problem.pdf (902.23 KB)

Hi, this problem is weird, since it should be working just fine. I am going to take a look on it and see what the problems is. About the references, attached is a report that you can give some time to see. Additional references are provided at the end of the file.

written
10 days ago by
Cassia

Great, thanks so much, now everyone here can benefit. I have been looking for exactly something like this for quite some time: LSM explained in a straightforward and accessible way. Looking forward to seeing the published paper!

written
10 days ago by
Alexander Niewiarowski

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