def hydrodynamics_only(boundary_conditions_fn,
mesh2d,
bathymetry_2d,
uv_init,
elev_init,
average_size,
dt,
t_end,
wetting_and_drying=False,
wetting_alpha=0.1,
friction='nikuradse',
friction_coef=0,
fluc_bcs=False,
viscosity=10**(-6),
sponge_viscosity=th.Constant(1.0)):
"""
Sets up a simulation with only hydrodynamics until a quasi-steady state when it can be used as an initial
condition for the full morphological model. We update the bed friction at each time step.
The actual run of the model are done outside the function
Inputs:
boundary_consditions_fn - function defining boundary conditions for problem
mesh2d - define mesh working on
bathymetry2d - define bathymetry of problem
uv_init - initial velocity of problem
elev_init - initial elevation of problem
average_size - average sediment size
dt - timestep
t_end - end time
wetting_and_drying - wetting and drying switch
wetting_alpha - wetting and drying parameter
friction - choice of friction formulation - nikuradse and manning
friction_coef - value of friction coefficient used in manning
viscosity - viscosity of hydrodynamics. The default value should be 10**(-6)
Outputs:
solver_obj - solver which we need to run to solve the problem
update_forcings_hydrodynamics - function defining the updates to the model performed at each timestep
outputdir - directory of outputs
"""
def update_forcings_hydrodynamics(t_new):
# update boundary conditions if have fluctuating conditions
if fluc_bcs:
in_fn, out_fn = boundary_conditions_fn(bathymetry_2d,
t_new=t_new,
state='update')
for j in range(len(in_fn)):
exec('constant_in' + str(j) + '.assign(' + str(in_fn[j]) + ')')
for k in range(len(out_fn)):
exec('constant_out' + str(k) + '.assign(' + str(out_fn[k]) +
')')
# update bed friction
if friction == 'nikuradse':
uv1, elev1 = solver_obj.fields.solution_2d.split()
if wetting_and_drying:
wd_bath_displacement = solver_obj.depth.wd_bathymetry_displacement
depth.project(elev1 + wd_bath_displacement(elev1) +
bathymetry_2d)
else:
depth.interpolate(elev1 + bathymetry_2d)
# calculate skin friction coefficient
cfactor.interpolate(2 * (0.4**2) / ((th.ln(11.036 * depth /
(ksp)))**2))
# choose directory to output results
ts = time.time()
st = datetime.datetime.fromtimestamp(ts).strftime('%Y-%m-%d %H:%M:%S')
outputdir = 'outputs' + st
# export interval in seconds
if t_end < 40:
t_export = 0.1
else:
t_export = np.round(t_end / 40, 0)
th.print_output('Exporting to ' + outputdir)
# define function spaces
V = th.FunctionSpace(mesh2d, 'CG', 1)
P1_2d = th.FunctionSpace(mesh2d, 'DG', 1)
# define parameters
ksp = th.Constant(3 * average_size)
# define depth
if wetting_and_drying:
H = th.Function(V).project(elev_init + bathymetry_2d)
depth = th.Function(V).project(
H + (0.5 * (th.sqrt(H**2 + wetting_alpha**2) - H)))
else:
depth = th.Function(V).project(elev_init + bathymetry_2d)
# define bed friction
cfactor = th.Function(P1_2d).interpolate(
2 * (0.4**2) / ((th.ln(11.036 * depth / (ksp)))**2))
# set up solver
solver_obj = th.solver2d.FlowSolver2d(mesh2d, bathymetry_2d)
options = solver_obj.options
options.simulation_export_time = t_export
options.simulation_end_time = t_end
options.output_directory = outputdir
options.check_volume_conservation_2d = True
options.fields_to_export = ['uv_2d', 'elev_2d']
options.solve_tracer = False
options.use_lax_friedrichs_tracer = False
if friction == 'nikuradse':
options.quadratic_drag_coefficient = cfactor
elif friction == 'manning':
if friction_coef == 0:
friction_coef = 0.02
options.manning_drag_coefficient = th.Constant(friction_coef)
else:
print('Undefined friction')
options.horizontal_viscosity = th.Function(P1_2d).interpolate(
sponge_viscosity * th.Constant(viscosity))
# crank-nicholson used to integrate in time system of ODEs resulting from application of galerkin FEM
options.timestepper_type = 'CrankNicolson'
options.timestepper_options.implicitness_theta = 1.0
options.use_wetting_and_drying = wetting_and_drying
options.wetting_and_drying_alpha = th.Function(V).interpolate(
th.Constant(wetting_alpha))
options.norm_smoother = th.Constant(1.0)
if not hasattr(options.timestepper_options, 'use_automatic_timestep'):
options.timestep = dt
# set boundary conditions
swe_bnd, left_bnd_id, right_bnd_id, in_constant, out_constant, left_string, right_string = boundary_conditions_fn(
bathymetry_2d, flag='hydro')
for j in range(len(in_constant)):
exec(
'constant_in' + str(j) + ' = th.Constant(' + str(in_constant[j]) +
')', globals())
str1 = '{'
if len(left_string) > 0:
for i in range(len(left_string)):
str1 += "'" + str(left_string[i]) + "': constant_in" + str(i) + ","
str1 = str1[0:len(str1) - 1] + "}"
exec('swe_bnd[left_bnd_id] = ' + str1)
for k in range(len(out_constant)):
exec(
'constant_out' + str(k) + '= th.Constant(' + str(out_constant[k]) +
')', globals())
str2 = '{'
if len(right_string) > 0:
for i in range(len(right_string)):
str2 += "'" + str(
right_string[i]) + "': constant_out" + str(i) + ","
str2 = str2[0:len(str2) - 1] + "}"
exec('swe_bnd[right_bnd_id] = ' + str2)
solver_obj.bnd_functions['shallow_water'] = swe_bnd
solver_obj.assign_initial_conditions(uv=uv_init, elev=elev_init)
return solver_obj, update_forcings_hydrodynamics, outputdir
def morphological(boundary_conditions_fn,
morfac,
morfac_transport,
suspendedload,
convectivevel,
bedload,
angle_correction,
slope_eff,
seccurrent,
mesh2d,
bathymetry_2d,
input_dir,
viscosity_hydro,
ks,
average_size,
dt,
final_time,
beta_fn,
surbeta2_fn,
alpha_secc_fn,
friction='nikuradse',
friction_coef=0,
d90=0,
fluc_bcs=False,
bed_form='meyer',
sus_form='vanrijn',
diffusivity=0.15):
"""
Set up a full morphological model simulation using as an initial condition the results of a hydrodynamic only model.
Inputs:
boundary_consditions_fn - function defining boundary conditions for problem
morfac - morphological scale factor
morfac_transport - switch to turn on morphological component
suspendedload - switch to turn on suspended sediment transport
convectivevel - switch on convective velocity correction factor in sediment concentration equation
bedload - switch to turn on bedload transport
angle_correction - switch on slope effect angle correction
slope_eff - switch on slope effect magnitude correction
seccurrent - switch on secondary current for helical flow effect
mesh2d - define mesh working on
bathymetry2d - define bathymetry of problem
input_dir - folder containing results of hydrodynamics model which are used as initial conditions here
viscosity_hydro - viscosity value in hydrodynamic equations
ks - bottom friction coefficient for quadratic drag coefficient
average_size - average sediment size
dt - timestep
final_time - end time
beta_fn - magnitude slope effect parameter
surbeta2_fn - angle correction slope effect parameter
alpha_secc_fn - secondary current parameter
friction - choice of friction formulation - nikuradse and manning
friction_coef - value of friction coefficient used in manning
d90 - sediment size which 90% of the sediment are below
fluc_bcs - if true this allows the user to impose boundary conditions which vary with time
bed_form - choice of bedload formula between 'meyer' (meyer-peter-muller) and 'soulsby' (soulsby-van-rijn)
sus_form - choice of suspended load formula between 'vanrijn' (van Rijn 1984) and 'soulsby' (soulsby-van-rijn)
diffusivity - diffusivity value in suspended sediment concentration equation. The default value (found by callibration) should be 0.15
Outputs:
solver_obj - solver which we need to run to solve the problem
update_forcings_hydrodynamics - function defining the updates to the model performed at each timestep
diff_bathy - bedlevel evolution
diff_bathy_file - bedlevel evolution file
"""
t_list = []
def update_forcings_tracer(t_new):
# update bathymetry
old_bathymetry_2d.assign(bathymetry_2d)
# extract new elevation and velocity and project onto CG space
uv1, elev1 = solver_obj.fields.solution_2d.split()
uv_cg.project(uv1)
elev_cg.project(elev1)
depth.project(elev_cg + old_bathymetry_2d)
horizontal_velocity.interpolate(uv_cg[0])
vertical_velocity.interpolate(uv_cg[1])
# update boundary conditions if have fluctuating conditions
if fluc_bcs:
in_fn, out_fn = boundary_conditions_fn(orig_bathymetry,
flag='morpho',
morfac=morfac,
t_new=t_new,
state='update')
for j in range(len(in_fn)):
exec('constant_in' + str(j) + '.assign(' + str(in_fn[j]) + ')')
for k in range(len(out_fn)):
exec('constant_out' + str(k) + '.assign(' + str(out_fn[k]) +
')')
# update bedfriction
hc.interpolate(th.conditional(depth > 0.001, depth, 0.001))
aux.assign(
th.conditional(11.036 * hc / ks > 1.001, 11.036 * hc / ks, 1.001))
qfc.assign(2 / (th.ln(aux) / 0.4)**2)
# calculate skin friction coefficient
hclip.interpolate(th.conditional(ksp > depth, ksp, depth))
cfactor.interpolate(
th.conditional(depth > ksp,
2 * ((2.5 * th.ln(11.036 * hclip / ksp))**(-2)),
th.Constant(0.0)))
if morfac_transport:
# if include tracer solver then update_forcings is run twice but only want to update bathymetry once
t_list.append(t_new)
double_factor = False
if suspendedload:
if len(t_list) > 1:
if t_list[len(t_list) - 1] == t_list[len(t_list) - 2]:
double_factor = True
else:
# if have no tracer then update_forcings is only run once so update bathymetry at each step
double_factor = True
if double_factor:
z_n.assign(old_bathymetry_2d)
# mu - ratio between skin friction and normal friction
mu.assign(th.conditional(qfc > 0, cfactor / qfc, 0))
# bed shear stress
unorm.interpolate((horizontal_velocity**2) +
(vertical_velocity**2))
TOB.interpolate(1000 * 0.5 * qfc * unorm)
# calculate gradient of bed (noting bathymetry is -bed)
dzdx.interpolate(old_bathymetry_2d.dx(0))
dzdy.interpolate(old_bathymetry_2d.dx(1))
# initialise exner equation
f = 0
if suspendedload:
# source term
# deposition flux - calculating coefficient to account for stronger conc at bed
B.interpolate(th.conditional(a > depth, a / a, a / depth))
ustar.interpolate(th.sqrt(0.5 * qfc * unorm))
exp1.assign(
th.conditional(
(th.conditional(
(settling_velocity / (0.4 * ustar)) - 1 > 0,
(settling_velocity / (0.4 * ustar)) - 1,
-(settling_velocity /
(0.4 * ustar)) + 1)) > 10**(-4),
th.conditional(
(settling_velocity / (0.4 * ustar)) - 1 > 3, 3,
(settling_velocity / (0.4 * ustar)) - 1), 0))
coefftest.assign(
th.conditional((th.conditional(
(settling_velocity / (0.4 * ustar)) - 1 > 0,
(settling_velocity / (0.4 * ustar)) - 1,
-(settling_velocity /
(0.4 * ustar)) + 1)) > 10**(-4),
B * (1 - B**exp1) / exp1,
-B * th.ln(B)))
coeff.assign(
th.conditional(coefftest > 0, 1 / coefftest, 0))
if sus_form == 'vanrijn':
# erosion flux - above critical velocity bed is eroded
s0.assign(
(th.conditional(1000 * 0.5 * qfc * unorm * mu > 0,
1000 * 0.5 * qfc * unorm * mu, 0) -
taucr) / taucr)
ceq.assign(0.015 * (average_size / a) *
((th.conditional(s0 < 0, 0, s0))**(1.5)) /
(dstar**0.3))
elif sus_form == 'soulsby':
ucr.interpolate(0.19 * (average_size**0.1) *
(th.ln(4 * depth / d90) / th.ln(10)))
s0.assign(
th.conditional((th.sqrt(unorm) - ucr)**2.4 > 0,
(th.sqrt(unorm) - ucr)**2.4, 0))
ceq.interpolate(ass * s0 / depth)
else:
print(
'Unrecognised suspended sediment transport formula. Please choose "vanrijn" or "soulsby"'
)
# calculate depth-averaged source term for sediment concentration equation
source.interpolate(-(settling_velocity * coeff *
solver_obj.fields.tracer_2d / depth) +
(settling_velocity * ceq / depth))
# update sediment rate to ensure equilibrium at inflow
sediment_rate.assign(ceq.at([0, 0]) / coeff.at([0, 0]))
if convectivevel:
# correction factor to advection velocity in sediment concentration equation
Bconv.interpolate(
th.conditional(depth > 1.1 * ksp, ksp / depth,
ksp / (1.1 * ksp)))
Aconv.interpolate(
th.conditional(depth > 1.1 * a, a / depth,
a / (1.1 * a)))
# take max of value calculated either by ksp or depth
Amax.assign(th.conditional(Aconv > Bconv, Aconv,
Bconv))
r1conv.assign(1 - (1 / 0.4) * th.conditional(
settling_velocity / ustar < 1, settling_velocity /
ustar, 1))
Ione.assign(
th.conditional(
r1conv > 10**(-8), (1 - Amax**r1conv) / r1conv,
th.conditional(r1conv < -10**(-8),
(1 - Amax**r1conv) / r1conv,
th.ln(Amax))))
Itwo.assign(
th.conditional(
r1conv > 10**(-8),
-(Ione + (th.ln(Amax) *
(Amax**r1conv))) / r1conv,
th.conditional(
r1conv < -10**(-8),
-(Ione + (th.ln(Amax) * (Amax**r1conv))) /
r1conv, -0.5 * th.ln(Amax)**2)))
alpha.assign(-(Itwo -
(th.ln(Amax) - th.ln(30)) * Ione) /
(Ione * ((th.ln(Amax) - th.ln(30)) + 1)))
# final correction factor
alphatest2.assign(
th.conditional(
th.conditional(alpha > 1, 1, alpha) < 0, 0,
th.conditional(alpha > 1, 1, alpha)))
# multiply correction factor by velocity and insert back into sediment concentration equation
corrective_velocity.interpolate(alphatest2 * uv1)
else:
corrective_velocity.interpolate(uv1)
if bedload:
# calculate angle of flow
calfa.interpolate(horizontal_velocity / th.sqrt(unorm))
salfa.interpolate(vertical_velocity / th.sqrt(unorm))
div_function.interpolate(th.as_vector((calfa, salfa)))
if slope_eff:
# slope effect magnitude correction due to gravity where beta is a parameter normally set to 1.3
# we use z_n1 and equals so that we can use an implicit method in Exner
slopecoef = (1 + beta *
(z_n1.dx(0) * calfa + z_n1.dx(1) * salfa))
else:
slopecoef = th.Constant(1.0)
if angle_correction:
# slope effect angle correction due to gravity
tt1.interpolate(
th.conditional(
1000 * 0.5 * qfc * unorm > 10**(-10),
th.sqrt(cparam / (1000 * 0.5 * qfc * unorm)),
th.sqrt(cparam / (10**(-10)))))
# add on a factor of the bed gradient to the normal
aa.assign(salfa + tt1 * dzdy)
bb.assign(calfa + tt1 * dzdx)
norm.assign(
th.conditional(
th.sqrt(aa**2 + bb**2) > 10**(-10),
th.sqrt(aa**2 + bb**2), 10**(-10)))
# we use z_n1 and equals so that we can use an implicit method in Exner
calfamod = (calfa + (tt1 * z_n1.dx(0))) / norm
salfamod = (salfa + (tt1 * z_n1.dx(1))) / norm
if seccurrent:
# accounts for helical flow effect in a curver channel
# again use z_n1 and equals so can use an implicit method in Exner
free_surface_dx = depth.dx(0) - z_n1.dx(0)
free_surface_dy = depth.dx(1) - z_n1.dx(1)
velocity_slide = (horizontal_velocity * free_surface_dy
) - (vertical_velocity *
free_surface_dx)
tandelta_factor.interpolate(
7 * 9.81 * 1000 * depth * qfc /
(2 * alpha_secc * ((horizontal_velocity**2) +
(vertical_velocity**2))))
if angle_correction:
# if angle has already been corrected we must alter the corrected angle to obtain the corrected secondary current angle
t_1 = (TOB * slopecoef * calfamod) + (
vertical_velocity * tandelta_factor *
velocity_slide)
t_2 = (TOB * slopecoef * salfamod) - (
horizontal_velocity * tandelta_factor *
velocity_slide)
else:
t_1 = (TOB * slopecoef *
calfa) + (vertical_velocity *
tandelta_factor * velocity_slide)
t_2 = ((TOB * slopecoef * salfa) -
(horizontal_velocity * tandelta_factor *
velocity_slide))
# calculated to normalise the new angles
t4 = th.sqrt((t_1**2) + (t_2**2))
# updated magnitude correction and angle corrections
slopecoef = t4 / TOB
calfanew = t_1 / t4
salfanew = t_2 / t4
if bed_form == 'meyer':
# implement meyer-peter-muller bedload transport formula
thetaprime.interpolate(
mu * (1000 * 0.5 * qfc * unorm) /
((2650 - 1000) * 9.81 * average_size))
# if velocity above a certain critical value then transport occurs
phi.assign(
th.conditional(thetaprime < thetacr, 0,
8 * (thetaprime - thetacr)**1.5))
# bedload transport flux with magnitude correction
qb_total = slopecoef * phi * th.sqrt(
g * (2650 / 1000 - 1) * average_size**3)
elif bed_form == 'soulsby':
abb.interpolate(
th.conditional(
depth >= average_size, 0.005 * depth *
((average_size / depth)**1.2) / coeff_soulsby,
0.005 * depth / coeff_soulsby))
ucr_bed.interpolate(
th.conditional(
depth > d90, 0.19 * (average_size**0.1) *
(th.ln(4 * depth / d90)) / (th.ln(10)),
0.19 * (average_size**0.1) * (th.ln(4)) /
(th.ln(10))))
s0_bed.interpolate(
th.conditional((th.sqrt(unorm) - ucr_bed)**2.4 > 0,
(th.sqrt(unorm) - ucr_bed)**2.4, 0))
qb_total = slopecoef * abb * s0_bed * th.sqrt(unorm)
else:
print(
'Unrecognised bedload transport formula. Please choose "meyer" or "soulsby"'
)
# add time derivative to exner equation with a morphological scale factor
f += (((1 - porosity) * (z_n1 - z_n) /
(dt * morfac)) * v) * fire.dx
# formulate bedload transport flux with correct angle depending on corrections implemented
if angle_correction and seccurrent is False:
qbx = qb_total * calfamod
qby = qb_total * salfamod
elif seccurrent:
qbx = qb_total * calfanew
qby = qb_total * salfanew
else:
qbx = qb_total * calfa
qby = qb_total * salfa
# add bedload transport to exner equation
f += -(v * (
(qbx * n[0]) + (qby * n[1]))) * fire.ds(1) - (v * (
(qbx * n[0]) +
(qby * n[1]))) * fire.ds(2) + (qbx *
(v.dx(0)) + qby *
(v.dx(1))) * fire.dx
else:
# if no bedload transport component initialise exner equation with time derivative
f = (((1 - porosity) * (z_n1 - z_n) /
(dt * morfac)) * v) * fire.dx
if suspendedload:
# add suspended sediment transport to exner equation multiplied by depth as the exner equation is not depth-averaged
qbsourcedepth.interpolate(source * depth)
f += -(qbsourcedepth * v) * fire.dx
# solve exner equation using finite element methods
fire.solve(f == 0, z_n1)
# update bed
bathymetry_2d.assign(z_n1)
if round(t_new, 2) % t_export == 0:
# calculate difference between original bathymetry and new bathymetry
diff_bathy.interpolate(-bathymetry_2d + orig_bathymetry)
diff_bathy_file.write(diff_bathy)
# choose directory to output results
ts = time.time()
st = datetime.datetime.fromtimestamp(ts).strftime('%Y-%m-%d %H:%M:%S')
outputdir = 'outputs' + st
# final time of simulation
t_end = final_time / morfac
# export interval in seconds
t_export = np.round(t_end / 100, 0)
th.print_output('Exporting to ' + outputdir)
x, y = th.SpatialCoordinate(mesh2d)
# define function spaces
P1_2d = th.FunctionSpace(mesh2d, 'DG', 1)
vectorP1_2d = th.VectorFunctionSpace(mesh2d, 'DG', 1)
V = th.FunctionSpace(mesh2d, 'CG', 1)
vector_cg = th.VectorFunctionSpace(mesh2d, 'CG', 1)
# define test functions on mesh
v = fire.TestFunction(V)
n = th.FacetNormal(mesh2d)
z_n1 = fire.Function(V, name="z^{n+1}")
z_n = fire.Function(V, name="z^{n}")
# define original bathymetry before bedlevel changes
orig_bathymetry = th.Function(V).interpolate(bathymetry_2d)
# calculate bed evolution
diff_bathy = th.Function(V).interpolate(-bathymetry_2d + orig_bathymetry)
# define output file for bed evolution
diff_bathy_file = th.File(outputdir + "/diff_bathy.pvd")
diff_bathy_file.write(diff_bathy)
# define parameters
g = th.Constant(9.81)
porosity = th.Constant(0.4)
ksp = th.Constant(3 * average_size)
a = th.Constant(ks / 2)
viscosity = th.Constant(10**(-6))
# magnitude slope effect parameter
beta = th.Constant(beta_fn)
# angle correction slope effect parameters
surbeta2 = th.Constant(surbeta2_fn)
cparam = th.Constant((2650 - 1000) * 9.81 * average_size * (surbeta2**2))
# secondary current parameter
alpha_secc = th.Constant(alpha_secc_fn)
# calculate critical shields parameter thetacr
R = th.Constant(2650 / 1000 - 1)
dstar = th.Constant(average_size * ((g * R) / (viscosity**2))**(1 / 3))
if max(dstar.dat.data[:] < 1):
print('ERROR: dstar value less than 1')
elif max(dstar.dat.data[:] < 4):
thetacr = th.Constant(0.24 * (dstar**(-1)))
elif max(dstar.dat.data[:] < 10):
thetacr = th.Constant(0.14 * (dstar**(-0.64)))
elif max(dstar.dat.data[:] < 20):
thetacr = th.Constant(0.04 * (dstar**(-0.1)))
elif max(dstar.dat.data[:] < 150):
thetacr = th.Constant(0.013 * (dstar**(0.29)))
else:
thetacr = th.Constant(0.055)
# critical bed shear stress
taucr = th.Constant((2650 - 1000) * g * average_size * thetacr)
# calculate settling velocity
if average_size <= 100 * (10**(-6)):
settling_velocity = th.Constant(9.81 * (average_size**2) *
((2650 / 1000) - 1) / (18 * viscosity))
elif average_size <= 1000 * (10**(-6)):
settling_velocity = th.Constant(
(10 * viscosity / average_size) *
(th.sqrt(1 + 0.01 * ((((2650 / 1000) - 1) * 9.81 *
(average_size**3)) / (viscosity**2))) - 1))
else:
settling_velocity = th.Constant(1.1 * th.sqrt(9.81 * average_size *
((2650 / 1000) - 1)))
# initialise velocity, elevation and depth
elev_init, uv_init = initialise_fields(mesh2d, input_dir, outputdir)
uv_cg = th.Function(vector_cg).interpolate(uv_init)
elev_cg = th.Function(V).interpolate(elev_init)
depth = th.Function(V).project(elev_cg + bathymetry_2d)
old_bathymetry_2d = th.Function(V).interpolate(bathymetry_2d)
horizontal_velocity = th.Function(V).interpolate(uv_cg[0])
vertical_velocity = th.Function(V).interpolate(uv_cg[1])
# define bed friction
hc = th.Function(P1_2d).interpolate(
th.conditional(depth > 0.001, depth, 0.001))
aux = th.Function(P1_2d).interpolate(
th.conditional(11.036 * hc / ks > 1.001, 11.036 * hc / ks, 1.001))
qfc = th.Function(P1_2d).interpolate(2 / (th.ln(aux) / 0.4)**2)
# skin friction coefficient
hclip = th.Function(P1_2d).interpolate(
th.conditional(ksp > depth, ksp, depth))
cfactor = th.Function(P1_2d).interpolate(
th.conditional(depth > ksp,
2 * ((2.5 * th.ln(11.036 * hclip / ksp))**(-2)),
th.Constant(0.0)))
# mu - ratio between skin friction and normal friction
mu = th.Function(P1_2d).interpolate(
th.conditional(qfc > 0, cfactor / qfc, 0))
# calculate bed shear stress
unorm = th.Function(P1_2d).interpolate((horizontal_velocity**2) +
(vertical_velocity**2))
TOB = th.Function(V).interpolate(1000 * 0.5 * qfc * unorm)
# define bed gradient
dzdx = th.Function(V).interpolate(old_bathymetry_2d.dx(0))
dzdy = th.Function(V).interpolate(old_bathymetry_2d.dx(1))
if suspendedload:
# deposition flux - calculating coefficient to account for stronger conc at bed
B = th.Function(P1_2d).interpolate(
th.conditional(a > depth, a / a, a / depth))
ustar = th.Function(P1_2d).interpolate(th.sqrt(0.5 * qfc * unorm))
exp1 = th.Function(P1_2d).interpolate(
th.conditional(
(th.conditional(
(settling_velocity / (0.4 * ustar)) - 1 > 0,
(settling_velocity /
(0.4 * ustar)) - 1, -(settling_velocity /
(0.4 * ustar)) + 1)) > 10**(-4),
th.conditional((settling_velocity / (0.4 * ustar)) - 1 > 3, 3,
(settling_velocity / (0.4 * ustar)) - 1), 0))
coefftest = th.Function(P1_2d).interpolate(
th.conditional((th.conditional(
(settling_velocity / (0.4 * ustar)) - 1 > 0,
(settling_velocity /
(0.4 * ustar)) - 1, -(settling_velocity /
(0.4 * ustar)) + 1)) > 10**(-4),
B * (1 - B**exp1) / exp1, -B * th.ln(B)))
coeff = th.Function(P1_2d).interpolate(
th.conditional(coefftest > 0, 1 / coefftest, 0))
if sus_form == 'vanrijn':
# erosion flux - above critical velocity bed is eroded
s0 = th.Function(P1_2d).interpolate(
(th.conditional(1000 * 0.5 * qfc * unorm * mu > 0, 1000 * 0.5 *
qfc * unorm * mu, 0) - taucr) / taucr)
ceq = th.Function(P1_2d).interpolate(
0.015 * (average_size / a) *
((th.conditional(s0 < 0, 0, s0))**(1.5)) / (dstar**0.3))
elif sus_form == 'soulsby':
if d90 == 0:
# if the value of d90 is unspecified set d90 = d50
d90 = th.Constant(average_size)
else:
d90 = th.Constant(d90)
coeff_soulsby = th.Constant((R * g * average_size)**1.2)
ass = th.Constant(0.012 * average_size * (dstar**(-0.6)) /
coeff_soulsby)
ucr = th.Function(P1_2d).interpolate(0.19 * (average_size**0.1) *
(th.ln(4 * depth / d90)) /
(th.ln(10)))
s0 = th.Function(P1_2d).interpolate(
th.conditional((th.sqrt(unorm) - ucr)**2.4 > 0,
(th.sqrt(unorm) - ucr)**2.4, 0))
ceq = th.Function(P1_2d).interpolate(ass * s0 / depth)
else:
print(
'Unrecognised suspended sediment transport formula. Please choose "vanrijn" or "soulsby"'
)
# update sediment rate to ensure equilibrium at inflow
sediment_rate = th.Constant(ceq.at([0, 0]) / coeff.at([0, 0]))
testtracer = th.Function(P1_2d).interpolate(ceq / coeff)
# calculate depth-averaged source term for sediment concentration equation
source = th.Function(P1_2d).interpolate(
-(settling_velocity * coeff * sediment_rate / depth) +
(settling_velocity * ceq / depth))
# add suspended sediment transport to exner equation multiplied by depth as the exner equation is not depth-averaged
qbsourcedepth = th.Function(V).interpolate(source * depth)
if convectivevel:
# correction factor to advection velocity in sediment concentration equation
Bconv = th.Function(P1_2d).interpolate(
th.conditional(depth > 1.1 * ksp, ksp / depth,
ksp / (1.1 * ksp)))
Aconv = th.Function(P1_2d).interpolate(
th.conditional(depth > 1.1 * a, a / depth, a / (1.1 * a)))
# take max of value calculated either by ksp or depth
Amax = th.Function(P1_2d).interpolate(
th.conditional(Aconv > Bconv, Aconv, Bconv))
r1conv = th.Function(P1_2d).interpolate(
1 - (1 / 0.4) * th.conditional(settling_velocity / ustar < 1,
settling_velocity / ustar, 1))
Ione = th.Function(P1_2d).interpolate(
th.conditional(
r1conv > 10**(-8), (1 - Amax**r1conv) / r1conv,
th.conditional(r1conv < -10**(-8),
(1 - Amax**r1conv) / r1conv, th.ln(Amax))))
Itwo = th.Function(P1_2d).interpolate(
th.conditional(
r1conv > 10**(-8),
-(Ione + (th.ln(Amax) * (Amax**r1conv))) / r1conv,
th.conditional(
r1conv < -10**(-8),
-(Ione + (th.ln(Amax) * (Amax**r1conv))) / r1conv,
-0.5 * th.ln(Amax)**2)))
alpha = th.Function(P1_2d).interpolate(
-(Itwo - (th.ln(Amax) - th.ln(30)) * Ione) /
(Ione * ((th.ln(Amax) - th.ln(30)) + 1)))
# final correction factor
alphatest2 = th.Function(P1_2d).interpolate(
th.conditional(
th.conditional(alpha > 1, 1, alpha) < 0, 0,
th.conditional(alpha > 1, 1, alpha)))
# multiply correction factor by velocity and insert back into sediment concentration equation
corrective_velocity = th.Function(vectorP1_2d).interpolate(
alphatest2 * uv_init)
else:
corrective_velocity = th.Function(vectorP1_2d).interpolate(uv_init)
if bedload:
# calculate angle of flow
calfa = th.Function(V).interpolate(horizontal_velocity /
th.sqrt(unorm))
salfa = th.Function(V).interpolate(vertical_velocity / th.sqrt(unorm))
div_function = th.Function(vector_cg).interpolate(
th.as_vector((calfa, salfa)))
if slope_eff:
# slope effect magnitude correction due to gravity where beta is a parameter normally set to 1.3
slopecoef = th.Function(V).interpolate(
1 + beta * (dzdx * calfa + dzdy * salfa))
else:
slopecoef = th.Function(V).interpolate(th.Constant(1.0))
if angle_correction:
# slope effect angle correction due to gravity
tt1 = th.Function(V).interpolate(
th.conditional(1000 * 0.5 * qfc * unorm > 10**(-10),
th.sqrt(cparam / (1000 * 0.5 * qfc * unorm)),
th.sqrt(cparam / (10**(-10)))))
# add on a factor of the bed gradient to the normal
aa = th.Function(V).interpolate(salfa + tt1 * dzdy)
bb = th.Function(V).interpolate(calfa + tt1 * dzdx)
norm = th.Function(V).interpolate(
th.conditional(
th.sqrt(aa**2 + bb**2) > 10**(-10), th.sqrt(aa**2 + bb**2),
10**(-10)))
if seccurrent:
# accounts for helical flow effect in a curver channel
free_surface_dx = th.Function(V).interpolate(elev_cg.dx(0))
free_surface_dy = th.Function(V).interpolate(elev_cg.dx(1))
velocity_slide = (horizontal_velocity * free_surface_dy) - (
vertical_velocity * free_surface_dx)
tandelta_factor = th.Function(V).interpolate(
7 * 9.81 * 1000 * depth * qfc / (2 * alpha_secc *
((horizontal_velocity**2) +
(vertical_velocity**2))))
t_1 = (TOB * slopecoef * calfa) + (
vertical_velocity * tandelta_factor * velocity_slide)
t_2 = ((TOB * slopecoef * salfa) -
(horizontal_velocity * tandelta_factor * velocity_slide))
# calculated to normalise the new angles
t4 = th.sqrt((t_1**2) + (t_2**2))
# updated magnitude correction and angle corrections
slopecoef = t4 / TOB
if bed_form == 'meyer':
# implement meyer-peter-muller bedload transport formula
thetaprime = th.Function(V).interpolate(
mu * (1000 * 0.5 * qfc * unorm) /
((2650 - 1000) * 9.81 * average_size))
# if velocity above a certain critical value then transport occurs
phi = th.Function(V).interpolate(
th.conditional(thetaprime < thetacr, 0,
8 * (thetaprime - thetacr)**1.5))
elif bed_form == 'soulsby':
if d90 == 0:
d90 = th.Constant(average_size)
coeff_soulsby = th.Constant((R * g * average_size)**1.2)
abb = th.Function(P1_2d).interpolate(
th.conditional(
depth >= average_size, 0.005 * depth *
((average_size / depth)**1.2) / coeff_soulsby,
0.005 * depth / coeff_soulsby))
ucr_bed = th.Function(P1_2d).interpolate(
th.conditional(
depth > d90, 0.19 * (average_size**0.1) *
(th.ln(4 * depth / d90)) / (th.ln(10)),
0.19 * (average_size**0.1) * (th.ln(4)) / (th.ln(10))))
s0_bed = th.Function(P1_2d).interpolate(
th.conditional((th.sqrt(unorm) - ucr_bed)**2.4 > 0,
(th.sqrt(unorm) - ucr_bed)**2.4, 0))
else:
print(
'Unrecognised bedload transport formula. Please choose "meyer" or "soulsby"'
)
# set up solver
solver_obj = th.solver2d.FlowSolver2d(mesh2d, bathymetry_2d)
options = solver_obj.options
options.simulation_export_time = t_export
options.simulation_end_time = t_end
options.output_directory = outputdir
options.check_volume_conservation_2d = True
if suspendedload:
# switch on tracer calculation if using sediment transport component
options.solve_tracer = True
options.fields_to_export = [
'uv_2d', 'elev_2d', 'tracer_2d', 'bathymetry_2d'
]
options.tracer_advective_velocity = corrective_velocity
options.tracer_source_2d = source
else:
options.solve_tracer = False
options.fields_to_export = ['uv_2d', 'elev_2d', 'bathymetry_2d']
options.use_lax_friedrichs_tracer = False
# set bed friction
if friction == 'nikuradse':
options.quadratic_drag_coefficient = cfactor
elif friction == 'manning':
if friction_coef == 0:
friction_coef = 0.02
options.manning_drag_coefficient = th.Constant(friction_coef)
else:
print('Undefined friction')
# set horizontal diffusivity parameter
options.horizontal_diffusivity = th.Constant(diffusivity)
options.horizontal_viscosity = th.Constant(viscosity_hydro)
# crank-nicholson used to integrate in time system of ODEs resulting from application of galerkin FEM
options.timestepper_type = 'CrankNicolson'
options.timestepper_options.implicitness_theta = 1.0
if not hasattr(options.timestepper_options, 'use_automatic_timestep'):
options.timestep = dt
# set boundary conditions
swe_bnd, left_bnd_id, right_bnd_id, in_constant, out_constant, left_string, right_string = boundary_conditions_fn(
orig_bathymetry, flag='morpho')
for j in range(len(in_constant)):
exec(
'constant_in' + str(j) + ' = th.Constant(' + str(in_constant[j]) +
')', globals())
str1 = '{'
for i in range(len(left_string)):
str1 += "'" + str(left_string[i]) + "': constant_in" + str(i) + ","
str1 = str1[0:len(str1) - 1] + "}"
exec('swe_bnd[left_bnd_id] = ' + str1)
for k in range(len(out_constant)):
exec(
'constant_out' + str(k) + '= th.Constant(' + str(out_constant[k]) +
')', globals())
str2 = '{'
for i in range(len(right_string)):
str2 += "'" + str(right_string[i]) + "': constant_out" + str(i) + ","
str2 = str2[0:len(str2) - 1] + "}"
exec('swe_bnd[right_bnd_id] = ' + str2)
solver_obj.bnd_functions['shallow_water'] = swe_bnd
if suspendedload:
solver_obj.bnd_functions['tracer'] = {1: {'value': sediment_rate}}
# set initial conditions
solver_obj.assign_initial_conditions(uv=uv_init,
elev=elev_init,
tracer=testtracer)
else:
# set initial conditions
solver_obj.assign_initial_conditions(uv=uv_init, elev=elev_init)
return solver_obj, update_forcings_tracer, diff_bathy, diff_bathy_file
def update_forcings_tracer(t_new):
# update bathymetry
old_bathymetry_2d.assign(bathymetry_2d)
# extract new elevation and velocity and project onto CG space
uv1, elev1 = solver_obj.fields.solution_2d.split()
uv_cg.project(uv1)
elev_cg.project(elev1)
depth.project(elev_cg + old_bathymetry_2d)
horizontal_velocity.interpolate(uv_cg[0])
vertical_velocity.interpolate(uv_cg[1])
# update boundary conditions if have fluctuating conditions
if fluc_bcs:
in_fn, out_fn = boundary_conditions_fn(orig_bathymetry,
flag='morpho',
morfac=morfac,
t_new=t_new,
state='update')
for j in range(len(in_fn)):
exec('constant_in' + str(j) + '.assign(' + str(in_fn[j]) + ')')
for k in range(len(out_fn)):
exec('constant_out' + str(k) + '.assign(' + str(out_fn[k]) +
')')
# update bedfriction
hc.interpolate(th.conditional(depth > 0.001, depth, 0.001))
aux.assign(
th.conditional(11.036 * hc / ks > 1.001, 11.036 * hc / ks, 1.001))
qfc.assign(2 / (th.ln(aux) / 0.4)**2)
# calculate skin friction coefficient
hclip.interpolate(th.conditional(ksp > depth, ksp, depth))
cfactor.interpolate(
th.conditional(depth > ksp,
2 * ((2.5 * th.ln(11.036 * hclip / ksp))**(-2)),
th.Constant(0.0)))
if morfac_transport:
# if include tracer solver then update_forcings is run twice but only want to update bathymetry once
t_list.append(t_new)
double_factor = False
if suspendedload:
if len(t_list) > 1:
if t_list[len(t_list) - 1] == t_list[len(t_list) - 2]:
double_factor = True
else:
# if have no tracer then update_forcings is only run once so update bathymetry at each step
double_factor = True
if double_factor:
z_n.assign(old_bathymetry_2d)
# mu - ratio between skin friction and normal friction
mu.assign(th.conditional(qfc > 0, cfactor / qfc, 0))
# bed shear stress
unorm.interpolate((horizontal_velocity**2) +
(vertical_velocity**2))
TOB.interpolate(1000 * 0.5 * qfc * unorm)
# calculate gradient of bed (noting bathymetry is -bed)
dzdx.interpolate(old_bathymetry_2d.dx(0))
dzdy.interpolate(old_bathymetry_2d.dx(1))
# initialise exner equation
f = 0
if suspendedload:
# source term
# deposition flux - calculating coefficient to account for stronger conc at bed
B.interpolate(th.conditional(a > depth, a / a, a / depth))
ustar.interpolate(th.sqrt(0.5 * qfc * unorm))
exp1.assign(
th.conditional(
(th.conditional(
(settling_velocity / (0.4 * ustar)) - 1 > 0,
(settling_velocity / (0.4 * ustar)) - 1,
-(settling_velocity /
(0.4 * ustar)) + 1)) > 10**(-4),
th.conditional(
(settling_velocity / (0.4 * ustar)) - 1 > 3, 3,
(settling_velocity / (0.4 * ustar)) - 1), 0))
coefftest.assign(
th.conditional((th.conditional(
(settling_velocity / (0.4 * ustar)) - 1 > 0,
(settling_velocity / (0.4 * ustar)) - 1,
-(settling_velocity /
(0.4 * ustar)) + 1)) > 10**(-4),
B * (1 - B**exp1) / exp1,
-B * th.ln(B)))
coeff.assign(
th.conditional(coefftest > 0, 1 / coefftest, 0))
if sus_form == 'vanrijn':
# erosion flux - above critical velocity bed is eroded
s0.assign(
(th.conditional(1000 * 0.5 * qfc * unorm * mu > 0,
1000 * 0.5 * qfc * unorm * mu, 0) -
taucr) / taucr)
ceq.assign(0.015 * (average_size / a) *
((th.conditional(s0 < 0, 0, s0))**(1.5)) /
(dstar**0.3))
elif sus_form == 'soulsby':
ucr.interpolate(0.19 * (average_size**0.1) *
(th.ln(4 * depth / d90) / th.ln(10)))
s0.assign(
th.conditional((th.sqrt(unorm) - ucr)**2.4 > 0,
(th.sqrt(unorm) - ucr)**2.4, 0))
ceq.interpolate(ass * s0 / depth)
else:
print(
'Unrecognised suspended sediment transport formula. Please choose "vanrijn" or "soulsby"'
)
# calculate depth-averaged source term for sediment concentration equation
source.interpolate(-(settling_velocity * coeff *
solver_obj.fields.tracer_2d / depth) +
(settling_velocity * ceq / depth))
# update sediment rate to ensure equilibrium at inflow
sediment_rate.assign(ceq.at([0, 0]) / coeff.at([0, 0]))
if convectivevel:
# correction factor to advection velocity in sediment concentration equation
Bconv.interpolate(
th.conditional(depth > 1.1 * ksp, ksp / depth,
ksp / (1.1 * ksp)))
Aconv.interpolate(
th.conditional(depth > 1.1 * a, a / depth,
a / (1.1 * a)))
# take max of value calculated either by ksp or depth
Amax.assign(th.conditional(Aconv > Bconv, Aconv,
Bconv))
r1conv.assign(1 - (1 / 0.4) * th.conditional(
settling_velocity / ustar < 1, settling_velocity /
ustar, 1))
Ione.assign(
th.conditional(
r1conv > 10**(-8), (1 - Amax**r1conv) / r1conv,
th.conditional(r1conv < -10**(-8),
(1 - Amax**r1conv) / r1conv,
th.ln(Amax))))
Itwo.assign(
th.conditional(
r1conv > 10**(-8),
-(Ione + (th.ln(Amax) *
(Amax**r1conv))) / r1conv,
th.conditional(
r1conv < -10**(-8),
-(Ione + (th.ln(Amax) * (Amax**r1conv))) /
r1conv, -0.5 * th.ln(Amax)**2)))
alpha.assign(-(Itwo -
(th.ln(Amax) - th.ln(30)) * Ione) /
(Ione * ((th.ln(Amax) - th.ln(30)) + 1)))
# final correction factor
alphatest2.assign(
th.conditional(
th.conditional(alpha > 1, 1, alpha) < 0, 0,
th.conditional(alpha > 1, 1, alpha)))
# multiply correction factor by velocity and insert back into sediment concentration equation
corrective_velocity.interpolate(alphatest2 * uv1)
else:
corrective_velocity.interpolate(uv1)
if bedload:
# calculate angle of flow
calfa.interpolate(horizontal_velocity / th.sqrt(unorm))
salfa.interpolate(vertical_velocity / th.sqrt(unorm))
div_function.interpolate(th.as_vector((calfa, salfa)))
if slope_eff:
# slope effect magnitude correction due to gravity where beta is a parameter normally set to 1.3
# we use z_n1 and equals so that we can use an implicit method in Exner
slopecoef = (1 + beta *
(z_n1.dx(0) * calfa + z_n1.dx(1) * salfa))
else:
slopecoef = th.Constant(1.0)
if angle_correction:
# slope effect angle correction due to gravity
tt1.interpolate(
th.conditional(
1000 * 0.5 * qfc * unorm > 10**(-10),
th.sqrt(cparam / (1000 * 0.5 * qfc * unorm)),
th.sqrt(cparam / (10**(-10)))))
# add on a factor of the bed gradient to the normal
aa.assign(salfa + tt1 * dzdy)
bb.assign(calfa + tt1 * dzdx)
norm.assign(
th.conditional(
th.sqrt(aa**2 + bb**2) > 10**(-10),
th.sqrt(aa**2 + bb**2), 10**(-10)))
# we use z_n1 and equals so that we can use an implicit method in Exner
calfamod = (calfa + (tt1 * z_n1.dx(0))) / norm
salfamod = (salfa + (tt1 * z_n1.dx(1))) / norm
if seccurrent:
# accounts for helical flow effect in a curver channel
# again use z_n1 and equals so can use an implicit method in Exner
free_surface_dx = depth.dx(0) - z_n1.dx(0)
free_surface_dy = depth.dx(1) - z_n1.dx(1)
velocity_slide = (horizontal_velocity * free_surface_dy
) - (vertical_velocity *
free_surface_dx)
tandelta_factor.interpolate(
7 * 9.81 * 1000 * depth * qfc /
(2 * alpha_secc * ((horizontal_velocity**2) +
(vertical_velocity**2))))
if angle_correction:
# if angle has already been corrected we must alter the corrected angle to obtain the corrected secondary current angle
t_1 = (TOB * slopecoef * calfamod) + (
vertical_velocity * tandelta_factor *
velocity_slide)
t_2 = (TOB * slopecoef * salfamod) - (
horizontal_velocity * tandelta_factor *
velocity_slide)
else:
t_1 = (TOB * slopecoef *
calfa) + (vertical_velocity *
tandelta_factor * velocity_slide)
t_2 = ((TOB * slopecoef * salfa) -
(horizontal_velocity * tandelta_factor *
velocity_slide))
# calculated to normalise the new angles
t4 = th.sqrt((t_1**2) + (t_2**2))
# updated magnitude correction and angle corrections
slopecoef = t4 / TOB
calfanew = t_1 / t4
salfanew = t_2 / t4
if bed_form == 'meyer':
# implement meyer-peter-muller bedload transport formula
thetaprime.interpolate(
mu * (1000 * 0.5 * qfc * unorm) /
((2650 - 1000) * 9.81 * average_size))
# if velocity above a certain critical value then transport occurs
phi.assign(
th.conditional(thetaprime < thetacr, 0,
8 * (thetaprime - thetacr)**1.5))
# bedload transport flux with magnitude correction
qb_total = slopecoef * phi * th.sqrt(
g * (2650 / 1000 - 1) * average_size**3)
elif bed_form == 'soulsby':
abb.interpolate(
th.conditional(
depth >= average_size, 0.005 * depth *
((average_size / depth)**1.2) / coeff_soulsby,
0.005 * depth / coeff_soulsby))
ucr_bed.interpolate(
th.conditional(
depth > d90, 0.19 * (average_size**0.1) *
(th.ln(4 * depth / d90)) / (th.ln(10)),
0.19 * (average_size**0.1) * (th.ln(4)) /
(th.ln(10))))
s0_bed.interpolate(
th.conditional((th.sqrt(unorm) - ucr_bed)**2.4 > 0,
(th.sqrt(unorm) - ucr_bed)**2.4, 0))
qb_total = slopecoef * abb * s0_bed * th.sqrt(unorm)
else:
print(
'Unrecognised bedload transport formula. Please choose "meyer" or "soulsby"'
)
# add time derivative to exner equation with a morphological scale factor
f += (((1 - porosity) * (z_n1 - z_n) /
(dt * morfac)) * v) * fire.dx
# formulate bedload transport flux with correct angle depending on corrections implemented
if angle_correction and seccurrent is False:
qbx = qb_total * calfamod
qby = qb_total * salfamod
elif seccurrent:
qbx = qb_total * calfanew
qby = qb_total * salfanew
else:
qbx = qb_total * calfa
qby = qb_total * salfa
# add bedload transport to exner equation
f += -(v * (
(qbx * n[0]) + (qby * n[1]))) * fire.ds(1) - (v * (
(qbx * n[0]) +
(qby * n[1]))) * fire.ds(2) + (qbx *
(v.dx(0)) + qby *
(v.dx(1))) * fire.dx
else:
# if no bedload transport component initialise exner equation with time derivative
f = (((1 - porosity) * (z_n1 - z_n) /
(dt * morfac)) * v) * fire.dx
if suspendedload:
# add suspended sediment transport to exner equation multiplied by depth as the exner equation is not depth-averaged
qbsourcedepth.interpolate(source * depth)
f += -(qbsourcedepth * v) * fire.dx
# solve exner equation using finite element methods
fire.solve(f == 0, z_n1)
# update bed
bathymetry_2d.assign(z_n1)
if round(t_new, 2) % t_export == 0:
# calculate difference between original bathymetry and new bathymetry
diff_bathy.interpolate(-bathymetry_2d + orig_bathymetry)
diff_bathy_file.write(diff_bathy)
# define mesh
mesh2d = th.Mesh("meander_test_3112_test.msh")
x, y = th.SpatialCoordinate(mesh2d)
# define function spaces
V = th.FunctionSpace(mesh2d, 'CG', 1)
P1_2d = th.FunctionSpace(mesh2d, 'DG', 1)
vectorP1_2d = th.VectorFunctionSpace(mesh2d, 'DG', 1)
# define underlying bathymetry
bathymetry_2d = th.Function(V, name='Bathymetry')
gradient = th.Constant(0.0035)
L_function = th.Function(V).interpolate(th.conditional(x > 5, th.pi*4*((th.pi/2)-th.acos((x-5)/(th.sqrt((x-5)**2+(y-2.5)**2))))/th.pi, th.pi*4*((th.pi/2)-th.acos((-x+5)/(th.sqrt((x-5)**2+(y-2.5)**2))))/th.pi))
bathymetry_2d1 = th.Function(V).interpolate(th.conditional(y > 2.5, th.conditional(x < 5, (L_function*gradient) + 9.97072, -(L_function*gradient) + 9.97072), 9.97072))
init = max(bathymetry_2d1.dat.data[:])
final = min(bathymetry_2d1.dat.data[:])
bathymetry_2d2 = th.Function(V).interpolate(th.conditional(x <= 5, th.conditional(y <= 2.5, -9.97072 + gradient*abs(y - 2.5) + init, 0), th.conditional(y <= 2.5, -9.97072-gradient*abs(y - 2.5) + final, 0)))
bathymetry_2d = th.Function(V).interpolate(-bathymetry_2d1 - bathymetry_2d2)
input_bathymetry_2d = th.Function(V).interpolate(bathymetry_2d)
initial_bathymetry_2d = th.Function(V).interpolate(bathymetry_2d)
# If the hydrodynamics simulation has already been run with the required parameters
# then hydro can be set to False. This is because we do not need to run the initial
# hydrodyamics simulation again as the previous run can be used to initialise
# the full simulation