pypesh Documentation

pypesh is a repository that attempts to find a solution to advection diffusion problem

\[0 = \Delta \varphi - \mathrm{Pe} (u \cdot \nabla \varphi),\]

with \(\varphi = 1\) for \(z \to \infty\) and \(\phi = 0\) on a surface of the sphere and \(\mathrm{Pe}\) denoting Peclet number. Final value determining the flux of \(\varphi\) onto the sphere is the Sherwood number defined as

\[\mathrm{Sh} = \frac{\Phi}{ 4 \pi D R}.\]

Where \(D\) is diffusion constant and \(\Phi\) is flux falling onto the sphere.

How to install

python3 -m pip install pypesh

and you’ll be good to go.

Package contents

Analytic

pypesh.analytic.clift_approximation(pe)

Analytic approximation to numerical solution of Clift et. al.

Parameters

pe (float) – Peclet number defined as R u / D.

Returns

Sherwod calculated from Clift et. al. approximation

Return type

float

Example

>>> import pypesh.analytic as analytic
>>> analytic.clift_approximation(1000)
6.800655008742168
pypesh.analytic.direct_impact(peclet, ball_radius)

Our approximation of Sherwood number for peclet and ball_radius

Parameters

  • peclet (float) – Peclet number defined as R u / D.

  • ball_radius (float) – Radius of big ball

Returns

Sherwod calculated our approximation

Return type

float

Example

>>> import pypesh.analytic as analytic
>>> analytic.direct_impact(1000, 1)
0.0
>>> analytic.direct_impact(1000, 0.9)
3.6249999999999982
pypesh.analytic.our_approximation(peclet, ball_radius)

Our approximation of Sherwood number for peclet and ball_radius

Parameters

  • peclet (float) – Peclet number defined as R u / D.

  • ball_radius (float) – Radius of big ball

Returns

Sherwod calculated our approximation

Return type

float

Example

>>> import pypesh.analytic as analytic
>>> analytic.our_approximation(1000, 1)
6.800655008742168
>>> analytic.our_approximation(1000, 0.9)
10.425655008742165
pypesh.analytic.sherwood_from_flux(flux, peclet)

sherwood is defined as flux_dimesional/(4 pi D R) = U R^2 flux / (4 pi D R) = flux*(Pe/4 pi)

Parameters

  • flux (float) – Flux passing the ceiling

  • peclet (float) – Peclet number defined as R u / D.

Returns

sherwood from dimensionless flux

Return type

float

Example

>>> import pypesh.analytic as analytic
>>> analytic.sherwood_from_flux(0.1, 1000)
7.957747154594768

Stokes Flow

For comparison check: https://en.wikipedia.org/wiki/Stokes%27_law

pypesh.stokes_flow.psi(r, z, ball_radius)

Given location of the tracer find streamfunction – Stokes flow around sphere of size ball_radius (stationary) and ambient flow u_inf = [0,0,1].

Location is measured from the centre in cylindrical coordinates [r, phi, z], z is parallel to ambient flow.

Compare: https://en.wikipedia.org/wiki/Stokes%27_law#Transversal_flow_around_a_sphere

Parameters

  • r (float) – distance from central axis of ball.

  • z (float) – height above the plane perpendicular to ambient flow, containing centre of sphere

  • ball_radius (float) – Radius of ball.

Returns

value of streamline at [r, 0, z]

Return type

float

Example

>>> import pypesh.stokes_flow as sf
>>> sf.psi(1, 1, 1)
0.05805826175840782
pypesh.stokes_flow.stokes_around_sphere_explicite(r, z, ball_radius)

Given location of the tracer find drift velocity – Stokes flow around sphere of size big_r (stationary) and ambient flow u_inf = [0,0,1].

Location is measured from the centre of the sphere.

Compare: https://en.wikipedia.org/wiki/Stokes%27_law#Transversal_flow_around_a_sphere

Parameters

  • r (any) – Radius

  • z (any) – Height

  • ball_radius (float) – Radius of ball.

Returns

Velocity in rho direction, Velocity in phi direction, Velocity in z direction

Return type

tuple

Example

>>> import pypesh.stokes_flow as sf
>>> sf.stokes_around_sphere_explicite(1, 1, 0.7)
(-0.14013972519640885, 0, 0.4583120114372806)
pypesh.stokes_flow.stokes_around_sphere_jnp(q, ball_radius)

Given location of the tracer find drift velocity – Stokes flow around sphere of size big_r (stationary) and ambient flow u_inf = [0,0,1].

Location is measured from the centre of the sphere.

Compare: https://en.wikipedia.org/wiki/Stokes%27_law#Transversal_flow_around_a_sphere

Parameters

  • q (jnp.array) – Lenght 3, cartesian position from centre of the sphere.

  • ball_radius (float) – Radius of ball.

Returns

Flow velocity in cartesian coordinates at q

Return type

jnp.array

Example

>>> import jax.numpy as jnp
>>> import pypesh.stokes_flow as sf
>>> posjnp = jnp.array([1,1,1])
>>> sf.stokes_around_sphere_jnp(posjnp, 0.9)
Array([-0.0948298, -0.0948298,  0.4803847], dtype=float32)
pypesh.stokes_flow.stokes_around_sphere_np(q, ball_radius)

Given location of the tracer find drift velocity – Stokes flow around sphere of size big_r (stationary) and ambient flow u_inf = [0,0,1].

Location is measured from the centre of the sphere.

Compare: https://en.wikipedia.org/wiki/Stokes%27_law#Transversal_flow_around_a_sphere

Parameters

  • q (np.array) – Lenght 3, cartesian position from centre of the sphere.

  • ball_radius (float) – Radius of ball.

Returns

Flow velocity in cartesian coordinates at q

Return type

np.array

Example

>>> import pypesh.stokes_flow as sf
>>> import numpy as np
>>> posnp = np.array([1,1,1])
>>> sf.stokes_around_sphere_np(posnp, 0.9)
array([-0.09482978, -0.09482978,  0.48038476])
pypesh.stokes_flow.streamline_radius(z, ball_radius, r_start=1)

Find the radius of streamline at heigh z, that goes through the position [r_start, 0, 0]

Location is measured from the centre in cylindrical coordinates [r, phi, z], z is parallel to ambient flow.

Parameters

  • z (float) – height above the plane perpendicular to ambient flow, containing centre of sphere

  • ball_radius (float) – Radius of ball.

  • r_start (float) – radius of searched streamline for z = 0

Returns

radius of streamline that pases [r_start, 0, 0]

Return type

float

Example

>>> import pypesh.stokes_flow as sf
>>> sf.streamline_radius(5, 0.8)
0.2710224007612492

Generate Mesh

For package: https://scikit-fem.readthedocs.io/en/latest/

pypesh.generate_mesh.gen_mesh(mesh=0.01, far_mesh=0.5, cell_size=1, width=10, ceiling=10, floor=10, save=False, show_mesh=False)

Generates mesh used then by sciki-fem to solve advection-diffusion in Stokes Flow.

Parameters

  • mesh (far) – Density of triangles in most interesting region

  • mesh – Density of triangles in region far away from axis, ceiling and sphere

  • cell_size (float, optional) – Scales cell size, keeping ratio of walls to ceiling

  • width (float, optional) – Width of cell

  • ceiling (float, optional) – Distance from 0 to upper boundary of cell

  • floor (float, optional) – Distance from 0 to lower boundary of cell

  • save (Bool, optional) – Default False, if True than mesh will be saved in /meshes/

  • show (bool, optional) – Default False, if True mesh is shown

Returns

skfem object, used by scikit-fem

Return type

<skfem MeshTri1 object>

Example

>>> import pypesh.generate_mesh as msh
>>> msh.gen_mesh()
Transforming over 1000 vertices to C_CONTIGUOUS.
Transforming over 1000 elements to C_CONTIGUOUS.
<skfem MeshTri1 object>
Number of elements: 90756
Number of vertices: 46225
Number of nodes: 46225
Named boundaries [# facets]: left [972], right [40], top [264], bottom [20], ball [396]

Finite Element Method

Using package: https://scikit-fem.readthedocs.io/en/latest/

pypesh.fem.get_mesh(peclet)

Loads mesh adequate to Peclet number

Parameters

peclet (float) – Peclet number defined as R u / D.

Returns

skfem object <skfem MeshTri1 object> used by scikit-fem, <skfem CellBasis(MeshTri1, ElementTriP1) object> boundaries of mesh

Return type

tuple

Example

>>> import pypesh.fem as fem
>>> fem.get_mesh(1000)
(<skfem MeshTri1 object>
Number of elements: 90756
Number of vertices: 46225
Number of nodes: 46225
Named boundaries [# facets]: left [972], right [40], top [264], bottom [20], ball [396], <skfem CellBasis(MeshTri1, ElementTriP1) object>
Number of elements: 90756
Number of DOFs: 46225
Size: 19603296 B)
pypesh.fem.sherwood_fem(peclet, ball_radius)

sherwood is defined as flux_dimesional/(4 pi D R) = U R^2 flux / (4 pi D R) = flux*(Pe/4 pi)

Parameters

  • peclet (float) – Peclet number defined as R u / D.

  • ball_radius (float) – Radius of the big ball.

Returns

Sherwood calcualted for selected peclet and ball radius using fem approach.

Return type

float

Example

>>> import pypesh.fem as fem
>>> fem.sherwood_fem(10000, 0.9)
54.93467214954524

Trajectories approach

Using package: https://pychastic.readthedocs.io/en/latest/

pypesh.trajectories.draw_trajectory_at_x(x_position, peclet, ball_radius, trials=5, floor_h=5, t_max=10)

Generate trajectories of particles in a simulation for certain x position amd returns whole trajectory.

Parameters

  • x_postition (float) – Radius where probability is evaluated (simulation initiation point)

  • peclet (float) – Peclet number defined as R u / D.

  • ball_radius (float) – Radius of the big ball.

  • trials (int, optional) – Default 5, Number of trajectories.

  • floor_h (int, optional) – Default 5, Initial depth for simulation

  • t_max (float, optional) – Default 5, time of simulation

Returns

ball_hit - jnp.array() of 1 and 0, if 1 trajectory within radius 1, 0 miss roof_hit - jnp.array() of 1 and 0, if 1 trajectory at the end above height 2, 0 miss something_hit - jnp.array() of 1 and 0, union of ball_hit and roof_hit trajectories - jnp.array() trials by 100*t_max by 3 with x(t), y(t), z(t) positions for each trial.

Return type

dict

Example

>>> import pypesh.trajectories as traj
>>> traj.draw_trajectory_at_x(0.1, 1000, 0.9, trials = 1, t_max = .1)
{'ball_hit': Array([False], dtype=bool), 'roof_hit': Array([False], dtype=bool), 'something_hit': Array([False], dtype=bool), 'trajectories': Array([[[ 1.01513535e-01,  1.41892245e-03, -4.98940468e+00],
        [ 1.03897616e-01,  9.31997492e-04, -4.97806931e+00],
        [ 1.04288198e-01, -1.80142978e-03, -4.97903204e+00],
        [ 9.97019112e-02, -3.12771578e-03, -4.96664095e+00],
        [ 1.02413967e-01, -2.95094796e-03, -4.96111631e+00],
        [ 1.01300426e-01,  6.96127070e-04, -4.94909143e+00],
        [ 1.03424884e-01, -6.30148593e-03, -4.94255972e+00],
        [ 1.03292465e-01, -5.54783177e-03, -4.93507099e+00],
        [ 9.94927734e-02, -4.31211060e-03, -4.92812681e+00],
        [ 9.83759165e-02, -7.30469078e-03, -4.92086792e+00]]],      dtype=float32)}
>>> traj.draw_trajectory_at_x(0.1, 1000, 0.9, trials = 2, t_max = .05)
{'ball_hit': Array([False, False], dtype=bool), 'roof_hit': Array([False, False], dtype=bool), 'something_hit': Array([False, False], dtype=bool), 'trajectories': Array([[[ 9.8736316e-02, -7.5448438e-04, -4.9963560e+00],
        [ 9.6689783e-02, -4.3136943e-03, -4.9961877e+00],
        [ 9.7639665e-02, -5.3856885e-03, -4.9852118e+00],
        [ 9.8702230e-02, -7.6030511e-03, -4.9785647e+00],
        [ 9.1684863e-02,  7.3614251e-04, -4.9669151e+00]],[[ 1.0136999e-01, -1.1048245e-02, -4.9944925e+00],
        [ 1.0074968e-01, -3.2547791e-03, -4.9900651e+00],
        [ 1.0723757e-01, -6.1367834e-03, -4.9799914e+00],
        [ 1.1393108e-01, -1.7973720e-03, -4.9786887e+00],
        [ 1.1445464e-01, -7.9164971e-03, -4.9739923e+00]]], dtype=float32)}
pypesh.trajectories.hitting_propability_at_x(x_position, peclet, ball_radius, trials=100, floor_h=5, partition=1, t_max=40)

Generate trajectories of particles in a simulation for certain x position. Than calculates the propability of hitting for this position.

Parameters

  • x_postition (float) – Radius where probability is evaluated (simulation initiation point)

  • peclet (float) – Peclet number defined as R u / D.

  • ball_radius (float) – Radius of the big ball.

  • trials (int, optional) – Number of trajectories.

  • floor_h (int, optional) – Initial depth for simulation.

  • parition (int, optional) – Default 1, if to expensive in RAM partition into ‘partition’ parts.

  • t_max (float, optional) – Default 40.0, select time of simmulation

Returns

value of propability

Return type

float

Example

>>> import pypesh.trajectories as traj
>>> traj.hitting_propability_at_x(0.1, 10**4, 0.9)
Array(0.82, dtype=float32, weak_type=True)
pypesh.trajectories.sherwood_trajectories(peclet, ball_radius, mesh_out=4, mesh_jump=6, trials=100, floor_h=5, spread=4, t_max=40.0, partition=1)

Calculates the distribution of probability of hitting as a function of radius, at depth floor_h. Addaptive sampling is implemented to ensure effective calculation. Position of greatest slope is assumed to be at streamline that pass position [1, 0, 0], then spread is scaled as sqrt(1/peclet) sim sqrt(D). Then integrates with weigth and finds the Sherwood number

Parameters

  • peclet (float, optional) – Peclet number defined as R u / D.

  • ball_radius (float) – Radius of the big ball.

  • mesh_out (int, optional) – Amount of samples outside the region of highest slope

  • mesh_jump (int, optional) – Amount of samples in the region of highest slope

  • trials (int, optional) – Number of trajectories per position, uncertainty of propability estimation is sqrt(trials)/trials.

  • floor_h (int, optional) – Initial depth for simulation

  • spread (int, optional) – How far in sqrt(1/peclet), mesh_out will reach

  • parition (int, optional) – Default 1, if to expensive in RAM partition into ‘partition’ parts.

  • t_max (float, optional) – Default 40.0, select time of simmulation

Returns

float - estimation of resulting sherwood number (flux/advective_flux), array - Radius samples, array - Probability values ordered as radius samples

Return type

tuple

Example

>>> import pypesh.trajectories as traj
>>> traj.sherwood_trajectories(10**6, 0.9)
(3646.3631062456793, array([0.10064565, 0.11064565, 0.12064565, 0.13064565, 0.13464565,
   0.13864565, 0.14264565, 0.14664565, 0.15064565, 0.16064565,
   0.17064565, 0.18064565]), array([1.        , 1.        , 0.98999995, 0.95      , 0.83      ,
   0.68      , 0.45      , 0.17999999, 0.02      , 0.        ,
   0.        , 0.        ], dtype=float32))
pypesh.trajectories.simulate_trajectory(drift, noise, initial, t_max, whole_trajectory=False, seed=0)

Simulates trajectories starting from initial conditions, affected by noise and moved via drift.

Parameters

  • derift (callable) – Function describing drift term of the equation, should return np.array of length dimension.

  • noise (callable) – Function describing noise term of the equation, should return np.array of size (dimension,noiseterms).

  • initial (jnp.array) – Array of positions where to start simulating

  • t_max (float) – Time of calculation

  • whole_trajectory (boole, optional) – Deafult False, if True than also returns whole trajectory (Warning expensive in memory)

  • seed (int) – Default 0, if diferent than 0, different random seed is used for SDE.

Returns

ball_hit - jnp.array() of 1 and 0, if 1 trajectory within radius 1, 0 miss roof_hit - jnp.array() of 1 and 0, if 1 trajectory at the end above height 2, 0 miss something_hit - jnp.array() of 1 and 0, union of ball_hit and roof_hit

Return type

dict

Example

>>> import pypesh.trajectories as traj
>>> import pypesh.stokes_flow as sf
>>> def drift(q):
...     return sf.stokes_around_sphere_jnp(q, 0.9)
>>> noise = traj.diffusion_function(100)
>>> init = traj._construct_initial_trials_at_x(0, 5, 3)
>>> traj.simulate_trajectory(drift, noise, init, 20)
{'ball_hit': Array([False,  True,  True], dtype=bool), 'roof_hit': Array([ True,  True,  True], dtype=bool), 'something_hit': Array([ True,  True,  True], dtype=bool)}

Visualisation

Visualisation of different results from code.

pypesh.visualisation.draw_cross_section_fem(peclet, ball_radius, downstream_distance=5, show=False, density=400, maximal_radius=1)

Draws cross section of hitting probability at selected height using scikit-fem.

Parameters

  • peclet (float) – Peclet number defined as R u / D.

  • ball_radius (float) – Radius of the big ball.

  • downstream_distance (float, optional) – Default 5, downstream distance in ball radius where distribution is plotted

  • show (bool, optional) – Default False, if True plot is shown

  • denstity (int, optional) – Default 400, how many points in result

  • maximal_radius (float, optional) – Default 1, maximal radius to place in result

Returns

density by 2 array with location probability pairs.

Return type

np.array

Example

>>> import pypesh.visualisation as visual
>>> visual.draw_cross_section_fem(1000, 0.9, density = 10)
array([[ 0.00000000e+00,  7.96836968e-01],
    [ 1.11111111e-01,  6.95108509e-01],
    [ 2.22222222e-01,  4.52912665e-01],
    [ 3.33333333e-01,  2.11422004e-01],
    [ 4.44444444e-01,  6.74661539e-02],
    [ 5.55555556e-01,  1.42087363e-02],
    [ 6.66666667e-01,  1.95186128e-03],
    [ 7.77777778e-01,  1.57053994e-04],
    [ 8.88888889e-01,  5.67654858e-06],
    [ 1.00000000e+00, -2.40005303e-07]])
pypesh.visualisation.draw_cross_section_traj(peclet, ball_radius, downstream_distance=5, show=False, mesh_out=4, mesh_jump=6, spread=4, trials=200, partition=1, t_max=40)

Draws cross section of hitting probability at selected height using pychastic.

Parameters

  • peclet (float) – Peclet number defined as R u / D.

  • ball_radius (float) – Radius of the big ball.

  • downstream_distance (float, optional) – Default 5, downstream distance in ball radius where distribution is plotted

  • show (bool, optional) – Default False, if True plot is shown

  • denstity (int, optional) – Default 10, how many points in result

  • maximal_radius (float, optional) – Default 1, maximal radius to place in result

  • mesh_out (int, optional) – Amount of samples outside the region of highest slope

  • mesh_jump (int, optional) – Amount of samples in the region of highest slope

  • spread (int, optional) – How far in sqrt(1/peclet), mesh_out will reach

  • trials (int, optional) – Default 200, Number of trajectories.

  • parition (int, optional) – Default 1, if to expensive in RAM partition into ‘partition’ parts.

  • t_max (float, optional) – Default 40.0, select time of simmulation

Returns

density by 2 array with location probability pairs.

Return type

np.array

Example

>>> import pypesh.visualisation as visual
>>> visual.draw_cross_section_traj(1000, 0.9)
array([[0.        , 0.77499998],
   [0.09137468, 0.67000002],
   [0.18274937, 0.47999999],
   [0.27412405, 0.315     ],
   [0.36549873, 0.155     ],
   [0.45687341, 0.04      ],
   [0.77310118, 0.        ],
   [1.08932895, 0.        ],
   [1.40555671, 0.        ]])
pypesh.visualisation.draw_distributions_fem(peclet, ball_radius, limits=[-2.5, 2.5, -2.5, 5], draw_streamline=False)

Draws distribution of concentration

Parameters

  • peclet (float) – Peclet number defined as R u / D.

  • ball_radius (float) – Radius of the big ball.

  • limits (list) – minimal radius, maximal radius, minimal height, maximal height

  • draw_streamline (boole, optional) – Default False, if True adds a streamline for pe -> infty

Return type

None

Example

>>> import pypesh.visualisation as visual
>>> visual.draw_distributions_fem(1000, 0.9)
pypesh.visualisation.visualise_trajectories(peclet, ball_radius, positions, t_max=20, downstream_distance=5, show=False)

Draws trajectories simulated by pychastic.

Parameters

  • peclet (float) – Peclet number defined as R u / D.

  • ball_radius (float) – Radius of the big ball.

  • positions (dict) – Keys: position where simulate, Values: how many times

  • t_max (float, optional) – Default 20, time of simulation

  • downstream_distance (float, optional) – Default 5, downstream distance in ball radius where distribution is plotted

  • show (bool, optional) – Default False, if True plot is shown

Returns

list of outcomes from pychastic for each position

Return type

list

Example

>>> visual.visualise_trajectories(1000, 0.9, {0.1: 2, 0.2: 1}, t_max=0.05)
[{'ball_hit': Array([False, False], dtype=bool), 'roof_hit': Array([False, False], dtype=bool), 'something_hit': Array([False, False], dtype=bool), 'trajectories': Array([[[ 9.74556133e-02,  3.57697834e-03, -4.99383068e+00],
    [ 1.03875704e-01,  1.23539870e-03, -4.99430752e+00],
    [ 1.00019522e-01,  2.17465451e-03, -4.99006510e+00],
    [ 9.64764059e-02,  3.37144663e-03, -4.98079920e+00],
    [ 9.23867375e-02,  9.56971012e-03, -4.97192192e+00]],
   [[ 1.00915655e-01, -3.44433682e-03, -4.99829912e+00],
    [ 9.91058275e-02, -7.94267934e-03, -4.98968840e+00],
    [ 1.03579685e-01, -5.71627077e-03, -4.98568869e+00],
    [ 1.08699612e-01, -5.76060778e-03, -4.97635365e+00],
    [ 1.07397184e-01, -8.04143026e-03, -4.96779823e+00]]],      dtype=float32)}, {'ball_hit': Array([False], dtype=bool), 'roof_hit': Array([False], dtype=bool), 'something_hit': Array([False], dtype=bool), 'trajectories': Array([[[ 1.9748163e-01,  3.5769783e-03, -4.9938278e+00],
    [ 2.0392781e-01,  1.2353971e-03, -4.9943018e+00],
    [ 2.0009771e-01,  2.1746524e-03, -4.9900560e+00],
    [ 1.9658074e-01,  3.3714436e-03, -4.9807873e+00],
    [ 1.9251731e-01,  9.5697045e-03, -4.9719067e+00]]], dtype=float32)}]
pypesh.visualisation.visualise_trajectories_with_streamplot(peclet, ball_radius, positions, t_max=20, downstream_distance=5, show=False, save='no')

Draws trajectories simulated by pychastic.

Parameters

  • peclet (float) – Peclet number defined as R u / D.

  • ball_radius (float) – Radius of the big ball.

  • positions (dict) – Keys: position where simulate, Values: how many times

  • t_max (float, optional) – Default 20, time of simulation

  • downstream_distance (float, optional) – Default 5, downstream distance in ball radius where distribution is plotted

  • show (bool, optional) – Default False, if True plot is shown

Returns

list of outcomes from pychastic for each position

Return type

list

Example

>>> visual.visualise_trajectories_with_streamplot(1000, 0.9, {0.1: 2, 0.2: 1}, t_max=0.05)
[{'ball_hit': Array([False, False], dtype=bool), 'roof_hit': Array([False, False], dtype=bool), 'something_hit': Array([False, False], dtype=bool), 'trajectories': Array([[[ 9.74556133e-02,  3.57697834e-03, -4.99383068e+00],
    [ 1.03875704e-01,  1.23539870e-03, -4.99430752e+00],
    [ 1.00019522e-01,  2.17465451e-03, -4.99006510e+00],
    [ 9.64764059e-02,  3.37144663e-03, -4.98079920e+00],
    [ 9.23867375e-02,  9.56971012e-03, -4.97192192e+00]],
   [[ 1.00915655e-01, -3.44433682e-03, -4.99829912e+00],
    [ 9.91058275e-02, -7.94267934e-03, -4.98968840e+00],
    [ 1.03579685e-01, -5.71627077e-03, -4.98568869e+00],
    [ 1.08699612e-01, -5.76060778e-03, -4.97635365e+00],
    [ 1.07397184e-01, -8.04143026e-03, -4.96779823e+00]]],      dtype=float32)}, {'ball_hit': Array([False], dtype=bool), 'roof_hit': Array([False], dtype=bool), 'something_hit': Array([False], dtype=bool), 'trajectories': Array([[[ 1.9748163e-01,  3.5769783e-03, -4.9938278e+00],
    [ 2.0392781e-01,  1.2353971e-03, -4.9943018e+00],
    [ 2.0009771e-01,  2.1746524e-03, -4.9900560e+00],
    [ 1.9658074e-01,  3.3714436e-03, -4.9807873e+00],
    [ 1.9251731e-01,  9.5697045e-03, -4.9719067e+00]]], dtype=float32)}]

Pe vs Sh

Final package pesh that calculates the value of \(Sh\) for given Peclet and radius of rigid ball.

pypesh.pesh.all_sherwood(peclet, ball_radius, mesh_out=4, mesh_jump=6, trials=100, floor_h=5, spread=4, t_max=40.0, partition=1)

Calculates the sherwood number using clift approximation, fem approach and trajectories approach.

Parameters

  • peclet (float, optional) – Peclet number defined as R u / D.

  • ball_radius (float) – Radius of the big ball.

  • mesh_out (int, optional) – For trajectories, amount of samples outside the region of highest slope

  • mesh_jump (int, optional) – For trajectories, amount of samples in the region of highest slope

  • trials (int, optional) – For trajectories, number of trajectories per position, uncertainty of propability estimation is sqrt(trials)/trials.

  • floor_h (int, optional) – For trajectories, initial depth for simulation

  • spread (int, optional) – For trajectories, how far in sqrt(1/peclet), mesh_out will reach

Returns

float - Clift et. al., float - our approximation, float - fem approach, float - fem approach different integral , float - trajectories approach

Return type

tuple

Example

>>> import pypesh.pesh as pesh
>>> pesh.all_sherwood(1000, 0.9)
(6.800655008742168, 10.425655008742165, np.float64(12.033892568100546), np.float64(11.720084145681978), 0)
pypesh.pesh.sherwood_heatmap()

Interpolate the sherwood number for given peclet and ball_radius and returns a function.

Returns

Gives sherwood number for given peclet and ball_radius

Return type

function

Example

>>> import pypesh.pesh as pesh
>>> import numpy as np
>>> interp = pesh.sherwood_heatmap()
>>> to_call = np.array([1000,0.9])
>>> interp(to_call)
array([12.03389257])