Computational Fluid Dynamics

Welcome to the Computational Fluid Dynamics(CFD) toolbox. This module explains how to use the Laminar Flow interface to model and simulate fluid mechanics for laminar, incompressible fluids.

The engineering community often uses the term CFD, computational fluid dynamics, to refer to the numerical simulation of fluids.

1. Features

  • 2D and 3D

  • Stokes flows

  • Laminar Incompressible flows

  • Steady and unsteady simulations

  • BDF time schemes up to order 4

  • Moving domain support using Arbitrary Lagrangian Eulerian (ALE) formulation

  • Support high order geometry including in context of moving domain

  • Stabilization pressure and advection dominated using Galerking Least Square

  • Boundary conbditions : no-slip, slip(symmetry), inflow(not necessarily aligned with an axis), pressure, outflow

2. CFD Toolbox

2.1. Models

The CFD Toolbox supports both the incompressible Navier-Stokes and the Stokes equations.

The fluid mechanics model (Navier-Stokes or Stokes) can be selected in json file:

Listing : select fluid model
"Models": { "equations":"Navier-Stokes" }

2.2. Materials

Table 1. Materials properties defined in the heat toolbox (symbols are given without components suffix)
Name Symbol Shape Description

density

rho

scalar

density

dynamic-viscosity

mu

scalar

dynamic viscosity

turbulent-dynamic-viscosity

mu_t

scalar

turbulent dynamic viscosity

consistency-index

mu_k

scalar

consistency index

power-law-index

mu_power_law_n

scalar

power law index

viscosity-min

mu_min

scalar

viscosity-max

mu_max

scalar

viscosity-zero-shear

mu_0

scalar

viscosity-infinite-shear

mu_inf

scalar

carreau-law-lambda

mu_carreau_law_lambda

scalar

carreau-law-n

mu_carreau_law_n

scalar

carreau-yasuda-law-lambda

mu_carreau_yasuda_law_lambda

scalar

carreau-yasuda-law-n

mu_carreau_yasuda_law_n

scalar

carreau-yasuda-law-a

mu_carreau_yasuda_law_a

scalar

2.3. Fields and symbols expressions

Table 2. Fields available
Name Description Shape

velocity

the velocity of fluid

vectorial

pressure

the pressure

scalar
Table 3. Symbols expressions by considering fluid as the keyword used with the toolbox
Symbol Expression Description

fluid_U_0

\(u_0\)

evaluate the component X of fluid velocity

fluid_U_1

\(u_1\)

evaluate the component Y of fluid velocity

fluid_U_2

\(u_2\)

evaluate the component Z of fluid velocity

fluid_U_magnitude

\(\lVert \boldsymbol{u} \rVert = \sqrt{\boldsymbol{u} \cdot \boldsymbol{u}}\)

evaluate the magnitude of fluid velocity

fluid_curl_U

\(\nabla \wedge \boldsymbol{u}\)

evaluate the curl of fluid velocity

fluid_curl_U_magnitude

\(\lVert \nabla \wedge \boldsymbol{u} \rVert\)

evaluate the magnitude of curl of fluid velocity

fluid_div_U

\(\nabla \cdot \boldsymbol{u}\)

evaluate the div of fluid velocity

The next step is to define the fluid material by setting its properties namely the density \(\rho_f\) and viscosity \(\mu_f\). In next table, we find the correspondance between the mathematical names and the json names.

Table 4. Correspondance between fluid parameters and symbols in JSon files
Parameter Symbol

\(\mu_f\)

mu

\(\rho_f\)

rho

A Materials section is introduced in json file in order to configure the fluid properties. For each mesh marker, we can define the material properties associated.

Listing : Materials section
"Materials":
{
    "<name>"
    {
        "markers":"[marker1,marker2]",
        "rho":"1.0e3",
        "mu":"1.0"
    }
}

2.3.1. Generalised Newtonian fluid

The non Newtonian properties are defined in cfg file in fluid section.

The viscosity law is activated by:

Table 5. Viscosity law
option values

viscosity.law

newtonian, power_law, walburn-schneck_law, carreau_law, carreau-yasuda_law

Then, each model are configured with the options reported in the following table:

Viscosity law options unit

power_law

power_law.k

power_law.n

dimensionless

dimensionless

walburn-schneck_law

hematocrit

TPMA

walburn-schneck_law.C1

walburn-schneck_law.C2

walburn-schneck_law.C3

walburn-schneck_law.C4

Percentage

g/l

dimensionless

dimensionless

dimensionless

l/g

carreau_law

viscosity.zero_shear

viscosity.infinite_shear

carreau_law.lambda

carreau_law.n

\(kg.m^{-1}.s^{-1}\)

dimensionless

dimensionless

carreau-yasuda_law

viscosity.zero_shear

viscosity.infinite_shear

carreau-yasuda_law.lambda

carreau-yasuda_law.n

carreau-yasuda_law.a

\(kg/(m \times s)\)

\(kg/(m \times s)\)

dimensionless

dimensionless

dimensionless

2.4. Boundary Conditions

We start by a listing of boundary conditions supported in fluid mechanics model.

2.4.1. Dirichlet on velocity

A Dirichlet condition on velocity field reads:

Dirichlet condition
\[\boldsymbol{u}_f = \boldsymbol{g} \quad \text{ on } \Gamma\]

or only a component of vector \(\boldsymbol{u}_f =(u_f^1,u_f^2 ,u_f^3 )\)

\[u_f^i = g \quad \text{ on } \Gamma\]

Several methods are available to enforce the boundary condition:

  • elimination

  • Nitsche

  • Lagrange multiplier

2.4.2. Dirichlet on pressure

\[\begin{align} p & = g \\ \boldsymbol{u}_f \times {\boldsymbol{ n }} & = \boldsymbol{0} \end{align}\]

2.4.3. Neumann

Table 6. Neumann options
Name Expression

Neumann_scalar

\(\boldsymbol{\sigma}_{f} \boldsymbol{n} = g \ \boldsymbol{n} \)

Neumann_vectorial

\(\boldsymbol{\sigma}_{f} \boldsymbol{n} = \boldsymbol{g} \)

Neumann_tensor2

\(\boldsymbol{\sigma}_{f} \boldsymbol{n} = g \ \boldsymbol{n}\)

2.4.4. Slip

\[\boldsymbol{u}_f \cdot \boldsymbol{ n } = 0\]

2.4.5. Inlet

The boundary condition at inlets allow to define a velocity profile on a set of marked faces \(\Gamma_{\mathrm{inlet}}\) in fluid mesh:

\[\boldsymbol{u}_f = - g \ \boldsymbol{ n } \quad \text{ on } \Gamma_{\mathrm{inlet}}\]

The function \(g\) is computed from flow velocity profiles:

Constant profile
\[\text{Find } g \in C^0(\Gamma) \text{ such that } \\ \begin{eqnarray} g &=& \beta \quad &\text{ in } \Gamma \setminus \partial\Gamma \\ g&=&0 \quad &\text{ on } \partial\Gamma \end{eqnarray}\]
Parabolic profile
\[\text{Find } g \in H^2(\Gamma) \text{ such that : } \\ \begin{eqnarray} \Delta g &=& \beta \quad &\text{ in } \Gamma \\ g&=&0 \quad &\text{ on } \partial\Gamma \end{eqnarray}\]

where \(\beta\) is a constant determined by adding a constraint to the inflow:

velocity_max

\(\max( g ) = \alpha \)

flow_rate

\(\int_\Gamma ( g \ \boldsymbol{n} ) \cdot \boldsymbol{n} = \alpha\)

Table 7. Inlet flow options
Option Value Default value Description

shape

constant,parabolic

select a shape profile for inflow

constraint

velocity_max,flow_rate

give a constraint wich controle velocity

expr

string

symbolic expression of constraint value

2.4.6. Outlet flow

Table 8. Outlet flow options
Option Value Default value Description

model

free,windkessel

free

select an outlet modeling
Free outflow
\[\boldsymbol{\sigma}_{f} \boldsymbol{ n } = \boldsymbol{0}\]
Windkessel model

We use a 3-element Windkessel model for modeling an outflow boundary condition. We define \(P_l\) a pressure and \(Q_l\) the flow rate. The outflow model is discribed by the following system of differential equations:

\[\left\{ \begin{aligned} C_{d,l} \frac{\partial \pi_l}{\partial t} + \frac{\pi_l}{R_{d,l}} = Q_l \\ P_l = R_{p,l} Q_l + \pi_l \end{aligned} \right.\]

Coefficients \(R_{p,l}\) and \(R_{d,l}\) represent respectively the proximal and distal resistance. The constant \(C_{d,l}\) is the capacitance of blood vessel. The unknowns \(P_l\) and \(\pi_l\) are called proximal pressure and distal pressure. Then we define the coupling between this outflow model and the fluid model by these two relationships:

\[\begin{align} Q_l &= \int_{\Gamma_l} \boldsymbol{u}_f \cdot \boldsymbol{ n }_f \\ \boldsymbol{\sigma}_f \boldsymbol{ n }_f &= -P_l \boldsymbol{ n }_f \end{align}\]
Table 9. Windkessel options
Option Value Description

windkessel_coupling

explicit, implicit

coupling type with the Navier-Stokes equation

windkessel_Rd

real

distal resistance

windkessel_Rp

real

proximal resistance

windkessel_Cd

real

capacitance

2.4.7. Implementation of boundary conditions in json

Boundary conditions are set in the json files in the category BoundaryConditions.

Then <field> and <bc_type> are chosen from type of boundary condition.

The parameter <marker> corresponds to mesh marker where the boundary condition is applied.

Finally, we define some specific options inside a marker.

Listing : boundary conditions in json
"BoundaryConditions":
{
    "<field>":
    {
        "<bc_type>":
        {
            "<marker>":
            {
                "<option1>":"<value1>",
                "<option2>":"<value2>",
                // ...
            }
        }
    }
}

2.4.8. Options summary

Table 10. Boundary conditions
Field Name Option Entity

velocity

Dirichlet

expr

type

number

alemesh_bc

faces, edges, points

velocity_x

velocity_y

velocity_z

Dirichlet

expr

type

number

alemesh_bc

faces, edges, points

velocity

Neumann_scalar

expr

number

alemesh_bc

faces

velocity

Neumann_vectorial

expr

number

alemesh_bc

faces

velocity

Neumann_tensor2

expr

number

alemesh_bc

faces

velocity

slip

alemesh_bc

faces

pressure

Dirichlet

expr

number

alemesh_bc

faces

fluid

outlet

number

alemesh_bc

model

windkessel_coupling

windkessel_Rd

windkessel_Rp

windkessel_Cd

faces

fluid

inlet

expr

shape

constraint

number

alemesh_bc

faces

2.5. Body forces

Body forces are also defined in the fluid section of BoundaryConditions in the json file.

"BoundaryConditions":
{
    "fluid":{
        "VolumicForces":
        {
            "<marker>":
            {
                "expr":"{0,0,-gravityCst*7850}:gravityCst"
            }
        }
    }
}
The marker corresponds to mesh elements marked with this tag. If the marker is an empty string, it corresponds to all elements of the mesh.

2.6. Post Processing

"PostProcess":
{
    "Exports":
    {
        "fields":["field1","field2",...]
    },
    "Measures":
    {
        "<measure type>":
        {
            "label":
            {
                "<range type>":"value",
                "fields":["field1","field3"]
            }
        }
    }
}

2.6.1. Exports for vizualisation

The fields allowed to be exported in the fields section are:

  • velocity

  • pressure

  • displacement

  • vorticity

  • stress or normal-stress

  • wall-shear-stress

  • density

  • viscosity

  • pid

  • alemesh

2.6.2. Measures

  • Points

  • Force

  • FlowRate

  • Pressure

  • VelocityDivergence

The following fluid variables are available

Name

Description

fluid_U_magnitude

magnitude of the velocity

fluid_Ux

x component of the velocity

fluid_Uy

y component of the velocity

fluid_Uz

z component of the velocity

fluid_P

pressure

You can use these expressions in Statistics

Points

In order to evaluate velocity or pressure at specific points and save the results in .csv file, the user must define:

  • "<tag>" representing this data in the .csv file

  • the coordinate of point

  • the fields evaluated ("pressure" or "velocity")

"Points":
{
  "<tag>":
  {
    "coord":"{0.6,0.2,0}",
    "fields":"pressure"
  },
 "<tag>":
  {
    "coord":"{0.15,0.2,0}",
    "fields":"velocity"
  }
}
Flow rate

The flow rate can be evaluated and save on .csv file. The user must define:

  • "<tag>" representing this data in the .csv file

  • "<face_marker>" representing the name of marked face

  • the fluid direction ("interior_normal" or "exterior_normal") of the flow rate.

"FlowRate":
{
    "<tag>":
    {
        "markers":"<face_marker>",
        "direction":"interior_normal"
    },
    "<tag>":
    {
        "markers":"<face_marker>",
        "direction":"exterior_normal"
    }
}
Forces

compute lift and drag

"Forces":["fsi-wall","fluid-cylinder"]

2.6.3. Export user functions

A function defined by a symbolic expression can be represented as a finite element field thanks to nodal projection. This function can be exported.

"Functions":
{
    "toto":{ "expr":"x*y:x:y"},
    "toto2":{ "expr":"0.5*ubar*x*y:x:y:ubar"},
    "totoV":{ "expr":"{2*x,y}:x:y"}
},
"PostProcess":
{
   "Exports":
   {
       "fields":["velocity","pressure","pid","totoV","toto","toto2"]
   }
}

2.7. Stabilization methods

2.7.1. GLS family

Galerkin leat-Square (GLS) stabilization methods can be activated from the cfg file by adding stabilization-gls=1 in the fluid prefix. Others options available are enumerated in the next table and must be given with the prefix fluid.stabilization-gls.

Table 11. GLS stabilzation options in CFG
Option Value Default value Description

type

gls,supg,gls-no-pspg, supg-pspg, pspg

gls

type of stabilization

parameter.method

eigenvalue,doubly-asymptotic-approximation

eigenvalue

method used to compute the stabilization parameter

parameter.hsize.method

hmin,h,meas

hmin

method used for evalute an element mesh size

parameter.eigenvalue.penal-lambdaK

real

0.

add a mass matrix scaled by this factor in the eigen value problem for the stabilization parameter

convection-diffusion.location.expressions

string

if given, the stabilization is apply only on mesh location which verify expr>0

2.7.2. CIP family

Documentation pending

2.8. Run simulation

The computational fluid dynamics applications available are

  • 2D CFD toolbox : feelpp_toolbox_fluid_2d

  • 3D CFD toolbox : feelpp_toolbox_fluid_3d

Here is an example of execution of the 2D CFD toolbox on 4 processors using the configuration script <myfile.cfg>

mpirun -np 4 feelpp_toolbox_fluid_2d --config-file=<myfile.cfg>