Boundary Conditions¶
For all boundaries, it is necessary to provide Nek5000 with the boundary condition type and, for some types, the boundary condition value.
Boundary condition values are assigned in the .usr file in the userbc subroutine (see userbc for details).
This section focuses on what boundary condition types are available in Nek5000 and how to assign them.
We first discuss how Nek5000 stores the boundary condition information in the code itself.
Boundary condition types are stored as a character identifier code on every face of every element for every solution field.
These are stored in memory in the cbc array, which is of type character*3.
It is declared in a common block in Nek5000/core/INPUT as cbc(6,lelt,0:ldimt1), which is then included as part of TOTAL in the user subroutines in the .usr file as well as any relevant source code files.
The cbc array is set at runtime either from information carried directly in the mesh file (.rea/.re2), from information provided in the .par file, or (for advanced users) directly in usrdat or usrdat2.
The general conventions for boundary condition identifiers are:
- uppercase letters correspond to boundary conditions which do not require any additional user input.
- lowercase letters correspond to user defined boundary conditions which require values to be set in the
userbcsubroutine in the user file. - lowercase letters ending with
l, e.g.vl, are specified in face-local coordinates, i.e. normal, tangent and bitangent directions. - lowercase letters including an
h, e.g.sh, are a “horizontal” variant where the normal velocity component is zero. This can be combined withlto produce a “horizontal, local” variant.
The available boundary condition types, along with the identifier codes, are described in the following sections for the fluid velocity and pressure, and the temperature and passive scalars.
Fluid Velocity and Pressure¶
Two kinds of boundary conditions are applicable to the fluid velocity: essential (Dirichlet) boundary conditions in which the velocity is specified, and natural (Neumann) boundary conditions in which the traction is specified. For segments that constitute the boundary \(\partial \Omega_f\), see Fig. 31, one of these two types of boundary conditions must be assigned to each component of the fluid velocity.
The fluid boundary condition can be all Dirichlet if all velocity components of \({\bf u}\) are specified, or it can be all Neumann if all three traction components (\(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\)) are specified on the boundary face. Where \({\bf \hat e_n}\) is the unit vector normal to the boundary face. It can also be mixed Dirichlet/Neumann if Dirichlet and Neumann conditions are selected for different velocity components. Neumann boundary conditions for velocity are assigned differently depending on if the no-stress or full-stress formulation is used. If the no-stress formulation is selected, then traction is not defined on the boundary, rather the individual components of the velocity gradient are defined. In this case, any Neumann boundary condition imposed must be homogeneous, i.e. equal to zero, and mixed Dirichlet/Neumann boundaries must be aligned with one of the Cartesian axes. These conditions are not required for the full-stress formulation, which assigns the components of the stress tensor directly.
In general, the boundary condition for pressure satisfies the following equation, unless explicitly specified in the sections below.
\[\nabla \cdot \frac{1}{\rho}\nabla p = -\nabla \cdot \frac{D \bf u}{D t} +\nabla \cdot \frac{1}{\rho}\left(\nabla \cdot \boldsymbol{\underline \tau}\right) + \nabla \cdot \bf f\]
where the stress tensor is given as
\[\boldsymbol{\underline \tau} \equiv \mu\left[\nabla {\bf u} + \left(\nabla {\bf u}\right)^T\right]\]
Inlet (Dirichlet), v¶
Standard Dirichlet boundary condition for velocity.
\[\begin{split}{\bf u} \cdot {\bf \hat e_x} &= u_x\\ {\bf u} \cdot {\bf \hat e_y} &= u_y\\ {\bf u} \cdot {\bf \hat e_z} &= u_z\end{split}\]
Where, \({\bf \hat e_n}\) is the unit vector normal to the boundary face, \({\bf \hat e_x}\), \({\bf \hat e_y}\), and \({\bf \hat e_z}\) are unit vectors aligned with the Cartesian axes and \(u_x\), \(u_y\), and \(u_z\) are set in the userbc subroutine in the user file.
Inlet (Dirichlet) - local vl¶
Standard Dirichlet boundary condition for velocity in local coordinates.
Where, \({\bf \hat e_n}\), \({\bf \hat e_t}\), and \({\bf \hat e_b}\) are the normal, tangent, and bitangent unit vectors on the boundary face, and \(u_n\), \(u_1\), and \(u_2\) are set in the userbc subroutine in the user file.
Outlet, O¶
The open (outflow) boundary condition arises as a natural boundary condition from the variational formulation of Navier Stokes.
| no-stress | full-stress |
|---|---|
| \(\nabla {\bf u} = 0\) | \(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n} = 0\) |
Where \({\bf \hat e_n}\) is the unit vector normal to the boundary face.
The userbc subroutine is not called for this boundary condition type.
Pressure outlet, o¶
Similar to a standard outlet, but with a specified pressure.
| no-stress | full-stress |
|---|---|
| \(\nabla {\bf u} = 0\) | \(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n} = 0\) |
Where \({\bf \hat e_n}\) is the unit vector normal to the boundary face and \(p_a\) is set in the userbc subroutine in the user file.
Outlet - normal, ON¶
Open boundary with zero velocity in the tangent and bitangent directions.
| no-stress | full-stress |
|---|---|
| \(\nabla {\bf u}\cdot{\bf \hat e_n} = 0\) | \(\left(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\right) \cdot {\bf \hat e_n} = 0\) |
| \({\bf u} \cdot {\bf \hat e_t} = 0\) | \({\bf u} \cdot {\bf \hat e_t} = 0\) |
| \({\bf u} \cdot {\bf \hat e_b} = 0\) | \({\bf u} \cdot {\bf \hat e_b} = 0\) |
Where \({\bf \hat e_n}\), \({\bf \hat e_t}\), and \({\bf \hat e_b}\) are the normal, tangent, and bitangent unit vectors on the boundary face.
If the surface normal vector is not aligned with a principal Cartesian axis, the full-stress formulation must be used.
The userbc subroutine is not called for this boundary condition type.
Pressure outlet - normal, on¶
Similar to an outlet - normal boundary, but with a specified pressure.
| no-stress | full-stress |
|---|---|
| \(\nabla {\bf u}\cdot{\bf \hat e_n} = 0\) | \(\left(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\right) \cdot {\bf \hat e_n} = 0\) |
| \({\bf u} \cdot {\bf \hat e_t} = 0\) | \({\bf u} \cdot {\bf \hat e_t} = 0\) |
| \({\bf u} \cdot {\bf \hat e_b} = 0\) | \({\bf u} \cdot {\bf \hat e_b} = 0\) |
Where \({\bf \hat e_n}\), \({\bf \hat e_t}\), and \({\bf \hat e_b}\) are the normal, tangent, and bitangent unit vectors on the boundary face, and \(p_a\) is set in the userbc subroutine in the user file.
If the surface normal vector is not aligned with a principal Cartesian axis, the full-stress formulation must be used.
Periodic, P¶
Where possible, one can effect great computational efficiencies by considering the problem in a single geometric unit and requiring periodicity of the field variables.
Where \(\boldsymbol{\delta}{\bf x}\) is the offset vector between two periodic faces.
The userbc subroutine is not called for this boundary condition type.
Periodic boundaries are a special case where the boundary condition is enforced on the mesh connectivity level.
To use periodic boundary conditions, the surface meshes must be conformal.
For third-party meshes they must also have a corresponding pair of boundary ID values which need to be provided during conversion, i.e. to exo2nek, gmsh2nek, or cgns2nek.
Additionally, the mesh must be at least 3 elements thick in the direction normal to the periodic boundaries.
Symmetry, SYM¶
Symmetric face or a slip wall.
| no-stress | full-stress |
|---|---|
| \({\bf u} \cdot {\bf \hat e_n} = 0\) | \({\bf u} \cdot {\bf \hat e_n} = 0\) |
| \(\nabla{\bf u}\cdot {\bf \hat e_t} = 0\) | \(\left(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\right)\cdot {\bf \hat e_t} = 0\) |
| \(\nabla{\bf u}\cdot {\bf \hat e_b} = 0\) | \(\left(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\right)\cdot {\bf \hat e_b} = 0\) |
Where \({\bf \hat e_n}\), \({\bf \hat e_t}\), and \({\bf \hat e_b}\) are the normal, tangent, and bitangent unit vectors on the boundary face.
If the surface normal vector is not aligned with a principal Cartesian axis, the full-stress formulation must be used.
The userbc subroutine is not called for this boundary condition type.
Traction, s¶
Full Neumann boundary conditions for velocity.
Where \({\bf \hat e_n}\) is the unit vector normal to the boundary face, \({\bf \hat e_x}\), \({\bf \hat e_y}\), and \({\bf \hat e_z}\) are unit vectors aligned with the Cartesian axes and \(tr_x\), \(tr_y\), and \(tr_z\) are set in the userbc subroutine in the user file.
The full-stress formulation must be used for this boundary type.
Traction - local, sl¶
Similar to traction, but in local coordinates.
\[\begin{split}p &= 0\\ \left(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\right)\cdot {\bf \hat e_n} &= tr_n\\ \left(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\right)\cdot {\bf \hat e_t} &= tr_1\\ \left(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\right)\cdot {\bf \hat e_b} &= tr_2\end{split}\]
Where \({\bf \hat e_n}\), \({\bf \hat e_t}\), and \({\bf \hat e_b}\) are the normal, tangent, and bitangent unit vectors on the boundary face, and \(tr_n\), \(tr_1\), and \(tr_2\) are set in the userbc subroutine in the user file.
The full-stress formulation must be used for this boundary type.
Traction - horizontal, sh¶
Similar to symmetry, but with specified non-zero traction in the tangent and bitangent directions given in Cartesian coordinates
\[\begin{split}{\bf u} \cdot {\bf \hat e_n} &= 0\\ \left(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\right)\cdot {\bf \hat e_x} &= tr_x\\ \left(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\right)\cdot {\bf \hat e_y} &= tr_y\\ \left(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\right)\cdot {\bf \hat e_z} &= tr_z\end{split}\]
Where \({\bf \hat e_n}\) is the unit vector normal to the boundary face, \({\bf \hat e_x}\), \({\bf \hat e_y}\), and \({\bf \hat e_z}\) are unit vectors aligned with the Cartesian axes and \(tr_x\), \(tr_y\), and \(tr_z\) are set in the userbc subroutine in the user file.
The full-stress formulation must be used for this boundary type.
Traction - horizontal, local, shl¶
Similar to symmetry, but with specified non-zero traction in the tangent and bitangent directions.
\[\begin{split}{\bf u} \cdot {\bf \hat e_n} &= 0\\ \left(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\right)\cdot {\bf \hat e_t} &= tr_1\\ \left(\boldsymbol{\underline \tau} \cdot {\bf \hat e_n}\right)\cdot {\bf \hat e_b} &= tr_2\end{split}\]
Where, \({\bf \hat e_n}\), \({\bf \hat e_t}\), and \({\bf \hat e_b}\) are the normal, tangent, and bitangent unit vectors on the boundary face, and \(tr_1\) and \(tr_2\) are set in the userbc subroutine in the user file.
The full-stress formulation must be used for this boundary type.
Wall, W¶
Dirichlet boundary condition corresponding to a no-slip wall.
\[\bf u = 0\]
The userbc subroutine is not called for this boundary condition type.
Other BCs¶
| Identifier | Description | Type | Note |
|---|---|---|---|
A |
Axisymmetric boundary | Mixed | Can only be used on face 1, treated as SYM, see below |
E |
Interior boundary | – | Denotes faces that connect adjacent elements |
' ' |
Empty | – | Treated as an interior boundary |
int |
Interpolated (NEKNEK) | Dirichlet | Interpolated from the adjacent overset mesh, see: Overlapping Overset Grids |
p |
Periodic | – | For periodicity within a single element |
mm |
Moving mesh | – | – |
ms |
Moving surface | – | – |
msi |
Moving internal surface | – | – |
mv |
Moving boundary | Dirichlet | – |
mvn |
Moving boundary, normal | Dirichlet | Zero velocity in non-normal directions |
For an axisymmetric flow geometry, the axis boundary condition (A) is provided for boundary segments that lie entirely on the axis of symmetry.
This is essentially a symmetry (mixed Dirichlet/Neumann) boundary condition in which the normal velocity and the tangential traction are set to zero.
This requires a 2D mesh where the x-axis is the axis of rotation.
Temperature and Passive Scalars¶
The three types of boundary conditions applicable to the temperature are: essential (Dirichlet) boundary condition in which the temperature is specified; natural (Neumann) boundary condition in which the heat flux is specified; and mixed (Robin) boundary condition in which the heat flux is dependent on the temperature on the boundary. For segments that constitute the boundary \(\partial \Omega_f' \cup \partial \Omega_s'\) (refer to Fig. 2.1), one of the above three types of boundary conditions must be assigned to the temperature.
The two types of Robin boundary condition for temperature are: convection boundary conditions for which the heat flux into the domain depends on the heat transfer coefficient \(h_{c}\) and the difference between the environmental temperature \(T_{\infty}\) and the surface temperature; and radiation boundary conditions for which the heat flux into the domain depends on the Stefan-Boltzmann constant/view-factor product \(h_{rad}\) and the difference between the fourth power of the environmental temperature \(T_{\infty}\) and the fourth power of the surface temperature.
The boundary conditions for the passive scalar fields are analogous to those used for the temperature field. Thus, the temperature boundary conditions and character identifier codes are identical for the passive scalar fields. The user can specify an independent set of boundary conditions for each passive scalar field.
Specified value (Dirichlet), t¶
Standard Dirichlet boundary condition for temperature and passive scalars. Used for inlets, isothermal walls, etc.
Where \(temp\) is set in the userbc subroutine in the user file.
Flux (Neumann), f¶
Standard heat flux boundary condition.
Where \({\bf \hat e_n}\) is the unit vector normal to the boundary face and \(flux\) is set in the userbc subroutine in the user file.
Insulated, I¶
Zero-Neumann boundary condition. Used for insulated walls, outlets, symmetry planes, etc.
Where \({\bf \hat e_n}\) is the unit vector normal to the boundary face.
The userbc subroutine is not called for this boundary condition type.
Newton cooling (convection), c¶
Robin boundary condition for a surface exposed to a fluid at given temperature and heat transfer coefficient.
Where \({\bf \hat e_n}\) is the unit vector normal to the boundary face, \(h_c\) is the convective heat transfer coefficient, and \(T_{\infty}\) is the ambient temperature.
The convective heat transfer coefficient and ambient temperature are set in the userbc subroutine in the user file.
Periodic, P¶
Periodic boundary conditions require that all fields in the simulation are periodic.
Where \(\boldsymbol{\delta}{\bf x}\) is the offset vector between two periodic faces.
The userbc subroutine is not called for this boundary condition type.
See the fluid velocity and pressure periodic boundary condition for more information.
Radiative cooling, r¶
Robin boundary condition for a surface where radiation heat transfer is significant.
Where \({\bf \hat e_n}\) is the unit vector normal to the boundary face, \(h_{rad}\) is the radiative heat transfer coefficient, and \(T_{\infty}\) is the ambient temperature.
The radiative heat transfer coefficient and ambient temperature are set in the userbc subroutine in the user file.
Other BCs¶
| Identifier | Description | Type | Note |
|---|---|---|---|
A |
Axisymmetric boundary | – | treated as I |
E |
Interior boundary | – | – |
' ' |
Empty | – | Treated as an interior boundary |
int |
Interpolated (NEKNEK) | Dirichlet | Interpolated from the adjacent overset mesh, see: Overlapping Overset Grids |
O |
Outflow | Neumann | Identical to I |
p |
Periodic | – | For periodicity within a single element |
SYM |
Symmetry | Neumann | Identical to I |
Setting Boundary Conditions Types¶
Assigning boundary condition types in Nek5000 is handled differently depending on if you are using a third-party meshing tool such as Gmsh, ICEM, Cubit, etc. and importing the mesh with exo2nek, gmsh2nek, or cgns2nek, or if you are using a Nek-native tool such as preNek or genbox (see Genbox).
In either case, the boundary condition types are set by assigning the corresponding character identifier code in the character boundary condition array, cbc.
The character boundary condition array itself is described here and the supported character codes were described in the sections above for momentum and temperature and passive scalars.
The differences between Nek-native tools and third-party meshing tools are only in how this array gets set.
For Nek-native tools, this array is read directly from the .rea or .re2 file, which is set based on input provided to the tool itself.
For third-party meshing tools, the boundary ID is set in the tool – e.g. as a sideset ID in ICEM – and this information is propagated to the .re2 (mesh) file.
The cbc array is later filled at runtime based on the boundary IDs.
The recommended method of setting the boundary condition type from the boundary ID is through the .par file.
This is done through the boundaryTypeMap key, which is available for the VELOCITY, TEMPERATURE, and SCALARXX directives.
By default, Nek5000 assumes the boundary IDs are sequential and start from 1.
If this is not the case, the optional boundaryIDMap key is available for the MESH directive.
See here for more information on the .par file.
A few simple examples of setting the BC types via the .par file for a mesh with boundary IDs assigned in a third-party mesher are below.
Warning
Setting the boundary condition types in the .par file is NOT supported in V19 or earlier versions.
In the simplest example, the mesh has 4 boundaries each with a sequentially numbered boundary ID.
| Boundary ID | Velocity | Temperature |
|---|---|---|
| 1 | v |
t |
| 2 | O |
I |
| 3 | W |
f |
| 4 | SYM |
I |
To set the boundary condition types, the boundaryTypeMap key is used in the .par file.
The boundaryTypeMap key is a comma-separated list of the boundary condition types to be assigned to the domain and is avaialble for the velocity, temperature and passive scalar fields.
The character identifiers can always be used for assignment.
Additionally, some of the common boundary types can be assigned using plain-English equivalents in the .par file only.
For a list of these see here.
By default, Nek5000 assumes the boundary IDs in your mesh start with 1 and are numbered sequentially.
Due to the sequential ordering of the boundary IDs in this example, these boundary types can be set using only the boundaryTypeMap keys in the VELOCITY and TEMPERATURE directives:
[VELOCITY]
boundaryTypeMap = v, O, W, SYM
[TEMPERATURE]
boundaryTypeMap = t, I, f, I
If your boundary IDs are not sequential or do not start with 1, they can be explicitly declared using the boundaryIDMap key in the MESH directive.
The boundaryIDMap key is a comma-separated list of integers corresponding to the boundary IDs in your mesh.
When using the boundaryIDMap key, Nek5000 makes no assumptions regarding the boundary ID values.
[MESH]
boundaryIDMap = 3, 4, 1, 2
[VELOCITY]
boundaryTypeMap = W, SYM, v, O
[TEMPERATURE]
boundaryTypeMap = f, I, t, I