# Boundary Conditions¶

The boundary conditions can be imposed in various ways:

• when the mesh is generated, e.g. with genbox, as is explained in Cylindrical/Cartesian-transition Annuli
• when an .rea file is read in prenek
• translated from side-set numbers in usrdat when using exotonek or similar

The general convention for boundary conditions is

• uppercase letters correspond to primitive boundary conditions, as given in Table 3, Table 5
• lowercase letters correspond to user defined boundary conditions, see Table 4 , Table 6
• lowercase letters ending with l, i.e. 'vl ', are specified in face-local coordinates, i.e. normal, tangent and bitangent directions.

Uppercase boundary conditions which require assigned values in the .rea file are considered legacy and are not recommended for use.

## Fluid Velocity¶

Two types of boundary conditions are applicable to the fluid velocity : essential (Dirichlet) boundary condition in which the velocity is specified and natural (Neumann) boundary condition in which the traction is specified. For segments that constitute the boundary $$\partial \Omega_f$$, see Fig. 17, 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 traction components $${\bf t} = [-p {\bf I} + \mu (\nabla {\bf u} + (\nabla {\bf u})^{T})] \cdot {\bf n}$$, where $${\bf I}$$ is the identity tensor, $${\bf n}$$ is the unit normal and $$\mu$$ is the dynamic viscosity, are specified; or it can be mixed Dirichlet/Neumann if Dirichlet and Neumann conditions are selected for different velocity components. If the nonstress formulation is selected, then traction is not defined on the boundary. In this case, any Neumann boundary condition imposed must be homogeneous, i.e. equal to zero. .. In addition, mixed Dirichlet/Neumann boundaries must be aligned with one of the Cartesian axes. For flow geometry which consists of a periodic repetition of a particular geometric unit, the periodic boundary conditions can be imposed, as illustrated in Fig. 17 . The available primitive boundary conditions for the fluid are given in Table 3 , with the user-specified boundary conditions in Table 4 .

Table 3 Primitive boundary conditions for velocity
Identifier Description Type Note
P Periodic Standard periodic boundary condition
p Periodic For periodicity within a single element
O Outflow Neumann Open boundary condition, zero pressure
ON Outflow, Normal Mixed Zero velocity in non-normal directions
W Wall Dirichlet No slip, $${ \bf{u} = 0}$$
SYM Symmetry Mixed
A Axisymmetric boundary
E Interior boundary
Table 4 User defined boundary conditions for velocity
Identifier Description Type Note
v Velocity Dirichlet Standard velocity boundary condition
vl Velocity, local Dirichlet Face-local coordinates (normal, tangnent, bitangent)
o Outflow Neumann Open boundary condition, specified pressure
on Outflow, Normal Mixed Zero velocity in non-normal directions
s Traction Neumann Specified traction in all directions
sl Traction, local Neumann Face-local coordinates (normal, tangent, bitangent)
sh Traction, horizontal Mixed Specified traction with zero normal velocity
shl Traction, horizontal, local Mixed Zero normal velocity, traction in tangent and bitangent
int Interpolated (NEKNEK) Dirichlet Interpolated from the adjacent overset mesh, see: Overlapping Overset Grids
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

The open(outflow) boundary condition (“O”) arises as a natural boundary condition from the variational formulation of Navier Stokes. We identify two situations

• In the non-stress formulation, open boundary condition (‘Do nothing’)

$[-p{\bf I} + \nu(\nabla {\bf u})]\cdot {\bf n}=0$
• In the stress formulation, free traction boundary condition

$[-p{\bf I} + \nu(\nabla {\bf u}+\nabla {\bf u}^T)]\cdot {\bf n}=0$
• the symmetric boundary condition (“SYM”) is given as

$\begin{split}{\bf u} \cdot {\bf n} &= 0\ ,\\ (\nabla {\bf u} \cdot {\bf t})\cdot {\bf n} &= 0\end{split}$

where $${\bf n}$$ is the normal vector and $${\bf t}$$ the tangent vector. If the normal and tangent vector are not aligned with the mesh the stress formulation has to be used.

• the periodic boundary condition (“P”) needs to be prescribed in the .rea or .re2 file since it already assigns the last point to first via $${\bf u}({\bf x})={\bf u}({\bf x} + L)$$, where $$L$$ is the periodic length.

• the wall boundary condition (“W”) corresponds to $${\bf u}=0$$.

For a fully-developed flow in such a configuration, one can effect great computational efficiencies by considering the problem in a single geometric unit (here taken to be of length $$L$$), and requiring periodicity of the field variables. Nek5000 requires that the pairs of sides (or faces, in the case of a three-dimensional mesh) identified as periodic be identical (i.e., that the geometry be periodic).

For an axisymmetric flow geometry, the axis boundary condition is provided for boundary segments that lie entirely on the axis of symmetry. This is essentially a symmetry (mixed Dirichlet/Neumann) boundary conditionin which the normal velocity and the tangential traction are set to zero.

For free-surface boundary segments, the inhomogeneous traction boundary conditions involve both the surface tension coefficient $$\sigma$$ and the mean curvature of the free surface.

## 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 condition menu will reappear for each passive scalar field so that the user can specify an independent set of boundary conditions for each passive scalar field.

Table 5 Primitive boundary conditions (Temperature and Passive scalars)
Identifier Description Type Note
P Periodic Standard periodic boundary condition
p Periodic For periodicity within a single element
I Insulated Neumann zero gradient
O Outflow Neumann Identical to “I”
SYM Symmetry Neumann Identical to “I”
A Axisymmetric boundary
E Interior boundary
Table 6 User defined boundary conditions for temperature and passive scalars
Identifier Description Type Note
t Temperature Dirichlet Standard Dirichlet boundary condition
f Flux Neumann Standard Neumann boundary condition
c Newton cooling Robin Specified heat transfer coefficient
int Interpolated (NEKNEK) Dirichlet Interpolated from the adjacent overset mesh, see: Overlapping Overset Grids
• open boundary condition (“O”)

$k(\nabla T)\cdot {\bf n} =0$
• insulated boundary condition (“I”)

$k(\nabla T)\cdot {\bf n} =0$

where $${\bf n}$$ is the normal vector and $${\bf t}$$ the tangent vector. If the normal and tangent vector are not aligned with the mesh the stress formulation has to be used.

• the periodic boundary condition (“P”) needs to be prescribed in the .rea file since it already assigns the last point to first via $${\bf u}({\bf x})={\bf u}({\bf x} + L)$$, where $$L$$ is the periodic length.

• Newton cooling boundary condition (“c”)

$k(\nabla T)\cdot {\bf n}=h(T-T_{\infty})$
• flux boundary condition (“f”)

$k(\nabla T)\cdot {\bf n} =f$

## Internal Boundary Conditions¶

In the spatial discretization, the entire computational domain is subdivided into macro-elements, the boundary segments shared by any two of these macro-elements in $$\Omega_f$$ and $$\Omega_s$$ are denoted as internal boundaries. For fluid flow analysis with a single-fluid system or heat transfer analysis without change-of-phase, internal boundary conditions are irrelevant as the corresponding field variables on these segments are part of the solution. However, for a multi-fluid system and for heat transfer analysis with change-of-phase, special conditions are required at particular internal boundaries, as described in the following.

For a fluid system composes of multiple immiscible fluids, the boundary (and hence the identity) of each fluid must be tracked, and a jump in the normal traction exists at the fluid-fluid interface if the surface tension coefficient is nonzero. For this purpose, the interface between any two fluids of different identity must be defined as a special type of internal boundary, namely, a fluid layer; and the associated surface tension coefficient also needs to be specified.

In a heat transfer analysis with change-of-phase, Nek5000 assumes that both phases exist at the start of the solution, and that all solid-liquid interfaces are specified as special internal boundaries, namely, the melting fronts. If the fluid flow problem is considered, i.e., the energy equation is solved in conjunction with the momentum and continuity equations, then only the common boundary between the fluid and the solid (i.e., all or portion of $$\partial \overline{\Omega}_f'$$ in Fig. 17) can be defined as the melting front. In this case, segments on $$\partial \overline{\Omega}_f'$$ that belong to the dynamic melting/freezing interface need to be specified by the user. Nek5000 always assumes that the density of the two phases are the same (i.e., no Stefan flow); therefore at the melting front, the boundary condition for the fluid velocity is the same as that for a stationary wall, that is, all velocity components are zero. If no fluid flow is considered, i.e., only the energy equation is solved, then any internal boundary can be defined as a melting front. The temperature boundary condition at the melting front corresponds to a Dirichlet condition; that is, the entire segment maintains a constant temperature equal to the user-specified melting temperature $$T_{melt}$$ throughout the solution. In addition, the volumetric latent heat of fusion $$\rho L$$ for the two phases, which is also assumed to be constant, should be specified.