Boundary Conditions - CFD#

This subsection defines the boundary conditions associated with fluid dynamics physics. Lethe supports the following boundary conditions:

none boundary condition (default).
noslip boundary conditions strongly impose the velocity on a boundary to be \(\mathbf{u}=[0,0]^T\) and \(\mathbf{u}=[0,0,0]^T\) in 2D and 3D respectively.
slip boundary conditions impose \(\mathbf{u} \cdot \mathbf{n}=0\), with \(\mathbf{n}\) the normal vector of the boundary. Imposing slip boundary conditions strongly is not trivial in FEM. We refer the reader to the deal.II documentation for explanations on how this is achieved.
partial slip boundary condition simulates an intermediary between slip and noslip boundary conditions, in which the fluid feels an attenuated stress due to the walls. The attenuation is controlled by the boundary layer thickness (m). The partial slip boundary condition introduces the “penalization factor” beta to the \(\mathbf{n}\) normal vector of the boundary, and the boundary layer thickness (m) as a parameter to calculate the shear stress at the boundaries.
periodic boundary conditions, in which fluid exiting the domain will reenter on the opposite side.
function where a Dirichlet boundary condition is set from an arbitrary function. These functions can be used to define all sorts of steady-state and transient velocity boundary conditions such as rotating walls. It is also possible to weakly impose a Dirichlet boundary condition. In this case, the type should be set to function weak. This will result in Nitsche method being used to weakly impose the boundary condition instead of it being strongly imposed by overwriting the values of the degrees of freedom. The function weak boundary type should only be used in very specific cases where the problem is very stiff, for example when it is fully enclosed and a non-trivial velocity profile is imposed. It can also be used to impose an outbound dirichlet boundary condition.
outlet where a do nothing boundary condition, which is a zero traction, is imposed when the fluid is leaving the domain (\(\mathbf{u} \cdot \mathbf{n}>0\)) and a penalization is imposed when the fluid is inbound. This is useful when turbulent structures or vortices are leaving the domain since it prevents the re-entry of the fluid. The boundary condition imposed is thus:

\[\nu \nabla \mathbf{u} \cdot \mathbf{n} - p \mathcal{I} \cdot \mathbf{n} - \beta (\mathbf{u}\cdot \mathbf{n})_{-} \mathbf{u} = 0\]

or in Einstein notation:

\[\nu \partial_i u_j n_j - p n_i - \beta ( u_k n_k)_{-} u_i = 0\]

where \(\beta\) is a constant and \((\mathbf{u}\cdot \mathbf{n})_{-}\) is \(\min (0,\mathbf{u}\cdot \mathbf{n})\). We refer the reader to the work of Arndt et al 2015 for more detail.

Finally, Lethe also supports not imposing a boundary condition on an ID. Not imposing a boundary condition is equivalent to the do nothing boundary condition (none), which results in a zero net traction on a boundary. This, in fact, imposes \(\int_{\Gamma}(-p\mathcal{I} + \mathbf{\tau}) \cdot \mathbf{n}=0\) where \(p\) is the pressure, \(\mathcal{I}\) is the identity tensor, \(\mathbf{\tau}\) is the deviatoric stress tensor and \(\Gamma\) is the boundary.

subsection boundary conditions
  set number                = 2
  set time dependent        = false
  set fix pressure constant = false
  subsection bc 0
    set id                 = 0
    set type               = function

    subsection u
      set Function expression = -y
    end
    subsection v
      set Function expression = x
    end
    subsection w
      set Function expression = 0
    end

    # Center of rotation used for torque calculation
    subsection center of rotation
      set x = 0
      set y = 0
      set z = 0
    end

    set periodic_id        = 1
    set periodic_direction = 0
    set beta               = 0
  end
  subsection bc 1
    set type = noslip
  end
end

number specifies the number of boundary conditions of the problem. Periodicity between 2 boundaries counts as 1 condition even if it requires two distinct boundary ids.

Warning

The number of boundary conditions must be specified explicitly. This is often a source of error.

Note

The index in subsection bc .. must be coherent with the number of boundary conditions set: if number = 2, bc 0 and bc 1 are created but bc 2 does not exist.

Likewise, if number = 2 and there is no subsection bc 0 explicitly stated, the boundary is still created, with none by default.

time dependent specifies if a boundary condition is time-dependent (true) or steady (false). By default, this parameter is set to false. This is here to improve the computational efficiency for transient cases in which the boundary conditions do not change.
fix pressure constant specifies if a zero pressure constraint should be applied on a single node of the coarse grid solver when using geometric multigrid preconditioning combined with the lethe-fluid-matrix-free solver. Essentially, this condition should be set to true whenever a user is using the lethe-fluid-matrix-free solver and simulating the flow within a closed domain (that is a domain on which all boundaries are either periodic or Dirichlet boundary conditions).
Each fluid dynamics boundary condition is stored in a bc # subsection :
- id is the number associated with the boundary condition. By default, Lethe assumes that the id is equivalent to the number # of the bc.
- type is the type of the boundary condition.
- The subsections u, v and w are used to specify the individual components of the velocity at the boundary using function expressions. These functions can depend on position (\(x,y,z\)) and on time (\(t\)).
- The center of rotation subsection is only necessary when calculating the torque applied on a boundary. See See Force and Torque Calculation for more information.
- periodic id and periodic_direction specify the id and direction of the matching periodic boundary condition. For example, if boundary id 0 (located at xmin) is matched with boundary id 1 (located at xmax), we would set id = 0, periodic_id = 1 and periodic_direction = 0.
- beta is a penalization parameter used for both the outlet, partial slip, and function weak boundary conditions. For the outlet boundary conditions beta should be close to unity, whereas beta of 10 or a 100 can be appropriate for the function weak boundary condition. For the partial slip condition, use high values of beta (i.e. > 50).
- boundary layer thickness (\(d_w\)) is the parameter applied to the partial slip boundary condition. It is used to estimate the tangential shear stress \(\tau_t = -\mu \frac{u}{d_w}\). For very high boundary layer thicknes, the boundary layer should behave exactly like the slip condition.

Caution

While using the lethe-fluid-sharp solver, it is wise to assign a weak type of boundary (outlet, partial slip, or function weak) to at least one boundary. The presence of particle(s) has a non-null contribution to the divergence of the problem, making it much harder for the linear solver to converge unless it is given some flexibility through of boundaries.

Caution

The lethe-fluid-matrix-free application does not support the pressure and partial slip boundary conditions.