Flow boundary conditions

Supported boundary conditions:

1. Free slip $\frac{\partial u_i}{\partial x_i}=0$ at the boundary $\mathrm{\Gamma }$

2. No slip $u_i=0$ at the boundary $\mathrm{\Gamma }$

Defining the boundary conditions

We have two ways of defining the boundary condition formulations: - VelocityBoundaryConditions, and - DisplacementBoundaryConditions. The first one is used for the velocity-pressure formulation, and the second one is used for the displacement-pressure formulation. The flow boundary conditions can be switched on and off by setting them as true or false at the appropriate boundaries. Valid boundary names are left and right, top and bot, and for the 3D case, front and back.

For example, if we want to have free free-slip in every single boundary in a 2D simulation, we need to instantiate VelocityBoundaryConditions or DisplacementBoundaryConditions as:

bcs = VelocityBoundaryConditions(;
    no_slip      = (left=false, right=false, top=false, bot=false),
    free_slip    = (left=true, right=true, top=true, bot=true),
)
bcs = DisplacementBoundaryConditions(;
    no_slip      = (left=false, right=false, top=false, bot=false),
    free_slip    = (left=true, right=true, top=true, bot=true),
)

The equivalent for the 3D case would be:

bcs = VelocityBoundaryConditions(;
    no_slip      = (left=false, right=false, top=false, bot=false, front=false, back=false),
    free_slip    = (left=true, right=true, top=true, bot=true, front=true, back=true),
)
bcs = DisplacementBoundaryConditions(;
    no_slip      = (left=false, right=false, top=false, bot=false, front=false, back=false),
    free_slip    = (left=true, right=true, top=true, bot=true, front=true, back=true),
)

Prescribing the velocity/displacement boundary conditions

Normally, one would prescribe the velocity/displacement boundary conditions by setting the velocity/displacement field at the boundary through the application of a background strain rate εbg. Depending on the formulation, the velocity/displacement field is set as follows for the 2D case:

Velocity formulation

stokes.V.Vx .= PTArray(backend)([ x*εbg for x in xvi[1], _ in 1:ny+2]) # Velocity in x direction
stokes.V.Vy .= PTArray(backend)([-y*εbg for _ in 1:nx+2, y in xvi[2]]) # Velocity in y direction

Make sure to apply the set velocity to the boundary conditions. You do this by calling the flow_bcs! function,

flow_bcs!(stokes, flow_bcs)

and then applying the velocities to the halo

update_halo!(@velocity(stokes)...)

Displacement formulation

stokes.U.Ux .= PTArray(backend)([ x*εbg*lx*dt for x in xvi[1], _ in 1:ny+2]) # Displacement in x direction
stokes.U.Uy .= PTArray(backend)([-y*εbg*ly*dt for _ in 1:nx+2, y in xvi[2]]) # Displacement in y direction
flow_bcs!(stokes, flow_bcs)

Make sure to initialize the displacement according to the extent of your domain. Here, lx and ly are the domain lengths in the x and y directions, respectively. Also for the displacement formulation it is important that the displacement is converted to velocity before updating the halo. This can be done by calling the displacement2velocity! function.

displacement2velocity!(stokes, dt) # convert displacement to velocity
update_halo!(@velocity(stokes)...)

Thermal Boundary Conditions

Thermal boundary conditions are collected in TemperatureBoundaryConditions. The same type is used in 2D and 3D.

Supported thermal boundary conditions:

1. No flux $\frac{\partial T}{\partial x_i}=0$ at the boundary $\mathrm{\Gamma }$

2. Constant temperature on the outer boundary $T=T_{\mathrm{\Gamma }}$ at the boundary $\mathrm{\Gamma }$

3. Constant heat flux in the pseudo-transient diffusion kernels $q_T=q_{\mathrm{\Gamma }}$ at the boundary $\mathrm{\Gamma }$

4. Periodic temperature $T({\mathrm{\Gamma }}_i)=T({\mathrm{\Gamma }}_j)$ across paired boundaries

5. Mask-based Dirichlet values inside the domain $T=f(x_i)$ at selected points in $\mathrm{\Omega }$

Face Names

In 2D, boundary tuples use left, right, top, and bot. In 3D, add front and back:

thermal_bc = TemperatureBoundaryConditions(;
    no_flux = (
        left = true,
        right = true,
        front = true,
        back = true,
        top = false,
        bot = false,
    ),
)

Faces omitted from no_flux, constant_flux, constant_value, or periodic are treated as inactive for that condition. The dimensionality is inferred from the longest tuple you provide, so a tuple with front or back creates a 3D boundary-condition set.

No-Flux Boundaries

Use no_flux to copy the adjacent interior temperature into the ghost layer. For example, this applies no-flux boundaries on the left and right sides of a 2D domain:

thermal_bc = TemperatureBoundaryConditions(;
    no_flux = (left = true, right = true, top = false, bot = false),
)

thermal_bcs!(thermal, thermal_bc)

Constant-Value Boundaries

Use constant_value for fixed-temperature outer boundaries. These values are applied by thermal_bcs! through the ghost-cell relation Tghost = 2 * Tboundary - Tinterior.

thermal_bc = TemperatureBoundaryConditions(;
    no_flux = (left = true, right = true),
    constant_value = (top = 273.0, bot = 1573.0),
)

thermal_bcs!(thermal, thermal_bc)

If no_flux and constant_value are both active on the same face, thermal_bcs! applies constant_value first and no_flux second.

Periodic Boundaries

Use periodic to copy the opposite interior temperature into the ghost layer. For example, this applies periodic temperature boundaries on the left and right sides of a 2D domain:

thermal_bc = TemperatureBoundaryConditions(;
    periodic = (left = true, right = true, top = false, bot = false),
)

thermal_bcs!(thermal, thermal_bc)

In 3D, include front and back when those faces should also be periodic:

thermal_bc = TemperatureBoundaryConditions(;
    periodic = (
        left = true,
        right = true,
        front = true,
        back = true,
        top = false,
        bot = false,
    ),
)

If multiple ghost-cell conditions are active on the same face, thermal_bcs! applies constant_value first, no_flux second, and periodic last.

Constant-Flux Boundaries

Use constant_flux to prescribe flux values in the pseudo-transient heat diffusion solver. These values are consumed by the PT compute_flux! kernels, not by thermal_bcs!.

thermal_bc = TemperatureBoundaryConditions(;
    no_flux = (left = true, right = true, front = true, back = true),
    constant_flux = (top = 0.0, bot = 0.03),
)

Mask-Based Dirichlet Conditions

Use dirichlet for fixed values inside the domain, selected by a mask:

thermal_bc = TemperatureBoundaryConditions(;
    no_flux = (left = true, right = true, top = false, bot = false),
    dirichlet = (; constant = 273.0, mask = mask),
)