Problem setup

Boundary conditions

Each boundary has exactly one type of boundary conditions. For periodic boundary conditions, the opposite boundary must also be periodic.

# IncompressibleNavierStokes.AbstractBC — Type.

abstract type AbstractBC

Boundary condition for one side of the domain.

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# IncompressibleNavierStokes.DirichletBC — Type.

struct DirichletBC{U} <: IncompressibleNavierStokes.AbstractBC

Dirichlet boundary conditions for the velocity, where u[1] = (x..., t) -> u1_BC up to u[d] = (x..., t) -> ud_BC, where d is the dimension.

When u is nothing, then the boundary conditions are no slip boundary conditions, where all velocity components are zero.

Fields

• u: Boundary condition

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# IncompressibleNavierStokes.PeriodicBC — Type.

struct PeriodicBC <: IncompressibleNavierStokes.AbstractBC

Periodic boundary conditions. Must be periodic on both sides.

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# IncompressibleNavierStokes.PressureBC — Type.

struct PressureBC <: IncompressibleNavierStokes.AbstractBC

Pressure boundary conditions. The pressure is prescribed on the boundary (usually an "outlet"). The velocity has zero Neumann conditions.

Note: Currently, the pressure is prescribed with the constant value of zero on the entire boundary.

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# IncompressibleNavierStokes.SymmetricBC — Type.

struct SymmetricBC <: IncompressibleNavierStokes.AbstractBC

Symmetric boundary conditions. The parallel velocity and pressure is the same at each side of the boundary. The normal velocity is zero.

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# IncompressibleNavierStokes.apply_bc_p! — Method.

apply_bc_p!(p, t, setup; kwargs...) -> Any

Apply pressure boundary conditions (in-place version).

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# IncompressibleNavierStokes.apply_bc_p — Method.

apply_bc_p(p, t, setup; kwargs...) -> Any

Apply pressure boundary conditions (differentiable version).

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# IncompressibleNavierStokes.apply_bc_temp! — Method.

apply_bc_temp!(temp, t, setup; kwargs...) -> Any

Apply temperature boundary conditions (in-place version).

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# IncompressibleNavierStokes.apply_bc_temp — Method.

apply_bc_temp(temp, t, setup; kwargs...) -> Any

Apply temperature boundary conditions (differentiable version).

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# IncompressibleNavierStokes.apply_bc_u! — Method.

apply_bc_u!(u, t, setup; kwargs...) -> Any

Apply velocity boundary conditions (in-place version).

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# IncompressibleNavierStokes.apply_bc_u — Method.

apply_bc_u(u, t, setup; kwargs...) -> Any

Apply velocity boundary conditions (differentiable version).

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# IncompressibleNavierStokes.boundary — Method.

boundary(β, N, I, isright) -> Any

Get boundary indices of boundary layer normal to β. The CartesianIndices given by I should contain those of the inner DOFs, typically Ip or Iu[α]. The boundary layer is then just outside those.

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# IncompressibleNavierStokes.offset_p — Function.

offset_p(bc, isnormal, isright)

Number of non-DOF pressure components at boundary.

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# IncompressibleNavierStokes.offset_u — Function.

offset_u(bc, isnormal, isright)

Number of non-DOF velocity components at boundary. If isnormal, then the velocity is normal to the boundary, else parallel. If isright, it is at the end/right/rear/top boundary, otherwise beginning.

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Grid

# IncompressibleNavierStokes.Dimension — Type.

struct Dimension{N}

Represent an N-dimensional space. Returns N when called.

julia> d = Dimension(3)
Dimension{3}()

julia> d()
3

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# IncompressibleNavierStokes.Grid — Method.

Grid(x, boundary_conditions; ArrayType) -> NamedTuple

Create useful quantities for Cartesian box mesh ``x[1] \times \dots \times x[d]with boundary conditionsboundary_conditions. Return a named tuple ([α]` denotes a tuple index) with the following fields:

• N[α]: Number of finite volumes in direction β, including ghost volumes

• Nu[α][β]: Number of u[α] velocity DOFs in direction β

• Np[α]: Number of pressure DOFs in direction α

• Iu[α]: Cartesian index range of u[α] velocity DOFs

• Ip: Cartesian index range of pressure DOFs

• xlims[α]: Tuple containing the limits of the physical domain (not grid) in the direction α

• x[α]: α-coordinates of all volume boundaries, including the left point of the first ghost volume

• xu[α][β]: β-coordinates of u[α] velocity points

• xp[α]: α-coordinates of pressure points

• Δ[α]: All volume widths in direction α

• Δu[α]: Distance between pressure points in direction α

• A[α][β]: Interpolation weights from α-face centers $x_I$ to $x_{I\pm h_{\beta }}$

Note that the memory footprint of the redundant 1D-arrays above is negligible compared to the memory footprint of the 2D/3D-fields used in the code.

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# IncompressibleNavierStokes.cosine_grid — Method.

cosine_grid(a, b, N) -> Any

Create a nonuniform grid of N + 1 points from a to b using a cosine profile, i.e.

$$x_i=a+\frac{1}{2}(1-\cos \left(\pi \frac{i}{n}\right))(b-a),i=0,\dots ,N$$

See also stretched_grid.

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# IncompressibleNavierStokes.max_size — Method.

max_size(grid) -> Any

Get size of the largest grid element.

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# IncompressibleNavierStokes.stretched_grid — Function.

stretched_grid(a, b, N) -> Any
stretched_grid(a, b, N, s) -> Any

Create a nonuniform grid of N + 1 points from a to b with a stretch factor of s. If s = 1, return a uniform spacing from a to b. Otherwise, return a vector $x\in {\mathbb{R}}^{N+1}$ such that $x_n=a+\sum _{i=1}^n{s}^{i-1}h$ for $n=0,\dots ,N$. Setting $x_N=b$ then gives $h=(b-a)\frac{1-s}{1-s^N}$, resulting in

$$x_n=a+(b-a)\frac{1-s^n}{1-s^N},n=0,\dots ,N.$$

Note that stretched_grid(a, b, N, s)[n] corresponds to $x_{n-1}$.

See also cosine_grid.

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# IncompressibleNavierStokes.tanh_grid — Function.

tanh_grid(a, b, N) -> Any
tanh_grid(a, b, N, γ) -> Any

Create a nonuniform grid of N + 1 points from a to b, as proposed by Trias et al. [11].

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Setup

# IncompressibleNavierStokes.Setup — Method.

Setup(
;
    x,
    boundary_conditions,
    bodyforce,
    issteadybodyforce,
    closure_model,
    projectorder,
    ArrayType,
    workgroupsize,
    temperature,
    Re
)

Create problem setup (stored in a named tuple).

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# IncompressibleNavierStokes.temperature_equation — Method.

temperature_equation(
;
    Pr,
    Ra,
    Ge,
    dodissipation,
    boundary_conditions,
    gdir,
    nondim_type
)

Create temperature equation setup (stored in a named tuple).

The equation is parameterized by three dimensionless numbers (Prandtl number, Rayleigh number, and Gebhart number), and requires separate boundary conditions for the temperature field. The gdir keyword specifies the direction gravity, while non_dim_type specifies the type of non-dimensionalization.

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Field initializers

# IncompressibleNavierStokes.random_field — Function.

random_field(setup; ...) -> Any
random_field(setup, t; A, kp, psolver, rng) -> Any

Create random field, as in [12].

• A: Eddy amplitude scaling

• kp: Peak energy wavenumber

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# IncompressibleNavierStokes.scalarfield — Method.

scalarfield(setup) -> Any

Create empty scalar field.

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# IncompressibleNavierStokes.temperaturefield — Function.

temperaturefield(setup, tempfunc) -> Any
temperaturefield(setup, tempfunc, t) -> Any

Create temperature field from function with boundary conditions at time t.

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# IncompressibleNavierStokes.vectorfield — Method.

vectorfield(setup) -> Any

Create empty vector field.

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# IncompressibleNavierStokes.velocityfield — Function.

velocityfield(setup, ufunc; ...) -> Any
velocityfield(setup, ufunc, t; psolver, doproject) -> Any

Create divergence free velocity field u with boundary conditions at time t. The initial conditions of u[α] are specified by the function ufunc(α, x...).

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