Note: Output is not generated for this example (to save resources on GitHub).

Unsteady actuator case - 3D

In this example, an unsteady inlet velocity profile at encounters a wind turbine blade in a wall-less domain. The blade is modeled as a uniform body force on a short cylinder.

We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.

using CairoMakie
using IncompressibleNavierStokes

Output directory

outdir = joinpath(@__DIR__, "output", "Actuator3D")

Floating point type

T = Float64

Backend

backend = CPU()
# using CUDA; backend = CUDABackend()

Reynolds number

Re = T(100)

A 3D grid is a Cartesian product of three vectors

x = LinRange(0.0, 6.0, 31), LinRange(-2.0, 2.0, 41), LinRange(-2.0, 2.0, 41)
plotgrid(x...)

Boundary conditions: Unsteady BC requires time derivatives

boundary_conditions = (
    # x left, x right
    (
        DirichletBC(
            (dim, x, y, z, t) ->
                dim == 1 ? cos(π / 6 * sin(π / 6 * t)) :
                dim == 2 ? sin(π / 6 * sin(π / 6 * t)) : zero(x),
        ),
        PressureBC(),
    ),

    # y rear, y front
    (PressureBC(), PressureBC()),

    # z rear, z front
    (PressureBC(), PressureBC()),
)

Actuator body force: A thrust coefficient Cₜ distributed over a short cylinder

cx, cy, cz = T(2), T(0), T(0) # Disk center
D = T(1)                      # Disk diameter
δ = T(0.11)                   # Disk thickness
Cₜ = T(0.2)                  # Thrust coefficient
cₜ = Cₜ / (π * (D / 2)^2 * δ)
inside(x, y, z) = abs(x - cx) ≤ δ / 2 && (y - cy)^2 + (z - cz)^2 ≤ (D / 2)^2
bodyforce(dim, x, y, z) = dim == 1 ? -cₜ * inside(x, y, z) : zero(x)

Build setup and assemble operators

setup = Setup(; x, Re, boundary_conditions, bodyforce, backend);
nothing #hide

Initial conditions (extend inflow)

ustart = velocityfield(setup, (dim, x, y, z) -> dim == 1 ? one(x) : zero(x));
nothing #hide

Solve unsteady problem

(; u, t), outputs = solve_unsteady(;
    setup,
    ustart,
    tlims = (T(0), T(3)),
    method = RKMethods.RK44P2(),
    Δt = T(0.05),
    processors = (
        rtp = realtimeplotter(;
            setup,
            plot = fieldplot,
            # plot = energy_history_plot,
            # plot = energy_spectrum_plot,
            nupdate = 1,
        ),
        # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20),
        # vtk = vtk_writer(; setup, nupdate = 10, dir = "$outdir", filename = "solution"),
        # field = fieldsaver(; setup, nupdate = 10),
        log = timelogger(; nupdate = 1),
    ),
);
nothing #hide

Post-process

We may visualize or export the computed fields (V, p)

Field plot

outputs.rtp

Export to VTK

save_vtk(state; setup, filename = joinpath(outdir, "solution"))

Copy-pasteable code

Below is the full code for this example stripped of comments and output.

using GLMakie
using IncompressibleNavierStokes

outdir = joinpath(@__DIR__, "output", "Actuator3D")

T = Float64

backend = CPU()
# using CUDA; backend = CUDABackend()

Re = T(100)

x = LinRange(0.0, 6.0, 31), LinRange(-2.0, 2.0, 41), LinRange(-2.0, 2.0, 41)
plotgrid(x...)

boundary_conditions = (
    # x left, x right
    (
        DirichletBC(
            (dim, x, y, z, t) ->
                dim == 1 ? cos(π / 6 * sin(π / 6 * t)) :
                dim == 2 ? sin(π / 6 * sin(π / 6 * t)) : zero(x),
        ),
        PressureBC(),
    ),

    # y rear, y front
    (PressureBC(), PressureBC()),

    # z rear, z front
    (PressureBC(), PressureBC()),
)

cx, cy, cz = T(2), T(0), T(0) # Disk center
D = T(1)                      # Disk diameter
δ = T(0.11)                   # Disk thickness
Cₜ = T(0.2)                  # Thrust coefficient
cₜ = Cₜ / (π * (D / 2)^2 * δ)
inside(x, y, z) = abs(x - cx) ≤ δ / 2 && (y - cy)^2 + (z - cz)^2 ≤ (D / 2)^2
bodyforce(dim, x, y, z) = dim == 1 ? -cₜ * inside(x, y, z) : zero(x)

setup = Setup(; x, Re, boundary_conditions, bodyforce, backend);

ustart = velocityfield(setup, (dim, x, y, z) -> dim == 1 ? one(x) : zero(x));

(; u, t), outputs = solve_unsteady(;
    setup,
    ustart,
    tlims = (T(0), T(3)),
    method = RKMethods.RK44P2(),
    Δt = T(0.05),
    processors = (
        rtp = realtimeplotter(;
            setup,
            plot = fieldplot,
            # plot = energy_history_plot,
            # plot = energy_spectrum_plot,
            nupdate = 1,
        ),
        # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20),
        # vtk = vtk_writer(; setup, nupdate = 10, dir = "$outdir", filename = "solution"),
        # field = fieldsaver(; setup, nupdate = 10),
        log = timelogger(; nupdate = 1),
    ),
);

outputs.rtp

save_vtk(state; setup, filename = joinpath(outdir, "solution"))

---

This page was generated using Literate.jl.