Operators

All discrete operators are built using KernelAbstractions.jl and Cartesian indices, similar to WaterLily.jl. This allows for dimension- and backend-agnostic code. See this blog post for how to write kernels. IncompressibleNavierStokes previously relied on assembling sparse operators to perform the same operations. While being very efficient and also compatible with CUDA (CUSPARSE), storing these matrices in memory is expensive for large 3D problems.

# IncompressibleNavierStokes.Offset — Type.

struct Offset{D}

Cartesian index unit vector in D = 2 or D = 3 dimensions. Calling Offset{D}()(α) returns a Cartesian index with 1 in the dimension α and zeros elsewhere.

See https://b-fg.github.io/2023/05/07/waterlily-on-gpu.html for writing kernel loops using Cartesian indices.

Fields

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# IncompressibleNavierStokes.Dfield! — Method.

Dfield!(d, G, p, setup; ϵ) -> Any

Compute the $D$-field (in-place version).

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# IncompressibleNavierStokes.Dfield — Method.

Dfield(p, setup; kwargs...) -> Any

Compute the $D$-field [6] given by

$$D=\frac{2|\mathrm{\nabla }p|}{{\mathrm{\nabla }}^{2}p}.$$

Differentiable version.

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# IncompressibleNavierStokes.Qfield! — Method.

Qfield!(Q, u, setup) -> Any

Compute the $Q$-field (in-place version).

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# IncompressibleNavierStokes.Qfield — Method.

Qfield(u, setup) -> Any

Compute $Q$-field [7] given by

$$Q=-\frac{1}{2}\sum _{\alpha ,\beta }\frac{\partial u^{\alpha }}{\partial x^{\beta }}\frac{\partial u^{\beta }}{\partial x^{\alpha }}.$$

Differentiable version.

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# IncompressibleNavierStokes.applybodyforce! — Method.

applybodyforce!(F, u, t, setup) -> Any

Compute body force (in-place version). Add the result to F.

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# IncompressibleNavierStokes.applybodyforce — Method.

applybodyforce(u, t, setup) -> Any

Compute body force (differentiable version).

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# IncompressibleNavierStokes.applypressure! — Method.

applypressure!(u, p, setup) -> Any

Subtract pressure gradient (in-place version).

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# IncompressibleNavierStokes.avg — Method.

avg(ϕ, Δ, I, α) -> Any

Average scalar field ϕ in the α-direction.

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# IncompressibleNavierStokes.convection! — Method.

convection!(F, u, setup) -> Any

Compute convective term (in-place version). Add the result to F.

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# IncompressibleNavierStokes.convection — Method.

convection(u, setup) -> Any

Compute convective term (differentiable version).

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# IncompressibleNavierStokes.convection_diffusion_temp! — Method.

convection_diffusion_temp!(c, u, temp, setup) -> Any

Compute convection-diffusion term for the temperature equation. (in-place version). Add result to c.

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# IncompressibleNavierStokes.convection_diffusion_temp — Method.

convection_diffusion_temp(u, temp, setup) -> Any

Compute convection-diffusion term for the temperature equation. (differentiable version).

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# IncompressibleNavierStokes.convectiondiffusion! — Method.

convectiondiffusion!(F, u, setup) -> Any

Compute convective and diffusive terms (in-place version). Add the result to F.

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# IncompressibleNavierStokes.diffusion! — Method.

diffusion!(F, u, setup) -> Any

Compute diffusive term (in-place version). Add the result to F.

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# IncompressibleNavierStokes.diffusion — Method.

diffusion(u, setup) -> Any

Compute diffusive term (differentiable version).

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# IncompressibleNavierStokes.dissipation! — Method.

dissipation!(diss, diff, u, setup) -> Any

Compute dissipation term for the temperature equation (in-place version). Add result to diss.

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# IncompressibleNavierStokes.dissipation — Method.

dissipation(u, setup) -> Any

Compute dissipation term for the temperature equation (differentiable version).

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# IncompressibleNavierStokes.dissipation_from_strain! — Method.

dissipation_from_strain!(ϵ, u, setup) -> Any

Compute dissipation term from strain-rate tensor (in-place version).

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# IncompressibleNavierStokes.dissipation_from_strain — Method.

dissipation_from_strain(u, setup) -> Any

Compute dissipation term $2\nu ⟨S_{ij}{S}_{ij}⟩$ from strain-rate tensor (differentiable version).

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# IncompressibleNavierStokes.divergence! — Method.

divergence!(div, u, setup) -> Any

Compute divergence of velocity field (in-place version).

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# IncompressibleNavierStokes.divergence — Method.

divergence(u, setup) -> Any

Compute divergence of velocity field (differentiable version).

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# IncompressibleNavierStokes.divoftensor! — Method.

divoftensor!(s, σ, setup) -> Any

Compute divergence of a tensor with all components in the pressure points (in-place version). The stress tensors should be precomputed and stored in σ.

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# IncompressibleNavierStokes.eig2field! — Method.

eig2field!(λ, u, setup) -> Any

Compute the second eigenvalue of $S^2+R^2$ (in-place version).

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# IncompressibleNavierStokes.eig2field — Method.

eig2field(u, setup) -> Any

Compute the second eigenvalue of $S^2+R^2$, as proposed by Jeong and Hussain [7].

Differentiable version.

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# IncompressibleNavierStokes.get_scale_numbers — Method.

get_scale_numbers(
    u,
    setup
) -> NamedTuple{(:uavg, :ϵ, :η, :λ, :Reλ, :L, :τ), <:Tuple{Any, Any, Any, Any, Any, Nothing, Nothing}}

Get the following dimensional scale numbers [8]:

• Velocity $u_{\text{avg}}=⟨u_i{u}_i{⟩}^{1/2}$

• Dissipation rate $ϵ=2\nu ⟨S_{ij}{S}_{ij}⟩$

• Kolmolgorov length scale $\eta =(\frac{{\nu }^3}{ϵ}{)}^{1/4}$

• Taylor length scale $\lambda =(\frac{5\nu }{ϵ}{)}^{1/2}{u}_{\text{avg}}$

• Taylor-scale Reynolds number $R{e}_{\lambda }=\frac{\lambda u_{\text{avg}}}{\sqrt{3}\nu }$

• Integral length scale $L=\frac{3\pi }{2u_{\text{avg}}^2}{\int }_0^{\mathrm{\infty }}\frac{E(k)}{k}\mathrm{d}k$

• Large-eddy turnover time $\tau =\frac{L}{u_{\text{avg}}}$

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# IncompressibleNavierStokes.gravity! — Method.

gravity!(F, temp, setup) -> Any

Compute gravity term (in-place version). add the result to F.

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# IncompressibleNavierStokes.gravity — Method.

gravity(temp, setup) -> Any

Compute gravity term (differentiable version).

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# IncompressibleNavierStokes.interpolate_u_p! — Method.

interpolate_u_p!(up, u, setup) -> Any

Interpolate velocity to pressure points (in-place version).

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# IncompressibleNavierStokes.interpolate_u_p — Method.

interpolate_u_p(u, setup) -> Any

Interpolate velocity to pressure points (differentiable version).

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# IncompressibleNavierStokes.interpolate_ω_p! — Method.

interpolate_ω_p!(ωp, ω, setup) -> Any

Interpolate vorticity to pressure points (in-place version).

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# IncompressibleNavierStokes.interpolate_ω_p — Method.

interpolate_ω_p(ω, setup) -> Any

Interpolate vorticity to pressure points (differentiable version).

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# IncompressibleNavierStokes.kinetic_energy! — Method.

kinetic_energy!(ke, u, setup; interpolate_first) -> Any

Compute kinetic energy field (in-place version).

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# IncompressibleNavierStokes.kinetic_energy — Method.

kinetic_energy(u, setup; kwargs...) -> Any

Compute kinetic energy field $k$ (in-place version). If interpolate_first is true, it is given by

$$k_I=\frac{1}{8}\sum _{\alpha }(u_{I+h_{\alpha }}^{\alpha }+u_{I-h_{\alpha }}^{\alpha }{)}^2.$$

Otherwise, it is given by

$$k_I=\frac{1}{4}\sum _{\alpha }((u_{I+h_{\alpha }}^{\alpha }{)}^2+(u_{I-h_{\alpha }}^{\alpha }{)}^2),$$

as in [9].

Differentiable version.

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# IncompressibleNavierStokes.laplacian! — Method.

laplacian!(L, p, setup) -> Any

Compute Laplacian of pressure field (in-place version).

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# IncompressibleNavierStokes.laplacian — Method.

laplacian(p, setup) -> Any

Compute Laplacian of pressure field (differentiable version).

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# IncompressibleNavierStokes.momentum! — Method.

momentum!(F, u, temp, t, setup) -> Any

Right hand side of momentum equations, excluding pressure gradient (in-place version).

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# IncompressibleNavierStokes.momentum — Method.

momentum(u, temp, t, setup) -> Any

Right hand side of momentum equations, excluding pressure gradient (differentiable version).

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# IncompressibleNavierStokes.pressuregradient! — Method.

pressuregradient!(G, p, setup) -> Any

Compute pressure gradient (in-place version).

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# IncompressibleNavierStokes.pressuregradient — Method.

pressuregradient(p, setup) -> Any

Compute pressure gradient (differentiable version).

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# IncompressibleNavierStokes.scalewithvolume! — Method.

scalewithvolume!(p, setup) -> Any

Scale scalar field with volume sizes (in-place version).

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# IncompressibleNavierStokes.scalewithvolume — Method.

scalewithvolume(p, setup) -> Any

Scale scalar field p with volume sizes (differentiable version).

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# IncompressibleNavierStokes.smagorinsky_closure — Method.

smagorinsky_closure(
    setup
) -> IncompressibleNavierStokes.var"#closure#224"

Create Smagorinsky closure model m. The model is called as m(u, θ), where the Smagorinsky constant θ should be a scalar between 0 and 1 (for example θ = 0.1).

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# IncompressibleNavierStokes.smagtensor! — Method.

smagtensor!(σ, u, θ, setup) -> Any

Compute Smagorinsky stress tensors σ[I] (in-place version). The Smagorinsky constant θ should be a scalar between 0 and 1.

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# IncompressibleNavierStokes.tensorbasis! — Method.

tensorbasis!(B, V, u, setup) -> Tuple{Any, Any}

Compute symmetry tensor basis B[1]-B[11] and invariants V[1]-V[5], as specified in [10] in equations (9) and (11). Note that B[1] corresponds to $T_0$ in the paper, and V to $I$.

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# IncompressibleNavierStokes.tensorbasis — Method.

tensorbasis(u, setup) -> Tuple{Tuple, Tuple}

Compute symmetry tensor basis (differentiable version).

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# IncompressibleNavierStokes.total_kinetic_energy — Method.

total_kinetic_energy(u, setup; kwargs...) -> Any

Compute total kinetic energy. The velocity components are interpolated to the volume centers and squared.

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# IncompressibleNavierStokes.vorticity! — Method.

vorticity!(ω, u, setup) -> Any

Compute vorticity field (in-place version).

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# IncompressibleNavierStokes.vorticity — Method.

vorticity(u, setup) -> Any

Compute vorticity field (differentiable version).

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# IncompressibleNavierStokes.laplacian_mat — Method.

laplacian_mat(setup) -> Any

Get matrix for the Laplacian operator.

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