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 [5] 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 [6] 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.applypressure Method

applypressure(u, p, setup) -> Any

Subtract pressure gradient (differentiable 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; use_viscosity) -> Any

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

Keyword arguments:

• with_viscosity = true: Include viscosity in the operator.

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IncompressibleNavierStokes.diffusion Method

diffusion(u, setup; kwargs...) -> 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 [6].

Differentiable version.

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IncompressibleNavierStokes.get_scale_numbers Method

get_scale_numbers(
    u,
    setup
) -> NamedTuple{(:uavg, :ϵ, :η, :λ, :Reλ, :L, :τ, :Re_int), <:NTuple{8, Any}}

Get the following dimensional scale numbers [7]:

• 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 [8].

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.qcrit Method

qcrit(u, setup) -> Any

Compute $Q$, the second invariant of the velocity gradient tensor.

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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#182"

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.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.unit_cartesian_indices Method

unit_cartesian_indices(D) -> Any

Get tuple of all unit vectors as Cartesian indices.

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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.lastdimcontract Method

lastdimcontract(a, b, setup) -> Any

Compute c[I] = sum_i a[I, i] * b[I, i], where c[:, i] and b[:, i] are tensor fields (elements are SMatrixes), a[:, i] is a scalar field, and the i dimension is contracted (multiple channels).

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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 [9] 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{Any, Any}

Compute symmetry tensor basis (differentiable version).

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