LES Models API

SmagorinskyModel{T, N, A, Arch}

Classic Smagorinsky (1963) subgrid-scale model.

Mathematical Formulation

The eddy viscosity is:

νₑ = (Cₛ Δ)² |S̄|

where:

• Cₛ is the Smagorinsky constant (typically 0.1-0.2)
• Δ is the filter width (grid spacing)
• |S̄| = √(2 S̄ᵢⱼ S̄ᵢⱼ) is the strain rate magnitude

Fields

• C_s::T: Smagorinsky constant
• filter_width::NTuple{N, T}: Filter width in each direction (Δx, Δy, ...)
• eddy_viscosity::A: Cached eddy viscosity field (Array or CuArray)
• strain_magnitude::A: Cached |S̄| field
• architecture::Arch: CPU() or GPU() architecture

Example

# Create model for 256³ domain with Δ = 2π/256
model = SmagorinskyModel(
    C_s = 0.17,
    filter_width = (2π/256, 2π/256, 2π/256),
    field_size = (256, 256, 256)
)

# Create GPU model
model_gpu = SmagorinskyModel(
    C_s = 0.17,
    filter_width = (2π/256, 2π/256, 2π/256),
    field_size = (256, 256, 256),
    architecture = GPU()
)

# Compute eddy viscosity from velocity gradients
compute_eddy_viscosity!(model, ∂u∂x, ∂u∂y, ∂u∂z, ∂v∂x, ∂v∂y, ∂v∂z, ∂w∂x, ∂w∂y, ∂w∂z)

# Access the result
νₑ = get_eddy_viscosity(model)

AMDModel{T, N, A, Arch}

Anisotropic Minimum Dissipation model (Rozema et al., 2015).

Mathematical Formulation

The eddy viscosity is:

νₑ = max(0, νₑ†)

where the predictor is:

νₑ† = -C (Δₖ² ∂uᵢ/∂xₖ ∂uⱼ/∂xₖ Sᵢⱼ) / (∂uₘ/∂xₙ ∂uₘ/∂xₙ)

Key features:

• Uses anisotropic filter widths Δₖ in each direction
• Automatically switches off in laminar/transitional regions
• Provides minimum dissipation required for subgrid energy transfer
• No explicit filtering or test-filtering required

Fields

• C::T: Poincaré constant (model constant)
• filter_width::NTuple{N, T}: Anisotropic filter widths (Δx, Δy, Δz)
• eddy_viscosity::A: Cached eddy viscosity field (Array or CuArray)
• eddy_diffusivity::A: Cached eddy diffusivity (for scalars)
• architecture::Arch: CPU() or GPU() architecture

Model Constant Recommendations

Example

# Create AMD model for anisotropic grid
model = AMDModel(
    C = 1/12,  # Spectral method
    filter_width = (2π/256, 2π/256, 2π/64),  # Anisotropic
    field_size = (256, 256, 64)
)

# Create GPU AMD model
model_gpu = AMDModel(
    C = 1/12,
    filter_width = (2π/256, 2π/256, 2π/64),
    field_size = (256, 256, 64),
    architecture = GPU()
)

# Compute eddy viscosity
compute_eddy_viscosity!(model, ∂u∂x, ∂u∂y, ∂u∂z, ∂v∂x, ∂v∂y, ∂v∂z, ∂w∂x, ∂w∂y, ∂w∂z)

References

Rozema, W., Bae, H.J., Moin, P., Verstappen, R. (2015). "Minimum-dissipation models for large-eddy simulation", Physics of Fluids 27, 085107.

compute_eddy_viscosity!(model::SmagorinskyModel, velocity_gradients...)

Compute eddy viscosity from velocity gradient components.

GPU-aware: Uses broadcasting for GPU arrays, optimized SIMD loops for CPU.

2D Case

compute_eddy_viscosity!(model, ∂u∂x, ∂u∂y, ∂v∂x, ∂v∂y)

3D Case

compute_eddy_viscosity!(model, ∂u∂x, ∂u∂y, ∂u∂z, ∂v∂x, ∂v∂y, ∂v∂z, ∂w∂x, ∂w∂y, ∂w∂z)

compute_eddy_viscosity!(model::AMDModel, velocity_gradients...)

Compute AMD eddy viscosity from velocity gradient components.

GPU-aware: Uses broadcasting for GPU arrays, optimized SIMD loops for CPU.

2D Case

compute_eddy_viscosity!(model, ∂u∂x, ∂u∂y, ∂v∂x, ∂v∂y)

3D Case

compute_eddy_viscosity!(model, ∂u∂x, ∂u∂y, ∂u∂z, ∂v∂x, ∂v∂y, ∂v∂z, ∂w∂x, ∂w∂y, ∂w∂z)

The AMD formula uses anisotropic scaling: νₑ† = -C (Δₖ² ∂uᵢ/∂xₖ ∂uⱼ/∂xₖ Sᵢⱼ) / (∂uₘ/∂xₙ ∂uₘ/∂xₙ)

compute_eddy_diffusivity!(model::AMDModel, velocity_gradients..., scalar_gradients...)

Compute eddy diffusivity for scalar transport using AMD model.

GPU-aware: Uses broadcasting for GPU arrays, optimized SIMD loops for CPU.

For a scalar field b with gradient ∇b, the eddy diffusivity is:

κₑ = max(0, κₑ†)

where: κₑ† = -C (Δₖ² ∂v/∂xₖ ∂b/∂xₖ) / (∂b/∂xₙ ∂b/∂xₙ)

For 2D flows, v is the vertical velocity component. For 3D flows (buoyancy-driven), w is the vertical velocity.

compute_sgs_stress(model::EddyViscosityModel, strain_components...)

Compute the deviatoric subgrid stress tensor:

τᵢⱼᵈ = -2 νₑ S̄ᵢⱼ

GPU-aware: Uses broadcasting which works for both CPU and GPU arrays.

2D Output

Returns (τ11, τ12, τ22).

3D Output

Returns (τ11, τ12, τ13, τ22, τ23, τ33).

get_eddy_viscosity(model::EddyViscosityModel)

Return the current eddy viscosity field.

get_eddy_diffusivity(model::AMDModel)

Return the current eddy diffusivity field (AMD model only).

get_filter_width(model::EddyViscosityModel)

Return the filter width(s).

mean_eddy_viscosity(model::EddyViscosityModel)

Compute the domain-averaged eddy viscosity.

max_eddy_viscosity(model::EddyViscosityModel)

Return the maximum eddy viscosity in the domain.

sgs_dissipation(model::EddyViscosityModel, strain_magnitude::AbstractArray)

Compute the subgrid-scale dissipation rate:

εₛₛ = 2 νₑ |S̄|²

GPU-aware: Uses broadcasting which works for both CPU and GPU arrays. Returns the dissipation field.

mean_sgs_dissipation(model::EddyViscosityModel, strain_magnitude::AbstractArray)

Compute domain-averaged SGS dissipation rate. GPU-aware: Uses broadcasting and sum() which work for both CPU and GPU arrays.

set_constant!(model::SmagorinskyModel, C_s::Real)

Update the Smagorinsky constant.

set_constant!(model::AMDModel, C::Real)

Update the AMD Poincaré constant.