Reference

ParticleField(maxparticles::Int, R=FLOAT_TYPE; <keyword arguments>)

Create a new particle field with maxparticles particles. The particle field is created with the default values for the other parameters.

Arguments

• maxparticles::Int: Maximum number of particles in the field.
• R=FLOAT_TYPE: Type of the particle field. Default is FLOAT_TYPE.

Keyword Arguments

• formulation::Formulation=ReformulatedVPM{FLOAT_TYPE}(0, 1/5): VPM governing equations. Default is reformulated to conserve mass for a tube and angular momentum for a sphere.
• viscous::ViscousScheme=Inviscid(): Viscous scheme. Note that with rVPM formulation, artificial viscosity is not needed for numerical stability, as is common in VPM.
• np::Int=0: Number of particles currently in the field.
• nt::Int=0: Current time step number.
• t::Real=0: Current simulation time.
• transposed::Bool=true: If true, the transposed scheme of the stretching term is used (recommended for stability).
• fmm::FMM: Fast multipole method tuning and auto-tuning settings.
• M::Array{R, 1}=zeros(R, 4): Auxilliary memory for computations. Should not be modified for most purposes.
• SFS::SubFilterScale=NoSFS{FLOAT_TYPE}(): Subfilter-scale turbulence model.
• kernel::Kernel=Kernel(zeta_gauserf, g_gauserf, dgdr_gauserf, g_dgdr_gauserf): Regularization scheme. Default is Gaussian smoothing of the vorticity field.
• UJ=UJ_fmm: Method used to compute the $N$-body problem. Default uses the fast multipole method to achieve $O(N)$ complexity.
• Uinf::Function=(t) -> [0.0,0.0,0.0]: Uniform freestream velocity function Uinf(t).
• relaxation::Relaxation=Relaxation(relax_pedrizzetti, 1, 0.3): Relaxation scheme to re-align the vorticity field to be divergence-free.
• integration::Tintegration=rungekutta3: Time integration scheme. Default is a Runge-Kutta 3rd order, low-memory scheme.
• useGPU::Int: Run on GPU if >0, CPU if 0. Default is 0. (Experimental and does not accelerate SFS calculations)

rVPM

Alias for the reformulated VPM formulation. Enforces conservation of mass and momentum for a spherical fluid element and is the default formulation (f=0, g=1/5)

cVPM

Alias for the classic VPM formulation.

formulation_tube_continuity

Alias for mass conserving tube formulation. Enforces conservation of mass for a vortex tube (f=1/2, g=0)

formulation_tube_momentum

Alias for momentum conserving tube formulation. Enforces conservation of momentum for a vortex tube (f=1/4, g=1/4)

Inviscid()

Creates an inviscid scheme with zero kinematic viscosity.

CoreSpreading(nu, sgm0, zeta::Tzeta=zeta_fmm; <keyword arguments>)

Creates a core spreading viscous scheme with the given parameters.

Arguments

• nu::Real Kinematic viscosity.
• sgm0::Real Core size after reset.
• zeta::Function = zeta_fmm Basis function evaluation method.
• beta::Real = 1.5 Maximum core size growth σ/σ_0.
• itmax::Int = 15 Maximum number of RBF iterations.
• tol::Real = 1e-3 RBF interpolation tolerance.
• iterror::Bool = true Throw error if RBF didn't converge.
• verbose::Bool = false Verbose on RBF interpolation.
• v_lvl::Int = 2 Verbose printing tab level.
• debug::Bool = false Print verbose for debugging.
• rbf::Function = rbf_conjugategradient RBF function.
• rr0s::Array{R, 1} = zeros(R, 3) Initial field residuals.
• rrs::Array{R, 1} = zeros(R, 3) Current field residuals.
• prev_rrs::Array{R, 1} = zeros(R, 3) Previous field residuals.
• pAps::Array{R, 1} = zeros(R, 3) pAp product.
• alphas::Array{R, 1} = zeros(R, 3) Alpha coefficients.
• betas::Array{R, 1} = zeros(R, 3)
• flags::Array{Bool, 1} = zeros(Bool, 3)

ParticleStrengthExchange(nu; <keyword arguments>)

Creates a particle strength exchange viscous scheme with the given parameters.

Arguments

• nu::Real Kinematic viscosity.
• recalculate_vols::Bool = true Whether to recalculate particle volumes.

`FMM(; p::Int=4, ncrit::Int=10, theta::Real=0.5, shrink_recenter::Bool=true,
      relative_tolerance::Real=1e-3, absolute_tolerance::Real=1e-6,
      autotune_p::Bool=true, autotune_ncrit::Bool=true,
      autotune_reg_error::Bool=true, default_rho_over_sigma::Real=1.0)`

Parameters for FMM solver.

Arguments

• p : Order of multipole expansion.
• ncrit : Maximum number of particles per leaf.
• theta : Neighborhood criterion. This criterion defines the distance where the far field starts. The criterion is that if θ*r < R1+R2 the interaction between two cells is resolved through P2P, where r is the distance between cell centers, and R1 and R2 are each cell radius. This means that at θ=1, P2P is done only on cells that have overlap; at θ=0.5, P2P is done on cells that their distance is less than double R1+R2; at θ=0.25, P2P is done on cells that their distance is less than four times R1+R2; at θ=0, P2P is done on all cells.
• shrink_recenter : If true, shrink and recenter multipole expansions to account for nonzero particle radius.
• relative_tolerance : Relative error tolerance for FMM calls.
• absolute_tolerance : Absolute error tolerance fallback for FMM calls in case relative tolerance becomes too small.
• autotune_p : If true, automatically adjust p to optimize performance.
• autotune_ncrit : If true, automatically adjust ncrit to optimize performance.
• autotune_reg_error : If true, constrain regularization error in FMM calls.
• default_rho_over_sigma : Default value for ρ/σ in FMM calls (unused if autotune_reg_error is true).

`noSFS = NoSFS{FLOAT_TYPE}()`

Alias for the no subfilter-scale model.

`SFS_Cs_nobackscatter = ConstantSFS(Estr_fmm; Cs=1.0, clippings=(clipping_backscatter,))`

Alias for the Constant SFS model with no backscatter.

`SFS_Cd_twolevel_nobackscatter = DynamicSFS(Estr_fmm, pseudo3level_beforeUJ, pseudo3level_positive_afterUJ; alpha=0.999, clippings=(clipping_backscatter,))`

Alias for the Dynamic SFS model with two levels and no backscatter. This is the recommended SFS model for high fidelity modeling.

`SFS_Cd_threelevel_nobackscatter = DynamicSFS(Estr_fmm, pseudo3level_beforeUJ, pseudo3level_positive_afterUJ; alpha=0.667, clippings=(clipping_backscatter,))`

Alias for the Dynamic SFS model with three levels and no backscatter. This is similar to the two level version but uses a lower value of alpha (0.667).

singular

Alias for the singular kernel.

gaussian

Alias for the Gaussian kernel.

gaussianerf

Alias for the Gaussian error function kernel.

winckelmans

Alias for the Winckelmans kernel.

norelaxation

Alias for the no relaxation scheme.

pedrizzetti

Alias for the Pedrizzetti relaxation scheme.

correctedpedrizzetti

Alias for the corrected Pedrizzetti relaxation scheme. Is a modification to the pedrizzetti relaxation that preserves the vortex strength magnitude.

euler(pfield::ParticleField, dt::Real; relax::Bool=false, custom_UJ=nothing)

Convects the pfield by timestep dt using a forward Euler step.

Arguments

• pfield::ParticleField The particle field to integrate.
• dt::Real The time step.
• relax::Bool Whether to apply relaxation (default: false).
• custom_UJ Optional custom function for updating U and J.

Steps the field forward in time by dt in a third-order low-storage Runge-Kutta integration scheme. See Notebook entry 20180105.

rungekutta3(pfield::ParticleField, dt::Real; relax::Bool=false, custom_UJ=nothing)

Steps the field forward in time by dt in a third-order low-storage Runge-Kutta integration scheme using the VPM reformulation. See Notebook entry 20180105 (RK integration) and notebook 20210104 (reformulation).

Arguments

• pfield::ParticleField The particle field to integrate.
• dt::R3 The time step.
• relax::Bool Whether to apply relaxation (default: false).
• custom_UJ Optional custom function for updating U and J.

UJ_fmm(pfield)

Calculates the velocity and Jacobian that the field exerts on itself through a fast-multipole approximation, saving U and J on the particles.

NOTE: This method accumulates the calculation on the properties U and J of every particle without previously emptying those properties.

UJ_direct(pfield)

Calculates the velocity and Jacobian that the field exerts on itself by direct particle-to-particle interaction, saving U and J on the particles.

NOTE: This method accumulates the calculation on the properties U and J of every particle without previously emptying those properties.