Internal Functions

Swalbe.equilibrium! — Method

equilibrium!(feq, height, velocity, gravity)

Calculation of the equilibrium distribution feq for the shallow water lattice Boltzmann method.

Arguments

feq :: Array{<:Number,3}: Equilibrium distribution function, to be calculated
height :: Array{<:Number,2}: The height field $h(\mathbf{x},t)$
velocity :: Array{<:Number,2}: x-component of the macroscopic velocity
gravity <: Number: Strength of the gravitational acceleration in lattice units

Examples

julia> using Swalbe, Test

julia> feq = zeros(10,3); ρ = ones(10); u = fill(0.1,10);

julia> Swalbe.equilibrium!(feq, ρ, u, 0.1) # Supply dummy u^2 as well.

julia> feq[:,1]
10-element Vector{Float64}:
 0.94
 0.94
 0.94
 0.94
 0.94
 0.94
 0.94
 0.94
 0.94
 0.94

julia> @test all(feq[:,1] .≈ 1 - 0.1/2 - 0.01)
Test Passed

source

Swalbe.equilibrium! — Method

equilibrium!(feq, height,velocityx, velocityy, vsquare, gravity)

Calculation of the equilibrium distribution feq for the shallow water lattice Boltzmann method.

Arguments

feq :: Array{<:Number,3}: Equilibrium distribution function, to be calculated
height :: Array{<:Number,2}: The height field $h(\mathbf{x},t)$
velocityx :: Array{<:Number,2}: x-component of the macroscopic velocity
velocityy :: Array{<:Number,2}: y-component of the macroscopic velocity
vsquare :: Array{<:Number,2}: Dummy array that is preallocated to be filled with the square of the velocity vector
gravity <: Number: Strength of the gravitational acceleration in lattice units

Mathematics

The detailed motivation and derivation can be found in the article of Salmon. Similar to the standard Navier-Stokes approximating lattice Boltzmann methods the equilibrium distribution feq is an expansion to second order in velocity u. If you want it is an slow mode part of the shallow water theory and thus the equilibrium is given as

$f_i^{\text{eq}} = h \bigg(1 - \frac{5}{6}g h - \frac{2}{3}u^2\bigg),\quad i=0 \newline f_i^{\text{eq}} = w_i h \bigg(g h + 3 \mathbf{c}_i\cdot\mathbf{u} + \frac{9}{2}(\mathbf{c}_i\cdot\mathbf{u})^2) + \frac{3}{2}u^2\bigg),\quad else$

where $g$ is the gravitational acceleration (in lattice units) and $w_i, \mathbf{c}_i$ are the weights and lattice velocities.

It has been shown that it is possible to get rid of the gravity driven term in the equilibrium distribution, thus

$f_i^{\text{eq}} = h \bigg(1 - \frac{2}{3}u^2\bigg),\quad i=0 \newline f_i^{\text{eq}} = w_i h \bigg(3 \mathbf{c}_i\cdot\mathbf{u} + \frac{9}{2}(\mathbf{c}_i\cdot\mathbf{u})^2) + \frac{3}{2}u^2\bigg),\quad else$

then of course the topography gradient has to be included as a force term.

Examples

julia> using Swalbe, Test

julia> feq = zeros(5,5,9); ρ = ones(5,5); ux = fill(0.1,5,5); uy = zeros(5,5);

julia> Swalbe.equilibrium!(feq, ρ, ux, uy, zeros(5,5), 0.1) # Supply dummy u^2 as well.

julia> feq[:,:,1]
5×5 Matrix{Float64}:
 0.91  0.91  0.91  0.91  0.91
 0.91  0.91  0.91  0.91  0.91
 0.91  0.91  0.91  0.91  0.91
 0.91  0.91  0.91  0.91  0.91
 0.91  0.91  0.91  0.91  0.91

julia> Swalbe.equilibrium!(feq, ρ, ux, uy, zeros(5,5), 0.0) # Supply dummy u^2 as well.

julia> @test all(feq[:,:,1] .≈ 1 - 2/3 * 0.01)
Test Passed

References

source

Swalbe.BGKandStream! — Method

BGKandStream!(fout, feq, ftemp, Fx, Fy, τ)

Performs a BGK collision operation with a WFM forcecorrection and a subsequent streaming of the resulting populations.

Arguments

fout :: Array{<:Number,3}: Streamed distribution after the collision processes
feq :: Array{<:Number,3}: Equilibrium distribution, computed with equilibrium!
ftemp :: Array{<:Number,3}: Temporary distribution from the time step before, only useful if $\tau \neq 1$
Fx :: Array{<:Number,2}: Sum of forces acting on the fluid in x-direction
Fy :: Array{<:Number,2}: Sum of forces acting on the fluid in y-direction
τ <: Number: Relaxtion time, if not supplied $\tau = 1$ assumed

Mathematics

The lattice Boltzmann equation in its discretized format is relatively simple to write down

$f_{\alpha}(\mathbf{x}+\mathbf{e}_{\alpha}\Delta t, t+\Delta t) - f_{\alpha}(\mathbf{x}, t) = -\frac{1}{\tau}(f_{\alpha}(\mathbf{x}, t) - f^{\text{eq}}_{\alpha}(\mathbf{x}, t)) + \Delta t \mathcal{S}_{\alpha},$

where the collision kernel is approximated with a BKG single relaxation time (SRT)

$\Omega_{\alpha} = \frac{1}{\tau}(f_{\alpha}(\mathbf{x}, t) - f^{\text{eq}}_{\alpha}(\mathbf{x}, t),$

and a source term $\mathcal{S}$ which is given by

$\mathcal{S}_{\alpha} = \frac{3 w_{\alpha}}{e_{\alpha x}e_{\alpha x}+e_{\alpha y}e_{\alpha y}}\mathbf{e_{\alpha}}\cdot\mathbf{F}_{\alpha} .$

The term $e_{\alpha x}e_{\alpha x}+e_{\alpha y}e_{\alpha y}$ is either zero for the zeroth population, 1 for the first four populations or 2 for the remaining ones.

Examples

julia> using Swalbe

julia> feq = ones(5,5,9); ftemp = zeros(5,5,9); fout = zeros(5,5,9);

julia> feq[1,1,:] .= 2.0 # To check the streaming process 
9-element view(::Array{Float64, 3}, 1, 1, :) with eltype Float64:
 2.0
 2.0
 2.0
 2.0
 2.0
 2.0
 2.0
 2.0
 2.0

julia> Swalbe.BGKandStream!(fout, feq, ftemp, zeros(5,5), zeros(5,5), 1.0)

julia> fout[:,:,6] # The value 2 should have moved one down and one to the right!
5×5 Matrix{Float64}:
 1.0  1.0  1.0  1.0  1.0
 1.0  2.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0

References