/
theory_relaxation.jl
405 lines (355 loc) · 13.1 KB
/
theory_relaxation.jl
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"""
$(SIGNATURES)
RHS-ODE of BGK equation
"""
function bgk_ode!(df, f::AA{T}, p, t) where {T}
M, τ = p
df .= (M .- f) ./ τ
end
"""
$(SIGNATURES)
RHS-ODE of ES-BGK equation
"""
function esbgk_ode!(df, f::AA{T}, p, t) where {T}
gs = p[1:end-1]
τ = p[end]
G = esbgk(f, gs...)
df .= (G .- f) ./ τ
end
"""
$(SIGNATURES)
Calculate collision time with variable hard sphere (VHS) model
"""
vhs_collision_time(ρ, λ, μᵣ, ω) = μᵣ * 2.0 * λ^(1.0 - ω) / ρ
"""
$(SIGNATURES)
For Rykov model prim[end] should be λₜ
"""
vhs_collision_time(prim::AV, muRef, omega) =
vhs_collision_time(prim[1], prim[end], muRef, omega)
# ------------------------------------------------------------
# ν model
# ------------------------------------------------------------
"""
$(SIGNATURES)
Regulator to construct ν-BGK model
_L. Mieussens and H. Struchtrup. Phys. Fluids, 16(8): 2797-2813, 2004._
"""
function νbgk_relaxation_time(τ::T, u, prim::AV, tp)::T where {T}
c = @. abs(u - prim[2])
return @. τ / mieussens_frequency(c, tp)
end
function νbgk_relaxation_time(τ::T, u, v, prim::AV, tp)::T where {T}
c = @. sqrt((u - prim[2])^2 + (v - prim[3])^2)
return @. τ / mieussens_frequency(c, tp)
end
function νbgk_relaxation_time(τ::T, u, v, w, prim::AV, tp)::T where {T}
c = @. sqrt((u - prim[2])^2 + (v - prim[3])^2 + (w - prim[4])^2)
return @. τ / mieussens_frequency(c, tp)
end
mieussens_frequency(η, ::Type{Class{1}}) = @. 0.431587 * η^1.791288
mieussens_frequency(η, ::Type{Class{2}}) = @. 0.0268351 * (1 + 14.2724 * η^2)
mieussens_frequency(η, ::Type{Class{3}}) = @. 0.0365643 * (1 + 10 * η^2.081754)
mieussens_frequency(η, ::Type{Class{4}}) = @. 0.1503991 * (1 + 0.92897 * η^4) # usually this isn't used
"""
$(SIGNATURES)
Regulator to construct ν-Shakhov model
_Yuan & Wu, J. Fluid Mech., 942, 2022._
``ν = A(p/μ)(νeq0(ξ)+2νeq0(0))``
"""
function νshakhov_relaxation_time(τ::T, u, prim::AV)::T where {T}
c = @. abs(u - prim[2])
return @. 1 / 0.0871 * τ / (νeq0(c) + 2 * νeq0(eps()))
end
νeq0(x) = 1.5 * (exp(-x^2) + sqrt(pi) / 2 * (1 / x + 2 * x) * erf(x))
# ------------------------------------------------------------
# AAP model
# ------------------------------------------------------------
"""
$(SIGNATURES)
Calculate mixture primitive variables from AAP model
"""
function aap_hs_prim(prim::AM, tau::AV, mi, ni, me, ne, kn)
mixprim = similar(prim)
if size(prim, 1) == 3
mixprim[1, :] = deepcopy(prim[1, :])
mixprim[2, 1] =
prim[2, 1] +
tau[1] / kn *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 2] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(prim[2, 2] - prim[2, 1])
mixprim[2, 2] =
prim[2, 2] +
tau[2] / kn *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 1] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(prim[2, 1] - prim[2, 2])
mixprim[3, 1] =
1.0 / (
1.0 / prim[end, 1] - 2.0 / 3.0 * (mixprim[2, 1] - prim[2, 1])^2 +
tau[1] / kn * 2.0 * mi / (mi + me) *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 2] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(
1.0 / prim[end, 2] * me / mi - 1.0 / prim[end, 1] +
2.0 / 3.0 * me / mi * (prim[2, 2] - prim[2, 1])^2
)
)
mixprim[3, 2] =
1.0 / (
1.0 / prim[end, 2] - 2.0 / 3.0 * (mixprim[2, 2] - prim[2, 2])^2 +
tau[2] / kn * 2.0 * me / (mi + me) *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 1] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(
1.0 / prim[end, 1] * mi / me - 1.0 / prim[end, 2] +
2.0 / 3.0 * mi / me * (prim[2, 1] - prim[2, 2])^2
)
)
elseif size(prim, 1) == 4
mixprim[1, :] = deepcopy(prim[1, :])
mixprim[2, 1] =
prim[2, 1] +
tau[1] / kn *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 2] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(prim[2, 2] - prim[2, 1])
mixprim[2, 2] =
prim[2, 2] +
tau[2] / kn *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 1] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(prim[2, 1] - prim[2, 2])
mixprim[3, 1] =
prim[3, 1] +
tau[1] / kn *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 2] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(prim[3, 2] - prim[3, 1])
mixprim[3, 2] =
prim[3, 2] +
tau[2] / kn *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 1] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(prim[3, 1] - prim[3, 2])
mixprim[4, 1] =
1.0 / (
1.0 / prim[end, 1] - 2.0 / 3.0 * (mixprim[2, 1] - prim[2, 1])^2 -
2.0 / 3.0 * (mixprim[3, 1] - prim[3, 1])^2 +
tau[1] / kn * 2.0 * mi / (mi + me) *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 2] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(
1.0 / prim[end, 2] * me / mi - 1.0 / prim[end, 1] +
2.0 / 3.0 * me / mi * (prim[2, 2] - prim[2, 1])^2 +
2.0 / 3.0 * me / mi * (prim[3, 2] - prim[3, 1])^2
)
)
mixprim[4, 2] =
1.0 / (
1.0 / prim[end, 2] - 2.0 / 3.0 * (mixprim[2, 2] - prim[2, 2])^2 -
2.0 / 3.0 * (mixprim[3, 2] - prim[3, 2])^2 +
tau[2] / kn * 2.0 * me / (mi + me) *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 1] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(
1.0 / prim[end, 1] * mi / me - 1.0 / prim[end, 2] +
2.0 / 3.0 * mi / me * (prim[2, 1] - prim[2, 2])^2 +
2.0 / 3.0 * mi / me * (prim[3, 1] - prim[3, 2])^2
)
)
elseif size(prim, 1) == 5
mixprim[1, :] = deepcopy(prim[1, :])
mixprim[2, 1] =
prim[2, 1] +
tau[1] / kn *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 2] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(prim[2, 2] - prim[2, 1])
mixprim[2, 2] =
prim[2, 2] +
tau[2] / kn *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 1] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(prim[2, 1] - prim[2, 2])
mixprim[3, 1] =
prim[3, 1] +
tau[1] / kn *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 2] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(prim[3, 2] - prim[3, 1])
mixprim[3, 2] =
prim[3, 2] +
tau[2] / kn *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 1] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(prim[3, 1] - prim[3, 2])
mixprim[4, 1] =
prim[4, 1] +
tau[1] / kn *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 2] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(prim[4, 2] - prim[4, 1])
mixprim[4, 2] =
prim[4, 2] +
tau[2] / kn *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 1] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(prim[4, 1] - prim[4, 2])
mixprim[5, 1] =
1.0 / (
1.0 / prim[end, 1] - 2.0 / 3.0 * (mixprim[2, 1] - prim[2, 1])^2 -
2.0 / 3.0 * (mixprim[3, 1] - prim[3, 1])^2 -
2.0 / 3.0 * (mixprim[4, 1] - prim[4, 1])^2 +
tau[1] / kn * 2.0 * mi / (mi + me) *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 2] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(
1.0 / prim[end, 2] * me / mi - 1.0 / prim[end, 1] +
2.0 / 3.0 * me / mi * (prim[2, 2] - prim[2, 1])^2 +
2.0 / 3.0 * me / mi * (prim[3, 2] - prim[3, 1])^2 +
2.0 / 3.0 * me / mi * (prim[4, 2] - prim[4, 1])^2
)
)
mixprim[5, 2] =
1.0 / (
1.0 / prim[end, 2] - 2.0 / 3.0 * (mixprim[2, 2] - prim[2, 2])^2 -
2.0 / 3.0 * (mixprim[3, 2] - prim[3, 2])^2 -
2.0 / 3.0 * (mixprim[4, 2] - prim[4, 2])^2 +
tau[2] / kn * 2.0 * me / (mi + me) *
(
4.0 * sqrt(2.0) / (3.0 * sqrt(π)) * prim[1, 1] / (ni + ne) / (mi + me) *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2])
) *
(
1.0 / prim[end, 1] * mi / me - 1.0 / prim[end, 2] +
2.0 / 3.0 * mi / me * (prim[2, 1] - prim[2, 2])^2 +
2.0 / 3.0 * mi / me * (prim[3, 1] - prim[3, 2])^2 +
2.0 / 3.0 * mi / me * (prim[4, 1] - prim[4, 2])^2
)
)
else
throw("AAP mixture: dimension dismatch")
end
return mixprim
end
"""
$(SIGNATURES)
Calculate mixture collision time from AAP model
"""
function aap_hs_collision_time(prim::AM, mi, ni, me, ne, kn)
ν = similar(prim, 2)
ν[1] =
prim[1, 1] / (mi * (ni + ne)) * 4.0 * sqrt(π) / 3.0 *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 1]) / (sqrt(2.0) * π * kn) +
prim[1, 2] / (me * (ni + ne)) * 4.0 * sqrt(π) / 3.0 *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2]) / (sqrt(2.0) * π * kn)
ν[2] =
prim[1, 1] / (mi * (ni + ne)) * 4.0 * sqrt(π) / 3.0 *
sqrt(1.0 / prim[end, 1] + 1.0 / prim[end, 2]) / (sqrt(2.0) * π * kn) +
prim[1, 2] / (me * (ni + ne)) * 4.0 * sqrt(π) / 3.0 *
sqrt(1.0 / prim[end, 2] + 1.0 / prim[end, 2]) / (sqrt(2.0) * π * kn)
return 1.0 ./ ν
end
"""
$(SIGNATURES)
Source term of AAP model in DifferentialEquations.jl
"""
function aap_hs_diffeq!(du, u, p, t)
if length(u) == 6
I₁, I₂, I₃, E₁, E₂, E₃ = u
w = [
I₁ E₁
I₂ E₂
I₃ E₃
]
elseif length(u) == 8
I₁, I₂, I₃, I₄, E₁, E₂, E₃, E₄ = u
w = [
I₁ E₁
I₂ E₂
I₃ E₃
I₄ E₄
]
elseif length(u) == 10
I₁, I₂, I₃, I₄, I₅, E₁, E₂, E₃, E₄, E₅ = u
w = [
I₁ E₁
I₂ E₂
I₃ E₃
I₄ E₄
I₅ E₅
]
else
end
τᵢ, τₑ, mi, ni, me, ne, kn, γ = p
τ = [τᵢ, τₑ]
# modified variables
prim = mixture_conserve_prim(w, γ)
mixprim = aap_hs_prim(prim, τ, mi, ni, me, ne, kn)
mixw = mixture_conserve_prim(mixprim, γ)
if length(u) == 6
du[1] = (mixw[1, 1] - I₁) / τᵢ
du[2] = (mixw[2, 1] - I₂) / τᵢ
du[3] = (mixw[3, 1] - I₃) / τᵢ
du[4] = (mixw[1, 2] - E₁) / τₑ
du[5] = (mixw[2, 2] - E₂) / τₑ
du[6] = (mixw[3, 2] - E₃) / τₑ
elseif length(u) == 8
du[1] = (mixw[1, 1] - I₁) / τᵢ
du[2] = (mixw[2, 1] - I₂) / τᵢ
du[3] = (mixw[3, 1] - I₃) / τᵢ
du[4] = (mixw[4, 1] - I₄) / τᵢ
du[5] = (mixw[1, 2] - E₁) / τₑ
du[6] = (mixw[2, 2] - E₂) / τₑ
du[7] = (mixw[3, 2] - E₃) / τₑ
du[8] = (mixw[4, 2] - E₄) / τₑ
elseif length(u) == 10
du[1] = (mixw[1, 1] - I₁) / τᵢ
du[2] = (mixw[2, 1] - I₂) / τᵢ
du[3] = (mixw[3, 1] - I₃) / τᵢ
du[4] = (mixw[4, 1] - I₄) / τᵢ
du[5] = (mixw[5, 1] - I₅) / τᵢ
du[6] = (mixw[1, 2] - E₁) / τₑ
du[7] = (mixw[2, 2] - E₂) / τₑ
du[8] = (mixw[3, 2] - E₃) / τₑ
du[9] = (mixw[4, 2] - E₄) / τₑ
du[10] = (mixw[5, 2] - E₅) / τₑ
end
return nothing
end