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Sutherlands Law for temperature dependent viscosity (#1808)
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* sample 2D

* bug fixes

* typo

* sutherlands 1d

* sutherlands 3d

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* variable mu

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* comment

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* Update examples/tree_2d_dgsem/elixir_navierstokes_taylor_green_vortex_sutherland.jl

* remove todo

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Co-authored-by: Andrew Winters <[email protected]>

* test vals

* non-functionalized mu

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* Update src/equations/compressible_navier_stokes.jl

Co-authored-by: Michael Schlottke-Lakemper <[email protected]>

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---------

Co-authored-by: Andrew Winters <[email protected]>
Co-authored-by: Michael Schlottke-Lakemper <[email protected]>
Co-authored-by: Jesse Chan <[email protected]>
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5 changes: 2 additions & 3 deletions examples/dgmulti_2d/elixir_navierstokes_lid_driven_cavity.jl
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Expand Up @@ -4,12 +4,11 @@ using Trixi
###############################################################################
# semidiscretization of the ideal compressible Navier-Stokes equations

# TODO: parabolic; unify names of these accessor functions
prandtl_number() = 0.72
mu() = 0.001
mu = 0.001

equations = CompressibleEulerEquations2D(1.4)
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu(),
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu,
Prandtl = prandtl_number())

# Create DG solver with polynomial degree = 3 and (local) Lax-Friedrichs/Rusanov flux as surface flux
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Expand Up @@ -5,12 +5,11 @@ using Trixi
###############################################################################
# semidiscretization of the compressible Navier-Stokes equations

# TODO: parabolic; unify names of these accessor functions
prandtl_number() = 0.72
mu() = 6.25e-4 # equivalent to Re = 1600
mu = 6.25e-4 # equivalent to Re = 1600

equations = CompressibleEulerEquations3D(1.4)
equations_parabolic = CompressibleNavierStokesDiffusion3D(equations, mu = mu(),
equations_parabolic = CompressibleNavierStokesDiffusion3D(equations, mu = mu,
Prandtl = prandtl_number())

"""
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Expand Up @@ -4,12 +4,11 @@ using Trixi
###############################################################################
# semidiscretization of the ideal compressible Navier-Stokes equations

# TODO: parabolic; unify names of these accessor functions
prandtl_number() = 0.72
mu() = 0.001
mu = 0.001

equations = CompressibleEulerEquations2D(1.4)
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu(),
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu,
Prandtl = prandtl_number())

# Create DG solver with polynomial degree = 3 and (local) Lax-Friedrichs/Rusanov flux as surface flux
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Expand Up @@ -4,12 +4,11 @@ using Trixi
###############################################################################
# semidiscretization of the ideal compressible Navier-Stokes equations

# TODO: parabolic; unify names of these accessor functions
prandtl_number() = 0.72
mu() = 0.001
mu = 0.001

equations = CompressibleEulerEquations2D(1.4)
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu(),
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu,
Prandtl = prandtl_number())

# Create DG solver with polynomial degree = 3 and (local) Lax-Friedrichs/Rusanov flux as surface flux
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5 changes: 2 additions & 3 deletions examples/p4est_3d_dgsem/elixir_navierstokes_blast_wave_amr.jl
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Expand Up @@ -5,12 +5,11 @@ using Trixi
###############################################################################
# semidiscretization of the compressible Navier-Stokes equations

# TODO: parabolic; unify names of these accessor functions
prandtl_number() = 0.72
mu() = 6.25e-4 # equivalent to Re = 1600
mu = 6.25e-4 # equivalent to Re = 1600

equations = CompressibleEulerEquations3D(1.4)
equations_parabolic = CompressibleNavierStokesDiffusion3D(equations, mu = mu(),
equations_parabolic = CompressibleNavierStokesDiffusion3D(equations, mu = mu,
Prandtl = prandtl_number())

function initial_condition_3d_blast_wave(x, t, equations::CompressibleEulerEquations3D)
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Expand Up @@ -5,12 +5,11 @@ using Trixi
###############################################################################
# semidiscretization of the compressible Navier-Stokes equations

# TODO: parabolic; unify names of these accessor functions
prandtl_number() = 0.72
mu() = 6.25e-4 # equivalent to Re = 1600
mu = 6.25e-4 # equivalent to Re = 1600

equations = CompressibleEulerEquations3D(1.4)
equations_parabolic = CompressibleNavierStokesDiffusion3D(equations, mu = mu(),
equations_parabolic = CompressibleNavierStokesDiffusion3D(equations, mu = mu,
Prandtl = prandtl_number())

"""
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Original file line number Diff line number Diff line change
Expand Up @@ -5,12 +5,11 @@ using Trixi
###############################################################################
# semidiscretization of the compressible Navier-Stokes equations

# TODO: parabolic; unify names of these accessor functions
prandtl_number() = 0.72
mu() = 6.25e-4 # equivalent to Re = 1600
mu = 6.25e-4 # equivalent to Re = 1600

equations = CompressibleEulerEquations3D(1.4)
equations_parabolic = CompressibleNavierStokesDiffusion3D(equations, mu = mu(),
equations_parabolic = CompressibleNavierStokesDiffusion3D(equations, mu = mu,
Prandtl = prandtl_number())

"""
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Expand Up @@ -5,7 +5,6 @@ using Trixi
###############################################################################
# semidiscretization of the compressible Navier-Stokes equations

# TODO: parabolic; unify names of these accessor functions
prandtl_number() = 0.72
mu() = 6.25e-4 # equivalent to Re = 1600

Expand Down
Original file line number Diff line number Diff line change
Expand Up @@ -4,12 +4,11 @@ using Trixi
###############################################################################
# semidiscretization of the ideal compressible Navier-Stokes equations

# TODO: parabolic; unify names of these accessor functions
prandtl_number() = 0.72
mu() = 0.001
mu = 0.001

equations = CompressibleEulerEquations2D(1.4)
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu(),
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu,
Prandtl = prandtl_number())

# Create DG solver with polynomial degree = 3 and (local) Lax-Friedrichs/Rusanov flux as surface flux
Expand Down
5 changes: 2 additions & 3 deletions examples/tree_2d_dgsem/elixir_navierstokes_shearlayer_amr.jl
Original file line number Diff line number Diff line change
Expand Up @@ -5,12 +5,11 @@ using Trixi
###############################################################################
# semidiscretization of the compressible Navier-Stokes equations

# TODO: parabolic; unify names of these accessor functions
prandtl_number() = 0.72
mu() = 1.0 / 3.0 * 10^(-5) # equivalent to Re = 30,000
mu = 1.0 / 3.0 * 10^(-4) # equivalent to Re = 30,000

equations = CompressibleEulerEquations2D(1.4)
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu(),
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu,
Prandtl = prandtl_number())

"""
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Original file line number Diff line number Diff line change
Expand Up @@ -5,12 +5,11 @@ using Trixi
###############################################################################
# semidiscretization of the compressible Navier-Stokes equations

# TODO: parabolic; unify names of these accessor functions
prandtl_number() = 0.72
mu() = 6.25e-4 # equivalent to Re = 1600
mu = 6.25e-4 # equivalent to Re = 1600

equations = CompressibleEulerEquations2D(1.4)
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu(),
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu,
Prandtl = prandtl_number())

"""
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@@ -0,0 +1,94 @@

using OrdinaryDiffEq
using Trixi

###############################################################################
# semidiscretization of the compressible Navier-Stokes equations

prandtl_number() = 0.72

# Use Sutherland's law for a temperature-dependent viscosity.
# For details, see e.g.
# Frank M. White: Viscous Fluid Flow, 2nd Edition.
# 1991, McGraw-Hill, ISBN, 0-07-069712-4
# Pages 28 and 29.
@inline function mu(u, equations)
T_ref = 291.15

R_specific_air = 287.052874
T = R_specific_air * Trixi.temperature(u, equations)

C_air = 120.0
mu_ref_air = 1.827e-5

return mu_ref_air * (T_ref + C_air) / (T + C_air) * (T / T_ref)^1.5
end

equations = CompressibleEulerEquations2D(1.4)
equations_parabolic = CompressibleNavierStokesDiffusion2D(equations, mu = mu,
Prandtl = prandtl_number())

"""
initial_condition_taylor_green_vortex(x, t, equations::CompressibleEulerEquations2D)
The classical viscous Taylor-Green vortex in 2D.
This forms the basis behind the 3D case found for instance in
- Jonathan R. Bull and Antony Jameson
Simulation of the Compressible Taylor Green Vortex using High-Order Flux Reconstruction Schemes
[DOI: 10.2514/6.2014-3210](https://doi.org/10.2514/6.2014-3210)
"""
function initial_condition_taylor_green_vortex(x, t,
equations::CompressibleEulerEquations2D)
A = 1.0 # magnitude of speed
Ms = 0.1 # maximum Mach number

rho = 1.0
v1 = A * sin(x[1]) * cos(x[2])
v2 = -A * cos(x[1]) * sin(x[2])
p = (A / Ms)^2 * rho / equations.gamma # scaling to get Ms
p = p + 1.0 / 4.0 * A^2 * rho * (cos(2 * x[1]) + cos(2 * x[2]))

return prim2cons(SVector(rho, v1, v2, p), equations)
end
initial_condition = initial_condition_taylor_green_vortex

volume_flux = flux_ranocha
solver = DGSEM(polydeg = 3, surface_flux = flux_hllc,
volume_integral = VolumeIntegralFluxDifferencing(volume_flux))

coordinates_min = (-1.0, -1.0) .* pi
coordinates_max = (1.0, 1.0) .* pi
mesh = TreeMesh(coordinates_min, coordinates_max,
initial_refinement_level = 4,
n_cells_max = 100_000)

semi = SemidiscretizationHyperbolicParabolic(mesh, (equations, equations_parabolic),
initial_condition, solver)

###############################################################################
# ODE solvers, callbacks etc.

tspan = (0.0, 20.0)
ode = semidiscretize(semi, tspan)

summary_callback = SummaryCallback()

analysis_interval = 100
analysis_callback = AnalysisCallback(semi, interval = analysis_interval,
save_analysis = true,
extra_analysis_integrals = (energy_kinetic,
energy_internal))

alive_callback = AliveCallback(analysis_interval = analysis_interval)

callbacks = CallbackSet(summary_callback,
analysis_callback,
alive_callback)

###############################################################################
# run the simulation

time_int_tol = 1e-9
sol = solve(ode, RDPK3SpFSAL49(); abstol = time_int_tol, reltol = time_int_tol,
ode_default_options()..., callback = callbacks)
summary_callback() # print the timer summary
Original file line number Diff line number Diff line change
Expand Up @@ -5,12 +5,11 @@ using Trixi
###############################################################################
# semidiscretization of the compressible Navier-Stokes equations

# TODO: parabolic; unify names of these accessor functions
prandtl_number() = 0.72
mu() = 6.25e-4 # equivalent to Re = 1600
mu = 6.25e-4 # equivalent to Re = 1600

equations = CompressibleEulerEquations3D(1.4)
equations_parabolic = CompressibleNavierStokesDiffusion3D(equations, mu = mu(),
equations_parabolic = CompressibleNavierStokesDiffusion3D(equations, mu = mu,
Prandtl = prandtl_number())

"""
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14 changes: 14 additions & 0 deletions src/equations/compressible_navier_stokes.jl
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Expand Up @@ -62,3 +62,17 @@ Under `GradientVariablesEntropy`, the Navier-Stokes discretization is provably e
"""
struct GradientVariablesPrimitive end
struct GradientVariablesEntropy end

"""
dynamic_viscosity(u, equations)
Wrapper for the dynamic viscosity that calls
`dynamic_viscosity(u, equations.mu, equations)`, which dispatches on the type of
`equations.mu`.
For constant `equations.mu`, i.e., `equations.mu` is of `Real`-type it is returned directly.
In all other cases, `equations.mu` is assumed to be a function with arguments
`u` and `equations` and is called with these arguments.
"""
dynamic_viscosity(u, equations) = dynamic_viscosity(u, equations.mu, equations)
dynamic_viscosity(u, mu::Real, equations) = mu
dynamic_viscosity(u, mu::T, equations) where {T} = mu(u, equations)
26 changes: 16 additions & 10 deletions src/equations/compressible_navier_stokes_1d.jl
Original file line number Diff line number Diff line change
Expand Up @@ -21,6 +21,9 @@ the [`CompressibleEulerEquations1D`](@ref).
Fluid properties such as the dynamic viscosity ``\mu`` can be provided in any consistent unit system, e.g.,
[``\mu``] = kg m⁻¹ s⁻¹.
The viscosity ``\mu`` may be a constant or a function of the current state, e.g.,
depending on temperature (Sutherland's law): ``\mu = \mu(T)``.
In the latter case, the function `mu` needs to have the signature `mu(u, equations)`.
The particular form of the compressible Navier-Stokes implemented is
```math
Expand Down Expand Up @@ -80,7 +83,7 @@ where
w_2 = \frac{\rho v1}{p},\, w_3 = -\frac{\rho}{p}
```
"""
struct CompressibleNavierStokesDiffusion1D{GradientVariables, RealT <: Real,
struct CompressibleNavierStokesDiffusion1D{GradientVariables, RealT <: Real, Mu,
E <: AbstractCompressibleEulerEquations{1}} <:
AbstractCompressibleNavierStokesDiffusion{1, 3, GradientVariables}
# TODO: parabolic
Expand All @@ -89,7 +92,7 @@ struct CompressibleNavierStokesDiffusion1D{GradientVariables, RealT <: Real,
gamma::RealT # ratio of specific heats
inv_gamma_minus_one::RealT # = inv(gamma - 1); can be used to write slow divisions as fast multiplications

mu::RealT # viscosity
mu::Mu # viscosity
Pr::RealT # Prandtl number
kappa::RealT # thermal diffusivity for Fick's law

Expand All @@ -103,16 +106,17 @@ function CompressibleNavierStokesDiffusion1D(equations::CompressibleEulerEquatio
gradient_variables = GradientVariablesPrimitive())
gamma = equations.gamma
inv_gamma_minus_one = equations.inv_gamma_minus_one
μ, Pr = promote(mu, Prandtl)

# Under the assumption of constant Prandtl number the thermal conductivity
# constant is kappa = gamma μ / ((gamma-1) Pr).
# constant is kappa = gamma μ / ((gamma-1) Prandtl).
# Important note! Factor of μ is accounted for later in `flux`.
kappa = gamma * inv_gamma_minus_one / Pr
# This avoids recomputation of kappa for non-constant μ.
kappa = gamma * inv_gamma_minus_one / Prandtl

CompressibleNavierStokesDiffusion1D{typeof(gradient_variables), typeof(gamma),
typeof(mu),
typeof(equations)}(gamma, inv_gamma_minus_one,
μ, Pr, kappa,
mu, Prandtl, kappa,
equations,
gradient_variables)
end
Expand Down Expand Up @@ -159,10 +163,12 @@ function flux(u, gradients, orientation::Integer,
# in the implementation
q1 = equations.kappa * dTdx

# Constant dynamic viscosity is copied to a variable for readability.
# Offers flexibility for dynamic viscosity via Sutherland's law where it depends
# on temperature and reference values, Ts and Tref such that mu(T)
mu = equations.mu
# In the simplest cases, the user passed in `mu` or `mu()`
# (which returns just a constant) but
# more complex functions like Sutherland's law are possible.
# `dynamic_viscosity` is a helper function that handles both cases
# by dispatching on the type of `equations.mu`.
mu = dynamic_viscosity(u, equations)

# viscous flux components in the x-direction
f1 = zero(rho)
Expand Down
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