Sutherlands Law for temperature dependent viscosity

examples/dgmulti_2d/elixir_navierstokes_lid_driven_cavity.jl

@@ -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

examples/dgmulti_3d/elixir_navierstokes_taylor_green_vortex.jl

@@ -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())
 
 """

examples/p4est_2d_dgsem/elixir_navierstokes_lid_driven_cavity.jl

@@ -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

examples/p4est_2d_dgsem/elixir_navierstokes_lid_driven_cavity_amr.jl

@@ -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

examples/p4est_3d_dgsem/elixir_navierstokes_blast_wave_amr.jl

@@ -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)

examples/p4est_3d_dgsem/elixir_navierstokes_taylor_green_vortex.jl

@@ -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())
 
 """

examples/p4est_3d_dgsem/elixir_navierstokes_taylor_green_vortex_amr.jl

@@ -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())
 
 """

examples/tree_1d_dgsem/elixir_navierstokes_convergence_periodic.jl

@@ -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
 

examples/tree_2d_dgsem/elixir_navierstokes_lid_driven_cavity.jl

@@ -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

examples/tree_2d_dgsem/elixir_navierstokes_shearlayer_amr.jl

@@ -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())
 
 """

examples/tree_2d_dgsem/elixir_navierstokes_taylor_green_vortex.jl

@@ -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())
 
 """

examples/tree_2d_dgsem/elixir_navierstokes_taylor_green_vortex_sutherland.jl

@@ -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

examples/tree_3d_dgsem/elixir_navierstokes_taylor_green_vortex.jl

@@ -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())
 
 """

src/equations/compressible_navier_stokes.jl

@@ -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 the opposite case, `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)

src/equations/compressible_navier_stokes_1d.jl

@@ -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
@@ -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
@@ -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
 
@@ -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
@@ -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)