PCNSFVFluidEnergyHLLC

Implements the fluid energy flux portion of the porous HLLC discretization.

Overview

This object implements the fluid energy equation portion of the porous HLLC implementation described in detail in PCNSFVHLLC.

Input Parameters

fpFluid properties userobject
C++ Type:UserObjectName
Controllable:No
Description:Fluid properties userobject
variableThe name of the finite volume variable this kernel applies to
C++ Type:NonlinearVariableName
Controllable:No
Description:The name of the finite volume variable this kernel applies to

Required Parameters

blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
boundaries_to_forceThe set of boundaries to force execution of this FVFluxKernel on.
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The set of boundaries to force execution of this FVFluxKernel on.
boundaries_to_not_forceThe set of boundaries to not force execution of this FVFluxKernel on.
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The set of boundaries to not force execution of this FVFluxKernel on.
force_boundary_executionFalseWhether to force execution of this object on the boundary.
Default:False
C++ Type:bool
Controllable:No
Description:Whether to force execution of this object on the boundary.
ghost_layers2The number of layers of elements to ghost.
Default:2
C++ Type:unsigned short
Controllable:No
Description:The number of layers of elements to ghost.
prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_point_neighborsFalseWhether to use point neighbors, which introduces additional ghosting to that used for simple face neighbors.
Default:False
C++ Type:bool
Controllable:No
Description:Whether to use point neighbors, which introduces additional ghosting to that used for simple face neighbors.

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime, system
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

Input Files

(modules/navier_stokes/test/tests/finite_volume/cns/straight_channel_porosity_step/hllc.i)
(modules/navier_stokes/test/tests/finite_volume/cns/mms/1d-with-bcs/varying-eps-hllc.i)
(modules/navier_stokes/test/tests/finite_volume/cns/mms/1d-with-bcs/porous-hllc.i)

(modules/navier_stokes/test/tests/finite_volume/cns/straight_channel_porosity_step/hllc.i)

p_initial=1.01e5
T=273.15
# u refers to the superficial velocity
u_in=1

[GlobalParams]
  fp = fp
  two_term_boundary_expansion = true
[]

[Mesh]
  [cartesian]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = 0
    xmax = 18
    nx = 180
  []
  [to_pt5]
    input = cartesian
    type = SubdomainBoundingBoxGenerator
    bottom_left = '2 0 0'
    top_right = '4 1 0'
    block_id = 1
  []
  [pt5]
    input = to_pt5
    type = SubdomainBoundingBoxGenerator
    bottom_left = '4 0 0'
    top_right = '6 1 0'
    block_id = 2
  []
  [to_pt25]
    input = pt5
    type = SubdomainBoundingBoxGenerator
    bottom_left = '6 0 0'
    top_right = '8 1 0'
    block_id = 3
  []
  [pt25]
    input = to_pt25
    type = SubdomainBoundingBoxGenerator
    bottom_left = '8 0 0'
    top_right = '10 1 0'
    block_id = 4
  []
  [to_pt5_again]
    input = pt25
    type = SubdomainBoundingBoxGenerator
    bottom_left = '10 0 0'
    top_right = '12 1 0'
    block_id = 5
  []
  [pt5_again]
    input = to_pt5_again
    type = SubdomainBoundingBoxGenerator
    bottom_left = '12 0 0'
    top_right = '14 1 0'
    block_id = 6
  []
  [to_one]
    input = pt5_again
    type = SubdomainBoundingBoxGenerator
    bottom_left = '14 0 0'
    top_right = '16 1 0'
    block_id = 7
  []
  [one]
    input = to_one
    type = SubdomainBoundingBoxGenerator
    bottom_left = '16 0 0'
    top_right = '18 1 0'
    block_id = 8
  []
[]

[Modules]
  [FluidProperties]
    [fp]
      type = IdealGasFluidProperties
    []
  []
[]

[Problem]
  fv_bcs_integrity_check = false
[]

[Variables]
  [pressure]
    type = MooseVariableFVReal
    initial_condition = ${p_initial}
  []
  [sup_vel_x]
    type = MooseVariableFVReal
    initial_condition = 1
    scaling = 1e-2
  []
  [T_fluid]
    type = MooseVariableFVReal
    initial_condition = ${T}
    scaling = 1e-5
  []
[]

[AuxVariables]
  [vel_x]
    type = MooseVariableFVReal
  []
  [sup_mom_x]
    type = MooseVariableFVReal
  []
  [rho]
    type = MooseVariableFVReal
  []
  [worst_courant]
    type = MooseVariableFVReal
  []
  [porosity]
    type = MooseVariableFVReal
  []
[]

[AuxKernels]
  [vel_x]
    type = ADMaterialRealAux
    variable = vel_x
    property = vel_x
    execute_on = 'timestep_end'
  []
  [sup_mom_x]
    type = ADMaterialRealAux
    variable = sup_mom_x
    property = superficial_rhou
    execute_on = 'timestep_end'
  []
  [rho]
    type = ADMaterialRealAux
    variable = rho
    property = rho
    execute_on = 'timestep_end'
  []
  [worst_courant]
    type = Courant
    variable = worst_courant
    u = sup_vel_x
    execute_on = 'timestep_end'
  []
  [porosity]
    type = MaterialRealAux
    variable = porosity
    property = porosity
    execute_on = 'timestep_end'
  []
[]

[FVKernels]
  [mass_advection]
    type = PCNSFVMassHLLC
    variable = pressure
  []

  [momentum_advection]
    type = PCNSFVMomentumHLLC
    variable = sup_vel_x
    momentum_component = 'x'
  []
  [eps_grad]
    type = PNSFVPGradEpsilon
    variable = sup_vel_x
    momentum_component = 'x'
    epsilon_function = 'eps'
  []

  [energy_advection]
    type = PCNSFVFluidEnergyHLLC
    variable = T_fluid
  []
[]

[FVBCs]
  [rho_left]
    type = PCNSFVStrongBC
    boundary = 'left'
    variable = pressure
    superficial_velocity = 'ud_in'
    T_fluid = ${T}
    eqn = 'mass'
  []
  [rhou_left]
    type = PCNSFVStrongBC
    boundary = 'left'
    variable = sup_vel_x
    superficial_velocity = 'ud_in'
    T_fluid = ${T}
    eqn = 'momentum'
    momentum_component = 'x'
  []
  [rho_et_left]
    type = PCNSFVStrongBC
    boundary = 'left'
    variable = T_fluid
    superficial_velocity = 'ud_in'
    T_fluid = ${T}
    eqn = 'energy'
  []
  [rho_right]
    type = PCNSFVStrongBC
    boundary = 'right'
    variable = pressure
    pressure = ${p_initial}
    eqn = 'mass'
  []
  [rhou_right]
    type = PCNSFVStrongBC
    boundary = 'right'
    variable = sup_vel_x
    pressure = ${p_initial}
    eqn = 'momentum'
    momentum_component = 'x'
  []
  [rho_et_right]
    type = PCNSFVStrongBC
    boundary = 'right'
    variable = T_fluid
    pressure = ${p_initial}
    eqn = 'energy'
  []

  # Use these to help create more accurate cell centered gradients for cells adjacent to boundaries
  [T_left]
    type = FVDirichletBC
    variable = T_fluid
    value = ${T}
    boundary = 'left'
  []
  [sup_vel_left]
    type = FVDirichletBC
    variable = sup_vel_x
    value = ${u_in}
    boundary = 'left'
  []
  [p_right]
    type = FVDirichletBC
    variable = pressure
    value = ${p_initial}
    boundary = 'right'
  []
[]

[Functions]
  [ud_in]
    type = ParsedVectorFunction
    value_x = '${u_in}'
  []
  [eps]
    type = ParsedFunction
    value = 'if(x < 2, 1,
             if(x < 4, 1 - .5 / 2 * (x - 2),
             if(x < 6, .5,
             if(x < 8, .5 - .25 / 2 * (x - 6),
             if(x < 10, .25,
             if(x < 12, .25 + .25 / 2 * (x - 10),
             if(x < 14, .5,
             if(x < 16, .5 + .5 / 2 * (x - 14),
                1))))))))'
  []
[]

[Materials]
  [var_mat]
    type = PorousPrimitiveVarMaterial
    pressure = pressure
    T_fluid = T_fluid
    superficial_vel_x = sup_vel_x
    porosity = porosity
  []
  [porosity]
    type = GenericFunctionMaterial
    prop_names = 'porosity'
    prop_values = 'eps'
  []
[]

[Executioner]
  solve_type = NEWTON
  line_search = 'bt'
  type = Steady
[]

[Outputs]
  [out]
    type = Exodus
    execute_on = 'final'
  []
  checkpoint = true
[]

[Debug]
  show_var_residual_norms = true
[]

(modules/navier_stokes/test/tests/finite_volume/cns/mms/1d-with-bcs/varying-eps-hllc.i)

[GlobalParams]
  fp = fp
[]

[Mesh]
  [cartesian]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = .1
    xmax = .6
    nx = 2
  []
[]

[Modules]
  [FluidProperties]
    [fp]
      type = IdealGasFluidProperties
    []
  []
[]

[Problem]
  fv_bcs_integrity_check = false
[]

[Variables]
  [pressure]
    type = MooseVariableFVReal
  []
  [sup_mom_x]
    type = MooseVariableFVReal
  []
  [T_fluid]
    type = MooseVariableFVReal
  []
[]

[ICs]
  [pressure]
    type = FunctionIC
    variable = pressure
    function = 'exact_p'
  []
  [sup_mom_x]
    type = FunctionIC
    variable = sup_mom_x
    function = 'exact_rho_ud'
  []
  [T_fluid]
    type = FunctionIC
    variable = T_fluid
    function = 'exact_T'
  []
[]

[FVKernels]
  [mass_advection]
    type = PCNSFVMassHLLC
    variable = pressure
  []
  [mass_fn]
    type = FVBodyForce
    variable = pressure
    function = 'forcing_rho'
  []

  [momentum_x_advection]
    type = PCNSFVMomentumHLLC
    variable = sup_mom_x
    momentum_component = x
  []
  [eps_grad]
    type = PNSFVPGradEpsilon
    variable = sup_mom_x
    momentum_component = 'x'
    epsilon_function = 'eps'
  []
  [momentum_fn]
    type = FVBodyForce
    variable = sup_mom_x
    function = 'forcing_rho_ud'
  []

  [fluid_energy_advection]
    type = PCNSFVFluidEnergyHLLC
    variable = T_fluid
  []
  [energy_fn]
    type = FVBodyForce
    variable = T_fluid
    function = 'forcing_rho_et'
  []
[]

[FVBCs]
  [mass_left]
    variable = pressure
    type = PCNSFVStrongBC
    boundary = left
    T_fluid = 'exact_T'
    superficial_velocity = 'exact_superficial_velocity'
    eqn = 'mass'
  []
  [momentum_left]
    variable = sup_mom_x
    type = PCNSFVStrongBC
    boundary = left
    T_fluid = 'exact_T'
    superficial_velocity = 'exact_superficial_velocity'
    eqn = 'momentum'
    momentum_component = 'x'
  []
  [energy_left]
    variable = T_fluid
    type = PCNSFVStrongBC
    boundary = left
    T_fluid = 'exact_T'
    superficial_velocity = 'exact_superficial_velocity'
    eqn = 'energy'
  []
  [mass_right]
    variable = pressure
    type = PCNSFVStrongBC
    boundary = right
    eqn = 'mass'
    pressure = 'exact_p'
  []
  [momentum_right]
    variable = sup_mom_x
    type = PCNSFVStrongBC
    boundary = right
    eqn = 'momentum'
    momentum_component = 'x'
    pressure = 'exact_p'
  []
  [energy_right]
    variable = T_fluid
    type = PCNSFVStrongBC
    boundary = right
    eqn = 'energy'
    pressure = 'exact_p'
  []
[]

[Materials]
  [var_mat]
    type = PorousMixedVarMaterial
    pressure = pressure
    superficial_rhou = sup_mom_x
    T_fluid = T_fluid
    porosity = porosity
  []
  [porosity]
    type = GenericFunctionMaterial
    prop_names = 'porosity'
    prop_values = 'eps'
  []
[]

[Functions]
[exact_rho]
  type = ParsedFunction
  value = '3.48788261470924*cos(x)'
[]
[forcing_rho]
  type = ParsedFunction
  value = '-3.83667087618017*sin(1.1*x)*cos(1.3*x) - 4.53424739912202*sin(1.3*x)*cos(1.1*x)'
[]
[exact_rho_ud]
  type = ParsedFunction
  value = '3.48788261470924*cos(1.1*x)*cos(1.3*x)'
[]
[forcing_rho_ud]
  type = ParsedFunction
  value = '(-(10.6975765229419*cos(1.5*x)/cos(x) - 0.697576522941849*cos(1.1*x)^2/cos(x)^2)*sin(x) + (10.6975765229419*sin(x)*cos(1.5*x)/cos(x)^2 - 1.3951530458837*sin(x)*cos(1.1*x)^2/cos(x)^3 + 1.53466835047207*sin(1.1*x)*cos(1.1*x)/cos(x)^2 - 16.0463647844128*sin(1.5*x)/cos(x))*cos(x))*cos(1.3*x) + 3.48788261470924*sin(x)*cos(1.1*x)^2*cos(1.3*x)/cos(x)^2 - 7.67334175236034*sin(1.1*x)*cos(1.1*x)*cos(1.3*x)/cos(x) - 4.53424739912202*sin(1.3*x)*cos(1.1*x)^2/cos(x)'
[]
[exact_rho_et]
  type = ParsedFunction
  value = '26.7439413073546*cos(1.5*x)'
[]
[forcing_rho_et]
  type = ParsedFunction
  value = '1.0*(3.48788261470924*(3.06706896551724*cos(1.5*x)/cos(x) - 0.2*cos(1.1*x)^2/cos(x)^2)*cos(x) + 26.7439413073546*cos(1.5*x))*sin(x)*cos(1.1*x)*cos(1.3*x)/cos(x)^2 - 1.1*(3.48788261470924*(3.06706896551724*cos(1.5*x)/cos(x) - 0.2*cos(1.1*x)^2/cos(x)^2)*cos(x) + 26.7439413073546*cos(1.5*x))*sin(1.1*x)*cos(1.3*x)/cos(x) - 1.3*(3.48788261470924*(3.06706896551724*cos(1.5*x)/cos(x) - 0.2*cos(1.1*x)^2/cos(x)^2)*cos(x) + 26.7439413073546*cos(1.5*x))*sin(1.3*x)*cos(1.1*x)/cos(x) + 1.0*(-(10.6975765229419*cos(1.5*x)/cos(x) - 0.697576522941849*cos(1.1*x)^2/cos(x)^2)*sin(x) + (10.6975765229419*sin(x)*cos(1.5*x)/cos(x)^2 - 1.3951530458837*sin(x)*cos(1.1*x)^2/cos(x)^3 + 1.53466835047207*sin(1.1*x)*cos(1.1*x)/cos(x)^2 - 16.0463647844128*sin(1.5*x)/cos(x))*cos(x) - 40.1159119610319*sin(1.5*x))*cos(1.1*x)*cos(1.3*x)/cos(x)'
[]
[exact_T]
  type = ParsedFunction
  value = '0.0106975765229418*cos(1.5*x)/cos(x) - 0.000697576522941848*cos(1.1*x)^2/cos(x)^2'
[]
[exact_eps_p]
  type = ParsedFunction
  value = '3.48788261470924*(3.06706896551724*cos(1.5*x)/cos(x) - 0.2*cos(1.1*x)^2/cos(x)^2)*cos(x)*cos(1.3*x)'
[]
[exact_p]
  type = ParsedFunction
  value = '3.48788261470924*(3.06706896551724*cos(1.5*x)/cos(x) - 0.2*cos(1.1*x)^2/cos(x)^2)*cos(x)'
[]
[exact_sup_vel_x]
  type = ParsedFunction
  value = '1.0*cos(1.1*x)*cos(1.3*x)/cos(x)'
[]
[eps]
  type = ParsedFunction
  value = 'cos(1.3*x)'
[]
[exact_superficial_velocity]
  type = ParsedVectorFunction
  value_x = '1.0*cos(1.1*x)*cos(1.3*x)/cos(x)'
[]
[]

[Executioner]
  solve_type = NEWTON
  type = Steady
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  nl_max_its = 50
  line_search = bt
  nl_rel_tol = 1e-12
  nl_abs_tol = 1e-12
[]

[Outputs]
  exodus = true
  csv = true
[]

[Debug]
  show_var_residual_norms = true
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2pressure]
    type = ElementL2Error
    variable = pressure
    function = exact_p
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2sup_mom_x]
    variable = sup_mom_x
    function = exact_rho_ud
    type = ElementL2Error
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2T_fluid]
    variable = T_fluid
    function = exact_T
    type = ElementL2Error
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/cns/mms/1d-with-bcs/porous-hllc.i)

eps=0.9

[GlobalParams]
  fp = fp
[]

[Mesh]
  [cartesian]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = .1
    xmax = 1.1
    nx = 2
  []
[]

[Modules]
  [FluidProperties]
    [fp]
      type = IdealGasFluidProperties
    []
  []
[]

[Variables]
  [rho]
    type = MooseVariableFVReal
  []
  [rho_ud]
    type = MooseVariableFVReal
  []
  [rho_et]
    type = MooseVariableFVReal
  []
[]

[ICs]
  [rho]
    type = FunctionIC
    variable = rho
    function = 'exact_rho'
  []
  [rho_ud]
    type = FunctionIC
    variable = rho_ud
    function = 'exact_rho_ud'
  []
  [rho_et]
    type = FunctionIC
    variable = rho_et
    function = 'exact_rho_et'
  []
[]

[FVKernels]
  [mass_advection]
    type = PCNSFVMassHLLC
    variable = rho
    fp = fp
  []
  [mass_fn]
    type = FVBodyForce
    variable = rho
    function = 'forcing_rho'
  []

  [momentum_x_advection]
    type = PCNSFVMomentumHLLC
    variable = rho_ud
    momentum_component = x
    fp = fp
  []
  [momentum_fn]
    type = FVBodyForce
    variable = rho_ud
    function = 'forcing_rho_ud'
  []

  [fluid_energy_advection]
    type = PCNSFVFluidEnergyHLLC
    variable = rho_et
    fp = fp
  []
  [energy_fn]
    type = FVBodyForce
    variable = rho_et
    function = 'forcing_rho_et'
  []
[]

[FVBCs]
  [mass_in]
    variable = rho
    type = PCNSFVHLLCSpecifiedMassFluxAndTemperatureMassBC
    boundary = left
    temperature = 'exact_T'
    superficial_rhou = 'exact_rho_ud'
  []
  [momentum_in]
    variable = rho_ud
    type = PCNSFVHLLCSpecifiedMassFluxAndTemperatureMomentumBC
    boundary = left
    temperature = 'exact_T'
    superficial_rhou = 'exact_rho_ud'
    momentum_component = 'x'
  []
  [energy_in]
    variable = rho_et
    type = PCNSFVHLLCSpecifiedMassFluxAndTemperatureFluidEnergyBC
    boundary = left
    temperature = 'exact_T'
    superficial_rhou = 'exact_rho_ud'
  []

  [mass_out]
    variable = rho
    type = PCNSFVHLLCSpecifiedPressureMassBC
    boundary = right
    pressure = 'exact_p'
  []
  [momentum_out]
    variable = rho_ud
    type = PCNSFVHLLCSpecifiedPressureMomentumBC
    boundary = right
    pressure = 'exact_p'
    momentum_component = 'x'
  []
  [energy_out]
    variable = rho_et
    type = PCNSFVHLLCSpecifiedPressureFluidEnergyBC
    boundary = right
    pressure = 'exact_p'
  []
[]

[Materials]
  [var_mat]
    type = PorousConservedVarMaterial
    rho = rho
    superficial_rhou = rho_ud
    rho_et = rho_et
    porosity = porosity
  []
  [porosity]
    type = GenericConstantMaterial
    prop_names = 'porosity'
    prop_values = '${eps}'
  []
[]

[Functions]
[exact_rho]
  type = ParsedFunction
  value = '3.48788261470924*cos(x)'
[]
[forcing_rho]
  type = ParsedFunction
  value = '-3.83667087618017*eps*sin(1.1*x)'
  vars = 'eps'
  vals = '${eps}'
[]
[exact_rho_ud]
  type = ParsedFunction
  value = '3.48788261470924*eps*cos(1.1*x)'
  vars = 'eps'
  vals = '${eps}'
[]
[forcing_rho_ud]
  type = ParsedFunction
  value = 'eps*(-(10.6975765229419*cos(1.2*x)/cos(x) - 0.697576522941849*cos(1.1*x)^2/cos(x)^2)*sin(x) + (10.6975765229419*sin(x)*cos(1.2*x)/cos(x)^2 - 1.3951530458837*sin(x)*cos(1.1*x)^2/cos(x)^3 + 1.53466835047207*sin(1.1*x)*cos(1.1*x)/cos(x)^2 - 12.8370918275302*sin(1.2*x)/cos(x))*cos(x)) + 3.48788261470924*eps*sin(x)*cos(1.1*x)^2/cos(x)^2 - 7.67334175236034*eps*sin(1.1*x)*cos(1.1*x)/cos(x)'
  vars = 'eps'
  vals = '${eps}'
[]
[exact_rho_et]
  type = ParsedFunction
  value = '26.7439413073546*cos(1.2*x)'
[]
[forcing_rho_et]
  type = ParsedFunction
  value = '1.0*eps*(3.48788261470924*(3.06706896551724*cos(1.2*x)/cos(x) - 0.2*cos(1.1*x)^2/cos(x)^2)*cos(x) + 26.7439413073546*cos(1.2*x))*sin(x)*cos(1.1*x)/cos(x)^2 - 1.1*eps*(3.48788261470924*(3.06706896551724*cos(1.2*x)/cos(x) - 0.2*cos(1.1*x)^2/cos(x)^2)*cos(x) + 26.7439413073546*cos(1.2*x))*sin(1.1*x)/cos(x) + 1.0*eps*(-(10.6975765229419*cos(1.2*x)/cos(x) - 0.697576522941849*cos(1.1*x)^2/cos(x)^2)*sin(x) + (10.6975765229419*sin(x)*cos(1.2*x)/cos(x)^2 - 1.3951530458837*sin(x)*cos(1.1*x)^2/cos(x)^3 + 1.53466835047207*sin(1.1*x)*cos(1.1*x)/cos(x)^2 - 12.8370918275302*sin(1.2*x)/cos(x))*cos(x) - 32.0927295688256*sin(1.2*x))*cos(1.1*x)/cos(x)'
  vars = 'eps'
  vals = '${eps}'
[]
[exact_T]
  type = ParsedFunction
  value = '0.0106975765229418*cos(1.2*x)/cos(x) - 0.000697576522941848*cos(1.1*x)^2/cos(x)^2'
  vars = 'eps'
  vals = '${eps}'
[]
[exact_p]
  type = ParsedFunction
  value = '3.48788261470924*(3.06706896551724*cos(1.2*x)/cos(x) - 0.2*cos(1.1*x)^2/cos(x)^2)*cos(x)'
  vars = 'eps'
  vals = '${eps}'
[]
[]

[Executioner]
  solve_type = NEWTON
  type = Steady
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  nl_max_its = 50
  line_search = none
  nl_rel_tol = 1e-12
  nl_abs_tol = 1e-12
[]

[Outputs]
  exodus = true
  csv = true
[]

[Debug]
  show_var_residual_norms = true
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2rho]
    type = ElementL2Error
    variable = rho
    function = exact_rho
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2rho_ud]
    variable = rho_ud
    function = exact_rho_ud
    type = ElementL2Error
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2rho_et]
    variable = rho_et
    function = exact_rho_et
    type = ElementL2Error
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

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