Equations¶

Keyword	Required	Default	Valid values
‘equations’	Yes	–	Any of equation sets detailed below.

The equations key selects the equation set to be solved as well as the finite volume solver or the high order strong form Discontinuous Galerkin/Flux Reconstruction solver.

The corresponding equation set options should also be defined in the control dictionary. The valid equation keys are shown in the table below.

Solver	Description	Compressible	High order	Incompressible
Euler	Use the Euler flow equations in the simulation. These are compressible, inviscid flow equations.	“euler”	–	–
RANS	Use the RANS (Reynolds-Averaged Navier-Stokes) equations in the simulation. These are compressible, viscid flow equations with additional terms included to account for the effect of turbulence on the flow without the expense of simulating the turbulent flow structures themselves.	“rans”	“dgrans”	Coming soon
Viscous	Use the viscous flow equations. These are compressible, viscid flow equations which do not model turbulence.	“viscous”	“dgviscous”	Coming soon
LES	Use the Large Eddy Simulation (LES) equations.	“les”	“dgles”	Coming soon

Example usage:

parameters = {
  ..
  # Use the euler equation set
  'equations' : 'euler',
  # Define the inputs for the 'euler' equation set
  'euler': {..}
  ..
}

Common Settings¶

The following parameters are common for all of the equation sets.

Keyword	Required	Default	Valid values
‘order’	Yes	–	‘first’ \| ‘second’ \| ‘euler_second’
‘first order cycles’	No	–	Positive integer
‘linear gradients’	No	False	True \| False
‘leastsq gradients’	No	False	True \| False
‘inviscid flux scheme’	Yes	“Roe”	“HLLC” \| “Rusanov” \| “Roe” \| “Roe low diss”
‘reconstruction’	No	“muscl”	“muscl” \| “umuscl” \| “lpumuscl”
‘roe low dissipation sensor’	No	–	“FD” \| “NTS” \| “DES” \| “NONE”
‘roe dissipation sensor minimum’	No	–	Float between 0.05 and 1.0
‘freeze limiter cycle’	No	–	Positive integer
‘relax’	Yes (for incompressible equation set)	0.5	Positive float
‘turbulence’	Yes (for ‘les’ or ‘rans’ equation set)	{}	See “turbulence”

order¶

Sets the order of the spatial discretisation. ‘euler_second’ is second order for the flow variables and first order for the turbulence.

first order cycles¶

Specifies the number of cycles the solver will run with first order spatial discretisation. This can be used to improve stability at start up.

Example usage: Run 500 cycles first order.

parameters = {
  ..
  'equations' : 'euler',
  'euler': {
    ...
    'first order cycles': 500,
    ...
  }
  ..
}

linear gradients & leastsq gradients¶

By default zCFD uses a Weighted Green-Gauss gradient calculation. This can be switched to a standard Green-Gauss by setting ‘linear gradients’ to True. Setting ‘leastsq gradients’ to True will set the gradient calculation to Least Squares.

inviscid flux scheme¶

Picks the scheme to use for the inviscid flux. HLLC is good when running explicit, Roe is best for implicit and Rusanov is good for getting an implicit simulation going. The Roe low diss scheme can be used with LES or Hybrid RANS/LES

roe low dissipation sensor¶

The sensor used to decide whether to use (and by how much) the Roe low dissipation scheme.

Setting	Notes
‘DES’	Uses the blending function from the selected DES model to turn on the Roe low dissipation scheme in LES regions
‘NONE’	Uses the value set on ‘roe dissipation sensor minimum’ everywhere
‘FD’	See Johnsen et al. JCP 229 (2010) pag. 1234
‘NTS’	See Xiao et al. INT J HEAT FLUID FL 51 (2015) pag. 141 https://doi.org/10.1016/j.ijheatfluidflow.2014.10.007

roe dissipation sensor minimum¶

Sets the minimum value that the dissipation term in the Roe scheme is multiplied by when using the ‘Row low diss’ scheme. This allows you control the stability of the low diss scheme (smaller values reduce the dissipation and hence the stability of the scheme)

freeze limiter cycle¶

If set the values of the limiter used in the MUSCL reconstruction are frozen at the specified cycle. This can be useful in the latter stages of a simulation if noise from the limiter is causing the residual convergence to stall.

relax¶

The relaxation factor used on the flow equations in the incompressible solver. Increasing this will speed up convergence at the expense of stability and vice versa.

turbulence¶

The turbulence dictionary is required for the “rans” and “les” equations sets.

Euler¶

Compressible Euler flow is inviscid (no viscosity and hence no turbulence). The compressible Euler equations are appropriate when modelling flows where momentum significantly dominates viscosity - for example at very high speed. The computational mesh used for Euler flow does not need to resolve the flow detail in the boundary layer and hence will generally have far fewer cells than the corresponding viscous mesh would have.

As well as the “common” settings, the following settings are available:

Keyword	Required	Default	Valid values	Description
‘precondition’	Yes	False	True \| False	Use low speed mach preconditioner

Example usage:

parameters = {
  ..
  # Use the euler equation set
  'equations' : 'euler',
  'euler' : {
            'order' : 'second',
            'first order cycles': 100,
            # Cycle on which to freeze the limiter values
            'freeze limiter cycle': 500,
            # Use low speed mach preconditioner
            'precondition' : True,
            'Inviscid Flux Scheme': 'HLLC',
          },
}

Viscous¶

The viscous (laminar) equations model flow that is viscous but not turbulent. The Reynolds number of a flow regime determines whether or not the flow will be turbulent.

The computational mesh for a viscous flow has to resolve the boundary layer so will generally be larger than that used for an Euler simulation. A viscous simulation will run faster than a RANS simulation using the same mesh since fewer equations are being modelled.

As well as the “common” settings, the following settings are available:

Keyword	Required	Default	Valid values	Description
‘precondition’	Yes	False	True \| False	Use low speed mach preconditioner

Example usage:

parameters = {
  ..
  # Use the viscous equation set
  'equations' : 'viscous',
  'viscous' : {
                'order' : 'second',
                'first order cycles': 100,
                # Use low speed mach preconditioner
                'precondition' : True,
                'linear gradients' : False,
                'Inviscid Flux Scheme': 'HLLC',
              },
  ..
}

RANS¶

The fully turbulent (Reynolds Averaged Navier-Stokes Equations)

As well as the “common” settings, the ‘RANS’ equation set requires the additional ‘turbulence’ dictionary, which can have the following parameters:

Keyword	Required	Default	Valid values	Description
‘les’	Yes	‘none’	‘none’ \| ‘DES” \| ‘DDES’ \| ‘IDDES’ \| ‘SAS’	Selects the hybrid RANS/LES model.
‘model’	Yes	–	‘sst’ \| ‘sas’ \| ‘sa-neg’ \| ‘sst-transition’	Selects the RANS turbulence model
‘approximate sa from sst results’	No		True \| False	Restart an SA-neg calculation from an SST solution
‘rotation correction’	No	False	True \| False	Adds the rotation correction term to SST or SA-neg
‘limit gradient k’	No		Float from 0.01 to 1	Activates the \(\tilde{\nu}\) gradient limiter in the SA-neg model
‘qcr’	No	Off	2000 \| 2020 \| 2021	Activates the non-linear QCR correction to the turbulent stress. Applicable to both SST and SA-neg

les¶

Selects the hybrid RANS/LES model in the RANS solver which is applied to the base model selected with the “model” key word.

Setting	Notes
‘none’	Solve the RANS equations with no hybrid RANS/LES model
‘DES’	Detached Eddy Simulation
‘DDES’	Delayed Detached Eddy Simulation
‘IDDES’	Improved Delayed Detached Eddy Simulation
‘SAS’	Scale Adaptive Simulation

model¶

Selects the RANS turbulence model.

Setting	Notes
‘sst’	The Menter Shear Stress Transport Turbulence Model (https://turbmodels.larc.nasa.gov/sst.html)
‘sa’	Spalart-Allmaras turbulence model (https://turbmodels.larc.nasa.gov/spalart.html)
‘sa-neg’	Negative Spalart-Allmaras turbulence model (https://turbmodels.larc.nasa.gov/spalart.html)
‘sst-transition’	The Langtry-Menter 4-equation Transitional SST Model (https://turbmodels.larc.nasa.gov/langtrymenter_4eqn.html)

limit gradient k¶

Activates the \(\tilde{\nu}\) gradient limiter in the SA-neg model. This helps eliminate gradient under/over shoot in regions of poor mesh quality.

The value should be between 0 and 1. 0 is off and 1 is maximum limiting.

qcr¶

Activates the non-linear QCR correction to the turbulent stress. Applicable to both SST and SA-neg. QCR has been shown to improve the prediction of separated corner flows, particularly on highly loaded wings. Three versions of the QCR model are available: QCR 2000 , QCR 2020 and QCR 2021.

Example RANS usage:¶

parameters = {
  ..
  # Use the viscous equation set
  'equations' : 'RANS',
  'RANS': {
              'order' : 'second',
              'first order cycles': 100,
              'freeze limiter cycle': 500,
              'linear gradients' : False,
              'leastsq gradients' : False,
              'Inviscid Flux Scheme': 'HLLC',
              'roe low dissipation sensor': 'DES',
              'roe dissipation sensor minimum': 0.05,
              'turbulence' : {
                                'model' : 'sst',
                                # No Hybrid RANS/LES
                                'les' : 'none',
                                'rotation correction': False,
                                'limit gradient k': 0.5
                                'qcr': 2020
                              },
             },
  ...
}

Advanced parameters for turbulence models¶

The following parameters can be used in the ‘turbulence’ dictionary to allow tuning of the turbulence models in zCFD. These are advanced parameters and shouldn’t normally need to be adjusted.

Keyword	Required	Default	Valid values	Description
‘betastar’	No	0.09	Positive number	Constant in the dissipation term on the k equation in the SST model
‘cdes_kw’	No	0.78	Positive number	The \(k - \omega\) part of the blended \(C_{DES}\) in SST
‘cdes_keps’	No	0.21	Positive number	The \(k - \epsilon\) part of the blended \(C_{DES}\) in SST
‘cd1’	No	20	Positive number	Constant in the \(f_{dt}\) blending function in IDDES
‘cd2’	No	3	Positive integer	Constant in the \(f_{dt}\) blending function in IDDES
‘production’	No	0	Positive integer	Selects the model for the production term in the SST model.
‘cdes’	No	0.65	Positive number	\(C_{DES}\) constant used for SA-neg based hybrid RANS/LES

‘production’ can have the following values:

Setting	Notes
0	Selects the vorticity based production term (SST-V) \(P_m=\mu_t\Omega_v^2-\frac{2}{3}\rho k \delta_{ij}\frac{\partial u_i}{\partial x_j}\)
1	Selects the strain based production term for incompressible flows \(P_m=\mu_t S^2\)
2	Selects the strain based production term for compressible flows \(P_m=\mu_t S^2 -\frac{2}{3}\rho k \delta_{ij}\frac{\partial u_i}{\partial x_j}\)
3	Selects the Kato-Launder source term \(P_m=\mu_t S\Omega_v -\frac{2}{3}\rho k \delta_{ij}\frac{\partial u_i}{\partial x_j}\)

LES¶

As well as the “common” settings, the ‘LES’ equation set requires the additional ‘turbulence’ dictionary, which can have the following parameters:

Keyword	Required	Default	Valid values	Description
‘les’	Yes	‘none’	‘none’ \| ‘WALE” \|	Selects the hybrid RANS/LES model.

les¶

Selects the model for the subgrid eddy viscosity in the LES solver: either “none” for implicit LES or “WALE” to use the Wall Adapted Local Eddy Viscosity model.

Example usage:

parameters = {
  ..
  # Use the viscous equation set
  'equations' : 'LES',
  'LES': {
              'order' : 'second',
              'first order cycles': 100,
              'freeze limiter cycle': 500,
              'linear gradients' : False,
              'leastsq gradients' : False,
              'Inviscid Flux Scheme': 'HLLC',
              'roe low dissipation sensor': 'DES',
              'roe dissipation sensor minimum': 0.05,
              'turbulence' : {
                                'les' : 'none',
                              },
             },
  ...
}

High Order¶

When using the high order strong form Discontinuous Galerkin/Flux Reconstruction solver the keys in the “common” equations dictionary are the same for each equation set except that the ‘linear gradients’ and ‘limiter’ keys are not valid and additional keys are available in the equations dictionary. For example all the available keys for ‘dgviscous’ are a combination of the ‘viscous’ keys and the keys shows below. The same logic applies to the ‘dgrans’, and ‘dgles’ equation sets.

These additional DG keys are:

Keyword	Required	Default	Valid values	Description
‘order’	Yes		0 \|1 \| 2 \| 3 \| 4	This parameter sets the order of the polynomials within each element used to define the numerical solution
‘freeze diffusion during rk stages’	Yes	True	True \| False	Fixes diffusion terms after first RK stage
‘inviscid flow’	Yes	False	True \| False	Defines if only advective terms should be computed
‘c11 stability parameter’	Yes	0	Number
‘ldg upwind parameter’	Yes	0.5	Number between 0 & 1
‘shock sensing’	Yes	False	True \| False
‘shock sensing variable’	Yes	density	“density” \| “temperature” \| “mach” \| “turbulence”
‘shock k’	No	1.0	Number
‘shock sensing viscosity scale’	No	1.0	Number

DG Nodal locations¶

In addition to the equation set the location of the DG solution points must be specified in the parameters dictionary. The definitions first need to be imported by including this import statement at the start of the control dictionary. The nodal locations are used in the generation of Lagrange polynomials used to define the solution within an element.

There are various sets of nodal locations defined in literature for use with high order methods. Varying the nodal locations can minimise aliasing errors within the DG method. It is recommended to use the default values.

Keyword	Required	Default	Valid values	Description
‘line’	Yes	line_gauss_lobatto	line_evenly_spaced \| line_gauss_lobatto \| line_gauss_legendre_lobatto	Specifies 1D points defining nodal locations.
‘tri’	Yes	tri_shunn_ham	tri_shunn_ham \| tri_evenly_spaced,	Specifies 2D triangle points defining nodal locations.
‘tetrahedron’	Yes	tet_shunn_ham	tet_shunn_ham \| tet_evenly_spaced	Specifies 3D tetra points defining nodal locations.
‘pyramid’	Yes	pyramid_gauss_legendre_lobatto	pyramid_evenly_spaced \| pyramid_gauss_legendre \| pyramid_gauss_legendre_lobatto	Specifies 3D pyramid points defining nodal locations.

Before you use the nodal locations the dictionaries must be imported by adding the following to your control file:

from zcfd.solvers.utils.DGNodalLocations import *

To use the default values:

from zcfd.solvers.utils.DGNodalLocations import *

parameters = {
..
    'Nodal Locations' : nodal_locations_default['Nodal Locations']
..
}

Or select from the available types

from zcfd.solvers.utils.DGNodalLocations import *

parameters = {
..
  'Nodal Locations' : {
                        'line':  line_gauss_legendre_lobatto,
                        'tetrahedron': tet_evenly_spaced,
                        'tri' : tri_evenly_spaced,
                        'pyramid': pyramid_evenly_spaced
                      },
..
}

Example “dgviscous”¶

from zcfd.solvers.utils.DGNodalLocations import *

parameters = {
..
    'equation': "dgviscous",
    'dgviscous' : {
               # Keys for viscous
               'precondition' : True,
               'Inviscid Flux Scheme': 'HLLC',

               # Additional DG Keys
               'order' : 2,
               'inviscid flow': False,
               'freeze diffusion during rk stages': False
               'c11 stability parameter': 0.0,
               'LDG upwind parameter': 0.5,
               'Shock Sensing': False,
               'Shock Sensing k': 1.0,
               'Shock Sensing Viscosity Scale': 1.0,
               'Shock Sensing Variable': 'density',
            },
    'Nodal Locations' : nodal_locations_default['Nodal Locations']
..
}