Large Eddy Simulation models. More...

Namespaces
module	enum_les
	Enumeration for Large Eddy Simulation models.

module	variables_turbulence_les
	Define variables associated to the Large Eddy Simulation models.

Functions
subroutine	mod_dynamic_sgs_constant::compute_uniform_dynamic_smagorinsky (velocity, strain_rate_magnitude, model_constant)
	Compute the dynamic Smagorinsky model constant using the Germano identity.

subroutine, public	mod_compute_filtered_fields::compute_filtered_scalar_field (filtered_field, field, filter_ratio)
	Apply an explicit filter to a scalar field using 1D mask.

subroutine	mod_compute_les_subgrid_viscosity::compute_les_subgrid_viscosity (velocity)
	Compute the subgrid scale dynamic viscosity.

double precision function	mod_wale_functions::compute_g_square (velocity_gradient, ii, jj)
	Compute the variables necessary to calculate the turbulent viscosity in the WALE model.

Detailed Description

Large Eddy Simulation models.

Function Documentation

◆ compute_filtered_scalar_field()

subroutine, public mod_compute_filtered_fields::compute_filtered_scalar_field	(	double precision, dimension(:,:,:), intent(out)	filtered_field,
		double precision, dimension(:,:,:), intent(in)	field,
		integer, intent(in)	filter_ratio )

Apply an explicit filter to a scalar field using 1D mask.

The multidimensional filter is constructed as the tensor product of a 1D discrete mask mask_filter given in module private variable. Available filter-to-grid ratio are Δ/Δx=1,2,4,8,16. Composite Simpson is using as quadrature rule with the following number of intervall used for each filter-to-grid ratioi (Δ/Δx), respectivelly (1,1,2,4,8). The filtering is performed successively along each spatial direction, resulting in an efficient implementation compared to full stencil naive implementation (avoiding cache misses).

For a 1D mask \( w \) of size \( N \), the 2D kernel is:

\[ W(i,j) = \frac{w_i \, w_j}{n} \,. \]

with \( n \) the norm of the kernel

\[ {\left(\sum_i w_i\right)^2} \,. \]

For a single 2Δ interval, the 1D Simpson weights are: \( w = \{1,4,1\} / 6. \) In 2D or 3D, the weights become the tensor product of the 1D Simpson weights. For instance, in 2D:

This produces the classical 3×3 kernel:

  !!   /  1   4   1  \
  !!   |  4  16   4  |  / 36
  !!   \  1   4   1  /
  !!

Warning: This filter assumes a uniform Cartesian grid; using it on non-uniform meshes may lead to incorrect results.

Parameters

[in,out]	filtered_field	Filtered field.
[in]	field	Input vector field to be filtered.
[in]	filter_ratio	Filter-to-grid ratio.

◆ compute_g_square()

double precision function mod_wale_functions::compute_g_square	(	double precision, dimension(:,:), intent(in)	velocity_gradient,
		integer, intent(in)	ii,
		integer, intent(in)	jj )

Compute the variables necessary to calculate the turbulent viscosity in the WALE model.

◆ compute_les_subgrid_viscosity()

subroutine mod_compute_les_subgrid_viscosity::compute_les_subgrid_viscosity ( type(t_face_field), intent(in) velocity )

Compute the subgrid scale dynamic viscosity.

◆ compute_uniform_dynamic_smagorinsky()

subroutine mod_dynamic_sgs_constant::compute_uniform_dynamic_smagorinsky	(	type(t_face_field), intent(in)	velocity,
		double precision, dimension(:,:,:), intent(in)	strain_rate_magnitude,
		double precision, intent(out)	model_constant )

Compute the dynamic Smagorinsky model constant using the Germano identity.

This routine evaluates the dynamic Smagorinsky coefficient \( C_s \) using the Germano-Lilly procedure. The model constant is obtained from:

\[ C_s^2 = \frac{\langle L_{ij} M_{ij} \rangle} {\langle M_{ij} M_{ij} \rangle} \]

where:

\( L_{ij} \) is the Leonard stress tensor;
\( M_{ij} \) is the model tensor;
\( \langle . \rangle \) is a spatial averaging.

The Leonard tensor is defined as:

\[ L_{ij} = \widetilde{u_i u_j} - \tilde{u}_i \tilde{u}_j \]

where:

\( \tilde{(\cdot)} \) denotes filtering at scale \( \tilde{\Delta} = 2\Delta \), and \( u_i \) is the implicit filtered velocity at scale \( Delta \).

The model tensor reads:

\[ M_{ij} = \widetilde{ \left( |S| S_{ij} \right) } - r^2 |\tilde{S}| \tilde{S}_{ij} \]

with:

\[ S_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)\ , \quad |S| = \sqrt{2 S_{ij} S_{ij}} \]

The filter ratio is:

\[ r = {\tilde{\Delta}}/{\Delta} = 2 \]

References

Germano, M., Piomelli, U., Moin, P., Cabot, W. H. (1991). "A dynamic subgrid-scale eddy viscosity model". Physics of Fluids A, 3(7), 1760–1765.
Lilly, D. K. (1992). "A proposed modification of the Germano subgrid-scale closure method". Physics of Fluids A, 4(3), 633–635.
Dairay, T. et al. (2017). "Numerical dissipation vs. subgrid-scale modelling for large eddy simulation." Journal of Computational Physics, 2017.