[ConvectionDiffusionApplication] Adding crosswind stabilization to Eulerian Convection–Diffusion element

[Marco1410]

📝 Description
This pull request introduces a crosswind stabilization term in the EulerianConvectionDiffusionElement of Kratos.

The implementation follows the discontinuity-capturing approach proposed in A discontinuity‑capturing crosswind‑dissipation for the finite element solution of the convection‑diffusion equation by Ramon Codina. The method adds a nonlinear diffusion in the direction orthogonal to the flow in order to reduce the spurious oscillations that may still appear when using streamline-based stabilization techniques.

The stabilization term can be activated through the element settings.
By default CROSS_WIND_STABILIZATION_FACTOR is zero.

 "transient_parameters" : {
        "dynamic_tau" : 1.0,
        "theta"       : 0.5,
        "cross_wind_stabilization_factor" : 0.0
   }

Additionally, a unit test has been included to verify the correct behaviour of the implementation.

To illustrate the effect of the stabilization, two example simulations are shown below:

• Without crosswind stabilization:
  Spurious oscillations appear near steep gradients.
  

• With crosswind stabilization:
  The oscillations are significantly reduced.
  

🆕 Changelog

• eulerian_conv_diff.cpp
• convection_diffusion_transient_solver.py
• test_eulerian_conv_diff.cpp

[rubenzorrilla]

I'm not fully convinced about the elemental cross wind factor calculation. Aside of this, the others are minors.

[rubenzorrilla]

Are you sure about this? Shouldn't it involve the viscosity besides the tau?

[Marco1410]

@rubenzorrilla, I use tau because I want to remove the contribution in the streamline direction.

The problem with the projected Péclet is that when the velocity and the gradient are aligned, the crosswind contribution disappears, even if oscillations are still present.

This is likely why the original formulation is implemented in this way. For this reason, I think it is better to keep the current approach, since it is more appropriate and robust.

[rubenzorrilla]

Suggested change

	u_proj = (conv/norm_grad) * (grad_phi_halfstep/norm_grad);
	u_proj = (conv/std::pow(norm_grad,2)) * grad_phi_halfstep;

Easier to follow

[rubenzorrilla]

I'd use a more self-descriptive variable name

[rubenzorrilla]

Suggested change

	// Projected velocity
	// Velocity projected in the direction of the solution gradient

[rubenzorrilla]

Suggested change

	// Projected Peclet
	// Peclet number projected in the direction of the solution gradient

[rubenzorrilla]

Shouldn't this include the source term and diffusion?

[rubenzorrilla]

Also the beta term in case we use a compressible flux.

[rubenzorrilla]

Minors. As a summary of our discussion, lets add the complete residual for the crosswind calculation and see what happens when avoiding the "custom" limiter stuff.

[rubenzorrilla]

Finally, we agree on having alpha_c = Variables.C as default, but we leave commented current option with the projected Peclet number and limiter. Lets put some comments explaining our discussion.