Computational Details

Computational approach

Alya is a parallel multi-physics/multiscale simulation code developed at the Barcelona Supercomputing Centre to run efficiently on high-performance computing environments.

The convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed in Charnyi et al.  (2017) and Jofre et al.  (2014), which conserves linear and angular momentum, and kinetic energy at the discrete level. Both second- and third-order spatial discretizations are used.

Neither upwinding nor any equivalent momentum stabilization is employed. In order to use equalorder elements, numerical dissipation is introduced only for the pressure stabilization via a fractional step scheme (Codina (2017)), which is similar to approaches for pressure-velocity coupling in unstructured, collocated finite-volume codes (Jofre et al. (2014)). The set of equations is integrated in time using a third order Runge-Kutta explicit method combined with an eigenvalue-based time-step estimator (Trias and Lehmkuhl (2011)). This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach (Lehmkuhl et al. (2019)). Thus, is an optimal methodology for high-fidelity simulations of complex flows as the ones required in the present project.

Spatial and temporal resolution, grids

A mesh resulting in approximately 250 million degrees of freedom (DoF). With a stretched grid, the maximum grid resolution in the duct centre is reported at ${\Delta z^{+}=11.6}$ , ${\Delta y^{+}=13.2}$ , ${\Delta x^{+}=19.5}$ . Correspondingly, the wall resolution (in terms of the first grid point) is reported as ${z^{+}=0.074}$ , ${y^{+}=0.37}$ in the spanwise and normal directions, respectively, is used. This resolution was deemed sufficient to compute the flow in the diffuser and is based on the previous work of Ohlsson et al. (2010). For the temporal resolution a third order explicit Runge Kutta method using a dynamic time stepping with CFL below 0.9 has been used.

Computation of statistical quantities

The statistical quantities are computed a posteriori from the velocity and pressure fields gathered during the additional 21 flowthrough-times. First, the averaged velocity and pressure are computed along with their gradients. Then the fluctuating pressure and velocity are computed as:

${p=P-{\overline {P}}}$

${u=U-{\overline {U}}}$

and their gradients as:

${p,k=P,k-{\overline {P}},k}$

${u,k=U,k-{\overline {U}},k}$

This allows to easily define the Reynolds stress tensor and the velocity triple correlation using the fluctuating velocity. The rest of the terms in the budget equations are computed in the usual way.

References

Charnyi, S., Heister, T., Olshanskii, M. A. and Rebholz, L. G. (2017): On conservation laws of NavierStokes Galerkin discretizations. In Journal of Computational Physics, Vol. 337, pp. 289-308.
Jofre, L., Lehmkuhl, O., Ventosa, J., Trias, F. and Oliva, A. (2014): Conservation properties of unstructured finite-volume mesh schemes for the Navier-Stokes equations. In Numerical Heat Transfer, Part B: Fundamentals, Vol. 54, no. 1, pp. 289-308.
Codina, R. (2001): Pressure stability in fractional step finite element methods for incompressible flows. In Journal of Computational Physics, Vol. 170, no. 1, pp. 112-140.
Trias, F. X. and Lehmkuhl, O. (2011): A self-adaptive strategy for the time integration of NavierStokes equations. In Numerical Heat Transfer. Part B, Vol. 60, no. 2, pp. 116-134.
Lehmkuhl, O., Houzeaux, G., Owen, H., Chrysokentis, G. and Rodriguez, I. (2019): A low-dissipation finite element scheme for scale resolving simulations of turbulent flows. In Journal of Computational Physics, Vol. 390, pp. 51-65.