Computational Details

This section provides details of the computational strategies employed to compute the present case. Firstly, details of the numerical solver are given. Then, information about the computational grid is provided. Finally, the statistical quantities and their computation are explained.

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. For this DNS, the data has been obtained using the incompressible Navier-Stokes solver of Alya since the flow is not subject to compressibility effects. The general code is described in Vazquezet al. (2016) while the latest numerical schemes for the incompressible flow solver are described in Lehmkuhl et al. (2019).

In a nutshell, the convective term is discretized using a Galerkin finite element (FEM) scheme recently proposed by Charnyi et al. (2017), which conserves linear and angular momentum, and kinetic energy at the discrete level (see Olshanskii and Rebholz (2020)). 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 (2001)), which is similar to the approach for the pressure-velocity coupling in unstructured, collocated finite-volume codes (see, for example, Jofre et al. (2014)). The set of equations is integrated in time using a third order Runge-Kutta explicit method (see Capuano et al. (2017)) combined with an eigenvalue-based time-step estimator (see Trias and Lehmkuhl (2011)). This approach has been shown to be significantly less dissipative than the traditional stabilized FEM approach (see 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

The computational grid resulted in about 62 million elements and approximately 250 million degrees of freedom (DoF). Details of the computational grid are provided in Fig. 7 and Fig. 8.

Figure 7: Details of the computational mesh. XY (top) and XZ (bottom) plane views in the entrance (left) and mid (right) sections of the diffuser. Only every 4th grid line is shown.

Figure 8: Details of the computational mesh in various cross-sections corresponding to: inlet duct (top left), diffuser expansion (top right), diffuser mid section (bottom left) and diffuser exit (bottom right). Only every 4th grid line is shown.

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. 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 was used with a dynamic time stepping that ensured a CFL below 0.9.

Computation of statistical quantities

The statistical quantities are computed a posteriori from the velocity and pressure fields gathered during an additional 21 flowthrough-times. In a first step, the time-averaged pressure and velocity fields are gathered among 1881 snapshots. These snapshots are considered to be loosely correlated (i.e., as independent experiments) hence a simple average is performed. Afterwards, the time-averaged pressure and velocity gradients are computed from the time-averaged pressure and velocity fields.

Then, the fluctuating quantities are obtained as:

Fluctuating pressure ${p=P-{\overline {P}}}$
Fluctuating velocity ${u=U-{\overline {U}}}$

along with their gradients as:

Gradient of fluctuating pressure ${p_{,k}=P_{,k}-{\overline {P}}_{,k}}$
Gradient of fluctuating velocity ${u_{,k}=U_{,k}-{\overline {U}}_{,k}}$

Using these quantities, the Reynolds stress tensor is easily recovered, along with the pressure autocorrelation, pressure-velocity correlation and triple velocity correlation:

Reynolds stresses ${R_{ij}^{*}=\rho {\overline {u_{i}u_{j}}}}$
Pressure autocorrelation ${\overline {pp}}$
Pressure velocity correlation ${\overline {pu_{i}}}$
Triple velocity correlation ${\rho {\overline {u_{i}u_{j}u_{k}}}}$

The Reynolds stress budget equation terms are recovered using the aforementioned quantities and their gradients using matrix and vectorial math:

Convection ${C_{ij}^{*}=(R_{ij}^{*}{\overline {u}}_{k})_{,k}}$
Production ${P_{ij}^{*}=-(R_{ik}^{*}{\overline {u}}_{j,k}+R_{jk}^{*}{\overline {u}}_{i,k})}$
Turbulent diffusion ${D_{ij}^{1*}=-\rho ({\overline {u_{i}u_{j}u_{k}}})_{,k}}$
Turbulent diffusion ${D_{ij}^{2*}=-\rho ({\overline {p(u_{i}\delta _{jk}+u_{j}\delta _{ik})}})_{,k}}$
Turbulent diffusion ${D_{ij}^{3*}=\mu ({\overline {u_{i}u_{j}}})_{,kk}}$
Pressure strain ${\Phi _{ij}^{*}={\overline {p(u_{i,j}+u_{j,i})}}}$
Dissipation ${\epsilon _{ij}^{*}=2\mu {\overline {u_{i,j}u_{j,k}}}}$

Finally, Taylor lengthscale and Kolmogorov length and time scales are recovered as:

Taylor microscale ${\eta _{T}={\sqrt {\left|10\nu {\frac {k}{\varepsilon +s}}\right|}}}$
Kolmogorov length scale ${\eta _{K}=\left|{\frac {\nu ^{3}}{\varepsilon +s}}\right|^{0.25}}$
Kolmogorov time scale ${\tau _{K}={\sqrt {\left|{\frac {\nu }{\varepsilon +s}}\right|}}}$

Where ${\varepsilon }$ stands for the dissipation in the budget equation for the turbulent kinetic energy ${k}$ . The absolute value is taken to avoid negative roots and ${s}$ is a small value to avoid numerically dividing by zero. For a more in-detailed explanation on how these quantities have been obtained, the reader is referred to this document (PDF).

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.
Vázquez, M., Houzeaux, G., Koric, S., Artigues, A., Aguado-Sierra, J., Arís, R., Mira, D., Calmet, H., Cucchietti, F., Owen, H., Taha, A., Burness, E. D., Cela, J. M., & Valero, M. (2016): Alya: Multiphysics engineering simulation toward exascale. In Journal of Computational Science, Vol. 14, pp. 15-27.
Olshanskii, M. A., & Rebholz, L. G. (2020): Longer time accuracy for incompressible Navier–Stokes simulations with the EMAC formulation. In Computer Methods in Applied Mechanics and Engineering, Vol. 372, pp. 113369.
Capuano, F., Coppola, G., Rández, L., & Luca, L. De (2017): Explicit Runge – Kutta schemes for incompressible flow with improved energy-conservation properties. In Journal of Computational Physics, Vol. 328, pp. 86–94.