Quantification of resolution

This section provides details of the solution obtained after running the diffuser in Alya. Firstly, details of the mesh resolution are given. Then, the obtained results are compared with those of the literature for the Stanford diffuser. A discussion on the closure of the Reynolds stress equations budgets follows and closes this section.

Mesh resolution

The mesh resolution is quantified by obtaining a relation between the mesh characteristic length ( ${\Delta }$ ) and characteristic lengths of the turbulence, i.e., the Taylor microscale ( ${\eta _{T}}$ ) and Kolmogorov length scale ( ${\eta _{K}}$ ). The former relation is shown in Fig. 10 while the latter is reported in Fig. 11. As it can be seen, both relations indicate that the resolution achieved by the present grid is at DNS level. In particular, it is commonly accepted that DNS is achieved when ${\Delta /\eta _{K}\leq 5}$ , as shown in Fig. 11.

Figure 10: Stanford double diffuser, Alya DNS-250M DoF, relation between the mesh size and the Taylor microscale.

Figure 11: Stanford double diffuser, Alya DNS-250M DoF, relation between the mesh size and the Kolmogorov length scale.

Solution verification

For verification of the quality of the results obtained in the present simulations, these are compared with those of the DNS of Ohlsson et al. (2010) and the experimental data of Cherry et al. (2008) (available in UFR 4-16 of the Wiki) and a check is made whether the residuals in the budget equations for the Reynolds stresses are sufficiently small.

Velocity profiles and RMS quantities

Figs. 15 and 16 presented in the next section (Statistical Data) show, respectively, the mean velocities and RMS values of the streamwise fluctuations along characteristic lines of the diffuser for the experimental data and both DNS simulations. Further, Figs. 19 and 20 show, respectively, a comparison of the contours of the streamwise mean velocity and its RMS fluctuations at selected cross sections of the diffuser. As can be seen, there is a fair agreement between the 2 numerical simulations providing confidence in the present simulations, and where these deviate locally from the results of Ohlsson et al (2010), the present results are closer to the experimental data of Cherry et al (2008) so that overall good agreement is obtained with these data. An impression of the fluctuations can be obtained from an animation that can be seen in this link. It should be noted that the inclusion of a roughness element in the long inlet duct in order to trigger turbulence has a negligible effect on the quality of the results.

Reynolds stress equations budgets

The Reynolds stress equations and the turbulent kinetic energy (TKE) equation budget terms are discussed in this section. The first step is to show that Alya can close the budgets. To this aim, the figures below show the turbulent kinetic energy equation budgets and the Reynolds stress streamwise direction budget in a channel flow at ${Re_{\tau }=180}$ . The results are compared with the DNS data of Hoyas and Jimenez (2008) (in circles) and Moser et al. (1999) (in triangles) that can be found freely available online. As it can be seen, the residuals close well within an order of magnitude with respect to the production. This proves that the methodology used in this entry is sound and that Alya can close the budgets if given the necessary grid resolution and integration time.

Figure 12: Channel flow

{Re_{\tau }=180}

, Alya DNS-8M DoF, TKE and stream-wise Reynolds stress budget terms comparing with Hoyas and Jimenez (circles) and Moser, Kim and Mansour (triangles).

TKE and Reynolds stress budgets are provided in Statistical Data for the inlet duct of the diffuser. These profiles have been obtained by extracting two vertical (along y direction) and two horizonal (along z direction) lines for a fixed x position. A total of 100 planes on the x position are obtained between ${x/h=-25}$ and ${x/h=5}$ . These lines are then averaged amongst the x planes and then averaged among their direction (i.e., one vertical line with the other and likewise for the horizontal line) to exploit the symmetry of the duct. Finally, one average over the mid span of the line is performed. These budgets are shown in Fig. 17 and Fig. 18. As it can be seen, the budgets close well enough at around 11.7% and 14.6% of the production for TKE for the vertical and horizontal lines respectively, and 10.4% and 12.2% of the production for the stream-wise Reynolds stress for the vertical and horizontal lines respectively.

Regarding the budgets at the diffuser area, results have been found to be more challenging. In that region massive separation occurs, and being in a full three-dimensional configuration, no symmetry can be exploited to diminish the time integration required to close the budgets. In addition to that, a "slow breathing" mechanism has been observed in the diffuser, associated with small frequencies that have only been represented 5 times during the integration time considered. The figures below clearly show that closing the budgets has not been achieved with the 21 flow-throughs of integration time.

Figure 13: Stanford double diffuser, Re=10000. TKE and stream-wise Reynolds stress budgets at the diffuser area (

{x/h=1}

).

Figure 14: Stanford double diffuser, Re=10000. TKE and stream-wise Reynolds stress budgets at the diffuser area (

{x/h=15}

).

Therefore, it is recommended for future campaigns to increase a bit the grid resolution but also largely increase the integration time considered. While a DNS at Re=10000 is attainable, massively separated flows still pose a challenge on computational effort due to the large integration time required to converge the solution in terms of the Reynolds stress budgets.

References

Hoyas, S., Jiménez J.(2008): Reynolds number effects on the Reynolds-stress budgets in turbulent channels. In Physics of Fluids Vol. 20.10, pp. 101511.
Moser, R. D., Kim, J., Mansour, N. N. (1999): Direct numerical simulation of turbulent channel flow up to Re_tau 590. In Physics of fluids Vol. 11.4 , pp. 943-945.

Contributed by: Oriol Lehmkuhl, Arnau Miro — Barcelona Supercomputing Center (BSC)

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Description

Computational Details

Quantification of Resolution

Statistical Data

Instantaneous Data

Storage Format