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Large-Eddy and Direct Numerical Simulations of the Bachalo-Johnson Flow with Shock-Induced Separation

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Abstract

The Bachalo-Johnson experiment on an axisymmetric bump has been a primary validation case for turbulence models in shock-boundary-layer interactions since the 1980’s. In the present work, Wall-Modelled Large-Eddy Simulations (WMLES) of this flow were conducted using Improved Delayed Detached-Eddy Simulation (IDDES) as the sub-grid-scale (SGS) and wall model, with a synthetic turbulence generator, expecting close enough agreement with experiment. However, the WMLES results are disappointing, even in terms of the shock position, even though the results from two grids with 4.7 × 108 and 1.6 × 109 cells respectively agree well with each other. This failure of grid refinement to warn of an inaccurate simulation is of great concern, and the reasons for it are explored. We then conducted a Direct Numerical Simulation (DNS) embedded in the LES over a reduced domain, with 8 × 109 grid cells. The DNS has a far more accurate shock position and overall pressure distribution. The skin friction in the favourable pressure gradient is also much higher than in the LES; thus, wide differences appear upstream of the shock wave, most probably caused by the rapid acceleration which leads to atypical shear-stress profiles. Other SGS models were tried, and performed worse than IDDES. The DNS essentially fulfils the initial expectations although in a reduced domain and provides data for turbulence-modelling studies, for instance by extracting an effective eddy viscosity from it. The most noticeable remaining disagreement with experiment is over the Reynolds shear stress.

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Acknowledgments

An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. We are grateful for the extensive help of Dr. R. Balakrishnan. Large resources were also provided by the Supercomputing Centre “Polytechnichesky” of the St.-Petersburg Polytechnic University. We benefited from generous and detailed discussions with D. Johnson and W. Bachalo (their model has been destroyed, but they have blueprints).

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Authors and Affiliations

  1. Boeing Commercial Airplanes, P.O. Box 3707, Seattle, WA, 98124, USA

    Philippe R. Spalart

  2. St.-Petersburg Polytechnic University, 29, Polytechnicheskaya str., St.-Petersburg, 195251, Russia

    Kirill V. Belyaev, Andrey V. Garbaruk, Mikhail L. Shur, Mikhail Kh. Strelets & Andrey K. Travin

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  1. Philippe R. Spalart
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  2. Kirill V. Belyaev
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  3. Andrey V. Garbaruk
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  4. Mikhail L. Shur
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  5. Mikhail Kh. Strelets
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  6. Andrey K. Travin
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Correspondence to Philippe R. Spalart.

Appendix: Effect of Azimuthal Domain

Appendix: Effect of Azimuthal Domain

Here, we address the issue of the narrower domain, which had to be accepted to make the DNS possible. The primary established criterion, in simulations with periodic conditions, is for two-point correlation coefficients to drop close enough to zero and remain there when approaching a separation between the points equal to half of the period. We present such results, and also apply the other criterion of a weak dependence of the statistical quantities on the period; this is a test we were in a position to conduct with IDDES.

Figure 21 compares the wall-pressure correlations in the DNS and in an IDDES with a wide domain but for the same separation. The agreement is quite good upstream of separation, although in the DNS the levels fall only into the [0.1, 0.2] range, so that a description of this period as “just sufficient” may be in order. At the same separations, the IDDES field is lower. By symmetry, the DNS field must have a zero derivative at 7.5 degrees, but the IDDES does not. Near x/c = 1 and beyond, the DNS field reveals a considerable deterioration, naturally attributed to the much increased thickness of the turbulent layer. The IDDES field has values of the order of 0.3 to 0.6 at 7.5 degrees, so that it could be predicted that the narrow-domain DNS would not approach zero. This feature persists even after reattachment, probably reflecting eddies with significant lateral correlation propagating even after the boundary-layer thickness comes down.

Fig. 21
figure 21

Maps of two-point correlation coefficient of surface pressure from DNS in azimuthal domain of 15° (a) and from IDDES on Grid 2 in azimuthal domain of 60° (b)

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The impression given by two-point velocity correlations in Fig. 22 is more favourable. First, the IDDES results on both Grid 1 and Grid 2 indeed approach zero over much of the interval. Second, the residual correlations at mid-period in the DNS are well under 0.1 upstream of the shock, and then a little in excess of 0.1 after reattachment. These results suggest that a period of 30o would be an excellent compromise. Thus, there is no suggestion that a period of 360o would be required by the physics. Recall the body radius of about 0.4c, and the boundary-layer thickness near 0.05c; this suggests that the lateral curvature, while not negligible, is not dominant. The largest disagreement is at x/c = 0.8, after separation.

Fig. 22
figure 22

Profiles of two-point correlations of streamwise velocity in the “hot points” from IDDES on Grids 1 and 2 and from DNS

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Finally, Fig. 23 measures the effect of the narrow domain on the IDDES, with the resolution of Grid 2. There is essentially no effect either on pressure or skin friction. We conclude that the reduction of the domain width is not causing drastic problems although of course, in the future, a domain of the order of 30o, and the extension of DNS all the way to the outflow boundary, will remain desirable.

Fig. 23
figure 23

Comparison of distributions of pressure and friction coefficients predicted by IDDES on Grid 2 in narrow and wide azimuthal domains

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Spalart, P.R., Belyaev, K.V., Garbaruk, A.V. et al. Large-Eddy and Direct Numerical Simulations of the Bachalo-Johnson Flow with Shock-Induced Separation. Flow Turbulence Combust 99, 865–885 (2017). https://doi.org/10.1007/s10494-017-9832-z

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