Numerical study of turbulent separation bubbles with varying pressure gradient and Reynolds number

Abstract

A family of cases each containing a small separation bubble is treated by direct numerical simulation (DNS), varying two parameters: the severity of the pressure gradients, generated by suction and blowing across the opposite boundary, and the Reynolds number. Each flow contains a well-developed entry region with essentially zero pressure gradient, and all are adjusted to have the same value for the momentum thickness, extrapolated from the entry region to the centre of the separation bubble. Combined with fully defined boundary conditions this will make comparisons with other simulations and turbulence models rigorous; we present results for a set of eight Reynolds-averaged Navier–Stokes turbulence models. Even though the largest Reynolds number is approximately 5.5 times higher than in a similar DNS study we presented in 1997, the models have difficulties matching the DNS skin friction very closely even in the zero pressure gradient, which complicates their assessment. In the rest of the domain, the separation location per se is not particularly difficult to predict, and the most definite disagreement between DNS and models is near reattachment. Curiously, the better models tend to cluster together in their predictions of pressure and skin friction even when they deviate from the DNS, although their eddy-viscosity levels are widely different in the outer region near the bubble (or they do not rely on an eddy viscosity). Stratford’s square-root law is satisfied by the velocity profiles, both at separation and reattachment. The Reynolds-number range covers a factor of two, with the Reynolds number based on the extrapolated momentum thickness equal to approximately 1500 and 3000. This allows tentative estimates of the improvements that even higher values will bring to the model comparisons. The solutions are used to assess models through pressure, skin friction and other measures; the flow fields are also used to produce effective eddy-viscosity targets for the models, thus guiding turbulence-modelling work in each region of the flow.

Spatio-temporal dynamics of turbulent separation bubbles

“…We perform direct numerical simulations of a turbulent boundary layer at Re θ = U o θ o /ν = 490, induced to separate via an APG generated by suitable application of a velocity boundary condition on the top of the computational domain. This Reynolds number is comparable to recent DNS of separating TBLs (e.g., Coleman et al (2018) at Re θ = 500 -1500 in the ZPG region, and Abe (2017) at 300, 600 and 900). The Reynolds number is limited by the resolution demands imposed by the very long separation bubble that develops in this flow and the particularly long simulation time required to characterize the low-frequency motion.…”

Measurements of pressure and velocity fluctuations in a family of turbulent separation bubbles

“…x/L p = 0.2, in a manner very similar to the Patrick (1987) Na & Moin (1998a), Abe (2017) (cases SB2 and LB) and Coleman et al (2018) (case A) for x/L p < 0.5 are relatively large compared to our data, to the Coleman et al (2018) case C and to Patrick (1987). This is attributed to a lower Reynolds number for those former cases.…”

“…Based on a comparison of the normalized separation lengths L b /θ 0 and aspect ratios H b /L b listed in table 1, and based on the pressure and velocity data from figures 2 and 3, we conclude that our Large TSB is probably geometrically closest to the Na & Moin (1998a) flow, despite a significant difference in Reynolds number. Similarly, our Medium TSB is probably the closest to the Coleman et al (2018) case C, with a factor of approximately 2 in Reynolds number and a notable difference in pressure distribution. Of course, two TSB flows would only be identical if the Reynolds numbers and pressure distributions were identical, or equivalently if the Reynolds numbers were identical and the same transpiration profiles were imposed at the same wall-normal distance.…”

“…Generally speaking, except for Perry & Fairlie (1975), most c p distributions appear to cluster between the Coleman et al (2018) cases A and C. The pressure distributions from Na & Moin (1998a), Abe (2017) (case LB) and Wu & Piomelli (2018) are very close, mostly because the latter authors designed their simulations to reproduce Na and Moin's results. The Coleman et al (2018) case A has comparable pressure gradients but imposed on a shorter distance. In contrast, their case C imposes smaller gradients but over a larger distance.…”

“…The bi-modal distribution of pressure fluctuations, with a first peak of the fluctuating pressure coefficient near separation and a second peak near reattachment, was confirmed (here, is the root mean square of the wall-pressure fluctuations and is the fluid density). The drop in observed near the middle of the separation bubble was attributed to the negative production rate of turbulent kinetic energy (TKE) at the top of the shear layer, which is caused by the switch between APG and FPG in a suction-and-blowing transpiration profile (see also the corresponding discussion in Coleman et al (2018)). Furthermore, both the first and second peaks of appear to depend on the size of the TSB and consequently on the exact streamwise pressure distribution and transpiration profile.…”