Introduction

The HiFi-TURB-DLR rounded step test case is a case in which a turbulent boundary layer flows over a flat plate before reaching a planar rounded backward step. The particular geometry generates an adverse pressure gradient which accelerates the boundary layer upstream the step before inducing an incipient separation above the smooth step, a flow physics that is well known to be not correctly predicted by state-of-the-art Reynolds-Averaged Navier-Stokes (RANS) models.

Review of previous studies

The design of the test case was inspired by the experimental work on the flow around an axisymmetric test body of Disotell et al. and Disotell et al. (2017). Disotell et al. focused their study of the possible separation and reattachment in the rear part of the body. The original test case is here revised as a 2D planar flow problem without any tunnel walls but with a far-field condition opposite to the solid wall. The reasons for using a planar geometry are thoroughly discussed in UFR 3-36 Test Case.

Description of the test case

Geometry and flow parameters

The test case is designed as a numerical experiment with the aim of comparing RANS results to DNS data. The flow problem was defined by DLR with the aid of RANS calculations to get a condition with incipient separation. Other cases with moderate and full separation were also designed by DLR, see Alaya et al. (2021), but for the present uDNS only the incipient condition for the Reynolds number Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {Re_H=78,490}} was considered, where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H} is the step height. The geometry of the curved step is shown in Fig. 2 and its definition is reported in UFR 3-36 Test Case.

Figure 2: Set-up for DNS Simulation

The simulation was performed by using the UNIBG in-house software MIGALE, see ‌Bassi et al. (2015), on a computational mesh made of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 15,016,384} hexahedral elements with quadratic edges, a wall-normal growth ratio of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ~1.2} and a first cell height of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {y^+ \approx 1}} , cf. DNS 1-5 Computational Details. The average time step size was Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\Delta t = 19/14000}} CTU, where the convective time unit (CTU) is defined as the ratio between Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {H}} and the freestream velocity Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {U_\infty}} . The turbulence statistics were collected for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 26} CTU.

Computation domain and boundary conditions

Although for DNS the inflow boundary conditions are different than RANS, they must guarantee the same boundary layer properties at a given point upstream of the rounded step, i.e., Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x/H = -3.51}} , to allow a valid comparison. At the reference position Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x_{ref}}} , the properties to be matched are the Reynolds number based on the momentum thickness Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {Re_{\theta} = ????}} and the Reynolds number based on the friction velocity Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {Re_{\tau} = ???}} . As a technique to promote the laminar-turbulent transition and reduce the upstream length of the domain, inspired by the work of Housseini et al. (2016) and Schlatter and Örlü (2012), a local tripping term was used. To define the mesh density and the computational domain size that ensure the target boundary layer integral parameters at the reference location Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x_{ref}}} , a precursory computational campaign for the turbulent flow over a flat plate was performed. According to the outcomes of this campaign, the inlet boundary is located at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x/H = -12.71}} , where the Blasius laminar velocity profile computed at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {Re_x=650,000}} , the uniform static pressure and the uniform total temperature (see UFR 3-36: Table 2) are imposed. At location Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x/H = -12.1}} , within the laminar boundary layer region, the tripping source term is activated to promote the transition to turbulence. This set-up corresponds to the laminar and turbulent distances Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {L_{l} = ???}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {L_{t} = ???}} , respectively, see Fig. 2. At the outlet boundary, placed at $x/H=23.95$ , the static pressure Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {P_{s,ref}}} is imposed, see UFR 3-36: Table 2. To mitigate spurious perturbations possibly originating at the outlet boundary, the mesh is coarsened in the streamwise direction for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x/H > 13.82}} . The upper boundary is a permeable far-field Riemann boundary condition located Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 179 H} from the no-slip adiabatic wall and computed via the exact Riemann solver. Finally, side planes are considered as periodic with a distance from each other of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\Delta z = 3 H}} .

References

Disotell, K. J. and Rumsey, C. L.: Modern CFD validation of turbulent flow separation on axisymmetric afterbodies.
Disotell, K. J. and Rumsey, C. L. (2017): Development of an axisymmetric afterbody test case for turbulent flow separation validation. NASA/TM-2017219680, Langley Research Center, Hampton, Virginia
Alaya, E., Grabe, C. and Knopp, T. (2021): Design of a parametrized numerical experiment for a 2D turbulent boundary layer flow with varying adverse pressure gradient and separation behaviour. DLR-IB-AS-GO-2020-109, DLR-Interner Bericht, DLR Institute of Aerodynamics and Flow Technology
Bassi, F., Botti, L., Colombo, A. C, Ghidoni, A. and Massa, F. (2016): On the development of an implicit high-order Discontinuous Galerkin method for DNS and implicit LES of turbulent flows. European Journal of Mechanics, B/Fluids, Vol. 55(2), pp. 367-379
S. Hosseini, R. Vinuesa, P. Schlatter, A. Hanifia and D. Henningson (2016): Direct numerical simulation of the flow around a wing section at moderate Reynolds number, International Journal of Heat and Fluid Flow, 61:117–128
Schlatter, P. and Örlü, R. (2012): Turbulent boundary layers at moderate Reynolds numbers: inflow length and tripping effects,Journal of Fluid Mechanics, 710:5–34