DNS 1-5 Description

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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., but for the present uDNS only the incipient condition for the Reynolds number ${Re_{H}=78,490}$ was considered, where $H$ 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 have been performed by using the UNIBG in-house software MIGALE [‌20], see DNS 1-5 Computational Details on a computational mesh made of $15,016,384$ hexahedral elements with quadratic edges concentrated near the wall region. The wall-normal growth ratio is approximatively $1.2$ with a first cell height of ${y^{+}\approx 1}$. The average time step size is ${\Delta t=19/14000}$ CTU, where the convective time unit (CTU) is defined as the ratio between ${H}$ and the freestream velocity ${U_{\infty }}$. Turbulence statistics have been collected for $26$ CTU.

Computation domain and boundary conditions

Although for the DNS the inflow boundary conditions are different than RANS, they must be properly set to guarantee that the boundary layer properties at a given point upstream of the step, here at ${x/H=-3.51}$, must be matched. in order to allow a comparison between RANS and DNS.

It is also possible to numerically trip the boundary layer from laminar to turbulent to generate the desired turbulent boundary layer. Hence, to ensure a comparison to the results achieved with RANS turbulence models, a reference position upstream of the adverse pressure gradient area is defined where boundary layer properties need to match between RANS and DNS computations to permit the comparison downstream in the region of interest.

The reference position is located at ${x/H=-3.51}$. Depending on the generation of turbulence at the inlet, the computational domain needs to be adapted to ensure the correct boundary layer properties at the reference position. If numerical tripping is performed, the laminar and turbulent distances need to be determined upstream of the reference position by precursor simulation as displayed in Fig. 2.

Figure 2: Set-up for DNS Simulation

At the reference position ${x_{ref}}$, the properties of the turbulent boundary layer are determined by the Reynolds number based on the momentum thickness ${Re_{\theta }=1496}$ and the Reynolds number based on the friction velocity ${Re_{\tau }=609}$ computed with SA-neg-RC-QCR-LRe model. To estimate the computational effort for DNS, the largest values of ${Re_{\theta }}$ and ${Re_{\tau }}$ are also given at the position ${x/H=10.5}$ by $4,624$ and $1,399$ respectively.

Amongst the different flow condition proposed, UNIBG focused the effort on the incipient separation configuration case with ${Re_{H}=78,490}$. A precursory computational campaign for the turbulent flow over a flat plate has been performed with the purpose of (i) assessing the effectiveness of the Synthetic Inlet Turbulence (SIT) injection strategy inspired by the work of Housseini et al. [‌18] and Schlatter and Örlü [‌19]; (ii) investigating the influence of the mesh density on the solution; (iii) defining the inlet boundary position and flow condition that ensure the target boundary layer integral parameters at the reference location ${x_{ref}}$ (see Fig. 3). According to the outcomes of this campaign, the inlet boundary is set at ${x/H=-12.71}$, where the Blasius laminar velocity profile computed at ${Re_{x}=650,000}$, the uniform static pressure and the uniform total temperature (see Table 2) are imposed. At location ${x/H=-12.1}$, within the laminar boundary layer region, the tripping source term is activated to promote the transition to turbulence [‌18][‌19]. The outlet boundary is positioned at ${x/H=23.95}$, with a pressure outflow condition having the same value of the inlet static pressure and at the end of a streamwise coarsened mesh region starting at ${x/H=13.82}$. Following the setup of the RANS computations, the no-slip adiabatic boundary condition is set while at the upper boundary, situated at ${179H}$ from the wall upstream the rounded step, the far-field boundary condition is imposed. Side planes, instead, are considered as periodic with a distance from each other of ${\Delta z=3H}$.

The simulation have been performed by using the UNIBG in-house software MIGALE [‌20], see DNS 1-5 Computational Details. The computational mesh is made of $15,016,384$ hexahedral elements with quadratic edges concentrated near the wall region. The wall-normal growth ratio is approximatively $1.2$ with a first cell height of ${y^{+}\approx 1}$. Time integration is performed with the fifth order – eight stages Rosenbrock scheme ROD5_1 [‌22] using a global time step adaptation strategy [‌23]. The corresponding average step size is ${\Delta t=19/14000}$ CTU, where the convective time unit (CTU) is defined as the ratio between ${H}$ and the freestream velocity ${U_{\infty }}$. Turbulence statistics have been collected for $26$ CTU.

References

1. Disotell, K. J. and Rumsey, C. L.: Modern CFD validation of turbulent flow separation on axisymmetric afterbodies.
2. 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
3. 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
4. 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

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Contributed by: Francesco Bassi, Alessandro Colombo, Francesco Carlo Massa — Università degli studi di Bergamo (UniBG)

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