Introduction

Give a brief overview of the test case. Describe the main characteristics of the flow. In particular, what are the underlying flow physics which must be captured by the computations ? Give reasons for this choice (e.g. poorly understood flow physics, difficulty to predict the flow with standard turbulence models, ...). Detail any case-specific data that needs to be generated.

Review of previous studies

Provide a brief review of related past studies, either experimental or computational. Identify the configuration chosen for the present study and position it with respect to previous studies. If the test case is geared on a certain experiment, explain what simplifications ( e.g. concern- ing geometry, boundary conditions) have been introduced with respect to the experiment in the computational setup to make the computations feasible and avoid uncertainty or ambiguity.

Description of the test case

A detailed self-contained description should be provided. It can be kept fairly short if a link can be made to an external data base where details are given. Then only the differences should be clearly indicated.

Geometry and flow parameters

Describe the general set up of the test case and provide a sketch of the geometry, clearly identifying location and type of boundaries. Specify the non-dimensional flow parameters which define the flow regime (e.g. Reynolds number, Rayleigh number, angle of incidence etc), including the scales on which they are based. Provide a detailed geometrical description, by preference in form of a CAD, or alternatively as lists of points and a description of the interpolation.

The test case is designed as a numerical experiment with the aim of comparing RANS results to DNS data. For the set-up of the DNS, the inflow boundary conditions are different, e.g. a recycling method can be used to generate the turbulent input or synthetic turbulence can be injected. 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 APG-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 smooth bump, 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 DNS have been performed by using the UNIBG in-house software MIGALE [‌20]. MIGALE is an implicit high-order discontinuous Galerkin solver for the compressible and incompressible Navier-Stokes equations. Godunov fluxes are treated with the exact solution of the local Riemann problems while viscous fluxes are handled by means of the BR2 scheme [‌21]. The time integration is performed with linearly implicit Rosenbrock type Runge-Kutta schemes with optimal stability properties up to order five. 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.

Boundary conditions

Specify the prescribed boundary conditions, as well as the means to verify the initial flow development. In particular describe the procedure for determining the in flow conditions comprising the instantaneous (mean and fluctuating) velocity components and other quantities. Provide reference profiles for the mean flow and fluctuations at in flow - these quantities must be supplied separately as part of the statistical data as they are essential as input for turbulence-model calculations. For checking purposes, these profiles should ideally also be given downstream where transients have disappeared; the location and nature of these cuts should be specified, as well as the reference result.

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

Front Page

Description

Computational Details

Quantification of Resolution

Statistical Data

Instantaneous Data

Storage Format