On Scale-Resolving Simulation of Turbulent Flows

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On arbitrary unstructured meshes these schemes do not provide high order of accu- racy remaining of the ..... Ffowcs Williams, J.E., Hawkings, D.L.: Sound generated by turbulence and surfaces in unsteady motion. Philos. Trans. R. Soc.
On Scale-Resolving Simulation of Turbulent Flows Using Higher-Accuracy Quasi-1D Schemes on Unstructured Meshes Alexey Duben and Tatiana Kozubskaya

1 Introduction Currently, the scale-resolving LES and hybrid RANS-LES approaches for turbulentflow simulation are mainly used on structured meshes and, correspondingly, exploit structured algorithms. On this way, it is easier to satisfy the requirements for scaleresolving approaches, which are high accuracy and minimization of numerical dissipation, while maintaining the stability of the numerical scheme for the approximation of convective fluxes. At the same time, the applicability scope of LES and especially hybrid RANS-LES approaches while using structured meshes is limited with regard to industrial problems. The use of unstructured meshes can improve the workability of the scale-resolving approaches and promote their implementation in massive industrial computations. A weak point of most high-accuracy unstructured algorithms, in particular based on Discontinuous Galerkin or k-exact FV schemes, is their high computational costs which prevents their wide use in industrial applications. As an alternative, we suggest to use higher-accuracy schemes based on quasi-1D reconstruction of variables. On arbitrary unstructured meshes these schemes do not provide high order of accuracy remaining of the second theoretical order, however their accuracy proves to be noticeably higher in terms of error values in comparison with most second-order schemes. Moreover, being applied to uniform grid-like meshes they naturally transform to high-order (up to the 5th–6th) finite-difference algorithms. And, finally, thanks to their quasi-1D nature, these schemes possess significantly lower computational costs in comparison with very high order algorithms. A goal of this paper is to adjust the higher-accuracy quasi-1D schemes to the turbulent flow simulations within scale-resolving approaches which are rather sensitive to such numerical issues as scheme dissipation, instability, mesh quality, etc. A. Duben (✉) ⋅ T. Kozubskaya Keldysh Institute of Applied Mathematics of RAS, Moscow, Russia e-mail: [email protected] © Springer International Publishing AG 2018 Y. Hoarau et al. (eds.), Progress in Hybrid RANS-LES Modelling, Notes on Numerical Fluid Mechanics and Multidisciplinary Design 137, https://doi.org/10.1007/978-3-319-70031-1_14

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2 Numerical Approach in General For turbulent flow modelling we use the most recent DES formulation [18]. It has strongly advanced in solving one of the fundamental problems of the original method which is the so-called grey area problem resulting in a delay of numerical transition from the RANS to LES solution in shear layers. The governing equations are approximated using the vertex-centered EBR schemes [1, 2] on unstructured meshes. In this paper we mostly consider hexahedral meshes. For time integration we use the second-order implicit method based on the Newton linearization and locally preconditioned BICG-stab linear-algebra solver. All the numerical techniques in use are implemented within in-house code NOISEtte [3, 7] for solving aerodynamics and aeroacoustics problems on unstructured meshes.

2.1 Quasi-1D Scheme Consider specifically the EBR schemes which underlie the basic numerical algorithm and mainly determines its properties. The feature of these schemes consists in the special approximation of convective fluxes, so we can describe them briefly as applied to the hyperbolic system 𝜕𝐐∕𝜕t + ∇ ⋅ 𝐅(𝐐) = 0 written with respect to conservative variables 𝐐 = (𝜌, 𝜌𝐮, E)T . Here 𝜌—density, 𝐮—velocity vector, p— pressure, 𝐈—identity matrix, 𝐅 = (𝜌𝐮, 𝜌𝐮𝐮 + p𝐈, (E + p) 𝐮)T is the flux vector. The general formulation of edge-based vertex-centered schemes can be represented as ) ( d𝐐 1 ∑ =− 𝐡 (1) dt i vi j∈N (i) ij 1

where vi —the volume of the dual cell built around vertex i, N1 (i)—a set of first-level neighbors of node i, 𝐡ij —the numerical flux which is calculated as 𝐡ij = 𝐅ij ⋅ 𝐧ij , 𝐧ij = ∫ 𝐧 ds, 𝐧—the normal to the cell surface. A key point is that both the approxi𝜕Cij

mations of flux vector 𝐅ij and normal vectors 𝐧ij defined at the interface between the nodes i and j are calculated at the edge ij midpoint only. According to the assumption of (1), a specific EBR scheme is defined by the method of calculating 𝐅ij on the mesh ij. We calculate 𝐅ij as a solution of approximate L∕R L∕R Riemann problem basing on the left and right states 𝐅ij or/and 𝐐ij which, in their turn, are determined with the use of quasi-1D reconstructions along the edge ij direction. The reconstructions are built in a way that they have to transform to the correspondent high order finite-difference approximations when applied to uniform grid-like (i.e. translationally symmetric) meshes. Further we denote an EBR scheme as EBRn scheme if its highest theoretical order (which is reachable on translationallysymmetric meshes) is equal to n.

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Fig. 1 EBR scheme stencil for unstructured isotropic triangular mesh

Figure 1 schematically shows the EBR5 scheme stencil we use in 2D case for the flux calculation on a triangular unstructured mesh. The boundaries of cells built around the vertices i and j, the formulas for calculating the reconstructed variables 𝐐Lij and 𝐐Rij involve the values in points denoted as i − 2, i − 1, j + 1, j + 2. These “auxiliary” values are determined with the help of linear interpolation on the correspondent edges which are intersected by the edge ij direction (see Fig. 1). In doing so, the six points {i − 2, i − 1, i, j, j + 1, j + 2} constitute the 1D-stencil for the approximation of flux 𝐅ij at the edge ij midpoint for EBR5 scheme. The stencil can be more compact, for instance, if you use only the four points {i − 1, i, j, j + 1} or be more extended along the edge ij direction. The 4-points stencil corresponds to the EBR3 scheme. In 3D, the reconstructions of left and right states are built in a similar way with replacing the intersected edges with the intersected faces. Basing on the reconstructed variables 𝐐Lij and 𝐐Rij we calculate the flux 𝐅ij using the hybrid version of Roe scheme1 [13] which can be written as , 𝐅ij = (1 − 𝜎) ⋅ 𝐅Cij + 𝜎 ⋅ 𝐅UPW ij ( )] ] [ ( ) [ | | C R L and 𝐅UPW are where 𝐅Cij = 0.5 𝐅 𝐐Lij + 𝐅 𝐐Lij = 𝐅 − 𝜎 − 𝐐 0.5 𝐐 𝐀 | | ij ij ij ij ij | | central-difference and upwind approximations, respectively, of the convective flux, 𝐀ij —the Jacobian matrix composed of Roe-averaged variables, 𝜎 ∈ [0, 1]—the weight coefficient. It is known that the scale resolving approaches for the correct simulations require an optimal balance between the numerical dissipation and instability. A specificity of the EBR schemes allows to provide this balance not only by tuning the weight 𝜎, but also by choosing the stencil width and thereby switching, for instance, from EBR3 to EBR5 schemes or back depending on the flow parameters. Moreover, it is possible to use the stencils of different width at different interfaces of the same cell.

2.2 Scheme for Arbitrary Non-isotropic Meshes Structured meshes with hexahedral elements are widely used for simulations of cases with essential influence of shear layers as most reliable for scale-resolving 1

Instead of the Roe scheme, some other Riemann solvers can be used.

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simulation. It is easy to adjust these meshes for such flow peculiarities as anisotropy of quasi-2D vortexes in the initial part of shear layers and isotropy in the areas of developed turbulence. So anisotropic, aligned in homogenous (mostly spanwise) direction, and predominantly isotropic mesh cells are used in corresponding domains. Researchers who use structured algorithms adjust computational meshes to the flow physics basing on their algorithm’s capabilities and limitations. As an example, there are some best practice guidelines [16] for accurate jet noise simulation in respect of increasing ratio of grid spacing in axial direction. Smooth curvature of grid lines is usually maintained to prevent undesirable numerical oscillations. DES approach as well as many other scale-resolving methods based on explicit LES model (unlike implicit LES, ILES, approaches) necessitate exploiting lowdissipative numerical scheme for convective fluxes [15]. According to the recent DES formulation [18], the acceleration of RANS-to-LES transition during the numerical simulation is gained by the reduction both of the model and scheme dissipation at the initial region of shear layers. But the diminishing of dissipation also leads to instabilities of the solution which can be intensified by the presence of strong gradients and shocks. So the numerical algorithm should balance between minimal dissipation and stability of the solution during scale-resolving simulation to achieve maximum accuracy for the mesh in use. Application of the original low-dissipative EBR scheme with extended stencil for simulation of jet on the structured hexahedral mesh resulted in instabilities of the solution in the initial shear layer regions near nozzle edge. Analysis of the solution revealed that the problem was related to quasi-1D property of the EBR scheme. In order to clarify the issue let’s look at the typical mesh pattern in the transverse section near the initial shear layer region and examine the quasi-1D scheme stencil (see Fig. 2). There are two sets of mesh lines: aligned along the nozzle edge (jet azimuthal direction) and along radial direction. The structured algorithm considers corresponding reconstruction directions which are “physically” reasonable. But extended quasi-1D scheme stencil in azimuthal direction which is plotted on Fig. 2 (including points i − 2, i − 1, i, j, j + 1, j + 2) has following weaknesses. Distances between points involved in the reconstruction are different. Moreover corresponding values of variables are computed using linear interpolation by definition of the EBR scheme. Finally, it’s essential that the values at i − 2 and/or j + 2 points could vary significantly because of the thin boundary layer. These properties cause instabilities while using low-dissipative schemes. Increasing the scheme dissipation is not the best solution because it could essentially influence the RANS-to-LES transition and subsequent accuracy of flow prediction. Note that moderate changes in mesh

Fig. 2 EBR scheme stencil in transversal section of anisotropic mesh near nozzle edge for jet case

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(curvature, stretching ratios) could result in significant accuracy and stability variation of quasi-1D unstructured scheme unlike corresponding structured. It makes the EBR scheme more sensitive to mesh quality than structured algorithms. In order to improve its robustness, an enhancement of the numerical algorithm is required. We developed a modification of the EBR scheme by implementing anisotropy which provides adaptive switching between stencil widths depending on the reconstruction direction, local mesh parameters and solution. Thus, it facilitates choosing appropriate quasi-1D scheme for the particular mesh edge to obtain maximum accuracy while maintaining stability of the solution. The scheme analyzes numerical stencils during initialization of the computation and sets appropriate stencil sizes depending on two criterions. The first one takes into account distance ratios inside extended scheme stencil. The second criterion is based on interpolation coefficients of the points involved in the reconstruction. Depending on the particular thresholds the EBR scheme switches between stencils of different size. This modification improves scheme stability in the initial region of shear layer and allows to avoid numerical oscillations in the areas of developed turbulence without increasing the scheme dissipation. Notice that switching to the compact stencil occurs only in the required directions of reconstruction.

3 Application and Validation The feasibility of developed numerical algorithm and its modifications is demonstrated on the predictions of two industry-oriented problems dealing with turbulent flows of different types (free and near-wall separated flows). The first problem is the simulation of immersed subsonic free round jet studied experimentally in [4, 5, 10, 11, 14, 20]. The second problem is the simulation of well-known case M219 [9] which represents open-type subsonic near-cavity flow. The common feature of these two problems is the presence and significant influence of the shear layers which essentially define simulated flow characteristics.

3.1 Computational Setup We consider the jet with Mach number Mjet = 0.9 and Reynolds number ReD = 1.1 × 106 based on nozzle diameter D. The computational setup including mesh and precomputed profiles of variables at nozzle exit is provided by Shur et al. [17, 19] from SPbPU. The results computed using Grid 2 (see [17, 19]) containing 4.1M nodes are considered. The comparison with the experimental data is given both for the aerodynamic [4, 5, 10, 11, 14] and far field acoustic [20] characteristics. The Ffowcs-Williams and Hawkings (FWH) method [8] with closed permeable control surfaces according to Shur et al. [16] is used for far field noise extrapolation. 300

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Fig. 3 Jet mesh in the longitudinal z∕D = 0 (left) and transverse x∕D = 0 (right) sections coloured by the type of EBR scheme stencil. Dash line at the right image corresponds to the nozzle edge

convective time units were accumulated to obtain both averaged data in the near field and far field noise. Figure 3 shows mesh patterns (every second point and edge is plotted) used for the jet case simulation. Here all edges are colored by either blue or red that corresponds to compact (EBR3) and extended (EBR5) scheme stencils. EBR3 means weighted 3rd order upwind and 4th order central-differences (CD) reconstruction. EBR5 hybridizes 5th order upwind and 6th order CD correspondingly. It can be seen that near the nozzle edge with anisotropic (coarse in azimuthal direction) cells the compact (EBR3) stencil is used. EBR5 scheme is used mostly in the streamwise axial direction due to very slow growth of corresponding cell size and mild curvature of mesh lines. Computational setup of M219 cavity flow with M = 0.85 and ReH = 1.37 × 106 (H—cavity depth) is detailed in Dankov et al. [6]. It follows recommendations from [12]. The mesh used for computations contains 4M nodes. The numerical results are compared with the experimental data [9] on pressure pulsations on cavity floor. Sample length of the signals of pressure pulsations for postprocessing was approximately 1 s. Figure 4 shows mesh patterns in the central longitudinal section with edges colored the same way as in Fig. 3. It is clearly seen that the extended reconstruction stencil is applied in zones downstream the leading edge, which are discretized by

Fig. 4 M219 cavity mesh in longitudinal z∕H = 0 section colored by type of EBR scheme stencil

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uniform cartesian mesh. Stretching or increasing mesh step in particular directions leads to switching to compact EBR3 scheme.

3.2 Results Averaged near field profiles of turbulence characteristics and far field 1/3 octave spectrums at different observer angles obtained from the jet case simulation are presented on Figs. 5 and 6 correspondingly. Along with experimental data the results of Shur et al. [19] marked as “NTS” are shown on the plots. They can be regarded as reference that means providing maximum accuracy of jet simulation on considered mesh. It is obvious that NOISEtte results are very well compared with both experimental and numerical which are obtained using high-accuracy structured algorithm. Crucially, application of unstructured quasi-1D approach doesn’t lead to far field noise contamination that can appear due to numerical oscillations caused by lowdissipative scheme instabilities. It is important because unlike turbulent flow characteristics acoustics is sensitive to non-physical waves that could occur in the solution. At the same time obtained results reveal plausible jet flow prediction which significantly depends on accurate simulation in the initial parts of shear layers and early RANS-to-LES transition. Results of the computation of the flow near M219 cavity are presented on Fig. 7. As well as for the jet case they are in good agreement with the corresponding experimental data [9], regarding both overall sound pressure level (OASPL) and its frequency bands distribution. Correct prediction of the Rossiter modes’ frequencies, levels and shapes exhibits accurate simulation of the flow physics.

Fig. 5 Jet case results: streamwise distributions of averaged jet centerline longitudinal velocity (top, left) and its rms (top, right) and resolved Reynolds stresses along jet lipline (bottom)

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Fig. 6 Jet case results: 1/3 octave SPL distrubutions in far field points at different observer angles

Fig. 7 M219 cavity case results: OASPL (left, top) and power spectra density in corresponding points on cavity floor. OASPL distribution is along the line y∕H = −1, z∕H = −0.25

4 Concluding Remarks The numerical algorithm based on the edge-based quasi-1D schemes possessing higher accuracy and moderate computational costs on unstructured meshes is considered. The issues related to the application of the low-dissipative EBR scheme version for the scale-resolving simulation on anisotropic meshes are discussed. Some solutions which improve the scheme robustness are proposed. They mostly relates to the adaptive anisotropy of the scheme depending on the flow and mesh properties. The

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feasibility of the developed numerical algorithm is demonstrated on simulations of the two cases dealing with separating turbulent flows. The computations performed show a strong sensitivity of the results of scaleresolving simulations to the numerical scheme in use and confirm a need in its careful adjustment. In this sense the “grey area” regions are of special attention. In these areas the scheme should be both accurate and stable enough to allow for a correct representation of arising small physical instabilities. The problems under consideration in the paper are proper for the most methods and codes that implement scaleresolving approaches to turbulent flow simulation on unstructured meshes. In particular, all of them, to a greater or lesser extent, should involve techniques providing an adaptive dissipation-instability balancing. Further improvements of unstructured numerical algorithm are still required. In future investigations we plan to develop the hybrid scheme mostly in the direction of adaptability improvement and equip it with the WENO techniques based on WENOEBR schemes, to extend its applicability to the problems with shocks. Acknowledgements The research is supported by Russian Science Foundation. The implementation of the DES approach [18] on unstructured meshes and the computations are performed within Project 14-11-00060. The development of adapting higher-accuracy algorithms is a part of Project 16-11-10350. The computations were carried out using “Lomonosov” (MSU) and “10P” (JSCC RAS) supercomputers.

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