# PAdDLES: p-Adaptive Discretisations for Large Eddy Simulation in Industrial Geometry

Turbulence is pervasive in turbomachinery, and predicting its effect on the flow and performance is a major challenge for CFD (Computational Fluid Dynamics) tools. RANS (Reynolds-averaged Navier–Stokes) methods, modelling the averaged impact of turbulence, are currently the industrial standard. Scale-resolving approaches such as DNS (Direct Numerical Simulation) and LES (Large Eddy Simulation) compute either all turbulent structures directly, or only represent the larger, and model the impact of the smallest structures. However even if LES is much less demanding than DNS, its application to most practical flows still requires computational resources orders of magnitude above what is currently available and therefore these approaches are not used for aerodynamic design, unless for very specific niche applications.

RANS is not suited for the prediction of noise, flow instabilities, transition, etc. Given the importance of these phenomena, the industrial use of at least LES will be indispensable to meet the projected goals of fuel efficiency and noise reduction, put forward by the aeronautic industry and regulatory bodies.

Up to now, DNS and LES of fundamental flows in academia have been performed using highly accurate structured mesh discretisations, such as finite differences or spectral methods, in order to cope with the important resolution requirements. These methods are, however, not applicable for the simulation of complex geometries such as encountered in industrial applications. For the latter reason, industry continues to use state of the art solvers, well suited for RANS approaches, for DNS and LES. Due to their low order of accuracy, very fine meshes then need to be used. Consequently industrial computations tend to be (severely) under-resolved.

There is a growing consensus that, in order to enable reliable industrial LES, new discretisation technologies will be needed that combine accuracy, massive parallelism and geometric flexibility. Recently, new discretisation techniques have emerged which provide high accuracy on unstructured, low quality methods. These methods also provide strong scalability up to very high core counts. This strong scalability allows to drastically reduce computational lead times, such that these become compatible with industrial design (fast science). Due to the flexibility of the discretisation allowing for both mesh size (h) and order (p) adaptation, one can furthermore expect to reduce operator dependence of the results (reliable analysis). The discontinuous Galerkin method (DGM) is arguably the most mature method in this class.

In the PAdDLES project a DGM solver is assessed for scale-resolving simulations. The main goal of the solver is the evaluation of whether p-adaptive DGM can be used for fast and reliable LES of turbomachinery flows. Due to the interesting characteristics of the discretisation scheme to dissipate the under-resolved scales, an implicit LES (ILES) model can be used in which the discretisation provides the adequate destruction of the turbulent structures which would decay beyond the mesh resolution. This approach is very promising as it does not require parameter tuning through dynamic procedures, which would be very hard in industrial geometry. For the same reasons, it is also quite controversial, such that a rigorous assessment and comparison to more standard explicit modeling approaches is required.

The first step of the project is therefore to investigate the quality of the ILES model on two canonical test cases: the channel flow and the 2D periodic hill flow. On the channel case, computations are performed from Re* _{τ}* = 180 to 950 [1]. Figure 1 shows the resulting velocity profiles obtained on the channel flow at several Reynolds numbers. Figure 2 presents

*Figure 1: LES of the turbulent channel flow at Re*

_{τ}= 180, 395, 590 and 950. Mean velocity profiles <u^{+}>. DGM (circle) compared to DNS (solid line) of Moser et al. (Re_{τ}= 180, 395 and 590) and Hoyas et al. (Re_{τ}= 950). The curves are shifted vertically by <Δu^{+}> = 2 for clarity.the averaged velocity fluctuation profiles. The flow over a 2D periodic hill benchmark was one of the benchmark cases during the second International Workshop on high order methods [2]. Both DNS (Re_{b} = 2800) and LES (Re_{b} = 10595) are also considered. Figure 3 shows the resulting volume rendering of vorticity at both Reynolds numbers.

A second part of the research concerns the demonstration of the advantages of highly scalable order adaptive DGM discretization. Two cases are considered. The first concerns DNS of the transitional low pressure turbine cascade T106C at Re_{2,is} = 8^{3}. This flow regime is notoriously difficult to predict. A second application is the LES of the JEAN nozzle (JEAN = Jet Exhaust Aerodynamics and Noise). The Reynolds number of the flow at the exit of the nozzle is Re = 5^{3} with a Mach number of M = 0.75.

At this moment in time, fixed order computations have been run with an interpolation order p=3. Figure 4 shows a snapshot of the resulting Mach number field for the T106C turbine case, whilst figure 5 shows an example of instantaneous flow field for the JEAN nozzle.

*Figure 2: Turbulent channel flow at Re*

_{τ}= 180 (DNS, top) and 950 (LES, bottom). RMS turbulent velocities: <u^{'}^{+}>*_{rms}(circle), <v^{'+}>*_{rms}(square) and <w^{'+}>*_{rms}(triangle) of DGM/ILES compared to DNS (solid line) of Moser et al. (Re_{τ}= 180) and Hoyas et al. (Re_{τ}= 950).Currently variable order computations are prepared for the T106C turbine blade. An adequate variation of the interpolation order across the domain allows to optimize computational cost. Typically, locally high order will be used for a better representation of the complex flow features near the trailing without significantly increasing the computational cost. The results on the T106C blade will be presented at the 2014 ASME Turbine Technical Conference and Exhibition (ASME Turbo Expo) [3].

*Figure 3: 2D periodic hill benchmark. Volume rendering of vorticity. Top: DNS at Re*

_{b}= 2800. Bottom: LES at Re_{b}= 10595.*Figure 4: T106C turbine blade. Instantaneous Mach number field.*

*Figure 5: JEAN nozzle benchmark. Mach number (top) and vorticity (bottom) fields.*

Project PAdDLES was made possible through the Partnership for Advanced Computing in Europe, PRACE. HPC system JUQUEEN of Jülich Supercomputing Centre serves as computing platform for this project.

The research was furthermore funded by the European Regional Development Fund under contract N° EP1A122030000102, and the FP7 European research project IDIHOM.

**References**

[1] C. Carton de Wiart, K. Hillewaert, L. Bricteux, and Winckelmans G. Implicit LES of free and wall bounded turbulent ows based on the discontinuous Galerkin/symmetric interior penalty method. Accepted with minor revisions for the publication in International Journal of Numerical Methods in Fluids (IJNMF), 2014.

[2] -. Second International Workshop on High-Order CFD Methods. Cologne, Germany,

27-28 May 2013.

[3] K. Hillewaert, C. Carton de Wiart, G. Verheylewegen, and T. Arts. Assessment of a high-order discontinuous Galerkin method for the direct numerical simulation of transition at low Reynolds number in the T106C high-lift low pressure turbine cascade. In Proceedings of the ASME Turbine Technical Conference and Exhibition, number GT2014-26739. ASME Turbo Expo 2014, 2014.

Koen Hillewaert

Argo team leader

Cenaero Headquarters

Rue des Frères Wright, 29 - B-6041 Gosselies/Belgium

e-mail: koen.hillewaert@cenaero.be