A finite element overlapping scheme for turbomachinery flows on parallel platforms
Introduction
Turbomachinery flow phenomena are extremely complex, and the reliable prediction of decelerating cascade flows must account for a number of challenging features such as impingement, separation, transition, rotation, tip and passage vortices. Traditionally in turbomachinery design procedures resort to extensive try-and-cut at the development stage to address these issues. High performance rotors and narrow design margins demand accurate flow analysis. To this end, computational fluid dynamics (CFD) stands as a potentially powerful tool to reduce development costs and times, relating continuous improvement to the physical comprehension of turbomachinery environment. Nonetheless, three-dimensional (3D) accurate predictions of viscous flow field are yet to be used directly in the design procedures. Considerable progress has been made in the last decade in developing appropriate CFD tools capable of predicting flow patterns and other aerodynamic properties of flows in turbomachinery components, and in particular in blade rows. Many issues relating to the numerical simulation and flow modeling have been faced enabling different aspects of solution quality, such as turbulence modeling and code numerics. In this regard, the turbulence models in use range from simple algebraic eddy viscosity models (EVM) (e.g. [1], [2], [3]), to one-equation Spalart–Almaras [4] or isotropic two-equation models with low-Reynolds extension [5], [6]. More advanced models, such as the non-equilibrium eddy-viscosity variants [7], [8], or second-moment closure (though applied only to 2D cascade flows [9]) are occasionally adopted. Quite recently, applications of large eddy simulation to turbine blade and tip clearance flows have appeared [10], [11].
Numerical schemes are usually based on finite-volume method (FVM) algorithms. In turbomachinery, finite-difference codes are seldom used [2], and, to the best of our knowledge, the use of finite element method (FEM) is not widely documented. A number of FEM applications have dealt with simple compressible flow configurations (e.g. [12], [13]), notwithstanding “research” FEM solvers have been rarely used in the context of demanding incompressible turbomachinery configurations (e.g. [14], [15], [16], [17]). Recently, the availability of high performance computing platforms has established in engineering research and practice. Most implicit parallel solution algorithms for Reynolds averaged Navier–Stokes (RANS) approaches have been tested for a wide range of fundamental classical and industrial flows, also including FEM applications [18], [19], [20], [21]. Again, the few works relating to turbomachinery are generally based on parallel algorithms which were developed on FVM, such as the pioneering study by Ecer et al. [22], or the recent work by Fritsch et al. [23].
In this context, the present paper introduces a parallel overlapping scheme, which have been implemented in an effective finite element RANS solver [8], [16], overcoming the limits inherent in the state-of-the-art FEM solver for turbomachinery flows in terms of both flow modeling and computational algorithm.
Under the modeling viewpoint, the proposed methodology adopts a non-linear EVM, considered as a fair baseline in turbomachinery simulation, as it includes provisions to account for curvature and non-equilibrium effects, and to attenuate stagnation-point inconsistent behaviour. This turbulence closure, still undergoing validation efforts in several flow problems (such as: external aerodynamics, environmental problems, etc.), has not yet been extensively applied in the context of demanding turbomachinery flows. Exceptions include work carried out by Leschziner and coworkers [7] and at our Department on hydraulic turbines [8]. The non-linear k–ε model [24] is considered as an effective compromise between the numerical robustness of standard EVM formulations and the greater potentiality of second-moment closures.
Under the computational viewpoint, we propose a stable higher-order FEM scheme, coupled with a parallel implicit solver based on an original Schwarz like domain decomposition (DD) algorithm. In such a framework, numerical instabilities are tackled adopting consistent Petrov–Galerkin (PG) formulations [25], [26], [27]. The achieved stability–accuracy compromise represents a key-feature of the method [17]. The solution methodology is based on an additive overlapping DD procedure, that practically lacks of solver sequential phases and permits a simple updating of the interface conditions. Communication costs (CCs), that affect adversely overlapping DD schemes, are limited by using an original algorithm labeled one level inexact explicit non-linear overlapping Schwarz method. This algorithm combines the intrinsic parallelism of Schwarz schemes [28], [29] with a convergence accelerator based on a condensed cycle technique (CCT). This technique, first proposed in [17], [30], merges the additive Schwarz iterations with the fixed point non-linear iterative cycle. Message passing operations are based on the message passing interface (MPI) libraries [31], preserving the solver portability among parallel architectures. In this work, the parallel RANS solver performance is evaluated on three platforms, namely CRAY T3E, IBM SP2 and SP3.
The paper is organized into five sections. In Section 2, the physical approach is introduced, while in Section 3 both the FEM and parallel DD algorithms are presented in their variational form, and the performance of parallel solver are discussed. In Section 4, comparative studies to assess the predicting capabilities of linear and non-linear turbulence closure on benchmark flows are discussed. Performance analyses for the tested parallel platforms are also presented. Conclusions are drawn in Section 5.
Section snippets
Fluid model formulations
The physics involved in the fluid dynamics of incompressible turbulent flows, in rotating and stationary reference frames, is modeled by a RANS approach. Each quantity U is then decomposed into its conventional average (denoted by an overbar) and the fluctuation with respect to the latter (denoted by a prime), as .
In addition to a standard two-equation EVM, the proposed methodology features a cubic k–ε model [24]. This adopts a non-isotropic constitutive relation in terms of the mean
Numerical formulation
In order to get a numerical solution of the boundary value problem presented in Section 2, a FEM weighted residual method was used. Such an approach, very popular in the context of scientific computing for a large class of flow regimes and pattern (e.g. [25], [26], [33], [34]), is able to offer invaluable advantages with respect to the computational industry standard for turbomachine aerodynamics. With reference to the numerical oscillations and instabilities that might be encountered when the
Introductory remarks
In this section, we report on the numerical performance of the presented FEM parallel overlapping algorithm in predicting valid flow behaviour pertinent to realistic, decelerating, stationary and rotating turbomachinery. The presented computations concern with two compressor blade rows: a double circular arc (DCA) blade cascade, and a 3D axial compressor rotor of non-free vortex (NFV) design. The turbomachinery flows examined were modeled as incompressible. The first test case is a severe
Conclusions
The paper investigated the predicting capabilities of a parallel FEM overlapping scheme developed for the purpose of solving turbomachinery CFD. This scheme addresses the use of cubic EVM on higher-order stable finite element spaces, which is shown to resolve important physical effects pertinent to the flow in decelerating blade rows.
The FEM formulation has been efficiently implemented on parallel architecture by use of a Schwarz additive DD algorithm. A quadratic PG method was developed and
Acknowledgements
The authors would like to acknowledge MURST (under the projects COFIN 1999, and Ateneo 2000). The authors are also grateful for the computational support provided by CINECA (under HPC Grant) and ENEA Frascati HPC.
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