ON THE USE OF NEWTON’S ROOTFINDING ALGORITHM FOR SOLVING THE RANS EQUATIONS Aldo Bonfiglioli Universit` a degli Studi della Basilicata, Potenza, Italy Abstract Viene descritta l’implementazione dell’algoritmo di Newton per risolvere i sistemi di equazioni algebriche non-lineari che si ottengo dalla discretizzazione delle equazioni della fluidodinamica. In particolare, vengono studiate le equazioni di Navier-Stokes mediale a la Reynolds-Favre (RANSE) in cui la turbolenza ` e descritta tramite il modello ad una equazione dovuto a Spalart e Allmaras. Le applicazioni considerate, entrambe bidimen- sionali, riguardano il flusso attraverso una schiera anulare di turbina, International Stan- dard Configuration 11 (STCF 11), ed il flusso esterno attorno al profilo RAE 2822. Ven- gono confermate le eccellenti qualit` a di convergenza dell’algoritmo che non solo consente di raggiungere la soluzione stazionaria in poche decine di iterazioni non-lineari, ma in- oltre permette di raggiungere livelli di convergenza dell’ordine dello zero macchina, dif- ficilmente ottenibili utilizzando metodi iterativi meno sofisticati. D’altro canto, vengono evidenziati alcuni limiti dell’implementazione descritta. 1 Introduction Despite a widespread feeling that only unstructured grid schemes provide sufficient ver- satility to handle complex geometries and allow automatic mesh adaptation, shade is still being cast about the ability of unstructured grid schemes to provide accurate results for boundary layer flows, especially if triangular/tetrahedral meshes are employed within the layer. Although the use of hybrid grids, with hexahedra being used in the near wall region, might somewhat alleviate the problem, it is questionable, for example, how to (automat- ically) construct hexahedral cells within detached shear layers, eventually rolling up into more complex fluid structures. Therefore, the right question to be posed is rather how to construct accurate unstructured schemes for high Reynolds’ number Navier-Stokes calcu- lation on truely unstructured grids. Two different strategies addressing this issue can be identified in the literature over re- cent years: one of these advocates the use of truely multidimensional physical models while retaining a low order (linear) representation of the dependent variables, the other still re- lies on the one-dimensional Riemann model commonly used in state-of-the-art CFD codes, but employs higher order polynomial representation of the unknowns. The Discountinuous Galerkin (DG) method belongs to this latter class and has gained considerable attention over recent years by showing impressively detailed results on coarse (by Finite Volume (FV) standards) meshes. The DG method inherits from the Finite Element (FE) method the flexibility in handling complex geometries and the ease in increasing polynomial order by swithching between different element types. It also borrows ideas from the FV setting, 1