On the modelling turbulent transition in turbine cascades with flow separation

Highlights

• Transition criterion for separation bubble has been re-formulated and can be used in 2D as well as in 3D.

• The criteria based transition model predicts shock wave – boundary layer interaction and its Reynolds number dependence.

• The local formulation transition models not yet reach reliability of the criteria based model.

Abstract

The work discusses performance, modification and implementation details of three intermittency based transition models. The intermittency is applied in the SST eddy viscosity turbulence model and in an explicit algebraic Reynolds stress model (EARSM). Experimental results of transonic flows through turbine cascades at different Reynolds numbers are used as test cases, where the interaction of shock wave with laminar or turbulent boundary layer can be distinguished and reproduced in simulations. Of the transition models tested, only the criteria based $\gamma -\zeta$ model captures well the influence of Reynolds number on the transition. The criterion for transition on separation bubble has been re-formulated so that the model is local in streamwise direction and applicable also in 3D flows. The model is still non-local in wall-normal direction but this is found acceptable considering its performance and the fact that its evaluation is made fully automatic in the framework of structured multi-block solver.

Introduction

The modelling of transition to turbulence in the framework of averaged Navier–Stokes equations is important topic of computational fluid dynamics. While many transition models have been proposed, the reliable prediction of various transition phenomena still remains subject of research. This is especially true for transition connected with the flow separation. So called short recirculation bubbles have only displacement effect on outer flow, but long bubbles may qualitatively change the flow-field and lead e.g. to stall in case of a wing. In any case, the separation changes forces and moments acting on the body and importance of its modelling is evident. For turbomachinery field, which is also subject of present paper, Mayle [1] provided comprehensive review of experimental data for, among other transition mechanisms, for transition on separation bubble. He also proposes empirical correlations for onset and length of transition.

In numerical modelling of transition, certain standard represents the two-equation $\gamma -R{e}_{\theta }$ model of Langtry and Menter [2] which became part of commercial CFD codes. The two transition model transport equations are coupled with the SST two-equation model. The set is able of predicting also separation bubbles laminar at separation, with transition occurring on the bubble and reattaching in turbulent state. Its performance on flow over NACA4215 airfoil with separation bubble has been studied in Genç et al. [3], Karasu et al. [4]. Including transition improves results over plain SST model. Comparison with the $k-k_l-\omega$ transition model (more on it below) then showed that the later model is superior. In recent study Karasu et al. [5] applied similar methodology to a 3D wing and studied influence of aspect ratio on separation. Again, the $k-k_l-\omega$ model has been found superior to transitional SST model. In 3D applications, the one additional transport equation less plays also a role in preferring the $k-k_l-\omega$ model.

The $\gamma -R{e}_{\theta }$ model has some principal disadvantages from physical point of view. The equation for intermittency γ has expected physical meaning but the second equation is purely ad hoc means of transporting the transition criteria parameter in complex geometries discretized on unstructured grids. The model has local formulation but the transition criteria are evaluated in the whole flow-field which, which necessarily leads to some ambiguity in to which boundary layer they are pertaining, which is the correct free-stream velocity in case of moving walls etc. Implementing new transition criteria seems not straightforward either. In recent development by Menter et al. [6] the second equation is removed and the model formally made Galilean invariant i.e. no more dependent of velocity relative to a wall. Possibility of implementing other transition criteria without re-calibrating the whole model seems questionable though. Nevertheless, this 1-equation model has been implemented and is tested in this study towards other two models and experimental data.

Promising example of a transition model with local formulation is the 3-equation $k-k_l-\omega$ model by Walters and Cokljat [7]. The concept of turbulent and laminar energy of velocity fluctuations is attractive from physical point of view but the actual performance still depends on many empirical functions and coefficients which are not in direct connection with empirical transition criteria. On the same concept, but without additional transport equation, is based the transition modified $k-\omega$ model of Kubacki and Dick [8]. The authors demonstrated its performance in turbomachinery flows with bypass transition and even transition on separation, in the later case however in the 3D unsteady RANS framework. Again, the model relies on calibration on selected test cases and does not contain transition criteria explicitly. The model is used in present tests in its newest form [9].

The user of mentioned models need not worry about the details of the model as long as he applies it within limits foreseen by the authors of the model. Otherwise a modification is necessary which again requires detailed knowledge of the model performance and role of model coefficients and switching functions, consequently going through all the test cases again and again. For example, the transitional $k-\omega$ model failed in the present study. In such situation, models based on empirical transition criteria are easier to work with. They have high reliability and can be easily extended by adding a new criterion if needed – examples are the transition with very low free-stream turbulence or transition on surfaces with various types of roughness. Therefore they are especially suitable for open, or academic codes. Unfortunately the criteria have non-local formulation. Here we need to distinguish between the dependency on the free-stream parameters (including boundary layer thickness) and the dependency on upstream history of the flow. The first makes the model non-local in wall-normal direction whereas the later non-local in streamwise direction. In the framework of a parallel multi-block solver considered here the work is naturally distributed block-wise and the first kind of non-local formulation is acceptable provided the whole thickness of the boundary layer is contained in same block. This can typically be full-filled in problems of external aerodynamics. The up-stream dependence, typical for transition criteria for separation bubble, on the other hand, is unacceptable. In this work it is resolved by re-formulating the criteria and in the end the model can be labeled block-local. At the same time in test cases considered here, the criteria-based model appears as most accurate and reliable of the 3 models tested. One probably is ready for performance compromises having to stick to an unstructured CFD solver, however with a structured solver one needs not to be. The inclusion of non-local model into unstructured solver is still possible by defining nodes on wall-normal lines in areas of transition and evaluating criteria on these lines, see Kožulović and Lapworth [10].

An empirical transition criteria based model can be formulated in different ways. Příhoda et al.use algebraic model with two-layer intermittency [11], [12]. Since transition is restricted to thin boundary layers and the criteria take care of non-local effects the algebraic form is adequate in many cases and the model performs well e.g. in turbomachinery applications [13]. The algebraic transition model has been coupled with the EARSM model and showed good results in flat plate test cases in [14]. A transport equation for intermittency has been used in Steelant and Dick [15] or Lodefier et al. [16] and Suzen et al. [17]. For unsteady phenomena as wake induced transition the intermittency has been split into near-wall and free-stream part with separate transport equations [18], [19]. The latest model has been shown by its authors performing well in bypass and wake induced transition, however the transition on separation bubble has been treated with less detail and no results shown. In this work we choose this model and introduce some modifications suitable for prediction of transition on separation bubble. The transition criterion for separation bubble is re-formulated in a way that makes it not only local in streamwise direction but also usable and robust in 3D where the original form is not applicable at all. The block-local implementation of the model is discussed in detail. The results then are shown in cases of shock wave-boundary layer interaction on a turbine blade where the model correctly predicts the Reynolds number dependence of the type of interaction (laminar or turbulent) and massive separation at off-design flow conditions.