Discontinuous Galerkin solution of the RANS and equations for natural and bypass transition
Introduction
In the last decades, many reliable and robust turbulence models [1], [2], [3], [4] have been developed and coupled with Reynolds Average Navier-Stokes (RANS) equations to accurately simulate a wide range of engineering flows. This enhanced modelling capability made Computational Fluid Dynamics (CFD) a tool commonly adopted for industrial analysis and design, and considered complementary to experimental investigations.
However, these turbulence models are not well suited for low to moderate Reynolds numbers flows, which are characterized by strong transitional phenomena. Laminar-turbulent transition is common in aerospace, turbomachinery, maritime, automotive, and cooling applications; and a wrong prediction of this flow feature can dramatically affect their overall performance. As a consequence, numerical tools able to accurately predict transitional flows are mandatory for the efficient design of many industrial applications.
The laminar-turbulent transition can be predicted with different approaches: (i) Direct Numerical Simulation (DNS) [5], (ii) Large Eddy Simulation (LES) [6], [7], [8], [9], [10] and (iii) RANS approach coupled with a transition model. However, DNS and LES computational cost for many industrial flows characterized by low to moderate Reynolds numbers is still too large for a routine use in industry, and only the RANS approach can represent a viable solution. Moreover, in many applications where the transition is enforced within a limited area of the flow by geometric discontinuities, pressure gradient and/or flow separation, an accurate prediction of the flow features can be guaranteed also by a RANS approach supplemented with a transition model.
Next generation CFD solvers will be based on innovative numerical technology and turbulence models. In this context, the greater accuracy and geometrical flexibility guaranteed by Discontinuous Galerkin (DG) methods in solving the RANS and transition model equations could represent an appealing solution to enhance the predicting capabilities of standard industrial CFD codes, without resorting to intensive computational approaches, such as DNS and LES.
DG methods [11], [12], [13], [14], [15], [16] have emerged as a promising technique, as they provide a very high accuracy together with the geometrical flexibility required by complex industrial configurations, possibly adopting non-conforming grids made of arbitrarily shaped cells. Although this higher accuracy requires an increased computational cost compared to standard finite volume methods, the compact stencil of the DG discretization, involving only one cell and its neighbours, makes the method very well suited for massively parallel computer platforms.
Transition models can be classified into three families: (i) low-Reynolds, (ii) non-local and (iii) local models. Low Reynolds turbulence models [17], [18] have shown some limitations in the prediction of the transition for general flow conditions. Non-local models are based on correlations [19], [20], which relate the momentum thickness Reynolds number to local free-stream conditions, such as turbulence intensity and pressure gradient. These models can be easily calibrated and can often capture correctly the major physical effects. Several correlations have been developed for different transition mechanisms, i.e. natural, bypass, separation induced and cross-flow transition. The main drawback lies in their non-local formulation, i.e. the information on the integral thickness of the boundary layer and the state of the flow outside the boundary layer are required. Finally, the local models are built on transport equations, that require only local variables. Among these models, we can distinguish two different types: empirical correlation models [21], [22] and phenomenological transition models [23] based on the concept of laminar kinetic energy (LKE). The development of phenomenological models, that aim at incorporating the physics of boundary layer transition, is an attractive but challenging research field as many mechanisms influencing the flow phenomenon are not yet fully understood. Many authors [24], [25], [26], following the seminal work of Walters [23], [27], proposed different variants of the three equation phenomenological transition model based on the concept of the LKE proposed by Mayle [28].
In the higher-order context, only few attempts have been made to simulate transitional flows with a RANS approach. In the DG framework, the predicting capability of a low Reynolds variant of the turbulence model was assessed in [29], while Crivellini et al. [30] investigated the Spalart-Allmaras model ability to capture transition. Lorini et al. [31] presented the first implementation of a non-local transition model in a DG solver, while Ferrero et al. [32] presented the implementation of a non-local transition model based on laminar kinetic energy. To date, local transition models have been mainly developed in the low-order finite volume (FV) framework, and few attempts have been made to extend these models to high-order solvers [33], [34].
The objective of this work is to develop and implement a phenomenological transition model based on the LKE concept into the high-order DG code MIGALE. The prediction capabilities of the transition model are assessed by computing turbulent flows experiencing natural and bypass transition. The model was validated by computing the flow (i) over flat plates with zero/adverse pressure gradients and different values of turbulence intensity, (ii) through a low pressure turbine cascade (MTU T106A) at Reynolds number and (iii) through the turbine cascade LS89 for different values of turbulence intensity.
The paper is organized as follows. Section 2 describes the transition model employed, while the descriptions of the DG spatial discretization and implicit time integration are given in Section 3. Section 4 is dedicated to the discussion of the numerical results. Finally, Section 5 gives concluding remarks.
Section snippets
Transition model
The transitional model presented in this section is based on the work of Walters and Cokljat [27], where the RANS system is closed with three additional equations for the turbulent kinetic energy k, the laminar kinetic energy kL, and the specific dissipation rate where ϵ is the dissipation.
The complete set of the RANS and model equations for a compressible flow can be written as
Space and time discretization
The governing equations for m variables in dimensions can be written in compact form aswhere is the unknown solution vector, are the convective and viscous flux functions, is the vector of source terms, and is the transformation matrix arising from the use of the primitive variables.
By multiplying Eq. (45) by an arbitrary test function and integrating by parts, we obtain the weak
Results
The proposed model has been validated by computing the flow over the flat plates of the ERCOFTAC T3 series, with zero and non-zero pressure gradients, investigated experimentally in the early 1990’s [45], [46]. These test cases are mainly characterized by bypass transition, due to the levels of freestream turbulence intensity, i.e., Tu1 ≥ 1%. To assess the model accuracy when dealing with natural transition, the Schubauer and Klebanoff (SK) flat plate has been also computed [47].
Conclusions
The implementation of a novel local transition model based on the LKE concept into a high-order accurate DG code has been here presented. The work represents one of the first attempts to introduce this class of models in a high-order numerical context.
Validation has been performed by computing the flow over a flat plate with zero/adverse pressure gradient and different boundary conditions. The performance has been also assessed by computing complex transitional turbomachinery flows, i.e., the
CRediT authorship contribution statement
M. Lorini: Conceptualization, Methodology, Software. F. Bassi: Conceptualization, Methodology, Software. A. Colombo: Conceptualization, Writing - review & editing. A. Ghidoni: Conceptualization, Methodology, Software, Supervision, Writing - review & editing. G. Noventa: Software, Validation, Visualization, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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