Evaluation of curvature correction methods for tip vortex prediction in SST kω turbulence model framework

https://doi.org/10.1016/j.ijheatfluidflow.2018.12.002Get rights and content

Highlights

  • Different curvature correction models are described, and incorporated into the SST kω model

  • By considering the classical Rankine vortex, the response of the curvature correction models in the viscous vortex core, and in the inviscid outside region are investigated.

  • It is shown that some of the curvature correction models are so sensitive to their clipping values, or their calibration parameters.

  • It is noted that the curvature correction models that can accurately predict the tip vortex flow fields, alter the boundary layer behaviour on the foil, and form a leading edge vortex.

Abstract

This paper presents and studies effects of curvature correction (CC) methods to improve two equation RANS simulations of tip vortex flows, exemplified using the SST kω turbulence model. Performance of the CC models is first evaluated in the classical Rankine vortex flow field, and then extended into the study of tip vortex flows over an elliptical foil. The results have been compared with experimental measurements in terms of the vortex strength and velocity field, and the importance of the turbulence closure in tip vortex simulations is highlighted.

Contribution of the CC models in different terms of the turbulent kinetic energy and specific dissipation transport equations are described, and it is discussed why a CC model may have mesh resolution dependent results. By considering the distribution of the CC function, it is shown that although some of the models can predict the location of the tip vortex core accurately, they still do not significantly improve the vortex prediction as the impact on the turbulent viscosity is wrong or not enough. It is further noted that as some of these models have been calibrated on specific vortex flows, they may not be completely applicable for other cases without recalibration. It is shown that some CC models provide accurate tip vortex predictions, primarily the ones based on the sensitization of the turbulent viscosity. Further, it is noteworthy that the successful models are active not only around the vortex, but also change the boundary layer characteristics on the foil, and the boundary layer separation lines, which consequently can provide the required momentum for the vortex core accelerated axial velocity.

Introduction

The tip vortex flow is an example of a flow with strong streamline curvature, a flow feature that is widely known to be inadequately represented by any linear eddy-viscosity model. It occurs wherever a flow passes over a lifting wing with a finite span. The pressure difference between the foil surfaces drives the fluid from the high pressure side on one surface to the low pressure side on the other. This makes the flow highly three-dimensional at the tip region creating a vortex pattern (Arndt, Keller, 1992, Arndt, Arakeri, Higuch, 1991). The vortex then moves downstream and roll up of the wing wake occurs until its circulation is nominally equal to that of the wing. This typically extends to a few wing spans downstream of the trailing edge. This proves to be a challenging flow field to study because of the presence of turbulence and the large gradients of pressure and velocity in all three directions, especially across the vortex core (Schot et al., 2014). Understanding tip vortex flows is essential to the solution of many engineering problems, including lift induced drag tip inefficiency, the overturning of small planes flown into the tip wake of larger aircraft, and marine propeller tip vortex cavitation (Pennings et al., 2015).

Even though computational resources have increased considerably during the last decades, the Reynolds-Averaged Navier-Stokes (RANS) have remained the most widely used approach in industrial CFD applications as it offers the most economic approach for computing complex turbulent industrial flows. The averaging adds a stress tensor to the original Navier–Stokes equations, the so called the Reynolds stress tensor, representing the action of turbulent structures on the average flow. The tensor can be modelled with either linear or non-linear eddy viscosity models, or in a more complex way by solving the Reynolds stress transport equations, so called Reynolds Stress Models (RSMs).

The most widely used approach is to use a linear eddy viscosity model, based on the Boussinesq approximation to form a linear relation between the Reynolds stress tensor and the flow mean strain rate tensor, where the coefficient of this relation is called the turbulent viscosity. The assumption that the Reynolds stress tensor is linearly proportional to the mean strain rate does not consider the near-wall anisotropy of normal stresses and limits the ability of the model to respond to sudden changes in mean strain rate, e.g. in flows with high streamline curvature, making the eddy-viscosity models insensitive to streamline curvature and system rotation (Spalart, 2015). This drawback, especially in fully three-dimensional turbulent structures where anisotropic turbulent quantities are of importance, limits the accuracy of the modelling (Iaccarino et al, 2006). RSMs, however, can account for streamline curvature effects in a systematic manner because of the presence of exact production terms containing mean flow gradients and the system rotation. They also contain the convective transport of the second moments and hence provide accurate means for predicting curved flows. The application of RSMs still is limited in complex industrial applications due to the excessive computational cost and numerical stiffness. This is the motivation to incorporate rotation and curvature effects into the scalar, eddy viscosity framework of RANS approaches (Arolla, Durbin, 2013b, Arolla, Durbin, 2013a).

To remedy some of the shortcomings of linear eddy viscosity models, several corrections have been proposed. The vast majority of curvature correction (CC) models are based on the modification of the turbulent length scale in strong streamline curvature regions aiming to provide an improved formulation or function to describe the turbulent length scale. As the turbulent viscosity is directly proportional to the turbulent length scale in the linear eddy viscosity assumption, the turbulent viscosity can be directly multiplied by the length scale modification functions. This approach, also called the bifurcation approach, sensitizes the turbulent viscosity coefficient (Cμ) to the streamline curvature (Durbin, 2011) through expressions involving invariants of the strain and vorticity tensors. As an alternative for eddy viscosity models using transport equations for turbulent quantities, it is also attractive to modify production or destruction terms in these transport equations to include streamline curvature effects there (Duraisamy and Iaccarino, 2005).

The earliest efforts incorporating curvature corrections were for the kϵ turbulence model through sensitizing the turbulent viscosity coefficient to the Richardson number of the flow (Leschziner, Rodi, 1981, Demuren, Rodi, 1986, Cheng, Farokhi, 1992), similar to the Bradshaw expression (Bradshaw, 1996). The primary studies were conducted on flows in curved channels, e.g. Ye and McCorquodale (1998); Pruvost et al. (2004), with focus on comparing the averaged velocity and turbulent kinetic energy between curvature corrected RANS and experimental data. CC models are usually tested at different Bradshaw–Richardson numbers, a number that represents the ratio of the shear strain rate tensor to the rotational rate tensor (Cazalbou et al., 2005). The adoption of the Bradshaw–Richardson curvature correction in other applications, e.g. plane jet in a weak or moderate crossflow conditions (Pathak et al., 2005), showed a slight improvement of the results comparing to the original kϵ model. As CC functions are usually calibrated based on the Reynolds stress analysis in one application, e.g. a curved channel or a backward facing step, their adoptions into other applications often lead to limited improvements (Isaev et al., 2014).

In order to derive a frame indifferent model, a Galilean invariant curvature model was proposed by Spalart and Shur (1997) and incorporated into the Spalart–Allmaras one equation model. The model proved to improve flow predictions in different applications, e.g. backward facing steps (Spalart and Shur, 1997) and curved channels (Shur et al., 2000), and motivated further developments, e.g. Zhang and Yang (2013); Yang and Tucker (2016).

Hellsten (1998) developed the use of the Richardson number and proposed a function incorporating the streamline curvature into the destruction term of the ω transport equation in the SST kω turbulence model, often denoted as the SST-RC in literature. The model was calibrated for a two-dimensional fully developed channel flow with a spanwise rotation. Later, Zhang et al. (2010) used the intrinsic mean spin tensor instead of the mean vorticity tensor in the SST-RC model (Hellsten, 1998) to consider the reference frame rotation. They observed better prediction by using this CC, especially in the prediction of flow fluctuations in a centrifugal impeller.

Pettersson et al. (1999) proposed a new approach to sensitize the turbulence closures to Galilean invariant forms of the strain rate and rotational rate tensors, previously proposed by Girimaji (1997), by mimicking the behaviour of a second moment closure in rotating homogeneous shear flows. They implemented a CC model into the v2f turbulence model and tested it on different cases highlighting the improvement of the results comparing to the original turbulence model. Later, inspired by Pettersson et al. (1999); Arolla and Durbin (2013b); Arolla (2013) proposed a simpler CC model based on flow analyses in rotating homogeneous shear flows. The proposed model incorporates the streamline curvature effects into the production term of the ω transport equation. The model was validated in various applications, including flows inside a rotating channel, over a rotating backward step, and in cyclones. Later, the turbulent time scale prediction was modified according to the near wall behaviour of a flow to improve the stability and accuracy of the model (Arolla, Durbin, 2013a, Arolla, Durbin, 2014).

Smirnov and Menter (2009) employed the CC of Spalart and Shur (1997) into the SST kω turbulence model, often called the SST-CC model in literature. They tested the model on a wide range of both wall bounded and free shear turbulent flows with system rotation and streamline curvature. They observed better predictions by employing the CC for cases including developed flows in plane rotating channels, developed flows in curved channels, two-dimensional flows in a duct with U turn, flows in a centrifugal compressor, and NACA 0012 wing tip vortex. However, the deficiency of the model in accurately predicting the accelerated vortex core axial velocity in the NACA 0012 wing tip vortex was also noted.

Ali et al. (2015) used the SST-CC model to assess flow predictions in centrifugal compressors. They observed similar predictions between the SST and SST-CC models in terms of velocity profiles at the mixing plane, later also reported by Tao et al. (2014). They highlighted the sensitivity of the production multiplier, fr1, to the curvature of the blade surfaces of the compressor, along with an abnormal behaviour of the CC function, fr1, where production values are insignificant.

Stabnikov and Garbaruk (2016) employed the CC function of the SST-CC only as a multiplier for the production term of the k transport equation. They observed that the modified curvature correction results in a better prediction than the original version for three - dimensional vortex flows and gives similar results for 2D boundary layer flows.

The main objective of the current study is to evaluate and analyze a wide range of CC models in a RANS modeling framework, where in here the SST kω model has been used. To our knowledge, a comprehensive comparison has not previously been done, within the framework of one code and on the same test cases. The investigated models cover both the sensitization approach and the transport equation modification approach. The response of the models are firstly analyzed in the classical Rankine vortex flow field. Considering the rotational and strain rate tensors of the Rankine vortex, the possible response of the models in increasing or decreasing the turbulent viscosity is discussed. Then, the study is extended to the analysis of the tip vortex flow on an elliptical foil where two different mesh resolutions are considered for the simulations. All of the CC models are evaluated in a medium mesh resolution, having 16 cells across the vortex core diameter. The evaluation contains comparisons of the axial velocity and turbulent properties such as the turbulent viscosity, turbulent kinetic energy, and turbulent dissipation. The study also includes the analysis of the different terms of the SST transport equations, especially the production terms, and their contributions to the tip vortex flow region. The results of Euler (inviscid) and laminar (i.e. without the use of a turbulence model) simulations are also presented to provide further insights towards the dependency of tip vortex prediction on the turbulent viscosity modelling.

We continue the paper by presenting the governing equations. Then, the elliptical foil geometry, computational domain, and mesh specifications are presented. First, the CC models are evaluated on the Rankine vortex flow field, with the detailed description of their responses inside the viscous core, and in the inviscid outside region. Then, the results of the flow simulations around the elliptical foil are presented, including the evaluation of different terms of the transport equations of the SST kω model, as well as the impact of CC adoptions on the turbulent properties of the flow. Finally, numerical predictions of the tip vortex velocity field are presented and compared with experimental PIV measurements.

Section snippets

Turbulence modelling

The CC models are implemented in OpenFOAM and used together with the SST kω model proposed by Menter (1993); Menter et al. (2003). The SST kω model adopts a blending function that switches from the kω model close to the wall to the kϵ model outside the boundary layer. To provide further simplicity, the ϵ transport equation is transformed into an ω transport equation by a variable substitution. The transformed equation is similar to the one in the standard kω model, but adds an additional

Rankine vortex

The Rankine vortex is a circular flow mimicking a vortex with a viscous core, in which an inner circular region (vortex core) is in solid body rotation, and in the outer (inviscid) region the tangential velocity is inversely proportional to the distance from the origin,uθ={r2Γvortexraa22rΓvortexr>a.The radial and axial velocities are zero, thus the velocity field only includes the tangential velocity, as illustrated in Fig. 1. Consequently, the vorticity is constant in the solid rotation

Curvature correction evaluation on Rankine vortex

In this section, responses of the CC models to the prescribed Rankine vortex velocity field are analyzed based on the distribution of η parameters, S and Ω fields. As the Rankine vortex velocity field is prescribed through two simple terms, one for the viscous vortex core region and the other for the inviscid outer region, the response of each CC model for each of these two regions can be analysed separately.

Conclusion

In this paper, RANS studies of tip vortex flows are presented by evaluating different curvature correction (CC) methods incorporated into the SST kω turbulence model. The investigation includes two types of CC approaches: sensitization of Cμ and modification of the turbulence transport equations production or destruction terms.

First, the response of the CC models to the prescribed Rankine vortex velocity field is evaluated, and then the CC models are used in the prediction of the tip vortex

Acknowledgements

Financial support for this work has been provided by Rolls-Royce Marine through the University Technology Centre in Computational Hydrodynamics hosted at the Department of Mechanics and Maritime Sciences at Chalmers. The simulations were performed on resources at Chalmers Centre for Computational Science and Engineering (C3SE) provided by the Swedish National Infrastructure for Computing (SNIC).

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