Improved Turbulence Prediction in Turbomachinery Flows and the Effect on Three-Dimensional Boundary Layer Transition †

For the numerical prediction of turbomachinery flows, a two-equation turbulence model in combination with a proper transition model to account for laminar boundary layers and their transition to turbulence is state of the art. This paper presents the ability of such a method (k-ω + γ-ReΘ) for turbulence prediction and the effect on three-dimensional boundary layer behavior. For this purpose, both applied models (turbulence and transition) are improved to better account for turbulence length scale effects and three-dimensional transition prediction (Bode et al., 2014 and 2016), since these are the main deficiencies in predicting such kinds of flows. The improved numerical method is validated and tested on existing turbine cascades with detailed experimental data for the viscous regions and additionally on a low-speed axial compressor rig where wake-induced transition takes place.


Introduction
Nowadays, the industrial design of turbomachines and their components will be done with threedimensional Navier-Stokes solvers (CFD). ese mostly RANS solvers are able to simulate multistage D blade passages with unsteady ow e ects. Hence these types of solvers are the key design tool for today and tomorrow [ ]. As an up-to-date numerical method for turbomachinery ows and their applications the two equation k-ω turbulence model a er Wilcox ( ) [ ] in combination with the γ-Re Θ transition model a er Menter and Langtry [ ] which is used to incorporate laminar boundary layers and their transition to turbulence is widely spread. Whilst these numerical methods reduced the need for experimental investigations, even so they need to be validated against highly accurate experimental data since the turbulence and transition models should be able to correctly reproduce the physical ow phenomena inside a turbomachine like the transitional process on the viscous surfaces or the secondary ow.
From a present-day perspective these above mentioned models are able to numerical predict the midspan boundary layer behavior on the airfoils. Nevertheless even in a simple cascade there are still uncertainties in the prediction of the three-dimensional boundary layer behavior on the airfoils and the sidewalls (hub and tip) and their interaction with the secondary ow phenomenas, cf. [ , ]. Furthermore, in a multistage component environment the interaction of these e ects are increased and the prediction accuracy of the downstream blade rows is highly dependent on the prediction of the upstream blade rows. us, an improved steady and unsteady numerical method is necessary for the design of new multistage turbomachines and their components. For example an increased prediction accuracy of the turbulent kinetic energy (turbulence intensity) and its dissipation will lead to an improved boundary layer transition prediction. is in turn leads to a be er prediction of the wake of the airfoils and hence more accurate ow condition for the downstream blade row. e γ-Re Θ transition model and its extinction to three-dimensional boundary layer transition a er Menter and Smirnov [ ] in combination with the SST model [ ] was already validated against general testcases and also successfully applied to three-dimensional turbomachinery ows in Bode et al. [ ] and showed its good agreement to experimental data. In the present paper the k-ω turbulence model a er Wilcox ( ) [ ] with a modi cation a er Bode et al. [ ] to improve the turbulence prediction in combination with the transition model a er Menter and Langtry [ ] and its extinction to threedimensional boundary layer transition a er Menter and Smirnov [ ] will be used to further improve the turbulence prediction and hence the transitional behavior and its impact on the loss prediction. erefore the CFD solver will be validated against testcases with increasing complexity and will be presented to show the ability of the used numerical method to accurately predict the turbulence and transitional behavior of steady three-dimensional single and multistage turbomachinery components.

. Numerical Method
An up-to-date numerical method, the parallel CFD-solver TRACE of DLR Cologne has been applied, cf. . Incorporation of Turbulence Length Scale E ects on Turbulence and Transition Prediction e validation of todays CFD-solvers especially on experimental cascade data with medium or high in ow turbulence intensity from 3 ≤ Tu ≤ 10% and in combination with moderate turbulence length scales ends up in unphysical too high the eddy viscosity leading to a wrong prediction of the turbulence and hence transitional ow. To avoid this behavior the CFD user o en changes the turbulence length scale to t the transitional data which is most probably wrong. Also the application of modi ed turbulence models sometimes leads to unphysical behavior around the leading edge and along more than 60% of the passages suction side where the eddy viscosity is damped to harsh. erefore the k-ω turbulence model a er Wilcox ( ) is modi ed, so that the "correct" behavior regarding overall characteristics and boundary layer development is given but the unphysical behavior of the eddy viscosity is reduced. For this reason, a criterion for the determination of viscous regions (boundary layers and wakes) has been developed as an additional element of the implemented approach (cf. [ ]).
is criterion is based on the large values of turbulent dissipation rate ω. It takes the relationship between the turbulent dissipation rate estimated from the k − ω turbulence model and the turbulent dissipation rate in the free stream of the ow estimated by the new approach. e e ect of the very high ratio in the boundary layer and wakes is used to separate them from the free stream.
( ) e time-scale bound is only applied in these viscous regions, e ectively preventing the eddy viscosity destruction in non-viscous areas by multiplying the time-scale bound by a factor b v , which is 1.0 in the boundary layer and the wake region and 0.1 in the free stream (cf. [ ]).  e applied grid (OH-structure) consists of . . nodes ( nodes in spanwise direction, nodes around the blade surface, nodes normal to the surface, half-span simulation) with a high low Reynolds resolution of the boundary layers. is results in an average dimensionless wall distance of y + ≈ 1.0 in a cell-centered scheme.
Inlet Flow Free stream ow conditions are derived experimentally and compared to the numerical ones at −1.0 · C a x ≈ at position IN of Slot A, B and C, cf. gure . In Moore [ ] detailed inlet velocity, turbulent kinetic energy coe cient and turbulent intensity pro les are given. Representative, gure shows that the prescribed inlet velocity, turbulent kinetic energy coe cient as well as the turbulent intensity matches the experimental ones.  e reason for that is seen in Figure . Here numerical results for the intermi ency at boundary layer edge are shown for both combinations. Starting with die suction side surface in Figure (a) and (b) it is well seen that die laminar region or transition location is more upstream predicted with the SST-CF compared to VB-CF resulting in more total pressure loss downstream of the cascade as seen in Figure . Contrary to that is seen in Figure (c) and (d) where the SST-CF gives also a smaller laminar region compared to the VB-CF results but this is closer the the experimental data in Figure . All in all both numerical combinations show adequate results in predicting the laminar turbulent transition process on suction side and sidewall of the Durham cascade compared to experimental data.    Fig. goes from the right to the le . e test rig is designed as a closed air loop and exists of inlet guide vanes, blades and vanes in each stage. e two stator and rotor rows as well as the inlet guide vanes are NACA pro led. e geometry of the test rig, with a blade height of mm and an axial distance between two blade rows of approximately mm, is chosen to enable equipping measurement probes without unduly a ecting the ow on the one hand, and obtaining a distinct quasi two-dimensional main ow region at mid span on the other hand. A detailed description of the test rig, the experimental data and their underlying post-processing is given by [?]. In order to investigate the rotor-stator interaction in the low-speed axial compressor, di erent steady and unsteady ow measurement techniques like surface-mounted hot lm sensors, split-bre probes and pneumatic probes have been deployed. e split-bre probes, which are used to measure characteristic ow values like turbulence intensity, velocity and ow angles, are located upstream and downstream of stator . It is possible to traverse the probes over the complete blade height and over a °circumferential arc behind each blade. e split-bre probe measurements, in conjunction with the surface-mounted hot-lm sensors, deliver the required experimental data to analyze both the in uence of rotor-wakes on the boundary layer development and the quality of its numerical prediction. In [?] the numerical prediction quality of the state-of-the-art turbomachinery design code TRACE has already been validated against the experimental data. Wol et al. [?] conducted steady and unsteady RANS simulations and showed that only the rst and last approx. 20% of the blade height are in uenced by secondary ow e ects. At mid span a two dimensional ow can be assumed. erefore, only 15% of the blade height at mid span can be considered in quasi three-dimensional (Q D) numerical simulations. e measurements and the numerical simulations have been conducted at the steady state rotor speed of rpm for three di erent operating points. e normalized operating parameters of the operating point near best e ciency (OPs ), the operating point near stability limit (OPs ) and the operating point near choke limit (OPs ) are given in Table . Figure .