Analysis of a curvature corrected turbulence model using a 90 degree curved geometry modelled after a centrifugal compressor impeller

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1 Analysis of a curvature corrected turbulence model using a 90 degree curved geometry modelled after a centrifugal compressor impeller K. J. Elliott 1, E. Savory 1, C. Zhang 1, R. J. Martinuzzi 2 and W. E. Lin 1 1 Department of Mechanical and Materials Engineering The University of Western Ontario, London, ON, Canada N6A 5B9 2 Department of Mechanical and Manufacturing Engineering University of Calgary, Calgary, AB, Canada T2N 1N4 kellio9@uwo.ca ABSTRACT The effects of curvature on turbulence quantities and the performance and functionality of a curvature corrected SST (SST-CC) model are investigated. Steady state simulations are run using a simplified geometry that is similar in curvature and Reynolds number to a typical centrifugal compressor design. Velocity, turbulence kinetic energy and Reynolds normal stress profiles, as well as production multiplier,, and eddy viscosity contours are compared between the RSM-SSG, SST-CC and SST models. Overall, the SST-CC model showed an appropriate sensitivity to curvature, however there are still questions to be answered regarding the effects of the term in certain regions. 1. INTRODUCTION Flow and surface curvature are always present in turbomachinery components. Some examples are the curved blades of an axial machine, or the axial to radial transition in a centrifugal machine. An important part of understanding the flow physics in these machines is identifying and knowing how to deal with curvature. Curvature introduces an extra level of complexity that can greatly affect the flow structure and turbulence quantities. This adds to the complexity of the flow field that is present in a turbomachine. Thus, to fully understand and quantify the effects of curvature on a turbulent flow, it is important to limit as many intricacies as possible to be able to isolate curvature effects. To eliminate other effects, this study considers a simplified geometry modelled after a centrifugal compressor impeller. The simplified geometry is designed with a similar curvature and operated at a similar Reynolds number to a centrifugal compressor currently being studied. With the simplified geometry, complexities such as developing flow, three dimensional blade curvature, high rotation rate, and a converging cross section are eliminated, which allows the focus to be directed towards the surface curvature. Curvature effects become particularly relevant when using numerical modelling techniques. This is the basis of curvature corrected turbulence models, which use various methods to account for the changes that arise due to curvature. The present analysis will focus on a specific curvature corrected model, specifically the SST-CC model of Smirnov and Menter [1], and examine how it performs in regards to curvature as compared to two other uncorrected models. The eventual intentions are to compare the simplified geometry, which resembles a blade passage of a centrifugal compressor, to the more complex compressor flow in terms of curvature parameters. 2. BACKGROUND A classical review by Bradshaw [2] describes the effects of curvature as an extra rate of strain to the already present principal strain. From a review of the literature, it has been found that the effect of this extra rate of strain depends on many factors that include: the magnitude of the curvature, the directionality of the curvature (convex (CVX) or concave (CCV)), the Reynolds number and the presence of pressure gradients [3-5]. All of these factors must be taken into consideration when analyzing a flow with curvature. Directionality of the curvature is a dominant factor. It has been well documented that a convex curvature will suppress (or

2 stabilize) the effects of turbulence, showing decreased shear stress, turbulence kinetic energy and turbulent mixing, whereas a concave curvature has the opposite, destabilizing effect, showing increases in those turbulence quantities [3]. The effects of curvature on turbulence and flow structure has been extensively studied using various simplified configurations such as 90 degree ducts, 180 degree U-turn ducts or rotating ducts using either numerical or experimental techniques. A summary of completed experiments using ducts can be found in [6]. In general, the experiments focus on investigating the flow characteristics relating to curvature, while the numerical studies have investigated the abilities of the most common turbulence models available today (,, SST, RSM) to predict the behavior of a curving flow. More recently, the numerical studies tend to focus on turbulence models that have been altered with curvature corrections. The and models have been corrected in different ways and have shown improvements that are competitive with more complex models such as Reynolds Stress Models (RSM), while still maintaining the simplicity of eddy viscosity models. Some examples of these corrections can be found in [7,8]. A curvature corrected version of the SST model has been recently developed [1] and has performed well for various test cases, including some curved ducts. This particular curvature correction is more attractive than the and corrections mentioned above since the SST model has been shown to perform well in centrifugal compressor flows [9], which ties in with relating these results to a compressor geometry. Smirnov and Menter [1] have effectively demonstrated the performance of the developed SST-CC (curvature corrected) model, however they do not provide an in-depth analysis of the different quantities affected by curvature, focusing primarily on mean flow field quantities. Thus, the present research attempts to further study the SST-CC model and improve the understanding of the flow physics by analyzing various turbulence and curvature related quantities in the SST turbulence model and comparing them to the SST-CC results. 3. TURBULENCE MODELLING Three different turbulence models are considered in this analysis: the SST, SST-CC and RSM-SSG models. Since there is no experimental data available, a comparison is made between the SST and SST-CC models to determine where differences occur, or in other words, where the curvature correction is applied. The RSM-SSG model acts as a guideline for this comparison due to its anisotropic nature and increased sensitivity to curvature as compared to eddy viscosity based models. The SST model uses the formulation in the freestream and the formulation in the near wall region, in combination with blending functions to connect the two domains. The transport equations for the SST model are given as [10]: ( ) (1) ( ) Where: (2) (3) Constants and details relating to Eqs. (1) (3) can be found in [10]. The SST-CC turbulence model was developed based on a correction [11] to the Spalart-Allmaras (S-A) one equation model. It consists of a multiplier to the production term,, in the and equations of the SST model (Eqs. (1) (2), respectively [10]) that is given by [1]: Where: * + (4), ( )- (5) The constants and are equal to 1.0, 2.0 and 1.0, respectively [1]. Note that the magnitude of is greater than 1 for concave curvatures (enhanced production) and less than 1 for convex curvatures (decreased production). In Eq. (5), the terms and are dependent on the strain rate tensor,, the rotation tensor,, the rotation rate of the system,, and a variable,, dependent on the strain and the turbulence eddy frequency,, given by [1]: (6)

3 [ ] (7) Where: (8) the equation for turbulent flow through a pipe, given by [13]: (16) (9) (10) The RSM-SSG model does not use the eddy viscosity assumption and instead solves transport equations (Eq. (11)) for the six individual Reynolds stresses, given, where is the production term and is the pressure-strain correlation term, given by Eqs. (12) (15) [12]. Figure 1: Left: Full geometry, Right: 10 degree section ( ) ( ) (11) (12) (13) [ ] (14) (15) The constants for Eqs. (11) (15) can be found in [12]. 4. GEOMETRY AND NUMERICAL METHOD The geometry studied in this work, shown in Fig. 1, was modelled after a centrifugal impeller. The full model is shown on the left, with the 10 degree curved portion of the full geometry shown on the right. A 10 degree section was chosen to roughly match the pitch of the centrifugal impeller passage, but also to drastically reduce the mesh complexity and the computational time involved in running the case. Periodic boundary conditions (shown in green in Fig. 1) were used to connect the full 360 degree model. Since the basis of this work was to isolate curvature effects, a long straight section was added to the inlet of the curved region (see Fig. 2) to ensure that the flow entering the curved region was fully developed. The required entrance length was approximated using Figure 2: Full computational domain To enforce similar flow conditions, the inlet boundary condition was set as a velocity inlet with a Reynolds number that matched that of the centrifugal compressor blade passage. The traditional Re formulation for turbomachinery is, where is the rotational speed, however the simplified geometry does not rotate, so this alternate Re was used. The outlet was set as a static pressure outlet, as was the outlet in the centrifugal compressor case. The geometry was meshed with a hexahedral mesh, using y + values close to 1 adjacent to the walls. The mesh was found to be fully independent after gradually increasing the number of elements over 5 different meshes in the curved region of the geometry. The meshes were refined until the differences in velocity, total pressure and TKE were less than 1%. Steady state simulations were run using the commercial software ANSYS CFX 13 [14], in which a coupled solver and a finite volume method are used. The default advection schemes are a first order accurate scheme for turbulence quantities and a

4 second order accurate scheme for the continuity, momentum and energy equations. 5. RESULTS AND DISCUSSION The SST-CC model is analyzed from different perspectives: first, the velocity and turbulence kinetic energy profiles are considered; second, the Reynolds normal stresses are examined, and finally, the parameter and eddy viscosity are investigated. The performance of the SST-CC model is measured relative to the RSM-SSG model, since the latter is more sensitive to curvature than the eddy viscosity based SST models. Throughout this section, the vertical axis,, represents the traverse from concave (zero) to convex (unity) curvature in the geometry, and all plots were taken in the periodic boundary condition plane. Fig. 3 shows a schematic of the plot locations, and the coordinate system used. ( = 0), fairly sizeable differences are still seen between the SST-CC and RSM-SSG models. Figure 4: Streamwise velocity profiles at 45 and 90 Figure 3: Schematic of plot locations and coordinate system used 5.1 Velocity and Turbulence Kinetic Energy (TKE) The velocity profiles at = 45 and 90 along the curve are shown in Fig. 4. Locations before 45 show minimal differences between the SST and SST- CC models, and, thus, are not shown for conciseness. In the 45 plot (Fig. 4a), it can be seen that even at this location, there is only a small difference between the SST and SST-CC models, and overall both match well with the RSM-SSG results. At 90 on the other hand (Fig. 4b), significant differences appear throughout the entire section, with the SST-CC matching the RSM-SSG velocity more closely than the SST model. That being said, on the concave side The turbulent kinetic energy profiles at the same two locations (45 and 90 ) are presented in Fig. 5. In the 45 plot (Fig. 5a), the SST-CC model matches the RSM-SSG model very well, indicating that it is behaving appropriately as compared to the original SST (uncorrected) model, based on the known curvature effects (that there is enhanced TKE near the concave side and suppressed TKE near the convex side). In the 90 plot (Fig. 5b), the SST-CC model is reacting to the curvature accordingly by showing the same trends as in the 45 case, however the differences are not as drastic. The RSM-SSG model is predicting a very high peak towards the convex side, which may be due to a potential onset of separation in this region. Thus, in terms of velocity and TKE, the SST-CC model appears to be behaving similarly to the RSM- SSG model. This is with the exception of the TKE at 90, where the RSM-SSG model shows unusual results on the convex side. All in all, these plots

5 suggest that the SST-CC is correctly accounting for the effects of curvature. 0.5, the SST and SST-CC models match well, suggesting that there is no curvature correction occurring here. In terms of matching the RSM-SSG results, the SST-CC model generally shows good agreement near the convex side, however near the concave side the it underpredicts the RSM-SSG peak for the in-plane ( and ) stresses and overpredicts the RSM-SSG peak for the out of plane ( ) stresses. Figure 5: TKE profiles at 45 and Reynolds Normal Stresses Eddy viscosity models, such as the SST models, assume local isotropy of the turbulent length scale and for this reason are known to perform poorly in flows with sudden changes in the mean strain rate, or when the flow and strain principal axes are not aligned, for example in flows with streamline curvature. The RSM-SSG model does not suffer from this problem because it does not make the local isotropy assumption. The poor performance in eddy viscosity models is particularly apparent in the Reynolds normal stresses [15]. In Fig. 6, the Reynolds normal stresses are plotted at the 45 section. From these plots it can be seen that the SST-CC model tends to show an improvement on the convex side in terms of peaks and/or curve shape and is consistent with predicted curvature trends, showing an increase in turbulent stresses near the concave side and a corresponding decrease on the convex side as compared to the SST model. It is also noteworthy that at the centre of the geometry, at = (c) Figure 6: Reynolds normal stresses at 45 :, and (c).

6 The same trends continue into the 90 section Reynolds normal stresses shown in Fig. 7; the SST- CC model shows the correct effects of curvature relative to the SST model. The stresses are unusual for the RSM-SSG model, showing a large peak on the convex side, which is clearly responsible for the same large peak on the TKE plot in Fig. 5b. Again, there is a potential onset of separation in this region, which could be causing this peak. The stresses predicted by the SST-CC model are generally closer to the RSM-SSG model on both the convex and concave sides. The stresses are predicted well by both the SST and SST-CC models, with both models predicted similar profiles. In a general sense, the SST-CC model is performing as expected, showing higher stresses near the concave side and lower stresses near the convex side, relative to the SST model. The SST-CC seems to match the RSM-SSG results better at the 45 location than the 90 location, which could be due to an interesting distribution in the curvature correction parameter, which will be discussed in the next section. Overall, the SST-CC model is predicting the correct trends in Reynolds stresses due to curvature effects as compared to the SST model, which suggests an improvement with the curvature correction addition. 5.3 Curvature Correction Parameter, As stated previously, the SST-CC model uses a production correction multiplier to either increase or decrease the production around concave and convex curvatures. Fig. 8 demonstrates that, qualitatively, the curvature correction is functioning as expected as it shows a large region of increased production near the concave surface, a region of decreased production near the convex surface and a multiplier near 1 (i.e. No correction) prior to the curved section of the geometry. There are several interesting areas to focus on in the contours. First, there are very sharp gradients in in the transition region from concave to convex curvature at the centre of the geometry. In this area, there is a rapid change from to side by side. Although this sharp gradient is not physically realistic, its appearance is likely due to the formulation of. Second, there is a large region near the concave side with maximum. This region is interesting since the limiter of 1.25 in the definition of has a strong effect in this region. It would be interesting to investigate the sensitivity of with respect to the limiter, specifically in these regions and in other similar regions. Third, there is an interesting region on the concave side towards the 90 section, where the value quickly changes from maximum to eventually reducing to a production reduction. (c) Figure 7: Reynolds normal stresses at 90 :, and (c). Looking back at the TKE and Reynolds stresses in the previous sections, the SST-CC did not match the RSM-SSG at the 90 section as well as at the 45 section. This sudden change in could have an

7 effect on these differences. One unanswered question is the reasoning behind this sudden change in in terms of flow field. Figure 8: Production multiplier f r1 (SST-CC) One other evaluation of the general qualitative performance of the curvature correction is in the prediction of the eddy viscosity, as shown in Fig. 9. This has been used previously to evaluate the curvature corrected S-A model [16] which is the basis of the correction for the SST-CC model. Figure 9: Eddy viscosity contours for the SST and SST-CC models. From Fig. 9b, it can be seen that the SST-CC model is suitably responding to the curved walls, in that there is an increased eddy viscosity region appearing near the concave wall, and a decreased region near the convex wall that appears roughly halfway up the curve. This is contrary to the SST prediction of the eddy viscosity in Fig. 9a, which does not show any sensitivity to curvature by predicting roughly the same eddy viscosity across the entire span. 6. CONCLUSIONS AND FUTURE WORK A simplified geometry, based on the impeller of a centrifugal compressor stage, was investigated to determine the ability of the SST-CC model to predict the effects of curvature. The model was evaluated by considering streamwise velocity, TKE, Reynolds normal stresses, the production multiplier,, and the eddy viscosity at two different locations along the curve: 45 and 90. The evaluation was made based on a comparison with the RSM-SSG model, which has an increased sensitivity to curvature. The following conclusions were made: The SST-CC mean streamwise velocity profile at 90 matched the RSM-SSG results better than the SST model; minimal differences were found at 45. SST-CC TKE profiles showed appropriate sensitivity to curvature, with increased TKE on the concave side and decreased TKE on the convex side, and matched well with the RSM-SSG results at 45. Reynolds normal stresses were predicted reasonably well by the SST-CC model in terms of curvature effects, however the SST- CC model tends to underpredict the in-plane normal stresses and overpredict the out of plane normal stresses on the concave side, with respect to the RSM-SSG model. Better agreement was found between the SST-CC and RSM-SSG models at 45 than at 90, which could be attributed to a rapid change in the parameter near the 90 section. Both the and eddy viscosity plots qualitatively showed the appropriate effects of the curvature correction, but there are some interesting regions in the plot that require further investigation. Overall, the SST-CC model showed sensitivity to curvature that is consistent with the literature. Future work will include relating this simplified geometry to experimental results of the centrifugal compressor stage after which it is modelled.

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