Research on Rotation/Curvature Correction Method of SA Turbulence Model for Numerical Simulation of Axial Compressor

The strong rotation effect exists in the flow of compressors, but it is rarely considered in the common turbulence models used in numerical simulation. In addition, when the widely used turbulence model is applied to the numerical simulation of compressor flow, it often needs to be modified. In order to evaluate the applicability of various turbulence model modification methods in axial compressor flow simulation, and explore suitable modification methods. Based on the numerical simulation software ASPAC for axial flow compressor developed by our research group, three turbulence model rotation/curvature correction methods were implemented based on SA turbulence model and applied to the flow simulation of Rotor 37. The results show that the modified models mainly affect the flow separation of the rotor suction surface. Specifically, the SA-Helicity model improves the prediction accuracy of the separation flow in the compressor. The total temperature and pressure calculated by SA-R and SA-RC models are on the small side. In this study, the applicability of rotation/curvature correction method of various SA turbulence models in compressor flow simulation is evaluated in detail based on transonic compressor, which provides a basis for the selection of turbulence models and the research of turbulence model and correction methods adapted to compressor simulations.


Introduction
As one of the important components of aeroengine, compressor mainly realizes intake compression through blade work.Higher pressure ratio and compression efficiency have always been the development direction of compressor, which has a direct impact on improving engine thrust and reducing fuel consumption.It is difficult to fully obtain the details of the flow field through experimental research, which is due to the complex flow phenomena of the compressor, such as boundary layer interference of the shock wave, tip leakage of rotor, wake disturbance of the upstream leaf row, and blade stall.At the same time, the experimental research of compressor has the characteristics of high cost and long cycle.In recent years, with the development of computer hardware and numerical solution methods, CFD methods have been increasingly applied to the research of compressors with relatively low cost and short cycle.At present, CFD, experimental research and theoretical analysis have become the three main complementary tools in the development of compressors.
Because of the unsteady high frequency pulsation in turbulent motion and the multiplicity in space scale, the accurate prediction of turbulence is very difficult.Compressible turbulent motion with inverse pressure gradient in compressors is accompanied by complex flow phenomena such as separation, shock wave and wake interference, so the prediction ability of turbulence model is challenged.Arima [1] et al. based on the self-developed solver used the low Reynolds number k-ε turbulence model to predict performance parameters such as the compression ratio of NASA Rotor67 and Rotor37, and found that the calculation results of Rotor67 were in good agreement with the experiment.However, the calculation of Rotor37 differs greatly from the experiment.Shabbir [2] et al. used B-L, k-ε and a rotation-corrected k-ε model to solve Rotor37 and made a detailed comparative analysis with the experimental results.They found that the modified k-ε model can obtain relatively good prediction results, but the three models still have large deviations in the vicinity of the hub and in the wake area.Similarly, for transonic compressor rotor Rotor37, Yin et al. [3] used the self-developed CFD solver to solve the problem respectively with SA, k-ε and k-ω turbulence model, and the final results show that the k-ω model has higher solution accuracy.Liu et al. [4] used Fluent to conduct a detailed numerical study on corner stall of cascade of low-speed PVD compressor, and compared RANS models including SA, standard k-ε, realizable k-ε, k-ω and SST.They also predict the corner stall of suction surface by Reynolds stress model.It is found that the prediction results of k-ε and Reynolds stress models are in good agreement with the experimental results, and the prediction results of corner stall of SA and SST models are too large.Suder et al. [5] analyzed in detail the tip leakage of the rotor of the transonic compressor by experimental and numerical method, and found that the shock wave in the channel interacts with the tip leakage flow, and the influence area extends to 10% of the span direction near the casing.Moreover, tip leakage flow leads to partial blockage of flow channel, increases the angle of attack of the blade, and causes to boundary layer separation at the tip gap under specific working conditions.
In addition, the flow in the compressor has another remarkable characteristic: due to the rotation of the system and the bending of the blade channel, the centrifugal force and Coriolis force have significant effects on the generation of flow field and turbulence.The research shows that the rotation of the system or the geometry of the flow channel in the flow field can lead to the bending of the flow line, which can strengthen or restrain the turbulent motion in different regions.Therefore, the vorticity viscosity model based on the isotropic turbulence hypothesis is not well adapted to the turbulence calculation under the influence of streamline bending and system rotation, which needs to be corrected.
A large number of experiments and DNS research results show that Coriolis force from the rotation of the system improves turbulent kinetic energy and turbulent shear stress of pressure surface, but turbulence generation of suction surface will be inhibited, resulting in a laminar flow trend [6]- [9].As for the influence of streamline bending on turbulence, researchers have conducted a large number of experimental and numerical studies based on curved tubes [10]- [12], and found that the turbulence pulsation is suppressed near the convex surface under the pressure gradient on the vertical wall.Near the concave surface, the turbulent pulsation is intensified.
As can be seen from the above review, due to the complexity of the flow in the compressor, no RANS turbulence model has shown obvious advantages in the prediction of all compressor types and various specific flow phenomena.In addition, the studies on the influence mechanism of rotation/curvature on turbulence generation and the verification of modified models are almost all based on simple rotating channels and U-shaped tubes, and there are still few researches on the applicability of the modified model in compressor flow simulation.The SA-Helicity correction proposed by Liu et al. [13] was based on the verification results of PVD cascade, which showed that the modified model could greatly improve the prediction accuracy of the corner stall of the suction surface.Zhang Qian et al. [14] applied the SA-Helicity model to the solution of Rotor67 and Rotor37, and found that the SA-Helicity model can effectively improve the stability of the numerical model and improve the simulation accuracy at near stall conditions.
At present, some methods for correcting rotating curvature based on SA turbulence model have been implemented, these correction methods have important significance in the systematic study of the applicability of axial compressor flow simulation.On the one hand, it lays a foundation for understanding the complex flow mechanism in compressor simulation, and also provides support for the selection of turbulence model and turbulence modeling suitable for compressor simulation.In this paper, three modified models, SA-R, SA-RC and SA-Helicity, are used to study the turbulence model of compressor based on self-developed solver.Combined with the experimental data, the prediction ability of different turbulence model modification methods to the overall performance and flow field details of the compressor was investigated in detail.

Numerical method
In this paper, ASPAC [15] [16], a three-dimensional structured grid solver independently developed by our research group, is used to solve dimensionless NS equations in Cartesian coordinate system.
Which V is control volume, S is enclosing the surface of the control body,  W is a vector of conserved variables. c F and  v F are convection flux and viscous flux respectively,  T S is the source term containing centrifugal force and Coriolis force.The dispersion of inviscid flux is obtained by using the Roe scheme of flux differential splitting, in which the left and right values of the flow field variables are obtained by MUSCL interpolation.Second order central difference is used for viscous flux.Time discretization is a hidden scheme, and LU-SGS iteration is adopted.

Physical model
Rotor37 is an inlet rotor developed by NASA Lewis Laboratory for the aspirated engine axial flow compressor project.The number of blades in the whole ring is 36, and the design speed is 17188.7rpm.Table 1 shows the main characteristic parameters of Rotor37 [17].Reid and Moore [18] carried out detailed experimental measurements on compressor inlet stage of Rotor37.
In order to reduce the amount of computational mesh, Rotor37 was calculated by taking only one blade channel to generate a hexahedral structured grid for steady solution.Tip clearance in the model is 0.0356cm.The simulation was carried out at the design speed.At the entrance, the total temperature and pressure distribution of Station 1 and the axial intake conditions are provided in the given literature [17].Spatial periodic boundary conditions are adopted on both sides of the channel.The pressure outlet is set at the outlet, the static pressure at the hub is given, and the static pressure distribution at the outlet radial is obtained through the radial balance equation.Then, by increasing the outlet pressure step by step, the compressor characteristic curve is simulated.3. SA turbulence model and its correction method

SA turbulence model
The SA turbulence model proposed by Spalart and Allmaras [19] has a good prediction effect for turbulent flows with adverse pressure gradient.In addition, the model has a small amount of computation, good robustness and low requirement on the accuracy of boundary layer mesh near the wall, so it is widely used for engineering solution SA turbulence model obtains the turbulence viscosity coefficient by solving the transport equation of turbulence viscosity coefficient variable: In the transport equation, the three terms on the right side of the equal sign are generation source term, diffusion term and dissipation term respectively.The turbulent viscosity is defined as follows In the turbulent generating source term Other model parameters and constant terms are defined in the original literature [19] and are not listed in detail here.

SA turbulence model
Rotation/curvature correction model 3.2.1.SA-RC correction.The SA-RC correction method was proposed by Spalart and Shur [20].This method introduces two dimensionless parameters * r and r to characterize the effects of curvature and rotational effects, respectively, and modifies the generation source term of SA turbulence model.RC indicates Rotation/Streamline Curvature.Based on the standard SA model, the generation source term is multiplied by the curvature correction coefficient 1 The two dimensionless parameters * r and r characterize the curvature of the streamline and the rotation of the system, respectively, and the expressions for the two are as follows: 0.5 0.5 2 The constant term in the correction function is: In Equation ( 8), the time derivative ij DS Dt of the strain rate tensor needs to be calculated, and the specific calculation method is shown in literature [21].

SA-R correction.
In the region dominated by the rotation effect, turbulence generation is inhibited, that is, turbulence generation term is reduced when the modulus of vorticity is larger than the modulus of strain rate tensor.Conversely, the turbulence generation term is improved.No correction is made in the boundary layer region that is close to the rotation rate tensor and strain rate tensor.Based on the above ideas, Darcles et al. [22] use the SA-R correction method of the SA turbulence model, where the model constant prod C is 2.0.
3.2.3.SA-Helicity correction.Liu et al. [13] modified the SA turbulence model by taking into account the Energy Backscatter mechanism and introducing Helicity.The verification results on the PVD cascade show that the modified SA turbulence model can significantly improve the prediction accuracy of corner stall of blade, total pressure correction coefficient and static pressure coefficient.The SA-Helicity correction method is based on the SA-noft2 turbulence model, that is, set 2 t f equal to 0 in equation ( 2), and use the Helicity relative density to correct the first term in the generation source term.
The model constants are

Results and discussion
Firstly, the solution results of standard SA and various modified models on Rotor37 standard model are compared with the experimental results.Then, the prediction ability of modified methods in compressor flow simulation is discussed in detail, and the influence of various modified methods on compressor flow is analysed.
In order to evaluate the calculation results of the overall performance parameters of Rotor37 by different SA turbulence model correction methods, Figure 2. and Figure 3. respectively show the distribution of total pressure and temperature under different flow conditions.From the prediction of total pressure, it can be seen that the calculated values of SA-R and SA-RC correction methods are lower than the standard SA model, and the deviation from the experimental results is larger than the standard SA model.The SA-Helicity correction method improves the calculation accuracy of total pressure effectively.In the prediction of total temperature, the SA-R model was in the best agreement with the experimental results, followed by the SA-RC and standard SA models, and the total temperature predicted by the SA-Helicity model was higher than the experimental value.The calculation results of adiabatic efficiency are shown in Figure 4.The efficiency of all models is lower than the experimental results, which is consistent with the results in other literatures [14] [23].The SA-Helicity model has the lowest adiabatic efficiency, followed by SA-RC and SA-R models, and the standard SA model has the highest adiabatic efficiency.In order to analyse the solution results of various SA turbulence models on the internal flow structure of the compressor in more detail, Figure 5. shows the distribution of turbulent viscosity perpendicular to the flow section at the position of 50% chord length under the peak efficiency condition.It can be seen that the turbulence viscosity of Sa-Helicity model is the highest, followed by SA-RC and standard SA models, and the turbulence viscosity of SA-R model is the lowest.It shows that the SA-Helicity model increases the generation rate of turbulence, while the SA-R model reduces the right term of the turbulence transport equation.The above conclusion that SA-Helicity model can increase turbulence viscosity is consistent with the results based on Rotor67 in literature [23] and [24].Figure 6.shows the limiting streamline distribution of suction surface of blade under peak efficiency condition.Combined with the pressure distribution, it can be seen that there is a shock wave in the middle of the blade.This flow phenomenon is captured by all models due to the separation of the shock wave/boundary layer interference flow.In addition, except for the SA-Helicity model, the other turbulence models showed obvious corner stall at the root of trailing edge.This may be due to the fact that the Helicity model increases the turbulent viscosity, thus increasing the mixing of the low velocity region with the main flow, providing kinetic energy for the fluid of boundary layer, and ultimately inhibiting the corner stall.8. show the spanwise distribution of total pressure and temperature under 98% choke mass flow and near stall condition, respectively.It can be seen that the calculations of the various models have the same conclusion under the two conditions: The total pressure and temperature predictions of the SA-Helicty model are in the best agreement with the experimental values, followed by the standard SA model.The deviations of total pressure and temperature calculated by SA-R and SA-RC models are larger than those of standard SA models.Based on the above analysis, except for the SA-Helicity model, it is the low turbulence viscosity that leads to corner stall in other models, resulting in excessive loss of total pressure in the radial direction.It is worth noting that the prediction of total temperature, total pressure and efficiency of almost all models near the casing showed large deviations (t The SA-Helicity model was also used only in the near stall condition, and the total pressure prediction near the casing was in good agreement with the experimental results).Suder et al. [5] conducted experimental and numerical studies on the tip leakage, and found that the tip leakage generated from the suction front edge interacts with the main flow, resulting in a region of intense shear motion, which separates the boundary layer of the casing.The complex flow state in this region poses a great challenge to the numerical model.

Conclusion
In this paper, the application of the rotation/curvature correction method of SA turbulence model to the flow simulation of Rotor37 is studied.Based on the simulation results of the compressor, the following conclusions are obtained: (1) The SA-Helicity correction method considers the Energy Backscatter mechanism.By introducing Helicity, and the turbulent viscosity is increased by introducing Helicity in the simulation of Rotor37, which can inhibit the corner stall at the root of the suction surface of compressor blades.This method improves the accuracy of the model and is recommended to be used.
(2) The total temperature and pressure calculated by SA-R and SA-RC models are lower than the standard SA model, and corner stall still occurs on the blade suction surface, so the correction effect of the model is not ideal.
The corner stall of suction surface of blade is one of the important causes of compressor total pressure loss.The Helicity correction can effectively improve the calculation accuracy of the SA model for the separation flow in this region.However, due to the complexity of tip leakage flow, the prediction error of all the modified models in this paper is still too large for the total pressure near the casing.Therefore, the flow mechanism and accurate modeling of tip leakage flow are worthy of further study.

Figure 1 .
Figure 1.(a) Computation grid for Rotor37 (b) grids in the tip gap.

Figure 2 .
Figure 2. Total pressure predictions of different modified SA models

Figure 3 .
Figure 3.Total temperature predictions of different modified SA models

Figure 4 .
Figure 4. Adiabatic efficiency predictions of different modified SA models

Figure 7 .
Figure 7. Spanwise distribution of pressure ratio and temperature ratio predicted with different modified SA models (98% choke mass flow)

9 Figure 8 .
Figure 8. Spanwise distribution of pressure ratio and temperature ratio predicted with different SA correction models (Near stall condition)