Profile Loss Analysis of Transonic Turbine Cascade with RANS and DDES*

（Key Laboratory for Thermal Science and Power Engineering of Ministry of Education,Tsinghua University,Beijing，China）

Abstract：With the increase of blade loading, the loss prediction model used in the design process needs refinement and improvement to meet the high-performance design.For the turbine design,most of existing profile loss models are developed for subsonic and transonic cases and their accuracy in high Mach number flow are limited.The primary research interest of this work is to study the flow mechanism of turbine cascade with high Mach number and the related profile loss.In this work,a transonic turbine cascade with strong shock wave is numerically studied with Reynolds Averaged Navier-Stokes(RANS).Also,to overcome the limitations of RANS modeling,Delayed Detached Eddy Simulation(DDES)type high-fidelity turbulence simulation is also conducted.Based on the numerical results,the primary loss sources, including the boundary layer loss, the trailing loss and the shock loss are analyzed and results from existing loss models are assessed.The results from current work may help to develop refinement profile loss model for the design of turbine cascade working in the high Mach number regime.

Keywords：Transonic Cascade,Boundary Layer Loss,Trailing Edge Loss,Shock Wave,RANS,DDES

1 Introduction

Gas turbine is widely used in aviation,shipbuilding,electric power,petroleum and other industrial fields,because of its large power,light weight,small size,good mobility and so on.In order to ensure turbomachine works steadily and reliably during the long-term re-use process,and maintains high power density and efficiency.Gas turbine design needs to consider economy,safety and other comprehensive requirements.The design method of the traditional gas turbine mainly depends on experimental measurement and the empirical formula derived from a large amount of experimental data.This method has the disadvantages of long cycle,high cost and poor portability.With the continuous improvement of the turbomachine,the requirements of the turbine components are increasingly harsh.The existing experience and semi-empirical design methods can no longer meet the design requirements of these components.So the original loss models also need to be continually refined to meet the requirements of high-performance turbomachinery.Ainley and Mathieson[1]developed several performance prediction methods of loss in machine,including profile loss,secondary loss and tip leakage loss.Some of these models are still in use today.Dunham and Came[2]proposed a correction method at different Reynolds numbers to the Ainley and Mathieson methods of turbine performance prediction.Denton[3]analyzed the loss generation in two-dimensional axial turbine cascade and lack of understanding of many loss generating mechanisms.Gong,Zhu,Zhang,et al.[4]studied three types of transonic blades at different attack angles and discussed the prediction method of the blade loss.And they put forward some constructive suggestions to the existing calculation methods of the shock waves.

With the continuous development of the gas turbine,the power density of the gas turbine is gradually increased.But its series is basically maintained constant,which leads to the increase of the enthalpy drop at each stage.That results in the transonic flow in the cascade and efficiency reduction of the gas turbine.Therefore,it is necessary to study the mechanism of loss inside transonic blades to improve the efficiency of gas turbine.Christopher[5]experimentally studied the loss of four different blades at different attack angles and outlet Mach numbers.Duan,Tan,et al.[6]simulated two types blades and analyzed the loss generation,the simulated results are in good agreement with available experiment data.

With the rapid development of Computational Fluid Dynamics(CFD),many realistic and complicated flows can be simulated by numerical calculation,and the numerical simu-lation results are in good agreement with the experimental results.The RANS model is called Reynolds Averaged Navier-Stokes and widely used currently.The control equations are calculated by statistical average.so that the turbulent pulsations at various scales need not be calculated but only the averaged motion is calculated.Thus the space and time resolution are decreased,the computation quantity is reduced.But the main disadvantage of the RANS is that it can only provide average information about turbulence,so the accuracy of the simulation has limitation for complex flow.The DDES model is called Delayed Detached Eddy Simulation type high-fidelity turbulence simulation.DDES is proposed on the basis of Detached Eddy Simulation(DES),which adopts RANS method near the wall and LES(Large Eddy Simulation)method away from the wall.This method can obtain more flow field structure than RANS under the condition of ensuring the accuracy of calculation.However,it requires more computational resources than RANS,but less than LES.In this paper,RANS and DDES are used to simulate the transonic blade and modified loss model is proposed,which help to better understand the loss of transonic flow.

2 Loss Models

There are many different definitions of loss coefficient to evaluate the loss,and the stagnation pressure loss coefficient is one of them and widely used for individual blade row,which is computed from

Where thept1is the inlet stagnation pressure,pt2andp2are the outlet stagnation pressure and static pressure at the measurement plane respectively,which is shown in Figure 4.In the turbine blade row,the entropy generation is employed as the metric to account for loss.So the most direct expression for the loss of the flow process is probably the entropy loss coefficient,for a perfect gas and adiabatic flow through a stationary blade row,which is formulated as

Where theT2is the outlet static temperature corresponding to the flow determined from the measurement at the measurement plane.Theht2andh2are the outlet stagnation enthalpy and static enthalpy at the measurement plane.

Kacker and Okapuu[7]defined the energy loss coefficientΔΦ2and proposed the total pressure loss coefficient

All of these are the definitions of overall flow loss.In order to study the mechanism of loss,the loss caused by irreversible entropy is divided into three parts:boundary loss,trailing edge loss,shock wave loss,secondary loss,tip-leakage loss and annulus loss.These losses are not mutually independent but affect each other.This paper only discusses the first three losses.

2.1 Boundary Layer Loss

The boundary layer loss is also called profile loss,generated in the blade surface boundary layers.It is due to the viscous dissipation in the boundary layers,which depends on the flow pattern(laminar or turbulent)in the boundary layers,surface roughness and surface pressure distribution.In transonic flow,it interacts with the shock wave as the Mach number increases,studied by Bian,Lin,et al.[8].

2.2 Boundary Loss Model

All of the entropy generation in boundary layer can be written by Denton[3]

Where thexis the blade surface distance,Csis the total length of the blade surface,mis the mass flow rate,u,ρ,Tare the free stream velocity,density and temperature outside the boundary layer,respectively.Cdis the dissipation coefficient.For the turbulent boundary layer,a reasonable approximation of the dissipation coefficient is 0.002,where the averageRθis in the range 500＜Rθ＜ 1 000,referred from Denton and Cumpsty[9].For the laminar boundary layers,the dissipation coefficient supposed by Ttuckenbrodt[10]can be calculated from

2.3 Trailing Edge Loss

When fluid flows in the cascade,the blade suction side and the pressure side are the different structures,resulting in the different expansions on the suction side and pressure side,the corresponding pressure and velocity of fluid vary greatly.Due to the thickness of the outlet side of the blade,the two fluid flows along the blade suction side and pressure side do not converge immediately after leaving the blade.Instead,vertical swirling regions(wake regions)form behind the exit side.Two parts of the flow exchange energy and become gradually uniform at the vortex area,resulting in a certain loss of energy.

Trailing edge loss model

The trailing edge loss is derived as[3],

Where the Δptis the stagnation drop from the upstream of trailing edge to the downstream of the trailing edge.ois the throat width,tis the thickness of trailing edge.δ*andθare the displacement and momentum thickness of boundary layer at upstream of trailing edge.Cpbis the base pressure coefficient,defined as

Wherepbcalled the base pressure,is the averaged pressure acting on the base of trailing edge.pref,uTEandρare the reference parameters at the upstream of the trailing edge.

2.4 Shock Wave Loss

Shock wave loss is a losses of energy due to the generation of a shock in supersonic flow.When a supersonic flow occurs in the transonic cascades,the supersonic flow creates a surge in the pressure increase area,causing a significant re-duction in the stagnation pressure and velocity of the flow,which results in energy loss.The emergence of diffuser may increase the thickness of the boundary layer,and may also cause the detachment of the boundary layer,resulting in an increase of the boundary layer loss.The shock wave loss is limited in transonic cascades.

Shock wave loss model

In transonic turbine,high stage pressure ratio is obtained,and so shock wave is occurred.The shock wave loss can be computed by Denton[3].

Where theγis the adiabatic exponent of gas,Cvis the specific heat capacity,Mis the upstream Mach number of the shock.

3 Transonic Numerical Simulation

3.1 Blade Design

The cascade coordinates from the Christopher[5]are used in this paper to design the blade.Using the method of spline curve interpolation,the whole blade is added more points.Meanwhile,the leading edge and the trailing edge were locally added more points.Among them,there are 1239 points on the pressure side,and the suction side has 1342 points.Adding more points is to make the blade curve more smoothly,as shown in Figure 1,the main geometric parameters are shown in Table 1.

Fig.1 Blade profile and points distribution schematic diagram

Tab.1 The main geometry parameters for blade

3.2 Boundary Conditions

The inlet and outlet section are set sufficiently far away from the blade to minimize their location effects.The length of the inlet section is designed according to the position of the measuring points arranged at the inlet of the test rig.For the outlet section,due to the transonic flow,the outlet is extended to about 5 times of the chord length to fully develop the downstream flow,Schlichting,Gersten,et al.[11].The Reynolds numberRe(defined by the outlet flow quantities and chord length)and the stagnation temperatureTtremain constant,then the inlet stagnation pressure,the outlet static pressure and temperature are calculated according to the different outlet Mach numbers.However,because the downstream measuring points arranged on the test rig is 0.5 times of the chord length,since the outlet is extended,numerical iteration is continued to find the boundary conditions of the actual outlet that meet the requirements.Obtained the boundary conditions are shown in Table 2.

The structured mesh of blade is generated by using Numeca AutoGid.And the total grid number is about 4 million,which is shown in Figure 2.The turbulence model is S-A(Spalart-Allmaras)turbulence model,andy+for the first cell adjacent to the wall is smaller than 1.The hub and shroud faces of blade are implemented as symmetric boundary conditions,and the surfaces along the height of the blades are periodic boundary conditions.The inlet boundary conditions are the stagnation temperature and pressure,the outlet condition is the static pressure.

Tab.2 Boundary conditions at different outlet Mach numbers

Fig.2 The mesh and boundary distribution of blade

All numerical simulations are conducted with the well proved in-house code by Su,Yamamoto,et al.[12],Su,Sasaki,et al.[13],Lin,Yuan et al.[14],Su and Yuan[15].The in house code is based on the multi-block structured mesh,and the integral form of N-S equation is solved with the finite volume method.For the DDES computation,the model de-veloped by Spalart,Deck,et al.[16]is used.In the DDES simulation,the fifth-order high resolution is used to reduce the numerical dissipation and multigrid accelerated implicit method is adopted in the unsteady simulations.

4 Results and Discussion

4.1 Loss Models Analysis

The results of numerical are calculated,the pitchwise variation of total pressure loss at different Mach number is obtained.Then the results are compared with the experimental results,which are shown in Figure 3.According to the loss models mentioned in the previous loss model section,the missed-out losses are calculated,and compared with the experimental data,as shown in Figure 4.

Fig.3 Pitchwise variation of total pressure loss.

Fig.4 Total pressure loss coefficients of prediction model and experimental results distribution at different outlet Mach number

In Fig.3,the abscissa is the ratio of the blade outlet coordinate to the pitch in the direction of the cascade(y/s),wherey/s=0.5 is the trailing edge.Comparing the distribution of numerical results and experimental results at different Mach number,we can find that the results of numerical calculation are higher than the experimental results on the pressure surface away from the trailing edge,and the deviation of numerical results on the suction side is reduced.Especially for the Mach number less than 1.21,the numerical results and experimental results are basically consistent.This shows that the numerical results are reliable at low Mach numbers.From the results of numerical and experimental,it can be found that the total pressure losses distribution at pitchwise is basically symmetrical,which also shows that the losses on the suction side and the pressure side are basically the same.However,the total pressure loss on the pressure side increases more slowly than that on the suction side,aty/s≈0.3-0.65.This is because the flow separation on the suction side is more easily,and as the Mach number increases,shock absorption occurs on the suction side and the shocks interfere with the boundary layer resulting in greater losses.

In Figure 4,the mix-out loss calculated by using the Denton defined total pressure loss coefficient and the experimental measurement values are roughly coincident with a total deviation of less than 5%,except for the Mach number of 0.92.This shows that Denton's calculation equation of the overall total pressure loss is feasible in transonic flow.However,it can be seen from the results which are calculated by Kacker and Okapuu,it is different from the experimental results.Especially,the calculated results at transonic flow are even more inconsistent with the experimental results.

4.2 Profile Loss Analysis

The entropy loss coefficients at different Mach number are also computed as shown in Figure 5.Combing Fig.5 and Fig.6,when the Mach number is 0.92,there is as phenomenon of local supersonic flow at the outlet of the suction,and no shock wave is generated in the cascade channel.The loss at this case is mainly due to the dissipation caused by the boundary layer and trailing edge losses.With the Mach number increased,the fluid velocity increases,the boundary layer thickness becomes thin,but the shock wave is not very strong at Mach number is 1.02 and 1.12,so the proportion of shock wave loss is not large.From the figure 6,it can be found that the Mach number from the 1.02 to 1.12,the overall entropy loss coefficient is reduced.As the Mach number continues to increase,the intensity of the shock wave gradually increases,and it can be seen that the shock position gradually moves from suction side to the trailing edge,the thick black solid line represents the shock wave in Figure 7.The area affected by the shock wave also gradually increases.Therefore,as the Mach number increases greatly,the shock wave loss is the mainly loss and the total entropy loss increases,but the magnitude of the increase is not large.It can be found from the Mach number counter plot that the interaction between the shock wave and the wake decreases gradually.Therefore,the overall increase in entropy loss is not large.

Fig.5 Entropy loss coefficients distribution at different outlet Mach number

Fig.6 Mach number counter distributions at different outlet Mach number

Fig.7 Instantaneous flow structure comparison by Q criterion

Fig.8 Instantaneous numerical schlieren comparison

5 Comparison of RANS and DDES

Figure 7 and Figure 8 are the instantaneous flow field of DDES and unsteady RANS(URANS),respectively.The Q-criterion was used to show the vortex structure of the flow field and globally colored with Mach number.The structures and numerical schlieren in the figures show that compared with the URANS results,the DDES method obtains a richer three dimensional vortex structure.The DDES method captures the formation,shedding,and blending of wake vortices.As can be seen from Figs.7 and 8,the wake vortex is formed by boundary layer from the suction side and pressure side falling off at the trailing edge,and grows in the process of moving downstream,accompanied by the formation of a small vortex.In the process of transporting downstream,it is mixed with the mainstream and gradually dissipated.At the same time,a positive shock wave and an expansion wave at the trailing edge appear on the suction side,and no interaction between the shock wave and the wake is observed in this case.

6 Conclusions

In this paper,the different loss models were discussed,the blade was simulated by RANS and DDES,and the numerical results were compared with the experimental results.At last,the loss generation in transonic turbine cascade was analyzed.The conclusions are as follows.

1.Through the calculation results of different loss models can be found,the calculation results of the total pressure loss coefficient defined by Denton have a good agreement with the experimental data at transonic flow.However,the results calculated by Kacker and Okapuu are greatly different from experimental data.

2.From the total pressure loss along the pitch direction,the total pressure loss coefficient is symmetrically distributed along the blade cascade,and the loss in the mainstream is very small.However,as approaching the trailing edge,the loss increases.Therefore,the method of the reducing losses is mainly to control the loss of the trail zone of the outlet section.

3.In subsonic flow,the losses in cascade are trailing edge loss and boundary layer loss.As the Mach number increases,the boundary loss decreases,the shock wave loss appears and continues to increase until the shock wave loss as the mainly loss.In transonic flow,the shock wave position moves to the trailing edge from the suction side,and the wake interacts with the shock decreases.But the overall entropy loss increases slightly,with the Mach number increasing.

4.The RANS method cannot capture the pressure wave formed by wake shedding.DDES finely captures the generation,development and dissipation process of the wake vortices.The DDES method has certain guiding significance for studying the mechanism of flow loss.