Rapid Aerodynamic Design of Prop-rotor Blade with Optimization

(School of Aerospace Engineering,Tsinghua University,China,)

Abstract：Aerodynamic design of convertible prop-rotor is a challenging and complex task. Instead of focus on a design condition, prop-rotor are requested to attain good performance in wide range of operating conditions. In present work, to further improve performance of prop-rotor under each operating condition, the concept of variable-pitch proprotor has been proposed. However, a compromising aerodynamic design of prop-rotor is inevitable, due to discrepancy of operating condition. Consequently, a multi- objective optimization is implemented with genetic algorithm. Factors which has significantly influenced on aerodynamic characteristics of prop- rotor are employed as design variable. To avoid expensively computational cost, two theories has been implemented and validated while it able to provide a promising result without high consuming of time. Eventually, several designed individuals within different performance has been analyzed and discussed. Subsequently a calculation of aerodynamic characteristic with highfidelity solver has been conducted as validation for designed individual.

Keywords：Aerodynamic Design, Optimization, Prop-Rotor, Preliminary Design,BEMT,Vortex Theory

Nomenclature

PLhovHovering power loading

FMFigure of merit

DLDisk load

ηpPropulsive efficiency

P0Profile loss coefficient

TStatic temperature

κFactor of induced loss

P∞Environmental pressure

T∞Environmental temperature

V∞Cruise velocity

CTThrust coefficient

CpPower coefficient

CiParameter of multinomial equation

Dimensionless radial position

ρDensity

VndAdvanced ratio

0 Introduction

Concept of tiltrotor aircraft has combined capability of helicopter and fixed wing aircraft.Its propulsive system relies on a pair of convertible rotor,due to advantage of aerodynamic characteristics of rotor,its propulsive system is able to provide capability of hovering in reasonable high efficiency.By contrast with the conventional helicopter,tiltrotor aircraft are able to cruise in high-speed within high propulsive efficiency.Therefore,it is an attractive approach to achieve both vertical and short take-off/landing(STOL).However,a significant challenge for convertible rotor aircraft which is to design a prop-rotor with good performance under each operating condition.By contrast,helicopter rotor and propeller has only operated optimally at a specified condition with narrow operate margin.When it operates under off-design condition,it will inevitably result in inferior performance.From an aerodynamic analysis of prop-rotor,induced loss dominates the hovering performance while profile loss plays a key role in propulsive efficiency.As a result,prop-rotor re-quests a large disk area and solidity to attain a good performance in hovering operation while prop-rotor requests a lower disk area to reduce profile loss and employed airfoil with high lift-drag ratio to provide demanding thrust as well as resist drag under propelling condition.Because of the discrepancy of design philosophy between hovering and propelling condition,a compromising design has to be conducted.

1 Related Work

In the design procedure of XV-15,McVeigh et al(1983)[1]firstly design an advanced composite replacement blade for each crucial flight condition by compromised chord and twist distribution of blades to achieve an acceptable hovering and propelling performance.Subsequently,Paisley et al(1987)[2]investigated in a derivative of the V-22 Ospre,who has implemented an aerodynamic optimization for design of blade shape to increase flying speed under operation of forward fly under consideration to maintain a good propelling performance.Alternatively,Liu et al(1990)[3]suggested a non-linear programming techniques as an approach for aerodynamic design of rotor blades.Due to multiple operating condition,aerodynamic design of prop-rotor is much more complex than either helicopter rotor or propeller.The problem of aerodynamic design is formed as a multi-objective optimization procedure.To further understand the problem of aerodynamic design of prop-rotor,a study has been conducted and a description of possible design parameters and its influence on aerodynamic performance of prop-rotor have been given by Leishman et al(2011)[4].Modern approach to conduct a multi-objective optimization are commonly employ a gradient-basted algorithm with linear superposition of multi-objective within weight coefficient to transform a multi-objection optimization to single objective optimizing problem.However,gradient-based algorithm commonly results in a local optimal for the non-convex problem.To tackle this problem,a type of stochastic optimizing method such as genetic algorithm has been adopted.Leusink et al(2013)[5]has implemented a multi-objective optimization by adopted genetic algorithm to further improve hovering and propulsive performance by optimized twist and chord distribution of 7A rotor.In addition,multi-objective optimizing approach will produce their optimal result as Pareto-optimal solution.It will allow designer to select and compare each compromised design from frontier-edge.

2 Numerical Approach

2.1 Genetic algorithm

Convertible rotor aircraft has a wide range of operating condition.For each operating state,which impose a very different inflow condition for the prop-rotor as well as blade loading.To achieve optimal performance on each states,which request a different aerodynamic design for the blade.Therefore,a compromising design has to be conducted to achieve lowest requirement of aerodynamic loading to the aircraft as well as maximum its efficiency at each operating states.This design problem leads to a multi-objective optimization problem.Since each objective are conflicted,solution of the optimization will result as a Pareto-Optimal Front.In present work,Fast and Elitist Multi-Objective Genetic Algorithm(NSGA-II)which proposed by Deb et al(2002)[6]is employed.

2.2 Aerodynamic solver

To avoid expensively computational cost of directly solve the RANS equation in the optimizing process,Blade Element Momentum Theory(BEMT)and Classical Vortex Theory are implemented for providing aerodynamic characteristics of prop-rotor and a comparison of accuracy are conducted.The BEMT are a theory which are combination of Blade Element and Momentum theory.The theory considers prop-rotor which consist of several blade element and its aerodynamic characteristics are determined by fluid flow through the blade element at each radial position.In addition,tip effect of prop-rotor is introduced with Prantal loss function and induced velocity to each blade element which are supplied by momentum theory.On the other hand,the Vortex Theory assume the rotor consist of a disk and awake with vortex filament.According to Biot-Savart Law,every vortex filament will induce a velocity and eventually implement an integration to vortex filament with Biot-Savart Law.An expression of induced velocity can be obtained.Therefore,the aerodynamic characteristics of prop-rotor can be calculated.

3 Challenges and Performance Metric

The challenge of aerodynamic design for convertible prop-rotor which is to maintain a good hovering performance as well as propulsive performance over a wide range of operating condition.Therefore,a multi-objective optimization has to be conducted.To further understand the designing philosophy of prop-rotor,an investigation of factors which will significantly influence the aerodynamic performance has been conducted.Conclusively,those factors can derive as

1）Solidity

2）Twist Distribution

3）Chord Distribution

4）Airfoil

5）Rotor Tip Speed

According to the definition of solidity which represent the produced thrust over the blade disk area.It has directly determined the thrust under each condition.Twist and Chord distribution has affected profile loss and induced loss respectively and an airfoil with high lift-drag-ratio are demanded.To avoid an additional reduction of rotor performance which induced by compressible effect,the tip speed of rotor has to design below the range of transonic Mach number.In this work,a further improvement of both performances under hovering and propelling condition has been implemented by genetic algorithm.

3.1 Hovering metric

Regrading to hovering operating state,Power Loading and Figure of Merit are commonly adopted as objective in optimizing procedure.By contrast between Power loading and Figure of Merit,Power Loading is an absolute metric of aerodynamic efficiency which straight forward measure thrust-power-ratio of rotor while Figure of Merit are the metric which measure between ideal and actual power requirement.Moreover,Power Loading can be derived as a form which consist of Figure of Merit and disk load as following:

From the equation,to attain a good hovering performance,a good Figure of Merit and low disk load was required.On the other hand,Leishman et al(2011)[4]indicated that a rotor within great hovering performance are not commonly operate within high Figure of Merit.Therefore,an absolute metric is employed as objective function for optimization in this work.

3.2 Propelling metric

A straight forward approach to measure propulsive efficiency is ratio of shaft power over propelling power,it can simply derive as following:

Subsequently,shaft power can be divided as Power which used for propelling,resist induced loss and profile loss.While propeller operate under high speed condition the propulsive efficiency can derive as following form:

From the equation,power used for resisting induced loss are diminishing within factor of.Therefore,profile loss is dominating propulsive efficiency.To further improve propulsive performance,profile loss will be the object to tackle.Alternatively,a more powerful metric to measure aircraft propulsive performance which is propulsive power loading.It considers propeller as a procedure which is inside the loop of aircraft design and generally consider the entire aircraft aerodynamic performance while aircraft are cruising.

4 Validation of Low-fidelity solver

In this part of work,a theoretical approach has been studied for avoiding expensively computational cost by directly solveRANS equation.A low-fidelity method is implemented for predicting aerodynamic performance of prop-rotor inside the loop of optimization.In our work,Vortex Theory and Blade Element Momentum Theory(BEMT)are implemented within Python.To further validate the accuracy of the low-fidelity methods,a validation which compare between calculation of low-fidelity method and experimental data are conducted.In the validation,the experimental data are chosen from Edwin et al(1938)[7]and the experimental dataset with 2 blades-RAF6-airfoil propeller are chosen.The comparing result have showed below:

In this validation,we were comparing thrust and power coefficient of propeller in various operating condition.From the figure,it indicated that Vortex Theory and BEMT has showed a promising prediction of thrust coefficient and both of the theory has a worse prediction in power coefficient,due to inaccuracy of prediction in drag.In the end,we decided to employ BEMT as solver for predicting aerodynamic performance in the optimizing process.

Fig.1 Comparison of thrust coefficient along advanced ratio

Fig.2 Comparison of power coefficient along advanced ratio

5 Validation of high-fidelity solver

In this section,a numerical solver which has been validated for providing an accurate calculation.A high-fidelity solver which is CFD++are used for validation by calculated the case of Caradonna Tung rotor and its computational result has compared with experimental data.The experimental dataset is collected from F.X.Caradonna et al(1981)[8].This article has contained various of dataset which has been conducted within different experimental setup and condition.To validate the solver,a dataset which is Caradonna rotor operate in hovering condition within 1250 RPM rotating speed and 8 deg of total pitch angle are chosen.The detail of the geometry has showed below:

Tab.1 Summary of geometrical and flow condition

In this validation,three set of structural gird within different number of mesh element has been generated for the calculation.These three set of mesh has shared a same topology and five type of boundary condition has been set for input,output,side boundary,blade and periodic boundary.The detail of the boundary setup has showed as

Fig.3 Experimental set-up of Caradonna Tung rotor

Fig.4 Computational domain and boundary

Fig.5 Surface mesh of Caradonna rotor

Fig.6 Ogrid for Caradonna rotor

In the calculation,Inflow boundary condition are given as total pressure and total temperature boundary condition while outflow boundary and side boundary are given as static pressure and temperature.The value of the total pressure,static pressure,total temperature and static temperature are given as ambient pressure and temperature which is 103027 Pa and 289.75 K respectively as show as Table.1.Therefore,the flow is induced by the work which was produced by the rotor,part of the work is used to produce thrust while the other are inducing flow.The fluid flow through the calculation domain within inflow and side boundary and leave the domain within outflow boundary and side boundary.Specifically,the single rotating frame within rotating speed 1250 RPM are employed and the RANS equation with SST turbulent model within wall function has been solved in the calculation.In addition,the steady RANS equation has been discreted with second order TVD scheme with minmoid limiter.Three set of mesh with 3,6 and 12 million of grid element are adopted for the validation.Moreover,thrust coefficient and pressure distribution in different radial position of blade has been compared for determining a promising element number of mesh for further calculation under consideration of both computational cost and accuracy.

Tab.2 Boundary condition for caradonna tung rotor

Fig.7 Independence analysis of grid

The graph which showed above have compared the capture of pressure distribution at radial position within 50%,60%,89%and 90%with 3,6,and 12 million of mesh element.In the first graph,it showed three mesh has little difference in prediction of pressure distribution while the other three of graphs indicated that the mesh within 6 and 12 million of element are able to accurately capture the peak of pressure distribution while the numerical dissipation are decreasing when the number of mesh element are increased.By contrast with thrust coefficient,the difference between 12 million and 6 million are little.In conclusion,the mesh within 6 million of element are appropriate for computation under the consideration of both computational cost and accuracy.

Fig.8 Pressure coefficient distribution at radial r/R=0.5,0.8,0.89,0.9

6 Procedure of Optimization

In this section,a pair of prop-rotor has been demanded for MAV to achieve capability of Short Take-Off and Landing(STOL).The requirement to the prop-rotor has been made and objective of optimization which is to improve hovering power loading and propulsive efficiency as much as possible.The detail of requirement has showed as

Tab.3 Design requirement for rapid aerodynamic design

Optimizing variable and original rotor

During optimizing process,a prop-rotor which is originally design for hovering state has been employed as original rotor.Radius of blades,chord distribution,twist distribution,hovering and propelling rotating speed in RPM are consider as design variable which have significantly influenced propulsive and hovering efficiency within those factors which has discussed in previous section.During the procedure of optimization,a multinomial equation is used to parameterize chord and twist distribution as

especiallyCiare design variable andis dimensionless radial position.In addition,to further improve both performance at each state,a variable pitch angle has been introduced to prop-rotor for further adapting difference of inflow angle between hovering and propelling state.The aerodynamic characteristic and geometry of original prop-rotor has provided as below:

Fig.9 Original rotor for the optimization

In this part of work,a problem of aerodynamic optimization with multi-objective has been solved with NSGA-II genetic algorithm and BEMT has been employed as a solver for calculating aerodynamic characteristic of prop-rotor.An absolute metric of performance which is hovering power loading and propulsive efficiency has been employed as objective of hovering and propelling state respectively.In the optimization,the rotor which showed above have been employed as original design,11 variables are involved in the op-timizing process and 120 individuals has been set for a generation in the NSGA-II algorithm.The population of individuals has been carefully decided to achieve diversity.Specifically,the flow graph of the optimizing process has showed below:

Tab.4 Performance of original rotor

Fig.10 Procedure of optimization

7 Constraint

The discrepancy between hovering and propelling under operating state,which have very different inflow condition and it requires different twist distribution for every airfoil element to work under angle of best Lift-Drag-ratio.Therefore,twist distribution of prop-rotor has to be compromised to attain a good performance under each design point.Due to discrepancy of inflow condition,root part of prop-rotor is inevitably work within either stalling or negative angle.To avoid negative thrust,a constraint of lift-coefficient is introduced which constrain every airfoil element to produce positive thrust,therefore further improve hovering performance as well as accelerate the optimizing process.

On the other hand,profile loss dominates propelling efficiency as we mention at previous part.A prior ideal for the design process is to require all airfoil element of prop-rotor work within high lift-drag ratio.However,this ideal will be an extremely strong constraint for prop-rotor and eventually none of the design individual will fulfil the condition while the discrepancy of inflow angle between hovering and propelling state are considerate.Therefore,a compromised strategy is to constrain medium and tip part of prop-rotor which produces large percentage of thrust over the total thrust to work within high lift-drag ratio and lower the restriction at root part of propeller as illustrate as Fig 11.

fig.11 Lift-drag ratio constraint

8 Result and Discussion

Result of multi-objective optimization

A prop-rotor with reasonable high hovering performance has been employed as initial geometry and a multi-objective optimization which consider both hovering and propelling performance was conducted with genetic algorithm.Some constraints are employed for either satisfying design objective or accelerate optimizing process.

Both figure which have showed above indicate hovering power loading and propulsive efficiency of each individuals along the optimizing process.From the graphs which showed a well converged in each objective.From figure 12 and 13,it indicates that hovering performance are sacrificed while the propelling performance are improving in the optimizing process.It illustrates each objective under hovering and propelling condition are conflicted.In the end,a collection of individual which contain all of qualified individual are constituted.An individual which are located at average level of both converged objectives are selected as design individual and the design individual has compared with two individuals which have best performance respectively under hovering and propelling state.

Fig.12 Converging history of propulsive efficiency

Fig.14 Geometry of selected blade

From figure,it indicated selected blade and the blade with high hovering power loading share a similar geometrical distribution.The only discrepancy these two blades is the rotating speed at propelling condition,it directly affect the descent of propulsive efficiency within profile loss whenas mentioned by Leishman et al(2011)[4].On the other hand,the blade with best propulsive performance have a large twist angle on the root part of rotor blade.In comparison with the other two blade,the twist distribution allows the blade work near to the angle of best liftdrag ratio on cruise condition.Therefore,it showed a better performance than other in the condition.

In this work,to further improve propulsive efficiency of prop-rotor over wide range of operating condition.We introduced a variable-pitch mechanism to allow prop-rotor to switch between each operating condition by change its total pitch angle.From the figure which showed above,it indicates the propulsive efficiency,thrust coefficient and power coefficient of selected blade which are calculated by BEMT.The selected blade is operating within different operating condition while it change its total pitch angle to attain a good performance under each operating condition.The blue line indicated the best propulsive performance and aerodynamic characteristics of the selected blade which are able to attain under different operating condition while its total pitch angle is changing.The graphs showed the mechanism of variable pitch angle are providing a sufficient way to achieve the design goal which require prop-rotor are able to attain good performance over wide range of operating condition.

9 Validation With CFD Analysis

In order to validate the selected optimized individual,a numerical solver which are directly solve the RANS equation are introduced and ANSYS ICEM CFD are employed for mesh generation.A mesh within 6 million elements are generated for validation.The mesh and detail of the computational domain have showed below.In this validation,five type of boundary condition are involved in each operating condition,single rotating frame and SST turbulence model with wall function has been employed.The steady RANS has been discretized with second order TVD scheme with minmod limiter.Regarding to the mesh,an O mesh has been generate around the blade,and the first level high of mesh has been set asy+=1while the Reynolds number at blade tip are 7.84×106in hovering state and 2.60×106in propelling state,it will ensure first level high of mesh along the radial position are below asy+=1.

In the calculation,we have conducted a simulation in each state to validate the optimized result.Specifically,the prop-rotor rotate with 3518.03 RPM and it will flip within 9.735 deg while it changes its operation between hovering and propelling condition.In the hovering state,the inflow boundary is given as total pressure and total temperature within ambient pressure 101325 Pa and temperature 287 K while outflow boundary condition is given with static pressure and temperature within ambient condition.On the other hand,the prop-rotor fly forward within 15 m/s,inflow,outflow and side boundary are given as static pressure 101325 Pa,temperature 287K and velocity condition.

In this case,a single rotating frame has been employed for calculation.The calculation has carried out for hovering and propelling condition.Since the blade rotate with 3518 RPM in hovering state,its Mach Number on blade tip achieveMtip=0.55,whileMtip=0.19under propelling condition and the selected blade rotate within 1139RPM.Table 7 which showed above has compared the prediction of thrust and shaft power which calculated by BEMT and high fidelity solver respectively.From the table,in both operating condition the prediction of thrust indicated an inconsistent result between low and high fidelity solver while shaft power is barely deferent.Nevertheless,the inconsistency of thrust,but the discrepancy of calculation between BEMT and CFD are around and below 10%.It is an acceptable result for the rapid design of prop-rotor in preliminary stage of design.

Tab.5 Comparison between selected blade,blade with best hovering and propulsive performance

Fig.15 Chord distribution of three selected blade

Fig.16 Twist distribution of three selected blade

Fig.17 Propulsive efficiency of selected blade work within different variable pitch angle in different advanced ratio

Fig.18 Thrust coefficient of selected blade work within different variable pitch angle in different advanced ratio

Fig.19 Power coefficient of selected blade work within different variable pitch angle in different advanced ratio

Fig.20 Computational domain and related boundary

Tab.6 Boundary condition set-up for hovering and cruise state

Tab.7 Comparison and error analysis of thrust and shaft power prediction

10 Conclusion

In this present work,a rapid aerodynamic design platform with optimizing approach is implemented for providing a preliminary design of prop-rotor while a demand of MAV to achieve STOL are proposed.To accomplish the task,NSGA-II genetic algorithm are employed for solving problem of multi-objective optimization.Blade Element Momentum Theory and Vortex Theory are validated and implemented for accelerating the optimizing process while modern CFD method are high consuming of time.From the evaluation of BEMT and Vortex Theory,which indicated that both theories are able to provided promising thrust prediction for the optimization and BEMT are much more robust than Vortex theory while the blade work under a stalling conditions.Therefore,BEMT has been selected as an aerodynamic solver for the optimization.To further understand the challenge of design,a study has been conducted and several design parameters which will significantly influence the aerodynamic performance of blade under each state has been introduced for the optimization as well as performance metric.A multi-objective optimization has been conducted for designing a proprotor,several constraints are introduced into the optimizing process for either to fulfil the requirement of design or accelerate the optimizing processby introduced the prior ideal from designer.A prop-rotor with good hovering performance has been used as initial geometry and the optimizing process showed a well converging in both objectives while the hovering performance are scarified to attain a good propulsive performance.The optimization showed a significant improvement of propulsive performance and it show a fast converged in the process.A design is selected from the collection of qualified design.To further validate the selected design,a validated CFD solver has been adopted for the validation.The validation indicated that the BEMT are able to provide a consistent prediction in shaft power while the prediction of thrust is relatively inaccuratebut it still acceptable for preliminary design.