Two－Way－Coupling Method for Rapid Aerothermoelastic Analyses of Hypersonic Wings

，，，＊

1.School of Aeronautics Science and Engineering，Beihang University，Beijing 100191，P.R.China；

2.Key Laboratory of Aircraft Advanced Design Technology（Behang University），Ministry of Industry and Information Technology，Beijing 100191，P.R.China；

3.Beijing Institute of Nearspace Vehicle′s Systems Engineer，Beijing 100076，P.R.China

0 Introduction

Hypersonic vehicles operate at a Mach number ranging from 5to 15，and they can fly in and across the atmosphere at hypersonic speeds［1］.Hypersonic vehicles can fly at high speeds，they are capable of long voyages，and they present good maneuverability and quick reactions.In addition，these vehicles can adapt to technologically advanced warfare and will be able to satisfy future high－speed transportation requirements.Therefore，research on hypersonic vehicles has become popular in recent years.

The aerodynamic shapes of most hypersonic vehicles include slender bodies，lifting bodies or waverider configurations.To reduce weight，the fuselage，wing and control surface always have large elasticity.Therefore，the aeroelastic effect of hypersonic vehicles must be considered.Hypersonic vehicles must fly at hypersonic velocity in the atmosphere，which causes a drastic aerodynamic heating effect on the vehicle surface.The structural stiffness and the aeroelastic response are influenced by the aerodynamic heating effect.Thus，owing to the impact of aerodynamic force and aerodynamic heating，hypersonic vehicles encounter an aero－thermal－structural coupling problem.During the design stage of this type of air vehicle，the influence of aerodynamic heating on the aeroelastic response must be considered［2］.

So far，relatively few studies have focused on the aerothermoelastic effect of hypersonic vehicles and the coupling relationship between the heating effect and static aeroelastic response［3］.In the past，researchers have typically focused on the aeroelastic response in the thermal environment［4］.Particularly，domestic researchers often assumed the temperature field in their calculations and neglected the variations in the structure temperature field induced by the aerodynamic heating of hypersonic flow［5］.In general，limited systematic research has been performed to develop an aerothermoelastic analysis method for hypersonic vehicles，and a uniform and rapid method for this problem has not yet been established.

Due to increasing demands in engineering fields，the concepts underlying a rapid aerothermoelastic analysis method are addressed here.Thus，strong couplings and weak couplings are explained，and one－and two－way－coupling methods are illustrated and compared.The findings indicate that the two－way－coupling method is appropriate for this problem.The aerothermoelastic analysis method proposed in this paper covers several disciplines，i.e.，aerodynamic calculation，aerodynamic heat flux calculation，heat transfer calculation，and static aeroelastic analysis.When introducing the calculation methods for each discipline，rapid calculation methods that are widely used in engineering are implemented to improve the computational efficiency.Based on the aforementioned concepts，a rapid aerothermoelastic analysis method is established.This method utilizes a two－way－coupling method and simultaneously considers the calculations required for each discipline，such as aerodynamic force，aerodynamic heat flux，heat transfer，and static aeroelastic response calculations.In the following sections，an aerothermoelastic simulation of the wing of a hypersonic vehicle is performed using this method.The simulation results obtained by the one－and two－way－coupling method are compared and analyzed.

1 Coupling and Calculation Methods for Each Discipline

1.1 Introduction and selection of coupling method

In general，the hypersonic aerothermoelastic problem can be divided into two categories conceptually：One is the aerodynamic heating problem，and the other is the aeroelastic problem.The aerodynamic heating problem contains the calculations of aerodynamic heat flux and heat transfer.After solving this problem，the temperature field of the structure can be obtained.The aerodynamic heat flux is a function of the surface temperature；therefore，the interaction between the aerodynamic heat flux and heat transfer must be considered.Furthermore，the surface geometrical shape of the vehicle can also influence the aerodynamic heat flux by affecting the aerodynamic force distribution.Thus，various coupling relationships are involved in the hypersonic aerothermoelastic problem，which are focused on by this paper.

1.1.1 Strongcouplingandweakcoupling

The hypersonic aerothermoelastic problem usually involves two different types of coupling relationships among the aerodynamic heat flux，aerodynamic force，inertial force，and elastic force.These two different types of coupling relationships are strong and weak couplings，as shown in Fig.1.

Fig.1 Strong and weak couplings in aerothermoelastic analyses

The strong couplings include the following three aspects：

（1）The coupling relationships among aerodynamic，inertial and elastic forces.

（2）The direct influence of the aerodynamic heating on stiffness and elastic force.

（3）The impact of aerodynamic force on aerodynamic heating.

The weak couplings include the following three aspects：

（1）The direct impact of aerodynamic heating on aerodynamic force.

（2）The direct impact of elastic force on aer－odynamic heating.

（3）The coupling relationship between aerodynamic heating and inertial force.

Since the weak couplings have little effect on aerothermoelastic properties，only the strong couplings are evaluated，as indicated by the solid lines in Fig.1［6］.Therefore，the following three assumptions are made to resolve the proposed problem：

（1）Aerodynamic heating does not impact aerodynamic force.

（2）Structural deformation does not directly impact aerodynamic heating.

（3）The coupling relationship between aerodynamic heating and inertial force does not exist.

Thus，in the rapid aerothermoelastic analysis method，only strong couplings should be considered.

1.1.2 One－way－couplingandtwo－way－coupling

methods

In coupling analyses，aerodynamic heating is considered to impact structural stiffness in two ways.On the one hand，the mechanical properties of the material will change with variations in the thermal environment.On the other hand，aerodynamic heating will generate external thermal stress on the structure.Furthermore，variation in the aerodynamic force induced by structural deformation can influence the aerodynamic heating.

Concerning the complexity of this problem，only the effects of aerodynamic heating on structural mechanical properties and aerodynamic force，in addition to that of structural deformation on aerodynamic force，have previously been studied.The feedback effect of aerodynamic force on aerodynamic heating has been neglected.In the one－way－coupling method，only the former effects are considered，whereas in the two－way－coupling method，all of the aforementioned effects are considered，as shown in Fig.2［7］.

In the two－way－coupling method，the feedback impact of aerodynamic force on aerodynamic heating is considered based on the one－way－coupling method，and it provides a more complex analysis that remains consistent with actual conditions.Thus，the two－way－coupling method can reduce errors.

Fig.2 One－and two－way－coupling methods

To analyze the aerothermoelastic problem of a hypersonic wing more effectively and obtain a better understanding of the coupling relationships of each discipline involved in the aerothermoelastic problem，the two types of coupling methods are applied for the same model，and the results are compared.

1.2 Calculation method for each discipline

1.2.1 Structuralstatic/dynamicequationina thermalenvironment

The static/dynamic equation of the structure of a hypersonic wing based on the finite element method（FEM）in the physical coordinate is expressed as follows［8］

where Msis the mass matrix，Fsthe load vector，qthe displacement vector，and（T）the stiffness matrix.

1.2.2 Aerodynamicforcecalculation

Here，aerodynamic force is calculated by the shock expansion method and local piston theory.First，the local airflow parameters are calculated by the shock expansion method.The method is illustrated as follows［9］.

When air flows past the rhombic wing，as shown in Fig.3，an oblique shock wave appears at the leading edge of the wing，and an expansion wave appears at the middle of the wing chord.

Oblique shock wave formulas are applied to calculate the airflow parameters after the wave，and the following is obtained.

Fig.3 Oblique shock wave and expansion wave

whereηrepresents the angle between the wedge leading edge and the horizontal plane，andβthe shock wave angle.Then，the following relationships can be obtained.

wherep1andp∞represent the pressure between and after the shock wave，respectively，andMa1andMa∞represent the Mach number between and after the shock wave，respectively.

The Prandtl－Meyer expansion wave formulas and isentropic formulas are applied to calculate the airflow parameters for two arbitrary Mach numbers，Ma1andMa2

The airflow parameters have been obtained.

Then，local piston theory can be applied to obtain the pressure coefficientcp［10］.

wherealocalrepresents the local sound velocity，andγis the specific heat ratio.

The meaning of the variables is shown in Fig.4.

Fig.4 Sketch map explaining local piston theory

1.2.3 Aerodynamicheatfluxcalculation

The Eckert reference temperature method，which is widely applied in engineering，is applied to the aerodynamic heat flux calculation.The theory of the Eckert reference temperature method is elaborated here［11］.When air flows past a wall，the temperature of the incoming flowTand other related parameters are corrected，and a reference temperatureT＊is used to replace the original temperatureT.Therefore，if the shear stress calculated by insertingT＊into the equation of incompressible fluid is equal to the shear stress calculated by the equation for compressible fluid，then the fluid friction equation of incompressible fluid can be used to calculate the compressible fluid value［7］.In the incompressible fluid calculation，the variables related to temperature must be obtained fromT＊.

whereTwis the wall temperature，andTrthe recovery temperature.In a turbulent flow，Trcan be calculated as

Once the reference temperatureT＊is obtained，cf，which is the coefficient of friction resistance in the boundary layer of turbulent flow，can be calculated［12］

whereRexis the local Reynolds number，Rex＝

The Stanton numberStrepresents the heat transfer coefficient of heat convection［13］，and it is equivalent toSt＝×0.82in turbulent flow.

Then，the heat flux can be defined as［7］

whereρUerepresents the mass of airflow，which flows past the unit area in the unit time，andcpthe constant－pressure specific heat of the air.

Based on the relational expression of isentropic flow，a one－to－one correspondence occurs between the local pressurepand the local parametersρ，Ma，vandT，separately.Therefore，a one－to－one correspondence occurs between the local pressurepand local heat fluxqaero.

High temperatures occur at the surface of hypersonic vehicles；therefore，thermal radiation must be considered.Assuming that the surface is not a black body，the radiation equation is as follows

whereσis the Stanford constant，which has a value of 5.669E－8W·m－2·T－4，andεthe radiation heat dissipation rate.

Finally，the heat flux conducting into the structure is

2 Analysis Framework

Based on the coupling relationships and calculation methods introduced in the previous sections，the rapid aerothermoelastic analysis method for hypersonic wings is presented here.In this method，only strong coupling is considered，and the two－way－coupling method is used.When each discipline is considered，the FEM is used to calculate the structural deformation and heat transfer，the shock expansion method and local piston theory are used to calculate the aerodynamic force，and the Eckert reference temperature method is used to calculate the aerodynamic heat flux.Fig.5 presents the framework of the hypersonic aerothermal－structural coupling analysis［14］.

The analysis steps are the following：

（1）At the beginning of the analysis，the initial parameters of the air flow，the initial structural parameters and the initial temperature field are given.The steady aerodynamic force is calculated by the shock expansion method and local piston theory.This steady aerodynamic force is loaded on the structure via the FEM，and then the deformation is also obtained via the FEM.Then，the deformation is utilized in the aerodynamic calculation mode.Solving iteratively until convergence，the static aeroelastic response of the structure at momenttis obtained.

（2）Based on the aerodynamic force distribution，the aerodynamic heat flux is calculated by the Eckert reference temperature method.The aerodynamic force is redistributed with structural deformation.When the two－way－coupling method is used，the redistributed aerodynamic force must be used.Otherwise，the redistribution of aerodynamic force can be neglected.

（3）The FEM is applied to analyze the heat transfer，and the temperature field of the structure at the next time stept＋Δtcan be obtained.The updated wall temperature，which is the initial condition of the aerodynamic heat flux calculation，can also be obtained.At the same time，the structure temperature field，which is the initial condition of the heat transfer calculation，can be obtained.

（4）Based on the temperature field obtained in the previous step，the FEM is used to perform a stress analysis for a structure in a thermal environment.Then，the stiffness matrix and thermal deformation of the heated structure are obtained.

（5）If the final time is reached，then the calculation ends.Otherwise，the stiffness matrix in the aeroelastic analysis and the structural deformation will be updated.The process returns to the aeroelastic calculation，where the aeroelastic response at moment is obtained，and the loop continues.

Aero－thermal－structural coupling is focused on in this paper；therefore，the aeroelastic analy－sis is considered to be completed instantaneously.Furthermore，since the aerodynamic heat flux calculation is performed via the Eckert reference temperature method，the method is fast.Thus，the time steps of the aerodynamic heat flux and heat transfer calculations can be equivalent to simplify the code.Therefore，the time step of each discipline in this paper can be equivalent.

Fig.5 Analysis framework

3 Calculation Model

The model proposed in this paper is the lowaspect－ratio wing of a hypersonic aircraft，and the wing span，chord length and thickness of the wing are shown in Fig.6［15］.

Hypersonic aircrafts usually encounter severe thermal environments in flight；therefore，a series of thermal protection measures are required to protect the wing structure，as shown in Fig.7.In the proposed model，7.6－mm－thick thermal protection shields are added on the upper and lower surfaces of the wing to achieve this goal［15］.

Fig.6 Configuration of the hypersonic aircraft wing

Fig.7 Scheme of thermal protection shields and structural layers on aircraft surface［14］

Table 1presents the material properties of the model，including the thermal and mechanical material property parameters of the three layers.″T－dep″indicates that the parameter depends on temperature，and″—″indicates that the parameter is not considered.

Table 1 Material property parameters of the three layers

The finite element model of the hypersonic wing used for the structural deformation calculation and the heat transfer calculation is shown in Fig.8.

Fig.8 Finite element model of hypersonic wing

4 Calculation and Results Analyses

4.1 Method verification

To verify the proposed method，a comparison between the results of this paper and those of Ref.［15］is provided in Table 2，which shows the lifting forces of four cases.

Table 2 Comparison between the results of previous studies and those of this study

The results obtained by the present method are consistent with the reference results，indicating that the method is feasible.Thus，an aerothermoelastic response analysis for the hypersonic wing will be performed using this method.

4.2 Example and result analysis

The static aerothermoelastic analysis of a wing under the aero－thermal－structural coupling frame is performed for a specific altitude，angle of attack，Mach number，and load factor.The specific parameter values used in the analysis are as follows：Ma＝8，H＝15km，α＝6°，longitudinal overload＝－1g，initial temperature of the structureTref＝293K，Prandtl numberPr＝0.86，and specific heat ratio of the atmosphereγ＝1.4.

The radiant emissivity of the surface layer of the structure，i.e.，the thermal protection shield，is 0.85；the time intervals for the aerodynamic heat flux and heat transfer calculations are both Δt＝ 4s；and the total calculation time isttotal＝600s.

Figs.9—12show several structural deformation and temperature field results for the structural layer，upper surface，and lower surface.Each figure contains the one－way－coupling result and the two－way－coupling result.

Because hypersonic vehicle experiments are difficult to conduct，the results of this paper are compared with those of Ref.［15］for verification.The comparison shows that the temperature field distributions of the wing surfaces between the two studies are consistent.

Fig.9 Deformation of the wing structure at 600s

Fig.10 Temperature field of the wing structural layer at 600s

From Figs.9—12，the differences between the results of the two methods can be identified.However，the differences are not very clear.In Table 3，the differences are more obvious.

Fig.11 Temperature field of the upper surface of the wing at 600s

Fig.12 Temperature field of the lower surface of the wing at 600s

Table 3shows that the different coupling methods primarily influence the surface temperature of the hypersonic vehicle.Although the maximum difference in surface temperature is close to 100K，if the thermal protection shields are effective，then the difference is eliminated，and the temperature fields of the structural layer in two situations are similar.

Table 3 Differences between the two methods

Fig.13shows the initial heat flux distribution at the lower surface of the wing；the temperature of the wing is 293K.Fig.14shows the heat flux distribution at the lower surface of the wing at 600s；the average temperature at 600sis 1 833 K.The comparison of the initial heat flux distribution with the heat flux distribution at 600s shows that the initial heat flux input is far larger than that at 600s.Therefore，the effect of thermal radiation is obvious based on Eq.（13）.

Fig.13 Initial heat flux distribution at the lower surface of the wing

Fig.14 Heat flux distribution at the lower surface of the wing at 600s

In previous research，only the change in stiffness was considered，whereas the impact of thermal deformation on aerodynamic force was neglected，which would generate considerable errors.

Thermal deformations are calculated in this paper.When the temperature is high，the thermal deformation is remarkable.In certain situations，the thermal deformation is even larger than the deformation induced by aerodynamic force，as shown in Figs.15，16.This finding also validates the rationality of considering the influence of thermal deformation on aerodynamic force in the analysis process presented in this paper.

Fig.15 Deformation caused by aerodynamic force

Fig.16 Deformation caused by thermal stress

5 Conclusions

The aero－thermal－structural coupling problems for hypersonic wings are discussed.The coupling and calculation methods are introduced and selected.Then，the aerothermoelastic analysis framework is presented，and an example is simu－lated.The deformation，temperature field，pressure distribution and heat flux distribution of a hypersonic wing while cruising are calculated.Furthermore，the results of the one－method and two－way－coupling method are compared.The following conclusions have been obtained：

（1）The high Mach numbers and severe thermal environments experienced by hypersonic vehicles cause obvious aero－thermal－structural coupling effects that should be considered when related issues are studied.

（2）The differences between the one－waycoupling method and the two－way－coupling method are in the surface heat flux calculations.However，the application of effective thermal protection shields can nearly eliminate the differences between the methods；thus，the final results are similar.

（3）Thermal deformation is obvious，and in certain situations，it is even larger than the deformation caused by aerodynamic force.As thermal deformation can obviously alter aerodynamic force，this factor should be considered in future studies.

Acknowledgements

This work was supported partly by the National Natural Science Foundation of China （Nos.11302011，11402013，11372023，11672018）；the National Key Research and Development Program （No.2016YFB0200703）；and the Fundamental Research Funds for the Central Universities（No.YWF－14－HKXY－006）.

［1］ DU Z L，WAN Z，YANG C.Robust aeroelastic design optimization of hypersonic wings considering uncertainty in heat flux［J］.Transactions of the Japan Society for Aeronautical ＆Space Sciences，2017，60（3）：152－163.

［2］ MCNAMARA J J，FRIDMANN P P.Aeroelastic and aerothermoelastic analysis of hypersonic vehicles：Current status and future trend：AIAA 2007－2013［R］.USA：AIAA，2007.

［3］ MCNAMARA J J，FRIDMANN P P，THURUTHIMATTAM B J.Aeroelastic and aerothermoelastic behavior in hypersonic flow［J］.AIAA Journal，2008，46（10）：2591－2610.

［4］ THURUTHIMATTAM B J，FRIEDMANN P P，POWELL K G.Aeroelasticity of a generic hypersonic vehicle：AIAA 2002－1209［R］.USA：AIAA，2002.

［5］ WU Z G，HUI J P，YANG C.Hypersonic aerothermoelastic analysis of wings［J］.Journal of Beijing University of Aeronautics and Astronautics，2005，31（3）：270－273.（in Chinese）

［6］ MCNAMARA J J，FRIDMANN P P.Aeroelastic and aerothermoelastic analysis in hypersonic flow：Past，present，and future［J］.AIAA Journal，2011，49（6）：1089－1122.

［7］ CULLER A J，MCNAMARA J J.Studies on fluidthermal－structural coupling for aerothermoelasticity in hypersonic flow［J］.AIAA Journal，2010，48（8）：1721－1738.

［8］ LI G S，WAN Z Q，YANG C.Integrated analysis of aerothermal－aeroelastic wings in hypersonic flow［J］.Journal of Beijing University of Aeronautics and Astronautics，2012，38（1）：53－58.

［9］ WU Z N， WANG B，ZHOU R.Aerodynamics（Volum 1）［M］.Beijing：Tsinghua University，2007：61－65.（in Chinese）

［10］MCNAMARA J J，CROWELL A R，FRIDMANN P P，et al.Approximate modeling of unsteady aerodynamics for hypersonic aeroelasticity ［J］.Journal of Aircraft，2010，47（6）：1932－1945.

［11］ECKERTE R G.Engineering relations for heattransfer and friction in high－velocity laminar and turbulent boundary－layer flow over surfaces with constant pressure and temperature［J］.Transactions of the ASME，1956，78（6）：1273－1283.

［12］BERTIN J J，CUMMINGS R M.Fifty years of hypersonics：Where we′ve been and where we′re going［J］.Progress in Aerospace Sciences，2003，39（6/7）：551－536.

［13］QIAN Y J.Aerodynamics ［M］.Beijing：Beihang University，2004：260－268.

［14］WAN Z Q，WANG Y K，LIU Y Z，et al.A highefficiency aerothermoelastic analysis method［J］.Science China，2014，57（6）：1111－1118.

［15］FALKIEWICZ N，CESNIK C E S，CROWELL A R，et al.Reduced－order aerothermoelastic framework for hypersonic vehicle control simulation［J］.AIAA Journal，2011，49（8）：1625－1646.