High-resolution simulation for rotorcraft aerodynamics in hovering and vertical descending flight using a hybrid method

Liangquan WANG,Guohua XU,Yongjie SHI

National Key Laboratory of Science and Technology on Rotorcraft Aeromechanics,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

1.Introduction

Rotorcraft plays an important role in aviation for its unique ability in hovering and vertical descending flight.However,an accurate prediction of the rotor flow field remains a challenging problem.The three-dimensional rotor wake is unsteady and complex.In a hovering condition,highly energetic tip vortices shed from the blade tip region,swirl downward,and then undergo vortex breakdown in the far wake.In descending flight,tip vortices constantly persist near the rotor disk and interact with the blades,which may cause a high level of fuselage vibration and remarkable induced power consumption.Moreover,when a rotorcraft is operated over certain ranges of descent rate,convection of the wake would be inhibited.It results in a doughnut-shaped ring around the rotor disk,which is known as the Vortex Ring State1(VRS,see Fig.1).The VRS is regarded as the roughest flight regime of a rotorcraft.According to statistical data,2at least 32 helicopter accidents were attributed to this dangerous regime between 1982 and 1997.

Over the past few decades,studies about the unsteady characteristics of rotor wake in hovering and descending flight have been focused on experiments3–11and qualitative analysis.12–15Meanwhile,relatively sparse numerical simulations have been paralleled.The aerodynamics near the blade surface,like compressible and viscous effects,are predicted well by conventional Eulerian-based CFD methods, but they are computationally expensive,and inherent numerical dissipation makes the rotor wake diffuse too early.16To date,conventional CFD methods are insufficient in far wake capturing of the rotor.

Lagrangian-based models can address the problem of nonphysical wake diffusion,and they are more computationally efficient than CFD.In recent years,coupled Eulerian/Lagrangian simulation methods have shown promise in rotor wake simulations and received much attention.17–20However,simple Lagrangian models(e.g.,prescribed wake models21and free wake models22)rely heavily on empirical parameters such as vortex core size and decay factor.Furthermore,they cannot provide detailed information of the wake structure.Due to the progress in wake modeling techniques,more advanced Lagrangian models have been developed.Those models are referred to as Vorticity Transport Models(VTMs).23–25They can explicitly conserve wake vorticity without any artificial dissipation and cancel the restriction of empirical parameters.

The main work of the present study is to couple a novel Lagrangian-based VTM model with an Eulerian-based CFD solver.The hybrid solver proposed in this paper combines the merits of CFD and the VTM.A CFD module is used to resolve the compressible blade near body aerodynamics and a VTM module is used to predict the complex wake convection.Details of the CFD/VTM hybrid solver are described in Section 2.Numerical simulations of rotorcraft in hovering and vertical descending flight are performed in Section 3.Results show good predictions of both the blade near body aerodynamics and the wake structure in hover.Induced in flow and time history of rotor thrust in descending flight are also investigated.Main conclusions are summarized in Section 4.

2.CFD/VTM model description

The computational zone is decomposed into two domains(Fig.2):the blade body- fitted Eulerian domain,which covers a relatively small region near the blade and follows a C-O topology grid(Fig.3),and the background Lagrangian domain,which employs a set of particles to model the wake vorticity.These two domains are solved by CFD and a VTM,respectively.Since the VTM only solves incompressible flow,the C-O grid of the Eulerian domain extends far enough(over two chord lengths)to ensure that outside the grid,air compressibility could be neglected.

2.1.CFD solution of blade body- fitted Eulerian domain

The simulation of the rotor blade near flow field follows the Eulerian description,which relates to the grid-based solutions of compressible Reynold Averaged Navier-Stokes(RANS)equations.RANS equations are solved in terms of the conservation forms of mass,momentum,and energy,and can be written in tensor form as26

where ρ,uj,p, τij,E,and qjare the fluid density,the velocity,the pressure,the stress tensor,the total energy per unit mass,and the heat flux,respectively.

In general,the blade rotates at a constant speed,so Eqs.(1)–(3)can be expressed as

where Q is the conservative variable vector,Fc,Gc,and Hcare the convective fluxes,and Fv,Gv,and Hvare the viscous fluxes as follows:

where u，v，w are the velocity components.

A structured,cell-centered finite volume scheme is employed for CFD calculation and to model the formation of blade tip vortices.The spatial discretization is based on Roe’s flux-difference splitting scheme.27Convective fluxes are evaluated at the face of the control volume by solution reconstruction from the cell center,e.g.,

where ARoeis the Roe matrix,and the Spalart-Allmaras oneequation model is chosen for turbulence modeling.

The dual time stepping technique is applied for unsteady simulation,and the implicit Lower-Upper Symmetric Gauss-Seidel(LU-SGS)scheme is adopted for pseudo time discretization.This discretization scheme is defined by

where D is the diagonal matrix,L the strictly lower triangular matrix,U the strictly upper triangular matrix,and R the residual.The implementation of the LU-SGS scheme consists of the following two steps:

Forward sweep:

Backward sweep:

Finally,the conservative variables are updated by

2.2.VTM solution of background Lagrangian domain

There are two categories of VTMs:one is grid-based and uses an adaptive Cartesian grid to model the rotor wake,23,24while in the other category,wake vorticity is represented by a set of viscous vortex particles.25In the current study,the grid-free Lagrangian description that utilizes vortex particles is adopted.The governing vorticity dynamics equation is written as follows:

The above equation is derived from incompressible Navier-Stokes equations and is written in a vorticity-velocity form.The first term on the right-hand side ω ·?u is the so-called vortex stretching term,which denotes the dilation and rotation effects of the vortex filaments within the flow.The second term ν?2ω is the viscous diffusion term,which expresses the effects of air viscosity.In Eq.(10),ω=?×u is the vorticity,denotes the material derivative,ν is the air kinematic viscosity coefficient,and the source S is the vorticity shed and trailed from each rotor blade segment as

where Γbis the blade bound circulation vector,and ubis the local velocity of the blade segment relative to the air.

Assume that at time t and position x,the vorticity of the ith particle is defined by

The velocity vector is computed by the application of the Poisson equation as

Substituting Green’s function G(x′)into Eq.(14)yields the generalized Biot-Savart equation28as

Green’s function is given by

where the error function is written as

The stretching term can be computed by substituting Eqs.(12)and(15)into ω ·?u,which gives rise to

where subscript ‘st” denotes the vortex ‘stretching term”.Eq.(18)represents a dense algebraic system.If the Lagrangian domain consists of N vorticity-containing particles,25the task of calculating all the pairwise interactions by invoking Eq.(15)on each particle is equivalent to the classical N-body problem.The CPU requirements will be of the order O(N2)if no accelerating technique is utilized.Such a direct summation approach would quickly become prohibitive as the number of particles N increases.To address this problem,the Fast Multipole Method(FMM)29based on kernel evaluations is implemented in the current study.It does not require multipole expansions and works well for highly non-uniform particle distributions.Furthermore,it reduces the complexity from O(N2)to asymptotic O(N),and remarkably accelerates the computation.

The Particle Strength Exchange(PSE)scheme30is selected to solve the viscous diffusion term.One significant advantage of PSE lies in that this scheme provides enough flexibility in dealing with viscous effects and does not require any viscous splitting.In addition,vorticity conservation is naturally satisfied when two particles ‘exchange” their strengths.This conservation property is crucial to simulation accuracy because a non-physical variation of vorticity can seriously influence the predicted wake structure.The rationale of the PSE scheme is to solve an approximate integral equation for the Laplacian operator,which can be expressed as follows:

where ζεis the PSE kernel,which is identical to the Gaussian distribution in Eq.(13).The approximate manipulation of the Laplacian operator avoids any numerical differentiation,and the integral operator can be discretized by midpoint quadrature.

In the present study,a predictor-corrector time marching scheme is used to determine the new position xn+1and vorticity of the particles ωn+1,and can be written as follows:

(1)Predictor step

(2)Corrector step

where subscript ‘vdt”denotes the ‘viscous diffusion term”.

2.3.Flowchart and coupling methodology

The flowchart of CFD/VTM is shown in Fig.4.After flow field initialization,the iterative procedure is repeated until converged results are obtained.An interpolation algorithm allowing data communication on the interface between CFD and the VTM is implemented.The key idea of this data exchange process is as follows:the VTM module provides outer velocity boundary conditions vinfor the CFD module,and receives the updated vortex sources from the CFD solution.The time marching scheme is given in Fig.5.The Lagrangian description enables the VTM to adopt a larger time-step size than that of CFD(i.e.,ΔtVTM＞ ΔtCFD).When conducting unsteady simulations,CFD corresponds to an azimuthal time step of 1°and the VTM has a 4°time step.

2.3.1.Transmission of vortex sources from CFD to VTM

In the current study,distributed airloads along the blade span are treated as lifting line vorticity source generators.This technique is referred to as ‘integrated vorticity source method”by He and Zhao.25Integrating the blade surface pressure calculated from CFD,we obtain the blade local lift vector Lb.Then the blade bound circulation can be determined by the Kutta-Joukowski theorem as

New viscous vortex particles are released into the flow field to account for blade bound circulation variations at each time step,and the vorticity strength of the new vortex particles is computed by Eq.(11).

Table 1 Caradonna-Tung rotor parameters.

2.3.2.Transmission of boundary conditions from VTM to CFD

The VTM specifies flow variables on the outer boundary of the CFD gird.31The velocity components u,v,w of the wake can be directly sent to CFD.One the other hand,since the VTM resolution is based on incompressible Navier-Stokes equations,fluid density and pressure cannot be specified like the velocity components.Considering that the flow is approximately is entropic for low-Mach number flow simulation,the density and pressure of the fluid are calculated through the following is entropic equations:

where γ is the ratio of specific heat coefficient,and Ma is the Mach number.

3.Numerical results and discussions

3.1.Code validation and wake kinematics

In the following subsections,the proposed CFD/VTM solver is validated against hover cases of the Caradonna-Tung model rotor,an OH-13E full-scale helicopter rotor,and a scaled V-22 tiltrotor.Simulation results are compared with available test data to show the capability of the presented hybrid solver.

3.1.1.Caradonna-Tung model rotor

The main parameters of this two-bladed model rotor are listed in Table 1.This rotor has no blade twist and the chord length c is constant.The rotor is operated at a collective pitch setting θ0of 8°with two different blade tip Mach numbers Matip,0.612 and 0.877.The near-body C-O type grid for the Eulerian domain consists of 225×43×74 grid points,where 225 points are used in the wrap around direction,43 points in the normal direction,and 74 points in the spanwise direction.The spacing between the outer boundary of the C-O grid and the blade surface is 3c.The number of vortex particles in the Lagrangian domain is about 52000.

In order to compare with the CFD/VTM coupling method,a full CFD simulation employing an overset grid system is also performed.Half of the overset grid system is shown in Fig.6.Blade chordwise pressure distributions at four representative spanwise locations(see Fig.7,r/R=0.68,0.80,0.89,and 0.96)are illustrated in Fig.8.The dots are from Caradonna and Tung’s experiment.32It can be seen that the results of both CFD/VTM(solid lines)and full CFD(dashed lines)methods are in good agreement with the experimental data,only with minor discrepancies at the blade leading edge.A Root Mean Square(RMS)analysis is carried out to evaluate the accuracy of these two methods,and the RMS is computed by

where ΔCpis the difference between the predicted data and the measured data in the experiment at each pressure transducer position.Results of the RMS analysis are shown in Fig.9.It is demonstrated that the predicted Cpdistributions of CFD/VTM are slightly better than those of the full CFD method.

Fig.10 presents the wake structure predicted by CFD/VTM.As shown in the 3D wake visualization(Fig.10(a)),concentrated tip vortices are preserved for a relatively long time.Both the near wake and the far wake could be clearly identified.In Fig.10(b)and(c),the rotor wake immediately contracts below the rotor disk plane,and demonstrates an ordered helicoidal structure in the first few revolutions.As the wake convects downstream further,the tip vortex filaments begin to bundle with each other.This mutual interaction phenomenon,or the so-called ‘vortex paring instability”,causes a breakdown of the vortex structure in the far wake.In contrast,only near-wake characteristics could be observed in the wake structure simulated by the full CFD method because of the well-known numerical dissipation (see Fig.11).The Lagrangian-based VTM model demonstrates its advantage on vortex tracking in the far wake.

Fig.12 shows the tip vortex evolution in the blade near body domain at three selected planes downstream from the trailing edge(spacing=0.2c,0.5c,1.0c).On Slice 1,both the CFD/VTM and full CFD methods capture the tip vortex very well,while on Slices 2 and 3,CFD/VTM preserves the vorticity better than the full CFD method.Since the VTM captures the far wake structure better than CFD,it provides a more accurate feedback to the blade near body domain,and hence results in less diffused tip vortices.

The simulation time of these two methods are documented in Table 2.Computations are carried out on Intel eight-core processors with a 3.4 GHz CPU speed.CFD/VTM and full CFD simulations are performed with the same blade bodyfitted C-O grid.The number of background grid nodes is about 5.14 million in full CFD simulation.As shown,it requires 28.7 h while CFD/VTM only needs 5.6 h.A noticeable reduction of the computation time is achieved by the CFD/VTM solver,which is mainly because that the wake structure is simulated by particles.A dense background gird33is no longer neededin this hybrid solver.In addition,a larger step size that the VTM allows is also conducive to improve the computational efficiency.

Table 2 Simulation time of the CFD/VTM coupling method and the full CFD method.

3.1.2.OH-13E helicopter rotor

To further validate the CFD/VTM hybrid solver,the downwash of a full-scale OH-13E rotor is calculated.The main characteristics of this rotor are given in Table 3.The rotor is operated at a tip Mach number of 0.403 with two different collective pitches,6.25°and 10.75°.Fig.13 shows the radial distributions of the down wash at four sampling stations below the rotor disk(Z=-0.1R,-0.3R,-0.5R,-0.7R),and videnotes the velocity of the down wash.As shown,the calculated data agrees well with the measured data.34The wake contracted near the radial position at r/R=0.8,and the downwash distributions are characterized by sharp velocity peaks at this location.Comparisons between the axial and radial components of tip vortex positions are given in Fig.14.As shown,correlations between the predicted and measured data are favorable.

Table 3 OH-13E rotor parameters.

3.1.3.Scaled V22 ‘Osprey”tiltrotor

Table 4 0.658-scale V22 rotor parameters.

Anothervalidation caseisa 0.658-scaleisolated V-22‘Osprey” tiltrotor.The rotor parameters are tabulated in Table 4.This scaled tiltrotor has a very high blade twist angle of-47.5°and with a precone angle of 2.5°.More information about this rotor con figuration can be seen in Ref.35Fig.15 compares the figure of merit when the tiltrotor is operated at various hover conditions,where CTand σ represent the rotor thrust and solidity,respectively.It can be clearly seen that CFD/VTM-predicted results match well with the test data.36A typical wake structure for this three-bladed model tiltrotor is shown in Fig.16.

3.2.Descending flight simulation

From Fig.17,it is seen that the non-dimensional thrust time histories of the two rotor con figurations are similar.However,the tiltrotor con figuration is less susceptible to the VRS with a higher vi0.At a low descent rate(μz＜ 0.4),the rotor thrust is not significantly in fluenced,and it is slightly higher than that in hover.As the descent rate increases,the rotor thrust starts to fluctuate,but a nearly converged solution can still be obtained(an incipient VRS).When the descent rate is higher than 1.0,a remarkable thrust loss occurs,which indicates the emergence of a deep VRS.

Fig.18 shows the normalized mean in flow of the rotor disk.As shown,the classical momentum theory is no more valid in descending flight.Predicted data matches well with the flight test of Taghizad et al.,9the wind tunnel test of Betzina,5and the empirical relation given by Castles and Gray.12Rotorinduced flow increases as the descent rate increases,and it grows faster than the opposing descent rate.Fig.19 shows the difference between hover and the VRS at a specific blade section.Compared with the hovering state,the high induced flow in the VRS dramatically reduces the effective Angle of Attack(AOA),and causes sudden loss of the rotor thrust.

Fig.20 shows the snapshots of particle distributions at different descent rates around the rotor disk.As shown,the vortex particles tend to move toward the disk plane,and hence the convection of vortices is inhibited.The accumulated vortices interact with rotor blades,which leads to a fluctuation of the rotor thrust.Furthermore,they can dramatically increase the induced velocity(as shown in Fig.18).At a descent rate of μz=1.2,most particles are transported above the rotor disk plane.Fig.21 shows the flow regime at this fully developed VRS.The wake structure is characterized by a toroidal‘vortex ring”.This‘vortex ring” lingers within the vicinity of the rotor disk(Fig.21(a)),and induces a large-scale recirculating flow(Fig.21(b)),which corresponds to the schematic of air flow shown in Fig.1.

4.Conclusions

An Eulerian-Lagrangian hybrid solver that combines a vorticity transport model with a CFD model has been developed in this paper.From numerical cases of hovering and vertical descending flight,the following conclusions can be drawn:

(1)In the hovering state,the blade surface pressure distribution,tip vortex position,induced velocity,and figure of merit predicted by the presented hybrid solver are in close agreement with experimental data,which indicates that it provides a high- fidelity analytical tool for rotorcraft application.

(2)The CFD/VTM coupling method demonstrates its exclusive capability in far wake modeling and vortex tracking.Rotor wake instabilities(i.e.,bundling of vortex filaments and wake breakdown in the far field)in hover that are hard to be captured by conventional CFD methods,are successfully simulated in the current study.In addition,since the far wake is modeled by viscous vortex particles,the computational cost of the hybrid solver is not expensive.The present coupling method will be added to the Chinese Laboratory of Rotorcraft Navier-Stokes(CLORNS)code in the near future.37

(3)In respect of vertical descending flight with a fixed collective pitch,the rotor thrust time history shows that the thrust starts to fluctuate at a moderate descent rate.A sudden loss of the thrust manifests the emergence of a deep VRS.Non-dimensional thrust behaviors of different rotor con figurations are similar.

(4)When the descent rate increases,the wake vorticity tends to move upward to the rotor disk plane.In contrast with the helicoidal wake structure in hover,the deep VRS is characterized by a toroidal vortex ring.This ring induces a largescale recirculating flow around the rotor disk,which dramatically reduces the blade effective angle of attack,and eventually causes the sudden loss of the rotor thrust.

Acknowledgements

This work was co-supported by the Funding of Jiangsu Innovation Program for Graduate Education of China(No.KYLX16_0389)and the Fundamental Research Funds for the Central Universities of China.

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