On Field Velocity Approach and Geometric

Conservation Law for Unsteady Flow Simulations

JayanarayananSitaraman ∗

Vishnu S. Iyengar † James D. Baeder ‡

Alfred Gessow Rotorcraft CenterDepartment of Aerospace Engineering

University of Maryland at College Park, MD 20742

Abstract

A technique to model unsteady flow phenomenonwhich involve simultaneous elastic motions of theboundaries and interaction with gust fields is dis-cussed. The field velocity approach is used to includethe effect of the gust fields caused by a vortex wakein to the flow computations. Elastic deformations ofthe boundaries causes changes in grid cell volumeswhich requires the rigorous enforcement of the Geo-metric Conservation Law. Mathematical modelingrequired to enforce conservation when performingsuch unsteady flow computations is detailed. Theapplication of the technique developed to a variety ofproblems from simple 2-D model problems to morecomplex realistic problems are detailed. The predic-tions obtained are validated with exact analyticalresults/ experimental data as appropriate. Overall,the field velocity approach together with a techniqueto enforce the Geometric Conservation Law is foundto provide a powerful tool for unsteady flow simula-tion.

Introduction

The field velocity or grid velocity approach is amethod for incorporating unsteady flow conditionsvia grid movement in computational fluid dynamicsimulations. This approach provides a uniquemethodology for directly calculating aerodynamic

∗Graduate Research Assistant†Graduate Research Assistant‡Associate Professor, Senior Member AIAA

responses to step changes in flow conditions. Phys-ically, the grid velocity can be interpreted as thevelocity of a grid point in the mesh during the un-steady motion of the boundary surface. For exam-ple, the simulation of a step change in angle of at-tack of an airfoil can be performed by incorporatinga step change in vertical grid velocity all over theflow domain. This method effectively decouples theinfluence of pure angle of attack from that of a pitchrate because the airfoil is not made to pitch, and be-cause the step change is enforced over the entire flowdomain uniformly [1]. A similar methodology canbe used for simulating responses of an airfoil to stepchanges in pitch rate and interaction with a travel-ing vertical gusts or convecting vortices [2, 3]. Thegrid velocity approaches are normally implementedwithout actually moving the grid. Rather, the timemetrics are modified to effectively simulate the mo-tion of the grid. But incorporation of the field veloc-ity approach without actually moving the grid doesat times violate the so called Geometric Conserva-tion Law (especially when there is large variation inthe grid velocities between successive grid points).The Geometric Conservation Law (GCL) is used tosatisfy the conservative relations of the surfaces andvolumes of the control cells in moving meshes. Pri-marily, the GCL states that the volumetric incre-ment of a moving cell must be equal to sum of thechanges along the surfaces that enclose the volume.Thomas and Lombard [4] were the first to recognizethe necessity of satisfying the geometric conserva-tion laws simultaneously with other physical conser-vations when solving moving mesh problems. Theyproposed a differential form of the GCL which needs

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16th AIAA Computational Fluid Dynamics Conference23-26 June 2003, Orlando, Florida

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Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

to be solved along with the conservative variables.More modern approaches ( [5, 6, 7]) efficiently com-pute the space and time metrics in a manner thatimplicitly guarantees the satisfaction of the GCL.One of the highly unsteady flow problems of prac-tical interest is computing the flow field around ahelicopter rotor blade undergoing aeroelastic defor-mations. The unsteadiness in flow field is primar-ily caused by the periodic blade motions and thewake induced inflow. Resorting to a multi-block,overset mesh type approach (which models all therotor blades) limits the practicality of such a cal-culation because of high computational overheads.Also one needs to carefully tune the grid spacingsto preserve the trailed vortices. But, use of a singleblock approach (which models a single rotor blade)requires the inclusion of the influences of a trailedvortex wake, whose geometry is computed from us-ing external calculation (usually a Lagrangian vor-tex lattice approach [8]). The field velocity ap-proach is used for wake inclusion in such cases. Inthis approach, there is a dynamic real movement ofthe mesh because of the aeroelastic blade deforma-tions and apparent movement of the mesh becauseof the use of the field velocity approach [9]. Thisposes a unique challenge for preserving the GCL.The computation of space and time metrics shouldbe performed consistently to maintain conservationand hence preserve accuracy.The objectives of this research effort are three fold.The first objective is to demonstrate the capabilitiesof the grid velocity approach for the simulation ofunsteady flow environments. The second objectiveis to develop methodologies for computing 4th orderaccurate space metrics and second order accuratetime metrics which satisfy the Geometric Conserva-tion Law. Finally the methodologies developed willbe applied to real helicopter rotor simulations andvalidated with flight test data in an effort to demon-strate the practical applicability.

Methodology

The computations are performed using the exten-sively modified Transonic Unsteady Rotor Navier -Stokes (TURNS) research code, which has been ap-plied to a variety of helicopter problems [10, 11]. Afinite difference upwind numerical algorithm is used.In this algorithm, the evaluation of the inviscidfluxes is based on an upwind-biased flux-difference

scheme originally suggested by Roe [12] and later ex-tended to three-dimensional flows by Vatsa et al. [13]The chief advantage of using upwinding is that iteliminates the addition of explicit numerical dissi-pation and has been demonstrated to produce lessdissipative numerical solutions. This feature, cou-pled with a fine grid description in the tip region,increases the accuracy of the wake simulation. TheVan Leer monotone upstream-centered scheme forconservation laws (MUSCL) approach is used to ob-tain second or third order accuracy, with flux lim-iters to be total variation diminishing scheme.The LU-SGS scheme suggested by Jameson andYoon [14, 15] is used for the implicit operator.Briefly, the LU-SGS method is a direct modificationof the approximate lower-diagonal-upper (LDU) fac-torization to the unfactored implicit matrix. The re-sulting factorization can be regarded as the symmet-ric Gauss-Seidel relaxation method. The LDU fac-torization yields better stability than the simple LUfactorization because the diagonal elements alwayshave the absolute value of the Jacobian matrices.Though the (LU-SGS) implicit operator increasesthe stability and robustness of the scheme, the use ofa spectral radius approximation renders the methodonly first order accurate in time. Therefore, in thisstudy a second order backwards difference in time isused, along with Newton-type sub-iterations to re-store formal second order time accuracy.Also the present numerical scheme employs a modi-fied finite volume method for calculating the metrics.Finite volume formulations have the advantage thatboth the space and time metrics can be formed ac-curately and free stream is captured accurately [7].Also it is to be noted that the computations includeaeroelastic deformation and prescribed induced in-flow variation. So, the space and time metrics needto be recomputed at each time step to maintain therequired spatial and temporal accuracy. The compu-tation of the space and time metrics will be discussedin detail in the following section.

Calculation of Space and Timemetrics

The strong conservation-law form of the Navier-Stokes equations in cartesian coordinates can bewritten as [16]

qt + fx + gy + hz = σx + θy + ωz (1)

2American Institute of Aeronautics and Astronautics

q = (ρ, ρu, ρv, ρw, ρE)T

E = e+u2 + v2 + w2

2

f = (ρu, p+ ρu2, ρuv, ρuw, ρuH)T

g = (ρv, ρvu, p+ ρv2, p+ ρvw, ρvH)T

h = (ρw, ρwu, ρwv, p+ρww2, ρwH)T H = E+p/ρ

where u, v, w are the velocity components in thecoordinate directions x,y,z; ρ is the density, p is thepressure, e the specific internal energy; and σ, θ, ωrepresent the viscous stress and work terms for eachcoordinate direction. Upon transforming to compu-tational coordinates ξ, η, ζ with aid of chain rule ofpartial derivatives, Eq. 1 becomes:

qτ + fξ + gη + hζ = σξ + θη + ωζ (2)

q = Jq

f = ξtq + ξxf + ξyg + ξzh σ = ξxσ + ξyθ + ξzω

g = ηtq + ηxf + ηyg + ηzh θ = ηxσ + ηyθ + ηzω

h = ζtq + ζxf + ζyg + ζzh ω = ζxσ + ζyθ + ζzω

Here terms of form ξx,y,z, ηx,y,z and ζx,y,z are thespace metrics, ξt, ηt and ζt are the time metrics inthe computational domain, and J is the jacobianof the inverse coordinate transformation (i.e J =det(∂(x,y,z)

∂(ξ,η,ζ) ).

Field Velocity Approach

Mathematically, the field velocity approach can beexplained by considering the velocity field V in thephysical cartesian domain. It can be written as

V = (u − xτ )i+ (v − yτ )j + (w − zτ )k (3)

where u, v and w are components of the velocityalong the coordinate directions and xτ , yτ and zτ arethe corresponding grid time velocity component. Forthe flow over a stationary wing, these componentsare zero. For a rotor blade rotating about the z-axis, both xτ , yτ have non-zero values owing to therotation of the mesh. Let the velocity induced bythe external potential be represented by a velocityfield (u′, v′, w′). Thus, the velocity field becomes

V = (u− xτ + u′)i+ (v− yτ + v′)j + (w− zτ +w′)k

The field velocity approach models this changed ve-locity field by changing the grid velocities. The mod-ified grid velocities are defined as

xτ i+ yτj+ zτk = (xτ −u′)i+(yτ − v′)j+(zτ −w′)k

Once the modified grid velocities are obtained, thegrid time metrics in the computational domain (ξt,ηt ζt) are computed as:

ξt = −(ξxxτ + ξyyτ + ξzzτ )ηt = −(ηxxτ + ηyyτ + ηzzτ )

ζt = −(ζxxτ + ζyyτ + ζzzτ )

Geometric Conservation Law

The geometric conservation law has the same form asthe mass conservation law (first component of vectorEq. 2) as it essentially represents the conservation ofcell volumes. One can easily obtain the differentialform of the GCL from the mass conservation equa-tion by setting ρ=1 and V = (u, v, w) = 0:

Jτ + (ζt)ξ + (ηt)η + (ξt)ζ = 0 (4)

The integral form of the GCL can be obtained in asimilar manner and can be stated as:

ν(t2)− ν(t1) =∫ t2

t1

∮S(t)

Vs. dSdt (5)

where ν(t2) and ν(t1) are the initial and final vol-umes and Vs is the moving velocity of the cell sur-face. Equations 4 and 5 need to be satisfied in orderto make the numerical discretization strictly conser-vative. Otherwise, artificial sources and sink can begenerated as a result of the numerical discretization.Thomas and Lombard [4] proposed the solution ofEq. 4 with the same differencing scheme as that usedfor solving the flow conservation equations to main-tain the GCL. But this adds to the computationaloverhead of the calculation.A more practical approach is to compute the Jaco-bian J by its geometric definition and calculate thetime metrics according to the Geometric Conserva-tion Law [7]. For example, for a first order accuratetime metric calculation one could find the time met-rics in the computational domain as:

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ζt = −VSξ

∆t

ηt = −VSη

∆t

ξt = −VSζ

∆t

Where VSξ , VSη and VSζ are the volumes swept bythe faces of the cell volume in the computationalcoordinate directions. This approach provides amethodology for implicitly satisfying the GCL with-out solving any additional equations. But, in apurely finite difference approach one is posed withthe problem of identifying cell volumes. In otherwords, a strategy needs to be identified to definebounded volumes around mesh points which can beused for the computations of the Jacobian, spacemetrics and time metrics in a consistent manner

Defining Cell Volumes

To preserve the overall order of accuracy of the com-putations, the space and time metrics need to becalculated to the same order of accuracy as that ofthe underlying spatial and temporal numerical dis-cretization. Using a classical finite difference ap-proach, one calculates the transformation matrix∂(x,y,z)∂(ξ,η,ζ) and inverts it to find the space metrics. Thejacobian of transformation is the determinant of thistransformation matrix. The order of accuracy of thespace metrics can be improved by using a largerfinite difference stencil while evaluating individualterms of the above matrix. But, for surface conform-ing mesh geometries, it is possible to create over-shoots and undershoots (for eg a the trailing edge ofan airfoil) when arbitrarily increasing the finite dif-ference stencil, which causes divergence of the flowsolution at times. Hence, a modified strategy, oftenreferred as the dual refined mesh approach, whichcomputes the space metrics and jacobian in a finite-volume like way is proposed.Figure 1 and 2 show the volumes around a meshpoint for 2-D and 3-D respectively. The open sym-bols represent the mesh points of the original meshin which the computations of flow variables are per-formed. The closed symbols show the refined meshgenerated by one dimensional interpolation in thedirection of the computational coordinates. The re-fined mesh points are generated using a quadratic

monotone interpolation scheme proposed by Sureshand Hyunh [17]. The monotonicity of the interpo-lation scheme prevents the generation of overshootsand undershoots in the refined mesh geometry. Also,the fourth order interpolation makes the calculationof the Jacobians and space metrics 4th order accu-rate spatially, even though the computations of theseare made in a finite-volume like way.Once, the refined mesh is obtained, a volume whichis bounded by 24 faces (8 edges in 2-D) can be de-fined around the mesh point. The Jacobian of thetransformation to the computational coordinates atevery mesh point can be evaluated by consistentlyevaluating the volumes of these bounded cells. TheJacobian(J) in Eq. 2 is that of the inverse transfor-mation and can be found by taking the inverse ofthe volumes calculated.

Evaluation of Space metrics

The space metrics are gradients to ξ = const, η =const and ζ = const surfaces scaled by the Jacobianof inverse transformation. The definition of thesesurfaces are known from the refined mesh which wascomputed while defining the volumes. The gradi-ents of these surfaces can be evaluated at every meshpoint to find the space metrics.For the computation of the right hand side of theflow conservation equations in the discretized form,one often needs the space metrics at the interfacepoints (ξ = i ± 1

2 ,η = j ± 12 , ζ = k ± 1

2 ). Theseare actually surface normals to the bounded cell de-fined around the mesh point (graphically describedin Fig 1 and 2) which can be evaluated by computingthe gradient vector to the appropriate surface.

Evaluation of Time metrics

The evaluation of the time metrics require compu-tation of volumes swept by the faces of the boundedvolume. The time metrics should have componentsbecause of both the actual deformation of the meshand the external velocity field, which is included us-ing the field velocity approach. Figure 3 illustratesthe methodology of computation. The time metricin a particular computational coordinate directionis calculated by accumulating the volume swept bythe face because of actual deformation and the ap-parent volume swept because of the field velocitycomponents.The temporal order of accuracy can be improved byusing the volumes swept at previous time steps and

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Baseline mesh

Cell Volume

Refined mesh

(i,j)

ξ=

η=const

const

η

ξ

(i−1,j)

(i+1,j)

(i, j−1)

(i,j+1)

Figure 1: Cell volume in two dimensions

ξ

ηζ

ξη

ζ=const

=const=const

(i,j−1,k)

(i,j+1,k)

(i,j,k−1)

(i+1,j,k)

(i−1,j,k)

(i,j,k+1)

(i,j,k)

Baseline mesh

refined mesh

Figure 2: Cell volume in three dimensions

devising an appropriate backward difference stencil.To make the scheme strictly conservative, the Jaco-bian is recalculated by including the change in vol-ume because of the apparent contribution from thefield velocity components. The space metrics are stillevaluated as surface normals to the actual deformedmesh, although they are rescaled by the modified Ja-cobian. Thus, this approach presents a unique wayof satisfying the GCL together with the use of thefield velocity approach.

Cell face at time=t+ t∆

Cell face at time=t

Cell face because of

field velocity componentsApparent movement due to

V2V1

Volume swept = V1 + V2

Figure 3: Evaluation of volumes swept by cell faces

Sample Results

Step Change in Angle of Attack

Computations are performed to obtain responsesof various airfoils to a step change in angle of at-tack using the field velocity approach. The indicialresponse is a combination of noncirculatory (wavepropagation) and circulatory effects. For compress-ible flow, the noncirculatory component exponen-tially decays and the circulatory component grows toa final asymptotic value. A schematic of the problemis illustrated in Fig 4(a). Figure 4(b) shows the liftand pitching moment responses obtained from theCFD computations. The responses obtained fromthe CFD computations clearly show this trend. Itis possible to obtain closed form analytical solutionsto both the normal force and pitching moment re-sponses for small times for a flat plate in linearizedcompressible flow [18]. The expressions of normalforce and pitching moment coefficients are given by:

Cn(s)α

=4

M∞

[1− 1−M∞

2M∞s

]

Cm(s)α

=1

M∞

[1− 1−M

2Ms+

(1− M∞

2

)s2

2M∞

]

In these equations the quantity s represents the non-dimensional time, which is defined as the distancetraveled by the airfoil in semi-chords. Figures 5(a)and 5(b) shows the comparison of the CFD resultswith the exact analytical solution for 4 different air-foils for the small non-dimensional times (i.e in theregion the exact solution is valid). The results showgood correlation with the exact solution for bothlift and pitching moment. The correlation seems to

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t=0, α=0 t > 0, α=1

(a) Illustration of step change inangle of attack

0 5 10 15 20 254

5

6

7

8

s=2V∞t/c

Cn/α

0 5 10 15 20 25−2

−1.5

−1

−0.5

0

0.5

s

Cm

/α

(b) Normal force and pitching moment responses for

SC1095 airfoil (M=0.5)

Figure 4: Response to step change in angle of attackusing field velocity approach

degrade slightly with increasing thickness and cam-ber. The NACA0006 airfoil, which is the closest toa flat plate approximation shows excellent correla-tion, while the NACA0015 airfoil shows differenceswith exact solution, especially at the higher Machnumber.A more detailed description and results of this ap-proach for extracting indicial responses to build bothlinear and non-linear kernels for reduced order aero-dynamics models are documented in previous re-search efforts [19, 20, 21].

Interaction of an Airfoil with a Travel-ing Gust

The next unsteady flow problem studied is the aero-dynamic response of an airfoil penetrating througha traveling vertical gust. The schematic of the prob-lem is illustrated in Fig 6(a). The parameter thatquantifies the relative convection speed of the gustis termed the gust speed ratio (λ = V

V +Vg, V=free

stream velocity, Vg=gust convection speed).The exact analytical solution for the response to atraveling vertical gust in compressible flow for smallperiods of time 0 ≥ s ≥ 2M

1+M was developed us-ing Evvard’s [22] theorem and verified against Leish-

0 0.2 0.4 0.6 0.8 10

3

6

9

12

15NACA0006

Cn/α

CFD (M=0.3) Exact (M=0.3)CFD (M=0.5) Exact (M=0.5)CFD (M=0.8) Exact (M=0.8)

0 0.2 0.4 0.6 0.8 10

3

6

9

12

15NACA0015

Cn/α

0 0.2 0.4 0.6 0.8 10

3

6

9

12

15SC1095

s=2V∞ t/c

Cn/α

0 0.2 0.4 0.6 0.8 10

3

6

9

12

15SC1095R8

s

Cn/α

(a) CFD vs exact analytical (Normal force)

0 0.2 0.4 0.6 0.8−3.5

−3

−2.5

−2

−1.5

−1

−0.5C

m/α

NACA0006

CFD (M=0.3) Exact (M=0.3)CFD (M=0.5) Exact (M=0.5)CFD (M=0.8) Exact (M=0.8)

0 0.2 0.4 0.6 0.8−3.5

−3

−2.5

−2

−1.5

−1

−0.5

Cm

/α

NACA0015

0 0.2 0.4 0.6 0.8−3.5

−3

−2.5

−2

−1.5

−1

−0.5

s=2V∞ t/c

Cm

/α

SC1095

0 0.2 0.4 0.6 0.8−3.5

−3

−2.5

−2

−1.5

−1

−0.5

s

Cm

/α

SC1095R8

(b) CFD vs exact analytical (Pitching moment)

Figure 5: Correlation between computed and exactanalytical results for step change in angle of attack

man’s [23] results from the reverse flow theorem bySingh [24]. The exact analytical solutions are as fol-lows:

Cl(s) =2s√Mλ3

[λ

λ+ (1− λ)M

] 12

(6)

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Cmle(s) =

−s2

4√M3λ2

[√2M + (1 −M)λλ+ (1 − λ)M

](7)

Figure 6(b) shows the numerical prediction of theaerodynamic responses of an airfoil to a travelingvertical gust. The field velocity approach which wasdescribed earlier is used to implement the unsteadyflow conditions for predicting the gust response. Theeffect of the gust is simulated by suitably modifyingthe field velocities in the vertical direction. At anyinstant of time the gust front can be calculated fromthe gust speed ratio. Thus, the field velocity valuesat all grid points behind the gust front are modi-fied to be equal to the gust magnitude to simulate apropagating gust.The initial flow conditions for the unsteady sim-ulation is generated by obtaining a steady stateflow solution around the airfoil. Once the initialsteady state is obtained, the gust is initiated 15chords upstream of the airfoil and made to convectat the desired speed representing the gust speed ratio(Vg = V (λ−1 − 1)). The instant of time s = 0 cor-responds to the instant where the gust front meetsthe airfoil leading edge.For moving gusts the rate of increase of the normalforce is larger for the faster moving gusts. This isbecause of the acceleration effects associated withthe impulsive change in the velocity field over a por-tion of the airfoil. This additional lift, which is thenon-circulatory lift, decays rapidly and thereafterthe total circulatory lift builds up slowly towardsthe steady state. The steady state magnitude is thesame for all cases and is determined by the effectiveangle of attack caused by the vertical gust veloc-ity. In the limiting case of infinite gust propagationspeeds, the lift response corresponds to an indicialchange in the angle of attack of equivalent magni-tude (Wagner [25] problem). The case of zero gustspeed or penetration of the airfoil in to a stationaryvertical gust is called the Kussner [26] problem.The comparison of numerical solutions and exact so-lutions for four different gust speeds are presentedin Figures 7(a) and 7(b). The numerical solutionsshow excellent correlation with exact analytical re-sults especially for the stationary gust case (λ=1);for the faster and slower convecting gust cases thereare some deviations from the exact solution. Lowerorder aerodynamic models can be constructed forthis problem by extracting the indicial responses for

g

V

V g

w

(a) Schematic of interaction with a vertical gust

0 5 10 15 200

2

4

6

8

10

s=2V∞ t/c

Cl(s

)/ra

d

0 5 10 15 20−0.1

−0.05

0

0.05

0.1

0.15

Cm

(s)/

rad

s

(b) Aerodynamic response to a vertical gust

Figure 6: Response to a traveling vertical gust usingfield velocity apporoach

aerodynamic load time history obtained the CFDcalculations [20, 3].

Airfoil Vortex Interaction

Airfoil Vortex Interaction is an unsteady two dimen-sional problem in which a vortex convects perpendic-ular to the vorticity vector and influences the flowaround and airfoil (Figure 8). It is the 2-dimensionalequivalent of the parallel blade vortex interactionphenomenon observed in helicopter flight. BladeVortex Interaction (BVI) is one of the loudest andmost annoying sources of the rotor noise. The BVInoise propagates out of plane, usually forward anddown making it the most audible helicopter noise toan observer on the ground.The aerodynamic generating mechanism of BVInoise can be explained briefly as follows: When thevortex is upstream of the blade it induces a down-wash, and after the vortex passes by the blade anupwash is induced. The time varying vertical veloc-ity change produced during the passage of the vortexchanges the local angle of attack and causes a cor-responding fluctuation in the blade loading. Thus,a sharp impulsive noise signature is created by the

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0 0.4 0.8 1.2 1.6 20

1

2

3

4λ=0.8

s

Cl(s

)

0 0.4 0.8 1.2 1.6 20

1

2

3

4λ=0.9

s

Cl(s

)

0 0.4 0.8 1.2 1.6 20

0.5

1

1.5

2

2.5

3

3.5λ=1.0

s

Cl(s

)

0 0.4 0.8 1.2 1.6 20

0.5

1

1.5

2

2.5

3

3.5λ=1.1

s

Cl(s

)

CFD Exact Analytical

(a) lift coefficient

0 0.4 0.8 1.2 1.6 2−0.3

−0.2

−0.1

0

0.1

0.2λ=0.8

Cm

(s)/

rad

0 0.4 0.8 1.2 1.6 2−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15λ=0.9

0 0.4 0.8 1.2 1.6 2−0.1

−0.05

0

0.05

0.1

0.15λ=1.0

Non dimensional time, s

Cm

(s)/

rad

0 0.4 0.8 1.2 1.6 20

0.02

0.04

0.06

0.08

0.1λ=1.1

Non dimensional time, s

CFD Exact Analytical

(b) pitching moment

Figure 7: Correlation between computed and exactanalytical results for interaction of an airfoil with atraveling gust (M=0.6)

large rate of change of loading with time.There are various ways of simulating the airfoil vor-tex interaction. One of the popular methodologiesis called the perturbation approach [27], in which aflow field is decomposed into two parts despite non-

linearity: one is a prescribed vortical disturbanceknown to satisfy the governing equations, and theother obtained from the solution of the governingequations. As a result, the vortex structure is freeof numerical diffusion. The second methodology isto use an accurate vortex-preserving Euler/Navier-Stokes solver [28], in which the vortex solution iscomputed directly from the numerical solution ofthe governing partial differential equations. Thismethodology requires a high resolution grid to pre-vent excessive numerical dissipation. The field veloc-ity approach discussed earlier provides yet anothermethodology for simulating this problem. The gridtime metrics are modified to include the velocity fieldinduced by the vortex.The results of the aerodynamic loads obtained fromall three methodologies are presented in Figure 9.The correlation of the results from all three ap-proaches are excellent. The field velocity approachpresents a computationally efficient way of simulat-ing the problem, as it does not require a high resolu-tion grid and is computationally less complex com-pared to the perturbation approach.The vortex preserving approach requires solutionadaptive grid generation to prevent numerical dissi-pation. An adaptive grid generation causes changesin the volumes of grid cells with each time step. Theresults presented below are from the work of LeiTang [28], which clearly illustrate the necessity ofsatisfying the GCL in such computations.Figure shows the unadapted and adapted mesh forthe vortex preserving approach. If the geometricconservation law is not satisfied, an artificial sourceterm is created which leads to the radiation of pres-sure waves from the leading edge of the airfoil. Fig-ure 11, which shows the pressure contours whenthe vortex is about 5 chords upstream of the air-foil, clearly illustrates this phenomenon. Satisfyingthe GCL eliminates these spurious source terms andproduces a smoother and more accurate solution asshown in Figure 12.

Aerodynamics and Acoustics in LowSpeed Helicopter Forward Flight

Simulation of a real helicopter rotor blade in forwardflight, as mentioned earlier, requires the use of thefield velocity approach to prescribe the effects of thetrailed wake from the other rotor blades. The rotorblade motions which consists of pitch, flap and lagalso needs to be included.

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g

vZ

��������

V

V

Γ

Xv

Point vortex

Figure 8: Schematic of Airfoil Vortex Interaction

−8 −6 −4 −2 0 2 4 6 8−0.25

−0.20

−0.15

Non dimensional time (s)

−0.10

−0.05

−0.15

0

0.05

0.10

Convecting VortexPerturbation approachField Velocity Apporach

CL

Figure 9: Blade Loading time history (M=0.6, missdistance(Zv=-0.25c))

(a) unadapted (b) adapted

Figure 10: Grids in Vortex region

−5 −4 −3 −2 −1 0 1 2 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

y

x

Figure 11: Non dimensionalized pressure contourswithout satisfying the GCL

−5 −4 −3 −2 −1 0 1 2 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

y

x

Figure 12: Non dimensionalized pressure contourswith the time metrics satisfying the GCL

The trailed vortex wake geometry can be obtainedusing a prescribed empirical formulae [29] or canbe predicted using a free wake analysis using a La-grangian vortex lattice method [8]. The vortex wakegeometry obtained consists of a description of thelocus of the vortex filaments in space at discrete az-imuthal positions of the rotor blade. Often, the az-imuthal discretization of the wake geometry is muchcoarser (∆ψ = 5o) than the azimuthal discretizationused for the flow computations (∆ψ = 0.25o). Thisnecessitates the use of an appropriate interpolationtechnique to obtain the vortex wake geometries atthe finer discretization level. In this case, the in-terpolation is performed spectrally, taking in to ad-vantage the periodicity of the wake with azimuthalangle.The field velocity approach requires the evaluation ofinduced velocities caused by the vortex wake systemat every grid point in the computational domain.This increases the computational complexity of theproblem, as it is a O(M ∗ N) computation, whereM is the number of vortex filaments and N is thenumber of grid points. For a typical computation,

9American Institute of Aeronautics and Astronautics

0 60 120 180 240 300 360−0.075

−0.05

−0.025

0

0.025

0.05

Azimuth

Cl*M

2

Baseline Fast method Fast method w/ Free wake

0 60 120 180 240 300 360−0.075

−0.05

−0.025

0

0.025

0.05

Azimuth

Cl*M

2

r/R=0.75

r/R=0.95 #1

#2

#3 #4

#R1

#R2

Figure 13: Computed lift vs azimuth for a BVI prob-lem for the OLS rotor

there would be a need to compute induced velocitiescaused by over 500 vortex filaments at each of thehalf a million grid points used. The increased com-putational complexity undermines the practicality ofthis approach. Therefore, a performance enhance-ment technique which maintains the necessary ac-curacy is required to perform routine computationsusing this methodology.The performance enhancement is obtained using fasthierarchical algorithms [30, 31]. Briefly, these algo-rithms use the recursive subdivision of the computa-tional domain and a delineation of the near-field andfar-field regions. Hence, for any grid point, the in-fluence because of all the vortex filaments in its nearfield is evaluated exactly, while the effects of vor-tex filaments in the far-field are evaluated using anappropriate interpolation or multipole expansion ofthe far field potential. Figure shows comparison ofthe aerodynamic lift obtained with and without theperformance enhancement. The run-times could bereduced by an order of magnitude using the perfor-mance enhancement with very little loss of accuracy.

The aerodynamic lift in Figure is obtained by sim-ulating the same conditions as wind tunnel experi-ments conducted by Splettoser et al. [32]. The ex-periments were conducted for a scaled AH-1 (OLS)rotor with a tip Mach number of 0.664 and an ad-vance ratio (ratio of free stream to tip Mach num-ber) of 0.164. Also, the rotor plane is tilted at anangle of 1 degree backward from the free stream di-rection. Rigid blade motions (pitch and flap) that

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

#4

#3

#2 #1

#R1

#R2

Figure 14: BVI locations at µ=0.164 for the OLSrotor

are consistent with the experiment are included inthe flow computations. The BVI locations for sucha flight condition is shown in Figure . These loca-tions are found using the empirical definition of thevortex wake. It is evident from Fig that all the BVIhotspots are well captured by the CFD simulation.Also, the differences between using a free wake basedvortex wake geometry and an empirical vortex wakegeometry are illustrated. The vortex wake geometryfrom free wake shows smoother BVI load peaks com-pared to the empirically predicted geometry. Thiscan be because of three reasons. They are (1) be-cause of the larger miss distance (vertical distancebetween the rotor blade and interacting vortices),(2) vortex core diffusion modeling in the free wakeanalysis and (3) inplane distortion of the wake whichwould generate slightly different BVI hotspots.Figure shows the differential blade surface pressurevariation close to the leading edge (x/c=0.03) pre-dicted by CFD analysis as compared to the exper-imental measurement by Splettstoesser et al. Theanalysis gives good overall correlation with the ma-jor features observed in the experiment. The phaseof the impulsive loadings are captured accurately.However, there are differences in the pressure deriva-tives, especially in the retreating side. The acousticsound pressure levels are very sensitive to the rate of

10American Institute of Aeronautics and Astronautics

0 60 120 180 240 300 3600

0.2

0.4

0.6

0.8

1

Azimuth

−∆C

pM2

CFD using field velocityExperimental

0 60 120 180 240 300 360−0.2

0

0.2

0.4

0.6

0.8

Azimuth

−∆C

pM2

r/R=0.91

r/R=0.975

Figure 15: Differential blade surface pressures at(x

c = 0.03) at two radial stations

change of pressure (pressure derivatives). The bladesurface pressures obtained from the CFD analysiswas used to find the acoustic pressures at a specificmicrophone location (placed in front of the rotor at30 degree out of plane and a distance 1.72 diame-ter below). An sound pressure level was computedusing the acoustic prediction code WOPWOP [33]which solves the linear Ffowcs-Williams-Hawkingsequations [34]. The differences in the lift deriva-tives are reflected in the correlation of acoustic soundpressure levels. The analysis does give very goodprediction to the peak pressure, which determinesthe noise level in a dB scale. Also, the acoustic anal-ysis does not include the thickness noise componentsat this stage. This might be one of the reasons forthe absence of negative peaks in the predicted soundpressure levels

High speed forward flight case for theUH-60A (Blackhawk)

For a high speed forward flight case, the elastic de-formations are more important than the influenceof the vortex wake. The wake becomes relativelyunimportant as the trailed wake is swept fartheraway from the rotor plane because of the higher freestream velocity. Therefore, the elastic deformationof the rotor blade needs to be incorporated accu-rately.The satisfaction of GCL is a necessity to preventspurious oscillations in a solution which has rela-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−20

0

20

40

t*

Pa

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−20

0

20

40

*

Pa

Experiment

Analysis

Mic #3 30o

1.72D

Figure 16: OLS acoustic pressure time history atmicrophone #3

tively large grid deformations. The flight condi-tions chosen for the results shown is that of theUH-60A helicopter in high speed forward flight withV=155kts and a thrust level CT=0.00666 (Figure17). This particular test case was chosen becauseof the noted modeling issues such as phase errorin the negative lift and error in the peak to peakmagnitudes of predicted pitching moments (Bous-man [35]). Also, a plethora of aerodynamic flighttest data (Kufeld et.al [36]) is available for this caseto validate the numerical solutions computed.Figure 18 and 19 show the sectional oscillatory aero-dynamic loads computed with and without satisfy-ing the GCL correlated against the flight test data.The sectional normal force does not show large dif-ferences when the GCL is not satisfied. But theeffects of spurious source terms are prominent in thesectional pitching moments, which are more sensi-tive to the variation of the surface pressure distri-butions. The sectional pitching moment computedwithout satisfying the GCL shows both a change inpeak to peak magnitude and the presence of highfrequency oscillations, compared to the smoother so-lution predicted with the GCL being satisfied. Thetest data, as expected, correlates better with the so-lution which satisfies the GCL.Figures 18 and 19 also demonstrate the superioraerodynamic predication capability of the presentapproach compared to the table-lookup based lin-earized aerodynamic prediction SC-hems used in

11American Institute of Aeronautics and Astronautics

routine comprehensive rotorcraft aeroelastic analy-sis. A detailed validation of the CFD based pre-diction scheme and analysis of the flow physics andvibratory loads are presented in one of our previousworks [37].

Summary, Conclusion and Fu-

ture Work

This paper reviews a unique methodology for com-puting the unsteady flow, namely the field velocityapproach. A technique is developed for rigorouslysatisfying the Geometric Conservation Law when us-ing deforming meshes and the field velocity approachis also detailed. Computations of cell volumes andspace metrics to a fourth order accuracy and con-sistent evaluation of time metrics for a strictly con-servative solution are discussed. Results presentedinclude representative two dimensional calculationsfor validating and evaluating the merits of the fieldvelocity approach. Results for more complex real-istic problems such as prediction of aerodynamicsand acoustics of helicopter rotors in forward flightare also presented.The following are the major conclusion drawn fromthis study:

• The field velocity approach is found to be an ac-curate and efficient way for modeling unsteadyflow problems. The results from the computa-tions using field velocity approach show goodcorrelation with both exact analytical results(2-D cases) and experimental data (3-D case).

• It was found that rigorously satisfying the Ge-ometric Conservation Law is required for prob-lems which have large grid deformations and/orlarge changes introduced in the time metrics be-cause of the field velocity approach

Future work will use the field velocity approach toexamine the flow features of the three dimensionalblade vortex interaction problem in more detail.Grid and time step dependence studies will be per-formed to ensure the accuracy and fidelity of thesolution. The acoustic computations would be per-formed once the high fidelity airloads are obtained.A comprehensive validation of sound pressure levelsat more microphone locations (both inplane and outof plane) are also intended.

155 kts

Tip vortices shed

Figure 17: UH-60A blackhawk in forward flight

0 60 120 180 240 300 360−40

−30

−20

−10

0

10

20

30

40

Azimuth

Nor

mal

forc

e (lb

/in)

Flight TestNo GCL With GCL Linear Aero

Figure 18: Sectional lift at r/R=0.775 for the UH-60A rotor (V=155 kts)

0 60 120 180 240 300 360−60

−50

−40

−30

−20

−10

0

10

20

30

40

Azimuth

Pitc

hing

Mom

ent (

lb−

in/in

)

Flight TestNo GCL With GCL Linear Aero

Figure 19: Sectional pitching moment at r/R=0.775for the UH-60A rotor (V=155 kts)

12American Institute of Aeronautics and Astronautics

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[18] Lomax, H., “Indicial Aerodynamics,” AGARDManual of Aeroelasticity, Pt. II, Nov. 1968,Chap. 6.

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[20] Singh, R. and Baeder, J. D., “The Direct cal-culation of Indicial Lift Response of a Wing

13American Institute of Aeronautics and Astronautics

Using Computational Fluid Dynamics,” Jour-nal of Aircraft, Vol. 35, No. 4, pp.465-471,1997.

[21] Lee, D., Leishman, J., G. and Baeder, J. D.,“A Nonlinear Indicial Method for Calculationof Unsteady Airloads,” 59th Forum of theAmerican Helicopter Society, Phoenix, Ari-zona, May 2003.

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[28] Tang, L., “Improved Euler Simulation of He-licopter Vortical Flows,” PhD Dissertation,University of Maryland, 1998.

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[31] Barnes, J., E. and Hut, P., ”A Hierarchi-cal Force Calculation Algorithm,” Nature, 324(1986) 446–449.

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14American Institute of Aeronautics and Astronautics