Unstructured Overset Grid Adaptation for Rotorcraft AerodynamicInteractions

Rajiv Shenoy ∗ Marilyn J. Smith †

School of Aerospace Engineering, Georgia Institute of TechnologyAtlanta, Georgia 30332-0150

A new adaptation strategy is presented that permits time-dependent anisotropic adaptation for dy-namic overset simulations. The current development permits adaptation to be executed over a periodictime window in a dynamic flow field so that an accurate evolution of the unsteady wake may be obtainedwithin a single unstructured methodology. Unlike prior adaptive schemes, this approach permits gridadaptation to occur seamlessly across any number of grids that are overset, excluding only the bound-ary layer to avoid surface manipulations. Demonstrations on rotor-fuselage interactions, including flowfield physics, time-averaged and instantaneous fuselage pressures, and wake trajectories are included.The ability of the methodology to improve these predictions without user intervention is confirmed,including simulations that confirm physics that have before now, not been captured by computationalsimulations. The adapted solutions exhibit dependency based on the choice of the flow field feature-based metric and the number of adaptation cycles, indicating that there is no single best practice forfeature-based adaptation across the spectrum of rotorcraft applications.

Introduction

The resolution of unsteady wake features is essential for amultitude of aeroelastic applications pertinent to rotorcraft.These include, but are not limited to modeling of aeroelas-tic rotor blades in forward flight, rotor-fuselage or rotor-rotor interaction, helicopter-ship interaction, and tail buf-fet. Other applications of interest can include wing-storeseparation, ship-wake interactions, and wind turbines. Thecurrent most popular computational fluid dynamics (CFD)approach to resolve these multiple reference frame applica-tions is via overset grids, where the moving body or compo-nent meshes are generally highly refined and overset on oneor more static background grids (Refs. 1–5). Despite theability of unstructured overset methods to model dynamicbodies, it does not address the issue of numerical dissipationthat can result in inaccuracy of the wake physics (Refs. 5,6).

Feature-based grid adaptation for unsteady problems hasbeen applied on single grids using various methodologies.Accurate predictions of hovering rotors in a single rotatingadaptive mesh have been performed by several researchers(Refs. 7–10). However, these scenarios can not be imme-diately applied to the prediction of rotors in forward flight,where adaptation is needed in both the background iner-

∗Graduate Research Assistant, Corresponding Author,rajiv.shenoy@gatech.edu, (404)894-3018

†Associate ProfessorPresented at the American Helicopter Society 67th AnnualForum, Virginia Beach, VA, May 3-5, 2011. Copyright c©2011 by the American Helicopter Society International, Inc.All rights reserved.

tial reference frame and the near-body rotating frame. Fur-ther, the interaction of rotors with non-moving bodies suchas fuselages, wind-tunnel struts, and other configurationalcomponents also require moving-grid capability to simu-late multiple motion frames. As an alternative to the oversetconfiguration, Park and Kwon (Ref. 11) have demonstratedan unstructured sliding mesh approach where a rotating gridcommunicates with a stationary background grid. Here, ar-ticulation of the rotor blades was made possible using griddeformation based on a spring analogy, and also appliesfeature-based grid adaptation. Another non-overset basedapproach has been described by Cavallo et al. (Ref. 12),which uses unstructured grid movement and deformation toenable moving body adaptation.

Past research efforts in overset adaptation have in manyinstances relied on an off-body (background) Cartesiangrid-based adaptive capability. Meakin (Ref. 13) presenteda grid component grouping algorithm with overset struc-tured grids using a method of adaptive spatial partition-ing and refinement and applied it to background Cartesiangrids. Variations of this technique have been subsequentlydemonstrated by Henshaw and Schwendeman (Ref. 14) andKannan and Wang (Ref. 15). Canonne et al. (Ref. 16)used an overset structured cylindrical grid topology to sim-ulate rotor motion in hover where the background grid isadapted. Hybrid-solver developmental efforts have focusedon rotor methodologies where two separate solvers are ap-plied in the near body and background regions, respec-tively. Duque et al. (Ref. 17) have employed a structurednear-body and unstructured wake grid approach to evaluaterotors. Here, isotropic adaptation was applied on the un-

structured background grid operating in a non-inertial refer-ence frame. This work recommended the use of anisotropicadaptation to accurately capture inherently anisotropic phe-nomena such as tip-vortices and to exploit computationallyefficiency of this grid adaptation technique. Park and Dar-mofal (Ref. 18) introduced a parallel anisotropic adaptationcapability for non-overset tetrahedral grids. This techniquehas been applied to investigate several applications such assonic-boom propagation (Refs. 19, 20), viscous transonicdrag prediction (Ref. 19), and re-entry vehicle configura-tions (Ref. 21). Recent development has focused on thecoupling of a body-fitted unstructured solver with a high-order Cartesian solver to propagate the wake in the midand far fields. Sankaran et al. (Ref. 22) and Wissink etal. (Refs. 23, 24) have successfully implemented automaticmesh refinement (AMR) in the Cartesian background solverto resolve the wake based on flow field features.

The effect of time-dependency on the flow field is an es-sential aspect in applying grid adaptation to study dynamicmoving bodies. Researchers (Refs. 12, 16) have shownthat adapting the solution at a given frequency (based onflow time) have proven effective, with increasing frequencyyielding higher accuracy. Investigations involving off-bodyCartesian-based adaptive mesh refinement (Refs. 14,15,22–24) have extended this rationale to adapt the solution a fre-quency comparable to that of the solver time step, allowingfor a coupled adaptive flow solver. However, this capabilityis not computationally efficient for tetrahedral unstructured-based methods as they do not have the advantage of octreedata structures to provide for the adaptive mechanics.

An alternative approach is thus required which addressesthe time-dependency issue without the computational over-head of frequent adaptation. Kang and Kwon (Ref. 8)present an adaptation technique that detects local maximaof a vortex core every five degrees and using a 3-D parabolicblended curve to represent the vortex core path. Park andKwon (Ref. 11) describe a ’quasi-unsteady’ adaptive pro-cedure for rotors in forward flight based on a time periodor window dependent on the blade passing frequency. Cellssatisfying an adaptation indicator are marked at each timestep within the window and adaptation is performed forthose cells at the conclusion of each window. Sterenborget al. (Ref. 25) describes a similar technique and applied itfor a fluid-structure interaction investigation. The extensionof a time-dependent feature-based adaptation methodologyfor anisotropic grids involving dynamic bodies is delineatedby Alauzet and Oliver (Ref. 26).

These successful implementations of grid adaptationprovide impetus for further investigation of this approachfor rotorcraft applications involving multiple grids. What iscommon across these prior overset-based efforts is that theadaptation is restricted to the off-body background meshes.Since vorticity originates on a viscous surface where thenear-body grid is employed, the full capability of the adap-tation cannot be exploited unless the adaptation can occur

across the meshes. In addition, for rotorcraft configurations,a time-dependent strategy is necessary to permit accurateand efficient simulation of the wake growth.

This paper presents a new adaptation strategy that per-mits time-dependent anisotropic adaptation for dynamicoverset simulations. Since an unstructured backgroundgrid can encompass many complex stationary bodies (e.g.fuselage, nacelle, and tower) the overset grid refinementcomplexity inherent in a structured grid-based approach isavoided. The current development dictates adaptation to beexecuted over a periodic time window in a dynamic flowfield so that an accurate evolution of the unsteady wake maybe obtained within a single unstructured methodology.

Two configurations that continue to elude accurate simu-lation in one or more areas with current CFD methods werechosen to evaluate the new adaptation strategy. These twoconfigurations fall under the category of rotor-fuselage in-teraction (RFI), an aeroelastic phenomenon that is charac-terized by an unsteady rotor wake impinging on an inertialfuselage. These two cases permit both inviscid and viscousoptions to be rapidly evaluated.

Methodology

Flow Solver

FUN3D, NASA’s unstructured RANS solver, was selectedas the flow solver with which to demonstrate the new adap-tation strategy. FUN3D utilizes an implicit, node-basedfinite volume scheme to resolve the RANS equations onunstructured, mixed-topology grids (Ref. 27).Both com-pressible and incompressible (Ref. 28) Mach regime ca-pabilities are available in the flow solver. Time-accuracyis achieved using a second-order backward differentia-tion formula (BDF). Roe’s flux difference splitting scheme(Ref. 29) is used compute the inviscid fluxes, while anequivalent central difference approximation is utilized to re-solve the viscous fluxes. A Gauss-Seidel strategy is usedto solve the resulting linear system of equations. FUN3Dhas available a plethora of turbulence methods available, ofwhich Menter’s kω-SST (Ref. 30) 2-equation model wasapplied in this effort. SUGGAR++ (Ref. 31) and DiRTlib(Ref. 32) provide overset capability with FUN3D and havebeen successfully used for compressible and incompress-ible rotorcraft applications (Refs. 3,5). In such simulations,the background grid, which consist of the fuselage and otherwind-tunnel static geometries up to the far-field boundaries,are assembled with finer near-body grids for each of themoving rotor blades.

Anisotropic Feature-Based Adaptation

FUN3D’s anisotropic tetrahedral adaptation capability(Refs. 18,33) formed the basis for the new adaptation strat-egy. This feature-based adaptation requires the identifica-tion of a feature, as well as an algorithm or key to define the

grid modification. In this effort, several different featureswere explored, including vorticity and pressure gradients.The vorticity adaptation key, Ke,ω , is similar to the one suc-cessfully applied by (Ref. 17), which scales vorticity (ω)with an edge length, `e. The formulation of the vorticity-based key for a given edge connecting nodes n1 and n2 iscomputed for the averaged vorticity magnitude across theedge,

Ke,ω = `e|ω|n1 + |ω|n2

2. (1)

A comparable pressure gradient key, Ke,p, was also definedas the magnitude of the pressure gradient ∆p scaled by thelength of the cell edge (`e) as

Ke,p = `e|∆p|. (2)

Using one of these keys, the normalized local adaptationintensity, I, is derived for each node as the maximum of theedge key, Ke, over all incident edges of a given node,

I = maxedges

(Ke

Kt

), (3)

where Kt is a user-specified tolerance. The new isotropicmesh spacing is calculated using an estimate of the spacingon the original mesh h0, a coarsening factor C (typicallyaround 115%), and the adaptation intensity by,

h1 = h0 min

(C(

1I

)0.2)

, (4)

Consequently, an anisotropic adaptation metric may be de-rived using a scalar quantity for the isotropic spacing anda Hessian to stretch the resulting mesh. The Hessian of aquantity (−) can be described as

H =

∂ 2(−)

∂x2∂ 2(−)∂x∂y

∂ 2(−)∂x∂ z

∂ 2(−)∂x∂y

∂ 2(−)∂y2

∂ 2(−)∂y∂ z

∂ 2(−)∂x∂ z

∂ 2(−)∂y∂ z

∂ 2(−)∂ z2

. (5)

Further details of the computation Hessian-based metricand its significance to the adaptation process are delineatedin (Ref. 33). The vorticity-based adaptation invokes thevorticity-magnitude Hessian to determine anisotropy, whilethe pressure gradient adaptation utilizes the Mach numberHessian, described in (Ref. 21).

Extension to Overset Grids

The overset grid adaptation capability does not restrictadaptation to any component grid. This enables each gridto evolve independently and, in general, ensures for an or-phan free composite grid. The current adaptation capa-bility for viscous flows is restricted only to nodes beyondthe boundary layer. FUN3D applies an adaptation software

module that computes the adaptation metric as well as han-dles all associated adaptive mechanics. Description of theparallelized adaptation mechanics, which include grid op-erations such as node insertion and removal by splittingor collapsing edges, edge and face swapping, and nodesmoothing are detailed in Ref. 33. The extension of thismethod to include overset adaptation requires communica-tion with DiRTlib (Ref. 32), the grid connectivity module,to assign a component mesh ID for each node in the com-posite mesh. The code performs adaptation over the entirecomposite grid system by tracking the component mesh IDfor all added nodes.

Since overset assembly of the component meshes is han-dled by a library outside of the FUN3D framework (SUG-GAR++) (Ref. 31), a generalized global index conventionwas requisite so that subsequent assembly of the adaptedgrid with its domain connectivity information would becompatible with the solution information. This process isrequired to perform valid solution transfers between the un-adapted and adapted grid systems. The convention requiresboth the flow solver and adaptation code to assign com-posite grid global indexes by arranging nodes in contigu-ous fashion by mesh ID over the list of component meshes.Nodes added due to adaptation are initially assigned newglobal indexes by appending them to the current global in-dex list. Node removal results in unused global indexes,which is handled by a reverse, global-index shifting proce-dure. In order to satisfy the condition of contiguous meshIDs, a new procedure was introduced to re-sort the globalindexes of the adapted grid system as illustrated in Fig. 1.After adaptation, the component meshes are then saved, andthe resultant domain connectivity information is obtainedby invoking SUGGAR++ for subsequent grid assembly.

(a) Node removal (struck through) and insertion

(b) Resorted global indexes

Fig. 1. Global index convention illustrated for an exam-ple overset grid system.

Time-Dependent Adaptation

Time-dependent adaptation is obtained using a method-ology based on that developed by Alauzet and Olivier(Ref. 26). The anistropic grid metric is computed for ev-ery grid node at a given time step and is progressively inter-sected over a selected time window such that the strongestrestrictive metric at each node is retained to form the time-dependent grid metric. The Hessian, H, is intersected intime by collecting N solution samples within a time win-dow w as given by,

|Hw,max|=N⋂

n=1

|Hw,n|, (6)

Using the resulting metric, a new adapted mesh may beobtained suitable for multiple time intervals characterizedby the flow phenomena obtained within the adaptive win-dow. For rigid-body rotorcraft simulations, such a win-dow is identified as the time corresponding 1/nblades rev-olutions. In this study, the solution is sampled at each timestep throughout the time window to obtain the metric inter-section. The present approach does not utilize a solutiontransfer capability; hence, the inherent solver accuracy isretained as a result of re-simulation of the unsteady physics.

Georgia Institute of Technology (GIT) RFIDemonstration Case

The GIT fuselage configuration is comprised by a cylin-drical fuselage and a hemispherical nose to permit easieridentification of RFI has been extensively evaluated in theHarper Wind Tunnel (Ref. 34). The rotor blades have a rect-angular planform with a NACA-0015 airfoil section. Therotor blades are nearly rigid which allow for CFD analy-ses that neglect structural deformations. Two advance ratios(µ = 0.10 and µ = 0.20) are selected for investigation andthe relevant blade angles and thrust are reported in Table1. Data from this effort include instantaneous and time-averaged pressures along the fuselage, as well as vortex be-havior via laser light sheets. The fuselage length is non-dimensionalized (x/R) by the rotor radius (R = 457mm) forease in presentation.

This model has been evaluated by numerous prior com-putational efforts with a variety of approaches, for exampleRefs. 1, 2, 35–37. O’Brien (Ref. 3) used this as a valida-tion case for his series of actuator to overset rotor modelsimplemented into FUN3D. Numerous details of the time-averaged fuselage pressure coefficient have not been cap-tured by these methods, in spite of the simplistic model ge-ometry. O’Brien (Ref. 3) noted that some time-averagedfeatures just aft of the rotor were captured when the en-tire model (rotor strut and hub) were included, which priorefforts neglected. However, the vortex interaction observedin the original experiments near the nose (x/R 0.3) (Ref. 34)

Table 1. Blade angles and resulting thrust obtained forGIT test cases. (Note: All angles are reported in de-grees.)

µ β1s β1s CT0.10 -2.02 -1.94 0.0090450.20 -2.62 -3.29 0.009950

has not been adequately resolved by any of these prior sim-ulations.

Simulations were computed using the compressible, in-viscid equation set as separation and other viscous effectsshould be minimal for the configurations chosen. Solu-tion advancement was performed with a time step equiv-alent to 1◦ azimuthal sweep. During each time-step, 25sub-iterations were used in conjunction with the temporalerror control option to ensure 1-2 orders of magnitude re-duction in residual. The metric intersection was performedover a time window corresponding to 180◦ blade sweep or180 steps after the solution became periodic (after 2 revo-lutions). In order to obtain valid comparisons with experi-ment, the thrust values obtained for these cases were ascer-tained to be within 1% of those listed in Table 1.

Fig. 2. Model of the GIT rotor-airframe configuration.

Several feature-based adaptations were evaluated for theGIT RFI model to determine the validity of the method, aswell as the appropriate flow field metric. The µ = 0.1 casewas examined first, and the most successful metrics weredetermined to be the vorticity magnitude and pressure gra-dient, as shown in Table 2. The pressure gradient adapta-tion (Scheme C), resulted in a much more efficient adapta-tion than the vorticity magnitude (Scheme B), as denotedby the mesh node count. Only minor differences betweenthese two metrics were observed for the time-averaged fuse-lage pressure coefficient peaks at the x/R = 0.5 and 2.0locations (Fig. 3) after one cycle. The result of the firstadaptation was to refine the initial vortex interaction at thenose (x/R = 0.1), so that the magnitude and pressure riseare more adequately captured. The effects of the vortexshed from the second blade, observed in the pressures along

Table 2. Summary of the different adaptation metricsfor the GIT configuration at µ = 0.1

Scheme Description Grid SizeA Initial Grid 2.03M nodesB Adapted to |ω| (1 cyc.) 6.68M nodesC Adapted to ∇p (1 cyc.) 2.92M nodesD Adapted to |ω| (2 cyc.) 12.8M nodesE Adapted to |ω| (1 cyc.) &

∇p (1 cyc.)8.86M nodes

x/R = 0.2− 0.5, are not captured with either the baselineor the single adaptation. The vortex-fuselage interaction atx/R = 2.3 is captured by all three meshes as a weak pres-sure pulse, and the magnitude does not change with gridadaptation, although a minor (x/R < 0.05) shift forward isobserved upon mesh adaptation.

Fig. 3. Comparison of the time-averaged pressure dis-tributions on the top centerline of the GIT airframe re-sulting from 1 adaptive cycle for µ = 0.1.

A second adaptation cycle was performed on the twometrics singly, as well as a cross adaptation (Table 2). Theresults of the second adaptation are much more significant(Fig. 4). For clarity, the double adaptation with pressuregradient is not shown as it is similar to that of the doubleadaptation with the vorticity magnitude (Scheme D). Whilethe double adaptation on the single metric results in a muchlarger mesh size, its influence on the fuselage pressure char-acteristics is minimal. Conversely, adaptation on the vortic-ity magnitude, followed by a second adaptation on the pres-sure gradient (Scheme E) yields significant improvement,and employs about 75% of the mesh required by SchemeD. While the pressure rise associated with the vortex fromthe second blade is greater than the experimentally mea-sured value, the fuselage-vortex interaction described byBrand (Ref. 34) and denoted by the sharp rise in pressurebetween x/R = 0.4−0.5 is captured.

Further examining the instantaneous flow field (Figs. 5- 9), the characterizations noted in the time-average pres-

sures are re-enforced in these data. For the first quarterrevolution where the rotor blades approach and pass overthe fuselage (Ref. 5), the single and double adaptationson vorticity magnitude (Schemes B and D) improve the in-stantaneous pressure prediction only slightly, and includea significant lag (x/R = 0.05− 0.1) in the location of thefuselage interaction at x/R = 0.4− 0.5. The mixed dou-ble adaptation (Scheme E) still encounters some issues withmagnitude (ψ = 30◦ and 60◦) and a shift in phase of thisvortex interaction, particularly at ψ = 90◦, however the re-sults are significantly improved over the single adaptation.It is also important to note that the magnitude and characterof the pressure rise resulting from the vortex-fuselage in-teraction, which is completely missed by the baseline andother adaptation schemes, are overall well-captured. Minordifferences are observed in the centerline pressure exclusiveof this vortex interaction.

The second quarter of the rotor revolution continuesoverall this trend, but with increasing differences with ex-periment as the two rotor blades move past the fuselageto their original positions. Again the vortex interactionat x/R = 0.4− 0.5 continues to be better captured withScheme E. The experimentally observed vortex-fuselage in-teraction between x/R = 0.7−0.9 during this quarter is notcaptured by the vorticity adaptation. The mixed vorticitymagnitude-pressure gradient adaptation appears to capturethis feature early and as a weaker interaction at ψ = 120◦

and 150◦. The significant over pressure near the nose(x/R = 0.2) at ψ = 180◦ when the blades are directly overthe fuselage is not predicted using any method, indicatingthat additional refinement and viscous effects may be im-portant at this location. Once again, however, it is evidentthat the mixed adaptation (Scheme E) preserves more of theflow features.

Fig. 4. Comparison of the time-averaged pressure dis-tributions on the top centerline of the GIT airframe re-sulting from 2 adaptive cycles for µ = 0.1.

To further understand the significance of the grid refine-ment in each scheme, the vortex behavior is examined inFigs. 7 – 9. The schemes A–E (Fig. 2), are plotted fromtop to bottom at each selected azimuthal location. It is

(a) ψ = 30◦

(b) ψ = 60◦

(c) ψ = 90◦

Fig. 5. Comparison of instantaneous pressure distri-butions on the top centerline of the GIT airframe forµ = 0.1 (first quarter revolution).

clear from scanning from top to bottom that the vortex coreis more crisply predicted after one adaptation (second andthird plots) and refined further after the second adaptation(fourth and fifth plots). In addition, weak vorticity interact-ing with the fuselage, spread over large areas, become fur-ther defined with the adaptation. In Fig. 9 (b), at ψ = 180◦

where the blade is present over the fuselage centerline, theblade-vortex interaction is clearly captured with the meshadaptation in the near-body and fuselage grids.

Differences between the various adaptation schemes canalso be discerned from the magnitude and shape contoursof the vorticity in these figures. For example, the shapeof the vorticity between x/R = 0.4− 0.5 where the firstfuselage-vortex interaction occurs is very different. Specif-ically, tracing Scheme E (the bottom plot) across Figs. 7– 9, it is possible to discern the path of the tip vortex as itleaves the blade, interacts with the blade root vortex, and fi-nally collides with and encompasses the fuselage centerline.

(a) ψ = 120◦

(b) ψ = 150◦

(c) ψ = 180◦

Fig. 6. Comparison of instantaneous pressure distri-butions on the top centerline of the GIT airframe forµ = 0.1 (second quarter revolution).

Brand (Ref. 34) reported that the tip vortex from the priorblade could be observed at x/R = 0.3 at ψ = 188◦, which iscomparable to the grid adaptation results in Fig. 9 (b). Asthe blade moves across the fuselage, Brand reports that thevortex weakens (remaining approximately stationary aboutx/R = 0.3), while the vortex sheet from the blade passageis swept downstream. By ψ = 225◦, Brand notes that ”...vortex sheet appears to generate a region of circulatory flowwith a sense opposite to the tip vortex ” and ”...for ψ = 224◦

where the vortex sheet is no longer visible, but the remnantof the streakline appears to swirl as it travels rapidly down-stream along the top of the airframe.” This correlates to theazimuths between Figs. 7 a) and b), where weak vorticityis observed at approximately x/R = 0.3, correlating to theremnants of the preceding blade’s tip vortex. The vortexsheet roll-up, which was experimentally observed to travelbetween x/R = 0.4−0.5 during this azimuthal time periodcan also been observed traveling downstream at ψ = 30◦

(or ψ = 210◦) and with a distinct rotation by ψ = 60◦ (orψ = 240◦), located in the same fuselage locations.

Brand’s observations describe the development of thesecondary, downstream fuselage-vortex interaction that arealso observed in Scheme E, unlike the other adaptationschemes. Unfortunately, further experimental evaluation ofthis vortex and its interaction with the fuselage were notprovided in Ref. 34. The track of the vortex is clearly ob-served throughout the azimuthal sequence after its devel-opment. The lack of correlation for the fuselage centerlinepressure between the experiment and Scheme E (Fig. 5)appears to be due to the height of the vortex above the fuse-lage, which can be attributed to the nature of the inviscidCFD calculations.

The ability of the new scheme to capture complex un-steady features in the flow field is confirmed in Fig. 10. Thetop figure (Scheme A) illustrates the baseline grid, and thedistinction between the background and blade grids can beeasily observed. After one adaptation (Scheme B), the bladevortex interaction is more more defined, and the featuresare preserved as they cross the blade boundaries. SchemeE, which includes a second adaptation cycle on a differentfeature metric, further illustrates the ability of the schemeto capture and preserve complex features.

Finally, the influence of the grid adaptation on the flowfield character is illustrated in Fig. 11, where Q-criterioniso-surfaces, colored by vorticity magnitude are presented.Not only are the vortex magnitudes preserved and tracksmore crisply defined with the grid adaptations, but a num-ber of distinct flow field patterns, absent with the baselinegrid (11 a) can be observed. The blade vortex interaction iscaptured and propagates over several revolutions, while thespiral wake shed from the fuselage is also now present.

For comparison, a higher advance ratio of µ = 0.20 (Ta-ble 1) was also examined. Similar results were observedfor the adaptation process, as illustrated for the single vor-ticity magnitude grid adaptation (Scheme B) and doublevorticity magnitude-pressure gradient adaptation (SchemeC). Figure 12 illustrates the mean pressures, which showslittle improvement with grid adaptation compared with ex-periment, especially at the x/R = 1 location. However, asnoted by Brand (Ref. 34) and confirmed computationally byO’Brien (Ref. 3), this deficiency in the pressure correlationis due to the intricacies of the hub pin rotation, which isnot modeled in this effort. A better evaluation of the abilityof the adaptation at this advance ratio is the examination ofthe unsteady peak maximum and minimums, shown in Figs.13 and 14, respectively. The grid adaptation clearly im-proves the prediction of the maximum pressure peaks nearthe fuselage nose (Fig. 13), but little change is observedelsewhere along the fuselage centerline. The unsteady peakminimums (Fig. 14) indicate that the double adaptation ofvorticity magnitude and pressure gradient improve signif-icantly the area of interaction of the prior tip vortex andfuselage, as similarly noted for the lower advance ratio case.

(a) ψ = 30◦ (b) ψ = 60◦

Fig. 7. Vortex behavior for the various adaptationschemes along the GIT fuselage symmetry plane at µ =0.1. (From top to bottom: Schemes A-E)

The interaction of the strut and hub model on the fuselageinterference effects becomes more important at this advanceratio, as indicated by the correlations here.

The efficacy of the overset, time-accurate grid adapta-tion capability has been demonstrated, but the solution de-pendency on the selection of the grid feature adaptationmetric is clear. In addition, the uncertainty in number ofadaptation cycles suggests that a method to define conver-gence (rather than additional adaptation cycles) needs to be

Table 3. Summary of the different adaptation schemesutilized for µ = 0.2

Scheme Description Grid SizeA Initial Grid 2.03M nodesB Adapted to |ω| (1 cyc.) 6.99M nodesC Adapted to |ω| (1 cyc.) &

∇p (1 cyc.)12.7M nodes

(a) ψ = 90◦ (b) ψ = 120◦

Fig. 8. Vortex behavior for the various adaptationschemes along the GIT fuselage symmetry plane at µ =0.1. (From top to bottom: Schemes A-E)

determined. The application of adaptation using adjointmethods may be a solution. Adjoint adaptation has beendemonstrated for steady, single grids by Park and associatedauthors (Refs. 18, 33, 38). Extension of adjoint adaptationto include overset, time-accurate simulation capability maybe warranted.

ROBIN Wake Visualization Demonstration Case

The second demonstration case is based on the ROtor BodyINteraction (ROBIN) configuration, developed by NASA,which has been extensively used in various experimentsand computational studies on rotor-fuselage interactionsand wake trajectories (Refs. 5, 39–42). This streamlinedslender ROBIN fuselage model, described by a set alge-braic equations at various fuselage locations, yields a sim-ple analytical definition for a fuselage geometry. An enginemount or ”doghouse” is a characteristic feature included inthe configuration. An internally mounted rotor system isutilized consisting of four blades that are fully-articulated

(a) ψ = 120◦ (b) ψ = 180◦

Fig. 9. Vortex behavior for the various adaptationschemes along the GIT fuselage symmetry plane at µ =0.1. (From top to bottom: Schemes A-E)

with a NACA-0012 airfoil section. The current effort fo-cuses on the set of experiments conducted by Ghee andElliott (Ref. 40) in the 14-by-22-Foot Subsonic Tunnel atNASA Langley Research Center using the 2-m rotor testsystem (2MRTS). One of the experimental wake visualiza-tion cases (Table 4) was employed here to evaluate the in-fluence of the adaptation process. Once again, the lengthdata have been nondimensionalized by the rotor radius (R)to facilitate ease in interpreting the simulations.

The compressible viscous option within FUN3D wasapplied to the ROBIN demonstration. Similar to the GITcomputations, the time step equivalency of 1◦ was chosenwith 25 sub-iterations. Ten turbulence sub-iterations werealso applied to converge the loosely-coupled Menter’s kω-SST turbulence model. Here the time window for the time-dependent metric corresponded to 90◦ blade sweep or 90steps. Trim angles that were previously obtained on thesame grid in Smith et al. (Ref. 5) were used to estimate therequired trim. The thrust values obtained were ascertained

(a) Scheme A

(b) Scheme B

(c) Scheme E

Fig. 10. Mesh colored by vorticity contours demonstrat-ing BVI-effect for the GIT configuration at µ = 0.1.

(a) Scheme A

(b) Scheme B

(c) Scheme E

Fig. 11. Q-criterion iso-surfaces colored by vorticitymagnitude for the GIT RFI configuration at µ = 0.1.

Fig. 12. Comparison of the time-averaged pressure dis-tributions on the top centerline of the GIT airframe atµ = 0.2.

Fig. 13. Comparison of the unsteady positive peak pres-sures on the top centerline of the GIT airframe at µ =0.2.

Fig. 14. Comparison of the unsteady negative peakpressures on the top centerline of the GIT airframe atµ = 0.2.

to be within 2% of the experimental values listed in Table4, which validate the correlations with experiment withoutfurther trimming the rotor.

The purpose of this demonstration case was to exam-ine the influence of grid adaptation on the prediction of thewake vortex trajectories. To examine this, a baseline gridwake trajectory was compared to a single adaptation basedon vorticity magnitude. The application of only a singleadaptation with the selection of the vorticity magnitude asthe adaptation metric was based on the prior GIT RFI simu-lations for the range of experimental data available for cor-relation. The initial grid had 14.4 M nodes and the vorticitybased adaptation increased the size to 17.6 M nodes.

Table 4. Relevant parameters for the ROBIN wake vi-sualization test case. (Note: All angles are reported indegrees.)

µ CT αs β0 θ0 θ1s θ1c0.23 0.0064 -3.0 1.5 6.5 -3.2 -1.1

Fig. 15. Computational model of the ROBIN 2MRTSconfiguration.

Vorticity tracks were extracted at different longitudinalslices spanning the rotor disk. Vortex cores are determinedby mapping local maxima of the normal vorticity compo-nent on each of these planes. There was no attempt madeto differentiate individual vortex structures like tip vorticesfrom these contour maps. The wake trajectories for the ad-vance ratio of 0.23 (Fig. 16) are observed to be very simi-lar to the experiment and the baseline grid trajectories. Themajor influence of the grid adaptation was not readily trans-parent, as it was apparent primarily in the process throughwhich the wake trajectories were numerically determined.As discussed with the GIT demonstration case, and illus-trated here via Q-criterion plots (Fig. 17), the effect of thegrid adaptation was to further refine the vorticity through-out the flow field. Thus, the refined strength and minimizedextent of the vortex path rendered the wake tracking muchsimpler and with fewer approximations.

Conclusions

A new grid adaptation strategy has been developed anddemonstrated to permit time-dependent adaptation acrossoverset, unstructured meshes. The strategy has been suc-cessfully demonstrated on two rotor-fuselage interactionconfigurations of interest to the rotorcraft community. Inparticular, the following observations can be made:

1. Overset grid adaptation to include any grid has beendemonstrated using a strategy to identify and track thegrid that each node is located within, along with anadaptation strategy that has been successful for singlegrids.

2. Demonstration on two rotor-fuselage interaction casesillustrate the solution dependency on the selection ofthe adaptation metric and the number of grid adapta-tions.

3. Flow field features and fuselage pressures require aminimum of two grid adaptation cycles. The most ac-

(a) y/R =−0.3

(b) y/R = 0.3

(c) y/R =−0.8

(d) y/R = 0.8

Fig. 16. ROBIN sideline vortex trajectories at µ = 0.23.

(a) Initial Grid

(b) Adapted Grid

Fig. 17. Q-criterion iso-surfaces colored by vorticitymagnitude for the ROBIN configuration at µ = 0.23.

curate adaptation metrics are vorticity magnitude fol-lowed by pressure gradient.

4. Given a sufficient baseline grid, wake trajectories ofthe tip vortex can be readily obtained with one adapta-tion cycle using vorticity magnitude as the metric.

5. The uncertainty associated with the selection of theadaptation metric and the number of adaptation cyclesimply that a more rigorous adaptation process, suchas adjoint adaptation, may be more appropriate thanfeature-based adaptation. Future efforts will focus onthe development of an adaptation process that will cir-cumvent the issues identified with feature-based adap-tation, as well as extension of the method to otherrotorcraft-related applications.

Acknowledgments

A portion of this research has been supported by the De-partment of the Navy, Office of Naval Research under grantN00014-09-1-1019, titled ”Deconstructing Hub Drag”. Dr.Judah Milgram is the technical monitor. Computationalsupport was provided through the DoD High PerformanceComputing Centers at ERDC through an HPC grant fromthe US Navy. Any opinions, findings, and conclusions orrecommendations expressed in this material are those of theauthor(s) and do not necessarily reflect the views of the De-partment of the Navy or the Office of Naval Research.

The authors would like to especially acknowledge andthank the NASA FUN3D development team, in particularDr. Michael Park and Dr. Eric Nielsen, who pioneered the

grid adaptation efforts within FUN3D. Without their dis-cussions, ideas, and suggestions, this effort would not havebeen possible.

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