A99-33463

AIAA 99-3226

Higher Order Discretization of Convective Fluxes of Navier-S tokes Equations: Helicopter Computations

Jasim U. Ahmad MCAT Inc., Army/NASA Rotorcraft Division NASA Ames Research Center Moffett Field, CA 94035, USA

17th Applied Aerodynamics Conference 28 June - 1 July, 1999 / Norfolk, VA

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H igher O rder D2scre titi& ion o f Convec tive F luxes o f Nav ie r -S tokes E q u a tions : H e l icopter C o m p u ta tions .- r,.-*,-.

Jas im A h m a d M C A T Inc., A r m y / N A S A R o torcraft Division,

A r n e s Resea rch C e n te , M o t-‘fe tt Fie ld, C A 9 4 0 3 5 - 1 0 0 0

A b s tract

This p a p e r p r e s e n ts a br ief descr ip t ion o f a fo u r th - o r d e r , c e n tra l -d i f ference a lgor i thm fo r th e Nav ier -Stokes e q u a tio n s . T w o m o d e l p r o b l e m s < a r e u s e d fo r q u a n tita tive tes t ing o f th e fo u r th - o r d e r a lgor i thm. It a lso exam ines th e e ffects o f far- f ie ld b o u n d a r y condi t ions a n d the i r n u m e r i - cal i m n l e m e n ta tio n o n th e R o w solver. T h e a lgor i thm is tested For a m o d e m he l icopter r o to r . Resul ts arc c o m p a r e d with e x p e r i m e n ta l d a ta .

In t roduct ion

Var ious numer i ca l schemes exist to d a y fo r r o torcraft sim u lat ions a n d l o a d predic t ion, h o w e v e r , accura te l o a d pred ic t ion is still a cha l l eng ing p r o b l e m . Hel icopter flo w - tie lds a r e h igh ly u n s te a d y , n o n l i n e a r , a n d th r e e - d i m e n - s ional . In b o th h o v e r a n d in fo r w a r d flig h t, th e ro to r b l ades interact wi th th e tip vor tex a n d w a k e s h e e t d e v e l o p e d by e i ther itself o r th e o th e r b l ades . This interact ion, k n o w n as b lade-vor tex in teract ion ( B V I), resul ts in u n s te a d y l o a d i n g o f th e b lades a n d can cause a dist inct ive acous tic s igna ture [I]. Pred ic t ion o f r o to r wakes a n d resul t ing B V I no i se is o n e o f th e difticult tasks fo r C F D algor i thms. Typical ly, th e c o m p u te d vert ical wakes di f fuse to o rap id ly d u e to numer i ca l d iss ipat ion. Accura te pred ic t ion o f B V I no i se requ i res th a t th e C F D sim u lat ion p rese rve a b l a d e ’s s h e d vor tex fo r severa l revo lu t ions b e fo r e b e i n g d iss ipated. M o r e o v e r , r o ta ry -w ing sim u lat ions o fte n requ i re m o v i n g g r id systems to sim u late th e u n s te a d y a n d ro ta tio n a l flo w - fie ld.

Fo r g e n e r a l r o torcraft p r o b l e m s , discret izat ion o f th e convect ive flux te rms in th e Nav ier -Stokes e q u a tio n s is very i m p o r ta n t s ince th e w a k e p r o p a g a tio n is p r e d o m i -

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n a tely a n inviscid p h e n o m e n o n . Fo r th is stu d y w e cons ider a fo u r th - o r d e r , s p a tia l -accura te a lgor i thm fo r th e convec- tive fluxes. A compar i son with a th i rd -o rde r u p w i n d s c h e m e is a lso i nc luded fo r subson ic pi tch ing airfoi l p r o tr- lems.

B e c a u s e he l icopter r o tors d o n o t typical ly exhib i t s t rong shock -domina te d flows fo r th e t ransonic flo w ove r th e b lades , a fo u r th - o r d e r c e n tral d i f ference s c h e m e is cho - s e n fo r th e s p a tia l discret izat ion. Bes ides h i g h e r accuracy, it is easy to i m p l e m e n t as a m o d ifca tio n to th e cur rent scc- o n d - o r d e r schemes . This stu d y app l ies th e fo u r th - o r d e r s p a tia l d i f ferenc ing s c h e m e to th e convect ive te rms on ly a n d d o e s n o t invest igate h i g h e r - o r d e r discret izat ion o f vis- cous te rms . R e c e n t stud ies [2 ] fo r tu r b u l e n t R o w over a i r - foi ls s u g g e s t th a t accuracy o f th e v iscous te r m is as i m p o % :n t as t!? ‘3 convect. , _ _ _ _ i l jp t-72; !mvevttr. w e a ~ s u r n e th a t th e s e e ffects a r e smal l fo r he l icopter p r o b l e m s w h e r e th e c o m p u ta tio n a l accuracy is d o m ina ted by th e ro to r w a k e resolut ion. This stu d y d id n o t invest igate th e var ious strat- eg ies fo r a d d i n g numer i ca l d iss ipat ion a n d s u b s e q u e n t elfects o n th e accuracy a n d c o n v e r g e n c e o f th e m e th o d .

Fo r te m p o r a l accuracy, w e u s e d a s e c o n d - o r d e r back- w a r d E u Ie r s c h e m e . T h e u s e o f subi terat ions a t e a c h tim e ste p is i m p o r ta n t fo r accura te sim u latiori. o f two m o d e l u n s te a d y p r o b l e m s . T h e first m o d e l prob lc \n is a n inviscid vor tex convect ion in a f reest ream. T h e s e c o n d is a Nav ier - S tokes sim u lat ion fo r a n osci l lat ing airfoi l . T h e s e m o d e l p r o b l e m s a r e c losely re la ted to typ ica l he l icopter p r o b l e m s in g e n e r a l , in th e seILse o f b o th ro ta tio n a l m o tio ,n . as wel l as fo r vor tex p r o p a g a tio n in th e he l icopter w a k e .

Spec ia l emphas i s is g i ven to th e speci f icat ion o f b o u n d - ary condi t ions a n d numer i ca l b o u n d a r y t reatment . M a n y p rev ious invest igat ions s h o w c o n flictin g results, especia l ly fo r h o v e r p r o b l e m s . T h e speci f icat ion o f th e far- f ie ld b o u n d a r y condi t ions is very i m p o r ta n t to avo id n&phys i - cd re flec t ion f rom th e b o u n d a r y th a t c a n a ffect th e so lu-

tio n in th e in ter ior o f th e d o m a in. Fo r m o v i n g a n d osci l lat ing bod ies such as he l icopters a n d osci l lat ing a i r - foils, p o o r cho ice o f b o u n d a r y condi t ions c a n cause con- v e r g e n c e difficulties. S o lut ions b a s e d o n inviscid character ist ics th e o r y s e e m s to b e a n a p p r o p r i a te cho ice

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for l?lost Of t!lL: i0lOr p~Oi$~ll~S~ Fonmilations based til’i vi>- ious equations [3] can also lze implemented in t-he noslip wal! of the rotcir blade as we!! ES at the far&Id. Details of the boundary conditions and various suggestions specific IO the rotor problems are made.

Algorithm

The basic, flowsolver is based on the widely-used OVERFLOW [41 code developed at NASA ,Ames Research Center. The governing equations in generalized coordinates can be written as:

$? + -N-2)- V(Q) + R(Q) (1) where,

P P“

Q - J-’ pv

[I P’V e

is the solution vector of conscmative variables for the mass, momentum, and energy equations. Variable

aIT. PW. E(Q) agi - 2 is the convective operator, V(Q) - -is the

hi thin-layer viscous operator, and

0 pS2

II

R(Q) - J-’ -puR (2) 0

0

is the z-coordinate rotational source term to convert the equations to steady state for simulating hovering prob- lems. For all the other cases R(Q) - 0. The details of the governing equations can be found c&where [5], except for the source ‘terms. The source term formulation for rotor- fixed coordinate can be found in Ref. 6.

OVERFLOW is written for an overset-grid scheme for complex geometries; however, the basic algorithms are also applicable for a single static grid. Rotary wing is a moving grid simulation problem and a number of modifi- cations to OVERFLOW are needed for moving body prob- terns. The code offers a number of user-specified solution algorithms. For higher-order spatial accuracy, third-order upwind-biased, ‘and 4th-order central difference schemes for the convective terms are implemented. Temporal dis- cretization of second-order backward Euler is used for time integration. Ideally, spatial and temporal accuracy should be compatible for wave-liki problems; however, this study only uses second-order temporal accuracy, which avoids stringent stability restriction of the numeri- cal scheme that require very small timestep.

In a third-order scheme 161. the nUmerica algorithm spatially differences the flux terms using a Roe upwind- biked s&me for all three coordinate dinzctions with higher-order MUSCL-type limiting to model shocks accu- rately. Details of this third-order algorithm are discussed elsewhere[6]. In what follows is the brief discussion of

‘L.,... : -.

fourth-order central diiference algorithm. The interior dif- ferencing of the convective operators are:

Cl a kr[‘li+ ljk-qi- I jkl

=2 - &.[2(qi+ ljk-qi-ljk)-(qi+2jk-qi-Zjk)l

‘Xqlijk - c1+c2

i - 2,3 , . . . . . . N-2 (3) In the above equation C, is the second-order accurate

term and C, can be thought as an additional term to sec- ond-order differencing to make it fourth-order accurate. This helps manipulate coding to the existing second-order scheme as well as at the overset-_tid interface. For the glo- bal scheme to be of specific order-accurate, the near bound~ary scheme can be one order less than the interior scheme. For the fourth-order spatial operator. the third- order boundary operator is:

Bl - ‘8[91jke40jkI 4 a -9[9?jk-40j~l

83 - 2k,jk-qOjkl

'.r'lOjk - 6Ay &Bl+B2+Bj] (4)

Third-order first interior node operator is: I1 - -2[‘?Ojk-‘fl jkl

I, - 6[q,jk-Yljkl

Similarly, the operators at N - I and at N. i For this study, dissipation and viscous operators are

unchanged from the current OVERFLOW operator.’ Also, some turbulence models, such as the Baldwin-Lomax model, require calculations of vorticity near the bound- aries. Attempts to improve the accuracy of the viscous operator and turbulence model have been investigated in Ref. 2.

Far-Field Boundary

Like any other robust numerical algorithm, boundary treatment is important for rotorcraft problems, especially the far-field. For accurate simulation, the computational domain should enclose a very large far-field region; how- ever, large computational regions require large numbers of grig points, which rapidly increase the computational

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resource r~Wlii lVXh2n~s. To limit the domain to a finite size. f;r-ii& b&druies should allow waves to travel freely in and out oi th:: finite domain without any re&tion of the outgoing tiaves.‘.4s it will be seen in a hover problem in later section, even a consistent boundary may lead to a solution other than the expected physical behavior. Usu- ally, the computational doma‘in is far enough away so that viscous effects can be neglected at the far-field and the boundary can be treated with a characteristic formulation. The proper us(: of this characteristic formulation seems to mimic the physical solution closely as indicated by many studies including for a flow over cylinder [7]. Detailed descriptions of characteristics formulation can be found elsewhere [8][9]. The descri@iou below is pertinent to the rotorcraft problem in general.

Boundary treatments are ditTerent for subsonic and supersonic flow. For supersonic outflow, all the informa- tion can be extrapolated from inside the domain because all the characteristics are outgoing. For supersonic inflow, the botmday points CNI be calculated from the external known quantities. For subsonic inflow however, one of the characteristics goes out of the computational domain. In the case of subsonic outflow, one acoustic wave propa- gates against the flow direction. At least one external con- dition is ,nccessary corresponding to this characteristics vanabie. As such, the specification of subsonic inflow and outflow is much more difficult than for supersonic flows. Helicopter flow is always subsonic at the external compu- tational boundaries.

Testing of the various options of characteristics bound- ary conditions is currently in progress. In the present study, we tested two options discussed below. The second c@nrl re:alllted in an ;IImost, non-reflective, smooth bound- ary.

1: Characteristics approach using Hiemann invariants.

Ricmann invariants arc:

Rl 2c

- vyq

R2 2c

- ‘01 + .(-I

(6)

(7)

For subsonic inflow, specify outgoing acoustic wave H , * ‘1, 9 and entropy s and extrapolate incoming acoustic wave R2 from inside.

For subsonic outflow specify R, , extrapolate R, , it,,

and entropy from inside. This procedure for the hover problem produces a discontinuous slip stream at the exit boundary. This will be discussed fully in the implementa- tion of the hover boundary.

2: At outflow, characteristics approach using Riemann invariants and using pressure at infinity

WC choose pressure as the prescribed variable at the outflow boundary, to be specified as P,, extrapolate I(,,

Ull p from inside. This with combination of option 1 at the inflow produces a smooth Row at the exit. Hover results of this present study were obtained using this boundary specification and the locations of the computed vortices closely follow measured locations. - ‘A*=:

Helicopter Hover Boundary Conditions

A hovering rotor flow-field is different from a fixed- wing in forward flight as the rotor rotation is in a quiescent air. As such, the specification of hover boundaries along with above ch,aracteristics an: tricky.

Usually at the lower boundary, 1-D momentum theory is used to prescribe an outflow ma. The outflow velocity is a function of the thrust of the rotor. With this specifica- tion of an outflow velocity, mass conservation into the computational domain has to be maintained. Flow into the computational domain is modeled by a point sink of mass [l]. The sink is located at the rotor hub and its strength is adjusted to conserve mass across the boundaries. These specifications of inflow and outflow velocities are com- bined with the various characteristics options described above. For the helicopter computation in this study, it will be shown that option 2 is the pKfcrred choice.

Computation of Model Probjems

Two model 2-D problems were tested. The first irob- tern is an inviscid simulation of vortex convection. The second one is a Navier-Stokes scheme involving moving and dynamically oscillating grids. These two problems are very closely related to the various physical characteristics of a typical helicopter flowfield. For the first case, the unsteady results are compared with the initial vortex shape and its rate of decay of the initial strength of the vortex. For the oscillating airfoil, this study compares the results with available experimental data.

1: Vortex convection in’a periodic domain.

This test cast is passive convection of a,vortex in a free stream and monitoring the rate of decay of the vortex. A perfect numerical method should convect the vortex in space with no dissipation or dispersion. This case tests the time-accurate free convection of a vortex How. Flow speci- fications are considcrcd for a compressible Euler equa- tions for a 2-D rectangular domain.The mean How is p-l,p- I , and horizontal flow with 14 - 1 , Y - 0. We add to (he mean flow Lan isentropic. vortex, i.e. pertur- bation in II ‘and w , and the temperature T - p/p. No entropy penurbation is added. The computational domain is taken as 10x 10 in length. The boundary is extended periodically in both directions. The convection speed is MO3 - 0.50. The exact solution of the Eule; equation with these initial and boundary conditions is simply the passive convection of vortex with mean velocity. The vortex would

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siinply cosscs the ~riodir: PBuadary and back to its origi- nal position in a pried. The periodic boundary condition aii,ows Us to compute the long time evolution of the vortex. The calcu!‘ations XC completely subsonic.

The initial vortex is positioned at the center of the domain. The grid iesolution is 81 x 81 and the time step d: is normalized by the freesa-eam speed of sound and the unit length of the domain.

Both the second-order and fourth-order schemes were tested in this calcdation. Figure 1 and 2 show the line piots of the deJ,lsity Variable at a cut SeCtiOn of z - 5 through the cmcr of the domain for various discretization options. The vortex core radius as seen in these figures HT~ one. The founh-order scheme convects the vortex much better than both second-order scheme and third-order

1.4

1.2 I

1 1

_-_-----__

P ::: -

0.4 -

0.2 - b

‘1 ‘\ / ------------

\ ‘1

/’ I \ ,

d --- 2nd order(CD) - 4th order(CD) .......... 3rd Order(ROE) ---- Initial

0 . 0 --.2

. 4'. 1 . L _ ( 6 8 10

X

Figure I. Density plot after first period.

1 ---mm

0.8 P 0.6

0.4 - 4th order(CD) .......... 3rd Order(ROE) ---- Initial

OL' ' - '. ' - ' - ' 0 2 4 6 8 JO

X

Figure 2. Density plot after fifth period.

0.4

0.2

- 4th ofdef(cD) --- 2nd order(CD) .-...... 3rd order(ROE)

l Experiment

ol--.*-*-‘-*-’ -2 0 2 4 6 8 10

0.04

0.03 i

Cd “. &.y

. ,_.” ” . . . ,.f:’

” . . ..” . . . . . . . ’ . . . . . _.

-0.01’ s - ’ . ’ ’ ’ ’ -2 0 2 4 6 8 IO

-0.02 - -2 0 2 4 6 8 10

Angle

Figure 3. Hysteresis loop of lift, drag, and pitching moment coefficients for attached flow of NACA 0015 air- foil oscillating at a - 4" + 4.2”sin(2kM,t + 1.5~).

Mm - 0.29 Re - 1.95~10: k - 0.1

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upwind scheme. In this computation, in one period vortex travels about 20 vortex core radii, and the vortex s:arted to dissipate significantly after 400 core ‘radii of tnversc.

2: Oscillating Airfoil. ~ --*,-:

The computation simulates unsteady 2-D. subsonic flows over an oscillating NACAOO15 airfoil. The follow- ing specifications are used for the simulation: Free sueam Mach number of M, - 0.29, Reynolds number

Re - 1.95~10~ based on chord length, reduced frequency

‘r 0.8

" 0.6

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.ot‘ k - r?.!, end the inst2fitaneous angle of attack of the pitching of the airfoil is a - 4“ + 4.2”sin(2kM,t + 1.5~). The mean angle of pitch is 4O f and this flow specification corresponds to an attached flow. Detailed description of this study can be found in an existing numerical computa- tions [IO]. The only turbulence model used in this study is Baldwin-Lomax model [ 1 I]. The experimental data of Piziali [12] gives detailed measurements of static and dynamic pressure distributions, and cycle-averaged lift, drag, and pitching moment coefficients.

Computations used a C-type grid of 259 x 60 points, 100 pbints are on the upper surface, 80 points on the lower surface, and 40 points along the wake cut. The farfield boundary extended 15 chord lengths away from the airfoil surface.. The present study took about 1440 steps for a pitching cycle. This time-step m imics a typical helicopter simulation. The characteristics boundary condition option 1 is used at the far-field.

Figure: 3 shows hysteresis loops of lift, drag, and pitch- ing moment coefficients. Both second and fourth-order centml dilfercnce schemes results are very close to each other and also to the experimental results. Recall that the “fourth-order” scheme only uses the higher-order differ- encing for the convective terms, not for the viscous terms. Aerodynamic I‘orces I‘or this oscillating airfoil case with subsonic attached flow are influenced by viscous effects, so it is not surprising that the purely second aider scheme and the scheme with fourth order convective fluxes give sim ilar rcsutts. On the other hand, the results from the third-order upwind scheme are quite different from the experimental values, especially for drag and pitching moment.

Com puted results of hovering rotor.

A flow simulation was perfom led for a UH60 model helicopter rotor, for which Lorber et al. [I21 obtained detailed measurements of pressure and blade loading in an wind tunnel. The UH60 is a four-bladed rotor system. The blades have a non-linear twist of 20 degrees from root to tip and an aspect ratio of 15.5. Two results with different wake grid densities arc presented for a hovering case with a collective angle of 10.47” , a hover-tip Mach numb& of 0.628, and a measured thrust of C,/a - 0.085 . Helicopter quantities mentioned here are: C, is the thrust, u is the rotor solidity ratio, CQ is the power coefficient, and FM is the fi-ore of merit.

As indicated before, the governing equations for a hov- ering rotor are solved as a steady-state problem with the addition of a source term . In addition, periodicity can also be used to reduce the size of the’computational domain. In general, a computational domain can be reduced by a fac- tor )I, where ,I is equal to the number of blades Fig. 4. shows the periodic planes of the computational domain.

The overset-grid system for this case consists of four

Mass Sink Inflow

Figun: 4. Rotor l~i;~ttc and wake grids for tlk UH-60 model rotor btadc.

different grids which ;Irc shown in Fig. 4. The first grid surrounds the rotor ht;~de with a C-H topology. The next two grids cover the btadc root and tip respectively. The fourth grid consisls 01‘ a Cartesian background grid that is used to capture [IN! rotor wake system. This Cartesian background grid consis& of a uniform ly-spaced “inner box” region, with stretching toward the outer boundaries. The firs1 Cartesian wake grid had a uniform spacing in the inner box of 0.3~ and the second wake grid had a uniform spacing of 0.2~ in rtic i nncr box.

Figure 5 compares lhc computed sectional thrust to the measurctl cxlwri1~11~;11 dala [ 12). Figure 6shows compari- sons bctwccn cornputcd prcssun: cocfticients at different radial location 01‘ rhc blade. Agreement between the com-

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_ . . ..r.P.n” yLa,I”.AI) 32 kc cxpcrimcnt is quite good 3t XI i~bo~c! station of 0.675 radial location. Near the tip, the lift pre- diction is higher than the experimental value. This may be due to the local inflow changes caused by the tip vortices in the rotor wake. Table 1 shows the computed and experi- mental performance value for this model rotor. For the finer mesh, the computational performance ssems to be approaching to the experimental values

0.6 - l Experiment

0.2~ Wake Grid ---- 0.3~ Wake Grid

0 0.2 0.4 0.6 0.6 1 r/R

Figure 5. Sectional blade loading for the UH-60 model rotor calculations.

-2.5

\ -1.5

CP -0.5

0.5

l ExperhBent - 02 Wake Wid --- 0.3~ Wake Grid

I r/R = 0.675

1.5 ' 0

-2.5 r

1

CP

r/R = 0.965

X/C

Figure 6. Chordwise pressure coefficient at two different radial locations.

Summary anti~C:onClusims

The study gives details of a high-order accuracy cen- tral-difference and an upwind scheme These schemes are tested for two model problems that are representative of . ‘c-c=’ the physical phenomena that are present in.many rotary- wing aerodynamic problems. For these types of rotary- wing problems involving dy~~m!r F!z!! Z! ~CITPP. r2?:ec- tion, the central difference algorithm is more suitable than its upwind counterpart. The founh-order central difference scheme showed better results than the second-order scheme for vortex convection, but not for the oscillating airfoil problem. For these model problems, the use of lim- iters in the upwind scheme can lead to substantial errors in the final solution.

For the 3-D hovering rotor simulation, the fourth-order central difference algorithm gives reasonable comparisons to the experimental data. C‘areful specification of the far- field boundary conditions is just as important to the accu- racy of the final result as the choice of discretization.’ Improper use of boundary treatment can lead to an ill- posed problem and subsequent numerical instabilities. The etfect of boundary conditions on ovemll flowfield conver- gence will be tested in the future.

Table 1: Computed and experimental performance ,data for the model rotor blade

Experiment

0.3c waki &d

0.2~ wake grid

References

c,/o CQ/t3 FM

0.0974 0,0083 0.74

0.0982 O.&JOY nr.0. r\ nn. U.IV

0.975 0.0087 0.71

[I1

PI

[31

r41

Ahmad, J.U., and Strawn, R.C., “Hovering Rotor and Wake Calculations with q Overset-Grid Navier-Stokes Solver,” American Helicopter Soci- ety 55th Annual Forum, Montreal, Canada, 1999. Zingg, D.W., De Rango, S., Nemec, M., and Pul- ham, T., “Compatison of several Spatial Discreti- zations for the Navier-Stokes Equations,” AlAA 99-3260, 14th AIAA Computational Fluid Dynam- ics Conference, Norfolk, VA., 1999. Hayder, M.E., and Turkel, E., “Nonreflecting Boundary Conditions for Jet Flow Computations,” AIAA Journal, Vol. 33, No. 12, Dec. 1995, pp 2261-2270.

Jespersen, D., Pulliam, T., and Buning, P., “Recent Enhancements to OVERFLOW,” AIAA 97-0644, Reno, NV, 1997.

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[61

r71

PI

[91

r101

[Ill

r121

[I31

pi.:!!iL??, VU a . . .., “EfR‘fideni So!ction ~~%dz for ,. . . ..-. Navier-Stokes I$uations,” Lecture Notes for :he von Karmman Institute for Fluid Dynamics Licture Series, Brussels. Jan. 1986. Srinivasan, G.R., Baeder, J.D., Obayashi, S., and McCroskey, W.J.; “Flowfeld of a Lifting Rorcr in Hover: A Navier-Stokes Simulation,” AIAA Jour- nal, Vol. 30, No. 10. pp. 2371-2378, Oct. 1992.

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