AIAA 95-3470-CP

PREDICTION OF NEAR- ANDFAR-FIELD VORTEX-WAKETURBULENT FLOWS

Osama A. Kandil, Tin-Chee Wong, and Ihab AdamOld Dominion University, Norfolk, VA 23529, USA

C. H. LiuNASA Langley Research Center, Hampton, VA 23681,USA

AIAA Atmospheric Flight

Mechanics Conference

Baltimore, MD, August 7-9, 1995

AIAA 95-3470-CP

Prediction of Near- and Far-Field Vortex-Wake Turbulent Flows

Osama A. Kandil1, Tin-Chee Wong2, and Ihab Adam3

Dept. of Aerospace EngineeringOld Dominion University, Norfolk, VA 23529, USA

andC. H. Liu4

Aerodynamic and Acoustic Methods BranchNASA Langley Research Center, Hampton, VA 23681, USA

ABSTRACT

Prediction of both the near- and far-�eld vortex-wake turbulent ows are presented. TheReynolds-averaged Navier-Stokes (NS) equations are solved using the implicit, upwind Roe-ux-di�erencing scheme. The turbulence models of Baldwin and Lomax, one-equation model of Spalartand Allmaras and two-equation shear stress transport model of Menter are implemented with theNS solver for turbulent-ow calculation. For the near-�eld study, computations are carried outon a �ne grid for a rectangular wing with a NACA-0012 airfoil section and a rounded tip. Thewing has an aspect ratio of 1.5 and is mounted inside a wind tunnel at an angle of attack of 10�.The focus of study is the tip-vortex development, the near vortex wake roll-up, and validating theresults with the available experimental data. For the far-�eld study, the computations of vortex-wake interaction with the exhaust plume of a single engine are carried out using overlapping zonalmethod for a long distance downstream of a Boeing 727 wing in a holding condition. The resultsare compared with those of an incompressible parabolized NS solver known as the UNIWAKE code.

INTRODUCTION

Recently, the volume of civil air transportusing subsonic aircraft has increased at an alarm-ing rate. With this increase in air tra�c, sev-eral hazardous e�ects have recently become ofprimary concern. First, the landing and take-o� operations safety at busy airports of smalland medium size aircrafts when they encounterhigh-intensity turbulent vortices emanating fromlarge aircraft. The trailing aircraft, under the in-

uence of those vortex trails, could su�er highrolling moments, loss of climb and structuraldamages. The vortices persist up to a few milesand several minutes before they decay.

Second, the adverse e�ects of the engineexhaust on the stratosphere and troposphereduring cruise and holding conditions. A complex

ow regime develop behind those aircraft whichinclude the exhaust jet plume and the wake vor-

tices that entrain the exhaust plumes and even-tually break-up producing exhaust-atmospheremixing region. Substantial adverse e�ects onthe stratosphere and troposphere are expectedwhen the new eet of High Speed Civil Trans-port (HSCT) is introduced in the early years ofthe next century. Recent research e�orts are cur-rently directed at understanding the adverse at-mospheric e�ects of exhaust products from sub-sonic and supersonic civil transport aircrafts.These e�orts include predicting the e�ects of ex-haust plume on the dynamical, chemical and ra-diative stratospheric processes. A recent NASAreport on these issues is published under the At-mospheric E�ects of Aviation Project (AEAP),Ref. 1.

The origins of these hazardous e�ects arethe vortex-wake ows and the engine jet exhaust

1Professor, Eminent Scholar and Chairman of the Dept. Associate Fellow AIAA.2Research Associate, Aerospace Engineering Department.3Research Assistant, Member AIAA4Senior Research Scientist, Associate Fellow AIAA.

copyright c1995 by Osama A. Kandil. Published by the American Institute of Aeronautics and Astronautics, Inc.

with permission.

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plume and its interaction with the vortex-wake

ows. The vortex-wake ows include the tip-vortex development and roll-up formation whilethe jet exhaust plume include exhaust prod-ucts, temperature �eld, and their uid mechan-ics. At some distance downstream, the vortex-wake ows entrain the exhaust plume and later-on the vortex-wake breaks up and dilutes the ex-haust plume in the stratosphere.

The literature shows some experimentaland computational investigations that modeland analyze the roll-up of a tip vortex, the wake-vortex interaction, and the merging and decay, aswell as the hazardous e�ects of these phenomenaon trailing aircraft. Hallock and Eberle2 gave acomprehensive review of the research on aircraftwake vortices in the United States through themid-1970's.

Mathematical models and computationalmethods were developed with inviscid analysis3�6. Although an inviscid model cannot describethe aging of the wake including its di�usion, it isstill capable of representing the wake shape andits dynamics. The mathematical models used inthe above references were based on the use ofthe point-vortex method to compute the motionof a �nite number of point vortices. The three-dimensional inviscid model which is based onthe nonlinear vortex-lattice method, was used tocompute the interference ow between the wingsand the vortex-wake ows and to examine thehazardous e�ects.7

Viscous modeling of trailing vortices wasstudied in Ref. 8. Viscous interactions of vor-tex wakes and the e�ects of background tur-bulence, wind shear, and the ground on two-dimensional vortex pairs with the Navier-Stokesequations were presented in Refs. 9 and 10, andthe computer program is known as UNIWAKE.The interaction, merging, and decay of vorticesin two- and three-dimensional spaces were stud-ied in Refs. 11 and 12. A comprehensive reviewon the subject of viscous vortical ows can befound in a book by Ting and Klein.13 To es-timate the e�ects of density strati�cation, tur-bulence, and Reynolds number on vortex wakes,an approximate model was recently developed byGreene.14 Later, Greene and his coworkers15 pre-sented selected results for di�erent aircraft vor-tices, including a juncture vortex, a lifting-wingvortex, and a wake vortex.

In recent papers by the presentauthors16;17, the compressible Reynolds-averaged NS equations were used to computeand analyze vortex-wake ows of isolated andinteracting wings. The emphasis of the paperwas to study the e�ects of the near-wake vor-tex ow on a small follower wing for two ow-interference cases. The ux limiter in the owsolver was turned on and o� to study it numer-ical di�usive e�ect. The solution obtained withthe full NS equations without a ux limiter gavethe least numerically di�used tip-vortex core incomparison with those solutions for which a uxlimiter was used.

The multidisciplinary interaction of theaerodynamics and rigid-body dynamics betweena single tip vortex and a trailing wing was com-putationally investigated by present authors18.The time-accurate solutions of the unsteadyReynolds-averaged NS and Euler equations forrigid wing rolling motion provided the growthrate of the vortex-core size and the rolling-motion response of the wing. The Baldwin andLomax turbulence model was used for this case.

Very recently, research interest has alsobeen focused on the near-�eld and far-�eldvortex-wake interaction with the engine exhaustplume including vortex-wake breaks up for bothsubsonic and high speed civil transport (HSCT)aircraft. Computational uid dynamics plays asigni�cant role in the prediction of the near-�eldand far-�eld vortex-wake ows. Once this is ac-complished, the next step is to include the ex-haust plume products and chemical reactions,and its interaction with the vortex-wake owsincluding vortex wake break-up.

Recently, more advanced turbulence mod-els became readily available for use with NSsolvers. In this paper, the algebraic Baldwinand Lomax (BL) turbulence model19, the one-equation Spalart and Allmaras (SA) model20,and the two-equation k! (KW) model developedby Menter21 are used to study the tip-vortex andwake ows and the interaction of a tip vortexwith the temperature �eld of an exhaust plumeof a Boeing 727. Three key ingredients are con-sidered for achieving accurate prediction of these

ows. These are the grid �neness, turbulencemodel and computational e�ciency. The re-sults using di�erent models are validated withthe available experimental data or the results of

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the UNIWAKE solver.

FORMULATIONS

Two sets of the NS equations are used forthis paper; a compressible set and an incom-pressible set. The compressible set is solvedusing a computer program known as FTNS3Dwhich is used in Ref. 18. This is the modi�ed ver-sion of the well known CFL3D code. The incom-pressible set is solved using a computer programknown as UNIWAKE which is used in Refs. 9and 22.

The FTNS3D solver, described in detail inRef. 18, uses an upwind, ux-di�erence splitting,�nite-volume scheme solving the unsteady, com-pressible, Reynolds-averaged NS equations. Forthe exhaust plume/tip vortex interaction case,the buoyancy body force caused by tempera-ture di�erence is added as a source term with-out any approximation in the NS equations. Forall results in this paper, upwind-biased spatialdi�erencing is used for the inviscid terms, and

ux limiter is not used. The viscous terms aredi�erenced using second-order accurate centraldi�erencing. The resulting di�erence equationsare solved implicitly in time with the use of thethree-factor approximate factorization scheme.The one- and two- equation turbulence modelsare decoupled from the NS equations and partialdi�erential equation(s) for turbulence model aresolved sequentially at each time step thereafter.

The UNIWAKE solver consists of fourcomputational modules: (1) Vortex Lattice: Aprogram to compute the lift circulation distribu-tion on an aircraft wing based on the given aero-dynamic parameters and wing planform shape.(2) Betz: A program to generate the initial posi-tion and strength of rolled-up trailing edge vor-tices, given the lift circulation distribution. (3)Wake: A program to merge and decay thesevortices downstream, interacting with engine jetexhaust temperature and chemical products, bysolving the incompressible parabolized NS equa-tions with fourth-order compact scheme in uni-formly Cartesian system. The second deriva-tives in the streamwise direction is neglectedin the governing equations. The e�ects of theturbulence are included through the algebraicReynolds stress turbulence model. (4) Pinch: Aprogram to follow the inviscid line vortex �la-ment interaction of these vortices to instability

and pinching, utilizing curved vortex elements.Recently, some aspects of the compressibility anddensity variation are taken into account in thelatest version of UNIWAKE. It should be notedthat the buoyancy body caused by temperaturedi�erence is based on the Boussinesq approxima-tions which are not valid for high temperaturedi�erences. The detail of the governing equa-tions and recent enhancements can be found inRef. 22.

BOUNDARY AND INITIALCONDITIONS

Boundary conditions are explicitly imple-mented. They include inow-outow conditionsand solid boundary conditions. At the inowboundaries, the velocity pro�les are either pre-scribed or interpolated from the experimentaldata, while the Riemann-invariant boundary-type conditions are used. Temperature distri-bution is speci�ed for the engine exhaust plumeproblem. At the outow boundaries, pressurepro�le either interpolated from the experimentaldata or extrapolated from interior domain, whilethe other variables are determined as part of thesolution. At the geometric plane of symmetry,periodic conditions is set. For tip-vortex andnear-wake ow case, the tunnel walls are treatedas inviscid surface, except for the root wall.

The initial conditions correspond to theuniform ow with no-slip and no-penetrationconditions are used.

RESULTS AND DISCUSSION

Near-Field Computation of the Tip-Vortex

A rectangular wing with a NACA-0012 air-foil section and a rounded wingtip is consid-ered. The wing has an aspect ratio of 1.5 andis mounted inside a wind tunnel at an angle ofattack of 10�. The experimental work23 was doneat the Fluid Mechanics Laboratory at NASAAmes research center. The ow is turbulent witha Reynolds number of 4:6 � 106, based on theroot-chord length of the wing (c), and the owMach number is 0.3.

A C-O grid is used with 197� 53� 97 gridpoints in the streamwise wraparound, normal,and spanwise directions, respectively. A typi-cal grid used in this study is shown in Fig. 1.

4

The computational domain of the grid is gen-erated based on the dimension of the test sec-tion and is nondimensionalized by the root-chordlength. The origin of the axis is located at thequarter chord of the wing, the upper and lowerwalls are x=c=0.3333 above and below the wing,and the far-side wall is x=c=1.0 from the mount-ing wall. The inow (x=c = �0:4) and outow(x=c = 1:42) conditions from the experimentaldata are imposed as the boundary conditions.The grid is clustered in the normal direction withthe spacing of 5�10�5 near the body and is alsoclustered at the leading and trailing edges of thewing. The mounting side and tip region of thewing are also clustered to have better ow�eldresolution.

The steady-state solutions of the near-�eldhave been obtained using two-level of multi-gridwith BL, SA and KW turbulence models. Thetypical convergent histories of the residual errorand lift coe�cient for the SA model is shown inFig. 2. The residual error drops about two orderof magnitude and lift coe�cient gets to within0.5% of its �nal values in 1800 cycles. The �-nal lift coe�cient is 0.52 at 2400 cycles. On theCray-YMP computer, a typical case takes about13 hours.

Comparison of surface-pressure coe�cients(Cp) with the experimental data at threeconstant spanwise stations (left column) androunded tip regions (right column) with BL, SAand KW models is shown in Fig. 4. The def-inition of the � (theta) at the wing-tip regionis shown in Fig. 3. The results with BL andSA model are better than those with KW modelon the wing surface. Obviously, there is mas-sively separated region between x=c = 0:4 andx=c = 0:7 at the wing-tip region. The KWmodelpoorly predicts the Cp particularly at the trail-ing edge of the wing.

The side-by-side comparison of the cross-

ow total-pressure contours (Cpt) at two chord-wise stations at x=c = 0:63 (on the wing) andx=c = 1:19 (near wake) are shown in Figures 5and 6. One can see the roll-up of the vortexaround the tip from the low surface to the uppersurface, which corresponds to the large pressuregradient at the wing-tip region. Then, the vor-tex moves upward and outboard as moving in thedownstream direction. The development of thewing tip-vortex shows evidence of a good qualita-

tive agreement with experimental data. The lo-cation of the tip-vortex using BL and SA modelsis in fair agreement with that of the experimen-tal data while the results of the KW model showthe vortex is located more outboard and closerto the surface. However, the results predicted bythe KW model show better comparison with ex-perimental data in the wake region at the chordstation x=c = 1:19. The vortex-wake structureof the KW model show less di�usion than thoseof BL and SA models. The close-up of the cross-

ow velocity magnitude contours (Vc), shownin Fig. 7, con�rms that results from the two-equation model predict tighter vortex core thanthose of the BL and SA models in the near wake.However, the tip-vortex core still shows di�usionin comparison with the experimental data due tothe lack of grid resolution in the core region.

Far-Field Computation of Tip-Vortex/Plume Interaction

For this case, a tip-vortex/plume interac-tion of a Boeing 727 wing is considered. Thestudy addresses the computation and analysisof the vortex-wake interaction with the exhaustplume for a long distance downstream of thewing. The tip-vortex of the Boeing 727 wingis assumed to be fully rolled-up and the genera-tion region is not included in the computation.The initial velocity and pressure pro�les are gen-erated using the vortex-lattice and Betz modulesof the UNIWAKE.

The tip-vortex ow is assumed fully turbu-lent with a Reynolds number of 1�106, based onthe half semi-span of the wing (s), and the owMach number is 0.3. The tip-vortex and exhaustplume are located at y=s = 0:76, z=s = 0:0,and y=s = 0:4, z=s = �0:1, respectively. Thepeak temperature at the center of the engine istwo times the ambient temperature. The inowcrossow velocity (Vc), and temperature distri-butions (T) at x=s = 0:0 are shown in Fig. 8.

The NS equations are used to computethe development of this vortex and its inter-action with the plume for a long distance upto x=s = 110. The computations of FTNS3Dsolver are carried out using an overlapping zonalmethod and the schematic sketch is shown inFig. 9. For each stage of computation, a �negrid zone is used. The downstream distance (a)and the overlapping or bu�er zone (b) should be

5

chosen such that the downstream e�ects can beminimized. In this study, the following values arechosen; a/s=8.0, b/s=2.0, and Xmax/s=110.0(equivalent to one mile behind the inow plane).A rectangular grid of 201� 41 � 51 grid pointsin x, y, and z directions, respectively, are used.

The computations are carried out start-ing from the inow station of x=s = 0:0 up tox=s = 110:0 using the FTNS3D using KWmodeland UNIWAKE solvers on the same grid resolu-tion in the crossow plane. The results of thecrossow T and Vc contours at selected chord-wise stations with FTNS3D (left column) andUNIWAKE (right column) solvers are shown inFigures 10-12. All the results are plotted in ref-erence to the origin and the corresponding axesat the inow section. The results show the evolu-tion of the tip-vortex interaction with the plumeas it is advanced in the downstream. The ve-locity �eld of the tip vortex induces the exhaustplume movement upward and wrapping aroundthe tip-vortex while cooling it down.

The results of the FTNS3D code show thattip-vortex center �rst moves upward and inward,and then descends as it advances in the down-stream direction. On the other hand, the UNI-WAKE results show the tip-vortex center con-tinuously descends and stays almost at the samelateral location of y=s = 0:76. Since the UNI-WAKE code is a space-marching NS solver, thestep size in the streamwise direction is deter-mined based on extrapolated changes in the owvariables at each station. The code also has adynamic upwash adjustment that seeks to retainthe perceived vortical mean location at the po-sition of z=s = 0:0. The upwash overcomes theinherent downwash of the typical vortical wakestructure. The discrepancies between the twosolvers are due to the dynamic upwash adjust-ment of the UNIWAKE code and the fully three-dimensional computations of the FTNS3D code.The results with the UNIWAKE solver showmore di�usion as compared with the FTNS3D re-sults. One of the numerical parameters known asthe turbulent macroscale (�) in the UNIWAKEcode has to be adjusted from the default valuesof 0.2 to 0.04 in order to obtain adequate less-di�used results. The larger values of � (resultsare not shown here) show even more di�usive ef-fect as compared with the FTNS3D results.

CONCLUDING REMARKS

The computational solution of the un-steady, compressible, Reynolds-averaged Navier-Stokes equations is used to predict the near- andfar-�eld vortex-wake turbulent ows. Three dif-ferent turbulent models have been used with theNS equations which include the Baldwin and Lo-max model, Spalart and Allmaras model andKW model of Menter. For the near-�eld vortex-wake application, the focus is directed on thedevelopment and roll-up of the tip vortex for asubsonic wing while for the far-�eld vortex-wakeapplication, the focus is directed on the interac-tion of a tip-vortex of a typical 727 Boeing wingwith the temperature �eld of an engine exhaustplume. The results of the �rst application havebeen compared with the available experimentaldata. The comparison shows that the computedresults with the BL and SA models are betterwith the experimental data than those resultswith the KW model. The results of the secondapplication are compared with those of the UNI-WAKE code. The UNIWAKE results show dif-ferent motion of the vortex center along with theexhaust plume as it advances downstream, thanthat of the full Navier-Stokes code. The di�er-ence is attributed to the approximations associ-ated with the UNIWAKE code.

ACKNOWLEDGEMENT

For the �rst three authors, this work is supportedby the NASA Langley Research Center underGrant No. NAG-1-994. The computational re-sources provided by the NAS Facility at Amesand the NASA Langley Research Center are ap-preciated and recognized.

REFERENCES

1. Stolarski, R. S. and Wesoky, H. L., Edi-tors, \The Atmospheric E�ects of Strato-spheric Aircraft: A Fourth Program Re-port," NASA Ref. Pub. 1359, January1995.

2. Hallock, J. N. and Eberle, W. R., \Air-craft Wake Vortices: A State-of-Art Re-view of the United States R&D Program,"FAA Rept. FAA-RD-77-23, Feb. 1977.

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3. Chorin, A. J. and Bernard, P. S., \Dis-cretization of a Vortex Sheet, with an Ex-amples of Roll-Up," Journal of Computa-tional Physics, Vol. 13, Nov. 1973, pp. 423{429.

4. Hackett, J. E. and Evans, M. R., \VortexWakes Behind High Lift Wings," Journalof Aircraft, Vol. 11, July 1974, pp. 397{400.

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6. Rossow, V. J., \Inviscid Modeling of Air-craft Trailing Vortices," Proceedings ofNASA Symposium on Wake Vortex Min-imization," Washington, D. C., Feb. 1976,pp. 4{54.

7. Kandil, O. A., Mook, D. T., andNayfeh, A. H., \Application of the Non-linear Vortex-Lattice Concept to Aircraft-Interference Problems," Advances in En-gineering Science, NASA CP-2001, Vol. 4,Nov. 1976, pp. 1321{1326.

8. McCormick, B. W., Tangler, J. L., andSherrieb, H. E., \Structure of Trailing Vor-tices," Journal of Aircraft, Vol. 5, No. 3,May{June 1968, pp. 260{267.

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12. Liu, C. H., Tavantzis, J., and Ting,L., \Numerical Studies of Motion andDecay of Vortex Filaments{Implementingthe Asymptotic Analysis," AIAA Journal,Vol. 24, 1986, pp. 1290{1297.

13. Ting, L. and Klein R., Viscous VorticalFlows, Lecture Notes in Physics, Vol. 374,Springer-Verlag Berlin Heidelberg, 1991.

14. Greene, G. C., \An Approximate Model ofVortex Decay in the Atmosphere," Jour-nal of Aircraft, Vol. 23, No. 7, July 1986,pp. 566{573.

15. Greene, G. C., Lamar, J. E., and Kuben-dran, L. R., \Aircraft Vortices: Juncture,Wing and Wake," AIAA Paper 88-3743,July 1988.

16. Kandil, O. A., Wong, T. C., and Liu,C. H., \Analysis and Computation of Trail-ing Vortices and Their Hazardous E�ects,"FAA International Symposium, Washing-ton, D.C., Oct. 29{31, 1991, Vol. 2,pp. 36.1{36.24.

17. Wong, T. C., Kandil, O. A., and Liu,C. H., \Computation of Wake-VortexFlows and Control of Their E�ects onTrailing Wings," AIAA 4429-92-CP, Aug.1992, Vol. 1, pp. 280{292.

18. Kandil, O. A., Wong, T. C., and Liu,C. H., \Turbulent Flow over a 747/747Generator/Follower Con�guration and itsDynamic Response," AIAA 94-2383, June1994.

19. Baldwin, B., and Lomax, H., \Thin LayerApproximation and Algebraic Model forSeparated Turbulent Flow," AIAA 78-257,Jan. 1978

20. Spalart P. R., and Allmaras, S. R., \A One-Equation Turbulence Model for Aerody-namics Flows," AIAA 92-0439, Jan. 1992.

21. Menter, F. R., and Rumsey, C. L., \Assess-ment of Two-Equation Turbulence Modelsfor Transonic Flow," AIAA 94-2343, June1994.

22. Quackenbush, T. R., Teske, M. E., and Bi-lanin, A. J., \Enhancement of UNIWAKEExhaust Dynamics and Chemistry," C.D.I.Report No. 94-04.

23. Chow, J. S., Zilliac, G. G., and Bradshaw,P., \Measurement in the Near-Field of aTurbulent Wingtip Vortex," AIAA Paper93-0551, Jan. 1993.

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Figure 1: Typical C-O mesh of AR = 1:5 rectangular wing.

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Figure 2: Typical convergent histories of Log(residual) and lift coe�cient; SA turbulence model.

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Figure 3: De�nition of � at the wing-tip region.

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Figure 4: Comparison of surface-pressure coe�cients with the experiment.

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CptX=0.63P 1.053

O 0.977

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J 0.600

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H 0.448

G 0.373

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D 0.146

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7 -0.308

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CptX=1.19P 1.053

O 0.977

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K 0.675

J 0.600

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H 0.448

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E 0.222

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M

N

N

NN

N O

O

O

O

O

O

Y

Z

X=0.63

0.4 0.6 0.8

-0.2

0.0

0.2

AA

B CCC D

DDD EEE FF

F

FFFF GGGGG HHHH

II

I JJ

KK

KKLL

LMMM

MM

M

M

N

N

NN

N

O

O

OO

O

O

O

O

O

O

P

P

Y

Z

X=1.19

0.4 0.6 0.8

-0.2

0.0

0.2

111

23345

5

6

677899 AAAB

B

B CC

C

C

CDD

DD

D

DD

E

EE

E

E FF

F

F

F

G

G

G

GG

G

H

HH

HH

H

II

I

I

I

I

I

JJ

J

JJ

J

KK

K

K

KLLL

L

L

L

L

L

M

M

M

MM

NN

NNN

OO

O P

P

P

PPP

P

Y

Z

X=0.63

0.4 0.6 0.8

-0.2

0.0

0.2

99 AAAA BB

B CCCCCCC DD

DDE

EEE FF

F GG

G

H

H

HH I

II

J

JJ

JJ

JK

KK

LL

L

LL

M

M

M NNNN

O

OO

O

O

O

O

O

O

P

P

PP

PPP

P

P

P

Y

Z

X=1.19

0.4 0.6 0.8

-0.2

0.0

0.2

1223333 4445

5

666

777

888

9

99A

ABB

B

B

CC

C

CC

C DDD

DEE

E

E

EF

F

F

FFG

G

GG

H

HHHHHH

I

III

JJ

JJJ

K

K

KK

K LL

LLL

MM

MMMM

MM

NN

N

N

N

O

O

OO PPP

P

P

P

Y

Z

X=0.63

0.4 0.6 0.8

-0.2

0.0

0.2

34

566 777

777 88

88 9

9

99 AAA BBB CCC

C DDD EEEE

E

EE

FF

F

FF

G

GGG

G

H

HHH I

I

II

J

J

J KKK

K

K

K

L

L

L

LL M

MM

M

M

N

NN

N

O

O

OO

O

O

OP

P

P

P

P

P

P

PPP

PP

P

PP

Y

Z

X=1.19

Figure 5: Comparison of crossow Cpt contours at two chordwise stations; top-experiment, second-BL, third-SA and bottom-KW.

10

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

7

9

A

BC

C

D

E

F

G

G

HH

I

IJ

JJ K

K

K

L

L

L

M

M

M

N

N

N O

O

O

Y

Z

Experiment

CptX=0.63P 1.053

O 0.977

N 0.902

M 0.826

L 0.751

K 0.675

J 0.600

I 0.524

H 0.448

G 0.373

F 0.297

E 0.222

D 0.146

C 0.070

B -0.005

A -0.081

9 -0.156

8 -0.232

7 -0.308

6 -0.383

5 -0.459

4 -0.534

3 -0.610

2 -0.685

1 -0.761

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

79

A

B

CD

E

FG

H

H

I

I

J

J

K

K

L L

M

M

N

N

N

O

O

O

O

O

O

O

O

OO

Y

Z

Experiment

CptX=1.19P 1.053

O 0.977

N 0.902

M 0.826

L 0.751

K 0.675

J 0.600

I 0.524

H 0.448

G 0.373

F 0.297

E 0.222

D 0.146

C 0.070

B -0.005

A -0.081

9 -0.156

8 -0.232

7 -0.308

6 -0.383

5 -0.459

4 -0.534

3 -0.610

2 -0.685

1 -0.761

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

11

2

3

44 5

6

67

8

88

9

9

9

A

A

AB

B CC

C D

D

E

E

E

F

F G

G

G

G

G

H

H

HHHI

I

I

I

J

J

J

J KK

K

LL

L

M

M

N

N

N

N

O

O

O

O

Y

Z

X=0.63

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

A

B CCD

DD E

EEE

FF

FF

FF GGG

GG HHH

I

I

I J

J

JJ

K

KKL

L

L

L

L

L

M

M

M

MM

N

N

N

O

O

O

Y

Z

X=1.19

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

1

23

3

44

4556

67

7

8

88 99

AA BB

B CC

D

DD

D

EE

EE

E

E

F

F

F

F

G

GG

G H

H

H

H

II

II

J

J K

K

K

K

LL

LL

L

M

M

M

M

N

N

N

OO

P

P

P

Y

Z

X=0.63

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

8999 AA

A

B

B

B

BC

CCCD

D

DE

E

EEE

F

F

FF

F G

G

G

G

H

H

HH

I

I

IJ

J

J

KK

K

K

L

L

L

M

M

M

M

MM

N

N

N O

O

P

P

Y

Z

X=1.19

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

1

1

2

22

2

33

334

4

5

55

6

677

8

8

8

9

9

A

A

B

B

B

B

CC

C

C

D

D

D

E

E

E

F

F

F

G

G

G

H

HII

IJJ

J

J

J

K

K

K

L

L

LL

L

MM

N

N

O

O

O

P

P

P

Y

Z

X=0.63

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

2

3466 777

77 88

8

89

999 AAA B

B

B CC

C

C DD

EEEE

E

EEF

FF

FF

G

GG

GG

H

HHH I

I

II

J

JJ

K

KK

KK

K

L

L

L

L

LM

M

M

MM

N

N

O

O

P

Y

Z

X=1.19

Figure 6: Close-up of crossow Cpt contours at two chordwise stations; top-experiment, second-BL,third-SA and bottom-KW.

11

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

4

5

6

7

7

7

8

8

8

9

9

9

A

A

A

A

B

B

B B

C

CC

C

D

D

D

D

EE

E

E

F

F

FF G

G

GH

H

H

H

I

I I

J

J

J

K

KK

K

L

L

M

N

N

Y

Z

Experiment

VcX=0.63P 1.072

O 1.027

N 0.983

M 0.938

L 0.894

K 0.849

J 0.805

I 0.760

H 0.716

G 0.671

F 0.627

E 0.582

D 0.538

C 0.493

B 0.448

A 0.404

9 0.359

8 0.315

7 0.270

6 0.226

5 0.181

4 0.137

3 0.092

2 0.048

1 0.003

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

7

7

8

8

8

9

9

A

A

A

A

B

B

B

B

C

C

C

C

D

D

D

D

D

E

E

E

E

F

F

F

F

F

G

G

G

G

H

H

HH

II

I

J

J

J

J

K

K

K

L

LM

Y

Z

Experiment

VcX=1.19P 1.072

O 1.027

N 0.983

M 0.938

L 0.894

K 0.849

J 0.805

I 0.760

H 0.716

G 0.671

F 0.627

E 0.582

D 0.538

C 0.493

B 0.448

A 0.404

9 0.359

8 0.315

7 0.270

6 0.226

5 0.181

4 0.137

3 0.092

2 0.048

1 0.003

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

11

1

2

22

3

3

4

4

4

4

5

5

5

6

66

6

67

7

7

777

88

8

8

8

8

9

9

99

9

9

9

A

A

A

A

A

A

B

BB

B

B

B

B

C

C

C

C

C

C

C

C

D

DD

D

DDD

E

E

E

E

EE

F

F

F

FF

F

F

F

F

GGG

G

GG

H

H

H

H

H

H

I

I

I

II

I

JJJ

J

J

J

JJ

K

K

K

K

K

L

LL

L

M

MM

N

N

O

O

P

P

P

Y

Z

X=0.63

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

7

8

8

9

9

9

A

A

A

A

B

B

B

C

C

C

C

C

D

DD

DD

D

E

E

EEE

E

E

E

EEE

F

FFF

F

FFF

G

G

GGGGG

G

HH

Y

Z

X=1.19

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

1

1 22

3

3

4

4

4

555

555

6

6

6

6

77

7

7

7

8

8

8

8

88

9

9

9

9

9

9

9

9

AA

A

AA

A

A

BB

B

B

B

B

C

CC

CC

C

D

DDD

D

DD

D

D

E

EE

E

E

E

E

F

F

F

F

F

F

G

G

G

G

GGG

G

H

H

H

HHH

H

II

III

J

J

JJ

J

KK

K

KKK

K

L

LL

L

M

MM

NNO

Y

Z

X=0.63

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

5

55 66

6666

7

7

77

77

8

8

88

8

8

9

9

9

99

A

A

A

A

A

B

B

B

B

B

BB

B

C

C

C

C

C

C

D

D

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D

D

DD

E

E

Y

Z

X=1.19

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

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0.00

0.05

2

2

3

3

44

4

4

555

5

5

5

6

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666

7

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77

7

888

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89

9

9

9

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9

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9

A

A

AAA

A

A

B

B

B

B

B

B

C

CC

C

C

D

D

D

D

DD

D

D

D

E

E

EEE

E

F

F

FF

GG

GG

HH

H

H

H

H

I

I

I

I

I

I

J

JJ

JJ

K

K

KK

L

L

LL

M

M

M

N

N

N

N

N

OO

O

P

Y

Z

X=0.63

0.60 0.65 0.70 0.75 0.80 0.85

-0.10

-0.05

0.00

0.05

26

6

7

7

8

888 9

9

99

A

AAA

A

B

B

BB

B

C

C

CC

C

C

C

CCDD

DD

DDD

E

EE

E

EEEF

Y

Z

X=1.19

Figure 7: Close-up of crossow velocity contours at two chordwise stations; top-experiment, second-BL, third-SA and bottom-KW.

12

0.0 0.5 1.0 1.5 2.0 2.5-1.0

-0.5

0.0

0.5

1.0

2

3

5

6

8

9

A

A

B

B

CD

D

E

E

E

F

F

F

G

G

H

H

I

I

I

J K

L

L

M

NO

Z/s

Y/s

X/s=0.0Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5-1.0

-0.5

0.0

0.5

1.0

2

3

5B

F

Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

Figure 8: Crossow velocity (dashed) and temperature (solid) contours at the inow section.

Xmax

1

2

3

4

5

6

nmax

n

stage

th

th

ststage

F -- Fine Grid

a

b

F

F

F

F

F

F

F

F

Figure 9: Schematic sketch of the overlapping zonal method.

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

2

3 4

7

8

8

9

A

AB

C

E

E

E

F

F

G

G

G

H

I

I

I

J

J

K

K

L L

M

Z/s

Y/s

X/s=14Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

1

2

3

4

5

7

8AD

G

Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

56

7

7

9

A

C

C

D

D

E

F

F

G

H

H

I

IJ

J

LL

M

M

Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

1

2

3

4

58

9

Z/s

Y/s

X/s=14Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

Figure 10: Comparison of crossow T and Vc contours at one downstream station; left-FTNS3D,and right-UNIWAKE.

13

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

6

7

8

9

9

A

A

B

B

C

C

D

DE

F

G

G

H

H

I

IJ

J

K K

L

M

N

Z/s

Y/s

X/s=24Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

1

2

3

4

5

6 A

Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5 59

9

A

B

B

CDE

E

F

G

H

H

I

I

J

J

KK

L

L

NO

Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

1

1

2

2

3

4

56

Z/s

Y/s

X/s=24Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

5

5

89

9

A

BC

D

EF

F

G

G

G

H

H

IJ

J

K

K

L

Z/s

Y/s

X/s=34Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

1

1

2 2

3

4

6

Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.53

46

7

8

9

A

B

B

C

D

E

F

F

F

G

G

G

H

I

I

J

J

KN

Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

1

1

2

2

3 6

Z/s

Y/s

X/s=34Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

25

6

7

9

B

C

D

E

F

F

G

G

H

H

I

I

J

L

L

M

N O

Z/s

Y/s

X/s=44Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

1

1

2

3

4

Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5-1.5

-1.0

-0.5

0.0

5

6

6

78

8

9

A

B

B

CD

D

E

E

F

F

GG

H

H

I

J J

L

M

Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5-1.5

-1.0

-0.5

0.0

11

2

2

34

Z/s

Y/s

X/s=44Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

6

6

7 7

7

8

A

A

B

B

C

C

D

D

D

E

F

FG

G

H

H

I

J

J

K

K

O

Z/s

Y/s

X/s=54Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

1

1

2 2

5

Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5

-1.5

-1.0

-0.5

0.0

3 4

5

8

8

8

A

A

C

C

D

D

D

E

EF

G

H

H

II

J

K

Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.5

-1.0

-0.5

0.01

1

2 23

4

Z/s

Y/s

X/s=54Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

Figure 11: Comparison of crossow T and Vc contours at four downstream stations; left-FTNS3D,and right-UNIWAKE.

14

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5 4

4

7

8

8

99

A

A

A

BB

C

D

E

F

F

F

G

G

HI

J

K

LN

Z/s

Y/s

X/s=60Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

1

1

23

Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5

-1.5

-1.0

-0.5

0.0

34

4

6

7

8

9

A

B

BC

C

D

D

D

E

F

F

G

G

H

H

I

J

J

L

Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.5

-1.0

-0.5

0.0

1

12

2

3

Z/s

Y/s

X/s=60Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

36

78

8

9

A

A

BC

D

D

D

EE

F

F

G

H

I

J

K

O

Z/s

Y/s

X/s=70Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

1

23

Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5

-1.5

-1.0

-0.5

0.0

4

5

5

67

8

9

A

A

B

C

C

D

D

E

E

F

FG

G

H

IJ

K

L

Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.5

-1.0

-0.5

0.0

1

1

2

3

Z/s

Y/s

X/s=70Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

4

4

56

6

7

7

8

8

9

9

AB

B

C

D

D

D

E

E

F

FG

H

I

J

M

N

Z/s

Y/s

X/s=82Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5

-1.0

-0.5

0.0

0.5

1

1

2

Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

0.0 0.5 1.0 1.5 2.0 2.5-2.0

-1.5

-1.0

-0.5

4

5

7

8

9

9

9

C

D

E

F

F

G

G

G

H

HI

J

J

Level Vc

P 0.0480

O 0.0462

N 0.0443

M 0.0425

L 0.0407

K 0.0389

J 0.0370

I 0.0352

H 0.0334

G 0.0315

F 0.0297

E 0.0279

D 0.0260

C 0.0242

B 0.0224

A 0.0206

9 0.0187

8 0.0169

7 0.0151

6 0.0132

5 0.0114

4 0.0096

3 0.0078

2 0.0059

1 0.0041

0.0 0.5 1.0 1.5 2.0 2.5-2.0

-1.5

-1.0

-0.5

1

1

1

1

2

23

Z/s

Y/s

X/s=82Level T

P 1.9800

O 1.9407

N 1.9015

M 1.8622

L 1.8230

K 1.7837

J 1.7444

I 1.7052

H 1.6659

G 1.6266

F 1.5874

E 1.5481

D 1.5088

C 1.4696

B 1.4303

A 1.3911

9 1.3518

8 1.3125

7 1.2733

6 1.2340

5 1.1947

4 1.1555

3 1.1162

2 1.0770

1 1.0377

Figure 12: Comparison of crossow T and Vc contours at three downstream stations; left-FTNS3D,and right-UNIWAKE.

15