NASA-TM-83448 19830020908NASA Technical Memorandum 83448

' Applications of the Contravariant Form' of the Navier-Stokes Equations

Theodore KatsanisLewis Research CenterCleveland, Ohio ................

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Prepared for theSixth Computational Fluid Dynamics Conferencesponsoredby the AmericanInstitute of Aeronautics and Astronautics

, Danvers, Massachusetts, July 13-15, 1983

• ,_ I 111111

I J/kSA .....

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NASA Technical Library

3 1176 01441 9767

APPLICATIONSOF THE CONTRAVARIANTFORMOF THE NAVIER-STOKESEQUATIONS

Theodore KatsanisNational Aeronautics and Space Administration

Lewis Research CenterCleveland, Ohio 44135

SUMMARY

__ The contravariant Navier-Stokes equations in weak conservation form are_ well suited to certain fluid flow analysis problems. Three-dimensional con-

travariant momentumequations may be used to obtain Navier-Stokes equations inweak conservation form on a nonplanar two-dimensional surface with varyingstreamsheet thickness. Thus a three-dimensional flow can be simulated withtwo-dimensional equations to obtain a quasi-three-dimensional solution forviscous flow. Whenthe Navier-Stokes equations on the two-dimensional non-planar surface are transformed to a generalized body-fitted mesh coordinatesystem, the resulting equations are similar to the equations for a body-fittedmesh coordinate system on the Euclidean plane.

Contravariant momentumcomponents are also useful for analyzing compres-sible, three-dimensional viscous flow through an internal duct by parabolicmarching. This type of flow can be efficiently analyzed by parabolic marchingmethods, where the streamwise momentumequation is uncoupled from the twocrossflow momentumequations. This can be done, even for ducts with a largeamount of turning, if the Navier-Stokes equations are written with contravar-iant components.

INTRODUCTION

The Navier-Stokes equations are often transformed to special coordinatesystems for computational purposes. However, if it is desired to have theequations in conservation form, it is generally necessary to use Cartesian mo-mentum components. On the other hand, there are cases where it is useful touse the contravariant momentumcomponents in weak conservation form. Two ex-amples are considered here: one example is a nonplanar flow surface embeddedin a three-dimensional flow field; the other example is three-dimensional flowin a turning duct. Both examples have applications in turbomachinery.

There are many instances of two-dimensional flow where the flow is notplanar; i.e., either the two-dimensional surface is curved or the streamsheetthickness is not constant (e.g., axisymmetric flow). In such cases the three-dimensional Navier-Stokes equations may be reduced to two dimensions. A gen-eral method for doing this is to transform the three-dimensional equation to anew coordinate system such that two coordinate lines lie on the surface of in-terest, with the third coordinate normal to the surface. The momentumequa-tions are then combined so as to obtain contravariant momentumcomponents.Thus two components of momentumlie on the surface and the third component isomitted; the equation will involve only the two contravariant components. Asan illustration, equations are presented here for viscous flow on a rotatingsurface of revolution. A flow solution on such a surface would be useful foranalyzing flow through a turbomachine.

Ducts of complexgeometrymay be analyzedefficientlyif the flow is of a"boundarylayer"type; i.e., flow with a predominantdirection(no reve-seflow), and negligiblestreamwisediffusion. This kind of flow can be analyzedby parabolicmarchingmethods. Patankarand Spalding (ref. 4), Briley(ref. 8), Robertsand Forester (ref. 7), Moore and Moore (ref.6), and othershave calculatedthree-dimensionalviscousflow in ducts by means of parabolicspace marching. For this purposethe momentumequationsare uncoupled. Thestreamwisemomentumequation is used to advancestreamwisevelocitycomponents,and cross-streammomentumequationsare used to advancethe cross-streamveloc-ity components. If a body-fittedcoordinatesystem is used, the contravariantcomponentsof the momentum equation (withcontravariantvelocitycomponentsasdependentvariables)providea naturalmethod of uncouplingthe streamwisemo-mentum equationfrom the cross-streammomentumequation. As an example,equa-tions are presentedfor flow througha rotatingpassage,e.g., a blade passagethrougha turbomachineblade row.

NAVIER-STOKESEQUATIONS

The two-dimensional,planar,nonsteadyNavier-Stokesequationsin conser-vation form are

Btq + _xE + _yF : BxR + ByS (1)

whereD

p pVx pVy

oVx 2 +q = E = PVx P F = °VxVy

pVy PVxVy oV2y+ p

e Vx(e + p Vy(e + p

o o "

_xyR = xx S =

_xy _yy

Vx_xx + Vy_xy + k BxT Vx<xy + VyTyy + k ByTm

Here p is density, V is velocity, p is pressure, e is total internal en-ergy per unit volume, T is temperature, and T is shear stress tensor. Thex and y subscripts refer to the x and y coordinates. For numerical so-lutions, the equation may be transformed to an arbitrary { - n coordinatesystem (ref. I). These are usually body-fitted coordinates, such that constantvalues of { or n correspond to the body surface. The transformed equationsin conservation form are

_t_-+ B_ . _n_ = _n_ (2)

2

whereB B

pv ovn

" I PVxVn + nxP

q =q _ =1 pVyV_oVxV_++ _;YP_:X F ='J pVyVn + nyP

(e + p)V_ (e + p)Vn

0

+n:nx xx xynx'_xy+ ny-Cyy

nxBx - nyBy

= +Vy_ +k _ TBx Vx'_xx xy x

= +kaTBy Vx.txy + VSyy y

The superscripts _ and n refer to contravariant velocity components,and J is the Jacobian of the transformation. The thin-layer approximationis used here (viscous derivatives in the streamwise direction are neglected).Note that the momentumcomponents (the second and third equations) are still xand y components. If { and n (either contravariant or covariant) momen-tum components are used the equation cannot be written in conservation form ifthere is any curvature of the { or n coordinate lines.

It is desired to obtain two-dimensional Navier-Stokes equations for non-planar surfaces (i.e., quasi-three-dimensional). The desired equations can beobtained from the three-dimensional version of Eq. (2), written with contra-variant momentumcomponents in a {, n, c coordinate system. The _ and ncoordinates lie in the desired nonplanar surface, and _ is chosen in a spe-cial way to be orthogonal to the surface. It is assumed that the chosen non-planar surface is a stream surface, i.e., the contravariant velocity component

= O. Whenthis is done the third momentumequation can be eliminated andthe two-dimensional Navier-Stokes equation for flow on a two-dimensional sur-face is obtained. Rather than give the general equation, an example will begiven which is of interest for turbomachinery flow analysis.

NAVIER-STOKESEQUATIONSONA ROTATINGSURFACEOF REVOLUTION

For analysis of flow through turbomachinery blade rows it is useful toconsider flow on a blade-to-blade surface as shown in figure I. The coordi-nates are m (physical distance) and o (radians). In addition the stream-sheet thickness h must be specified, as shown in figure 2. The thicknessh and the radius r are both functions of m only. The three-dimensional

3

contravariantmomentum equationsare obtained in m, e, n coordinates,wheren is normal to the blade-to-bladesurface. The n coordinateis definedsuchthat the metric coefficientis l/h, so that not only is the normal velocityzero, but also the derivativein the n directionis zero. The surfaceofrevolutionis rotatingat a constantrotationalvelocity m. The relativevelocityis W; m and e subscriptswill denote physical (not covariant)velocities.

The Navier-Stokesequationobtainedis

atq' + amE' + arF' = amR' + arS' + K' (3)

where

° °"m °"_

q':_ °% °"m%e " Wm(e+ p) We(e + p) + turf• ml

R' I Tmm S' I Tme= T1 - rj1

nl) eo

+ k am'[ ee oWm_mm + Wo_me Wm_mo + WB< + k/r aw

I

0

1 sin e (p%2+ p-le° ) -r amhp

K' - rjI sin e(Tn_ - pWBWm)

0

J1 = 1/rh is the Jacobian and m and B momentumcomponentsare used.Except for the sourceterm, K', Eq. (3) is similarto Eq. (I). Eq. (3) can nowbe transformedto body-fittedcoordinates,similarto Eq. (2). The transformedequation is

G _,++_,: +_,+_, (4)at ' + au n n

where

4

m

Wn

q, ,W W_ + _mpm _, = _ PWmWn+ nmP

(e + p)W_ (e + p)WnJ

"o

+ net /r_, = nmTmm me _, K'

+ nOT : T22nm<me Be/r

nmBm + nOB°/r

Bm = WmTmm + WoTmo + k BmT

+ k/r B TBe -- Wm_me+ WeTee e

V# = _mWm + {e_/rlcontravariant relative_ : nmWm + ne_ Ir velocitycomponents

J = JIJ2 is the Jacobian (x, y, z) . ({, n, n), J2 is the Jacobian(m, e) . ({, n), and m and e momentumcomponents are used. Equation (4)uses m and e momentumcomponents, and is similar to Eq. (2), except forthe source term K' which is required because of the curvature of the m ande coordinates. The techniques and computer codes for solving Eq. (2) (ref. I)should be able to be extended to Eq. (4).

THREE-DIMENSIONALNAVIER-STOKESEQUATIONSFORA ROTATINGCOORDINATESYSTEM

An example of a viscous, three-dimensional internal flow that can be ana-lyzed by parabolic space marching is flow through a turbomachinery blade row.The desired equations are obtained by transforming the Navier-Stokes equationsin cylindrical (r, e, z) coordinates to a body-fitted (_, n, _) coordinate sys-tem. Then the contravariant components of the momentumequations are obtained.These equations involve only the contravariant velocity components. (The co-variant components of momentumhave a simpler form, but involve both covariantand contravariant velocity components (ref. 5).) This permits the streamwisemomentumcalculation to be uncoupled from the cross-stream momentumcalcula-tion. Continuity is satisfied indirectly by modifying an inviscid pressure

field to correctfor viscouseffects. Modifyingthe streamwisepressuregra-dient will affect streamwisevelocity componentsso that global continuityissatisfied. Modifyingthe pressureover a cross sectionwill affect the cross-stream velocitiesso that local continuitycan be satisfied. The steady-statethree-dimensionalNavier-Stokesequationsfor a rotatingcoordinatesystem(transformedto a body-fitted({, n, €) coordinatesystem,and with contravar-iantmo_ntum components)are:

B E + B F + B G = B S + B T + K (5)n € n

wherem _ m D

pW_ pWn p_

_w_w_ _wnw_ oW_w_

: _ oN_wo : .wnw" _ : .w_w_oW_, _wnw_ p_

_W_ I .pWnl L.pW_ I .

o on + _OT_/r + n { + €/r + { €_rTr _zTz _r_r _OTO z_z

] n + no_ /r + nzT z nrTr= " nrtr : nOTB nzlz

n + Ce_/r + n { + €/r + {CrTr CzTz CrTr COT0 zTZ

+C BB/r +€ BznrBr + nobo/r + nzBz CrBr B Z

0

pGp - g_ g_ /r + /r2_p_ - g_nPn - P_ - _r_oo _8_re

_pG gn_p_ gnnPn gn_Pc nr_oe no_re= .... /r + /r2

np

pG p gC_p_ gCn _ g_€ /r + /r2- - Pn Pc - Cr_eo €o're

0

: + {oWo/r +{W_ _rWr zWz

wn Contravariantrelativevelocitycomponets

6

n : nrT + not + nzT etcTr rr ro rz' "

Br = WrTrr + WoTrB + WzTrz + k BrT, etc.

I - e+Pp _rVe = CpT'- _rVo,Rothalpy

G_p, Gnp G are functionsof velocityand geometry' {p

The flow is in the { direction,and n, _ are the cross-streamdirections.The thin layer approximationis used.

CONCLUDINGREMARKS

Two examplesof applicationsof the contravariantNavier-Stokesequationshave been presented.

The first exampleshows that the contravariantNavier-Stokesequationscan be used to obtain two-dimensionalequationsfor a rotating surfaceofrevolutionwith varyingstreamsheetthickness. The equationsare similar inform to the conservation-lawform of the Cartesiantwo-dimensionalNavier-Stokes equations. The methods alreadydevelopedfor solvingthe Cartesiantwo-dimensionalequationscan thus be used for analyzingflow on a blade-to-blade surfaceof a turbomachine. Other applicationsfor nonplanarsurfacesseem possible.

The second example is for three-dimensionalviscousflow througha rota-ting duct. The contravariantform of the Navier-Stokesequationsprovides amethod for uncouplingthe streamwisemomentumfrom the cross-streammomentum.This makes it easier to use parabolicmarchingmethods for turbomachinery....blade passageswith a large amountof turning. A similartechniquecould beused with other types of internalpassages.

REFERENCES

1. Steger,J. L., "ImplicitFinite DifferenceSimulationof Flow aboutArbitraryTwo-DimensionalGeometries,"AIAA Journal,Vol. 16, No. 7,July 1978, pp. 679-686.

2. Pulliam,T. H. and Steger,J. L., "ImplicitFinite DifferenceSimulationsof Three-DimensionalCompressibleFlow," AIAA Journal,Vol. 18, No. 2, Feb.1980, pp. 159-167.

3. Katsanis,T., "FORTRANProgramfor CalculatingTransonicVelocitieson aBlade-to-BladeStream Surfaceof a Turbomachine,"NASA TN D-5427,1969.

• "A CalculationProcedurefor Heat,4. Patankar,S. V. and Spalding,D. B ,Mass and MomentumTransfer in Three-DimensionalParabolicFlows,"InternationalJournalof Heat and Mass Transfer,Vol. 15, Oct. 1972,pp. 1787-1806.

• 5. Walkden, F., "The Equationsof Motion of a Viscous,CompressibleGasReferredto an ArbitrarilyMoving CoordinatingSystem,"Royal AircraftEstablishment,RAE-TR-66140,ARC R/M-3609,1969.

6. Moore, J. and Moore, J. G., "Calculationof Three-Dimensional,ViscousFlowand Wake Developmentin a CentrifugalImpeller,"PerformancePredictionofCentrifugalPumps and Compressors,edited by P. Cooper, S. Gopa]akrishnan,C. Grennan, and J. Switzer,ASME, 1979, pp. 61-67.

7. Roberts,D. W. and Forester,C. K., "ParabolicProcedurefor Flows in Ductswith ArbitraryCross Section,"AIAA Journal,Vol. 17, No. 1, 1979, pp.33-40.

8. Briley,W. R., "NumericalMethod for PredictingThree-DimensionalSteady" Journalof ComputationalPhysics,Vol 14 No. 1,ViscousFlow in Ducts, • ,

1974, pp. 8-28.

Blade-tobladesurface

Figure1. - Blade-to-bladesurfaceof revolutionshowingm - 9coordinates.

//

FF

/

J/\

\

h

/Blade-to-bladestreamsurfaceJ

Figure2. -FlowIn a mixed-flowstreamchannel.

1. ReportNo. 2. GovemmentAccessionNo. 3. Reciplent'sCatalogNo.

NASATM-834484. Title and Subtitle 5. Report Date

Applications of the Contravariant Form of the Navier-Stokes Equations 6. Performing Organization Code

505-31-0A

7. Author(s) 8. Performing Organization Report No.

Theodore Katsanis E-174910. Work Unit No.

9. PerformingOrganizationNameandAddress11. Contract or Grant No.

National Aeronautics and Space AdministrationLewis Research CenterCl evel and, Ohio 44135 13.Type of Report andPeriodCovered

Technical Memorandum12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration t4 Sponsoring Agency CodeWashington, D.C. 20546

15. Supplementaw Notes

Prepared for the Sixth Computational Fluid Dynamics Conference sponsored by theAmerican Institute of Aeronautics and Astronautics, Danvers, Massachusetts,July 13-15, 1983.

16. Abstract

The contravariant Navier-Stokes equations in weak conservation form are wellsuited to certain fluid flow analysis problems. Three-dimensional contravariantmomentumequations may be used to obtain Navier-Stokes equations in weak conser-vation form on a nonplanar two-dimensional surface with varying streamsheet thick-ness. Thus a three-dimensional flow can be simulated with two-dimensional equa-tions to obtain a quasi-three-dimensional solution for viscous flow. When theNavier-Stokes equations on the two-dimensional nonplanar surface are transformedto a generalized body-fitted mesh coordinate system, the resulting equations aresimilar to the equations for a body-fitted mesh coordinate system on the Euclideanplane. Contravariant momentumcomponents are also useful for analyzing compreso-amble, three-dimensional viscous flow through an internal duct by parabolic march-ing. This type of flow can be efficiently analyzed by parabolic marching methods,where the streamwise momentumequation is uncoupled from the two crossflow momen-tum equations. This can be done, even for ducts with a large amount of turning,if the Navier-Stokes equations are written with contravariant components.

17. Key Words (Suggested by Authors)) 18. Distribution Statement

Navier-Stokes equations Unclassified - unlimitedComputational fluid dynamics STARCategory 02TurbomachineryQuasi-three-dimensional

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