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Turbulent flow calculations in S-shaped

diffusing ducts using a viscous marching

technique

S.C.M. Yu*

Fluids Section, Department of Mechanical

Engineering, Imperial College of Science,

Technology and Medicine, University of London,

ABSTRACT

Three-dimensional turbulent flow characteristics in S-shaped diffusing ducts arecomputed using a parabolised form of the Navier Stokes equations incorporatedwith an algebraic mixing length turbulence model. Elliptic effects of the pressurefield are accounted for by a potential flow solution plus a viscous correctionfactor. Iterative calculation methods are therefore not necessary. Thus, a finegrid density can be implemented in the transverse plane to obtain spatialresolution and as a consequence, wall functions are not required in thecalculations. Extensive comparison between velocity measurements andcomputed solutions are reported and the limitations of using this approach areconsidered.

INTRODUCTION

The performance of an intake to a gas turbine engine is characterized by itsrecovery of free stream total pressure, its distribution of total pressure and theamount of swirl in the air flow at the entry to the engine compressor face. Thefirst quantity is linked closely to the thrust of the engine and the latter two tooperational conditions and in particular to the compressor stall limit. The S-shaped diffuser is common in today's jet aircraft, as for example in the BritishAerospace Tornado and Jaguar. Subsonic flow in typical intake ducts involvesthe generation of secondary flows which will lead to significant flow distortionsat the engine face and important viscous effects with the boundary layerthickness often comparable to the duct radius.

* Present Address : School of Mechanical and Production Engineering,Nanyang Technological University, Singapore.

Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

306 Computational Methods and Experimental Measurements

Solving the Navier Stokes equations for three-dimensional viscous flows incurved ducts has been shown to be possible such as by, Humphrey, Taylor andWhitelaw [1] but requires a large amount of computer storage and run time. Theneglect of the streamwise diffusion terms and the separate definition of thestreamwise and transverse pressure gradients lead to parabolic equations whichhave led to reasonable agreement between solutions and experiments in the flowsin mildly curved ducts, as shown in the calculations by Patankar and Spalding[2]. However, according to Briley and McDonald [3], they do not enable thestreamwise momentum balance to be influenced by the experimentally observeddistortions of the static pressure field and are therefore not appropriate in thecalculation of the typical intake geometries since they involve comparativelystrong curvature.

The abandonment of the separate definition of pressures and neglect of thestreamwise diffusion terms in the Navier Stokes equations leads to partiallyparabolic equations which have been shown to be capable of calculating theflows in strongly curved ducts (with no significant streamwise separation), asdemonstrated by Launder, Choi and lacovides [4] using this approach.However, solution of partially parabolic equations requires global iterations andthree-dimensional storage for the pressure field, thus large computer storage andrun times are required though less than those required to solve the Navier Stokesequations.

It is clear from the above that it is desirable to solve equations whichrepresent strong curvature by an economical solving method. The approximateform of Navier Stokes equations of Briley and McDonald [3] permits a forward-marching integration as with parabolic equations and is appropriate to stronglycurved ducts. Since three-dimensional storage of the pressure field and theiterative calculation methods are not necessary, a fine grid density can beimplemented in the transverse plane to obtain spatial resolution and, as aconsequence, wall functions are not required in turbulent flow calculations. Asshown by Launder [5], calculations to the walls lead to better agreement withmeasurements than those which use wall functions.

The approach of Briley and McDonald [3] has been applied successfully tocurved ducts whose geometries can be fitted conveniently with orthogonalcoordinates system. However, the geometries encountered in typical intake ductscannot be represented easily by orthogonal coordinates, and this led to anextension of this approach using nonortriogonal coordinates as by Anderson [6].

The following sections describe briefly the governing equations whichinclude the main assumptions, the coordinate system and the turbulence model.Computed results will then be presented and discussed in relations to theexperimental measurements. This paper ends with a list of more importantconcluding remarks.

GOVERNING EQUATIONS

The flow is computed by a single sweep spatial marching procedure whichsolves an approximate^ form of the Navier-Stokes equations. Velocities aredivided into primary, Up, and secondary components, Us, the former representsthe main flow velocity and is aligned parallel to the reference line which canrepresent the geometry of the duct and normal to the transverse surface whilethe latter represents the velocities in the transverse plane, i.e.


Computational Methods and Experimental Measurements 307

U = Up + Ug (1)

with the primary flow velocity represented as

where \\ is the directional vector perpendicular to the centre-line direction andnormal to the transverse plane. In general orthogonal coordinates, the primarymomentum equation, with the neglect of the streamwise diffusion terms,i]-V(|iV- U), can be written as

Mi= ir [ (pU- V)U + V(P + Py) - pF ] =0 (3)

where V(P + Py) is the pressure, and pF is force due to the viscous stresses.

The streamwise pressure gradient is represented by three-dimensionalpotential flow solutions plus a one-dimensional correction for viscous blockageeffects and pressure loss. For internal flows, Pv is also determined to ensurethat an integral mass flux condition is satisfied throughout the curved duct such

JpUpdA = Constant (4)A

The streamwise vorticity equation is derived from the transverse momentumequations. It may be written as:

Ms = i?M2 + i]M] (5)

where Mi andMg are the momentum equations in the transverse directions. Thetransport of the streamwise vorticity, QI, is then given by

il-VxMs = 0 (6)

The streamwise diffusion terms in Eq.(6) are neglected in a manneranalogous to the streamwise momentum equation. The secondary flow velocity,Us and Eq. (1), is derived from scalar and vector surface potential denoted as 0and (p respectively and can be written as

— V x

with p the density, and po a constant reference density, where Vg is the surfacegradient operator which is given for orthogonal coordinates by

Vg = V- h (ii-V ) (8)

The overall velocity decompositon of Eq. (1) becomes



U = U + [Vs<|)+Vxii<p] (9)

Equations relating <j> and q> with Up, p, and the secondary flow vorticitycomponent H] can be derived using Eq. (9) .

From continuity,

V U = V upij + V • Vg()) + [V • — V x i% cpl (10)r

and from the definition of the streamwise vorticity,

O] = ir V x U= i] V x upi] + [i]- V x Vg^ + ij-V x ̂ V x ij (11)

The last term of Eq.(lO) and the second term of Eq.(ll) are zero by vectoridentity and can be written as:

9-9s4) = - V . U p (12)

and

y- i i - V x U p = i i - V x V x i i ( p (13)

In the present investigation, energy equations are not being used since theflows considered are mainly liquid flow experiments. For turbulent flow, analgebraic mixing length turbulence model is used. The mixing length / isdetermined from the empirical relationship of McDonald and Camarata [81 forequilibrium turbulent boundary layers which can be written as

/ (y) = 0.095btanh[ky/(0.095bl ' # (^

where 65 is the local boundary layer thickness, k is the von K arm an constant,taken as 0.43, y is distance from the wall, and D is a sublayer damping factordefined by

D = l / 2 [ + - f ] / < y (15)

Here p is the normal probability, y+ = y (i/p)̂ /̂(u/p), i is local shear stressy+=23, GI = 8, y is the distance to the nearest wall and 85 and Cf are computedfrom a momentum integral solution for flow in a straight pipe of circular crosssection with the same area distribution as the case being evaluated.

The wall damping function used by Briley and McDonald [31 and that of VanDriest [9] are plotted against y+ in figure 7b. The distributions of the dampingfactors are very different and that of Van Driest influences the flow beyond y+ =100 whereas that of Briley and McDonald [31 stops at around y+ = 40. Since theVan Driest's damping factor has been widely adopted in turbulent flowcalculations, its application is also considered below.



The above group of equations are transformed into a non-orthogonal body-fitted coordinate system. They are then solved by forward marching in thestream wise direction from an initial station where the flow profiles are known,by an implicit finite-difference technique. Details of the solution algorithm can befound in Yu [14].

Flare approximationIn some cases, a small region of streamwise recirculation can arise in curvedducts and the streamwise velocity components would become less than zero,i.e. pun< 0. The presence of the negative velocity components would beunstable for a forward marching technique. Improved stability can be achievedby adding small artificial convection terms to the finite difference equations.This technique is known as the FLARE approximation after Flugge-Lotz andReyhner [7]. For small regions of reversed primary flow the technique permitsthe calculations to proceed downstream beyond reattachment, confining theapproximation to the small separated region.

RESULTS

This section reports the calculated results and compares them with the velocitymeasurements obtained by Rojas, Whitelaw and Yianneskis [10], Whitelaw andYu[ll&13].

S-Diffuser of square to rectangular cross-sectionVelocities were measured in a square to rectangular cross-section S-duct byRojas, Whitelaw and Yianneskis [10] with two symmetrical inlet conditions atReynolds numbers of 790 and 40,000 corresponding to laminar and turbulentflows. The boundary layer thicknesses at the inlet station, Xn=0.0 and definedby 0.99Umax, were approximately 0.25 and 0.15 of the inlet hydraulic diameterrespectively/The S-duct comprised two 22.5 degree bends of 280 mm meanradius of curvature and with the cross-section of the bends expanded linearlywith downstream distance on both curved surfaces. The inlet cross-section wassquare (40 mm x 40 mm), and the exit cross-section after two 22.5 degrees ofturning, was rectangular (40 mm x 60 mm) with an exit-to-inlet area ratio of1.5. The ratio of the overall duct length to the centre-line displacement was 5.2.

The computational domain contained half of the cross-section at each stationsince there was symmetry above and below the plane of curvature at Z*=0.0. A50 x 50 half cross-sectional computational mesh was used in the transverseplane with the grid compressed near the wall where the change of velocitygradient was sudden. The streamwise step size was about one quarter of thehydraulic diameter which corresponded to 1.125 degrees of turning within theS-duct, i.e. 62 grid was used. The initial station was set at one hydraulicdiameter upstream of the start of the bend where the boundary layer region wasfitted with l/7th power law and parabolic profile for turbulent and laminar flowsituations respectively.

Contour plots of computed and measured streamwise mean velocities (U/Ub)are presented in figures 1 and 2 respectively. In the laminar flow case, thecalculated flow at the inlet of the first bend, Xy=0.0 and figure l(i), initiallyaccelerated slightly towards the inner wall, and the maximum velocity coremigrated towards the outer wall by the end of the first bend at XH= 2.5, figurel(ii), with the accumulation of low momentum fluid around R*=0.6 to 1.0



caused by the steady strengthening of the secondary flow as in themeasurements. The reversal of curvature in the second bend did not cause thedisplacement of the maximum velocity region back to the outer wall of thesecond bend and a negative 3U/3Z* gradient appeared at region R*=0.5 to 1.0at the exit plane X%=5.5, figure l(iii)b, as in measurements. A calculated plot ofthe vector cross-flow velocities at the exit, figure l(iii)a , shows a region oflarge values in the region bounded by Z*= -0.4 to -0.8 and R*= 0.4 to 0.8 andthis coincides with the measured region of high radial velocities. The turbulentflow case of figure 2(i to v) resembles the laminar flow field but the higherReynolds number, coupled with the thinner inlet boundary layer, have resultedin weaker secondary flows and thus give rise to a more uniform distribution ofthe stream wise mean velocity contours at downstream locations. The bulk flowstarts to shift towards the outer wall of the first bend at Xn=1.65, figure 2(ii)and by Xy=2.5, figure 2(iii), there is accumulation of low momentum fluid inthe plane of symmetry, signified by the negative 3U/3Z* gradients near theregion R*=1.0 but has not been shown by the calculation although thedisplacement of the maximum velocity region has. Inside the second bend atXn=3.85, figure 2(iv), the negative 9U/9Z* gradient emerged in the calculationbut is not as steep as in the measurements. By the exit plane and Xy=5.5, figure2(v), the region of negative 3U/3Z* gradient caused by the secondary flow hasextended over most of half of the duct cross-section from R*=0.5 to 1.0, butonly a minor effect is shown in the calculation. The bulk features for both floware represented well by the calculations though discrepancies existed in theturbulent case when compared to the laminar flow and were concentrated in theregion from R*=0.5 to 1.0. This may also due to the uncertainties in theturbulence model and the possible numerical errors associated with finite gridsize used.

S-Diffuser of square to rectangular cross-section with asymmetric inletconditionsReference 11 reports the velocity measurements obtained in the S-Diffuser ofRojas, Whitelaw and Yianneskis [10] but with three asymmetric inlet conditionsat a Reynolds numbers of 40,000. The three inlet conditions corresponded to aside-slip (case 1), zero angle of attack (with effects of the fueslage) (case 2) andlow angle of attack (case 3) if the S-shaped duct is considered as an intakepassage for jet aircraft. Again, the velocity information at about one hydraulicdiameter upstream of the inlet were used as initial conditions for thecalculations. Because of limited spaces, only the first two cases are shown here.The prescribed boundary conditions for the first two cases were similar to thosefor the calculations of the previous section, with symmetry of the flow aboveand below the plane of curvature at Z*=0.0. A 50 x 50 x 62 grid is also used forthe present investigation.

Streamwise mean velocity (U/Ub) contours at successive stations are shownin figures 3 and 4 for cases 1 and 2 respectively. Acceleration of the maximumvelocity core near the inside wall at the inlet, Xy=0.0 and figure 3(ii), in case 1was more prominent in the calculation than in the experiment. As aconsequence, the calculated flow shows the maximum velocity region shifted tothe outside wall by the end of the first bend at Xy=2.5 figure 3(iii), while in theexperiment it was located at around the centre-line region. Accumulation of lowmomentum fluid near the inside wall is indicated by the presence of the negative3U/3Z* as in experiment. At the exit, Xy=5.5 and figure 3(iv), the calculatedmaximum velocity region is shifted further towards the inside wall of the secondbend and the negative 3U/3Z* is extended between R*=0.5 to 1.0 showing a



substantial non-uniform distribution of the contours. With the maximumvelocity region located near the outside wall of the first bend at Xy=0.0, figure4(i), the secondary flow generated along the first bend in case 2 would beweaker than in case 1 so that at Xy=2.5, and figure 4(ii), no negative sign of3U/9Z* appeared. At the exit, Xn=5.5 and figure 4(iii), the contours show thatthe maximum velocity region is located near the inside wall and a minor non-uniform distribution of the contours has appeared near the outside wall and isless severe as in case 1. Comparing the exit streamwise mean contours andradial mean velocity profiles of these two cases with those of a symmetrical inletcondition of Rojas et al [10], figure 5 (a to c), the calculations have indicated thesame trend as that of the experiments: the non-uniform distribution of thestreamwise mean velocity contours near the outside wall is largest in case 1resulting from the presence of a large pair of local contra-rotating vortices,figure 5 (a & b). When this pair of vortices is absent, the correspondingredistribution of the contours became less severe as in case 2, figure 5c.

Circular cross-sectional S-shaped diffuser : Royal Aerospace Establishment (0.3length offset) S-ductThe detailed velocity measurements of Whitelaw and Yu [13] in the scaledmodel of the RAE 2129 (0.3 x Length Offset) S-Duct are one of the fewexperiments obtained in duct of circular cross-section using combined laser-Doppler anemometry and refractive index matching technique. Detailedgeometry of the duct can be found in reference 13. Velocity information at onehydraulic diameter upstream of the inlet of the duct were used as initialconditions for the calculations. The boundary layer thickness at this station isabout 0.2 of the inlet duct diameter. Special attention was paid to the innersurface of the first bend and the outer surface of the second bend at which theincipient separation was observed in the experiment. More near wallmeasurements were obtained in this case. Thus, initial studies were carried outto assess the effect of the wall functions of Eq. (15) and that of the Van Driest'sand four grid densities (25 x 25 x 62, 50 x 50 x 62, 50 x 50 x 123 and 80 x 80x 62) on an half cross-sectional plane.

The computed results at one hydraulic diameter downstream of the exit areshown in figure 6 with the two wall functions and the different grid densities.Two comments can be made from this series of tests: first the damping functionof Van Driest leads to better agreement with measurements and second theresults appeared to show an independence of results of the grid densities of 50 x50 x 62, 50 x 50 x 152, 80 x 80 x 62 cases and the 50 x 50 x 62 grid waschosen for further calculation. Profiles of mean velocity at the symmetry planeare shown in figure 7a and these calculated results were obtained with the wallfunctions of Van Driest. The streamwise separation regions calculated appearfurther downstream than in the experiments so that separation appears betweenstation 7 to 9 compared with the measured location between stations station 6 to10. The calculated streamwise (U/Ub) velocities at stations 3, 5, 7, 9, 11 and13 are shown in contour form in figure 8 together with the correspondingmeasurements. Figure 9 shows the calculated vector plots of transversevelocity.

Initially, the flow exhibit an inward acceleration in response to the favourablepressure gradient at station 3 and figure 8(i) where the magnitude of crossstream velocity, was similar (O.lUb), as expected. The migration of the coretowards the outer wall of the first bend is also evident between station 5 to 7,figures 8(ii to iii), but negative values of the gradient 3U/3Z* do not appear as



in the experiments. The corresponding calculated radial components are alsoshown to have the same order of magnitude as in the experiments up to station7. The negative value of the gradient 3U/3Z* calculated in the region aroundR*= - 1 .0 and Z*= 0.0 is evident at station 9 and figure 8(iv). At the exit plane,the measured negative gradient 9U/9Z* extends towards the inner wall alongthe plane of symmetry in experiment as a consequence of the growth of the localcontra-rotating vortices pair at region around R*=-0.8, see figure 8(v). Thecorresponding calculated radial components are of lower magnitude aroundR*=-0.8 compared to the experiments. At one hydraulic diameter downstreamof the exit, figures 8(vi), the pair of contra-rotating vortices on the bottom of theduct appear in calculated results but with relatively lower magnitude ofsecondary velocity than that of the experiment, some 25% lower, also see figure

DISCUSSION

The most significant difference between solutions and measurements inturbulent flows is concentrated in the region along the outside wall of thesecond bend where the measurements show negative 3U/9Z* gradientsemerging faster than in the calculations. This delay in the calculations has beenshown to be the main cause of disagreement at downstream locations. In thecalculation of Anderson [6] for the flows in the S- shaped diffusing duct ofVakili, Wu, Liver and Bhat [12] using a 50 x 50 transverse grid on a half plane,similar to that of the present studies, and this gave only minor improvements onthe results with a 25 x 25 grid. When a finer grid was implemented close to thewall, with the position of the first node changed from y+ = 8.5 to 0.5, animprovement was observed with the position of the separation region wellrepresented. Anderson [6] concluded that this was due to the improved wallgrid resolution which included the consideration of the vorticity components inthe wall region in the calculations and the generation of secondary flows wasessentially an inviscid process. However, in the present studies, the first nodewas located at around y+= 0.15 in a 50 x 50 transverse grid and a furtherincrease to a 80 x 80 transverse grid brought no differences to the calculatedresults even with the position of the first node reduced to below y+= 0.1. It istherefore unlikely that major differences stem from the grid resolution near thewall, but rather from the turbulence model which has the limitation imposed bythe viscosity assumption in an algebraically specified mixing length. Curvatureeffects due to the extra-strain rate are not considered although the ratio ofboundary-layer thickness to radius is of order of 0.036 in the S-duct of Rojas etal [10] and is larger in the RAE 2129 duct. Furthermore, analysis of theReynolds stress equations by Whitelaw and Yu [13] showed that regions ofpositive and negative production of turbulent kinetic energies exist along thesecond bend, and are also not represented by the turbulence model. Theinfluence of the Reynolds stresses at the boundary layer region is furthersubstantiated by the tests of two wall damping functions. As shown in thedistribution of the wall functions used by Briley and McDonald [3] and that ofVan Driest in figure 7b, that latter changes slowly from y+ = 100 to the wallwhile the former changes abruptly at y+ - 40. The consequence for the meanvelocity profile at the corresponding position is that the Van Driest' s appeared tohave a smaller magnitude of the Reynolds stresses than that given by Briley andMcDonald [3] in the region from y+ = 100 to the wall. This caused earlieroccurrence of the stream wise separation in the calculations of the RAE S-ductand, as a consequence, better agreement with measurements. The use of the



Van Driest wall function leads to better agreement with measurements becauseof the effects of Reynolds stresses distribution and thus, suggesting betteragreement with measurements may be achieved through modifications of theexisting turbulence model.

CONCLUSIONS

Several conclusions can be drawn from the present studies:

1) The appropriateness of the assumptions made in the group ofapproximate equations, such as the case of a potential pressure plus a one-dimensional viscous correction to represent the streamwise pressure field, wasfurther confirmed by comparing the solutions and measurements of the laminarflow results of the square to rectangular S-duct of Rojas, Whitelaw andYianneskis [10].

2) The turbulent flow calculations are less satisfactory than those for thelaminar case. Although the displacement of the maximum velocity region tothe inner wall of the second bend was demonstrated well at the exit, the non-uniform distribution of the streamwise mean velocity contours at the outer wallcaused by the presence of a pair of contra-rotating vortices was underestimatedin the calculation. The turbulence model, rather than the wall grid nodedistribution, is thought to be the source of this discrepancy.

3) A different wall damping factor, similar to that proposed by Van Driest,was introduced in assessing the results for the RAE 2129 S-duct and gave betteragreement with the experimental results because of the earlier streamwiseseparation. This difference may be due to a more extended treatment of thedamping function from y+ = 40 to 100 and hence lead to higher vorticity in themean velocity profile and is therefore more likely to experience a streamwiseseparation. This also implies that the calculated results were very sensitive tothe near wall Reynolds stresses treatment.

4) The level of agreement with measurements may be improved byreplacing the existing mixing length/eddy viscosity turbulence model by a moresophiscated one, such as the Reynolds stress model, which would be capable tocapture the effects of streamline curvature to the distribution of Reynoldsstresses.

ACKNOWLEDGEMENT

The author would like to thank Prof. J. H. Whitelaw in reading the draftmanuscript of this paper. The financial support from the Royal AerospaceEstablishment, Ministry of Defence (U.K.) is grateful acknowledgementtogether with many useful discussions with Dr. J. E. Flitcroft.



REFERENCES

[1] Humphrey, J. A. C., Taylor, A. M. K. P. and Whitelaw, J. H. 'Laminar flow in asquare duct with a strong curvature.' J. Fluid Mech., 83, pp509, 1977.

[2] Patankar, S. V. and Spalding, D. B.,'A calculation procedure for heat, mass, andmomentum transfer in three-dimensional parabolic flow.' Int. J. Heat Mass Transfer, 15,ppl787, 1973.

[3] Briley, W. R. and McDonald, H.,Three-dimensional viscous flow with largesecondary velocity.' J. Fluid Mech. 144, pp47, 1984.

[4] Launder, B. E., Choi, Y. D. and lacovides, H.'Numerical computation of turbulentflow in a square-sectioned 180 degree bend.' J. of Fluids Engng, 111, pp59, 1989.

[5] Launder, B. E. 'Numerical computation of connective heat transfer in complexturbulent flow: time to abandon wall functions ?' Int. J. of Heat Mass Transfer, 27,pp!485, 1986.

[6] Anderson, B. H. The aerodynamic characteristics of vortex ingestion for the F/A - 18inlet duct.'29th Aerospace Sciences Meeting, AIAA Paper no. 91-0130, 1991.

[7] Flugge-Lotz, I. and Reyhner, T. A. 'Interaction of a shock wave with a laminarboundary layer.'Int. J. of non-linear Mechanics, 3 , 1968.

[8] McDonald, H. and Camarata, F. J.'An extended mixing length approach forcomputing the turbulent boundary-layer development.Proceedings of Stanford conferenceon computation of turbulent boundary layers, Vol. I, Published by Stanford University,1969.

[9] Van Driest, E. R.'On turbulent flow near a wall.'J. Aero. Sci. , 23, pp!007, 1956.

[10] Rojas, J., Whitelaw, J. H. and Yianneskis, M. 'Flow in sigmoid diffusers ofmoderate curvature.' Fourth Symposium on Turbulent Shear Flows, Karlsruhe,Germany, 1983.

[11] Whitelaw, J. H. and Yu, S.C.M.' Flow characteristics in an S-shaped duct withasymmetric inlet conditions.'llth Australasian Conf. of Fluid Mechanics, Hob art,Australia, 1992.

[12] Vakili, A. D., Wu, J. M., Liver, P. A. and Bhat, M. K.'Experimental Investigationof secondary flows in a diffusing duct with vortex generators'. NASA NAG3-233, 1986.

[13] Whitelaw, J. H. and Yu, S.C.M. Turbulent flow characteristics in an S-shapeddiffusing duct.' Proc. 6th. Int. Symp. Appl. of L.D.A. to Fluid Mechanics, Lisbon,Portugal, 1992. (full length of the paper will appear in Experiments in Fluids)

[14] Yu, S. C. M. 'Flow Characteristics in S-shaped Diffusing Ducts.' PhD thesis,University of London, 1992.



Q.O 0.2 0.4 0.6 0-8 >>0

Q.Q 0.2 0.4 0.6 0.6

Figure 1 Contours of streamwise mean (U/Uy) and radial mean (V/Uy) velocitycomparison, laminar flow, Rojas et al [lo],(bottom- experiment, top- calculation)



0.0 0.? 0.4 Q.6 0.8 1.0(1) X -0 *"

0.0 0.2 0-4 0.6 0.8 1.0

0.0 0.2 0.4 0-6 0.8 I .0

Figure 2Contours of stream wise mean (U/Uy) velocitycomparison, turbulent flow, Rojas et al [10],(bottom- experiment, top- calculation)

0.4 0.6 0.6(v) X,, = 5.5



R*=0.0 0.1 0.3 0.5 0.7 0.9 . 1.0

Figure 3 Contours of streamwise mean (U/Ub) velocity comparison, turbulent flow,Case 1, Whitelaw and Yu [11], (bottom- experiment, top- calculation)

R*=0.0 0.1 0.3 0.5 0.7 0.9 1.0

Figure 4 Contours of streamwise mean (U/Ub) velocity comparison, turbulent flow,Case 2, Whitelaw and Yu [11], (bottom- experiment, top- calculation)



Z*=0.0.,

R*=o.o o.

z*=o.o

07 09 -0 R*=00 0. 0.3 0.5

Figure 5 Stream wise mean (U/Uy) velocity contours and Radial mean (V/Uy)velocity profiles comparison at the exit plane, X%=5.5, with uniform and asymmetricinlet conditions (Rojas, Whitelaw and Yianneskis [10], Whitelaw and Yu [11])(bottom- experiment, top- calculation)



McDonald's Van Driest's

Z» 0.0 0.5 i.o (a) 50x50% 123 L' 0.0 0.5 I o (b) 50x50x123

Figure 6 Contours of streamwise mean (U/Ub) velocity contours at one diameterdownstream of the exit plane with different grid densities and wall functions,Whitelaw and Yu [13], (right - calculation, left - experiment)

(a) 11 12 13

(b)

Figure 7 (a) Profiles of streamwise mean velocity (U/Ub) comparison at the symmetry plane,Whitelaw and Yu [13]

7 (b) Distribution of the wall damping function(McDonald's , Van Driest's )



Z* 0.0 OJ 1-0(i) Station 3 (Inlet)

R*0.0

Z- 0.0 0.5(ii) Station 5

Z* 0.0 0.5

(v) Station 11 (Exit)

L" 0.0 0.5

(vi) Station 13

Figure 8 Contours of streamwise mean velocity (U/Ub) comparison, turbulent flow,Whitelaw and Yu [13] (right - calculation, left - experiment)



(i) Station 3 (Inlet) Scale - 0.05Ub 00 Station 5

(v) Station 11 (Exit)

Figure 9 Calculated transverse vector plot, turbulent flow, Whitelaw and Yu [13]


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