1-s2.0-0045793095000144-main.pdf

Commuters & Fluidr Vol. 24. No. 7. m. 81 l-832. 1995

Pergamor~ 0045-7930(95)ooo14-3 Copyright 0 1995 I%vier Science Ltd

Printed in Great Britain. All rights reserved 004s7930195 $9.50 + 0.00

COMPUTATIONAL MODELLING OF THR.EE-DIMENSIONAL IMPINGING JETS WITH AND WITHOUT CROSS-FLOW USING SECOND-MOMENT

CLOSURE

M. A. LESCHZINER and N. Z. INCE Department of Mechanical Engineering, University of Manchester Institute of Science & Technology,

Manchester M60 IQD, U.K.

(Received 14 September 1994; in revised form 6 March 1995)

Abstract-Computational solutions are presented for three twin-jet configurations which are dominated by flow features arising from normal impingement on a flat plate and jet-jet interaction. Two cases are incompressible, one is subjected to cross-flow, provoking a ground vortex, and another is transonic. All three flows are closely associated with VSTOL operation very close to the ground at low aircraft speed, in which high-speed wall jets arising from impingement collide to form strong fountains. The solutions have been obtained with a conservative FV strategy combining higher-order discretisation and a pressure-correction algorithm, the latter originally devised for incompressible flow and extended to allow the capture of shocks. Unusually, the study investigates the performance of second-moment (Reynolds- stress-transport) closure for 3D jets; indeed, it appears to be the first including the application of this type of model to transonic 3D impinging jets. Comparisons are presented between computational solutions and experimental data, and these demonstrate, particularly for the incompressible cases for which the experimental database is much more extensive, that second-moment closure returns a superior representation of both jet and fountain behaviour relative to the k-f eddy-viscosity model.

1. INTRODUCTION

The description of single or multiple jets impinging on a solid surface and subjected simultaneously, to cross-flow is of central importance to the operational characteristics of VSTOL aircraft operating in ground proxirnity. Of principal interest are the effects of ground impingement and any cross-flow-induced ground vortices on the aircraft and on ground-based structures, the degree to which impingement-generated fountains arising from jet-jet interaction affect the aircraft- especially in the context of hot-gas reingestion-and the interplay between the jet and the cross-flow in so far as it influences the aerodynamic field around the aircraft.

The exceptional complexity of the strain and turbulence fields provoked by the combination of impingement, jet collision and cross-flow poses considerable challenges to any modelling strategy, whether based on simulation or on a RANS approach. Severe curvature is present in the shear layers at the jet and fountain edges close to the primary (jet-on-ground) and secondary (wall-jet-on- wall-jet) impingement zones, and this can have profound consequences for the turbulence level in the shear layers. The impingement process gives rise to large normal strains which generate turbulence at a rate dictated by the level of normal-stress anisotropy, especially close to the wall. Structural features of the fountains, especially their strength and spreading rate, respond sensitively to turbulence transport in the wall jets which collide to form the fountains. The wall jets themselves arise due to the interaction between an inner, highly anisotropic boundary layer, in which the length scale of the energetic eddies is rather small, with an outer shear layer, in which scales are considerably larger and less anisotropic. Their collision gives rise to the highly-curved and sheared fountain-base flow, in which the turbulence structure has a significant influence on the behaviour of the fountain. If -the jet is under-expanded, its structure is affected by shock and expansion waves which may amplify turbulence and invalidate the scaling laws pertaining to its incompressible counterpart. Finally, any cross-flow gives rise to jet deflection and a ground vortex on the impingement wall, with consequent flow curvature modifying turbulence transport and entrainment characteristics, aga.in via a mechanism linked to curvature-anisotropy interaction.

811

812 M. A. Leschziner and N. Z. Ince

From a numerical point of view, the principal challenge is to adequately support widely disparate scales on what is-unavoidably, in a three-dimensional context-a rather coarse computational mesh. In particular, there is a need to support the relatively thin, curved and highly sheared jet and fountain edges and to resolve the structure of any shock-related features, whose orientation and spatial location are quite different from those associated with shear. The difficulties introduced through the above disparity of scales are further aggravated by the unconfined nature of the flow and the influence of far-field processes on the jet-fountain structure, which demand the extension of the solution domain to regions far removed from the jets. Clearly, these conflicts and constraints demand the use of accurate, non-diffusive approximation schemes, but these give rise to stability problems which are considerably aggravated by the highly non-linear and coupled nature of the equations constituting the type of turbulence model needed to resolve the aforementioned physical processes.

With a few LES and other specialised approaches set aside [l-4], the large majority of computational studies directed towards three-dimensional jet flows of the present type have employed finite-volume procedures which incorporate eddy-viscosity models [5-161, almost all of the two-equation k-c variety. Numerical accuracy apart, the principal question arising in this context is whether an isotropic-viscosity closure is an adequate modelling framework. There is now a considerable body of evidence-having emerged, principally, from two-dimensional studies [17, 18]-to suggest that this approach is afflicted by serious, inherent weaknesses which manifest themselves most prominently in recirculating, swirling, rotating and other highly curved flows. The crucial mechanism misrepresented by any (linear) eddy-viscosity model is turbulence anisotropy and, related to it, the attenuation or amplification of turbulence due to curvature-related strain. Arguably, the only fundamentally sound route to resolving this interaction is via second-moment closure, and there is ample evidence to demonstrate that this fundamental strength translates to superior predictive performance in a wide range of two-dimensional flows. Second-moment-closure studies of 3D flows, of whatever type, are still rare, but most-including those concerned with jets [19-24]-give reason to conclude that this predictive superiority extends, at least in some measure, to the three-dimensional environment.

The present paper reports the outcome of parts of a wide-ranging research programme focusing on second-moment-model predictions of three-dimensional single and multiple jets, with and without cross-flow and impingement; other parts concerned with the behaviour of free jets in cross-flow (without impingement) are reported elsewhere [19,24]. While results emerging from the entire spectrum of efforts are not entirely conclusive or consistent in all respects, they do nevertheless provide clear indications that second-moment closure is superior to isotropic-viscosity modelling. In specific terms, second-moment closure returns an improved representation of the structure of the impingement-induced fountain, a more realistic response of turbulence to compressive strain, and levels of normal-stress anisotropy and associated turbulence-curvature interaction which are both qualitatively correct and broadly in line with experimental observations.

2. COMPUTATIONAL AND MODELLING FRAMEWORK

The calculations reported herein have been performed using a general 3D-BFC finite-volume code developed by Lin and Leschziner [19] for incompressible flow and extended, in the context of the present study, to compressible conditions along lines explained below. The parent algorithm solves the Reynolds and continuity equations for the primitive variables (V, V, W, P) in an iterative or-in compressible flow-time-marching manner by way of a pressure-correction algorithm over a staggered cell arrangement. Convection is approximated by a bounded variant of the quadratic scheme QUICK of Leonard [25] in which oscillations are removed by means of the LODA algorithm of Zhu and Leschziner [26]. Convergence is monitored by reference to the absolute sums of residuals for mass and momentum, normalised by associated jet-inlet fluxes.

The extension to compressible conditions is based on the retarded-pressure approach of McGuirk and Page [27]. Briefly, this entails the following five principal steps:

(i) The discretised momentum equations are formulated to describe the flux (density-weighted) variables instead of their primitive counterparts.

(ii) (iii)

(iv)

(v)

Computational modelling of three-dimensional impinging jets 813

The solution is marched in real time space rather than in iterative space. In supersonic zones, the discrete pressures used in the momentum equations are not those residing at the physical locations straddling the velocities-as is the case in incompressible conditions--but are retarded by means of the Mach-number-dependent monitoring func- tion. For 1:D conditions, the retardation may be expressed as

Fi=Pi+/@,-Pi_,) (1)

where i identifies a nodal location and

,u =Max{Ci,k[ 1 -($$r]} (2)

k, A4,r= O(1).

The pressure-correction equation is expressed in terms of perturbations of retarded pressure, from which the retarded-pressure field is then obtained. The real pressure, at computational nodes, is evaluated from an inverted version of the relationship linking the retarded pressure to its real counterpart; the density field is simply obtained from the real pressure.

In effect, the above retardation strategy mimics the hyperbolic character of the (inviscid) conservation law in supersonic conditions, and secures numerical stability in transonic conditions via the introduct.ion of a dissipative mechanism around shocks.

With the above five modifications introduced, the pressure-correction strategy, as formulated for incompressible conditions, is retained essentially unchanged. In particular, the algorithm degener- ates to the basic incompressible method at low Mach numbers, and so retains all the favourable properties associated with it when acoustic processes are insignificant.

The code inco:rporates the standard k+ turbulence model of Jones and Launder [28] and the second-moment closure of Gibson and Launder [29]. Both models were used in conjunction with log-law-based wa.11 laws. For incompressible flow, the latter model may be represented compactly by the following equations written in Cartesian-tensor form:

- Du,uja

( _kau,rri

Dt = dx, csukuI 5 ax, > +P,+&j-_581c (3)

in which

Pijs - 1

-_-au, --au, uik z + ujuk z

1

is the rate of stress production,

Aj=Q$l+A/Z+@,+@z

represents pressu:re-strain interaction,

&, =F (

u,u,-fsl+

(4)

(5)

(6)

is the Rotta term,

4i,?= -c2(p(i-;a~pkk)

is the isotropisation-of-production term, and

(7)

(-

3_ 3_ 4$ = c; UkU,,,nkn,,, 6#--u %n n.--ukujn&

>

k32 2 klkJ - k 2

(8) Cl%

4;2 = c; $km2nknm d,- 3 #ik2nknj- 2 4jk2nknf

>

-$

represent wall-related influences on the pressure-strain process, in which nk E wall-normal unit vector and x, = mu-ma1 distance from the wall.


The above wall-related fragment 4;. was originally formulated by reference to near-wall shear flows and is known to respond wrongly to wall-normal straining associated with impingement and reattachment. Specifically, it amplifies rather than attenuates the excessive tendency of&z to drive near-wall turbulence towards the isotropic state as the wall is approached. Recognising this, Craft and Launder [30] devised the following corrected version which gives the requisite response in shear as well as normal strain:

(10)

where a,,,, is the anistropy structure tensor:

- - 2 &. Ul %?I % = k 3 (11)

The performance of the above corrected form will be contrasted with that of the original Gibson-Launder proposal in one of the applications to follow.

-. The set of equation for uiuj is closed by the rate-of-dissipation equation:

(12)

The constants appearing in the above set are as follows:

Cl cz c, c; c; c: CT c: CI ct C,I CZ

1.8 0.6 0.22 0.5 0.3 0.044 0.08 0.6 2.5 15 1.9 1.45

In compressible conditions, the mean density is included in the flux terms, and all mean-flow and turbulence properties are interpreted as being mass-averaged, in which case the equations are formally (almost) identical to their incompressible counterparts. In general, mass-averaging leads to additional density-related fragments, but these are of little importance in weakly supersonic conditions. What may be far more influential, however, is the effect of variable density on the adequacy of the closure assumptions which all arise from the incompressible environment. Various proposals exist in the literature for compressibility corrections to the turbulence-transport equations; none are entirely transparent or well validated beyond simple-shear conditions. In some test calculations, not included herein, one correction proposed by Sarkar et al. [30] for simple jet flows has been investigated but found to return unrealistically strong modifications to the flow in the highly turbulent subsonic fountain. In view of this experience, we refrain from using any of the corrections proposed.

3. CASES EXAMINED AND RELATED COMPUTATIONAL DETAILS

Results are reported here for the three flows sketched in Fig. 1: an incompressible twin impinging jet examined by Saripalli [32], a similar arrangement of under-expanded air jets measured by

Fig. 1. Geometries investigated.

Computational modelling of three-dimensional impinging jets

II,,, I, I I , , 1111, I, , I , , 111,,,,I, , , 111*,,,, I , , 111*,,, , I , , II, I, I , I , , , -1 0 11(1,1,1 I , I I! ,111, ,I, I L,,,,,,, I I I

815

I Y/O

8

I V/V ,.f Y/O 5 0. 30

non-tapering nozzle

- * O t v/v,., tapering nozzle

Fig. 2. Saripallis jets-velocity fields with and without pipe-tapering.

Abbott and White [33], and an incompressible twin impinging jet subjected to weak cross-flow studied experimentally by Barata et al. [34]. For the first and third configurations, extensive LDA data are available. while for the second, the data comprise of static and dynamic pressure data, augmented by Sch.lieren photographs.

In Saripallis case, two water jets, 9D apart, were discharged at a height of 30 above the impingement wall. The actual jet pipes were submerged to a depth of 24 jet diameters below the free water surface and contained a tapered discharge section incorporating a 16: 1 area contraction. In contrast, most calculations were performed with uniform-diameter jet pipes and a computational entrainment boundary placed at 9 jet diameters above the discharge. Across this boundary, and any other of the s#ame type in later jet configurations, the total pressure was held invariant, a zero-normal-gradient condition was imposed for the velocity component parallel to the boundary, and the normal component was computed from the associated momentum equation applicable at the boundary. If the normal velocity so computed was directed into the domain, the value of any scalar property transported with the entrained mass flux (e.g. stresses, turbulence-energy dissipation) was assumed negligibly small, while a zero-gradient condition was imposed if the velocity was directed outwards. This treatment was applied locally as part of the iterative solution sequence. Test calculations were performed with the pipe taper included and excluded, and with the entrainment boundary placed between 3 and 24 diameters above the jet discharge. The outcome of one such test, with the upper boundary placed at 9 diameters above the discharge plane, is shown in Fig. 2. In no case was the observed sensitivity of the solution to these variations more than very marginal. Conditions at the jet exit were derived, as far as possible, from the experimental data. Within the present (Cartesian framework, the jet exit was represented by a castellated contour, with careful attention being paid to maintaining the correct discharge area, mass-flow rate and momentum flux. The rate of turbulence-energy dissipation was prescribed so as to achieve a discharge eddy viscosity 50 times the laminar one. For Reynolds-stress calculations, isotropy conditions were assumed (i.e. equality of normal stresses), and the shear stress was, consistently,


taken to be zero. At far-field boundaries, invariant total-pressure conditions were assumed. Finally, at the impingement plate, log-law-based wall laws were used. Calculations presented below were obtained with a mesh containing 55 x 52 x 31 nodes covering one-quarter of the physical domain, with symmetry conditions prescribed across the planes interfacing other quadrants.

Abbott and Whites compressible jets discharge from circular holes, 4.50 apart and 2.70 above the impingement plate, at nozzle pressure ratios (NPR = Ptot,jet/Pambient) between 2.6 and 3.5. The flat plate containing the holes runs parallel to the impingement wall. No information is provided on jet-exit conditions. Thus, the practice adopted here was to prescribe the appropriate stagnation pressure and to infer the jet velocity from isentropic relationships coupled with a zero-order extrapolation of nodal pressures located closest to the jet exit. This gave an exit Mach number very close to unit, as it should be. Isotropic turbulence was assumed at the jet exit, with turbulence energy prescribed to be 1% of the mean value. The rate of dissipation was determined in a manner identical to that adopted for Saripallis jets. Along outlet planes, the measured ambient static pressure was prescribed. Results reported below were obtained with a mesh containing 95 x 80 x 50 nodes.

Barata et al., [21] water jets, 5D apart, were injected into a tunnel of height 5D at a jet-to-cross- flow-velocity ratio R = 30, and the measurements were performed with LDA techniques. Boundary conditions were taken, where possible, from the experimental data; conditions which could not be determined from the data were prescribed along lines described above in relation to Saripallis case. Following grid-dependence tests, the final computations contained herein were made with a grid consisting of 61 x 46 x 51 lines covering one-half of the actual twin-jet arrangement.

Numerical accuracy is an important consideration in a study concerned with assessing the relative performance of turbulence models, but poses inevitable difficulties in physically complex 3D flows resolved with computationally demanding models. For the present flows, a total of 11 strongly coupled partial differential equations were solved, and resource requirements were extremely high, in terms of both storage and CPU times. Numerical accuracy was decisively promoted by the use of a quadratic approximation for convection, which is third-order accurate for uniform meshes. Limited grid-dependence studies were performed for all flows with sequences of three grids, with the range of total number of nodes varying roughly by a factor 4. In addition, tests were performed in which the grid-line distribution was varied, for a given total number of grid nodes, so as to increase the resolution in demonstrably sensitive regions. These studies led to the use of different final grids for different flows, the finest one containing 380,000 nodes. Nevertheless, numerical errors cannot be claimed to be vanishingly small, as is readily achieved in 2D flows. The tests performed allow the statement to be made that the numerical error in maximum mean-flow properties is of order 2-3%, while the maximum error in the Reynolds stresses is judged to be lower than 5%.

Computing times varied greatly from one case to another, with the Reynolds-stress-model calculations for the transonic jets being the most resource intensive at around 20 hours on a Cray YMP(l). Calculations were performed on a variety of machines including a Fujitsu VPl200, a Cray YMP and an Alliant FX2808. Typically, a Reynolds-stress-closure calculation was initiated from a converged eddy-viscosity-model solution, and the ratio of CPU times of the former relative to the latter was of order 2.5.

4. RESULTS AND DISCUSSION

4. I. Saripaliis incompressible twin jet

The availability of detailed LDA data for velocity as well as Reynolds stresses makes this case a particularly valuable one as a basis for assessing the performance of alternative turbulence models. Computational solutions for this case are contrasted in Figs 3-7 with the above experimental data across lines traversing both jet and fountain and lying in the symmetry plane which bisects both jets-plane A-A in the inset in Fig. 3. The coordinates x, x and y used in profile plots are also identified in the inset.

Profiles of the principal (downward-directed) jet velocity component (V) at different heights above the impingement plane are shown in Fig. 3. Both the k-c and the Reynolds-stress models


- OS8

___ ____ K-E

-1 0 I X/O -I 0 I Xl0 -- -0. 2 -0.2

-0. 4

I== -0.4

-0. 6 -0.6

-0.8 -0.8

-la0 v/v

-1.0 ,.t

Fig. 3. Saripallis jets-profiles of principal jet velocity component.

(identified by IXM) return variations which are similar in shape and broadly in accord with the experimental data. There are, however, some significant quantitative differences between the solutions. In the circular jet, following discharge, the Reynolds-stress model predicts a considerably narrower shear layer, which agrees closely with the measured variation. This difference reflects, principally, a lower level of entrainment into the jet from quiescent flow regions lying above the jet exit. In particular, the amount of fluid drawn downwards close to the jet pipe at the discharge height is much lower. Although the Reynolds-stress model is known to return a (self-similar) spreading rate for the round jet that lies well below that obtained with the k-c model, as well as being closer to the measured level, the difference observed here does not seem to be clearly related to that predictive characteristic. Nor does it appear to be linked in any obvious way to any secondary strain (say, one associated with curvature), which would be expected to provoke the most marked differences in model performance. Such differences would rather be anticipated as the shear layer at the edge of the jet is deflected sideways while the jet approaches the impingement region. Reference to the profile at Y/D = 0.2 in Fig. 3 does not, however, provide clear evidence of a curvature-induced attenuation in the spreading rate, although it ought to be noted that the profile considered is of the velocity component normal to the plate only, and this does not reflect well

- DSM/qk _ _____ ___ K-E,,+

u/v

0. 75

0. 50

0. 25

u/v

I ,.I

0. 75

0.50

-0.; --- ----- --.w

-0.50 -0.75 I=- Y/O IO. 30

0. 75

0. 50 0. 25

-0.25 -0.50 -0.75

Fig. 4. Saripallis jets-profiles of transverse jet velocity component.


Ia n/V, 0.30

- ml 0.25

r

Y/D = 0.50 _____._ K-E

0.20 ODPT

-2 -I 0 1 2

-2 -1 0 I 2

Gl"'/v,*~

0. 30

0. 25 I Y/D E 0.10

0.20

-2 -I 0 1 2

ta 2/v ,.t 0.30

0.25 Y/O = 0.10

0.20

0. 15

0. IO

0.05

0. 00

-2 -I 0 I 2

67) "2/v ,.I 0. 30

0. I

0. 25 Y/D = 0.05

20

0. I5

0. 10

0.05

0.00

-2 -I 0 1 2

m "*/v 0.02 '*I

T 0.01

0.00

-0.01

-2 -I 0 1 2

m "94,*~ 0.02

T 0.01

0.00

-0.01

Fig. 5. Saripallis jets-pro&s of Reynolds stresses across jet.

the shear strain in the impingement region. On the other hand, the shear-stress profiles in Fig. 5, particularly that at Y/D = 0.1, do point to such an attenuation.

Additional insight into the structure of the jet within the impingement region is offered by profiles of the lateral velocity component (U) at different heights above the impingement plane. These are given in Fig. 4, and suggest, consistently with the profiles in Fig. 3, that the Reynolds-stress model

Computational modelling of three-dimensional impinging jets 819 - 0% --___-__ K-E

ODPT

0. I

0.0

Fig. 6. Saripallis jets-profiles of principal fountain velocity component.

returns a considerably thinner shear layer than does the k-c model in the outer part of the wall jet, formed by the defiected shear layer of the circular jet following impingement. This particular structural difference may be claimed to be indicative of the ability of the Reynolds-stress model, in contrast to the kr form, to capture curvature-induced turbulence attenuation. The comparisons of stresses given in Fig. 5 may be said to support this conclusion: as is clearly seen, the Reynolds-stress model returns a superior representation of the separation between the normal stresses, as well as a lower shear stress in the curved jet portion close to impingement.

The correct representation of the fountain structure may be claimed to be a far more challenging task than that of resolving the jet, for this structure is the product of history, reflects the transport processes along a route involving unconfined spread, two impingement-induced deflections and near-wall effects imparted on the flow in the wall-jet phase. Hence, the fountain is only loosely anchored to the jet-discharge conditions, so that its structure is largely born out of internal processes. This fountain structure is conveyed in Fig. 6 by profiles of the principal velocity component at different heights above the impingement plate. These are supplemented by the variation of the fountains spreading rate, given in Fig. 7 in terms of the fountains half width. As seen, the Reynolds-stress model returns a representation of the fountains structure which is superior to that predicted by the k-c model in several respects. Close to the fountains base, the stress model pred.icts thinner and steeper layers-one consequence being that the velocity dip at

xl,*0 - DSM 1.25 ____ K_E

0 1 2 3 Y/O

Fig. 7. Saripallis jet-fountain spreading rate.

820 M. A. Imchziner and N. 2. Ince

the centre-line, associated with fluid stagnation at the base, is captured. Here again, the thinner shear layer may be partly attributed to curvature effects, but also reflects the thinner wall jet upstream of the fountain. Further downstream, the Reynolds-stress model returns a higher rate of spread and, with it, a more rapid reduction in peak velocity. One process affecting the spreading rate is curvature-inducted turbulence amplzjication in the fountains inner curved shear layer (i.e. the separated near-wall layer of the preceding wall-jet). Potentially more influential, however, are differences in the rate of spread of the fountain in the spanwise (2) direction. Finally, some of the fluid entrained into the fountain is highly turbulent (see Fig. 2), due to internal circulation, and differences in the level of turbulence within the entrained fluid provoke minor differences in the spreading rate. Hence, it would appear that there is no single mechanism responsible for the predictive improvements returned by the Reynolds-stress model. This is in contrast to many two-dimensional flows in which it is often possible to attribute, unambiguously, differences in turbulence-model performance to the influence of streamline curvature on the level of turbulence activity in curved shear layers.

4.2. Abbott and Whites compressible twin jet

The scarcity of experimental data for this case, in particular the absence of accurate and detailed velocity data, inevitably puts any turbulence-model assessment for the present compressible flow on a considerably less secure footing than that supporting the earlier case. It must also be acknowledged that, while the numerical mesh used here is significantly finer than the one adopted for the incompressible jet, resolution may not be as good in certain regions, principally because of the steeply-varying shock-induced flow features.

A factor aggravating the uncertainties noted above is that Reynolds-stress-model predictions yielded, in contrast to those made with the k-~ model, unsteady solutions, associated mainly with stand-off shock oscillations. This observation is, in itself, a significant outcome of the study and reflects important fundamental differences between the turbulence models examined. The key issue here appears to be the response of turbulence generation to normal straining (compression and expansion) and resulting level of turbulence transport.

Figure 8 compares typical contour plots of turbulence energy predicted for NPR = 3.3 by the k+ and Reynolds-stress models. These demonstrate that the levels and distributions of the turbulence energy inside the jet differ greatly between the two models. Specifically, the stress model returns considerably lower values in inner parts of the jet, which then rise towards the outer shear layer and the impingement region. To understand the origin of these differences, it is necessary to refer to the exact rate of turbulence-energy generation, the first three terms of which are:

p,=-2u:~-2u:~-_2u:~+.... I 2 3

(13)

If, for the sake of transparency of argument, incompressibility and isotropy of turbulence are assumed, then the above is clearly seen to return a vanishing level of turbulence generation. However, when the normal stresses are replaced by the stress-strain relations, as is done in any eddy-viscosity formulation, then the three terms in (13) are additive, involving the square of strain rates, and the result is a serious over-estimation of turbulence production and hence turbulence transport (for a broader discussion, see Ref. [18]). This difference in mechanism is particularly

0 1 2 3 4 x/o 0 1 2 3 4 X/D

Fig. 8. Transonic jets, NPR = 3.3 contours of turbulence energy.


important in the present flow, for the jet contains-as will be seen shortly-shock cells which involve several successive regions of elevated normal strain. The eddy-viscosity model thus generates high levels of turbulence energy which suppresses any unsteadiness and acts to smear the stand-off shock. It is interesting to remark in this context that observations arising from calculations of vortex shedding from bluff bodies with eddy-viscosity and Reynolds-stress models (Franke and Rodi [35]) are qualitatively consistent with the present behaviour. In that case, impingement on the windward face causes eddy-viscosity models to generate high levels of turbulence which is then convected downstream and seriously inhibits transient shedding.

With the above: constraints and uncertainties noted, the main objectives pursued here are then, first, to convey, qualitatively, the present status in modelling under-expanded 3D impinging jets; second, to identify, as best as possible, turbulence-model performance by reference to the limited data available; and third, to contrast certain predicted features with corresponding observations made in the inco:mpressible case in order to highlight model consistency.

Overall views of the flow fields predicted with the Reynolds-stress model for NPR = 2.6, 3.0 and 3.3 are given in Fllg. 9. These show computed Mach contours (around the middle of the oscillatory period) in comparison with hand-drawn abstractions obtained from Schlieren photographs which identify the shock. cells and the stand-off shock. The number and size of the shock-cells observed vary with the NPR value, and this sensitivity is broadly returned by the calculations. Moreover, the stand-off shock is clearly recognizable and has been captured at the experimentally recorded elevation above the impingement plate.

Figure 10 compares the Mach-contour fields predicted with the kr and Reynolds-stress models at NPR = 3.3, and these serve to illustrate that the latter model returns a sharper stand-off shock-a difference that is brought out particularly well in Fig. 11 which provides a comparison of static pressure variations along the jet axis. The origin, at Y = 0, is the impingement plate, and the triangular symbols identify grid-line positions. As seen, the shock is credibly captured within 2 or 3 internodal intervals. The maximum Mach number, 2.2, is reached in the centre of the first shock cell, while that just ahead of the stand-off shock is about 1.7, reducing to about 0.7 across the shock.

When computal:ional solutions are contrasted with experiments, the uncertainty arising from the transient features predicted by the Reynolds-stress model must be kept in mind. An impression of the degree of this uncertainty can be gleaned from Fig. 12. This shows predicted variations of impingement-plate pressure in comparison with experimental data for NPR = 3.0. The r.h.s. plot (b) shows solutions returned roughly at the extrema of an oscillation, while plot (a) arises from the k+ model. Figure 13 gives similar pressure plots for NPR = 2.6 and 3.3, pertaining roughly to the mid-point of oscillation. Agreement with reality is generally satisfactory for both models, but it must be recognised that the impingement and, perhaps to a lesser extent, the fountain-base pressure are largely dictated by convection processes, as opposed to diffusive transport.

Comparisons between predicted and experimental variations of static and total pressures across two fountain traverses at NPR = 3.0 are shown in Fig. 14. Here, the assumption has been made that the measured static pressure is unaffected by turbulence contributions, while the total pressure has been taken to comprise the mean and the turbulence component 2/3k. The figure includes two sets of curves fo:r the Reynolds-stress model, reflecting solutions at oscillation extrema and identifying the sensitivity of the fountain structure to shock oscillations. Supplementing these results are variations of dynamic pressure along the fountain centre-line, shown in Fig. 15. Here, the DSM variation is that arising roughly at the mid-point of the oscillation period. As seen, the level of unsteadiness is not negligible, and this poses obvious difficulties in assessing model performance by reference to the experimental data. However, the comparisons suggest, albeit vaguely, that the Reynolds-stress model returns, above the immediate fountain-base region, a less pointed and somewhat weaker fountain than that predicted by the k-c model; this behaviour is also observed for other NPR values and is qualitatively consistent with that found in Saripallis incompressible jet.

As regards agreement with the experimental data, it appears that the peak total pressure returned by the stress model is reasonably well predicted, while the k+ variant returns excessive values on the centre-line but closer agreement with experiment in the outer fountain region. The low levels of static pressure in the central portion of the fountain and of total pressure in its outer part

822 M. A. Leschziner and N. Z. lnce

2.5

0

\

* 2.0

I.5

1.0

0.5

0.0

0.

0.

r, 5 X/D

4 5 X/D

0 1 2 3 4 5 x/D

Fig. 9. Transonic jets-contours of Mach number in comparison with sketches derived from Schlieren photographs at NPR = 2.6, 3.0 and 3.3.


k-e

A M.

-1.6

- 2.2

2.5

0 \ w 2.0

1.5

1.0

0. 5

0. 0

0 1 2 3 4 x/o

Fig. 10. Transonic jets, NPR = 3.3 contours of Mach number predicted by the k-t and Reynolds-stress models.

returned by the Reynolds-stress model may signify higher frictional losses in the fountain base in which turbulence levels are very high. Since the peak dynamic pressure at the centre-line is correct, as is conveyed by Fig. 15, the centre-line Mach number is likely to be close to the experimental value. Here again, however, the transient nature of the flow and the consequent uncertainties in the above comparisons must be recalled.

4.3. Barata et al.s incompressible twin jet in cross-Jlow

The main kinematic features of this flow are conveyed by Fig. 16 which shows predicted velocity fields across planes A-A, B-B and C-C cutting through the jet and fountain regions, as identified in the inset in Fig. 16. The impingement is seen to result in a strong wall jet which interacts with the cross-flow to form a ground vortex. A comparison between k+ and Reynolds-stress solutions shows the latter to feature a significantly shorter, more rounded ground vortex. This prominent difference was also noted in an earlier study of a single jet in cross-flow 1201, where it was found that the result arising from the Reynolds-stress closure was in much closer agreement with experimental observations. The fountain is highly three-dimensional, spreading due to the combined action of pressure and diffusion. Moreover, the fountain itself impinges on the upper


I ____ K-E

0 I 2 Y/D

Fig. 11. Transonic jets, NPR = 3.3, variations of static pressure along jet-centre line.

wall, giving rise to secondary wall jets, from which fluid is then entrained into both the fountain and the main jets. While the ground vortex is evident in the jet plane, it cannot be readily identified in the fountain plane B-B, as the fountain tends to lift the vortex and diffuse it in the vicinity of the upper wall.

Figure 17 contains comparisons of profiles of the principal (vertical) velocity component along three different lines lying in plane C-C (see inset in Fig. 16) which cut across one jet and one (symmetric) half of the fountain. The two Reynolds-stress models contrasted in this figure and all to follow differ in respect of the wall-related correction to the pressure-strain model dr*: DSM (s.W.) denotes the original Gibson-Launder proposal, equation (9), while DSM (n.W.) identifies the Craft-Launder form (10). All three turbulence models give a fair representation of the measured flow, although there are indications that the second-moment variants return excessively narrow and pointed profiles as the impingement region is approached, suggesting an insufficient spreading rate. A lower spreading rate than that predicted by the k-c model is expected, for the impingement process results in strong turbulence-stabilising curvature in the shear layer surrounding the jet. In the first configuration examined in this paper, that of Saripalli, similar differences between k-c and stress-closure solutions were noted, although, in that case, the divergence in the spreading rate was attributed, principally, to differences in entrainment rates well upstream of the impingement zone.

1.0 1. 0 ml

0. 9 0. 9 0. 8 _____ K-E 0. 8 0. 7 0. 7

;t 0. 6 If 0.5 ? OS6

T 0.4 If 0.5

5 0.4 ;1 0.3 k 0.2

5 0. 3 i 0.2

9 0. 1 4 0. 1 0. 0 0. 0

-0. 1 -0. 1

-0. 2 -0. 2

0 1 2 3 4 5 6 0 1 2 3 4 5 6 Fig. 12. Transonic jets, NPR = 3.0, variations of static pressure on impingement wall at two time levels.


Profiles of normal Reynolds stresses in the jet are shown in Fig. 18. The measurements indicate that the interaction between the jet and the fountain provokes a significant asymmetry in the profiles of the Reynolds normal stresses, and it is evident that the Reynolds-stress model returns a far better representation of this asymmetry, as well as of the level of normal-stress anisotropy. Of particular significance in the context of the present study is the observation that the new wall-reflection model results in a noticeable reduction in wall-normal intensity and corresponding enhancement of t.he wall-parallel component, due to the correct sensitivity of that wall-reflection model to wall-normal strain. It is noted, however, that the effects of the wall-reflection model do not extend significantly beyond the region close to the wall and do not significantly affect the mean-flow characteristics.

The ability of the models to predict the fountain structure is conveyed in Figs 19-22 which show, -- respectively, profiles of streamwise velocity (U), upward velocity (V) and Reynolds stresses (u*, o*) and i;;, all along plane B-B identified in the inset in Fig. 16(b). While both the eddy-viscosity model and Reynolds-stress variants return fair representations of the fountain-shear stress excepted- there are clear indications that the latter modelling route is superior. Thus, Fig. 19 shows that the Reynolds-stress cllosures correctly predict the level of deflection of the fountain by the cross-flow. This is also implied in Fig. 20 by the lower level of discrepancy in the amount by which the computed V-profiles are displaced relative to the measured ones. These same models also yield closer agreement with experiment in respect of the normal stresses, Fig. 21, although the level of 2 is too high close to the fountain base. Introducing the new wall-reflection term is seen to result in an increase in the level of wall-normal stress 7 and a slight reduction in 2 in the fountain-base region. It might be supposed that this (desirable) behaviour is consistent with the earlier observation of an attenuation in 2 at the primary jet-impingement region, for here the wall-normal strain is expansive rather than compressive. In fact, the sensitivity displayed here,

1.0 0. 9 0. 8 0. 7

-I 0. 6 9% 0. 5 t- 0. 4 -3 0. 3 z 0.2 4 0. I

0. 0 -0. 1 -0. 2

- DSM

NPR = 2.6

I. 0

0.9 0.8 0. 7

-, 0.6 y1 0.5 $0.4

10.3 2 0.,2 4 0. I I/

4 5 6

- DSM -___-- K-E

@ Exp.

NPR = 3.3

0 1 2 3 4 5 6

Fig. 13. Transonic jets-variations of static pressure on impingement wall at NPR = 2.6 and 3.3.


4 -4

8 ' -6

-8

Pd p, I ___ -

t+OSAT --- - hp. @ *

Y/D = 1.25

0.00 0.25 0.50 0.75 1.00 1.25 X/O

8,

;I 6.. I 4.. Y/O = 1.50

0.00 0.25 0.50 0.75 1.00 1.25

X/D

8

-0

4:

6

4 Y/O = 1.75

B -6 . . 8)

-8 .,

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

8

6

4 Y/D = 1.50

2

0

-2

-4

-6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

8

6

-8 1 -8 1

0.00 0.25 0.50 0.75 1.00 1.25

X/D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Fig. 14. Transonic jets--variations of static and dynamic pressures across fountain at NPR = 3.0.

while advantageous, is not an intentional feature of the model which has been specifically formulated to be insensitive to wall-normal strain, whether compressive or expansive.

A puzzling degree of disagreement is observed in Fig. 22 between the calculated and the measured shear-stress profiles. This behaviour is inconsistent with other flow features in two important respects. First, the level of the normal stresses is, if anything, too high, especially close to the


0.0 0.5 I. 0 I. 5 2. 0 2. 5 y nl.,

Fig. 15. Transonic jets, NPR = 3.0, variation of maximum fountain dynamic pressure predicted by the k* and Reynolds-stress models.

1

2 cl S

3

4

5

-16 -12 -8 -4 0 4 8

- - _ ...a ,,,., *. . . . * - - - .,.. ,,,,,, *a.. . - _. .,.I## ,,,,,, a.. . n _ - _ . , .,*a - _ _. \.,,l . - _ _ .,r,, _ _ _ .._r,, - - _ . . .#./ . .

XI0 -6 -4 0 4 a 12 16

10 8 6 4 2 0 2 4 6 6

Fig. 16. Baratas jets-velocity fields across jet and fountain centre-planes (a) plane A-A; (b) plane B-B; (c) plane C-C.


1.00

0.75

0.50

s 0.2s

:- 0.00

2 3 4 z/o

osn (Il.!&) osn (e.Y.1 K-E

MPT 0.50

r 0.25

z- 0.00

-0.25

1 2 3 4 Z/D

Fig. 17. Baratas jets-profiles of main velocity component across jet plane C-C.

impingement point, and this would be expected to be accompanied by elevated shear-stress values. Second, reference to Fig. 20 suggests that the spreading rate of the fountain is somewhat over-estimated, a defect compatible with excessive shear-stress levels. All models produce far too low values of stress, however, although the second-moment variants tend to return somewhat better agreement. These serious differences and inconsistencies are not understood, at present, and give some reason to doubt the validity of the measured shear-stress data.

50 Y/D = 4.00 t

0 1 2 3 4 Z/D

Fig. 18. Baratas jets-profiles of main normal stress 2 across jet plane C-C.


-0.2 I-

-0. 3 X/D

-0.4

-8 -6 -4 -2 0 2 4 6 0

0. 4 Y/D = 4.75

0. 3

0. 2 :

>- 0.1 2

0. 0 !

0. 4

0. 3

0. 2

co.1

>oo .

-0. 1

-0. 2

-0. 3

-0. 4

-8 -6 -4 -2 0 2 4 6 0

- osn h-l. u. I - - - . - osn (0. u. ) . . . . . . . . . . K-E s;+%$yr+ 0 WT X,o~

-0 -6 -4 -2 0 2 4 6 0

Fig. 19. 13aratas jets-profiles of streamwise (17) velocity component across fountain plane B-B.

5. CONCLUSIONS

Calculations have been presented for incompressible and transonic twin-impinging jets pertaining to VSTOL applications. To the writers knowledge, these are virtually unique in that they have been obtained with full Reynolds-stress closure in combination with a higher-order approximation for convection within a unified numerical framework based on a pressure-correction approach. Emphasis has been placed on examining the predictive performance of the Reynolds-stress model relative to the k-4 eddy-viscosity variant.

-0 -6 -4 -2 0 2 4 6 0

-0.3

-0. 4

-0 -6 -4 -2 0 2 4 6 8

X/D -0.4 J

-0 -6 -4 -2 0 2 4 6 8

Fig. 20. Baratas jets--profiles of main (V) velocity component across fountain plane B-B.


In incompressible cases, clear evidence has emerged of the Reynolds-stress model returning a more realistic representation than that predicted by the k-c variant: in accordance with measurements, the jet is more focused, while the fountain is broader and less intense in terms of its maximum velocity. Moreover, the stress field is more realistic, in particular the level of normal-stress anisotropy.

When the jets are under-expanded, there arises a complex shock-cell structure and a stand-off shock, both of which the calculations resolve fairly well. The high levels of normal straining associated with the above features appear to be responsible for strong differences in turbulence model performance. Thus, the k-~ model is much more sensitive to normal straining and returns far higher levels of turbulence energy in the shock-cell region. This, in turn, results in smeared stand-off shocks and convergence to a steady-state solution. In contrast, the Reynolds-stress model

50 Y/D = 3.00 __

40 ..

*s >-30 ..

I> -* 20 . 0

-6 -4 -2 0 2 4 6 x/o

50 t Y/O = 4.75

50 ,_ Y/O = 4.00

40 .. N $30 ..

13 l 0 20 ..

________------__

-6 -4 -2 0 2 4 6 X/D

ml tn. u. 1 -.-.- Em (8. u. ) __________ K-E

50 t

Y/O I 4.75

-6 -4 -2 o 2 4 6 x/o

DSII tn. Y. ) -.-._ ml (8. Y. I _--__-__-_ K-E

OE%PT

-6 -4 -2 0 2 4 6 X/D

Fig. 21. Baratas jets-profile of normal stresses across fountain plane B-B.

.P


10 T 831

10

5

0

-5

-10 b... ____.~__:~~o Y/D = 3.00 0 o 000 X/D i . -5 -10 : :A0 . . Y/D = 4.00 0 0 0 0 X/D

-8 -6 -4 -2 0 2 4 6 8 -0 -6 -4 -2 0 2 4 6 0

10 - OSM h. h. I

Ns 0 -.-__ DSM (8. U. 1 2- 5 . . 0 0 _ ___ _ ____ _ K-E

1; 00 0 0 0 EXPT

0 0 00 - 0 .- 0

0 0 cl0

-5 .. 0000

Y/D = 4.75

-10 J X/D

-8 -6 -4 -2 0 2 4 6 8

Fig. 22. Baratas jets-profiles of shear stress across fountain plane B-B.

returns lower turbulence levels, much sharper shocks and unsteady features associated with shock oscillations. This unsteadiness has, unfortunately, clouded the comparisons with the limited experimental data available. There are indications, however, that some important predictive characteristics returned by the Reynolds-stress model in the incompressible fountain carry over to the compressible c,ase: here too, the model returns a broader fountain, though it is unclear whether this is consonant ,with reality.

Acknowledgements-The authors are grateful to Dr G. Page for his assistance and advice on the incorporation of his Pressure Retardation method into the authors code. Parts of the study documented in this paper were supported by British Aerospace (Military Aircraft Ltd). Some of the calculations were performed on the Amdahl VP1 100 and VP1200 computers at the Manchester Computer Centre and on the Cray Y-MP computer at the Rutherford Appleton Laboratory using resources granted to the investigators by the Science and Engineering Research Council.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

REFERENCES

R. A. Childs and D. Nixon, Unsteady three-dimensional simulations of a VTOL upwash fountain. AIAA Paper 86-0212, 24th Aerospace Science Meeting, Reno, Nevada (1986). M. H. Rizk and S. Menon, An investigation of excitation effects on a row of impinging jets using large eddy simulations. Paper AIAA-88-0043, AIAA 26th Aerospace Sciences Meeting, Reno, Nevada (1988). S. L. V. Coelho and J. C. R. Hunt, The dynamics of the near field of strong jets in cross-flows. J. Fluid Mech. 200, 95 (1989). M. Dormieux, P. Guillen and R. Abgrall, Numerical simulation of transverse jet flows by a non-reactive two species multidomain Euler flow solver. AIAA Paper 90-0126, 28th Aerospace Sciences Meeting, Reno, Nevada (1990). R. W. Claus and S. P. Vanka, Multigrid calculation of a jet in cross flow. AIAA Paper 90-0444,28th Aerospace Sciences Meeting (1990). S. V. Patankar, D. K. Basu and S. A. Alpay, Prediction of the three-dimensional velocity field of a deflected jet. J. Fluids Engng 99, 758 (1977). G. Bergeles, A. D. Gosman and B. E. Launder, The turbulent jet in a cross stream at low injection rates: three-dimensional numerical treatment. Numerical Heat Transfer 1, 217 (1978). A. J. White, The prediction of the flow and heat transfer in the vicinity of a jet in cross flow. ASME paper SO-WA/HT-26, Winter Annual Meeting, Chicago, IL (1980). W. Rodi and S. K. Srivatsa, A locally elliptic calculation procedure for three-dimensional flows and its application to a jet in cross-flow. Comp. Meths. Appl. Mech. Engng 23, 67 (1980). W. P. Jones and J. J. McGuirk, Computation of a round tubulent jet discharging into a confined cross-flow. Turbulent Shear Flows 2, (Edited by L. J. S. Bradbury, F. Durst, B. E. Launder, F. W. Schmidt and J. H. Whitelaw), pp. 233-245. Springer ( 1980). A. 0. Demuren, Numerical calculations of steady three-dimensional turbulent jets in cross flow. Compur. Meths. Appl. Mech. Engng 37, 309 (1983).


12. R. I. Sykes, W. S. Lewellen and S. F. Parker, On the vorticity dynamics of a turbulent jet in a cross-flow. J. Fluid Mech. 168, 393 (1985).

13. W. Van Daslem, Study of V/STOL flows using the fortified Navier Stokes scheme. Computational Fluid Dynamics (Edited by de Vahl Davis and Fletcher), pp. 725-735. Elsevier. (1988).

14. J. M. M. Barata, D. F. G. Durao and J. J. McGuirk, Numerical study of single impinging jets through a cross-flow. J. Aircraft 26, 1002 (1989).

15. J. J. McGuirk, G. J. Page, N. Sakellariou, J. E. Flitcroft, W. Abbott, D. R. White and P. Tattersall, Prediction and measurement of jet flow field features for ASTOVL aircraft. Proc. R. Aeronaut. Sot. Inr. Power Lift Conf., London (1990).

16. E. Savory, N. Toy, J. J. McGuirk and N. Sakellariou, An experimental and numerical study of the velocity field associated with a jet in a crossflow. Engineering Turbulence Modelling and Experiments (Edited by Rodi and Ganif), uu. 165-174. Elsevier (1990).

17. B E. Launder, Second-moment closure: present . . and future? Int. J. Heat Fluid Flow 10, 282 (1989). 18. M. A. Leschziner, Modelling engineering flows with Reynolds-stress turbulence closure. J. Wind Engng Znd. Aerodynam.

35, 21 (1990). 19. C. A. Lin and M. A. Leschziner, Computation of three-dimensional injection into swirling flow with second-moment

closure. Proc. 6th Int. Co@ on Numerical Methods in Laminar and Tubulent Flows, Swansea, pp. 1711-1725 (1989). 20. N. Z. Ince and M. A. Leschziner, Computation of three-dimensional jets in crossflow with and without impingement

using second-moment closure. Engineering Turbulence Mode/ring and Experiments (Edited by Rodi and Ganic), pp. 143-153. Elsevier (1990).

21. A. 0. Demuren, Calculation of 3D turbulent jets in crossflow with a multigrid method and a second-moment closure model. Engineering Turbulence Modelling and Experiments (Edited by Rodi and Ganic), pp. 155-163. Elsevier (1990).

22. J. Alvarez and W. P. Jones, Computation of a jet discharging into a cross-flow with a second-moment turbulence closure and a low-diffusive convection-discretisation scheme. Engineering Turbulence Modelling and Experiments 2 (Edited by W. Rodi and F. Martelli), pp. 239-250. Elsevier (1993).

23. N. Z. Ince and M. A. Leschziner, Calculation of single and multiple jets in cross-flow with and without impingement using Reynolds-stress transport closure. Paper 23.1, AGARD Symposium on Computational and Experimental Assessment of Jeis in Cross Flow, Winchester, U.K. (1993).

24. N. Z. Ince and M. A. Leschziner, Modelling three-dimensional jets in cross-flow with and without impingement using forms of second-moment closure. AIAA Paper 93-4862-CP (1993).

25. B. P. Leonard, A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Camp. Meths. Appl. Mech. Engng 19, 59 (1979).

26. J. Zhu and M. A. Leschziner, A local oscillation-damping algorithm for higher-order convection schemes. Camp. Meths. Appl. Mech. Engng 67, 355 (1988).

27. J. J. McGuirk and G. J. Page, Shock capturing using a pressure-correction method. AIAA J. 28, 1751 (1990). 28. W. P. Jones and B. R. Launder, The prediction of laminarisation with a two-equation model of turbulence. Znt. J. Heat

Mass Transfer 15, 301 (1972). 29. M. M. Gibson and B. E. Launder, Ground effect on pressure fluctuations in the atmospheric boundary layers. J. Fluid

Mech. 86, 491 (1978). 30. T. J. Craft and B. E. Launder, A new model of wall-reflection effects on the pressure-strain correlation and its

application to the turbulent impinging jet. J. AIAA 30, 2970 (1992). 31. S. Sarkar, G. Erlebacher and M. Y. Hussaini, Compressible homogeneous shear; simulation and modelling. Proc. 8th

Symp. on Turbulent Shear Flows, Munich, pp. 22-2.1-21-2.6 (1991). 32. K. R. Saripalli, Laser droplet velocimetry measurements in 3-D impinging twin-jet fountain flows. Turbulent Shear

Flows 5 (Edited by F. Durst, B. E. Launder, F. W. Schmidt and J. H. Whitelaw), pp. 146-168. Springer (1987). 33. W. A. Abbott and D. R. White, The effect of nozzle pressure ratio on the fountain formed between two impinging

jets. Royal Aerospace Establishment Technical Memorandum, P1166 (1989). _ --

34. J. M. M. Barata, D. F. G. Durgo and M. V. Heitor, Velocity characteristics of multiple impinging jets through a cross flow. J. Fluids Engng 114, 231 (1992).

35. R. Franke and W. Rodi, Calculation of vortex shedding past a square cylinder with various turbulence models. Proc. 8th Symp. on Turbulent Shear Flows, pp. 20.1.1-20.1.6 (1991).

Documents

1-s2.0-0045793095000144-main.pdf