Direct Numerical Simulation of the Three...

Flow, Turbulence and Combustion 71: 203–220, 2003.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

203

Direct Numerical Simulation of theThree-Dimensional Transition to Turbulencein the Transonic Flow around a Wing

S. BOURDET1, A. BOUHADJI2, M. BRAZA1 and F. THIELE3

1Institut de Mécanique des Fluides de Toulouse, Unité Mixte de Recherche CNRS/INPT UMRNo. 5502, Allée du Prof. Camille Soula, 31400 Toulouse, France; E-mail: [email protected] of Mechanical Engineering, University of Victoria, Canada3Hermann-Föttinger Institut für Strömungsmechanik, Technische Universität Berlin, Germany

Received 4 October 2002; accepted in revised form 30 October 2003

Abstract. The three-dimensional transition to turbulence in the transonic flow around a NACA0012wing of constant spanwise section has been analysed at zero incidence and within the Reynoldsnumber range [3000, 10000], by performing the direct numerical simulation. The successive stagesof the 2D and 3D transition beyond the first bifurcation have been identified. A 2D study has beencarried out near the threshold concerning the appearance of the first bifurcation that is the von-Kármán instability. The critical Mach number associated with this flow transition has been evaluated.Three other successive stages have been detected as the Mach number further increases in the range[0.3, 0.99]. Concerning the 3D transition of this nominally 2D flow configuration, the amplificationof the secondary instability has been studied within the Reynolds number range of [3000, 5000]. Theformation of counter-rotating longitudinal vorticity cells and the consequent appearance of a large-scale spanwise wavelength have been obtained downstream of the trailing-edge shock. A vortexdislocation pattern is developed as a consequence of the shock-vortex interaction near the trailingedge. The subcritical nature of the present 3D transition to turbulence has been proven by means ofthe DNS amplification signals and the Landau global oscillator model.

Key words: buffeting, DNS, instability, transition, transonic flow, wing.

1. Introduction

The understanding of the transition to turbulence for 3D flows is of a major in-terest in aerodynamics because of the amplification of instability modes that aredangerous for the flight behaviour and because of the drastic changes that theyproduce in the amplitudes and the mean values of the global and local parameters.The contribution of compressibility effects on the 3D transition to turbulence isnot yet fully understood. There is a need for investigating the physical mechanismsleading to the appearance of the successive stages of the 3D transition of a nom-inally 2D flow configuration. Rodriguez [21] has depicted the existence of a vonKármán instability in the transonic cylinder’s wake, persisting in the high Reynolds

204 S. BOURDET ET AL.

number range and coexisting with the fine-scale turbulence motion downstream ofthe shock wave.

In previous experimental studies, inherently unsteady quasi-periodic flow phe-nomena and the development of the von Kármán instability in the transonic regimepast aerofoils have been examined in the higher Reynolds number range. In thiscontext, the experimental studies of McDewitt et al. [17] are reported, focusedon transonic flows around a circular-arc airfoil within the Mach number range[0.75, 0.78] and Reynolds number range [106, 1.7×107]. By means of visualisationtechniques and pressure measurements, these studies have reported the existenceof a Mach number range where pronounced unsteady phenomena are developed.The two shocks formed at the suction and pressure sides oscillate alternatinglywith a well distinct frequency and they produce a quasi-periodic, alternating shock-induced separation zone. The experimental studies by Seegmiller et al. [24] con-firmed the existence of these phenomena and quantified the mean velocities, tur-bulent kinetic energy and Reynolds stresses by using Laser-Doppler anemometry.Furthermore, Seegmiller [23] provided measurements of phase-averaged velocitiesin the near region, made available to the European Community BRITE-EURAMresearch program ETMA [9]. A synthesis of CFD studies devoted to this kind offlow has been carried out by Dervieux et al. [10] and has shown the presence of analternating vortex pattern downstream of the shock and in the near wake.

There exist also numerical studies for flows around airfoils in the transonicregime [8, 16]. Although these studies report the existence of this kind of unsteadyphenomena, they investigated rather numerical aspects of these transonic flows.The origins of this kind of instability and the physical mechanisms related to thedevelopment of the first steps towards turbulence are not yet known in this categoryof flows, where the role of compressibility effects seems to be decisive, from themoderate to the high Reynolds number range. The present study aims at analysingthe physical mechanisms leading to the appearance of the von Kármán instability,and of the buffeting phenomenon in the transonic flow regime and investigating theway of establishment of the three-dimensionality under the shock-vortex interac-tion. This study has been carried out for a wing flow configuration (a NACA0012one), due to the importance of transonic flow phenomena to the onset of turbulentmotion in aeronautical applications. Prior to the 3D transition phenomena in flowsaround a wing, we have studied the successive steps of 2D transition in the tran-sonic flow regime, as a continuation of the studies by Bouhadji and Braza [2–4]in our research group. The method used is the Direct Numerical Simulation basedon the complete 3D viscous, unsteady compressible Navier–Stokes equations. TheMach number range of the study is [0.2, 1.0] and the Reynolds number range is[3000, 10000]. Before examining the onset of three-dimensionality and the sec-ondary instability under the compressibility effects, distinct successive steps ofthe 2D transition to turbulence are analysed by means of the 2D Navier–Stokessimulation in the transonic regime.

THREE-DIMENSIONAL TRANSITION TO TURBULENCE IN THE TRANSONIC FLOW 205

2. Governing Equations and Numerical Method for Compressible Flowsaround Bodies

The complete time-dependent Navier–Stokes equations have been solved in threedimensions under a conservative form, in a general non-orthogonal curvilinearcoordinates system. The Roe upwind scheme [22] has been used to discretise theconvection and pressure terms because of their hyperbolic character. The MUSCLapproach by van Leer [27] has been employed in order to increase the spatialaccuracy from the first to second order. This scheme is known to be particularlywell adapted to capture shock waves, providing good accuracy and stability. Thisscheme has been used without limiter. A careful grid refinement has been per-formed to avoid any spurious wiggle oscillation. However, in the higher Reynoldsnumber ranges, the use of limiters is recommendable to ensure monotonicity withthe use of reasonable grid sizes. The diffusion terms have been discretised by cen-tral differences and the temporal terms using an explicit, second order of precisionin time, four-stage Runge–Kutta scheme [13]. The use of the explicit scheme is ap-propriate in the present Reynolds number range, because the organised modes de-veloped in the transition process are low frequencies. However, an implicit schemeis preferable to investigate the high Reynolds number transition, especially theboundary-layer transition. The computational domain is a C-Type grid, consistingof 3 × 106 grid points for the 3D simulations, as well as [2.104, 105] for the 2Dcases. The grid resolution has been carefully studied to provide grid-independentsolutions [2]. Typical grid sizes used are 249 × 79, 369 × 49, 801 × 160 and300 × 120 × 80. A distance of ten chord-lengths separates the leading edge fromthe outflow boundary and there are seven chordlengths between the airfoil and theouter boundary. In the 3D simulations the spanwise length is four chords.

2.1. THE BOUNDARY CONDITIONS

Freestream conditions have been imposed at the outer boundaries, except for thedownstream one, where a first order extrapolation has been used. In 3D simula-tions the side boundary conditions are of Neumann type. On the airfoil surfaceNeumann conditions are used for the temperature (adiabatic wall), density and en-ergy. Pressure has been computed from the momentum equation which was solvednumerically with adherent condition for the velocities. Along the wake line, theboundary values of the velocity, pressure and density have been computed by aver-aging the mentioned variables from adjacent lines above and below the wake line.Detailed numerical tests have been carried out [2] to ensure that the boundary con-ditions and the computational domain size do not produce any spurious effect. Thenumerical characteristics of the solver and its behaviour in respect of the boundaryconditions have been reported in detail in [3–6]. We have ensured that the outerboundary was positioned far enough from the airfoil to match with simulationsthat used non-reflecting boundary conditions [1, 14]. Concerning the boundaryconditions at the spanwise sides of the computational domain, a detailed study


Figure 1. Streamlines around the trailing edge of the airfoil, 2D simulation, M∞ = 0.2,Re = 104 (quasi-incompressible flow). Two symmetrical re-circulation zones appear. Thereis no instability developed in the wake. (Original in colour)

has been carried out in our research group for incompressible flows, comparingthe periodic boundary conditions to the Neumann (zero gradient) ones. Indeed, theperiodic conditions need to be applied on a multiple of the predominant large-scalewavelength expected to appear along the span, in order to not truncate it. Therefore,an a-priori need of knowledge of this wavelength is needed. This comment hasbeen detailed in [18] for the incompressible cylinder wake. To our knowledge,there are no other studies yet in the transonic regime giving an assessment of theexpected spanwise organised wavelength. The Neumann conditions specify thatthe flow properties are similar on the two last adjacent lines at each spanwise end-section. This property is satisfied given a fine spanwise grid and these conditionsallow a spontaneous appearance of the physically expected spanwise wavelengthcharacterising the secondary instability. These conditions do not ‘favourise’ or‘force’ the appearance of a specific predominant spanwise wavelength as the pe-riodic conditions do. They need of course finer grids than the periodic boundaryconditions.

The numerical solver has been developed in our research group in IMFT, in theframework of the Ph.D. research of A. Bouhadji.

3. Compressibility Effects in the Wake Transition

3.1. THE SUCCESSIVE STEPS TO TRANSITION, SELECTED BY THE 2D FLOW

SYSTEM

At low Mach numbers corresponding to an almost incompressible flow, the regimeremains stable at zero incidence, as it has been reported in a number of exper-imental and numerical studies (see the collected flow visualisations reported byvan Dyke [26] (picture by P. Bradshaw, no. 72 p. 41 and by Head, no. 86 p. 51)[25, 20]. In the present study, at M = 0.2 and Re = 10000 (incompressible flow),the flow reaches naturally a steady state after a long transient phase (Figure 1).The predicted flow configuration consists of two symmetrical recirculation zones


formed downstream of the airfoil. The separation point is located at x/c = 0.76.By increasing the Mach number to 0.3, a slight oscillation of the lift coefficient isobtained; however the flow regime remains quasi-steady. At M = 0.35, the flow isno longer symmetric and steady. A weak oscillation appears in the lift coefficientand in the wake. The shape of the symmetric u-velocity profiles along the verti-cal axis downstream of the airfoil show two inflexion points as confirmed by thevanishing points of the vertical velocity shear (∂u/∂y). Therefore, the influence ofsmall perturbations will cause the appearance of a von Kármán instability, whenthe critical number (in this case the Mach number) will become higher than acritical value. This has been confirmed by the stability analysis solving the Orr–Sommerfeld equation, on the basis of the present Navier–Stokes mean solution.A preferential frequency mode is found to amplify at Mach number 0.35. Thedimensionless frequency value is found 2.384. The amplification of this instabilityis clearly developed by the Navier–Stokes solution in the Mach number range (0.2,0.35) as a first transition step due to the rise of the compressibility effects. Thecorresponding frequency is found 2.375, in very good agreement with the stabilitystudy.

The behaviour of the present system is completely different than the sameReynolds number flow in the incompressible regime, where no amplification of avon Kármán instability would occur around a lifting body at zero incidence. In thepresent case, due to the Mach number increase, the hyperbolic character of the flowis more and more strengthened compared to the elliptic one of an incompressibleflow. Therefore, the existing small perturbations produced by the truncation and theround-off errors in the numerical model travel more easily downstream towardsthe wake, because the convection velocity is increased under constant Reynoldsnumber conditions. Therefore, the inflectional shape of the mean velocity profilesin the wake are more easily destabilised than in the incompressible symmetric caseand they give rise to the von Kármán instability, called mode I in the present study.

A detailed numerical study has been carried out near the threshold involvingcomplete simulations over long time durations for a considerable number of nom-inal Mach values. The amplification envelopes of the signals (Figure 2) have beenused to evaluate the coefficients of the Stuart–Landau global oscillator model andthe critical Mach number for the appearance of the von Kármán instability [15, 19].The Stuart–Landau equation is presented in the following:

dA

dt= σA − 1

2l |A|2 A, (1)

where A is the complex envelope amplitude of the oscillations as a function of theperiod. In the present study, the vertical velocity component near the rear axis isconsidered to evaluate the amplitude oscillations, because it is the most significantquantity of the symmetry breaking. σ is the relative growth rate, σ = σr + iσi

where σr is the amplification rate in the linear regime and σi the angular frequency.l is the Landau constant. The real part of the model is considered in the present


Figure 2. Top: V -velocity component at x/c = 1.29, y/c = 0.01, showing the amplificationof the von Karman instability. Bottom: Growth rate, σr , of the von Karman instability in thetransonic regime; critical Mach number: Mcr = 0.236. (Original in colour)

study involving the absolute values of the amplification. Equation (1) is written asfollows:

d |A|dt

= σr |A| − 1

2lr |A|3 , (2)

where d |A| /dt is the evolution of the amplitude versus the period. The termσr |A| represents the linear part (exponential growth) of the instability and theterm −(1/2)lr |A|3 contains the non-linear effect on the mode. Equation (2) canbe integrated analytically as follows:

|A| = σr

{1

2lr (1 − e−2σr t ) + σr

|A0| e−2σr t

}−1/2

, (3)

where |A0| is the initial condition: |A| (t = 0). In order to avoid problems with thedetermination of the initial condition in the general case, where there is a transientphase before the established one due to the initial conditions of the simulation,


Equation (2) is discretised as follows, according to the general way of procedure inthe afore-mentioned studies:

|A|n+1 − |A|n�t

= σr |A|n − 1

2lr |A|3n . (4)

It is known that the sign of lr is related to the supercritical (lr > 0), or subcritical(lr < 0) nature of the instability near the threshold [11]. In the present case, giventhe evaluation of |An| by the direct numerical simulation and adopting the Stuart–Landau equation, the sign of lr can be given directly by the relation:

�n = |A|n+1 − γr |A|n|An|3

with γr = 1 + σr�t. (5)

By first considering the linear growth of the amplitude of the instability, withoutthe nonlinear term lr |A| in Equation (2), the amplification rate σr can be evaluatedand plotted versus the nominal Mach number.

This exponential growth coefficient σr can be written as a function of the criticalReynolds number near the threshold, σr = k(M − Mcr), where M is the upstreamMach number. According to Provansal, k represents a reduced frequency. This re-lation provides the critical Mach number evaluation for the appearance of the vonKármán instability, by performing a number of complete numerical simulationsnear the threshold and by evaluating for each one the amplification rate. The vari-ation of this rate as a function of the upstream Mach number gives the assessmentof the critical Mach number (Figure 2, bottom), Mcr = 0.236 in the present study.

At Mach number values 0.5 and 0.7 the von Kármán instability becomes morepronounced, as can be seen from the increased amplitudes of the lift coefficientoscillations (Figures 3a and 3b). The flow is organised according to an alternatingvortex pattern in the wake, similar to the one appearing in incompressible flows,essentially for flows past bluff bodies at low Reynolds numbers. At higher Machnumber values (M = 0.75 and M = 0.80), one may expect that mode I becomesmore pronounced; however, this is strikingly not the case, since another unsteadyphenomenon appears beyond the persistence of mode I: a new instability develops,having a much lower frequency and called mode II. This instability creates a dras-tic increase of the lift coefficient amplitude (Figures 3c and 3d). The two modesinteract non-linearly and lead to the appearance of additional predominant frequen-cies, as seen in the spectra of Figure 3c. This mode is a buffeting phenomenon. Itis associated with an alternating formation of large-scale pockets of accelerated,slightly supersonic flow near the leading edge and along the half chord distanceapproximately, on the two sides of the airfoil (Figure 4). The quantification ofthe flow parameters in the suction and pressure sides at two symmetric locations,shows a phase opposition by 180 degrees. The appearance of mode II instabilitycorresponds to a buffeting phenomenon, inducing a strong low frequency vibra-tion as reported in the high Reynolds number range [24]. This phenomenon is aninviscid oscillation triggered by the forcing action of mode I. It is striking that


Figure 3. Lift coefficient (CL) evolution versus time showing successive steps of the transi-tion. Spectra of CL, quantifying the frequencies of mode I and mode II, as a function of Machnumber, M∞: (a) 0.5, (b) 0.7, (c) 0.75, (d) 0.8, (e) 0.85, (f) 0.9, (g) 0.95, (h) 0.98.


Figure 4. Development of the buffeting phenomenon at M∞ = 0.8, Re = 104. Oscillation ofthe two supersonic regions at phase opposition by 180◦. (Original in colour)


Figure 5. Iso-mach contours at M∞ = 0.9, Re = 104. The shock wave inhibits the per-turbation from Mode I to travel upstream. (Original in colour)

mode II vanishes at higher Mach numbers (see lift coefficient at M = 0.85, Figure3e) and the flow is only governed by the mode I instability. Therefore, a furthersuccessive stage of the flow transition is reached. The vanishing of mode II isdue to the progressive appearance of shock waves around the airfoil trailing edge,that inhibit the influence of mode I to travel upstream, towards the leading edgeregion (Figure 5). By increasing further on the Mach number, a fourth transitionregime occurs: one may expect that the oscillations of mode I would increase toreach a saturation state, as in the case of incompressible flows. However, theseoscillations drastically decrease as the compressibility effects increase, to vanishtotally beyond M = 0.95. The flow field shows that a delta-shape strong supersonicregion is formed and the shock wave is swept downstream, as the Mach numberincreases from 0.9 to 0.95. The von Kármán pattern persists only downstream ofthe shock, whereas the upstream region turns back to a steady state. Indeed, asthe hyperbolic character of the flow is more and more strengthened, the impactof mode I instability is progressively moved downstream of the shock, that actsas a barrier inhibiting the perturbations to travel upstream. Therefore, due to thecompressibility effect, the absolute nature of mode I instability is progressivelytransformed to a convective one. Although the impact of mode I is no more visiblein the very near region (see the lift coefficient displaying a practically steady statebehaviour beyond M = 0.9), the trace of mode I instability persists and it is sweptfurther downstream in the wake, following the shock movement.

Figure 6 (top) shows the relation between the dimensionless fundamental fre-quency and the Mach number, according to the mentioned successive steps. It isfound that the mode I slightly decreases as a function of the Mach number for


Figure 6. Strouhal number (top) and drag coefficient Cd (bottom) versus Mach number.Drastic increase of Cd and decrease of St when the Mach number traverses the buffetingrange.

M < 0.7. Afterwards, the onset of mode II is associated with a decrease of themode I frequency. A partial explanation of this behaviour is that the externallysupplied energy, proportional to U 2

0 is attributed not only to sustain mode I but alsomode II amplification pattern within the Mach number range [0.7, 0.85]. For thesereasons, the mode I frequency increases again beyond M = 0.85, where mode IIhas vanished. However, a stability analysis would be needed to explain completelythis behaviour through a dispersion relation. Moreover, the present study providesthe drag coefficient evolution as a function of the Mach number. A drastic increaseof the drag coefficient is depicted in the specific range of the transonic regime,Figure 6 (bottom) corresponding to the buffeting mode II. This is a consequence ofthe previously mentioned changes related to mode I and mode II instabilities, thatfundamentally modify the near-wall dynamics. This rise of drag is a consequenceof the pressure drag variations in respect of the mode II instability and it can bealso simulated by an Euler approach. However, the instability process performsdrastic changes in the separated boundary layer that thickens considerably andbecomes unsteady, as well as in the shock-boundary layer interaction region. TheNavier–Stokes approach provides a more accurate evaluation of the drag increasewith respect to both, inviscid and viscous effects. The drag coefficient increase isan important feature induced by the compressibility effects on the flow transition.


Figure 7. DNS, transonic flow around a wing of constant NACA0012 section, α = 0◦,M∞ = 0.85, Re = 5000. Left: Iso-Mach surfaces, purple: M = 1.04, cyan: M = 0.55.Shock-boundary layer and shock-vortex interaction. Three-dimensional modification of thesecond shock near the trailing edge. Spanwise large-scale undulation in the near wake andvortex dislocation, as a consequence of the secondary instability. Right: streamwise vorticitycounter-rotating cells. (Original in colour)

It justifies the priority interest of these studies concerning the unsteady transonicregime in aeronautics nowadays.

Summarising the above successive changes, four important transition steps aredeveloped as Mach number increases. The role of Mach number is found decisive totrigger, to sustain and to inhibit the primary instability and to mark the successiveappearance of the described regimes.

3.2. 3D TRANSITION FEATURES

Among the identified regimes, one of the most important is related to the mode Iinstability, that is very pronounced in the vicinity of M = 0.85. Therefore, thisMach number value is selected to study the onset of three-dimensionality from anominally bi-dimensional flow configuration of a NACA0012 wing of constantsection along z. This study is carried out by Direct Numerical Simulation, forReynolds numbers 3000 and 5000. At Reynolds number 3000, the route to tur-bulence is firstly marked by the amplification of a secondary instability, as canbe seen in the time-dependent evolution of the w component at selected spanwisepositions (Figure 9 top). Beyond the drastic increase of the amplitude versus time,it is found that the frequency of this oscillation is regularly organised at the dimen-sionless value of 1.044. This fluctuating evolution is developed additionally andat a different frequency than the primary instability of the mode I. This quantifiesthe first steps of the filling up of the energy spectrum by more than one incom-mensurate frequency. These 3D effects are more pronounced at Reynolds number5000 and they lead to a significant decrease of the instantaneous local parameters(u and v velocity evolution versus time) and on the mean global ones (drag, lift and


Figure 8. Evolution versus time of the u and v velocity component and pressure coefficient(Cp) upstream of the trailing edge (x/c = 0.885, y/c = 0.0992, z/c = 0.675), in 3D and 2Dconfigurations. M = 0.85, Re = 5000.

pressure coefficients) and of their amplitudes, compared to the 2D results (Figure8). The birth and the progressive development of the streamwise vorticity lead tothe appearance of a large-scale spanwise undulation of the von Kármán vortexrows (Figure 7 left). In addition, the raise of streamwise vorticity occurs and isprogressively organised according to well distinct counter-rotating cells (Figure 7right). The streamwise wavelength characterising the spanwise undulation of thevon Kármán vortex rows is found λz = 2.40c. This is the characteristic wavelengthof the secondary instability developing as a successive step after the primary in-stability analysed in the previous section. Therefore, the first 3D change of theflow transition to turbulence is performed in a similar way as in incompressibleflows past bluff bodies [18]. It is worthwhile qualifying the nature of this secondaryinstability that appears downstream of the shock, by considering the Stuart–Landauglobal oscillator model described previously.

The present direct numerical simulation allows quantifying the amplitude vari-ations versus time and therefore evaluating the coefficients of this global instabilitymodel. The Reynolds number value of 3000 is chosen because it is near thethreshold. The evolution of the w velocity versus time is considered, because this


Figure 9. Top: Amplitude evolution versus period of the secondary instability. w velocitycomponent versus time at z/c = 1.665, origin of the reference frame is on the left side,referring to the streamwise direction. Middle: Evaluation of the linear growth coefficient σr .Bottom: Evaluation of the non-linear coefficient �n, according to the Landau model.

component clearly illustrates the onset of the secondary instability. Figure 9 showsthis evolution at the point (x/c = 1.17, y/c = 0.05, z/c = 1.665). The curveformed by the small circles shows the envelope of the instability amplification andtherefore the amplitude variation versus the period. The amplification rate in thelinear regime is determined. The rate |A|n+1 / |A|n, that is a first order approxima-tion for γr (5), has been plotted versus time, Figure 9 (middle). It is shown that apractically constant rate is attained in the immediate upstream vicinity of the sat-uration phase of the instability process. This ensures the validity of the first-orderapproximation taken for the assessment of γr according to the work by Hendersonand Barkley [12], as well as the fact that the first stage of the amplification of modeI corresponds indeed to an exponential growth. This is described by the linear partof the Stuart–Landau equation. The function �n (5) is shown on Figure 9 (bottom).The values of �n near the saturation region are positive. Therefore, the nature of thesecondary instability yielding the large-scale spanwise undulation, is subcritical. Inthe same way, we have shown that mode I instability is supercritical (Figure 9).

3.3. THE NATURAL VORTEX DISLOCATIONS PATTERN

The instantaneous iso-Mach field is shown in Figure 7 (left). The violet configura-tions clearly show the trailing edge shocks. Behind them, the originally rectilinearvon Kármán vortex rows are subjected to the large-scale spanwise undulation, as


mentioned above. It is clearly shown that the adjacent downstream vortex rowforms a local junction with the previous one and it displays a break in the continuityof its ‘spinal column’. This fundamental modification, occurring in the spanwisestructure of the already undulated main vortex rows in the near wake is a naturalvortex dislocation. A detailed analysis of this kind of structures in the space-timecan be found in [7]. In the present study, the vortex dislocation cells are clearlyformed (Figures 7 and 9) in a natural way, e.g. without any forcing, by means ofthe complete system of the Navier–Stokes approach.

4. Conclusions

The present study has identified the successive steps of the transition to turbulencein viscous transonic flows around bodies. It has provided an analysis of the com-pressibility effects on the instability mechanisms and on the birth of the turbulentmotion in the present category of flows, strongly marked by a shock-vortex inter-action. It is recalled that the transonic regime transition mechanisms have receivedlittle investigation up to now, compared to the incompressible bluff body flows.

− According to the present study, four successive transition steps have beendepicted related to the onset of mode I and mode II instabilities and to theirsuppression.

1. The increase of the compressibility effects from the low Mach numberregime to the intermediate one, M ∈ [0.3,0.35] triggers the mode I in-stability, that is found to be an absolute instability of a supercriticalnature.

2. The increase of the Mach number in the range [0.5, 0.8] sustains the modeI and gives rise to the mode II, related to a buffeting phenomenon.

3. The Mach number increase towards the upper transonic range leads to astrong interaction of the shock waves with the mode I. This is progres-sively swept in the wake, due to the barrier action of the shock waves toany perturbation travelling upstream. This shock-vortex interaction leadsto the progressive transformation of the absolute character of the mode Iinstability to a convective one and to the suppression of mode II.

4. A further increase of the Mach number from 0.9 to 0.99 sweeps com-pletely the mode I in the wake. The regime becomes steady in thenear-wall region.

− The present study has quantified the critical Mach number associated withthe appearance of mode I instability, by means of the Stuart–Landau globaloscillator model.


− A triple role of the compressibility effects with respect to the flow transition isdetected within the whole transonic regime: to trigger, to sustain and to inhibitthe primary instability in the near region.

− The first step of the 3D transition is a regular amplification of the transversevelocity component, followed by the amplification and organisation of thestreamwise vorticity according to counter-rotating cells. The secondary in-stability is therefore fully developed and displays a drastic modification ofthe originally rectilinear alternating vortex rows according to a large-scaleundulation downstream of the trailing-edge shock.

− The nature of the developed secondary instability under the compressibilityeffects of the transonic regime is subcritical.

− The spanwise undulated vortex rows display a vortex dislocation structure inthe near wake along the span, as a further transition step.

This final transition step is associated with a reduction of the amplitude variationsversus time and to a decrease of the drag and lift coefficients, compared to theirvalues issued from a 2D computation. Therefore, a reliable evaluation of the globalparameters is only achievable by performing a 3D simulation, within the presentReynolds and Mach number range.

Acknowledgements

This work has been carried out in the research group EMT2 (EcoulementsMonophasiques, Transitionnels et Turbulents) of the Institut de Mécanique desFluides de Toulouse. The solver has been fully parallelised by A. Mango and G. Ur-bach of the CINES (Centre Informatique National de l’Enseignement Supérieur).Part of this work is carried out on the basis of CPU allocations of the two nationalcomputer centers of France CINES and IDRIS (Institut du Développement et deRessources en Informatique Scientifique).

Part of this work has been carried out in the context of the CEC researchprogramme UNSI (Unsteady Viscous Flows in the Context of Fluid-StructureInteraction), No. BRPR-CT97-0583, ended in December 2000. The authors arealso very thankful to the UNSI partnership: DASA-M (Coordinator), Alenia, Bae,CASA, Dassault, SAAB, NUMECA, DERA, DLR, FFA, ONERA, IRPHE, TUB,UMIST. Part of the database of the present results has been put in the ERCOFTACData-Base, coordinated by Professor P. Voke, University of Surrey.

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