1American Institute of Aeronautics and Astronautics

Unsteady 3-D Navier-Stokes Simulations on Characteristic Frequency and LengthScales in Transonic Diffuser

Debasish Biswas*Toshiba Corporation

Research and Development CenterKawasaki, Japan

e-mail: debasish.biswas@toshiba.co.jp

Abstract

An improved low-Reynolds k-τ turbulent model developed by theauthor is assessed against experimental data of Sajben on the self-excited shock oscillation phenomena. The whole diffuser modelconfiguration including the suction slot located at certain axiallocation around the bottom and side walls to remove boundary layer,are included in the present computation model. The time-mean andunsteady flow characteristics in this transonic diffuser as a functionof flow Mach number and diffuser length are investigated in detail.The results of study showed, that in the case of shock-inducedseparation flow, the length and thickness of the reverse flow regionof the separation-bubble change, as the shock passed through itscycle. The instabilities in the separated layer, the shock /boundarylayer interaction, the dynamics of entrainment in the separationbubble, and the interaction of the travelling pressure wave with thepressure fluctuation region caused by the step-like structure of thesuction slot play very important role in the shock-oscillationfrequency.

Introduction

Unsteady transonic flows in diffuser have become increasinglyimportant, because of its application in new propulsion systems. Inthe development of supersonic inlet, air breathing propulsionsystems of aircraft and missiles, detail investigations of these typesof flow behavior are very much essential. In these propulsionsystems, naturally present, self-sustaining oscillations, believed to beequivalent to dynamically distorted flow fields in operational inlets,were found under all operating conditions. The investigations arealso relevant to pressure oscillations known to occur in ramjet inletsin response to combustor instabilities. The unsteady aspects of theseflows are important because the appearance of undesirablefluctuations generally impose limitation on the inlet performance.Test results of ramjet propulsion systems have shown undesirablehigh amplitude pressure fluctuations caused by the combustioninstability. The pressure fluctuations originated from the combustorextend forward into the inlet and interact with the diffuser flow-field.The frequencies of oscillation in the range of 100-500 Hz areconsidered as most troublesome. The root-mean-square (RMS)amplitude of oscillation can reach up to 20 percent of the meanpressure in the combustor. Presently some engines are built withover-relaxed diffusers from the point of view of performance.However, it is found that for some configurations such diffusers mayexhibit self-sustained oscillations, which impose major limitationson vehicle performance. Depending on different parameters such asthe diffuser geometry, the inlet/exit pressure ratio, the flow Machnumber, there may occur different complicated phenomena. Themost important characteristics are the occurrence of shock inducedseparation, the length of separation region downstream of the shocklocation, and the oscillation of shock location as well as theoscillation of the whole downstream flow. Sajben et al. (1-4)

experimentally investigated in detail the time mean and unsteadyflow characteristics of super-critical transonic diffuser as a functionof flow Mach number upstream the shock location and diffuserlength. The flows exhibited features similar to those in supersonicinlets of air-breathing propulsion systems of aircraft and missiles. Intheir experiments, two parameters were used to define a particulartest condition. One is the local Mach number at the edge of the topwall boundary layer immediately upstream of the shock and theother is the diffuser length. On the basis of spark schlierenphotograph, high speed schlieren movies, and oil flow patterns onthe walls, they reported that the flow always assumed one of twodistinct modes. For flow less than a critical value, the flow wasattached everywhere in the channel. The boundary layers on the topand bottom wall remained relatively thin, and an inviscid region ofconstant mean total pressure (the core flow) continued past the endof the un-extended channel. For flow Mach number above thecritical value, shock induced separation along the top wall wasobserved. The bottom-wall boundary layer remained attached (notconfirmed in detail). Both boundary layers thicken rapidly andmerge well inside the un-extended diffuser. The behavior of theunsteady quantities in this diffuser depends strongly on whether themean flow is attached or separated, and is radically different for eachcondition.

Starting from the end of 1980's, many numerical simulations werecarried out by several research groups. Liou and Hsieh's team (5-8)performed numerical simulations using the diffuser configuration ofSajben. In their numerical simulation, unsteady Reynolds averagedN-S equations with a k-ω type turbulence model are solved. Theyperformed the simulations in two steps. As a first step, steady stateequations are solved to obtain information regarding mean flow field.In the second step, unsteady equations were solved to obtaininformation regarding time dependent flow behavior. Theycompared their results with the experimentally measured shockmotion power spectral density distribution, and the amplitude andphase of pressure fluctuation. Hsieh et al.(7) studied the effect ofdiffuser length on the frequency of the self-excited shock oscillationin transonic diffuser. They reported that the location of thedownstream boundary has a strong effect on the frequency ofoscillation. The oscillation frequency decreased from 300 to 210 Hzas the diffuser length (length/throat height) increased from 8.6 to12.08. This means that the frequency decreased with the increase ofdiffuser length. However, for l/h=14.4, the trend is reversed and thefrequency of oscillation increased to 310 Hz. According to Hsiehet al., the results of their study suggest that the mechanism causingself-excited oscillation changed from viscous and convective wavedominated mode to inviscid acoustic wave dominated mode. Also,their computed amplitude of pressure fluctuation was quite higher ascompared to experimentally measured amplitude.

Robinet et al. (9) carried out some numerical simulation based onlinear stability analysis to study the physical characteristics of self-

* Senior Research Engineer

34th AIAA Fluid Dynamics Conference and Exhibit28 June - 1 July 2004, Portland, Oregon

AIAA 2004-2715

Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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sustained shock oscillations in transonic diffuser configuration ofSajben. In their study, information regarding the mean flow behaviorwas obtained by solving Reynolds averaged steady-state N-Sequations. Their stability approach was limited to the core region,where the viscous effect could be neglected. They compared theircomputed results on the dependence of the frequency shockoscillation on the diffuser length with the experimental data. Theirapproach based on the small perturbation technique to explain theself-sustained oscillation in transonic diffuser could predict theexperimental data up to the diffuser length of 12. For diffuser lengthabove 14, their linear stability approach failed to predict theexperimental data.

Main reason behind the discrepancy between the experimental andnumerical results is that the configuration of the transonic diffuserused by Sajben is not simulated properly in the numerical study. Inthe actual diffuser configuration, there is suction slot around thebottom and both the side walls upstream of the exit. The suction slotremoves the boundary layer and thus significantly changes the flowfield between the suction slot and the exit plane. Since the flow issubsonic in the entire diffuser downstream the shock, the effects ofsuction slot in the experiment obscures the true location of thephysical downstream boundary. However, no one included thesuction slot in the diffuser configuration used in their numericalstudy. Also, all the groups carried out numerical simulation bysolving 2-D Reynolds averaged N-S equation using the assumptionthat the flow is two-dimensional in the mid span plane of the diffuser.The assumption is true for flow Mach number below a critical value.However, in their numerical study they all choose a flow Machnumber, which is well above the critical Mach number. Under thatcondition, the flow undergoes shock-induced separation along thetop wall. Sajben et al. in their work reported some results on timemean Mach number distribution at the exit plane. The results showedthat a high degree of two-dimensionality (attached flow) remains forthe flow Mach number below a critical value, but three-dimensionaleffects becomes prominent in the case of separated flow, when theflow Mach number exceeds the critical value. In the case ofseparated flow, the presence of secondary flow gives rise to aninflux of low speed fluid from the sides toward the center causingthe boundary layer to thicken more rapidly on the centerline. For theflow Mach number condition used by all the research groups in theirnumerical study, the assumption of two-dimensionality is not validany more.

In fact, in simulating certain physical phenomena, eitherexperimentally or numerically, one needs to carefully consider theproper geometric configuration and the flow conditions, which aregenerating the flow phenomena to be studied. In the present work,the exact three-dimensional configuration of the Sajben's transonicdiffuser including the suction slot around the bottom and both theside walls upstream of the exit is considered to numerically study thephysical characteristics of self-sustained shock oscillations intransonic diffuser. Also, since under the present flow conditions,shock-induced separation, the boundary layer growth downstream ofthe shock location, near-wall turbulence activity play a veryimportant role in the self-sustained shock oscillation phenomena,emphasis is put in physics-based-modeling of the turbulencephenomena. In Sajben's experiment, it is observed that for flowMach number above a critical value, the flow experienced shock-induced separation along the top wall. After the onset of separation,the bubble enlarged quickly along the top wall. The length of theseparation region depends on the shock strength. The boundary layerof both top and bottom wall thicken rapidly downstream the shockand merge well inside the un-extended diffuser. This process resultsin dramatic reduction of core flow length. In order to correctly

predict flows of this type that include a complex, separated flowcharacterized by strong viscous-inviscid interaction, in the modelingof turbulence, transition from laminar to turbulent in the boundarylayer, shock boundary layer interaction, and shock inducedseparation are considered. Attentions are also paid to consider theeffect of free-stream turbulence and the pressure gradient on theabove mentioned phenomena. In order to demonstrate theperformance of the physics-based models they need to be assessedagainst reliable measurements. In this study Sajben's experimentaldata on self-sustained shock oscillations in transonic diffuser areused to test the performance of the turbulence model developed bythe author.

An increasing number of practical; engineering calculations oftransitional and turbulent flows have been based on two-equations-low-Reynolds number version of the k−ε turbulence model. Formany technologically important turbulent flows, two-equationmodels represent a nice compromise between zero or one-equationmodels and second order closures. The k-ε model is reasonably wellbehaved and has been applied to solve a variety of engineeringproblems with a moderate amount of success. However, manyimportant technological applications require the integration ofturbulence models directly to a solid boundary, particularly inproblems where wall transport properties are needed or where thereis flow separation. The problem of developing low-Reynolds-number near wall corrections to the k-ε model that can be robustlyand accurately integrated to a solid boundary remains unresolved sothat models along alternative lines continue to be proposed (10).Most of these near-wall k-ε models involve an excessive amount ofad hoc empiricisms and are numerically stiff in turbulent boundarylayer flows. This motivated some researchers to pursue alternativetwo-equation models based on a modeled transport equation for theturbulent time scale. The most notable example is the k-ω model ofWilcox and Traci(11) and Wilcox(12) , where modeled transportequations for the turbulent kinetic energy K and reciprocal turbulenttime scale ω are solved. There is considerable evidence that the k-ωmodel is more computationally robust than the k-ε model for theintegration of turbulent flows to a solid boundary.

In the author's previous work(13), the near wall asymptotic oftwo-equation turbulence models is examined from a basic theoreticalstandpoint. It is found that k-ε model has two major problemsassociated with it. The first arises from the lack of natural boundaryconditions for the turbulent dissipation rate. This caused modelers touse a variety of derived boundary conditions that are eitherasymptotically inconsistent (e.g., the boundary condition ofvanishing normal derivative of dissipation) or numerically stiff (e.g.,the boundary condition that ties the dissipation to higher orderderivatives of turbulent kinetic energy). The second problem whichcan be the source of substantial inaccuracies and numerical stiffness,is tied to the fact that the balance of terms at the wall in the modeleddissipation rate transport equation depends on higher order co-relations whose models have considerable uncertainties.

It is demonstrated in the previous work (13) that both of theseproblems can be largely alleviated by solving a modeled transportequation for the turbulent time-scale τ=k/ε , since 1) near the wallτ=y2/2ν, which provides the needed natural boundary conditions and,2) the balance of terms at the wall in the modeled transport equationfor τ involves only exact viscous terms. The k-τ model developed bythe author(13) was obtained by including this exact viscous term andby substituting wall-damping functions obtained by an asymptoticanalysis using the results of turbulent channel flows. The turbulencemodel was proposed to reproduce the wall-limiting behavior of

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turbulence, the correct stream-wise blending between a pre-transitional pseudo-laminar boundary layer and post-transitionalfully turbulent boundary layer under the experimental effect of free-stream turbulence and pressure gradient.

In order to solve the problem of flow separation with sufficientaccuracy, turbulence models need to be integrated directly to a solidboundary. Integration of turbulence models directly to solidboundary can be performed by introducing some damping functionsin the standard form of turbulence model, which take into accountthe near-wall low-Reynolds number effect. In most of the wellknown existing low-Reynolds number two-equations turbulencemodels, the low-Reynolds number damping functions are defined byassuming that the approximations that have been devised to handlenear-wall low-Reynolds flows are equally valid for low-Reynoldsnumber transitional flows. That is why in most of the existingmodels, the low-Reynolds damping functions are defined in terms ofsome local functions like flow normal distance y or y+ rather thansome quantity which takes into account the effect of flow field atany location. However, Biswas(14) in his work indicated that this isnot true. Physically, near correct predictions of transitional boundarylayer can be achieved by defining the low-Reynolds damping factorsas some quantity (for example, turbulence Reynolds number), whichis only a rather general indicator of the degree of turbulent activity atany location in the flow rather than a specific function of locationitself.. Based on the understanding of characteristics of differentlow-Reynolds number functions, an improved low-Reynolds two-equations turbulent model proposed by the author (13), in whichthese functions satisfy both the wall limiting effect and theasymptotic behavior in the region away from the wall is used.

A comprehensive comparison between computation andexperimental data regarding time mean and unsteady flow behavior(variation of separation pocket and re-circulation region, the lengthof core flow region, frequency of oscillation and the fluctuatingpressure and velocity with flow Mach number) is carried out.. Theresults helped to have a good understanding of who does what, howand why from the visualization mean and unsteady flow behavior,which was not possible to obtain due the limitation of measurementtechnique.

Computation Method

In the present section governing equations, turbulence model,numerical methodology, experimental and numerical modelconfiguration and boundary conditions will be discussed.

Governing Equations The basic equations used in the present study are the three-dimensional, compressible time-dependent Navier-Stokes equationsfor the conservation of mass, momentum and energy, together with alow-Reynolds number version two-equation turbulent modeldeveloped by the author for the turbulence kinetic energy and thespecific dissipation rate of turbulence kinetic energy. In addition, theequation of state for a perfect gas is assumed. There are, therefore,seven equations for the seven variables ρ, u, v, w, e, k and τ. All theequations are written in conservation form. In order to havesufficient grid resolution in the near-wall viscous layer region, bodyfitted co-ordinates are used. That means all the equations are writtenin general curvilinear co-ordinates.

Turbulence Model Based on usual Einstein summation convention applies to

repeated indices the Reynolds stress tensor jiij uu ′′−=τ can be

written as follows;

∂∂

+∂∂

+−=i

j

j

itijij x

uxu

K νδτ 3/2 (1)

εν µ /2KCt = (2)

In the above equations jiuuK ′′= 2/1 is the turbulent kinetic

energy, ijji xuxu ∂′∂∂′∂= /νε is the turbulent dissipation rate,

and C� is a dimensionless constant at high Reynolds number. In two-equation models transport equations are solved for any two linearlyindependent variables constructed from K and ε. In the K-ε model,modeled transport equations for K and ε are solved; In the K-ωmodel, modeled transport equations for K and the reciprocalturbulent time scale ω=ε/K are solved; and in the K-τ model,modeled transport equations for K and the turbulent time scaleτ=K/ε are solved. In low-Reynolds-number version K-τ model, theeddy-viscosity near the wall is taken to be of the form

τν µµ KfCT = (3)

Where, f� is the damping function to take into account of the near-wall effect and sufficiently far from the wall it assumes a value of1.To obtain the turbulent viscosity, we need to solve the transportequations for turbulent kinetic energy and the turbulent time scalegiven below.

∂∂

+

∂∂+−

∂∂

=ik

T

ij

iij x

Kx

Kxu

DtDK

σνν

ττ (4)

)1()1( 221 −+∂∂

+= fCxu

KC

DtD

j

iijτττ

ji

T

ji

T

xxxxK

K ∂∂

∂∂

+−

∂∂

∂∂

++ ττ

σνν

ττ

σνν

ττ 21

22

∂∂

+

∂∂+

i

T

i xxτ

σνν

τ 2

(5)

In equations (3) to (5) some empirical constants and functions areintroduced to incorporate the near-wall effects into the standard two-equation turbulent model. The purpose of these functions in differentmodels is to provide a somewhat similar kind of modifying influenceon the standard two-equation model. Modeling of the empiricalconstants and damping functions in terms of satisfying the wall-nearand asymptotic behavior is discussed in detail in the refs.(13) and(14). Therefore, it will not be discussed in the present paper, only abrief outline of modeling is given in this section. The low-Reynoldsnumber damping factor fµ and f2 are defined by considering thelimiting behavior of wall turbulence and the relationship betweentwo length scales. One very near the wall, which is the Taylor micro-scale, and the other in the fully turbulent region away from the wall(large-scale energy containing eddies). The length scale for largescale eddies is defined as:

ε/5.1KCl ddh = (6)However, near the wall, where the dissipation of turbulent kineticenergy is very large, the length scale is represented by the Taylormicro-scale defined as:

εν /1 KCl ldl = ; 21 =lC (7)These two length scale can be related as :

4American Institute of Aeronautics and Astronautics

etdl

dh Rll

∝ ; νε

2KRet = (8)

This Eq.(8) means that near the wall where, 0→etR , ldl

becomes larger than ldh and away from the wall where Ret becomesvery large, ldh dominates. Thus Ret serves as a link between these twolength scales of dissipation. In the present model fµ is defined as afunction of Rett by the following relation:

+×

−−=et

et

RR

f 5.181150

exp1µ (9)

In order to satisfy the limiting behavior at the wall 22 yf ∝

and free turbulence as well, the following equation for f2 isproposed:

−

−−=10

exp15.6

exp3.012

2eyet RR

f (10)

Where, 2yyKRey ∝=ν

(11)

The numerical values in equations.(9) and (10) are decided on thebasis of algebraic stress model and comparison of prediction withexperiment, respectively.

Numerical Methodology In the present work, unsteady three-dimensional, compressibleNavier-Stokes equations along with the transport equations forturbulent kinetic energy and turbulent time scale written ingeneralized curvilinear co-ordinate are solved using an efficientimplicit approximate factorization (IAF) finite difference scheme ofsecond order accuracy in time. In this approach, fully implicit finitedifference equations are solved performing several iterations at eachtime step to make factorization errors zero. Also, to make the presentapproach robust and to capture the shock structure better, third orderaccurate TVD (Total Variation Diminishing) scheme based onMUSCL-type (Monotone Upwind Scheme for Conservative Laws)approach with Roe's approximate Riemann solver is used. Theviscous fluxes are determined in central differencing manner.

Experimental and numerical diffuser model The transonic diffuser model used in the experiment of Sajben isshown in fig.1. The diffuser model is a convergent-divergent channelwith a flat bottom and a contoured top wall. The channel height atthe minimum area cross-section (throat) was 44 mm, the exit tothroat area ratio was 1.52, the divergent length to throat height ratiowas 7.2, and the throat cross-sectional aspect ratio was 4.0. Anarbitrarily selected location within the constant area sectionupstream of the throat (x/h=-4.04) was designed as a nominal inletsection. The nominal exit section was at x/h=8.65, within theconstant area section downstream of the divergent portion. Side- andbottom-wall boundary layer growth was limited by three sets offorward facing (ram) suction slots. The ratios of slot areas to the pre-slot channel cross-sectional areas were also given in fig.1. The massflow removed at each slot was monitored separately, in all tests, andthe percentages mentioned in fig.1 were nearly equal to the fractionof mass flow removed at each slot. The third slot located at x/h=9.8,is just downstream the exit. Since this slot is located upstream of theexhaust, the removal of boundary layer thus significantly changedthe downstream flow-field. In fig.2 is presented the diffuser modelused in the present computations. All the slots are included in this

computation model. The ratios of slot areas to the pre-slot channelcross-sectional areas are kept same as those of the experimentalmodel. Because in these models, the low-Reynolds number dampingfactors are introduced to incorporate the near wall viscous effectsinto the standard two-equation turbulent model. In this study, for allthe test cases, near the body surface region grids are distributed verydensely to resolve the viscous effects properly and the y+corresponding to minimum grid spacing at the wall is kept to a valueless than equal to 1. To resolve the stream-wise changes in the flowdirection maximum grid spacing along the flow direction is kept to avalue of less than 2mm. Grid independency tests were carried out todecide value of Optimum grid points. For the reference test case(x/h=14.4), 491 grid points along the flow direction, 91 grid pointsalong the height and 121 grid points along the span direction areused. Since in the present study, the time-mean and unsteadycharacteristics of transonic diffuser are investigated as a function ofshock Mach number and diffuser length. Five cases of flow Machnumbers are considered. Since the flow always assumed one of twodistinct modes depending on the flow Mach number, one casecorresponds to transitional Mach number, and two cases each forflow Mach number above and below that transition value areconsidered. Effects of three diffuser lengths x/h=14.4, 18.4 and 22.5on the flow characteristics are investigated. Information regardingdiffuser and flow conditions of the test cases are presented inTable.1.

Table.1 Test cases used to assess the experiment-------------------------------------------------------------------------------Flow Mach number Case-1 Case-2 Case-3------------------------------------------------------------------------ 1.189 x/h=14.4 x/h=18.4 x/h=22.5 1.247 x/h=14.4 x/h=18.4 x//h=22.5 1.273 x/h=14.4 x/h=18.4 x/h=22.5 1.317 x/h=14.4 x/h=18.4 x/h=22.5 1.347 x/h=14.4 x/h=18.4 x/h=22.5--------------------------------------------------------------------------

Boundary Conditions Since in the self-excited oscillation in transonic diffuser, thedominant phenomena are the interaction between the acoustic waves,which travels upstream and downstream between the terminal shockand the channel exit, and the convective waves (or interface waves)introduced by the interaction of separated boundary layer andterminal shock. Therefore, in the computation of such unsteady flowphenomena, non-reflective boundary conditions need to be used atboth the inlet and outlet boundary. In the present study,characteristics boundary condition is used at the inlet and a non-reflective type exit boundary condition reported in ref.(15) is used.At the inlet boundary, for subsonic flow conditions, the value of thetotal pressure, total temperature and the flow incident angle (zero forthe present study) are specified from the available flow conditions. Aprocedure that uses a characteristic boundary condition byextrapolating from the interior to the boundary the upstream-runningRiemann invariant R- , based on the total velocity w, is adopted here.According to this procedure, R w a− = − −( / ( ))2 1γ (12) Where, a is the sonic velocity. Total temperature T0 andisentropic relations are then used to determine inlet speed win;

5American Institute of Aeronautics and Astronautics

wR C T R

inp

in

=− + + − −

+

− −( ) ( ) ( )( )

( )

γ γ γγ

1 4 1 2 1

10

2 (13)

To specify the inlet turbulent kinetic energy and the time scale ofturbulence based on turbulence dissipation rate, the turbulenceintensity corresponds to a value for good wind tunnel and the timescale obtained from length scale is used. At the wall surface of the channel no slip conditions are applied.The boundary conditions for the pressure are of Neumann type. Theturbulence kinetic energy k and the time scale of turbulence are setequal to zero at the wall surface. Since the flow at the outlet boundary is subsonic, the first fourcharacteristics are outgoing and are obtained by implicitextrapolation of inner grid location, so only the fifth characteristicvariable need to be set. The average change in the characteristic isdetermined to achieve the specified average exit pressure from thelocal equilibrium conditions. The turbulent kinetic energy and thetime scale of turbulence at the exit boundary are implicitlyextrapolated from the interior points.

Results and Discussions

Results of the present study will be described and discussed in thissection. In the first half time mean flow characteristics, dependenceof flow behavior on the flow Mach number and the diffuser lengthwill be investigated in detail. In the later half, unsteady flowcharacteristics and the possible link of core flow, shock inducedseparation, etc., with the self-excited oscillation phenomena will bediscussed. In the computation the fraction of mass flow removed ateach slot is set equal to the experimental value by specifying therequired pressure differential at each slot. In the case of computingthe flow in extended duct, the ratio of inlet total pressure and staticpressure at the exit plane is set equal to the experimental value bychanging the ratio of total pressure to the static pressure at thechannel end. In the experiment also, higher plenum pressure wereused for extended diffuser configuration to maintain the samepressure distribution (and Mach number) at the test section, becauseof additional pressure drop along the extra length. In fig.3(a) and (b)are presented the spark schlieren photograph for flow Mach numberof 1.235 and 1.353, respectively. In fig.4(a) to (e) are presented thecomputed time mean Mach number distribution at the diffuser midspan, for flow Mach number of 1.189, 1.247, 1.273, 1.317, and 1.347,respectively. In fig.5(a) to (e) are also plotted the time mean Machnumber distribution at the diffuser mid span for flow Mach numberconditions same as that of fig.4, along with iso-surface of therespective Mach number to have a good understanding of the three-dimensional shock structure. It can be seen from both the sparkschlieren photograph and the computed Mach number distribution,that the shock become highly curved in the transverse direction, andthe degree of shock curvature depends strongly on the flow Machnumber. For low flow Mach number, the shock takes a single normalshock pattern, and no oblique shock is emanating from the edge ofthe boundary layer. However, for higher flow Mach number, theshock is of Lambda type. The boundary layer at the top wall thickenas the normal shock is hitting the top wall, two oblique shocksemanating from the edge of the top wall boundary layer joined thenormal shock slightly away from the wall, forming a lambda typeshock. Also, it can be observed from the Mach number distributionpresented in fig.4 and 5 that the flow assumed one of two distinctmodes. For flow Mach number less than 1.273, the flow is attachedevery where in the diffuser. The boundary layers on the top andbottom wall remain relatively thin. For flow Mach number greater

than 1.273, the flow undergoes shock-induced separation along thetop wall. After the onset of separation, the bubble enlarges quickly tocover a quite long region along the top wall. The bottom-wallboundary layer remains almost attached, with only a very small re-circulation flow for highest flow Mach number due to the presenceof step-like configuration of the suction slot, just downstream of thenominal exit station. In fig.6 is plotted the measured and computedvariation of shock location at the top wall and at the mid-streamregion (the line represents the experiments and symbol represents thecomputations). In all the x-y graphical representation of the results,the symbol represents the result of computations and the linerepresents the experimental data. The difference between the twolocations provides a measure of the degree of shock curvature. Themeasured variation of shock curvature with the flow Mach number ispredicted very well. In fig.7 is presented the measured and computed variation of time-mean location of shock, separation and reattachment points with theflow Mach number (the line represents experiments and symbolrepresents computations). It can be observed here that after the onsetof separation, the bubble enlarges quickly to cover about four throatheights in length. The maximum bubble length for the highest flowMach number in the present study is about 6. This experimentallyobserved flow separation phenomena are predicted very well by thepresent method. In fig.8(a) to (c) and fig.9(a) to (c) are presented the measured andcomputed Mach number contours at the exit plane for flow Machnumber of 1.235, 1.271, and 1.353 in the case of experiment, and forflow Mach number of 1.247, 1.273, and 1.347 in the case ofcomputations. In the experiments were made with 11-prong rake at17 span-wise locations, therefore plotting of contour required a greatdeal of extrapolation and it is not possible to obtain the wall nearflow behavior correctly. However, overall contour resolution is quitegood. The results indicate a trend away from two-dimensionality forhigher flow Mach number. It can be seen from both the measuredand computed results that in the case of separated flow, the presenceof secondary flow, gives rise to an influx of low speed fluid from thesides toward the center causing the boundary layer thicken morerapidly on the centerline. In the case of attached flow, the flow fieldis quite uniform. Since the diffuser model consisted of a flat bottomand a contoured top wall, the results of computation showed that theflow is stretched towards upper corner and there exist secondarycorner vortices. Because insufficient data due to measurementlimitation, corner vortex is not quite clear from the measuredcontour. In fig.10(a), (b) are presented the classical schematicrepresentation of flow patterns in the supercritical transonic diffuser.Flow patterns for weak shock and pressure gradient inducedseparation is presented in fig.10(a), and the flow patterns for strongshock and shock induced separation is presented in fig.10(b). Infig.11(a) to (e) are presented the graphical representation of thecomputed time-mean core flow for flow Mach number of 1.189,1.273, 1.273, 1.317, and 1.347, respectively. In fig.11 are presentedthe computed mean total pressure distribution at the mid-span of thetransonic diffuser by limiting the total pressure value betweenmaximum equivalent to the inlet total pressure and minimumequivalent to the 2% of the post shock total pressure. It can beobserved from this figures that for flow Mach number less than1.273, the boundary layers on the top and bottom walls remainrelatively thin, and the inviscid region of constant mean totalpressure (the core flow) continues past the end of the unextendedchannel. For flow Mach number above and equal to 1.273, both thetop and bottom wall boundary layer thicken rapidly and merge wellinside the unextended diffuser. It can be observed here that themaximum reduction of total pressure occur just behind the normalportion of the shock. It is very difficult to make visualize such flow

6American Institute of Aeronautics and Astronautics

pattern on the basis of experimental technique. The behavior ofunsteady quantities in the present transonic diffuser depends stronglyon whether the mean flow is attached or separated. In fig.12 ispresented the variation of measured and computed time-mean coreflow length as a function of flow Mach number. The time-mean coreflow length is defined as the stream-wise distance between the time-mean mid-stream shock locatio and the point where the top andbottom wall boundary layers merge. According to Sajben, the end ofcore flow region was determined by computing the vertical meantotal pressure profiles in the plane of symmetry of the diffuser modelat various stream-wise locations. The most upstream location wherethe computed maximum total pressure in the vertical profile wasdropped by more than 2% of the post-shock total pressure wasdecided as the end of the time-mean core flow. It can be seen fromfig.12, that there occurs appreciable reduction in core flow lengthwhen the flow Mach number becomes greater than 1.27. The rapidboundary layer growth aided by shock-induced separationdownstream the strong shock under this flow Mach numbercondition is expected to result in such an abrupt reduction in coreflow length. The present computation model predicted thisexperimentally observed physical flow behavior very well. All the results discussed so far was for the diffuser lengthx/h=14.4. In this section some results on time-mean flow behaviorfor extended diffuser model will described and discussed to havesome understanding of the effect of diffuser length on the flowbehavior. In fig.13(a) and (b) are presented the time-mean Machnumber distribution at the diffuser mid-span for the flow Machnumber of 1.247 and diffuser length corresponds to x/h=18.4 andx/h=22.5, respectively. In fig.14(a) and (b) are presented the time-mean flow Mach number distribution at the diffuser mid-span for theflow Mach number of 1.347 and diffuser length corresponds tox/h=18.4 and x/h=22.5, respectively. It can be seen from fig.13 and14 that there occur no appreciable change in time-mean flowbehavior with the extension of the diffuser length for both the flowMach number conditions. As it was observed in fig.4(b), (e) andfig.5(b), (e), the time-mean Mach number distribution of fig.13 and14 indicate that the shock become highly curved in the transversedirection, and the degree of shock curvature depends strongly on theflow Mach number. For flow Mach number of 1.247 (fig.13), theshock takes a single normal shock pattern, and no oblique shock isemanating from the edge of the boundary layer. However, for flowMach number of 1.347 (fig.14), the shock is of Lambda type. Theboundary layer at the top wall thicken as the normal shock is hittingthe top wall, two oblique shocks emanating from the edge of the topwall boundary layer joined the normal shock slightly away from thewall, forming a lambda type shock. Also, it was observed in theMach number distribution presented in fig.4 and 5, fig.13 and 14also show that the flow assumed one of two distinct modes. For flowMach number of 1.247, the flow is attached every where in thediffuser. The boundary layers on the top and bottom wall remainrelatively thin. For flow Mach number of 1.347, the flow undergoesshock-induced separation along the top wall. After the onset ofseparation, the bubble enlarges quickly to cover a quite long regionalong the top wall. The bottom-wall boundary layer remains almostattached, with only a very small re-circulation flow due to thepresence of step-like configuration of the suction slot, justdownstream of the nominal exit station. It is quite obvious not tohaving any appreciable change in time-mean flow behavior with theextension of diffuser length. Because following the experimentalprocedure, in the case of computing the flow in extended diffuser,the ratio of inlet total pressure and static pressure at the exit plane isset equal to the experimental value by changing the ratio of totalpressure to the static pressure at the channel end. In the experiment,higher plenum pressure were used for extended diffuserconfiguration to maintain the same pressure distribution (and Mach

number) at the test section, because of additional pressure drop alongthe extra length.

Unsteady Flow Characteristics

In this section results on unsteady flow quantities for differentflow Mach number and diffuser length will be described anddiscussed in detail. These quantities are the distribution of staticpressure fluctuation in the diffuser, velocity fluctuation distributionin the diffuser, time varying characteristic of separation bubble,spectral analysis of pressure fluctuation and the frequency offluctuation. In fig.15(a) to (e) are presented instantaneous Machnumber distribution in the transonic diffuser for flow Mach numberof 1.189, 1.247, 1.273, 1.317, and 1.347, respectively. It can beobserved from fig.15(c) to (e) that the separated shear layer is highlyunstable. Also, in the separation flow field, the reverse flow regionof the separation bubble is observed to experience considerablelength and thickness change with the entrainment of neighboringfluid due to the movement of shock through its cycle. In fig.16(a) to (e) are presented the distribution of velocityfluctuation in the transonic diffuser for flow Mach number of 1.189,1.247, 1.273, 1.317, and 1.347, respectively. It can be observed fromthis figure that the velocity fluctuation assumes one of the twodistinct modes. For flow Mach number below 1.273, the fluctuationlevel is very low and the effect of shock boundary layer interactionis quite weak. However, for flow Mach number above 1.273, whenthe flow experienced shock-induced flow separation, the fluctuationlevel is very large due to the strong interaction of shock andboundary layer and dynamics of entrainment in the separationbubble. However, one thing, that is common in all the flow Machnumber conditions, is region of strong fluctuation near the suctionslot due to its step like configuration. In fig.17(a) to (e) are presented the distribution of static pressurefluctuation in the transonic diffuser for flow Mach number of 1.189,1.247, 1.273, 1.317, and 1.347, respectively. Two iso-surfaces areconsidered to have a good understanding of the three-dimensionalcharacteristics of fluctuation. In these figures, for all the flow Machnumber conditions, the red color represents the positive fluctuationand the blue color represents the negative fluctuation. However thelevel of iso-surface is reduced with the decrease in flow Machnumber. It can be observed from this figure that the level of pressurefluctuation is quite large and the distribution of fluctuation along thechannel is quite periodic for flow mach number above 1.273.Whereas, for flow Mach number below 1.273 the level of fluctuationis low and the distribution of fluctuation along the channel is quiteflat. In all the flow Mach number conditions, there exists a curtain ofpressure fluctuation near the suction slot caused by step-likeconfiguration of suction slot. In fig.18(a) to (g) are plotted the measured and computed top wallstatic pressure fluctuation intensities, normalized with the localdynamic head, as a function of flow Mach number. The rms valuesgiven represent primarily the high frequency contributions andtherefore characterize the local state of boundary layer, not the lowfrequency, large scale oscillations related to the shock motion. Adramatic change in the vicinity of flow Mach number of 1.27 isassociated with the onset of separation and abrupt reduction of coreflow length. The three locations nearest the shock, are the moststrongly affected, undergoing an almost discontinuous change inmagnitude as separation occurs. A sharp increase in the locations atx/h=2.43 begins for flow Mach number above 1.27, associated withthe influence by the shock due to its unsteady motion over thatlocation. In fig.18 it can be noticed static pressure fluctuationintensities at x/h=6.37, 8.65, and 10.95 is large as compared to theother location for flow Mach number below 1.27. Since theselocations are near the suction sot location, the presence of suction

7American Institute of Aeronautics and Astronautics

slot is expected to influence the pressure fluctuation phenomenon.This fact is evident from the results presented in fig.17. In Table.2 are presented the measured and predictedfrequencies of shock oscillation as a function flow Mach number anddiffuser length. The result of computations showed the dependencyof the self-sustained shock oscillation behavior on flow Machnumber and diffuser length similar to those observed in theexperiment. The frequencies of shock oscillations are a strongfunction of both Mach number and diffuser length. For weak shocksand short channel lengths, the spectra are bimodal. If the shockstrength is maintained low but diffuser length is increased, thefrequencies of these peaks decrease, and an additional peak appears.If the shock strength is increased sufficiently to induce separation,only one peak occurs for all the cases of diffuser lengths. In the caseof weak shock, where shock/boundary layer interaction is lessdominant, the mechanism responsible for the oscillations is theacoustic wave propagation along the channel. However, thephenomenon is different than the conventional acoustic wavepropagation in a channel without any time delay. In this case, theinteraction of upstream travelling pressure wave with pressurefluctuation curtain caused by the step-like slot configuration, and theinteraction between the shock and the pressure wave resulted in timedelay or phase shift. This in turn resulted in increase in time taken bythe pressure wave to make a round trip in the channel. In the case ofseparated flow, the mechanism responsible for the pressurefluctuation phenomenon is given below. The combined effects ofinstabilities in the separated shear layer, the shock/boundary layerinteraction, the dynamics of entrainment of separation bubble, theinteraction of pressure wave with the curtain of pressure fluctuationnear the suction slot are responsible for the mechanism.

Table.2 Frequencies of shock Oscillation as function of Flow Mach number and Diffuser Length--------------------------------------------------------------------------Flow Mach number x/h=14.4 　x/h=18.4 x/h=22.5-------------------------------------------------------------------------- Exp Pred Exp Pred Exp Pred--------------------------------------------------------------------------Attached: 1.189 60 62 50 48 55 53 230 232 180 178 170 169 - - - - 250 249 1.247 95 94 65 67 63 65 232 231 190 192 175 176 - - - - 252 254Transition 1.273 139 140 163 165 169 170Separated 1.317 216 215 217 220 236 235 1.347 196 195 234 235 231 230--------------------------------------------------------------------------

As typical results of the spectral analysis of shock oscillationspectra, in fig.19(a) to (c) are presented power spectral densitydistribution for flow Mach number 1.347 (diffuser length x/h=14.4) ,and for flow Mach number of 1.247 (diffuser length x/h=18.4 and22.5). These shock oscillation power spectra show well-definedpeaks.

Conclusions

The results of the study led to the following conclusions. The flow in transonic diffuser assumes one of two distinct modes.For flow Mach number below 1.27 the core flow is quite long and itextends beyond the un-extended channel. However, for flow Mach

number above 1.27, shock-induced flow separation along the topwall is short and it covers a distance well inside the un-extendeddiffuser. The unsteady flow behavior depends strongly on whether themean flow is attached or separated. In the shock-induced separation flow field, the length andthickness of reverse flow region of the separation bubble changeswith time as the shock passed through its cycle. The step-like configuration of suction slot results in a curtain ofpressure fluctuation. The instabilities in the separated shear layer, theshock/boundary layer interaction, the dynamics of entrainment in theseparation bubble, and the interaction of propagating pressure wavewith the curtain of pressure fluctuation are responsible for the shockoscillation phenomenon. For flow Mach number below 1.27, the phenomenon is differentthan the conventional acoustic wave propagation in a channelwithout any time delay. In this case, the interaction of upstreamtravelling pressure wave with pressure fluctuation curtain caused bythe step-like slot configuration, and the interaction between theshock and the pressure wave resulted in time delay or phase shift.This in turn resulted in increase in time taken by the pressure waveto make a round trip in the channel.

References

1 Sajben, M.,Bogar, T. J. and Kroutil, J. C., "Forced OscillationExperiments in Supercritical Diffuser Flows," AIAA J., Vol.22, No.4,April 1984, pp.465-474.2 Bogar, T. J., Sajben, M. and Kroutil, J. C., "CharacteristicFrequencies of Transonic Diffuser Flow Oscillations," AIAA J.,Vol.21, No.9, Sep 1983, pp.1232-1240.3 Salmon, J. T., Bogar, T. J. and Sajben, M., "Laser DopplerVelocimeter Measurements in Unsteady, Separated, TransonicDiffuser Flows," AIAA J., Vol.21, No.12, Dec 1983, pp.1690-1697.4 Sajben, M., Bogar, T. J. and Kroutil, J. C., "Forced OscillationExperiments in Supercritical Diffuser Flows with Applications toRamjet Instabilities," AIAA Paper No.81-1487 (1981).5 Coakley T. J., Hsieh T., "Downstream Boundary Effects on theFrequency of Self-Excited Oscillations in Transonic Diffuser Flow,"AIAA paper 87-0167, Jan, 1987.6 Hsieh T., Bogar T. J., Coakley T. J., "Numerical Simulation andComparison for Self-Excited Oscillations in Diffuser Flow," AIAAJ., Vol.25, July, 1987, pp. 936-9437 Hsieh, T., "Unsteady Separated Boundary Layer in a TransonicDiffuser Flow with Self-Excited Oscillations," AIAA Paper 86-1037,May, 1986.8 Hsieh, T., Bogar, J., and Coakley, T., "Numerical Simulation andComparison with Experiment for Self-Excited Oscillations inDiffuser," AIAA Paper 85-1475, July 1985.9 Robinet, J. C., and Casalis, G., "Shock Oscillations in DiffuserModeled by a Selective Noise Amplification," AIAA J.,Vol.37,No.4,April 1999, pp.453-459.10Patel, V. C., et al, "Turbulence model for Near-wall Low-ReynoldsNumber Flow- A Review", AIAA. J, Vol.23, No.9, 1985, pp.1308-1319.11 Wilcox, D.C., and Traci, R.M., "A Complete Model ofTurbulence," AIAA Paper 76-351, July 1976.12 Wilcox, D.C., "Reassessment of the Scale-Determining Equationfor Advanced Turbulence Models," AIAA Journal, Vol.26, No.11,1988, pp.1299-1310.13 Biswas, D., et. al., "Application of a Two-dimensional Low-Reynolds Number Version Turbulent Model to TransitionalBoundary Layer Flows," AIAA Paper 2002-274814Biswas, D., et. al., "Calculation of Transitional Boundary Layerswith an Improved Low-Reynolds Number Version k-� Turbulence

8American Institute of Aeronautics and Astronautics

model," ASME Journal of Turbomachinery, Vol.116, 1994, pp.765-773.15 Erdos, J. I., et. a., "Numerical Solution of Periodic TransonicFlow Through Fan Stage," AIAA J., Vol.15, No.11, pp.1559-1568.

Fig.1 Experimental diffuser model

(a)

(b)Fig.2 Experimental diffuser model

　　　　　(a)

(b)Fig.3 Spark schlieren photograph

(a)

(b)

(c)

(d)

(e)Fig.4 Computed Mean Mach number distribution

(a)

(b)

(c)

(d)

(e)Fig.5 Mean Mach number distribution (with iso-surface)

9American Institute of Aeronautics and Astronautics

0.0

0.5

1.0

1.5

2.0

2.5

3.0

1.15 1.20 1.25 1.30 1.35

mach-number

dis

tanc

e

top-wallmainstream

Fig.6 Shock location at mid-stream and top wall

0.0

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2.0

3.0

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5.0

6.0

7.0

1.25 1.27 1.29 1.31 1.33 1.35

mach-number

dist

ance

shock-locationreatatchmentseparation系列4

Fig.7 Location of shock, separation and re-attachment

(a)

(b)

(c)Fig.8 Measured Mach number contour

(a)

(b)

(c)Fig.9 Computed Mach number contour

(a)

(b)Fig.10 Schematic representation of flow pattern

(a)

(b)

(c)

(d)

(e)Fig.11 Computed total pressure distribution

10American Institute of Aeronautics and Astronautics

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34 1.36

mach-number

dis

tance

shock-locationcore end系

Fig.12 Length of inviscid core flow

(a)

(b)Fig.13 Mach number for extended diffuser (M=1.247)

(a)

(b)Fig.14 Mach number for extended diffuser (M=1.347)

(a)

(b)

(c)

(d)

(e)Fig.15 Instantaneous Mach number distribution

(a)

(b)

(c)

(d)

(e)Fig.16 Velocity fluctuation distribution (iso-surface)

(a)

(b)

(c)

(d)

11American Institute of Aeronautics and Astronautics

Fig.17 Pressure fluctuation distribution (iso-surface)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1.15 1.20 1.25 1.30 1.35

mach number

rms

press

ure

2.43

0.00

0.01

0.01

0.02

0.02

0.03

0.03

0.04

0.04

0.05

1.10 1.15 1.20 1.25 1.30 1.35 1.40

mach number

rms

press

ure

3.13

0.00

0.01

0.02

0.03

0.04

0.05

0.06

1.10 1.15 1.20 1.25 1.30 1.35 1.40

mach number

rms

press

ure

4.61

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

1.10 1.15 1.20 1.25 1.30 1.35 1.40

mach number

rms

press

ure

6.37

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1.10 1.15 1.20 1.25 1.30 1.35 1.40

mach number

rms

press

ure

8.65

0.00

0.01

0.02

0.03

0.04

0.05

0.06

1.10 1.15 1.20 1.25 1.30 1.35 1.40

mach numberrm

s pr

ess

ure

10.95

0.00

0.01

0.01

0.02

0.02

0.03

0.03

0.04

1.10 1.15 1.20 1.25 1.30 1.35 1.40

mach number

rms

press

ure

14.12

Fig.18 Unsteady top wall static pressure

(a) (b)

(c)Fig.19 Typical power spectral distribution

(e)

(f)

(g)

0 100 200 300 400 500 6000

50

100Case 1

frequency[Hz]

DSP

0 100 200 300 400 500 6000

50

100

150

200

250

frequency[Hz]

Case 2

DSP

0 100 200 300 400 500 6000

50

100

150

200

250

frequency[Hz]

Case 3

DSP

(a)

(b)

(c)

(d)