Numerical Investigations of a Transitional Flat-Plate Boundary Layer with Impinging Shock Waves Alessandro Pagella, Ulrich Rist, Siegfried Wagner Institut f ¨ ur Aerodynamik und Gasdynamik Universit ¨ at Stuttgart Germany Abstract The transitional behaviour of a flat-plate bound- ary layer with shock-boundary layer interaction has been investigated for both small-amplitude and weakly non-linear disturbances at with constant wall temperature and with in- sulated wall, respectively. The separation bubble induced by the impinging shock wave remained steady in both situations. In the linear case, com- pressible linear stability theory results have been calculated and compared with direct numerical simulations. The two results agree well, if non- parallel effects already present in the same bound- ary layer without impinging shock wave are con- sidered. Maximum amplification rates were in- creased and shifted to lower disturbance frequen- cies. In the weakly non-linear case, fundamental, subharmonic and oblique scenarios were simu- lated. It was found, that in all three cases the - modes, the so-called vortex- or streak modes were strongly amplified downstream of reattachment. This might be triggered by a G ¨ ortler-type instability, which could be caused by the concave curcature of the boundary layer at reattachment. However, this statement remains to be proved. List of Symbols ” ” indicates dimensional free-stream values amplitude ratio of any flow variable skin friction coefficient , specific heats at constant pressure, volume energy dimensional disturbance frequency disturbance frequency fundamental disturbance frequency frequency mode in Fourier space spanwise Fourier mode dimensional reference length Mach number pressure Prandtl number global Reynolds number local Reynolds number time temperature Sutherland temperature wall temperature streamwise velocity component wall-normal velocity component spanwise velocity component streamwise coordinate wall-normal coordinate spanwise coordinate streamwise amplification rate streamwise wave number spanwise wave number thermal conductivity coefficient specific heat ratio viscosity density shock angle with respect to 33.1
Transcript
Numerical Investigations of a Transitional Flat-PlateBoundary Layer with Impinging Shock Waves
Alessandro Pagella, Ulrich Rist, Siegfried WagnerInstitut fur Aerodynamik und Gasdynamik
Universitat StuttgartGermany
Abstract
The transitional behaviour of a flat-plate bound-ary layer with shock-boundary layer interaction hasbeen investigated for both small-amplitude andweakly non-linear disturbances at Ma = 4:8 withconstant wall temperature and Ma = 4:5 with in-sulated wall, respectively. The separation bubbleinduced by the impinging shock wave remainedsteady in both situations. In the linear case, com-pressible linear stability theory results have beencalculated and compared with direct numericalsimulations. The two results agree well, if non-parallel effects already present in the same bound-ary layer without impinging shock wave are con-sidered. Maximum amplification rates were in-creased and shifted to lower disturbance frequen-cies. In the weakly non-linear case, fundamental,subharmonic and oblique scenarios were simu-lated. It was found, that in all three cases the (0; k)-modes, the so-called vortex- or streak modes werestrongly amplified downstream of reattachment.This might be triggered by a Gortler-type instability,which could be caused by the concave curcature ofthe boundary layer at reattachment. However, thisstatement remains to be proved.
List of Symbols
”1” indicates dimensional free-stream valuesA=A0 amplitude ratio of any flow variablecf skin friction coefficient
cp, cv specific heats at constant pressure, volumee energyf� dimensional disturbance frequencyF disturbance frequency = (2�f�L)=(u1Re)F0 fundamental disturbance frequencyh frequency mode in Fourier spacek spanwise Fourier modeL dimensional reference lengthMa Mach numberp pressurePr Prandtl numberRe global Reynolds number = (�1u1L)=�1Rx local Reynolds number =
pxRe
t timeT temperatureTs Sutherland temperatureTw wall temperatureu streamwise velocity componentv wall-normal velocity componentw spanwise velocity componentx streamwise coordinatey wall-normal coordinatez spanwise coordinate��i streamwise amplification rate�r streamwise wave number� spanwise wave number# thermal conductivity coefficient� specific heat ratio� viscosity� density� shock angle with respect to x
33.1
!x x-component of the vorticity obliqueness angle of the disturbances
Introduction
In trans-, super- and hypersonic flight condi-tions, shock-boundary layer interactions are om-nipresent. Research on the properties of suchflows began as early as mankind started to de-velop the knowledge and technology for flying be-yond the sound barrier in the late 1930’s andearly 1940’s. In 1946, Ackeret, Feldmann andRott (Ref 1) and Liepmann (Ref 15) performed thefirst systematic experimental studies, and a lot ofwork has been done in the late 1940’s and 1950’sby scientists in countries of both the then East andWest.
Figure 1: Schlieren photograph of T-38 shockwaves at Ma = 1:1, 13000 feet. Photo is courtesyof NASA Dryden Research Center Photo Collec-tion. NASA Photo: EC94-42528-1, December 13,1993. Photo by: Dr. Leonard Weinstein
Figure 1 shows a Schlieren photograph of a T-38experimental aircraft at low supersonic speed. Theimage gives an idea of the importance of shock-phenomena in such flows. For example, shock-boundary layer interaction plays an important rolein terms of the performance of jet-intakes, whichhave to decelerate the incoming flow from super-to subsonic speed. On the wings of modern pas-senger aircraft flying at high subsonic speed, lo-cal supersonic flow regions occur under certain
flight conditions. These are terminated by an al-most vertical shock, which influences the local flowproperties. In internal flows like turbomachinery,shock-boundary layer interactions are present aswell, as it can be seen in Fig 2.
Figure 2: Photograph of a turbine rotor (Ref 9).
After the second half of the 1950’s and in the1960’s, technical progress seemed to have no lim-its. Because of the ability to perform space flights,the possible speed for technical applications ex-tended to the hypersonic range, which made theprediction of thermal and pressure loads an issue,to which the structure is exposed during re-entry.In Fig 3, the four shadowgraphs represent early re-entry vehicle concepts. It was found, that a bluntbody produces a shock wave in front of the vehi-cle that shields the vehicle from excessive heating,reducing the thermal load compared to a pointeddesign. Because of the higher temperatures in hy-personic flight conditions, real-gas effects becomeimportant, too.
The authors would like to refer to more thoroughdiscussions of the properties of flows with shock-boundary layer interaction, which can be founde.g. in (Ref 4), (Ref 5) and (Ref 6). Althougha lot of problems have been solved during thelast decades of research in this particular field,a number of questions still remain unansweredwhile seemingly clarified problems led to new is-sues. In his review about achievements and un-resolved problems of shock-boundary layer inter-actions, Dolling (Ref 6) named one of those yetunresolved issues as the transitional behaviour of
33.2
Figure 3: Photographs of early re-entry designconcepts. Picture is courtesy of NASA, GRINDataBase Nr. GPN-2000-001938.
such flows. The work, done for this paper intendsto contribute some investigations to this particularfield.
The transition process for compressible flows canbe classified into several merging stages, similarto incompressible flows. The first phase of tran-sitional development in a low free-stream distur-bances environment, the amplification of distur-bances with small amplitudes was pioneered byLees and Lin (Ref 13) in 1946 and later extendedby Mack (Ref 16). The next phase of the tran-sition process, which follows the linear regime,can be explained in a similar but more compli-cated manner compared to incompressible flows.In incompressible boundary layers, after the two-dimensional Tollmien-Schlichting waves, which areexponentially amplified have reached sufficientlylarge amplitudes, a non-linear, three-dimensionaldevelopment takes place. Small-amplitude three-dimensional disturbances become amplified and aperiodic structure appears in spanwise direction.This stage is called secondary instability, whichcan be quantitatively described by Floquet the-ory (Ref 12). Eventually, so-called lambda vorticesform in a later development. Locally, high shearstresses occur, which are later breaking into sev-eral small structures, resulting in turbulent spots.These move downstream, grow and finally developinto the fully turbulent flow. A good insight into
hypersonic transition is given by Saric, Reshotkoand Arnal (Ref 17). However, transition undersuper- and hypersonic conditions is far from bee-ing understood and results of transional flows withshock-boundary layer interactions are not knownto the authors.
Numerical Scheme
Governing equations The numerical scheme isbased on the complete, three-dimensional, un-steady, compressible Navier-Stokes equations forcartesian cordinates in conservative formulation:
@�
@t+r � (�u) = 0 (1)
@(�u)
@t+r � (�uu) +rp =
1
Rer � � (2)
@(�e)
@t+r � (p+ �e)u
=1
(�� 1)RePrMa2r � (#rT ) + 1
Rer � (�u) ; (3)
where
� = �
�(ru+ruT )� 2
3(r � u)I
�(4)
with the velocity vector u = [u; v; w]T .
The energy is calculated as
e =
Zcv dT +
1
2(u2 + v2 + w2): (5)
The fluid is a non-reacting, ideal gas with constantPrandtl number Pr = 0:71 and specific heat ratio� = cp=cv = 1:4, with cp and cv as the specific heatcoefficients at constant pressure and volume, re-spectively. Viscosity � for temperatures above theSutherland temperature Ts is calculated by Suther-land’s law, for temperatures below Ts with the re-lation �=�1 = T=T1. The thermal conductivitycoefficient # is proportional to the viscosity. In oursimulations, all lengths are made nondimensionalwith a reference length L, which appears in the
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global Reynolds number Re = �1 � u1 � L=�1 =105. A local Reynolds number, which is used forthe presentation of the results, is defined as Rx =px �Re. The specific heat cv is normalised with
u21=T1 (with T1 giving the free-stream tempera-
ture) and time t is normalised with L=u1, whereu1 is the free-stream velocity. Density �, temper-ature T and viscosity � are standarized by theirrespective free-stream values.
Discretisation Time integration is performed atequidistant time steps with a standard Runge-Kutta scheme of fourth-order accuracy. In stream-wise direction, compact finite differences of fourth-order accuracy are applied, which are in a split-type formulation in order to have some dampingproperties with regard to small-scale numerical os-cillations, which occur at the high gradients result-ing from the shock. In the split-type formulation,the weighting of the numerical stencil alternateseach Runge-Kutta step from downwind to upwindand vice versa. If a stronger shock is applied,the damping characteristic of the split-type formu-lation is not sufficient enough. In this case, an im-plicit filter of fourth-order accuracy (Ref 14) is ap-plied to filter the variables of the solution vectoreach physical time step in streamwise direction. Inwall-normal direction split-type finite differences offourth-order accuracy are used to calculate con-vective terms, while viscous terms are calculatedby fourth-order central differences. In spanwise di-rection we have periodic boundaries, which allowto apply a spectral approximation with Fourier ex-pansion. A more complete description of the dis-cretisation can be found in (Ref 7).
Boundary and Initial Conditions At the free-stream boundary, a characteristic boundary con-dition (Ref 10) and, more recently, a non-reflectingboundary condition (Ref 18) is applied. The shockwave is introduced by holding the flow-variablesconstant in a limited area at the free-streamboundary, according to the Rankine-Hugoniot re-lations after the shock and the initial free-streamconditions before the shock. The inflow quanti-ties at the inflow boundary result from the solutionsof the compressible boundary layer equations andare held constant during the simulation. At the
wall, a no-slip condition and vanishing normal ve-locities are assumed. Disturbances are introducedat the disturbance strip with simulated blowing andsuction, given by the following equation:
with �2� � � � 2� and � = �2� at the begin-ning and the end of the disturbance strip, respec-tively. In our modal discretization in spanwise di-rection, k indicates the spanwise Fourier modes,with k = 0 meaning a two-dimensional distur-bance. The disturbance frequency F determinesthe streamwise wave number �r via the disper-sion relation of the disturbances. The spanwisewavenumber is �. Thus, the obliqueness angle is given by tan = (k�)=�r. The wall temperaturecan be chosen to remain constant or adiabatic.
Results
Unperturbed Base Flow In this paper, twocases of unperturbed, two-dimensional base-flowswith shock-boundary layer interaction are consid-ered. The first case is a Ma = 4:8 boundary layerwith a free-stream temperature of T1 = 55:4K andan impinging shock wave with a shock angle of� = 14o relative to the x-axis. The wall tempera-ture is held constant at Tw = 270K, which is equalto the adiabatic wall temperature of the same flowwithout shock. For this case, linear stability theoryresults and direct numerical simulations with smallperturbation amplitudes will be shown in the nextsection.
Figure 4 displays the density gradient field and se-lected streamlines near shock-impingement. Theshock causes the boundary layer to thicken. If acertain shock strength is exceeded, the boundarylayer is not able to react to the pressure gradi-ent of the shock without separating. This is thecase in our two base-flow configurations. The im-pinging shock wave enters into the boundary layer.Until it ends at the sonic line as an almost verti-cal shock, its shock-angle becomes increasinglysteeper because of the deceleration inside theboundary layer from the supersonic free stream
33.4
Figure 4: Density gradients j @�@yj and selected streamlines for Ma = 4:8, free stream temperature T1 =
55:4K and constant wall temperature Tw = 270K with a shock angle of � = 14o.
velocity to the sonic line. At the sonic line, it isreflected as a system of expansion waves.
Separation can be identified by taking a look at theskin friction, which is given in Fig 5 for the caseof Fig 4. The negative skin-friction, which rangesfrom Rx � 1240 to Rx � 1470 indicates the sepa-ration bubble. The separation bubble in the base-flow remains steady for all cases investigated here.Its shape can be judged from the correspondingstreamlines in Fig 4. Near separation and reat-tachment additional compression waves coalesceto the separation and reattachment shock well out-side the boundary layer, respectively. However,these compression waves are relatively weak hereand therefore can not be detected in Fig 4 but itcan be seen in Fig 16, which shows an unsteadycase with a stronger shock therefore beyond thescope of the present paper but briefly discussed inthe section, were planned future work is described.
Rx
c f
1000 1200 1400 1600
0
0.0005
0.001
Figure 5: Skin friction cf versus streamwise lo-cation Rx for Ma = 4:8, free stream tempera-ture T1 = 55:4K and constant wall temperatureTw = 270K with a shock angle of � = 14o.
The second case presented here is a boundarylayer at Ma = 4:5 with a free-stream temperatureof T1 = 61:58K and insulated wall. For this case,the behaviour of the boundary layer with larger dis-turbances, which exceed the linear disturbances ofthe former case at Ma = 4:8 will be shown. Theshock angle of the impinging shock wave here is� = 15o, which again produces a steady separa-tion bubble.
c f0
0.0005
0.001cf
Rx
Tw
[K]
800 1000 1200270
272
274
276
Tw
Figure 6: Skin friction cf and wall temperature Twversus streamwise location Rx for Ma = 4:5, freestream temperature T1 = 61:58K and insulatedwall. Shock angle � = 15o.
In Fig 6, the skin-friction coefficient cf and the walltemperature distribution Tw is given for this sec-ond case. The skin-friction coefficient indicatesthat the separation bubble at the wall ranges fromRx � 900 to Rx � 1060. At separation and reat-tachment, the wall-temperatrue rises. However,the total temperature rise from the flow before theinteraction region to the flow behind the interactionregion is only in the order of � 5K. An increas-ing Ma-number and/or increasing shock strengthwould also increase the magnitude of the temper-
33.5
ature rise. The trend of the wall-temperature distri-bution is similar to the shape of the wall-pressuredistribution, which has a plateau inside the sepa-ration bubble as well.
Influence of Small Disturbances Based on thelinear stability theory for compressible flows, whichwas put into its present formulation by (Ref 16),results of the boundary layer with Ma = 4:8 and� = 14o will be shown here. For this purpose, inthe case with shock, local temperature and mean-velocity profiles and their respective first and sec-ond derivatives were extracted from the results ofthe direct numerical simulation of the base flow.Then, these local data-sets were fed into the lin-ear stability equations. The results are shownin Fig 7 and in Fig 8, with the amplification rate��i = @ lnA(x)
A0
=@x , where A(x)=A0 is the ampli-tude ratio of any flow variable.
In flows at higher Ma-numbers, several instabilityregions with respect to the disturbance frequencyare present. The first two such regions are ofgreat importance for the stability behaviour. Theone at lower frequencies, the so-called first modehas smaller amplification rates than the secondor first Mack mode at higher frequencies. In thecase without shock in Fig 7 at Rx = 1400, the firstmode ranges from F � 0 to a F slightly smallerthan F = 10�4. Towards higher frequencies agap without amplification is followed by the sec-ond mode instability region, which expands up toF � 1:6 � 10�4. As one can see here, the max-imum amplification rates of the second mode aresignificantly larger than those of the first mode.
The shock wave, that hits the boundary layer atRx � 1360 causes the amplification rates of thesecond mode to rise, while it shifts to lower fre-quencies within the interaction region. The cleardistinction between the two instability regions asseen in the case without shock is lost. Thefirst mode seems to vanish near shock impinge-ment, while new instabilities are formed at frequen-cies above the present second mode close to theshock-impingement.
A comparison between linear theory resultsand direct numerical simulations in which small-amplitude disturbances are introduced at the dis-
0
0
00
Rx
F
1200 1400 16000
5E-05
0.0001
0.00015
0.0002
0.00025
0.00030.00750.00500.00250.0000
no shock -αi
≤
Figure 7: Linear stability for two-dimensional dis-turbances. Shown are results for the case withoutshock. Rx is the streamwise location, while F rep-resents the disturbance frequency.
turbance strip are shown in Fig 9. The maxi-mum disturbance amplitudes of the direct numeri-cal simulation are obtained with a Fourier analysisin time over the last disturbance period of the sim-ulation.
In Fig 9, results for two flow-variables (p and T )are shown. The maximum disturbance ampli-tude for the temperature typically describes thedisturbance behaviour for disturbance amplitudes,which are located in a more distant position tothe wall. On the other hand, the wall pressureamplitudes represent disturbance amplitudes ator near the wall. That way, the disturbance be-haviour of the boundary layer can qualitatively bedescribed with the two above quantities p and T ,considering their particular distance to the wall. Ingeneral, disturbance amplitudes in a further dis-tant location from the wall show better agreementwith linear stability theory than wall-near ampli-tudes, because at or near the wall, non-paralleleffects become larger. However, non-parallel ef-fects, which are not considered in linear stabilitytheory are present at more distant positions fromthe wall, too. They are responsible for the differ-ences between linear theory and direct numericalsimulation in both cases with and without shock inthe maximum temperature disturbance amplitude-curves in Fig 9. To summarize, the results of the di-rect numerical simulation with small perturbationswere validated by compressible linear stability the-
33.6
0
0
0
σ=14o
000
0
Rx
F
1200 1400 16000
5E-05
0.0001
0.00015
0.0002
0.00025
0.0003 0.00750.00500.00250.0000
-αi
≤
Figure 8: Linear stability for two-dimensional dis-turbances. Shown are results for the case withshock (shock angle � = 14o). Rx is the stream-wise location, while F represents the disturbancefrequency.
ory despite considerable non-paralle effects, whichare already present in a boundary layer withoutshock and excluded from standard linear stabilitytheory. This also applies for three-dimensional dis-turbances (not shown here) with smaller oblique-ness angles, while for larger obliqueness angles,the agreement with linear theory increasingly de-teriorates. This is also known for flows of that kindwithout shock-boundary layer interaction.
Influence of Larger Disturbances In this sec-tion, the behaviour of a boundary layer in the pres-ence of larger amplitudes compared to the linearcase in the section before is investigated. The con-ditions were chosen to generate only weakly non-linear behaviour which can be classified as sec-ondary instability. A good insight into the prop-erties of this transitional stage, based on sec-ondary instability theory calculations for the flowcase presented in this section without impingingshock wave can be found in (Ref 8). In the fol-lowing, (h; k) represents a mode of the frequencyh�F0, where F0 is the fundamental disturbance fre-quency and a spanwise wave number k � �, with �as the fundamental spanwise wave number. Thefollowing figures in this section show the resultsof a time-wise Fourier analysis of the last simu-lated disturbance period. Three non-linear scenar-
ios are considered. The fundamental case, wherea two-dimensional primary disturbance wave (1; 0)and a three-dimensional secondary disturbance(1; 1) are introduced at the disturbance strip withthe same disturbance frequency. In the subhar-monic case, a primary disturbance wave (1; 0) andsecondary disturbance wave (1=2; 1) with half ofthe frequency of the primary wave are introducedinto the boundary layer, while in the oblique sce-nario, the boundary layer is perturbed with onethree-dimensional wave (1; 1). The modes (1;�1)and (1=2;�1) are perturbed as well in the fun-damental, subharmonic and the oblique scenario,respectively, but not explicitly discussed. Due tosymmetry assumptions, they are equal to their cor-responding counterparts (1; 1) and (1=2; 1).
The upper picture in Fig 10 shows results for thetemperature fluctuations of the subharmonic sim-ulation. Here, F0 = 1 � 10�4 and the oblique-ness angle � = 25. For comparison, the dottedlines with symbols refer to simulations without im-pinging shock-wave. Behind the interaction regionthe disturbance amplitudes of all modes are largerthan the corresponding values in the case withoutshock. The amplitudes of the (0; 2) mode, which interms of the spanwise wave number k�� is the firststreak- or vortex mode created in the subharmonicscenario, becomes significantly larger than in thecase without shock. Its increase also exceeds therise of the primary and secondary disturbance am-plitudes of the same case. The lower picture inFig 10 shows the phase velocities of the primaryand secondary waves of the temperature distur-bance with and without shock wave. The phasevelocities indicate, whether the primary and sec-ondary disturbances synchronize, which is a re-quirement for resonance. As can be seen, the twowaves synchronise in both cases with and with-out shock, therefore the resonance condition is ful-filled. The synchronisation in the case with shockoccurs downstream the separation bubble. Be-cause of the effects induced by the shock, down-stream Rx � 820 the disturbance waves are decel-erated, then sharply accelerated at Rx � 970 andagain decelerated from Rx � 1000 to Rx � 1180,from where they run downstream synchronously.
The fundamental case is represented in Fig 11.Again, F0 = 1 � 10�4 and the spanwise wave num-ber of the secondary wave is � = 25. As in the
33.7
linear theory, no shock
Rx
max
.dis
t.am
plitu
de
600 800 1000 1200 1400 160010-9
10-8
10-7
10-6
10-5
10-4
DNS, with shock
linear theory, with shock T
pDNS, no shock
Figure 9: Comparison between linear stability theory results and direct numerical simulations results ofthe wall-pressure amplitude and the maximum temperature disturbance. Ma = 4; 8 and � = 14o.
subharmonic case, the first vortex mode, which is(0; 1) here starts to sharply rise at Rx � 1000.
Figure 12 shows the temperature disturbance am-plitudes for the so-called oblique scenario. Thisscenario is characterised by the interaction of twowaves (1; 1) and (1;�1). The fundamental distur-bance frequency and spanwise wave number arethe same like in the fundamental and subharmoniccases. As before, the vortex mode (0; 2) increasesintensely from Rx = 1000 downstreamward andsoon exceed the amplitude of (1; 1). Like the (0; 2),the other higher modes, which are generated bythe oblique scenario, such as (1; 3) and (0; 4), in-crease in a similar manner. However, their initialmagnitude is smaller than (1; 1) or (0; 2), there-fore the amplitude at the outflow does not reachthe magnitude of either (1; 1) or (0; 2). The simi-lar growth of the vortex modes in all perturbationscenarios investigated suggests that the growth ofthe (0; k)-modes downstream shock-impingementis independent from the applied scenario.
The left picture in Fig 13 depicts vorticity !x andstreamlines of the single (0; 2)-mode atRx = 1100.It indicates, that this modes correspond to a vortex.
However, the sum of the vortex mode amplitudesis still not large enough to produce a vortex inthe total flow, as can be seen from the stream-lines in Fig 14, which are not influenced. The re-sults of Fig 14 were obtained by adding the modes(0; 0),(0; 2) and (0; 4) directly generated by theoblique disturbance scenario to the undisturbed
base flow. Possible higher modes, such as (0; 6),(0; 8), etc. could be neglected, because their mag-nitude is very small.
z
y
0 0.1 0.2
0.1
0.2
0.3
0.4
0.5
0.00180.0004
-0.0011-0.0025
Rx=1100, σ=15o
Base flow + (0,0) + (0,2) + (0,4)
ωx
Figure 14: vorticity component !x of the modes(0; 0), (0; 2), (0; 4) added to the base flow at Rx =1100 with selected streamlines for the oblique sce-nario.
The maximum of the wall-normal velocity of thebase flow at Rx = 1100 is around 10�2, while themagnitude of the disturbance flow is � 10�5, as
Figure 10: Modal representation of maximum temperature disturbance amplitudes for the subharmoniccase.
can be seen in the right picture in Fig 13, whichgives the disturbance velocity amplitudes for thefirst, therefore largest vortex mode (0; 2).
In the case without shock, represented by Fig 15,vortices are also present in the (0; 2) mode.However, the amplification of the (0; k)-modes inthe case without shock is much smaller com-pared to the case with shock downstream shock-impingement. Also, the velocity profiles for thecase without shock, which are given by the rightpicture in Fig 15 have a different shape than thevelocity profiles for the case with shock in the rightpicture of Fig 13, especially the wall-normal veloc-ity v. The position of the vortices with respect tothe wall-normal distance y differs as well. Whilein the case with shock, the vortex-cores are lo-cated between the first and second maximum ofthe streamwise disturbance velocity w, the vortex
cores in the case without shock are in a farther po-sition from the wall, after the second maximum ofw. This and the smaller amplification rates mightindicate, that in the case with shock, a differentvortex-development mechanism is present than inthe case without shock.
As an explanation for the occurrence of suchvortices in shock-boundary layer interactions, aGortler instability is typically mentioned in the lit-erature, e.g. (Ref 2, 3, 11). However, this is dif-ficult to prove in a quantitative manner. In Fig 4,we clearly observe the concave curvature of theboundary layer near reattachment, which mighttrigger a Gortler instability (note that the figure isstretched in y-direction, thus not giving the physi-cal scale).
33.9
Rx
|T’ m
ax|
700 800 900 1000 1100 120010-6
10-5
10-4
10-3
10-2
primary (1,0)secondary (1,1)(0,1)
Fundamental, β=25, σ=15o
Figure 11: Modal representation of maximum temperature disturbance amplitudes for the fundamentalcase.
Conclusions and Future Research It wasfound, that for small disturbance amplitudesmaximum amplification rates were increased andshifted to lower frequencies. The linear stabilitytheory results agreed well with the correspond-ing direct numerical simulations. In the weaklynon-linear case, a strong increase of the so-calledstreak- or vortex modes (0; k) could be observedat and downstream reattachment independentlyof the disturbance scenario applied. A possibleexplanation for suchlike behaviour could be foundin a Gortler-type instability mechanism, due to theconcave curvature of the boundary layer in thatflow-region. This, however, remains to be provedin a quantitative manner.
For the future, further research will be carried outin the non-linear regime with higher Ma-numbersand stronger shocks. Stronger shocks eventuallyyield unsteady separation bubbles, such as shownin Fig 16 for a two-dimensional simulation at Ma =6. It is also intended to investigate possible controlof the flow by means of specific large-amplitudedisturbances.
Acknowledgement The authors would like tothank the Deutsche Forschungsgemeinschaft forsupporting this research within Sonderforschungs-bereich 259.
References
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[2] D. Aymer de la Chevalerie, L. De Luca, andG. Cardone. Gortler-type vortices in hyper-sonic flows: The ramp problem. ExperimentalThermal and Fluid Science, 15:69–81, 1997.
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[6] D.S. Dolling. Fifty years of shock-wave/boundary-layer interaction research:What next? AIAA J., Vol. 39, No. 8:1517–1531, 2001.
(dotted lines with symbols denote reference case without shock)
Rx
|T’ m
ax|
700 800 900 1000 1100 120010-8
10-7
10-6
10-5
10-4
10-3
10-2
(1,1)(0,2)(1,3)(0,4)
Oblique, β=25, σ=15o
Figure 12: Modal representation of maximum temperature disturbance amplitudes for the oblique case.
[8] N.M. El-Hady. Secondary instability of high-speed flows and the influence of wall coolingand suction. Phys. Fluids, A 4 (4):727–743,April 1992.
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