R E S E A R C H A R T I C L E
Particle image velocimetry measurements of a shock
wave/turbulent boundary layer interaction
R. A. HumbleÆ F. Scarano Æ B. W. van Oudheusden
Received: 29 September 2006 / Revised: 14 March 2007 / Accepted: 27 April 2007 / Published online: 23 June 2007 Springer-Verlag 2007
Abstract Particle image velocimetry is used to investi-gate the interaction between an incident shock wave and a turbulent boundary layer at Mach 2.1. A particle response assessment establishes the fidelity of the tracer particles. The undisturbed boundary layer is characterized in detail. The mean velocity field of the interaction shows the inci-dent and reflected shock wave pattern, as well as the boundary layer distortion. Significant reversed flow is measured instantaneously, although, on average no re-versed flow is observed. The interaction instantaneously exhibits a multi-layered structure, namely, a high-velocity outer region and a low-velocity inner region. Flow turbu-lence shows the highest intensity in the region beneath the impingement of the incident shock wave. The turbulent fluctuations are found to be highly anisotropic, with the streamwise component dominating. A distinct streamwise-oriented region of relatively large kinematic Reynolds shear stress magnitude appears within the lower half of the redeveloping boundary layer. Boundary layer recovery towards initial equilibrium conditions appears to be a gradual process.
1 Introduction
The interaction between a shock wave and a turbulent boundary layer (SWTBLI) creates a series of complicated
flow phenomena that have long been a problem area of modern high-speed fluid dynamics. The interaction embodies all of the problems associated with compress-ibility, flow separation and turbulence, which present for-midable challenges to experimentalists and theoreticians alike. Numerous efforts over the decades have therefore sought to gain a better understanding of its complex behaviour. SWTBLIs are typically characterized as com-pression ramp or incident shock wave interactions. The former case has been extensively studied for a wide variety of flow conditions and configurations (e.g. Settles and Dodson 1991). The salient features of this type of inter-action can be found in references given by Dolling (2001) and Smits and Dussuage (2006). These studies have shown that when the boundary layer separates, the shock foot and interaction zone undergo an unsteady motion at frequencies much lower than those of the incoming boundary layer. A variety of authors have tried to correlate this unsteady shock wave motion with upstream conditions (Andreopo-ulos and Muck1987), as well as the internal dynamics of the separated flow region itself (Dolling and Murphy
1983). Beresh et al. (2002), using conditional analysis, showed that the low-frequency motion could be related to the fullness of the instantaneous velocity profile of the incoming boundary layer. Yet in the conditional sampling analyses performed by Thomas et al. (1994), no discern-able statistical relationship could be found between the upstream boundary layer and shock wave motion. The precise mechanisms involved in the low-frequency dynamics are still therefore not fully understood.
In comparison to the compression ramp interaction, the latter case of incident shock wave interaction has received less attention. Studies mapping the mean flow properties as functions of Mach number and Reynolds number, as well as the incident shock wave strength and state of the incoming R. A. Humble (&) F. Scarano B. W. van Oudheusden
Faculty of Aerospace Engineering,
Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
boundary layer have been conducted (e.g. Holder et al.
1955; Chapman et al.1958; Green1970). The unsteadiness properties of this type of interaction, however, have been less well documented. Yet, behaviour similar to the com-pression ramp case has been found, such as low-frequency motion of the reflected shock wave when the interaction involves boundary layer separation (Dupont et al.2006).
In general, experimental studies of SWTBLIs have been hampered by the limitations of the experimental techniques used (Dolling2001). Whilst hot-wire measurements, wall pressure measurements and laser Doppler velocimetry have been indispensable in providing detailed information on the nature of SWTBLIs, without whole-field quantitative information, an instantaneous velocity characterization of the flowfield cannot be made. Furthermore, whilst numer-ical simulations of these flows have achieved some degree of success, being able to predict the mean flow properties reasonably well, the accurate prediction of the associated turbulence properties still remains problematic (Knight and Degrez 1998). Recently, however, large eddy simulation (LES) and direct numerical simulation (DNS) have been applied to the SWTBLI problem with significant success (e.g. Garnier and Sagaut2002; Pirozzoli and Grasso2006). Advances in laser and digital imaging technology have led to the improvement of nonintrusive, planar diagnostic tools, such as particle image velocimetry (PIV) in partic-ular. This technique is capable of performing direct instantaneous velocity flowfield measurements, making it suitable to investigate large-scale unsteady flow phenom-ena. Together with the ability to acquire large amounts of data, this technique offers the opportunity to investigate the spatial structure of SWTBLIs. From an instantaneous and a statistical point of view, PIV has historically found wide-spread application as a standard diagnostic tool in low-speed incompressible flows (Raffel et al.1998). Efforts to extend the technique into the high-speed compressible flow regime became possible with the introduction of high-en-ergy short-pulsed lasers, short interframe transfer CCD cameras, as well as developments in image interrogation methods (Scarano and Riethmuller2000). Despite efforts being hindered by the technical difficulties associated with optical diagnostics in supersonic wind tunnels, namely, flow seeding, illumination and imaging, PIV has been ap-plied to a variety of high-speed flow problems of practical interest, including SWTBLIs (e.g. U¨ nalmis et al. 2000; Beresh et al.2002; Hou et al.2003). These investigations, however, have typically considered ramp or blunt-fin configurations. Comparatively few PIV studies have con-sidered the impinging shock wave interaction (e.g. Haddad
2005). The need for a better understanding of this type of flow, as well as the potential of nonintrusive measurement techniques, provide the impetus for the application of PIV to this flow problem.
The subject of the present paper is to report on the application of PIV to the interaction between an incident shock wave and a turbulent boundary layer. A particle response assessment is first presented, which establishes the fidelity of the tracer particles under measurement conditions. The undisturbed boundary layer is then char-acterized in detail, in terms of its mean velocity and tur-bulence properties. Mean and instantaneous whole-field velocity measurements of the interaction region are ob-tained, from which inferences about the turbulence prop-erties are made. These results may be useful for analytical and computational modelling purposes.
2 Apparatus and experimental technique
2.1 Flow facility
Experiments were performed in the blow-down transonic-supersonic wind tunnel (TST-27) of the High-Speed Aerodynamics Laboratories at Delft University of Tech-nology. The facility generates flows in the Mach number range 0.5–4.2, in a test section of dimensions 280 mm· 270 mm. The Mach number was set by means of a continuous variation of the throat section and flexible nozzle walls. Small deviations in Mach number were cor-rected for by automatic fine adjustment of the choke. The tunnel operates at unit Reynolds numbers ranging from 30· 106to 130· 106m–1, enabling a blow-down operat-ing use of the tunnel of approximately 300 s.
Two types of experiment were conducted in the present study. First, the undisturbed boundary layer was charac-terized in detail, followed by an experiment to characterize the interaction. The boundary layer on the top wall of the wind tunnel was investigated in both cases. The boundary layer developed on a smooth surface under nearly adiabatic flow conditions for a development length of approximately 2 m. Upon entering the measurement domain, the bound-ary layer thickness was d99= 20 mm. A thick boundary layer is advantageous for PIV studies, since it provides an increase in the scales of the mean and fluctuating flowfield, enabling them to be better resolved. The experimental conditions are summarized in Table 1.
and spanned 96% of the test section. A schematic repre-sentation of the experimental apparatus is shown in Fig.1.
2.2 PIV technique
Two-component PIV was employed in the present study. Flow seeding constitutes one of the most critical aspects of PIV in high-speed flows. Titanium dioxide (TiO2) particles (Kemira UV-TITAN L830) were adopted with a nominal crystal size of dp= 50 nm and bulk density of qb= 200 kg/m3. The effective particle size is approxi-mately 400 nm (see ‘‘Particle response assessment’’). A high-pressure cyclone pressurized at 1,000 kPa generated the seeded stream, which was introduced into the settling chamber of the wind tunnel through a 2D rake distributor. The seeding rake spanned 26· 30 cm2 with six vertical airfoil-type bars, each with six orifices. Hot-wire ane-mometry measurements performed in the freestream of the facility revealed no noticeable difference in the mean velocity field, and only a 0.2% increase in turbulence intensity (~1%) as a result of the seeding device. The seeded flow was illuminated by a Big Sky CFR PIV-200 double-pulsed Nd:Yag laser, with a 200 mJ pulsed energy and a 7 ns pulse duration at wavelength 532 nm. Tunnel
access for the laser light was provided by a probe inserted 70 cm downstream of the shock generator. The probe shaped the light beam into a light sheet approximately 1.5 mm thick inside the test section. The laser pulse sep-aration in the boundary layer and interaction experiments was 0.6 and 2 ls, respectively, which gave particle dis-placements of approximately 0.3 and 1 mm, respectively, in the freestream flow. These correspond to 26 and 11 pixel displacements, respectively. Particle images were recorded by a PCO Sensicam QE 12-bit Peltier-cooled CCD camera with frame-straddling architecture and a 1376· 1040 pixel sized sensor. The sensor was cropped to 1376 · 432, given the large aspect ratio of the investigated flow region, and at the same time to achieve an increased recording rate of 10 Hz. A narrow-band-pass 532 nm filter was used to minimize background ambient light. In the boundary layer experiment, the camera was rotated 90 to maximize the spatial resolution. Table2 summarizes the PIV recording parameters.
The optical settings result in a particle image diameter ds for the boundary layer and interaction experiments of ds= 16 lm (2.4 pixels) and ds= 11 lm (1.7 pixels) respectively. Data sets of 500 and 1,500 image pairs were acquired respectively. Both sets of recorded images were interrogated using the WIDIM algorithm, as described by Scarano (2002). This method is based upon the deforma-tion of correladeforma-tion windows with an iterative multi-grid scheme, which is particularly suited for highly sheared flows. Image pairs in the boundary layer and interaction experiments were interrogated using windows of size 61· 7 and 21 · 17 pixels respectively, with an overlap factor of 75%.
2.3 Particle response assessment
The fidelity of the tracer particles was evaluated by con-sidering their dynamic response when passing through a steady oblique shock wave (OSW). The OSW generated in Table 1 Experimental conditions
Parameter Test case Undisturbed boundary layer Interaction experiment M¥ 2.05 2.07 U¥(m/s) 505 518 d99(mm) 20 20 d* (mm) 3.9 4.4 h (mm) 1.3 1.4 us(m/s) 19.4 19.4 cf 1.6· 10–3 1.52· 10–3 P0(kPa) 226 276 T0(K) 278 286 Reh 3.96· 104 4.92· 104
Fig. 1 Schematic representation of the experimental setup
Table 2 PIV recording parameters
Parameter Test case
Undisturbed boundary layer Interaction experiment Field of view, W· H (mm2) 5· 16 129· 40 Interrogation volume (mm3) 0.7· 0.08 · 1.5 1.9· 1.6 · 1.5 Digital resolution (pix./mm) 86 11
Recording distance (cm) z0= 15 z0= 60
Recording lens f = 105 mm f = 60 mm
f-number f#= 8 f#= 8
the freestream of the interaction experiment was used for such an assessment. The PIV measurement returns the velocity spatial distribution, making it possible to extract a velocity profile across the OSW. The velocity component normal to the shock wave was considered for this purpose. Figure2 shows the distribution of the mean normal velocity along with the shock-normal abscissa s.
To assess the spatio-temporal response of the particles, the profile of the velocity is shown against s in Fig.3, where s = 0 denotes the shock wave position. Here un1and
un2 are the upstream and downstream mean velocity,
respectively, normal to the shock wave. Observe an appreciable distance before the particle velocity down-stream of the shock wave reaches its reference value. The effects of a finite spatio-temporal resolution are also evi-dent, where it can be seen that the velocity begins to de-crease approximately one-quarter of a window size upstream of the shock wave as a result of the averaging effect intrinsic to the PIV interrogation method. The par-ticle relaxation time sp was obtained with an exponential curve fit of un ¼ unð Þ and yielded ss p= 2.1 ls, corre-sponding to a frequency response fp= 476 kHz. This value of sp is consistent with previous OSW particle response assessments reported by Scarano and van Oudheusden (2003) using similar particles.
The present particle response behaviour can be com-pared with a modified Stokes drag law, valid for small spherical particles. Given the relatively small particle Mach number and Reynolds number, the following drag relation to determine spapplies (Melling1986)
sp¼ dp2
qb
18lfð1þ 2:7KndÞ ð1Þ
where Kndis the Knudsen number based upon dp, and lfis the fluid viscosity. An expression for the Knudsen number in terms of the Mach number and Reynolds number is provided by 1.26c(MDu/Red) (Schaaf and Chambre1958), where c is the ratio of specific heats, taken as c = 1.4 for air. The Mach number MDuis based uponDu, the maximum particle slip velocity, which occurs downstream of the shock wave. This was determined to be MDu = 0.38. The downstream Reynolds number based upon dp was deter-mined to be Red 1. Using Eq. 1, this results in a relax-ation time sp of less than 1 ls. The discrepancy between this result and the measured result is ascribed to particle agglomeration, a phenomenon which introduces an uncer-tainty on the effective particle size and hence response characteristics. Inserting the experimentally determined sp back into Eq. 1, gives an effective particle agglomerate size of dp 900 nm, a value not dissimilar to the dp= 400 nm size found from electron scans of the porous agglomerates, as reported by Schrijer et al. (2006).
The particle dynamic effects can be further parameter-ized by the Stokes number St, defined as the ratio between spand a time scale of the flow sf. For accurate flow tracking at the time scale represented by sfit is necessary to meet the criterion that St << 1. Assuming an outer flow time scale of d/U¥, then this gives sf= 38 ls. The correspond-ing Stokes number is therefore St 0.06, which is of the same order as that reported by Urban and Mungal (2001) in their high-speed turbulent shear layer experiments.
3 Results and discussion
3.1 Undisturbed boundary layer properties
The van Driest effective velocity concept is used to give a suitable description of the boundary layer velocity profile Fig. 2 Distribution of un=un1 across the OSW. Shock-normal
abscissa s is shown in yellow
within the log-law region (White 1991). The nondimen-sional velocity u+and length scale y+normalized with the friction velocity usare defined as
uþ¼ u us ; yþ¼usy vw ; us¼ ffiffiffiffiffiffis w qw r ð2Þ where v is the kinematic viscosity, s is the shear stress, q is the fluid density and the subscript w denotes the wall condition. The mean experimental velocity profile u yð Þ determined from the boundary layer experiment is transformed into an effective velocity ueq using the van Driest compressibility transformation, given for an adiabatic flow by ueq¼ Ue a sin 1 a u Ue ¼ us 1 jln y þþ B where a¼ 1 Te Taw ð3Þ Here T is the temperature with constants j = 0.41 and B = 5.0. The subscripts e and aw denote the boundary layer edge and adiabatic wall conditions respectively. The right-hand side of Eq. 3 is the ordinary incompressible form of the law-of-the-wall; the left-hand side is the effective velocity. The corresponding law-of-the-wall fit for the present experimental data is shown in Fig.4. The statistical uncertainty associated with the mean velocity due to the limited number of realizations is <1%U¥. The experi-mental effective velocity profile coincides with the theo-retical profile when a friction velocity of us= 19.4 m/s is assumed. The corresponding skin friction coefficient determined from cf= 2us2qw/qeUe2 gives cf= 1.6 · 10–3, which agrees to within 10% of the van Driest II skin friction formula for a flat plate.
Within the logarithmic region, there is excellent agree-ment between the experiagree-mental data and the van Driest effective velocity. Spalding (1961) has provided a single composite formula for the entire wall-related region given by yþ¼ uþþ ejB ejuþ 1 juþ1 2 ju þ ð Þ21 6 ju þ ð Þ3 ð4Þ A departure of the experimental data from the single composite formula can be observed for approximately y+< 30. Here, the lower edge of the interrogation window becomes influenced by the presence of the wall. The closest point to the wall, however, lies within the viscous sublayer (y+< 5). To the author’s knowledge, PIV mea-surements within the viscous sublayer of a supersonic boundary layer have never been reported. A wake com-ponent, characteristic for turbulent boundary layers, can also be identified. The Coles wake parameter G, which is used to help describe the deviation of the outer layer profile from the law-of-the-wall was determined to be G = 0.45, which is in reasonable agreement with the value of 0.55 commonly admitted for zero-pressure-gradient incom-pressible boundary layers when Reh> 5,000 (Cebeci and Cousteix 1999). It should be remarked, that G varies with boundary layer history and somewhat with Mach number. The variation of the streamwise <u¢> and vertical <v¢> turbulence intensity, as well as the kinematic Reynolds shear stress u0v0are shown in Fig.5, where <> denotes the
root-mean-square quantity. The statistical uncertainty due to the limited number of realizations for the turbulence intensity and kinematic Reynolds shear stress is approxi-mately 3 and <10% respectively. Symbols are drawn at the
first data point and subsequently at data points that are at least 3% of the figure height distance from the previously plotted data point. The compressible momentum thickness is chosen to scale the wall-normal coordinate because it can be determined more accurately than the boundary layer thickness. The variation of <u¢> compares favourably with turbulence measurements made within a variety of super-sonic boundary layers (e.g. Petrie et al. 1986; Johnson
1974), as well as those obtained by means of PIV (Hou et al. 2002). Note that the turbulence properties do not attain their freestream values because the complete boundary layer is not resolved.
3.2 Mean flow properties of the interaction
To first give a general description of the interaction, the mean flow topology is shown in Fig.6. Mean velocity streamlines are displayed with mean vertical velocity contours in order to qualitatively illustrate the important flow features. The origin of the reference coordinate sys-tem, in these and subsequent results, is located on the tunnel wall, with x measured in the downstream flow direction from the extrapolated wall impact point of the incident shock wave and y normal to the wall. Spatial coordinates are normalized with the undisturbed boundary layer thickness. The streamlines verify a uniform outer flow upstream, and illustrate the distortion of the flowfield as a result of the interaction process. Regions of flow compression typically appear as densely spaced vertical velocity contours, whereas sparsely spaced vertical veloc-ity contours typically indicate regions of flow expansion. The incident shock wave can be seen to enter the boundary layer, where it begins to curve in response to the decreasing local Mach number. It reflects from the sonic line as an expansion fan, as labelled in Fig.6. Observe the com-pression waves generated within the incoming boundary layer approximately two boundary layer thicknesses up-stream of the extrapolated wall impact point of the incident
shock wave. These compression waves coalesce as they leave the boundary layer to form the reflected shock wave. The flow undergoes a recovery process farther down-stream. Subsonic fluid close to the wall, which has passed through the interaction begins to contract, causing the outer fluid to move back towards the wall. Although difficult to discern, a gradual recompression process takes place far-ther downstream, as fluid is slowly turned back towards the streamwise direction.
An instantaneous PIV recording from the interaction experiment is depicted in Fig.7. It shows some nonuniform seeding concentration. The incident and reflected shock waves can be visualized, whereas the boundary layer is highlighted by a comparatively lower seeding level. Tur-bulent activity within the downstream boundary layer can also be observed, as well as the intermittent nature of the boundary layer edge. Laser light reflections were mini-mized during the experiments by illuminating almost tan-gent to the wall.
The mean flow behaviour is described by the average streamwise velocity field in Fig.8. Velocity vectors are under-sampled (showing 1 in 22 in the streamwise direc-tion for clarity). The incident and reflected shock waves are visible as a sharp flow deceleration and change of direction for the first, whereas the reflected shock wave exhibits a somewhat smoother spatial variation of the velocity due to its unsteady nature and the averaging effect. From the mean velocity vectors, no reversed flow can be detected, although it appears that the flow is close to separation. Downstream of the interaction, the distorted boundary layer appears to increase in thickness and develops with a relatively low rate of recovery.
3.3 Two-dimensionality of the interaction
To examine the effects of spanwise nonuniformities that are often present in nominally two-dimensional flows, a multi-planar assessment of the interaction region was car-ried out within the range –2.5£ z/d £ 2.5, in increments of z/d = 0.5 (i.e. 9 planes). A total of 50 images were acquired at each spanwise location. Recording and interrogation settings were the same as those used in the interaction experiment. Figure9 shows three isosurfaces of mean
Fig. 6 Mean flow topology. Mean velocity streamlines are shown
Mach numbers 0.5, 1.0, and 1.5, determined using the adiabatic flow assumption. Also shown are under-sampled velocity vectors (showing 1 in 30 in the streamwise direction for clarity). It can be seen that a relatively small change in Mach number occurs within the spanwise region considered. The rapid dilation of the subsonic layer is evident, as it responds to the adverse pressure gradient imposed by the incident shock wave. The outermost isosurface of Mach 1.5 shows the displacement of the outer layer of the incoming boundary layer, as well as the reflected shock wave pattern farther downstream.
Figure10shows a rendered representation of the mean flow organization within the interaction. Flooded contours of streamwise velocity are shown, illustrating the low-speed velocity region. Mean stream tubes are also shown within the lower part of the boundary layer. The variation of the streamwise velocity around these tubes illustrates that slight three-dimensional effects exist. However, they seem characteristic of the fluid dynamic processes present and not due to the sidewall boundary layers. (Note that the test section width–boundary layer thickness aspect ratio is 14:1.) The measured flow properties show an appreciable
deviation from the centre-line values at distances from the centre-line greater than 30% of the test section width. This behaviour is ascribed to the lower measurement confidence level due to the finite size of the incoming seeded flow.
3.4 Instantaneous flow properties of the interaction
The instantaneous velocity fields reveal several interesting features associated with the unsteady behaviour of the interaction. Figure11 illustrates two fields of the instan-taneous streamwise velocity, which typify the dynamical events that take place. The time that elapses between consecutive recordings (10 Hz framing rate) is significantly greater than any characteristic flow time scale, leading to the measurement of uncorrelated velocity fields. It can be Fig. 8 Mean streamwise velocity distribution u=U1: Velocity
vectors show 1 in 22 in the streamwise direction
Fig. 9 Spanwise survey of interaction. Mean Mach number isosur-faces of 0.5, 1.0 and 1.5 are shown along with velocity vectors showing 1 in 30 in the streamwise direction
Fig. 10 Mean flow organization of the interaction. Mean stream tubes are shown flooded with mean streamwise velocity
seen that the outer freestream flow remains steady and there is no appreciable motion of the incident shock wave. The global structure of the interaction region, however, varies considerably in time. A thickening of the upstream approaching flow can be observed. In Fig.11b, fluid close to the wall is redirected upstream, leading to the formation of a separated flow region. This configuration forces fluid to detach and move away from the wall. The separation bubble length is found to vary within the range between 0 and 2d, with the velocity in the reversed flow region often attaining a value 10% of U¥. However, recall that the average velocity field shows that the boundary layer re-mains attached, indicating that reverse flow occurs only instantaneously.
After inspection of numerous realizations (not shown here for brevity), the interaction can be characterized, on an instantaneous basis, as containing irregularly shaped layers of relatively uniform streamwise velocity, most readily observed in the velocity vectors within the rede-veloping boundary layer. The term layer is used here to emphasize that whilst they are defined instantaneously, they typically extend across the measurement domain. The interaction contains a high-velocity outer layer (typi-cally u/U¥> 0.5), and a low-velocity inner layer (typically u/U¥< 0.5). They therefore loosely correspond to the supersonic and subsonic parts of the boundary layer respectively. The outer layer comprises most of the incoming boundary layer and includes fluid that is lifted above the interior fluid near the wall. It retains most of its streamwise velocity throughout the interaction. In contrast, a noticeable reduction in streamwise velocity occurs within the inner layer. This layer contains values of the same order as found within the near-wall region of the incoming boundary layer. It grows rapidly as it enters the first part of the interaction, often reaching its maximum thickness when it intersects with the incident shock wave.
These layers are typically separated by a thin region of relatively high shear. The interface is therefore a region of relatively large spanwise vorticity. Observe how the outer fluid often penetrates deep into the boundary layer in Fig.11b. The interface therefore has an irregular and intermittent nature, which is a particularly dominant fea-ture of the redeveloping boundary layer. The supposition of smaller scales is evident by the jagged edges of the inter-face between the high- and low-speed boundaries. It is interesting to observe, that whilst the subsequent reat-tachment process takes place within a relatively short dis-tance, the overall velocity deficit within the inner layer persists much farther downstream. This behaviour is substantiated by the turbulence properties presented in ‘‘Turbulence flow properties of the interaction’’.
Furthermore, by inspection of Fig.11, and other real-izations, it can be inferred that when the reversed flow
region expands, the reflected shock wave is often displaced away from the wall; and when it contracts, the reflected shock wave is brought closer to the wall. This mechanism has also been shown by Erengil and Dolling (1993) to be associated with the large-scale motion of the shock wave, upon examining a hypersonic two-dimensional compres-sion ramp interaction. No clear quantitative relationship could be formulated, however, based upon the present data. Overall, it is now clear that the mean flow organization is a somewhat simplified representation, since it is constructed from a statistical analysis of an instantaneous flowfield that is highly fluctuating and significantly more complex.
3.5 Turbulence flow properties of the interaction
Figure12a and b show the spatial distributions of <u¢> and <v¢> respectively. These results reflect the mixing that takes place within the interaction and the distributed nature of the turbulence. Note that <v¢> is scaled three times as sensitive as <u¢>. A substantial increase in <u¢> occurs throughout the interaction, initiating itself within the re-flected shock foot region, and reaching a maximum value of approximately 0.2U¥ beneath the incident shock wave. These results are comparable to the laser velocimetry measurements of Rose and Johnson (1975), Moderass and Johnson (1976) and Meyer et al. (1997), as well as the LES computations of Garnier and Sagaut (2002), which have all considered an incident shock wave interacting with a flat plate turbulent boundary layer. Maximum levels of <u¢> are over 300% greater than maximum levels of <v¢>,
indicating that appreciable turbulence anisotropy is present. Since the upstream flow is often lifted and turns around the bubble, whereas in other instances it remains fully at-tached, it can be inferred that the locus of large <u¢> within the first part of the interaction is a result of the averaging of this intermittently separated flow. Farther downstream, <u¢> can be seen to rapidly decay. At the downstream edge of the measurement domain, the familiar near-wall peak of <u¢> is now just beginning to reappear, indicating that the boundary layer is recovering.
In the case of <v¢>, significant fluctuations can be ob-served across the reflected shock wave, highlighting its unsteady behaviour. The incident shock wave appears as a relatively steady feature, except for the part that penetrates the boundary layer. Within the redeveloping boundary layer, elevated levels of <v¢> are broadly distributed across the lower half of the boundary layer. Whilst <u¢> rapidly decreases in this region, it can be seen that the elevated levels of <v¢> persist downstream. This behaviour is associated with the redistribution of turbulent kinetic en-ergy, mainly through the pressure–strain correlation terms (Ardonceau et al. 1980). The different turbulence evolu-tions of <u¢> and <v¢> can be readily understood when one considers the production term associated with each com-ponent. Following along the lines of turbulence studies concerning transonic SWTBLIs (De´lery and Marvin1986), consider first the production term of the streamwise com-ponent transport equation, written for an incompressible flow for simplicity as
Pu¼ 2u0v0
ou oy 2u
02ou
ox ð5Þ
It should be noted that there is an appreciable variation of mean density across the undisturbed boundary layer in the present study ðq=qe 0:57 at Me¼ 2:1Þ and so only a
general discussion will be given. In the first part of the interaction, the strain rateou=oy within the boundary layer is typically large. Furthermore, it is generally accepted that u0v0\0 when ou=oy > 0: (The reader can confirm this by
looking ahead at Fig.13.) Withou=ox a necessarily large negative value in this region since the flow is strongly decelerating, the production term of the streamwise turbulence intensity is essentially the sum of two large positive terms. This explains its substantial increase in the first part of the interaction. Consider now the production mechanism for the vertical component given by
Pv¼ 2u0v0 ov ox 2v 02ov oy 2u 0v0ov oxþ 2v 02ou ox ð6Þ
The reader will notice that the production mechanism for the vertical component contains terms, which are less
important than those occurring in the streamwise compo-nent transport equation. Here,ov=oy can be replaced with ou=ox, since the incompressible continuity equation is essentially satisfied for weakly compressible flows at moderate Mach number (M¥< 2). This was verified by considering the spatial distribution of these derivatives, where it was found that incompressible continuity was generally satisfied except in the immediate vicinity of the shock waves. If ov=ox is considered small throughout the interaction, then withou=ox being typically negative, it can be deduced that only the second term in Eq. 6 is important (and actually tends to decrease the production of the ver-tical component in the first part of the interaction behind the reflected shock foot as shown). Farther downstream, the flow begins to accelerate, and ou=ox becomes positive. This leads to the relatively slow production of the vertical turbulence intensity farther downstream. It is now clear that the typical boundary layer assumption of a sufficiently small wall-normal pressure gradient (¶p/¶y 0) may no longer be valid in the interaction, since there is an appre-ciable variation of the vertical velocity fluctuations normal to the wall.
The increased level of fluctuations along the incident and reflected shock waves in the freestream (approximately 4 and 7%U¥respectively) is typically encountered in these experimental conditions and is ascribed to the combined effect of decreased measurement precision and to small fluctuations of the shock wave position. The reflected shock wave exhibits a relatively higher level of velocity fluctuations, which is ascribed to its unsteady motion. Interestingly, an increased level of <u¢> can be observed at the tip of the incident shock wave, indicating that it undergoes an increased motion in this region. This con-firms the observations made in the DNS of an incident SWTBLI performed by Pirozzoli and Grasso (2006), where it was observed that coherent structures propagate in this region leading to an increased oscillatory motion at the shock wave’s tip. A weak feature immediately upstream of the incident shock wave (roughly parallel to it) can also be Fig. 13 Kinematic Reynolds shear stress distribution u0v0=U2
observed. This is due to optical aberration effects intro-duced by the inhomogeneous index of refraction field of this compressible flow (Elsinga et al.2005).
Consider now the Reynolds shear stress distribution. Such measurements are principally carried out to aid the modelling of turbulent effects by computational methods. They are of particular importance in the validation of tur-bulence closure models, since theoretical efforts are gen-erally hampered by the difficulties of representing the turbulence terms in the time-averaged equations. For compressible flows, the Reynolds shear stress is conven-tionally expressed by qu0v0; when the density fluctuations
are ignored. In this paper, the kinematic term u0v0=U2 1 is
regarded as being representative of the Reynolds shear stress. The spatial distribution of kinematic Reynolds shear stress u0v0=U2
1 is shown in Fig.13.
Initially moderate levels ofu0v0are present within the
undisturbed boundary layer. A substantial increase in magnitude occurs within the incident and reflected shock foot regions. This increase is expected, since it is known that supersonic flow, which undergoes a compression is associated with turbulence augmentation. There appears to be a systematic change of kinematic Reynolds shear stress farther downstream. The redeveloping boundary layer can be characterized by the presence of a distinct streamwise-oriented region of relatively large kinematic Reynolds shear stress magnitude in its lower part. Note the over-whelmingly negative values in this region, indicative of slower moving (u¢ < 0), upward-oriented (v¢ > 0) fluid, and/or faster moving (u¢ > 0), downward-oriented (v¢ < 0) fluid, relative to the mean flow. As noted by Ardonceau (1983), who studied the structure of turbulence in SWT-BLIs, these large kinematic Reynolds shear stresses imply the existence of large-scale eddies, consistent with the instantaneous results of the present study, and also indi-cated by the recovery of the boundary layer velocity profile with downstream development. The recovery of the tur-bulence properties, however, appears to be a gradual pro-cess with the present measurement domain insufficient to observe the boundary layer returning to its initial equilib-rium conditions.
4 Conclusions
This paper has reported on the application of PIV to the interaction between an incident planar shock wave and a turbulent boundary layer. A particle response assessment established that the fidelity of the tracer particles was consistent with similar studies. The experimentally inferred porous agglomerate size agreed with the electron scans reported in literature. The mean velocity profile and de-duced skin friction coefficient of the undisturbed boundary
layer showed good agreement with theory. The interaction was characterized by the mean velocity field, which showed the incident and reflected shock wave pattern, as well as the boundary layer distortion.
The unsteady flow properties were inspected by means of instantaneous velocity fields. The global structure of the interaction region varied considerably in time. Patches of reversed flow were frequently observed. Although signifi-cant reversed flow was measured instantaneously, on average, no reversed flow was observed. The interaction could be characterized instantaneously as exhibiting a multi-layered structure, namely, a high-velocity outer re-gion and a low-velocity inner rere-gion. These two layers were separated by an interface containing relatively high shear. The mean flow is therefore a somewhat simplified representation of the interaction.
The streamwise and vertical turbulence components evolved differently throughout the interaction. The turbulent fluctuations were found to be highly anisotropic, with the streamwise component dominating. The highest turbulence intensity occurred in the region beneath the impingement of the incident shock wave. An increased level of <u¢> was observed at the tip of the incident shock wave, indicating that it undergoes an increased motion in this region. A distinct streamwise-oriented region of relatively large kinematic Reynolds shear stress magnitude appeared within the lower half of the redeveloping boundary layer. Boundary layer recovery was observed to initiate downstream of the inter-action. The recovery towards the initial equilibrium condi-tions, however, appeared to be a gradual process.
Acknowledgments This work is supported by the Dutch Technol-ogy Foundation STW under the VIDI-Innovation Impulse program, grant DLR.6198.
References
Andreopoulos J, Muck KC (1987) Some new aspects of the shock wave/boundary layer interaction in compression flows. J Fluid Mech 180:405–428
Ardonceau PL (1983) The structure of turbulence in a supersonic shock wave/boundary layer interaction. AIAA J 22(9):1254– 1262
Ardonceau P, Lee DH, Alziary de Roquefort T, Goethals R (1980) Turbulence behaviour in a shock-wave/turbulent boundary layer interaction. AGARD CP-271, paper no. 8
Beresh SJ, Clemens NT, Dolling DS (2002) Relationship between upstream turbulent boundary-layer velocity fluctuations and separation shock unsteadiness. AIAA J 40(12):2412–2422 Cebeci T, Cousteix J (1999) Modeling and computation of boundary
layer flows. Horizon, Long Beach
Chapman DR, Kuehn DM, Larson HK (1958) Investigation of separated flows in supersonic and subsonic streams with emphasis on transition. NACA report 1356
Dolling DS, Murphy MT (1983) Unsteadiness of the separation shock wave structure in a supersonic compression ramp flowfield. AIAA J 21(12):1628–1634
De´lery J, and Marvin JG (1986) Shock-wave boundary layer interactions. AGARDograph 280
Dupont P, Haddad C, Debie`ve JF (2006) Space and time organization in a shock-induced separated boundary layer. J Fluid Mech 559:255–277
Elsinga GE, van Oudheusden BW, Scarano F (2005) Evaluation of aero-optical distortion effects in PIV. Exp Fluids 39:246–256 Erengil ME, and Dolling DS (1993) Physical causes of separation
shock unsteadiness in shock wave/turbulent boundary layer interactions. AIAA paper 93-3134, Orlando, Florida
Garnier E, Sagaut P (2002) Large eddy simulation of shock/boundary-layer interaction. AIAA J 40(10):1935–1944
Green JE (1970) Interactions between shock waves and turbulent boundary layers. Prog Aero Sci 11:253–340
Haddad C (2005) Instationnarite´s, mouvements d’onde de choc et tourbillons a grandes echelles dans une interaction onde de choc/ couche limite avec de´collement. PhD thesis, L’Universite´ de Provence, France
Holder DW, Pearcy HH, Gadd GE (1955) The interaction between shock waves and boundary layers. ARC technical report Hou YX, Clemens NT, Dolling DS (2002) Development of a
multi-camera PIV imaging system for studies of shock/boundary layer interactions. AIAA paper 2002-3232, St Louis, Missouri Hou YX, Clemens NT, Dolling DS (2003) Wide-field PIV study of
shock-induced turbulent boundary layer separation. AIAA paper 2003-0441, Reno, Nevada
Johnson DA (1974) Turbulence measurements in a Mach 2.9 boundary layer using laser velocimetry. AIAA J 12(5):711–714 Knight DD, Degrez G (1998) Shock wave boundary layer interactions in high speed flows: a critical survey of current numerical prediction capabilities. Advisory report 319, AGARD 2:1.1–1.35 Melling A (1986) Seeding gas flows for laser anemometry. In: Proceedings on the conference of advanced instrumentation for aero engine components. AGARD CP-399, 8.1
Meyer MJ, Butler TA, Bowersox RDW (1997) Compressible turbulence measurements in a supersonic boundary layer with impinging shock wave interactions. AIAA Paper 97-0427, Reno, Nevada
Moderass D, and Johnson DA (1976) Investigation of shock-induced separation of a turbulent boundary layer using laser velocimetry. AIAA paper 76-374, San Diego, California
Petrie HL, Samimy M, Addy AL (1986) Compressible separated flows. AIAA J 24(12):1971–1978
Raffel M, Willert C, Kompenhans J (1998) Particle image velocime-try: a practical guide. Springer, Heidelberg
Rose WC, Johnson DA (1975) Turbulence in a shock-wave boundary-layer interaction. AIAA J 13(7): 884–889
Pirozzoli S, Grasso F (2006) Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at M = 2.25. Phys Fluids 18:065113
Scarano F (2002) Iterative image deformation methods in PIV. Meas Sci Technol 13:R1–R19
Scarano F, Riethmuller ML (2000) Advances in iterative multi-grid PIV image processing. Exp Fluids 29:S051
Scarano F, van Oudheusden BW (2003) Planar velocity measure-ments of a two-dimensional compressible wake flow. Exp Fluids 34:430–441
Schaaf SA, Chambre PL (1958) Fundamentals of gas dynamics. Princeton University Press, Princeton
Schrijer FFJ, Scarano F, van Oudheusden BW (2006) Application of PIV in a Mach 7 double-ramp flow. Exp Fluids 41:353–363 Settles GS, Dodson LJ (1991) Hypersonic shock/boundary layer
database. NASA CR 177577
Smits AJ, Dussuage JP (2006) Turbulent shear layers in supersonic flow, 2nd edn. American Institute of Physics, New York Spalding DB (1961) A single formula for the law of the wall. J Appl
Mech 28:455–457
Thomas FO, Putman CM, Chu HC (1994) On the mechanism of unsteady shock oscillation in shock/wave turbulent boundary layer interaction. Exp Fluids 18:69–81
U¨ nalmis O¨H, Hou YX, Bueno PC, Clemens NT, Dolling DS (2000) PIV investigation of role of boundary layer velocity fluctuations in unsteady shock-induced separation. AIAA paper 2000-2450, Denver, Colorado
