Shock Waves DOI 10.1007/s00193-016-0656-x
ORIGINAL ARTICLE
Characterization of an incipiently separated shock wave/turbulent boundary layer interaction A.-M. Schreyer1,2 · J.-P. Dussauge3 · E. Krämer1
Received: 29 January 2015 / Revised: 8 February 2016 / Accepted: 8 April 2016 © Springer-Verlag Berlin Heidelberg 2016
Abstract The turbulence structure in a shock wave/ turbulent boundary layer interaction at incipient separation was investigated in order to get insight into turbulence generation and amplification mechanisms in such flow fields. The flow along a two-dimensional 11.5◦ compression corner was studied experimentally at a Mach number of M = 2.53 and with a momentum-thickness Reynolds number of Reθ = 5370. From hot-wire boundary layer traverses and surface heat-flux density fluctuation measurements with the fastresponse atomic layer thermopile, the turbulence structure and amplification was described. Space–time correlations of the mass-flux fluctuations across the boundary layer and the surface heat-flux density fluctuations were measured to further characterize the development of the turbulence structure across the interaction. The large-scale boundary layer structures are concealed by shock-related effects in the strongly disturbed shock-foot region. Shortly downstream, however, large-scale structures dominate the signal again, just as in the incoming flow. A mechanism explaining this behavior is suggested. Keywords Shock wave/boundary layer interaction · Turbulence structure · Supersonic compressible flow · Atomic layer thermopile Communicated by H. Kleine.
B
A.-M. Schreyer
[email protected]
1
Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, Stuttgart, Germany
2
Present Address: Institute of Fluid Mechanics, Technische Universität Braunschweig, Braunschweig, Germany
3
IUSTI UMR 7343, Aix-Marseille Université, Marseille, France
1 Introduction Shock wave/turbulent boundary layer interactions (SWTBLIs) occur in flow fields relevant for numerous aerospace applications: transonic airfoils, the supersonic inlets of airbreathing SCRamjet engines, rocket engine nozzles, as well as the external flow around super- and hypersonic vehicles. Due to strong pressure and heat loads associated with separation and the highly unsteady flow field, the vehicle structure and performance can be severely affected. Despite decades of ongoing research activities (see, for example, [1–4]), the behavior of such an interaction flow field, as well as the governing mechanisms of the associated phenomena, is still not fully understood [5,6]. Fluid motion in a turbulent boundary layer partly consists of temporally and spatially organized coherent structures. Near the wall, the bursting phenomenon describes the production and transport of turbulent energy [7]. In the outer regions, large-scale flow structures consisting of packets of hairpin vortices and leaning towards the wall in the main flow direction at angles around 45◦ –60◦ are present throughout the entire boundary layer (see, for example, [8–12]). This structure is similar in incompressible and compressible turbulent boundary layers. The behavior of boundary layer turbulence across the interaction with a shock wave has been the subject of numerous detailed studies. The turbulent fluctuation levels are strongly amplified across the interaction [4,5,13,14]. For a fully separated compression corner interaction, the turbulence-structure development has been studied by Selig et al. [14], who investigated a 24◦ compression corner interaction at Mach 2.84 and a unit Reynolds number of Re∞ /m = 6.5 × 107 by means of hot-wire measured massflux fluctuations and wall-pressure signals, and by Ardonceau [13], who studied a 18◦ compression corner at Mach 2.25 and
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Re∞ /m = 1.1×107 by means of laser Doppler anemometry and hot-wire anemometry. In those studies, the occurring increases in turbulence level are much stronger than predicted by rapid distortion theory and thus by the interaction with the shock alone [5]. The direct amplification of the incoming perturbations in the compression is thus not the major factor of amplification. Several effects potentially contributing to turbulence amplification or generation were identified: large-scale perturbations produced in the separated region or the mixing zone over the separated region are one possible influencing factor [5,14–16]. Frequencies around 10 kHz, which is about one order of magnitude larger than the shock oscillation frequency, should then make a relevant contribution to the signal energy. Another contribution may come from Taylor–Görtler vortices produced by the concave streamline curvature along and downstream of the interaction zone, as observed by Selig et al. [14] and Smits and Dussauge [5]. These unsteady counter-rotating vortices cause a mixing process between the high-momentum fluid in the outer region of the boundary layer and the low-momentum fluid in the near-wall region, re-energizing the inner region of the boundary layer that was retarded in the separation zone. This means that the turbulence amplification should be mostly independent of the unsteady motion of the shock [14]. On the other hand, several researchers suspected an influence of the low-frequency shock motion on downstream turbulence [4]. Ardonceau [13] explained the observed turbulence amplification with quasi-periodic counter-rotating spanwise vortices emerging from the shock-foot region and convecting downstream, causing an energy transfer between the mean flow and turbulence structures by mixing high- and low-momentum fluid. The development of the large-scale structures across an interaction has only been investigated in a small number of experimental studies [13,14]. More detailed descriptions of the structures downstream of the interaction are available in more recent experimental studies and numerical studies based on direct numerical simulations (DNS) [15,16]. Souverein et al. [17] studied an oblique shock reflection with incipient separation at Mach 1.7. The particle image velocimetry (PIV) experiment revealed the existence of large eddies resembling Kelvin–Helmholtz structures. Moreover, Piponniau et al. [18] observed the presence of intermediate frequencies, which are related to Kelvin–Helmholtz type structures, in the mixing zone downstream of the shock in a Mach 2.3 shock reflection. Such eddies, probably related to the presence of separation pockets, can be in competition with possible Görtler-type vortices in the case of compression ramp interactions. Wu and Martin [16] described the turbulence structure across a 24◦ compression corner interaction with separation at Mach 2.9 and with Reθ = 2400 from DNS data and
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Pirozzoli and Grasso [15] investigated an impinging shock interaction with separation at Mach 2.25 and Reθ = 3725 also by means of DNS. Wu and Martin [16] observed the typical coherent structures in the turbulent boundary layer upstream of the interaction. In the interaction near-field, the structures decrease in size and are more chaotic. Only far downstream (≈7δ), the structures start to regain their original size and shape after the compression through the shock. The large structures also influence the shape of the shock wave: it curves upstream when the structures pass through, which causes spanwise wrinkling [16]. Furthermore, Pirozzoli and Grasso [15] report the generation of large vortical structures in the region of the average separation point, which resemble hairpin structures. These structures then lift away from the wall and decay slowly while transported downstream. For the present study, the turbulence-structure development in an incipiently separated compression–corner interaction has been studied. Space–time correlations from multi-point measurements in the boundary layer are a common means for the investigation of large-scale structures, and usually the velocity, wall pressure and wall shear-stress signals are used [5]. Brown and Thomas [9] investigated organized structures in a turbulent boundary layer using an array of hot wires and wall shear-stress probes (hot-films). For the first time, they described the characteristic response in the wall shear-stress signal to organized structures passing by. In this study, an 11.5◦ compression corner interaction was investigated experimentally at Mach 2.54. Surface heat-flux density fluctuations and mass-flux fluctuations across the boundary layer were measured, and the characteristic frequencies occurring in the flow field were investigated based on a correlation study of the data. The results of this study will be presented and discussed here, and the behavior of large-scale boundary layer structures across the incipiently separated interaction will be described. An explanation of the observed amplification and subsequent relaxation of turbulent fluctuations in the interaction near-field will be proposed based on prevailing and emerging vortical structures and separation pockets. For reasons that will be explained later (Sect. 4.3), the low-frequency shock motion will not be discussed.
2 Experimental setup and type of interaction The experiments were carried out in the medium-sized supersonic wind tunnel, an indraft facility at the Institute of Aerodynamics and Gas Dynamics (IAG) in Stuttgart, Germany. Since the air reservoir is not pressurized, stagnation pressure and temperature are determined by ambient conditions and cannot be chosen independently. The test-section flow was generated with a half nozzle. The size of the test section is 150 mm × 200 mm and the
Characterization of an incipiently separated shock wave/turbulent boundary layer
Fig. 1 Compression corner model installed in the wind-tunnel test section. Two hot-wire probes are installed in the holding device and two ALTP sensors are flush-mounted into the model surface, respectively. Figure from Schreyer [19] Table 1 Parameters of the incoming turbulent boundary layer M∞
U∞ ( ms )
δ0 (mm)
θ0 (mm)
Reθ
2.53
562
7.5
0.57
5370
measurements were taken at the spanwise location along the center line. The present experiments were conducted at a nominal Mach number of M∞ = 2.54, and the turbulent boundary layer on the floor of the measurement chamber was investigated. A compression ramp with an inclination angle of θ = 11.5◦ was installed onto the test section floor (see Fig. 1), generating a shock wave/turbulent boundary layer interaction with an inviscid pressure rise of p2 / p1 = 2.04 (where p1 and p2 are the static pressure up- and downstream of the oblique shock, respectively) and incipient separation. The conditions in the incoming flow upstream of the interaction are summarized in Table 1, where U∞ is the mean free-stream velocity, δ0 and θ0 are the boundary layer and momentum thicknesses in the undisturbed boundary layer, respectively, and Reθ is the momentum-thickness Reynolds number. The boundary layer characteristics were measured at a distance of 3.33δ0 upstream of the ramp corner. There is no direct evidence of the existence of a separated zone. However, different criteria to assess the separation state appear in the literature (see Settles et al. [2] and Clemens and Narayanaswamy [6]), so we applied a number of different criteria carefully in order to assess the separation state in our experiment: (1) appearance of a triple inflection point in the surface pressure distribution [2,20], (2) mean location of the oblique shock pushed upstream from the ramp corner, as observed from schlieren visualizations [2,6], (3) behavior of the friction coefficient along the interaction [2,4], and (4) the rate of decay of the wake parameter downstream of the ramp corner [2]. All criteria gave indications that a very small zone of mean separation has started to develop, which is why we concluded that the interaction cannot be considered to be attached anymore, but is incipiently separated.
Fig. 2 Surface pressure distribution profiles along the model centerline. S and R mark the separation and reattachment points, respectively. The dashed line marks the inviscid pressure rise level. The standard deviation in the surface pressure, determined from repeated readings, is well below 0.1 %
In a first step, the first appearance of a triple inflection point in the surface pressure distribution was applied as criterion for the onset of separation, as suggested by Kuehn [20] and discussed in detail based on validation with numerous interaction cases by Settles et al. [2]. In Fig. 2, the measured static wall-pressure distribution pw along the model centerline in the streamwise direction x, normalized with the wall pressure pw,0 in the incoming boundary layer, is plotted along with a spline interpolation. The ramp corner location corresponds to x = 0. In spite of the limited number of data points, the shape of the profile suggests the existence of inflection points. The inflection points were determined from the pressure distribution according to Kuehn [2,20], and S and R mark the inferred separation and reattachment points, respectively, according to the same author. With the barely distinguishable, yet present, pressure plateau and in agreement with the extensive studies of Settles et al. [2], the interaction was judged to be incipiently separated based on this first criterion. The second, more qualitative indication for incipient separation is that the mean location of the oblique shock begins to be pushed upstream from the ramp corner (see Clemens and Narayanaswamy [6], Settles et al. [2]), which is visible in the schlieren visualization of the compression–ramp interaction presented in Fig. 3, where we marked the mean shock location with a dashed line. The third and fourth indicators, based on the behavior of the friction coefficient along the interaction and the rate of decay of the wake parameter downstream of the ramp corner, respectively, will be discussed in Sect. 4.2.1. Mass-flux fluctuations and wall-pressure fluctuations were measured at several distinct locations throughout the inter-
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Fig. 3 Schlieren optic visualization of the compression–ramp interaction, θ = 11.5◦ . The mean shock location is highlighted with a dashed line Table 2 Measurement locations of the centerline of the interaction region in the streamwise direction x Location
Description
(1)
x1 = −3.33δ0
Incoming undisturbed boundary layer
(2)
x2 = −0.67δ0
Mean shock position at the wall
(3)
x3 = 3.33δ0
Downstream of the ramp corner
(4)
x4 = 5.47δ0
Further downstream
The ramp corner is located at x = 0 mm
action region, as summarized in Table 2. The locations at significant points throughout the interaction—the incoming undisturbed boundary layer (1), the shock-foot region (2), shortly downstream of the ramp corner (3), and several boundary layer thicknesses downstream of the interaction region (4)—were chosen based on a previously documented detailed turbulence study of the interaction region (see [19]). At each measurement location, an atomic layer thermopile (ALTP) sensor (see Sect. 2.2) could be flush-mounted into the model surface to measure the surface heat-flux density fluctuations, and at the same streamwise location, the mass-flux fluctuations across the boundary layer could be measured by means of a hot-wire boundary layer traverse. An image of the experimental setup for one measurement location is shown in Fig. 1, which also shows additional ALTP and hot-wire sensors. 2.1 Hot-wire measurements Time-resolved mass-flux fluctuations were measured with a single hot wire oriented normal to the flow. To obtain sensitivity solely to mass-flux fluctuations, a relatively high overheat ratio of τ = 0.95 was chosen. The hot wire had a length and diameter of l = 1.25 mm and d = 5 µm, respectively, resulting in a length-to-diameter ratio of l/d = 250. Gold-plated tungsten wire was spot welded to the tips of the prongs of commercially available DANTEC 55P11 probes, customized
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for the application in supersonic flow by shortening and reinforcing the prongs with epoxy resin fillings. A single inclined wire (Dantec 55P12) was used to measure the time-averaged value of the mass-weighted turbulent shear stress (ρu) v . Calibrations and further data analysis were performed following the routines of Smits and Muck [21]. The measurements were performed with the custombuilt Cosytech 2.0 constant-temperature anemometer, that was developed for hot-wire measurements in short-duration ground-testing facilities. The system was described in detail by Weiss [22]. The bridge was optimally balanced at the location of maximum mass-flux fluctuations in the incoming boundary layer (normalized distance from the wall δy ≈ 0.7), in order to obtain the largest possible bandwidth of about 100 kHz. The hot-wire signal was high-pass filtered at 1 kHz. This is sufficient to examine the turbulence structure, especially the large-scale eddies at frequencies ≥10 kHz, which we studied in this case. The output voltages of both sensors, the hot wire and the ALTP, were sampled and digitized simultaneously using a transient recorder with 14-bit resolution at a sampling frequency of 2 MHz. In thin turbulent shear layers, fluctuation intensities measured by means of hot-wire anemometry can be underestimated mainly due to bandwidth limitations, as discussed by Gaviglio and Dussauge [23]. This occurs when the bandwidth is less than 4Ue /δ, where Ue is the mean streamwise velocity at the edge of the boundary layer and δ the boundary layer thickness. They proposed a method to correct for errors induced by the compensation circuit:
ec
2
em
2
= 1 + 4π 2
2
g0 g∞
2 N2
1 , ω02
2
where ec and em are the corrected and measured hot-wire signals, respectively, g0 and g∞ are the amplifier gains for low and high frequencies, ω12 is the time constant of the hot 0
wire, and N is an equivalent frequency, deduced from the radius of the curvature at the top of the auto-correlation curve [23]. This correction method was applied on the rms turbulence data presented in this article. For the uncertainty estimate, we followed the considerations of Smits and Dussauge [5,24], and took uncertainties in the calibration and measurement procedures, possible drift in the calibration constants, and end-conduction effects into account. We assumed wire sensitivity to mass-flux fluctuations only, neglecting the small contribution of totaltemperature fluctuations, which introduces a systematic error, as do the effects of the limited spatial and temporal resolution of the wire. For the variance of the streamwise velocity fluctuations, we estimate an error between 15 and 30 %, increasing towards the wall. For measurements with
Characterization of an incipiently separated shock wave/turbulent boundary layer
the inclined wire, the uncertainty is even higher [21], between 20 and 40 %.
Table 3 Turbulence levels in the freestream of the wind-tunnel facility determined from hot-wire measurements at a bandwidth of approximately 100 kHz and pressure measurements
2.2 The atomic layer thermopile
ρu (%)
p (%)
u (%)
T (%)
ρ (%)
0.52
0.97
0.17
0.25
0.61
Surface heat-flux density fluctuations were measured with a novel high-frequency response sensor, the atomic layer thermopile (ALTP). The sensing element consists of a thin active film of YBCO (yttrium–barium–copper–oxide YBa2 Cu3 O7−x ) crystals with anisotropic structure, where well-conducting copper oxide layers alternate with insulating YBCO layers. Each pair of copper oxide and YBCO layers forms an atomicsized thermocouple, so that the crystal consists of a serial connection of such thermocouples [25]. Absorption of convective and/or radiative heat transfer gives rise to a temperature gradient across the YBCO film, which induces a thermoelectric voltage signal (transverse Seebeck effect) [26]. Its transversal component depends linearly on the temperature difference between the front and back sides of the active sensor element, and is furthermore a function of the thickness and length of the YBCO film, as well as the appropriate Seebeck coefficient. This linear behavior sustains over more than 11 orders of magnitude of temperature gradients [26]. The active film of the sensor applied in the present experiment had a thickness between 500 nm ≤ δF ≤ 700 nm and an active area of 1 mm × 2 mm. A SiO/SiO2 (silicon oxide) coating is deposited onto the film surface to prevent degradation of the sensor sensitivity, as well as to allow sensor calibration at a well-defined wavelength of 10 µm, at which the protective coating causes a sharp absorption peak. In this configuration, the sensor has a temporal resolution of 1 MHz [25]. With the inevitable spatial integration due to the sensor size, an equivalent cut-off frequency of ≈280 kHz could be obtained for the current measurements. The sensor was placed in a cylindrical ceramic (MACOR) housing to insulate the sensor from the surrounding metallic experimental model. A one-dimensional heat flux in the direction perpendicular to the surface of the active film was thus ensured. This single sensor module was then flushmounted into the model surface (see Fig. 1). The ALTP signal was amplified by means of custom-built miniature low-noise amplifiers. To obtain optimal amplification, the fluctuating component of the signal is amplified separately from the overall signal. In the AC-branch, where the fluctuating component of the signal is provided, a highpass filter at 1 kHz is implemented.
measurements were analyzed with the approach according to Laufer [27]. The absolute values of the turbulent fluctuations of the mass flux ρu, static pressure p, streamwise velocity u, temperature T , and density ρ (in % of the mean quantity), are given in Table 3 with the assumption of sources convected at 0.4Ue . These fluctuation levels are comparable with (or better than) those of other supersonic blow-down and indraft facilities. In order to estimate the influence of the noise on data measured in the boundary layer, the power spectral density of the wind-tunnel noise measured in the incoming flow (x = −3.33δ0 ) is compared with measurements at two different locations in the boundary layer: at δy0 = 0.4 in the incoming undisturbed boundary layer and around the location of the shock foot (x = −0.63δ 0 ) (see Fig. 4). For δ frequencies f < 4 kHz f Ue < 0.056 , wind-tunnel noise dominates the signal. Above this frequency range, however, there is no influence. The noise will therefore not disturb the signal in the frequency range of interest for this study, the turbulent fluctuations f ≈ O(100) kHz and large-scale turbulent boundary layer structures f ≈ O (10–50) kHz [28]. Low-frequency shock motion O(1 kHz) is not discussed in
2.3 Wind-tunnel noise assessment The noise level and frequency content in the experimental facility was assessed by means of hot-wire and pressurefluctuation measurements in the incoming flow. The hot-wire
Fig. 4 Power spectral density of the wind-tunnel noise in the incoming flow (x = −3.33δ0 ) at δy0 = 1.4, compared with PSDs measured inside the boundary layer ( δy0 = 0.4) at x = −3.33δ0 and x = −0.63δ0
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this article because it is not of concern for the subject of interest: large turbulent structures and the evolution of such structures and boundary layer turbulence across the interaction.
(a)
3 Data analysis and post-processing From the hot-wire mass-flux measurements, the power spectral distribution across the boundary layer was investigated to determine the dominant frequency ranges in the regions of interest. For the detection of coherent structures, the signal correlation between the mass-flux fluctuations across the boundary layer and the wall heat-flux density fluctuations q was investigated. The cross-correlation coefficient was determined according to Rρu,q (y, t) q (xq , y = 0, z q , t0 )ρu (xρu , y, z ρu , t0 − t) = , (ρu)2 q 2
(b) (1)
where q (xq , y = 0, z q , t0 )ρu (xρu , y, z ρu , t0 − t) is the covariance and (ρu)2 and q 2 are the respective variances of the sensor time signals.
4 Experimental results In order to provide a basis for the discussion of turbulence production and amplification mechanisms in SWTBLI flow fields in Sects. 4.3 and 4.4, first the incoming turbulent boundary layer and the interaction will be characterized in Sects. 4.1 and 4.2, respectively. The behavior of the turbulent quantities across the interaction was observed in detail by means of hot-wire and Pitot probe measurements. 4.1 Incoming boundary layer The mean streamwise velocity profile measured in the incoming undisturbed boundary layer is shown in Fig. 5 in outer and inner scaling. The mean velocity profile made dimensionless with the mean velocity at the boundary layer edge Ue and the boundary layer thickness δ is compared with previous flat-plate boundary layer results at similar Mach numbers in Fig. 5a. The measured profile agrees well with the reference data. The profile transformed according to van Driest and made dimensionless with the friction velocity u τ and ν/u τ (ν is the viscosity) is plotted in Fig. 5b and compared with the log-law: U+ =
1 ln y + + B κ
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(2)
Fig. 5 Mean streamwise velocity profile in the incoming boundary layer a compared with literature data and b transformed according to van Driest. Flat-plate turbulent boundary layer profiles from Coles [29] and Kistler [30] are added for comparison
with the von Kármán constant κ = 0.4, constant B = 5.1, and the mean streamwise velocity U + and distance from the wall y + in inner variables. The temperature profile according to Walz [31] was applied for the van Driest transformation. The friction coefficient Cf = 0.00199 was obtained from the Clauser chart method, and the accordingly transformed velocity profile (denoted with CC) is in good agreement with the log-law for y + > 47 as expected for a turbulent flat-plate boundary layer. The velocity fluctuations in the incoming boundary layer corrected according to Gaviglio and Dussauge [23] (correction of 42 % at δy0 = 0.5) are shown in Fig. 6 and are compared with literature data (Klebanoff [32], Elena and Lacharme [34], Smits et al. [33]). For δy0 ≥ 0.3 the agreement is good. In the near-wall region, the turbulent fluctuations are underestimated due to limited hot-wire resolution. The set-
Characterization of an incipiently separated shock wave/turbulent boundary layer
Fig. 6 Hot-wire measured velocity fluctuations in Morkovin scaling (ρw is the fluid density at the wall) compared with results of Klebanoff [32], Smits et al. [33], and Elena et al. [34]
ting of the anemometer was chosen to optimize its response in the middle of the layer [24,33]. Close to the wall, it is known that the wire thermal inertia increases (see, among others, Morkovin [35] for general properties, and Smits and Dussauge [24] for practical examples), so that the system optimized in the middle of the layer tends to be undercompensated, and therefore delivers reduced signal rms values. The local Mach number is always ≥1.3 for all shown measurement points, so that no transonic calibration is needed for the wire heat transfer [36,37].
Fig. 7 Friction coefficient Cf along the interaction zone, determined with the Clauser chart method. The mean separation and reattachment points determined from the surface pressure distribution are marked with S and R, respectively
4.2 Boundary layer measurements across the interaction 4.2.1 Surface measurements Assuming that the inner layer can quickly adjust to perturbations, the wall stress can be estimated with the Clauser chart method by fitting the near-wall mean velocity to the standard log law [4]. The resulting distribution of the skinfriction coefficient along the surface is presented in Fig. 7. As expected from previous studies [2,4], a sudden decrease in wall shear can be observed in the central interaction zone. Of course this method has difficulties in measuring a negative friction coefficient, so that the minimum of Cf measured at x/δ0 ≈ 0.05 should be considered with caution. However, a rough extrapolation of the two branches of the Cf distribution in Fig. 7 supports the interpretation of the presence of a separated zone in the range suggested by Kuehn’s criterion (separation and reattachment locations S and R, respectively, as determined from Fig. 2 are marked). To characterize the evolution of the strength of the wake, Coles’ wake parameter, which is determined from the maximum deviation of the velocity profile in inner scaling from the
Fig. 8 Coles’ wake parameter 2Π κ along the interaction region of the 11.5◦ compression ramp compared with reference cases [2]
log law, is presented in Fig. 8 for several locations along the interaction. For comparison, the wake parameter downstream of the ramp corner for attached (8◦ ), incipiently separated (16◦ ) and fully separated interactions (20◦ ) studied by Settles et al. [2] at a Mach number of 2.85 are also shown. In the present study, the wake strength increases when approaching the ramp corner, which suggests an increasingly large velocity deficit between the inner log-law region and the edge of the boundary layer [2]. Downstream of the reattachment point, a continuous decrease in wake strength can be observed, which is in good agreement with the findings of Settles et al. [2]. In their experiment, the decay of the wake parameter downstream of the ramp corner is very flat for the attached case (8◦ ), and a significant decrease in wake strength
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(a) 0.25
x = −3.33 δ (corr) 0
x = −0.63 δ0 (corr) x = 3.33 δ (corr)
0.2
0
x = 7.33 δ (corr) 0
(ρu)rms ρU
only occurred once the flow started to separate and the slope of the decay increased with ramp angle. Despite the difference in Mach number between both experiments, the decay rate of the present 11.5◦ compression ramp interaction is in between the 8◦ and 16◦ cases investigated by Settles et al. [2], but it is closer to the incipient 16◦ case in the sense that a significant decrease in wake strength can be observed, especially over a distance of 4δ0 directly downstream of the ramp corner. In the present case, the wake parameter decreases from 2Π κ ≈ 13–3.8 over a distance of 7δ0 , which is nearly an equally strong as in Settles’ 16◦ case, while the wake parameter in Settles’ attached case varies only between 5 and 2 over the same distance. A quadratic regression was performed on the data, and is shown in Fig. 8 to visualize this comparison more clearly. This behavior provides further evidence that the present case is not attached anymore, but incipiently separated.
0.15
0.1
0.05
0 0
0.2
0.4
0.6
0.8
1
(b) −1.6
x = −3.33 δ
0
x = −0.63 δ0
−1.4
4.2.2 Observed turbulence behavior in the boundary layer
x = 3.33 δ
0
x = 7.33 δ
0
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τ/τw
−1.2
In the boundary layer, a strong amplification across the interaction was observed in all turbulent quantities, as expected from similar studies reported in the literature [4,5,13,14]. The intensities of the streamwise components, such as the mass-flux fluctuations (ρu) (see Fig. 9a) and the Reynolds stress component u 2 (not shown), show a further increase after the strong initial amplification. Then, the turbulent intensities stagnate on the amplified level for the observed downstream distance, and no boundary layer relaxation could be observed. Of course, with an extent of 7.33δ0 , the investigated downstream region was restricted to the interaction near-field. For the wall-normal fluctuation components [v and (ρu) v ] on the other hand, a relaxation process starts immediately downstream of the strong initial amplification. The Reynolds shear stresses ρu v [shown in Fig. 9b normalized with the local fluid density at the wall (ρw ) and u 2τ ] returned to almost equilibrium level within the observed downstream region. This behavior was also previously observed by Ardonceau [13], and according to Smits and Muck [4], the different amplification rates of the respective turbulent intensities also indicate a change in turbulence structure across the interaction. The findings encouraged some assumptions concerning the turbulence structure and, based thereupon, also on the mechanisms for turbulence production and amplification within a shock wave/turbulent boundary layer interaction. The streamwise velocity-fluctuation component remains in an amplified state for a considerable downstream distance before returning to equilibrium conditions. Ardonceau [13] suggested that turbulent eddies, which require a long distance to dissipate and thus for the turbulent structure to reorganize, may explain this long-lasting amplified state. This
1.2
y/δ
−1 −0.8 −0.6 −0.4 −0.2 0 0.2
0.4
0.6
y/δ
0.8
1
1.2
Fig. 9 a Mass-flux fluctuation profiles (rms) at several measurement positions along the interaction from hot-wire measurements. b Turbulent shear stresses ρu v across the interaction region normalized with friction velocity u 2τ and ρw from measurements with an inclined wire
interpretation is confirmed by the results of Wu and Martin [16]. Furthermore, Souverein et al. [17] observed large Kelvin–Helmholtz type eddies arising in the mixing layer downstream of the reflected shock foot in different cases of reflected shock interactions at Mach 1.7–2.3. Those structures coincided with an increased u fluctuation level. 4.3 Energy content and characteristic frequencies in the boundary layer signal Figure 10 shows the power spectra of the hot-wire measured mass-flux fluctuations normalized to unity for measurement locations (1)–(4). In the upstream boundary layer [location (1), shown in Fig. 10a], the main part of the energy is contained between frequencies of 5–50 kHz, corresponding to dimensionless frequencies of f n = f ·δ/Ue = 0.067– 0.667,
Characterization of an incipiently separated shock wave/turbulent boundary layer
(a)
(b)
(c)
(d)
Fig. 10 Power spectra (normalized to unity) for the mass-flux signal measured with a hot wire for four streamwise locations on the model centerline along the SWTBLI region. a Incoming undisturbed boundary layer b Mean shock position at the wall. The local shock location
for c x = 3.33δ0 and d x = 5.47δ0 is at δy0 = 1.75 and δy0 = 3.22, respectively, and thus outside of the presented regions. Figure adapted from Schreyer [19]
respectively, with the respective local boundary layer thickness δ and velocity Ue at the boundary layer edge. Several researchers [12,38] observed large-scale turbulent structures with characteristic frequencies of the same order of magnitude in similar boundary layers. The high energy content in this frequency range thus suggests the presence of such structures also in the currently investigated boundary layer. A similarly large energy content in the same frequency range is observed well downstream of the interaction (see Fig. 10d), as well as a strong signal coherence for the frequency range in question (see Fig. 12d). The large-scale structures are thus also present downstream of the interaction. This was observed before by Ardonceau [13], Wu and Martin [16],
and Pirozzoli and Grasso [15] for separated SWTBLI. Wu and Martin [16] observed that the coherent structures in the incoming boundary layer decrease in streamwise extent across the interaction, probably due to compression, yet, they still remain in the boundary layer for a considerable downstream distance. From the high degree of qualitative similarity between the power spectra ahead and several boundary layer thicknesses (x = 5.47δ0 ) downstream of the interaction, shown in Fig. 10a, d, respectively, it can be inferred that the structures seem to not even loose their coherence under the strong perturbation experienced during the interaction with the shock wave.
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The intensity peak below 3 kHz ( f n ≤ 0.04) occurring equally strong for all measurement locations in the x- and y-directions in Fig. 10a–d is identical to the low-frequency wind-tunnel noise (see Fig. 4), as also observed in the experiment of Selig et al. [14]. It was checked that in the frequency range in question, the frequency spectrum at δy0 = 1.4 outside of the incoming boundary layer is very similar to spectra inside the boundary layer, even at the mean shock-foot location, as discussed previously in Sect. 2.3. The similarity of the signals in the freestream and in the boundary layer for these low frequencies makes it reasonable to assume that it corresponds to a contribution from outside the boundary layer and independent of the interaction. Since the phenomena of interest for this study occur at frequencies at least one order of magnitude above this noise signal, no detrimental effect on the results is expected. The observed dominant frequencies are also about one order of magnitude larger than the characteristic frequency of the large-scale shock motion typical for the present flow field [5,13,14]. Moreover, it is not likely that the low-frequency motion of the shock will generate structures in the same frequency range [18]; therefore, the behavior of this frequency range will not be taken into account in the present analysis.
4.4 Signal correlation and suggested turbulence-amplification and production mechanisms The presence and evolution of large-scale structures is explored by two-point measurements between the instantaneous wall heat flux q and the mass flux (ρu) in the flow, measured simultaneously and at the same respective streamwise survey stations. We thus explore the links between the events at the wall and in the boundary layer and derive properties of the large turbulent structures from these. Having in mind that the pressure, friction, and heat flux at the wall are strongly interconnected, the present measurements are a variant of the attempts of Brown and Thomas [9] and Selig et al. [14]. It is therefore expected that the correlation between these quantities will decrease with normal distance from the wall. Figures 11 and 12 show isocontours of the correlation coefficient and the coherence, respectively, between the wall heat flux and the mass flux in the boundary layer for four streamwise measurement locations. The correlation between the signals is relatively low in general, yet significant, maximum absolute values of 0.5 are reached. The strongest correlations or anticorrelations occur in the near-wall region ( δy0 ≤ 0.4) and outside of the boundary layer. To reduce noise in the representations, correlation values between −0.1 < R < 0.1 were filtered out. Note that the gray-scale values do not correspond to the same correlation levels in all images.
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Figure 11a presents the correlation signal in the incoming boundary layer (x = −3.33δ0 ). Note that with the sign convention used for the wall heat flux, the correlation is negative: positive ρu fluctuations are statistically associated with negative heat-flux fluctuations. Negative time delays correspond to events in the downstream flow, as can be seen from the definition given in Eq. 1. In the external flow (y/δ0 ≈ 1.2) upstream of the interaction, a zone of significant positive correlation with the events at the wall exists. Such behavior has also been observed in the experiment of Selig et al. [14] in a fully separated interaction. This correlated zone is most probably linked to the wind-tunnel noise, which dominates the signal in the outer flow and can be felt across the entire boundary layer. As discussed in Sect. 2.3, the spectra in the outer flow and at the wall are identical for the corresponding very low frequencies f ≤ 3 kHz (see Fig. 4). Although these fluctuations have a rather modest rms level (u /U ≈ 0.2 %), they are concentrated in a very narrow frequency band and are felt down to the wall. This means that in this frequency band, the events at the wall are under statistical dependence of the conditions imposed by the external flow. Otherwise, most of the energy in the boundary layer is distributed over larger frequencies. The wind-tunnel noise is thus easily distinguishable from the signal that we are interested in. Downstream of the ramp corner (x = 3.33δ0 , Fig. 11c), the correlation pattern resembles the pattern in the incoming flow (Fig. 11a): a zone of strong correlation for slightly negative t. There is also a zone of significantly positive correlation in the outer flow (y/δ = 1.15), which coincides with a strong coherence (≈0.9) for low frequencies ≤3 kHz (see Fig. 12a), and can thus be attributed to low-frequency wind tunnel noise as in the incoming boundary layer. In the shock-foot region (x = −0.63δ0 , Figs. 11b, 12b, respectively), this correlation pattern and the coherence for low frequencies vanishes inside the boundary layer δy0 ≤ 0.5. We think that this is due to signal domination by the strong disturbance from compression waves and shock-induced flow separation, the later being probably intermittent. The strong anti-correlation peak occurring at δy0 ≈ 1.1 (see Fig. 11b) is most likely linked directly to the shock wave. 4.4.1 Large-scale turbulent structures Upstream of the interaction (see Fig. 11a), there is a zone of strong correlation starting just above the wall for negative time delays, representing events located downstream of the ALTP transducer at the wall. The obvious interpretation is that it corresponds to a lump of fluid coming from the wallnear region and convected downstream: the signature (for the present measurement technique) of the large energetic turbulent eddies. A corresponding significant signal coherence for the appropriate higher frequencies [O (5–50) kHz,
Characterization of an incipiently separated shock wave/turbulent boundary layer
(a)
(b)
(c)
(d)
Fig. 11 Cross-correlation coefficient isocontours of the hot-wire signal across the boundary layer and the ALTP signal at the wall. Isocontours are presented for four streamwise locations along the SWTBLI region. Hot-wire and ALTP sensors are always located at coinciding
streamwise locations along the model centerline. The local shock locations are at δy0 = 1.75 for c x = 3.33δ0 and at δy0 = 3.22 for d x = 5.47δ0 . Figure adapted from Schreyer [19]
f n = 0.067–0.667] (see Fig. 12a) is also observed. Furthermore, there is a zone of significant correlation (≈−0.2) in the outer region of the boundary layer around 0.5 ≤ y/δ0 ≤ 0.65 for −270 ≤ t ≤ −70 µs. This corresponds to a streamwise distance of ≈14δ0 over which the fluctuating velocities are correlated and leave a characteristic footprint in the wall heat-flux signal—the pattern is understood as the signature of coherent structures in the incoming turbulent boundary layer that are convected downstream for distances of several boundary layer thicknesses before decaying. This behavior was previously described by Favre et al. [39]. For location (2), where the wall transducer is located at the mean position of the shock foot, the correlation signal has two main differences (see Fig. 11b): first, the overall level of correlation is weaker than that upstream, the signature of the large-scale eddies has disappeared, and the correlation pattern between the wall and the external upstream flow is
strongly disturbed. For this location, the hot wire is generally located in the upstream flow since it is traversed perpendicular to the wall, while the wall transducer is installed at the mean location of the shock foot. As the shock moves up- and downstream, the wall transducer senses either the upstream flow or the downstream, incipiently separated layer. The correlation pattern is thus a mixed signal and shows the influence of the shock motion. The low level of correlation is consistent with the findings of Dupont et al. [40], showing negligible correlation between the wall pressure at the shock foot and in the upstream flow. Second, a small zone of slightly positive correlation (attributed to events located upstream, t > 0) is formed close to the wall. This could be a contribution of locally and instantaneously developing separation pockets and the pulsation of these intermittent separation bubbles. Presumably, new large-scale structures are also forming in this zone, as has been observed in similar cases (see Sou-
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(a)
(b)
(c)
(d)
Fig. 12 Coherence spectra across the boundary layer for four streamwise locations along the SWTBLI region. Hot-wire and ALTP sensors are always located at coinciding streamwise locations along the model centerline. Adapted from Schreyer [19]
verein et al. [17], Piponniau et al. [18]). For the present case, a contribution of this mechanism is suggested by the energetic range extending up to f n = 0.6 around the location of maximum mass-flux rms ( δy ≈ 0.7, see Fig. 10c). This frequency corresponds to a Strouhal number of St ≤ 0.8, which is in the typical range for vortex shedding in these flow cases. Downstream of the ramp corner, the correlation pattern approaches the pattern in the incoming boundary layer. First at a slightly lower magnitude at x = 3.33δ0 (Fig. 11c), and further downstream at x = 5.47δ0 , the patterns look rather similar to that observed in the incoming flow (compare Fig. 11a, d), both in shape and magnitude, except for the absence of the correlated zone at t < 0. In general, the energy content for the frequencies associated with large-scale structures is increased across the shock
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(see, for example, Fig. 10c, where the magnitude for the frequency range 10–50 kHz is increased for 0.5 ≤ y/δ ≤ 0.9 compared with the incoming flow.) In the higher frequency range (40–70 kHz), this energy increase can be associated with the direct amplification of turbulence through the shock, and with newly generated structures of Kelvin–Helmholtz type that evolve in the mixing zone downstream of the shock foot. Such structures have been observed with a similar range of characteristic frequencies in reflected shock interactions by Souverein et al. [17]. They can be assumed to exist in the current flow field as well due to the typical similarity of compression corner and reflected shock interactions [3]. In the frequency range associated with larger structures (5–30 kHz), a contribution is expected from the gain of energy experienced by the structures coming from the incom-
Characterization of an incipiently separated shock wave/turbulent boundary layer
ing boundary layer due to the mixing process between highand low-momentum fluid caused by the vortices shed in the mixing zone. The frequencies characteristic for the large-scale turbulent structures have disappeared from the coherence spectrum for the shock-foot region (see Fig. 12b). Due to the strong similarity between the signals in the incoming boundary layer and downstream of the interaction in the corresponding frequency range, it is not probable that the structures get completely destroyed when passing the shock wave. Also, in the related frequency range, a considerable energy content is observed in the mass-flux power spectra even in the shock-foot region (see Fig. 10b), although at a larger distance from the wall than in the incoming flow due to the thickening of the boundary layer. It is thus more plausible that the correlation signal is
merely dominated by a spurious signal due to shock passage and the strong disturbance from the shock wave, as well as the separation bubble. These conclusions are backed up by correlation measurements shown in Fig. 13, where the wall signal in the incoming boundary layer was correlated with boundary layer traverses at four different streamwise locations. For easier comparison, we present the temporal delays corrected for the contribution due to the distance between the sensors by applying Taylor’s hypothesis, i.e., we subtracted |t = (xHW − xALTP )/Ue | from the measured delay t. The characteristic signal for the large-scale structures can be observed for all four locations, even in the shock-foot region. Using Taylor’s hypothesis and assuming that the convection velocity Uc of the large-scale vortical structures
(a)
(b)
(c)
(d)
Fig. 13 Cross-correlation coefficient isocontours of the hot-wire signal across the boundary layer and the ALTP signal at the wall. Isocontours are presented for four streamwise locations along the SWTBLI region. The ALTP sensor is always located at the same location
x = −3.33δ0 in the incoming boundary layer, while the hot-wire profiles are measured at different streamwise locations along the model centerline. The local shock locations are at δy0 = 1.75 for c x = 3.33δ0 and at δy0 = 3.22 for d x = 5.47δ0 . Figure adapted from Schreyer [19]
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A.-M. Schreyer et al. Table 4 Average structure angle Θ of the large-scale eddies in the boundary layer at a distance from the wall of δy = 0.5, where δ is the respective local boundary layer thickness Streamwise location x
−3.33δ0
3.33δ0
5.47δ0
Average structure angle Θ
81◦
83.5◦
72.5◦
is approximately equivalent to the respective local mean streamwise velocity component U , the average structure angle Θ can be estimated from the time delay τmax for which maximum correlation occurs, and the separation y in the wall-normal direction between the ALTP and hot-wire sensors [10]: Θ = tan
−1
y U τmax
.
The results for three locations across the interaction are summarized in Table 4. The structures are leaning towards the wall in the downstream direction. Compared with literature data, a value of 81◦ in the incoming boundary layer is high, but consistent with the findings of Spina et al. [11] and Alving et al. [41] from two-point hot-wire measurements in zero pressure gradient boundary layers. Downstream of the interaction, the structure angle slightly increases at first (83.5◦ at x = 3.33δ0 ) and then decreases considerably when coming closer to equilibrium conditions (to 72.5◦ at x = 5.47δ0 ). 4.4.2 Differences between incipiently separated and fully separated interactions In the vicinity of the separated zone, Selig et al. [14] reported zero correlation across the entire boundary layer in their fully separated 24◦ compression corner interaction at Mach 2.84. In the current incipiently separated experiment, the correlation map in the zone affected by the shock-position intermittency appears deeply disturbed, with remains from the signal in the incoming boundary layer. It is possible that the correlation signal in this region in fully separated interactions is dominated by shock-related effects to an even larger extent than in the present case, due to the larger shock strength and much larger separated zone. Large-scale structures would then still be present, yet not sensed by means of surface–boundary layer correlation measurements. However, since also the turbulence behavior is different in attached and fully separated cases—where turbulent memory effects were observed in the former, but not in the latter (see Mikulla and Horstman [42])—and incipient separation represents an intermediate state between those cases, this behavior may also be an indicator of changing flow characteristics. Mikulla and Horstman [42] assumed that either turbulent memory is destroyed in the separation process or that a new boundary layer forms during reattach-
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ment. In the case of an interaction with incipient separation, the observed signal could thus reflect a combination of turbulence residuals from the incoming boundary layer, the formation of a new boundary layer, and of course the direct disturbance due to the shock wave, such as vortices shed in the shock-foot region. Another interesting aspect is the shift of the coherence maxima towards lower frequencies in the shock-foot region. This may just be due to the fact that the signal is dominated by the motion of the small separation bubble, as discussed earlier, but not to the same extent over the entire frequency range. Yet, it might also indicate a local increase in turbulent structure length scale, a phenomenon that Smits and Muck [4] observed previously from the dip of the mean van Driest transformed velocity profiles below the log-law.
5 Conclusions To gain insight into the turbulence production and amplification mechanisms in incipiently separated shock wave/turbulent boundary layer interactions, the flow field along a two-dimensional 11.5◦ compression corner geometry was investigated at Mach 2.54. Large-scale structures were studied by means of two-point correlation measurements with a wall transducer and a hot wire traversed across the boundary layer. Based on the observed turbulence behavior, the spectral content of the mass-flux fluctuation signal across the boundary layer, as well as the evolution of the space–time correlation between the signals, a mechanism describing turbulence amplification and generation along the interaction flow field was suggested. A large part of the signal energy in the incoming boundary layer is for frequencies in the order of O (10–50 kHz), f n = 0.134–0.667, representing large-scale turbulent structures, which is in agreement with results in the literature for comparable boundary layers. At the mean shock-foot location, the traces of these structures are masked by the oscillatory motion of the shock and the intermittent separation bubble. Further downstream, they again dominate the coherence signal, although slightly altered: in the downstream flow, the energy content for the frequencies in question is increased, and the frequency range is widened towards higher frequencies (≤70 kHz). We explained this with Kelvin–Helmholtz type vortices arising in the mixing layer and being convected downstream. It was also proposed that these vortices create an energy exchange between the outer flow and the wall-near region, transporting high-momentum fluid from the outer flow into the inner region of the boundary layer in exchange for low-momentum fluid, which would add energy to the large-scale coherent structures from the incoming boundary layer.
Characterization of an incipiently separated shock wave/turbulent boundary layer
Around the location of the shock foot, a distinct family of time-delayed twin correlation peaks appears across the whole boundary layer. This correlation pattern indicates the direct influence of the shock that oscillates over a streamwise distance of about δ around its mean location. In the same flow region, a correlated zone arises close to the wall and disappears shortly downstream. Intermittent separation pockets are a possible candidate for this. The suggested mechanism is probably a feature of incipiently separated interactions, since for fully separated interactions, zero correlation in the direct vicinity of the interaction was reported in the literature. Turbulence downstream of incipiently separated interactions could thus be a combination of turbulence residuals from the incoming boundary layer, the formation of a new boundary layer, and of course the direct disturbance due to the shock wave, such as vortices shed in the shock-foot region. Acknowledgments This work was supported by the German Research Foundation (DFG) within the framework of the GRK 1095 “AeroThermodynamic Design of a Scramjet Propulsion System for Future Space Transportation Systems”. Uwe Gaisbauer and Lars Werner at IAG are gratefully acknowledged for their support during the measurement campaign.
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