Vol.:(0123456789) Flow, Turbulence and Combustion (2022) 109:667–696 https://doi.org/10.1007/s10494-022-00339-5 1 3 Particle Image Velocimetry Measurements in Accelerated, Transonic Wake Flows Judith Richter1  · Charalampos Alexopoulos1 · Bernhard Weigand1 Received: 6 October 2021 / Accepted: 23 June 2022 / Published online: 5 August 2022 © The Author(s) 2022 Abstract This paper reports on particle image velocimetry (PIV) measurements in compress- ible accelerated wake flows generated by two different central injector types, which are mounted in a convergent-divergent nozzle. The injectors differ by the extent of their trail- ing edge located either in the subsonic (injector A) or supersonic flow region (injector B). In addition, the undisturbed nozzle flow without injector is studied as a reference case. The PIV results reveal typical wake flow structures expected in subsonic (injector A) and supersonic (injector B) wake flows. They further show that the Reynolds stresses Rexx and Reyy significantly decay in all three cases due to the strong acceleration throughout the noz- zle. Interestingly, in the case of injector A, the flow stays non-isotropic with Reyy > Rexx also far downstream in the supersonic flow region. These measurements were motivated by the lack of velocity data needed to validate numerical simulations. That is why this paper additionally contains results from (unsteady) Reynolds-averaged Navier-Stokes ((U) RANS) simulations of the two wake flows investigated experimentally. The URANS simu- lation of the injector A case is able to accurately predict the entire flow field and periodic fluctuations at the wake centerline. However, in the case of injector B, the RANS simula- tion underestimates the far wake centerline velocity by about 4%. 1 Introduction Efficient and rapid mixing of two or multiple substances has been of scientific interest throughout the last decades motivated by various industrial applications, such as premix- ing the inflow for e.g. annular combustion chambers, (sc)ramjets, and chemical reactors. In many of these applications, continuous parallel mixing of fast flowing gases is imple- mented by utilizing a central injector. This involves a wake flow downstream of the injec- tor’s trailing edge, whose flow structures decisively determine the time and quality of mix- ing. That is why many researchers have studied such wake flows for many years. However, investigations have been conducted mostly for purely sub- or supersonic wake flows. In the case of subsonic flow, the wake that develops downstream of the central injector is * Judith Richter judith.richter@itlr.uni-stuttgart.de 1 Institute of Aerospace Thermodynamics, University of Stuttgart, Pfaffenwaldring 31, 70569 Stuttgart, Germany http://orcid.org/0000-0002-9237-1823 http://crossmark.crossref.org/dialog/?doi=10.1007/s10494-022-00339-5&domain=pdf 668 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 dominated by large-scale alternating vortices (see Fig. 1a) known as the von Kàrmàn vor- tex street (Roshko 1953; Gerrard 1966). In particular, while the shear layer rolls up at one side, the shear layer from the other side is drawn across the wake centerline, thus, forming a subsequent vortex. As a consequence, the supply of vorticity to the growing vortex is reduced (Gerrard 1966). Various studies have shown that the vortex shedding frequency scales with the main flow velocity over a wide Reynolds number range (Roshko 1954; Goldstein 1950). Furthermore, normalized velocity profiles were shown to attain self-sim- ilarity in the far wake regardless of the geometry of the wake generator (Wygnanski et al. 1986; Wohler et al. 2014; Richter et al. 2019). This also holds under varying pressure gra- dients (Liu et al. 2002; Beuting et al. 2018a). In the case of supersonic wake flow, compressibility stabilizes turbulent diffusion and, thus, restrains the wake growth rate (Papamoschou and Roshko 1988; Barre et al. 1994; Smits and Dussauge 2006). It has been experimentally demonstrated that compressibility effects have an attenuating impact on the growth rate of the shear layers and the turbulent structures (Papamoschou and Roshko 1988; Barre et al. 1994). In addition, the character- istics of supersonic flow are fundamentally different from those of subsonic flow regarding the direction of the information flows and the reaction to changes in pressure as sketched in Fig. 1b. In particular, expansion fans are formed at the injector’s trailing edge followed by an expansion and recompression zone, from which two oblique shock waves leave (Naka- gawa and Dahm 2005; Amatucci et al. 1992). Despite these huge physical differences both, subsonic and supersonic wake flows, have in common that the flow profiles attain a self- similar state at a sufficient distance downstream from the wake generator (Nakagawa and Dahm 2006). Whereas many wake flow investigations have been conducted in either subsonic or supersonic co-flows, wake flows in transonic co-flows as well as in strongly acceler- ated co-flows, that pass through the transonic flow regime, are still largely unexplored. Few studies have been conducted focusing on the transonic flow around turbine blades. These studies, though, have investigated the influence of the vortex shedding frequency on redistribution of pressure and temperature within the wake near field (Carscal- len et  al. 2009, 1996). Results have shown that the strength and the point of vortex shedding depend on the injection quantity and that vortex shedding can be even com- pletely suppressed by sufficient injection (Motallebi and Norbury 1981). More recent, Fig. 1 Typical flow structures of a subsonic wakes (Injector A) and b supersonic wakes (Injec- tor B) (a) (b) 669Flow, Turbulence and Combustion (2022) 109:667–696 1 3 investigations on transonic wakes have been performed in order to develop a shock-wave flow reactor to synthesize nanoparticles (Wohler et al. 2014; Grzona et al. 2009; Chun 2009; Winnemöller et al. 2010, 2015). In such reactors, the precursor gas is injected in the subsonic region of a Laval nozzle. The cold mixing of the precursor with the gas- dynamically cooled co-flow avoids uncontrolled nanoparticle growth. The decomposi- tion of the precursor is then activated globally by a shock wave and the associated jump in temperature. Previous laser-induced fluorescence (LIF) studies in our lab have dem- onstrated that, despite the extreme conditions within a strongly accelerated transonic nozzle flow, the self-similarity of the injectant concentration is preserved regardless of the injection angle, the shape of the central injector (with/without ramps), the flow Reynolds number as well as the imposed pressure gradient (Wohler et al. 2014; Beuting et  al. 2018a; Chun 2009). Furthermore, different measurement techniques have been applied in order to provide an experimental data set that can be used for the validation of numerical simulations (Richter et al. 2019; Beuting et al. 2018b; Richter et al. 2018). This includes measurements of wake growth rates, concentration profiles, shedding fre- quency, Mach number, temperature and the identification of zones of micro/macro mix- ing. Recent numerical studies showed that unsteady Reynolds-averaged Navier-Stokes (URANS) simulations, incorporating the shear stress transport model by Menter (1994), can be used to predict such wake flows. In particular, numerical results obtained with URANS showed good agreement with laser-induced thermal acoustic (LITA) measure- ments of the temperature and Mach number distribution of an initially subsonic wake, which undergoes strong acceleration through a convergent-divergent nozzle (Richter et al. 2018). However, the same investigation revealed that the predicted velocity defect in the near field was about 17% smaller compared to LITA. This points to the complex- ity of the vortex formation mechanism and the associated lack of URANS to correctly predict the base pressure. This assumption was underpinned by an underestimation of the shedding frequency by 15% . In addition, numerical investigations have shown that the vortex formation mechanism strongly depends on the free-stream Reynolds number as well as the amount of injectant added to the wake (Richter et  al. 2018). Thus, the accuracy of the experimental results used as validation data in numerical investigations is of central importance. This paper aims to complement the existing data of transonic wake flows by perform- ing 2D particle image velocimetry (PIV) experiments in the center-plane of two dif- ferent transonic wakes that undergo strong acceleration in a convergent-divergent noz- zle. The measured velocity field and turbulence statistics contribute to better understand such wake flows. Further, the complete data set (including the results of the present PIV experiments together with former LIF and LITA measurements) is valuable for the detailed validation of numerical simulations of future studies. In the present study, PIV experiments were carried out at the University of Stuttgart in a flow channel involv- ing a convergent-divergent nozzle and optical access throughout the whole flow regime (subsonic–transonic–supersonic). Two different central injectors were utilized as wake generators to cover both cases sketched in Fig.  1: (initially) sub- and purely super- sonic wake flow. In addition, an experiment was also carried out without an injector in order to accurately investigate the undisturbed nozzle flow as a reference case. Fur- ther, numerical results of (U)RANS simulations using the k-�-SST turbulence model are provided as an example on how the obtained data may be used for validation. Thus, the focus of the present paper remains on the experimental findings because a detailed numerical study would be a topic for itself. 670 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 2 Experimental Methods 2.1 Experimental Setup 2.1.1 Test Facility Experiments were carried out at the supersonic test facility of the Institute of Aerospace Thermodynamics (ITLR) at the University of Stuttgart. The experimental test facility is depicted in Fig.  2. It consists of a screw compressor, which supplies air into a channel passing first an air dryer and three electrical heaters. Downstream of the flow channel, the exhaust air is discharged into the environment through a large chimney. Four compressed air tanks with a total capacity of 8 m 2 at 100 bar serve as an emergency air supply so that the heaters can be shut down in a controlled manner if the screw compressor fails. The sys- tem can compress ambient air up to 10bar with a maximum flow rate of 1.45 kg s−1 while the air can be heated up to 1500 K. A vortex mass flow meter (Endress + Hauser, Prow- irl 77H DN 100, accuracy < 1% of the measured value) and a pressure sensor (Omega, PAA21-C-10, accuracy < 0.5% of full scale), which measure the air mass flow rate and the total pressure at the inlet of the test section, are also installed. 2.1.2 Flow Channel The very same flow channel that was used for the present PIV investigations has already been used in previous studies (Richter et al. 2019; Beuting et al. 2018a, b; Richter et al. 2018). The modular flow channel is depicted in Fig. 3, for further details see (Richter et al. (2019)). It consists of three modules of rectangular cross-section with a constant width of 40 mm and a total length of 665 mm. The first module connects the heater with the opti- cally accessible test section (modules 2 and 3). It is equipped with an seeding port that was utilized to add the particles required for the PIV measurements to the main flow (details see Sect. 2.3). A calibrated thermocouple (Type-K, accuracy ±1.5K ), which is positioned Fig. 2 ITLR supersonic test facility Fig. 3 Modular, rectangular flow channel with a convergent- divergent nozzle designed for transonic mixing investigations 671Flow, Turbulence and Combustion (2022) 109:667–696 1 3 at the flow channel centerline, was used to measure the total temperature of the main flow. A wire mesh is located between module 1 and module 2 to dissolve large-scale turbulent structures evolving upstream from the heater or the injection device. For this purpose, a grid with 0.75 mm wire diameter and 2.0 mm mesh size is utilized. The second module comprises (optionally) a central injector and the convergent-divergent nozzle with a nozzle throat height of 26.3 mm. The nozzle is designed to accelerate air to a Mach number (defi- nition see Eq. 6) of M = 1.7 , for details on the nozzle geometry see Richter et al. (2018). Module 2 further provides an additional access port, which was not used for the present experiments. The third module has a constant height of 35.4 mm and is a planar exten- sion of the second module that allows for the observation of the mixing layer development. Both, modules 2 and 3, provide optical access through quartz glass windows from all four sides. Furthermore, static pressure taps (Scanivalve, DSA 3016, accuracy < 0.05% of full scale), are installed at the top wall to measure the wall pressure distributions in flow direc- tion throughout the three modules. Subject of the present investigation are transonic wakes generated by the central injec- tor. Two different geometries of such are considered in this study. Both central injectors are drop-shaped and extend over the entire width of the channel. They vary only in the extension of their trailing edges as sketched in Fig.  4: Injector  A extends to the nozzle entry, 42 mm upstream of it’s throat, whereas injector B reaches down into the supersonic flow region, 10 mm downstream of the nozzle’s critical cross-section. Both trailing edges measure hITE = 5mm in height and contain four exit holes of 2.5 mm in diameter with 4.8 mm distance from center to center. The injectant is fed laterally through the injector hold- ing plates. 2.1.3 Flow Conditions In all three cases, oil particles were added to the main flow through the seeding port located in the first module. The total temperature of the main flow was set to T0,main = 380K at the outlet of heater 3 (see Fig. 2). The total pressure was p0,main = 2.5bar for the cases of Fig. 4 Geometry and position of the investigated wake generators/central injectors, as well as the location of the coordinate system in the center of the nozzle throat plane. All dimensions in mm 672 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 undisturbed nozzle flow and with injector A, whereas for the case of injector B was set to p0,main = 3bar so that the main mass flow rate ṁmain was maintained constant to compen- sate for the narrowed nozzle throat due to the extended trailing edge. The total temperature of the injectant T0,inj was equal to T0,main . In addition, the injectant mass flow rate ṁinj was adopted from LIF measurements (see Beuting et al. (2018b)). All flow conditions of the three cases investigated are summarized in Table 1. 2.2 Schlieren Imaging In order to visualize the dominant wake flow structures, Schlieren imaging, which has been widely used to resovle such phenomena, was applied for the cases of injector A and B. The Schlieren experimental set up was identical to the one used in previous experiments (Rich- ter et  al. 2017) consisting of a point light source (green LED) triggered by a driver cir- cuit (Menser et al. 2015), two achromatic lenses (focal length f = 1000mm ), an adjustable aperture, and an off-the-shelf single-lens reflex camera. The driver enabled an emission of a high-power short-time light pulse, where a light pulse of Δt = 5�s showed best results in terms of light output as well as blurring. 2.3 Particle Image Velocimetry (PIV) PIV was used to determine the velocity fields of all three cases (without and with injec- tor  A/B) in the mid-x-y plane of the flow channel, between the second and third injec- tor bore, at z = 0mm . The experimental setup is shown in Fig. 5: The beam of a pulsed Nd:YAG laser (New Wave Research, Gemini PIV 200-15, wavelength � = 532 nm) was Table 1 Flow conditions of the three cases investigated: undisturbed nozzle flow (nozzle), Injector A wake flow (InjA), and Injector B wake flow (InjB) T0,main p0,main ṁmain ṁinj T0,inj (K) (bar) (gs−1) (gs−1) (K) Nozzle 380 2.5 515 – 380 InjA 380 2.5 515 0.5 380 InjB 380 3.0 515 0.5 380 Fig. 5 Schematic (top view) of the PIV setup 673Flow, Turbulence and Combustion (2022) 109:667–696 1 3 formed into a laser light sheet that measured Δx ≈ 70mm in width and Δz < 0.5mm in thickness. This light sheet was aligned along the mid-x-y plane as sketched in Fig. 6. Since the light sheet is deflected by the strong curvature of the nozzle, a correction lens was used to expose the subsonic flow region sufficiently. The camera used was a 16-bit imager sCMOS from LaVision with a 2560 × 2160 pixel image resolution and a maximum capture rate of 50 Hz whereas, the particles were pro- duced of Di-Ethyl-Hexyl-Sebacat (DEHS) using an aerosol generator (Topas ATM 210). According to the manufacturer, the particle diameter is dP < 1𝜇m . To distribute the par- ticles evenly over the entire channel height, two circular injection pipes were used as sketched in Fig. 7 that inject the aerosol in the mid-x-y plane of module 1 through seven small slits each. The vortex shedding at the tubes causes very good mixing of the aerosol with the main flow. All PIV measurements and evaluations were performed with the commercial software DaVis 8 from LaVision (LaVision 2011). The time difference between the two light pulses was chosen between Δt = 1...3�s depending on the expected flow velocity, which corre- sponds to a mean particle shift of 3 to 10 pixel. At each measurement position, 1000 indi- vidual measurements were taken at 2Hz. The vector calculation was done by cross-corre- lation functions with iterative reduction of the interrogation windows from 64 × 64 pixel to 32 × 32  pixel with two iterations, 90% overlap, and automatic weighting. In channel coordinates, the final interrogation window measures 1.86 × 1.86mm2 and the resolution of the data points is 0.19 mm. For further post-processing, only measuring points with a cross-correlation peak ratio of CPR ≥ 100 were considered and outliers were automatically eliminated by |𝜑 − �̄�| < 2𝜎�̄� , where � is either u or v, “ ̄ ” denotes a time-averaged quantity, and � is the standard deviation. The resulting velocities were averaged over 250 individual measurements at each measurement point. Surplus measurement points were neglected to Fig. 6 3D view of the light sheet illuminating the wake flow 674 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 provide the same sample size everywhere because a varyinng sample size makes the inter- pretation of the statistical error difficult. 2.3.1 PIV Particle Response Time How well the PIV particles follow the investigated flow is commonly evaluated by the Stokes number which is a dimensionless measure defined as the ratio of the relaxation time of a particle �P and the characteristic time scale of the main flow �main . Particle motion studies in turbulent compressible shear layers have shown that if St < 0.1 the particles follow all small-scale movements (Krstić 2006; Samimy and Lele 1991). In supersonic flows, the discontinuity of the flow velocity across an oblique shock wave (OSW) can be utilized to accurately determine �P (Raffel et al. 2018; Scarano and Oudheus- den 2003; Menter 1997; Ragni et al. 2011). As is common knowledge, the velocity deceler- ates abruptly across an OSW. Due to their inertia, the particles cannot depict this disconti- nuity of the flow velocity and, thus, react position- and time-delayed. In order to evaluate these delays, the distribution of shock-normal velocity un perpendicular to the shock wave, along the direction s, is required. This is illustrated in Fig. 8 by means of an OSW that is located downstream of injector  B. Figure  8 also includes the corresponding normalized velocity ũn along the curve s as well as over the elapsed time  t. An exponential curve fit (solid lines) yields a particle relaxation length �P = 0.71mm and time �P = 2.51�s , which is similar to literature data (Scarano and Oudheusden 2003; Ragni et al. 2011) of similar cases. The flow time scale �main must be considered separately for injector A/B. In the case of injector A, the vortex shedding frequency, which measures 7.3kHz (Richter et al. 2019), defines �main,A = 137�s . In the case of injector  B, the time scale of a free shear layer � = 10�∕Δu (Samimy and Lele 1991) can be utilized (Scarano and Oudheusden 2003), where a worst case scenario is chosen with � = 0.73mm being half the shear layer thickness (1)St = �P �main , Fig. 7 Injection pipes used for PIV seeding. All dimensions in mm 675Flow, Turbulence and Combustion (2022) 109:667–696 1 3 at the recompression neck and Δu = (459 − 70)ms−1 being the velocity difference at the point of injection. Accordingly, 𝜏main,B < 18.8𝜇s . Adopting this assessment, StA = 0.02 and StB < 0.13 and, thus, all flow structures of the present PIV measurements can be resolved. 2.3.2 PIV Measurement Uncertainty Estimation In order to evaluate the measurement uncertainty both, the systematic and statistical error, have to be addressed. For the current case, the systematic error is mainly affected by com- mon parameters of the PIV setup such as seeding density, out of plane motion, sound to noise ratio and correlation window size. According to Raffel et al. (2018) two methods to quantify the systematic error have become established in literature: The first one requires knowledge of the effect of all individual parameters on the overall uncertainty. The second one considers the correlation functions only, which are a result of all parameters and their contribution to the uncertainty. The latter is based on the cross-correlation peak ratio CPR that is the ratio of the height of the largest to the second largest correlation peak. Figure 9 shows the results of two example correlation functions of two image pairs each (interrogation window 32 × 32  pixel) with different CPR. Charonko and Vlachos (2013) showed that for CPR ≥ 2 the root mean square displacement error already drops significantly. They studied the rela- tion between CPR and the displacement error u d⃗ by means of three artificial PIV signals Fig. 8 Particle response across a planar oblique shock wave (OSW): Velocity field of the shock-normal velocity compo- nent un (top) and corresponding normalized velocity ũn over the shock-normal abscissa s as well as elapsed time t = s∕un and their exponential curve fits (bottom) 676 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 representing different flow phenomena and deduced a relation u d⃗ (CPR) . For the present investigations, only results with CPR ≥ 100 were considered for post-processing. This strict criterion could be applied with about 50% scrap and was chosen to guarantee the best possible measurement accuracy. For the current case, the formulation by Charonko and Vlachos yields an systematic error of < 0.5% (a minimum total shift of 3 pixels as a basis). The number of individual measurements considered for averaging has a significant influence on the accuracy of the time-average, i.e. the statistical error. How large the sample size NSP has to be chosen in order to achieve the desired accuracy depends on the temporal fluctuations of the measured flow quantities and therefore depends on the flow phenomenon under investigation. In order to estimate the NSP required in each case, Uzol and Camci (2001) proposed the following method: From the total individual measurements NSP,max of a flow quantity φ , 100 randomly selected, statistically independent average val- ues from NSP ≤ NSP,max∕2 samples are taken to calculate a time-average �̄�: Then the dispersion of the calculated mean values is a measure of the accuracy of the time-average due to the selected NSP . This procedure is repeated for a number of chosen NSP. In order to investigate how NSP affects the time-averaging of the present PIV measure- ments, Eq. (3) was evaluated by means of the velocity components u and v as well as the Reynolds stresses (Eq. 8) Rexx and Reyy with NSP,max = 500 and NSP = 10, 25, 50, 100, 250 . Three different horizontal lines ( y = 0, 2.5, 5mm ) within the nozzle ( −45mm < x < 45mm ) were considered in order to identify the worst case scenario, which was located at the cen- terline ( y = 0mm ). The normalized results are shown in Fig. 10, where only the dispersions of v̄ and Reyy are plotted because the fluctuations in y-direction were dominant (cf. Fig. 17) and, thus, dictate the required sample size. As expected, the dispersions of v̄ and Reyy decrease with increasing NSP . This means that the statistical error of the present PIV measurements can be reduced by choosing an appro- priate large NSP . In the case of injector A v̄ scatters strongly close to the trailing edge. This directly results from the periodic character of the wake, which requires a larger NSP to accu- rately predict the time-average of the measured velocities. As the flow accelerates throughout (2)�̄�NSP = 1 NSP NSP∑ i=1 𝜑i (3)�̄�NSP,rms = √√√√√ 1 100 ( 100∑ j=1 �̄�NSP ,j − �̄�NSP,max )2 Fig. 9 Typical cross-correlation functions with a unique solution (left) and increased signal-to- noise with still strong correlation (right), evaluated by means of the cross-correlation peak ratio CPR 677Flow, Turbulence and Combustion (2022) 109:667–696 1 3 the nozzle, the influence of NSP quickly diminishes. Besides, the choice of NSP = 250 sig- nificantly reduces the statistical errors. On the contrary, the scattering of Reyy is independent of the x-position. Nevertheless, the statistical error can be also reduced to a similar low level when considering 250 samples for time-averaging. The main findings from the injector A wake flow also apply to the injector B wake flow. As a result, the choice of NSP = 250 yields very accurate time-averaged flow properties ( ̄u , v̄ , Rexx and Reyy ) with statistical errors of < 2% . Thus, the total measurement accuracy (system- atic and statistical error) of the present PIV measurements accounts to < 2, 5% everywhere for both, the flow velocities as well as the Reynolds stresses. This corresponds to a total measure- ment uncertainty of < 3, 5ms−1 for ū and v̄ composed of < 2.5ms−1 systematic and ≈ 1ms−1 statistical error. Please note that the measured standard deviations of the flow velocities not only originate from the statistical error, but also from turbulent and periodic oscillations of the wake flow, i.e. (4)𝜎�̄� = 𝜎�̄�,statistical + 𝜎�̄�,periodic + 𝜎�̄�,turbulent . Fig. 10 Sample size validation based on the v velocity compo- nent and Reynolds stress Reyy for a Injector A and b Injector B (a) (b) 678 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 Since 𝜎�̄� was at least one magnitude larger than 𝜎�̄�,statistical in all of our measurements, 𝜎�̄� is a sound measure of the turbulent and periodic flow oscillations in this case. That is why in the following figures, the plotted error bars represent these flow variations, while the meas- urement accuracy is not shown because it is comparatively low. 3 Numerical Methods Numerical investigations were performed with ANSYS CFX 19.0 (ANSYS 2019) for all three cases by solving the (U)RANS equations using the k-�-SST turbulence model developed by Menter (1994). Unsteady simulations were only feasible for the case of injec- tor A, because the flow structures in the supersonic wake of injector B feature significantly smaller time scales. Subsequently, with the computational resources available for the pre- sent study, no time-resolved simulation was possible for the injector B case. Figure 11 illustrates the hybrid meshes enclosing injector A and B, respectively. The mesh of the undisturbed nozzle flow case is similar to the injector B case but without the grid refinement in the wake flow region. The numerical domain was limited ( −180mm ≤ x ≤ 200mm ) in flow direction, which represents module  2 and approxi- mately one third of module  3. In addition, in the timely-resolved configuration with injector A, only a single injection bore of the central injector is considered in z-direction ( Δz = 4.8mm ) to reduce the computational cost (Richter et al. 2019). For the cases with Fig. 11 Sketch of the 3D numerical domain including the positions of applied boundary conditions and a section plane of the hybrid mesh of Injector A and Injector B. All dimensions in mm 679Flow, Turbulence and Combustion (2022) 109:667–696 1 3 injector B and without injector, the half channel ( Δz = 20mm ) could be considered as the simulations were considered stationary. A grid sensitivity study was conducted to define an appropriate number of nodes Nnodes for each case that involved two steps: Firstly, the number of nodes clustered in the wake region was increased by factor 2 and 4. Secondly, the influence of the wall resolution at the central injector was studied in terms of the dimensionless wall distance y+ 1,inj = u∗y1∕� , where u∗ is the friction veloc- ity, � the kinematic viscosity an y1 the height of the first cell near the wall. In addition, the time step �URANS was doubled to ensure an appropriate time resolution (applies only to the injector  A setup). The resulting Nnodes , y+1,inj and �URANS of this assessment are summarized in Table 2. All of these yield an overall relative error < 1% of the velocity field compared to the respective next closest setup. Wall functions were applied every- where, where y+ 1,inj > 30. All flow channel and injector walls were treated as no-slip, smooth walls. Periodic boundary conditions were applied to the sidewalls of the injector A setup, and the center planes of both other setups were defined as symmetry planes. Profiles of total pressure, turbulence intensity, and eddy viscosity ratio were applied as channel inlet boundary conditions, together with an inlet (constant) total temperature. 3D RANS simulations that covered module 1 were performed to provide the required profile data (Richter et al. 2019). In this preliminary simulations, the boundary conditions of the experimental flow conditions ( ṁmain , T0,main ) and wall static pressure at the test section inlet ( pmain ) were applied together with an inlet turbulence intensity Tu of 5% and an inlet eddy vis- cosity ratio �t∕� between the turbulent and molecular dynamic viscosity of 100 to deter- mine the subsonic flow conditions at the channel inlet. At the channel outlet, the super- sonic boundary condition of ANSYS CFX was applied, where all dependent variables are extrapolated from the flow conditions upstream (ANSYS 2019). At the injector inlet, ṁinj was set to 1/4 (injector A) or 1/2 (injector B) of the experimental condition (assum- ing even mass flow distribution over all four injector bores) and T0,inj = 380K. 4 Results and Discussion The following chapter is organized in four sections: First, the general flow features are presented in Sect.  4.1. This includes the characteristic flow properties, the character- istic flow structures and the wall static pressure distribution of all three cases. Sec- tions 4.2, 4.3 and 4.4 comprise a detailed discussion of the PIV measurements, where a separate section is devoted to each of the topics of undisturbed nozzle flow, injector A wake flow and injector B wake flow. Table 2 Details ob the numerical setups of the undisturbed nozzle flow (nozzle), Injector A (InjA), and Injector B (InjB) Method Δz Nnodes y+ 1,channel y+ 1,inj �URANS (–) (mm) (×106) (–) (–) (�s) Nozzle RANS 20 11.37 > 30 – – InjA URANS 4.8 5.35 > 30 < 1 0.5 InjB RANS 20 34.39 > 30 > 30 – 680 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 4.1 General Flow Features 4.1.1 Flow Properties The appropriate definition of characteristic dimensionless numbers such as Reynolds ( Re ) and Mach number ( M ) is essential because strong acceleration causes large gradients within the nozzle. In the present study, Re and M were obtained either at the boundary layer edge above the injector’s trailing edge (index: ITE) or at the nozzle exit (index: NE). Re is calculated from Eq. (5), where � , ū and � are the density, averaged velocity and dynamic viscosity, respectively. L refers to a characteristic length that is the height of the injector’s trailing edge hITE = 5mm or the hydraulic diameter at the nozzle exit dh,NE = 9.39mm . Equation (6) defines M , where ū and c are the averaged velocity and the speed of sound. Table 3 summarizes Re and M of all the three cases investigated obtained from results of the corresponding (U)RANS simulations. In the case of injector B, the height of the noz- zle throat is decreased due to the extended trailing edge, and as a consequence, MNE is increased. 4.1.2 Flow Visualization (Schlieren Imaging) Short-time illuminated Schlieren imaging was performed in order to identify the wake flow structures, sketched in Fig. 1, for the cases of injector A/B. Even though the photographs reveal nothin unexpected, Fig. 12 is used to briefly discuss the physics of the wake flows as a basis of the following discussions: The Schlieren image of the injector A wake flow nicely visualizes the vortices that shed from the trailing edge. These oscillate at a frequency of 7.3kHz (Richter et  al. 2017) and, thus, can only be resolved if the illumination time is sufficiently short. The vortices merge with the background within only few millimetres downstream of the injector trailing edge because the density fluctuations, which are visualized by the Schlieren technique, quickly dissolve as the wake develops. In addition, the Schlieren image of injector A nicely pictures some of the general nozzle features: starting at the nozzle throat, expansion fans evolve that should be ideally cancelled out by the noz- zle contour when hitting the walls. However, a weak shock system downstream of the (5)Re = 𝜚ūL 𝜂 (6)M = ū c Table 3 Reynolds numbers ( Re ) and Mach numbers ( M ) at the injector’s trailing edge position (index ITE) and at the nozzle exit (index NE) of the three cases investigated: undisturbed nozzle flow (nozzle), Injec- tor A (InjA), and Injector B (InjB) ReITE ×105 MITE ReNE ×105 MNE Nozzle – – 2.17 1.70 InjA 0.63 0.35 2.17 1.70 InjB 1.70 1.43 2.25 1.95 681Flow, Turbulence and Combustion (2022) 109:667–696 1 3 nozzle in the second viewing window ( x > 100mm ) proves that this is not the case. Nevertheless, the oblique shocks are weak and the nozzle is well designed. In the Schlieren image of injector  B the flow characteristics of a supersonic wake flow, cf.  Fig.  1b, are most dominant. The large dark colored areas at the upper/lower corners of the trailing edge nicely depict the expansion fans. The two visible lines, which start from the recompression neck at about x = 20mm , hit the channel walls and continue within the second window ( x > 100mm ), display the two oblique shock waves and their reflections. The Schlieren image also clearly pictures the wavy structure of the highly turbulent shear layers in the first window ( x > 20mm ). However, in the sec- ond window these coherent structures have been dissolved. Apparently, these vortices decay into small-scale eddies at higher frequencies that the used Schlieren setup cannot capture. 4.1.3 Wall Pressure Distribution Figure  13 depicts the static wall-pressure distribution normalized by the inlet total pressure p0,main (see table 1). The results nicely reveal the strong acceleration through the convergent-divergent nozzle for all three cases. In the case of injector B, the flow is comparatively further accelerated, due to the narrowing of the nozzle throat (see Table  3). Furthermore, the reflection of shock waves originating from the recompres- sion neck at the channel walls (see Fig. 12) causes periodic pressure conditions, which are clearly observed in Fig.  13. The positions of the local maxima of static pressure coincide with the crossing points of the reflected shock waves at the center line (see Fig. 12b). At these crossing points, the flow is locally compressed, which explains the local increase in static pressure. Figure 13 also reveals that the simulations are able to predict the pressure conditions very well throughout the whole flow channel. In addition, calculations assuming 1D, isentropic flow conditions (entropy S = const. ) identify MS=const and uS=const through the nozzle to provide expected conditions for a plausibility check and normalization of velocities: (a) (b) Fig. 12 Short-time illuminated Schlieren images of both investigated injector configurations, a Injec- tor  A and b Injector  B. Injector trailing edges are marked white at the very left edges. In the area 40mm ≤ x ≤ 100mm no windows are present (see Fig. 3) 682 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 From this assessment, a nozzle exit Mach number of MNE = 1.6 is calculated, which is slightly lower than the nozzle designed Mach number of MD = 1.7 . However, previous LITA measurements (Richter et  al. 2018) have proven that MD is reached at the nozzle center line. This difference can be justified by losses in p0,main , so that p∕p0 is actually higher than assumed. 4.2 Undisturbed Nozzle Flow 4.2.1 Time‑Averaged Centerline Velocity PIV measurements were carried out in the undisturbed convergent-divergent nozzle flow to study the main flow conditions without injector and to provide reference data. Figure 14 shows the measured velocities ūCL and v̄CL along the nozzle centerline (index: CL) from the current PIV experiments and RANS simulations together with previous findings from LITA meas- urements (Richter et al. 2018). Strong acceleration through the convergent-divergent nozzle flow can be clearly observed. The LITA results seem to be slightly shifted upstream, which might be due to difficulties in the alignment of the flow channel. Overall, excellent agreement between the two experimental methods (LITA, PIV) as well as the RANS simulations is seen. However, the streamwise velocity at the nozzle exit is about 10% higher than it was expected based on the 1D isentropic calculations. As mentioned before, this can be attributed to losses (7a) ( p0 p ) � �−1 S=const. = ( T0 T ) S=const. = 1 + � − 1 2 M2 S=const. (7b)uS=const. = MS=const. √ �RTS=const. Fig. 13 Normalized wall static pressure distribution of the undisturbed nozzle flow (Nozzle), Injector A (InjA) and Injector B (InjB) from measurements (EXP) and numerical simulations ((U) RANS) Nozzle throat 683Flow, Turbulence and Combustion (2022) 109:667–696 1 3 in total pressure, which falsify the assesment based on the isentropic flow assumption (as dis- cussed above), and to boundary layer effects, which reduce the effective cross-section. 4.2.2 Turbulence Statistics Figure 15 depicts the normalized Reynolds stresses (8)Rexx = u� rms uS=const , Reyy = v� rms uS=const Fig. 14 Measured (PIV, LITA) and predicted (RANS) velocity components ūCL and v̄CL along the nozzle centerline ( y = z = 0mm ) of the undisturbed nozzle flow. The bars denote velocity fluctuations due to turbu- lent oscillations. For details on the reference data from laser-induced thermo acoustic (LITA) measurements see (Richter et al. 2018) Fig. 15 Measured, normalized specific Reynolds stresses Rexx and Reyy of the undisturbed noz- zle flow along the channel center line ( y = z = 0mm ). PIV data are recorded with optimized time differences between the light pulses Δt depending on the mean flow velecoity. For details on the reference data from laser-induced thermoacoustic (LITA) measure- ments see Richter et al. (2018) 684 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 derived from PIV as well as LITA measurements (Richter et al. 2018), where u� rms and v� rms are the root mean square of the fluctuating (periodic and turbulent) part of the velocity with 𝜑� = 𝜑 − �̄� , and uS=const is the local velocity from 1D isentropic calculations (Eq. 7). Since uS=const ranges between 100 to 500ms−1 the optimum time difference between the two light pulses Δtopt depends on the position within the nozzle. In particular, Δtopt = 1�s applies to the nozzle exit area, whereas for the nozzle entry Δtopt = 3�s . That is why Fig. 15 contains results from three different PIV measurements with the respective Δtopt , which are sepa- rated by vertical lines. Both Reynolds stresses Rexx and Reyy strongly decay due to acceleration. Only exception is a local increase at about 27mm . The results from numerical simulation (not shown here) reveal that at this location the expansion fans, which originate at the nozzle throat, cross the center line. In addition, results from LITA measurements confirm the trend of decaying Rexx but show lower fluctuations through the nozzle flow. The LITA technique is based on the transport of an induced density grating and, thus, it is very susceptible to turbulence fluctuations. For that reason, it is prone to underpredict a high turbulence level. 4.2.3 Velocity Profiles Normalized velocity profiles are shown in Fig. 16 for six different streamwise positions, where x = 0 indicates the nozzle throat position. The results nicely show that, due to the strong acceleration from sub- to supersonic speed, a velocity deficit develops at the nozzle centerline. This velocity deficit, originating from the acceleration, is much stronger than the expected wake deficit. Due to the superposition of both, the velocity deficit caused by flow acceleration and the wake deficit generated by the central injector, the assessment of the wake deficit based on the velocity field was not possible. To thouroughly investigated the evolution of the wake deficit and the wake growth rate, LIF measurements exploiting a fluorescent injectant were performed in the past. The interested reader may thus be referred to former publications (Richter et  al. 2019) that address this topic in detail, whereas the present paper focuses on the wake velocity field only. Furthermore downstream, the flat profile at the nozzle exit ( x = 40mm ) indicates a well-designed nozzle. In addition, the predicted (RANS) velocity profiles agree very well with the measurements (PIV), meaning that all simulated values are within the range of turbulent fluctuations visualized as bars. As discussed above, the plotted error bars are a measure of the periodic and turbulent flow oscillations u� = u� periodic + u� turbulent since they are at least one magnitude larger than the measurement uncertainty (cf. Eq. 4). Here, in the absence of vortex shedding, they rep- resent u� turbulent only. These turbulent fluctuations decrease significantly while the flow is accelarated to supersonic speed. This is in accordance with the drop of Rexx and Reyy (see Fig. 15). 4.3 Injector A Wake Flow 4.3.1 Time‑Averaged Velocity Figure 17 displays measured (PIV) and predicted (URANS) time-averaged velocities ūCL , v̄CL at the wake centerline of injector A within the nozzle and the first window of module 3. The steep increase of ūCL clearly evidences the strong acceleration within the nozzle and how the wake deficit, caused by injector A, decays quickly within less than 22mm . In par- ticular, ūCL is initially reduced by 83.5ms−1 at the first measuring point ( x = −36.5mm ) 685Flow, Turbulence and Combustion (2022) 109:667–696 1 3 compared to the case of the undisturbed nozzle flow (plotted in grey) but catches up at x = −20mm , from where the two courses with/without injector match. Downstream of the nozzle, within module 3 ( x > 75mm ), ūCL slightly decays by about 25ms−1 ( −2.5% ) within 150mm . This deceleration can be accounted to a boundary layer thickening causing an adverse pressure gradient in combination with the constant cross- section of module  3 and to the weak shock system present throughout the entire flow channel in module 3 (see Fig. 12a). The former accounts for less than half of the decel- eration, as indicated by the displacement thickness �∗ of the numerical simulation, where �∗ increases from 0.28mm at the nozzle exit ( x = 45mm ) to 0.49mm at the end of the numerical domain ( x = 200mm ). The latter also provides the explanation for the observed local oscillations of ūCL. Since the used PIV setup could not capture the second viewing window at once, different colors used in Fig. 17 refer to different measurement positions, which nicely Fig. 16 Undisturbed nozzle flow: Profiles of time-averaged absolute velocity ūabs∕uS=const. normalized w.r.t. the isentropic flow velocity (Eq. 7) at dif- ferent x-positions within the convergent-divergent nozzle. The bars denote velocity fluctuations due to periodic and turbulent oscillations 686 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 overlap with deviations < 1% . This proves that the flow is quasi stationary because the measurement positions where recorded at different times (separated by several minutes). For the same reason, it further shows that the PIV measurements were repeatable. The URANS simulation were able to accurately predict ūCL (error < 1% ) within the nozzle for x > −20mm , where the wake deficit has already disappeared. However, upstream of this point, closer to the injector’s trailing edge, the URANS simulation underestimates the wake deficit (error < 10% , see zoom in Fig. 17). Unfortunately, PIV measurements were not able to capture the recirculation zone in the near field of the injector A with the current PIV setup. This is due to the following two limitations: First, since the particles are fed into the main and not into the injector flow, the unmixed region in the vicinity of the trailing edge is not seeded. The injector flow was deliber- ately not provided with particles because the particle mass flow could not be controlled and, thus, could affect the wake flow in an uncontrolled manner. Second, the injector is made of copper and causes tremendous refractions so that the trailing edge was heavily overexposed. Within module  3 ( x > 75mm ) the predicted ūCL is slightly higher than the measured and does not exhibit any oscillations. The authors contribute this to the reduced numerical domain in z-direction omitting the side walls and manufacturing inaccuracies of the nozzle contour. Predicted (URANS) and measured (PIV) vertical velocity fluctuations v� CL are depicted as bars in the second plot of Fig. 17. They are initially large due to the vor- tex street that forms at the injector  A trailing edge (see Figs.  1 and  12a) and thus induces considerable periodic velocity fluctuations in y-direction. These fluctuations Fig. 17 Injector A wake flow: Measured (PIV) and predicted (URANS) time-averaged velocity components ūCL , v̄CL and Reynolds stresses Rexx , Reyy along the channel centerline ( y = z = 0mm ). The bars refer to the standard deviations 𝜎ū , 𝜎v̄ originating from turbulent and periodic oscillations of the wake flow. Different colors refer to different (overlapping) measurement positions 687Flow, Turbulence and Combustion (2022) 109:667–696 1 3 quickly decay while the initially large vortices decompose into smaller eddies and the flow accelerates to supersonic speed. URANS simulation is able to correctly predict the amplitude of v� CL , whereas it slightly underestimates u� CL . Note that in the case of URANS simulations u� CL and v� CL only depict periodic fluctuations. From the fact that, despite this limitation, the velocity fluctuations are well predicted, the authors conclude that the periodic fluctuations are dominant while the turbulent fluctuations are com- paratively minor. This has been observed before, e.g. in the periodic wake of a square cylinder (Ke 2019). A quantitative analysis of the share of periodic and turbulent fluc- tuations, however, would require time-resolved experimental data (e.g. from high-speed PIV or hot wire anemometry) to apply a Fourier transformation. Further, the contour plot of the entire velocity field (Fig. 18) shows that not only the centerline velocities, but also the entire flow field is very well predicted by the URANS simulation with an overall deviation of < 5% (Fig. 18c). The only difference would be that the flow field of the URANS (Fig. 18b) is perfectly symmetric, whereas local fluc- tuations were present in the experiment (Fig. 18). 4.3.2 Turbulence Statistics Normalized Reynolds stresses ( Rexx , Reyy ) also depict the impact of the strong accel- eration on the flow: Rexx and Reyy are initially extremely high due to periodic velocity fluctuations induced by the von Kàrmàn vortex street. Both rapidly decay by more than one order of magnitude while the flow is accelerated to supersonic speed. Such rapid decay in turbulence intensity may cause a laminarization process in the wake and inhibit (a) (b) (c) Fig. 18 Injector A wake flow: Contour plots of time-averaged absolute velocity ūabs : a PIV measurements, b URANS simulation, and c) relative deviation between PIV and URANS. The black frames outline the geometries of the two side windows; the injector trailing edge is sketched in gray 688 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 the mixing (Winnemöller et al. 2015). If laminarization occurs, this can be evaluated by means of the acceleration parameter Laminarisation becomes of significance if K > 2 × 10−6 ; for K > 1 × 10−5 the reversion to a laminar flow is considered to be complete (Yang and Tucker 2016). For the current cases, Kmax ≈ 1.8 × 10−6 within the convergent-divergent nozzle, thus, the flow stays turbulent. Nevertheless, this causes a strong decay in turbulence level. In the second viewing window, despite the strong acceleration, Reyy remains about twice as high as Rexx . This is remarkable because it seems that Reyy withstands, meaning that the vortex shedding causes a non-isotropic flow field far downstream in the super- sonic flow region. From this the authors conclude that the streamwise acceleration pre- liminary acts on the streamwise normal stresses, whereas the lateral normal stresses are less affected. This is in accordance with the rapid distortion theory (RDT) (Narasimha and Sreenivasan 1979), which applies to initially isotropic flows under a strong favora- ble pressure gradient. Here, turbulent eddies are “distorted”, resulting in a decrease of the normal stresses in the direction of the acceleration, while the lateral normal stresses may be further enhanced. However, in the current situation this effect is partly covered by the von Kàrmàn vortex street entering the convergent divergent nozzle, which pro- vides already dominant normal stresses at the inlet of the nozle. 4.3.3 Velocity Profiles Figure  19 depicts predicted (URANS) and measured (PIV) velocity profiles at differ- ent streamwise positions. An overall very good agreement between URANS simula- tions and PIV results is seen, meaning that all simulated values are within the spatial uncertainty of the PIV setup (marked by the line thickness). The only exception is the measurement position closest to the injector trailing edge, at x = −35mm , where the predicted deficit velocity deviates about 10% . This underpins our previous conclusions from LITA measurements (Richter et al. 2019). More specifically, LITA results showed that the centreline Mach number deficit was underestimated by the URANS simula- tion by about 17% . From this, one can conclude that the onset of vortex shedding is not entirely captured by our URANS. This shortcoming could neither be improved with a twofold finer grid resolution nor with halving the time step, both of which were inves- tigated with a grid sensitivity study. Nevertheless, a much higher grid resolution and a much smaller time step combined with the appropriate computer resources maybe could improve the results. A comparison with the velocity profiles of the undisturbed nozzle (see Fig. 16) shows that the wake deficit vanishes latest at the nozzle throat ( x = 0mm ). In addition, Fig. 19 nicely displays how the velocity profile flips due to the strong acceleration, indicating that the velocity deficit is much stronger than the wake deficit. Subsequently, an assess- ment of the wake deficit based on velocity profiles is highly difficult. That is why we assessed intensity profiles of LIF instead in our previous studies (Richter et  al. 2019; Beuting et al. 2018a, b; Richter et al. 2016). From these we were able to prove the self- similarity of the wake flow. (9)K = 𝜂 𝜌ū2 CL dūCL dx . 689Flow, Turbulence and Combustion (2022) 109:667–696 1 3 4.4 Injector B Wake Flow 4.4.1 Time‑Averaged Velocity Figure  20 displays the measured (PIV) and predicted (RANS) velocities ūCL and v̄CL in the wake of injector  B. Unfortunately, no PIV results could be obtained upstream of x = 25mm for the same reasons discussed above for injector A (main flow seeding and light refraction), thus, missing the first 15mm of the wake. That is why the recir- culation zone, which typically forms in the very vicinity of a blunt body’s trailing edge in supersonic co-flow (see Fig.  1b), was not captured. In addition, the measured ūCL at x > 25mm does not undergo further acceleration, meaning that the wake deficit has already catched up. This indicates that the recirculation zone must be short and, thus, sufficient mixing already took place until the first PIV measurement point only 15 mm Fig. 19 Injector A wake flow: Profiles of time-averaged velocity ūabs∕uS=const. normalized w.r.t. the isentropic flow velocity (Eq. 7) at different x-positions within the convergent-divergent nozzle. The bars denote velocity fluctuations due to periodic and turbulent oscillations 690 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 downstream of the injector trailing edge. This assessment agrees with the position of the recompression neck at x < 20mm (see Fig. 12b) and also with previous LIF results (Beuting et al. 2018b), which found that the zone of incomplete mixing extends 16 mm downstream of the point of injection. In addition, the fact that only the main flow was seeded with particles also underpins that, wherever PIV results can be obtained, suffi- cient mixing between the main and injector flow must have been completed. The measured ūCL-distribution in the second window ( x > 75mm ) is shaped by the recompression wave intersections, which cause alternating abrupt deceleration fol- lowed by a continuous re-acceleration. Here, the positions of locally minimum ūCL mark the positions where the reflected shock waves cross the channel center line. These agree with the shock wave cross sections in the Schlieren image (see Fig. 12b, 100mm < x < 250mm ). Different colors, as in the case of injector A, refer to different measurement positions. Again, the results from different positions overlap and, thus, verify the stationary flow field and repeatable PIV measurements. In the absence of vortex shedding, error bars of ūCL and v̄CL represent turbulent fluctuations only in the wake of injector  B. Nevertheless, the fluctuations are clearly enhanced by the tubulent character of the wake within the nozzle; at the nozzle exit they are of about the same magnitude as in the case of injector A. However, in contrast to the injector A wake, the turbulence level decays further from about 0.02 to 0.01 within the second viewing window. Figure 20 further reveals that the RANS simulation fails to correctly predict the cen- terline velocity: The predicted ūCL underestimates the PIV measurements by a constant Fig. 20 Injector B wake flow: Measured (PIV) and predicted (RANS) time-averaged velocity components ūCL , v̄CL and Reynolds stresses Rexx , Reyy along the channel centerline ( y = z = 0mm ). The bars refer to the standard deviations 𝜎ū , 𝜎v̄ originating from turbulent and periodic oscillations of the wake flow. Different colors refer to different (overlapping) measurement positions 691Flow, Turbulence and Combustion (2022) 109:667–696 1 3 offset of about 20m/s in module 3 ( x > 75mm ). The contour plot of the absolute veloc- ity ūabs (Fig. 21) shows that the mixing between the main and the injected flow is appar- ently underestimated. This large deviation results from differences in the vertical mean flow velocity  v̄ itself. This is a typical shortcoming of the RANS approach, which is known to suppress most of the unsteady structures of the flow (Luo et  al. 2015). As a consequence, the high frequent turbulent structures that strongly enhance the mix- ing at the shear layer edge are suppressed in the simulation. This shortcoming becomes particularly apparent at the nozzle exit ( x > 25mm ), where high Reynolds stresses of comparable magnitude with the case of injector A point to the existence of small-scale turbulent structures. 4.4.2 Turbulence Statistics The turbulent character of the wake is also reflected in the Reynolds stresses Rexx , Reyy . Both are elevated near the injector trailing edge and decay to < 2% in the supersonic flow region of module 3. Interestingly, Rexx , Reyy are level in this case. This seems to result from the supersonic wake characteristics that do not favour a direction of disturbances (no peri- odic shedding). 4.4.3 Velocity Profiles Figure 22 depicts velocity profiles at different streamwise positions in the injector B wake. The wake deficit quickly decays as the flow is accelerated within the nozzle but remains, though very small, throughout the test section. Furthermore, as stated earlier, RANS (a) (b) (c) Fig. 21 Injector B wake flow: Contour plots of time-averaged absolute velocity ūabs : a PIV measurements, b URANS simulation, and c) relative deviation between PIV and URANS. The black frames outline the geometries of the two side windows; the injector trailing edge is sketched in gray 692 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 simulation overpredicts the wake deficit. However, outside of the wake the velocities pro- files coincide, meaning that all simulated values are within the range of turbulent fluctua- tions visualized as bars. 5 Summary and Conclusions PIV measurements were conducted to experimentally investigate the velocity profiles and turbulent statistics of two different transonic, accelerated wakes generated by cen- tral injectors. The investigated injectors differed by the extent of the trailing edge that was either located upstream (injector A) or downstream (injector B) of the throat of a convergent-divergent nozzle. The main difference between the two wake flows is that the (initial) subsonic wake of injector A is characterized by periodic vortex shedding Fig. 22 Injector B wake flow: Profiles of time-averaged velocity ūabs∕uS=const. normalized w.r.t. the isentropic flow velocity (Eq. 7) at different x-positions within the convergent-divergent nozzle. The bars denote velocity fluctuations due to periodic and turbulent oscillations 693Flow, Turbulence and Combustion (2022) 109:667–696 1 3 and an increased turbulence level in the y-direction, whereas the supersonic wake of injector B is characterized by oblique shock waves and their periodic interaction with the wake. This leads to a very different behavior of the wake: whereas the induced crossflows of the vortex street help to rapidly reduce the wake deficit downstream of injector A, the periodic shock wave/wake interaction downstream of injector B locally narrows the wake, so that the wake deficit persists far downstream. In addition to the two wake flows, the undisturbed nozzle flow (without injector/ wake) was also investigated as a reference case. Furthermore, (U)RANS simulations using the k-�-SST model were performed to identify to what extend this model is able to correctly predict the time-averaged wake flow field. The major conclusions can be summarized as follows: Undisturbed nozzle flow – Rexx and Reyy strongly decay from about 8% to < 2% due to acceleration to supersonic speed. – Profiles of ūabs show that a velocity deficit at the nozzle centerline develops also in the absence of a wake generator. – RANS simulation is able to accurately predict the velocity field. Injector A wake flow – The strong acceleration attenuates the wake deficit, which only remains for a dis- tance of 4hITE . At this point the wake centerline velocity catches up with the center- line velocity of the reference case (undisturbed nozzle flow). – The flow stays non-isentropic also in the wake far field, where Reyy > Rexx. – URANS results are able to qualitatively capture the periodic vortex shedding but deviate from the experimental results near the injector trailing edge. Apart from this, the time-averaged velocity field is very well predicted and may be used for, e.g., evaluating the flow conditions and boundary layer growth. Injector B wake flow – The wake deficity decays quickly while the main flow is accelerated but consists downstream of the convergent-divergent nozzle, throughout the investigated super- sonic flow region ( > 380hITE). – ūCL is shaped by the shock wave interactions, which cause alternating deceleration and re-acceleration. – Conventional RANS approach suppresses most of the mixing of the injectant with the co-flow resulting in an underestimation of the centerline velocity by about 4% . Apart from this, the time-averaged velocity field is very well predicted and may be used for, e.g., evaluating the flow conditions and the positions of the shock waves. The present results complement the existing data of transonic wake flows by the velocity field and turbulence statistics. Together with previous experimental investigation (Chun 2009; Wohler et al. 2014; Beuting et al. 2018a, b; Richter et al. 2019), which include measurements of wake growth rates, concentration profiles, shedding frequency, Mach number, temperature and the identification of zones of micro/macro mixing, the pre- sent results provide valuable validation data for numerical investigations. In the future, 694 Flow, Turbulence and Combustion (2022) 109:667–696 1 3 detailed large eddy simulations of the present cases (injector A/B) are planned. These simulations can then provide further inside into the physics of injectors A and B wake flow behavior. Supplementary Information The online version contains supplementary material available at https:// doi. org/ 10. 1007/ s10494- 022- 00339-5. Acknowledgements Many thanks goes to Aleksandra Stajic who supported the PIV measurements during her summer school at ITLR. The computer resources for this project have been kindly provided by the Stein- buch Centre of Computing (SCC) at the Karlsruhe Institute of Technology (KIT). Author Contributions JR conducted the PIV experiments, processed the data, created the plots and dia- grams, and wrote the first draft of the manuscript. BW supervised the research and contributed to the inter- pretation of the data. CA contributed to the writing of the paper and created Figs. 11 and 12. All authors commented on previous versions of the manuscript and read and approved the final manuscript. Funding Open Access funding enabled and organized by Projekt DEAL. The authors kindly acknowledge the financial support of this work by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) through the research project “Experimental and Numerical Mixing Investigations in a Compressible Nozzle Flow” (WE 2549/31-3). Data Availability All processed, time-averaged PIV data are available for download as supplemental material. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com- mons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. References ANSYS, Inc. 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