Direct velocity measurements in high-temperature non-ideal vapor flows

Compressible flows of fluids operating at thermodynamic conditions close to the vapor saturation line and the critical point are of interest in various applications in the oil and gas sector, in the chemical process industry, and in the energy field, including power systems as organic Rankine cycles (ORCs), supercritical carbon dioxide cycles (sCO\(_2\)), steam Rankine cycles and high temperature heat pumps. These are vapor or supercritical fluid flows, occurring at moderate to high temperature, including supercritical states, in components which are typically unconventional turbomachines, valves, heat exchangers, and pipelines. Such flows belong to the domain of non-ideal compressible fluid dynamics (NICFD), which also includes near-critical and two-phase fluid flows.

In the last two decades, considerable advancements were made in the thermodynamic modeling of fluids at highly non-ideal conditions, where non-ideal refers to deviation of the fluid behavior from the ideal gas law \(Pv=RT\) (Span et al. 2001). Also, high-fidelity computational fluid dynamics (CFD) tools specifically conceived for highly non-ideal flows (Guardone and Vigevano 2002; Colonna and Rebay 2004; Cinnella and Congedo 2005; Vitale et al. 2015), and implementing state-of-the-art thermodynamic models (Span and Wagner 2003; van der Stelt et al. 2012; Thol et al. 2016), were developed and largely applied to perform fundamental studies and to design and analyze performances of the above described components.

Conversely, experiments required to verify theoretical findings and assess the accuracy of CFD tools are relatively recent, as a consequence of the hostile environment in which measurements have to be performed, which requires dedicated facilities to run the tests and specifically developed measurement techniques. Temperature measurements are obtained through thermocouples or resistance thermometers (depending on the required time resolution), while directional pressure probes provide, once calibrated, static and total pressure at the measuring point and the flow velocity via the fluid thermodynamic model, if the total temperature is measured. The stagnation pressure is of relevance if an evaluation of losses is required, as it is the case of components as turbomachines, valves, and pipelines. Differently from the perfect gas case, in non-ideal flows directional probes require a calibration procedure which is both fluid specific and thermodynamic condition specific. As an alternative, numerical calibration procedures can be adopted. Such techniques are under development, but not yet available if a high degree of accuracy is required. However, the pressure field can be measured for isentropic flows as the total pressure is conserved and the static pressure is measured through wall pressure taps. This is for instance the case of nozzle flows and uniform flows before/past oblique shocks and expansion fans, where the non-ideal pressure field was successfully measured by the authors in previous experimental campaigns (Spinelli et al. 2018; Zocca et al. 2019). As it is well known, the velocity field can be inferred from pressure and temperature field, by resorting to the thermodynamic model of the fluid. Indeed, since the velocity is obtained indirectly, its uncertainty is larger than the one of pressure and temperature. The uncertainty of the thermodynamic model should also be considered, even though it is negligible with respect to the one of operating conditions, if state-of-the-art thermodynamic models are adopted, see Congedo et al. (2011, 2013), Geraci et al. (2016) and Merle and Cinnella (2015).

In the field of non-ideal compressible flows, the above considerations raised the interest toward measurement methods which do not require calibration within reference flows, and specifically toward optical techniques such as schlieren imaging, laser Doppler velocimetry (LDV) and particle image velocimetry (PIV).

The schlieren technique (Spinelli et al. 2016, 2017; Zocca et al. 2018, 2019) is of relatively simple application in high-temperature non-ideal flows, but it is essentially qualitative, even though it can be exploited to measure the local Mach number, from Mach lines, and shock wave/expansion fan geometry (position, slope) (Cammi et al. 2021). However, this method can be used only in supersonic flows and with limited accuracy; moreover, to retrieve the velocity field from the Mach number one requires the application of the thermodynamic model, and thus the knowledge of at least two thermodynamic properties at the measuring point. Contrarily, both PIV and LDV techniques guarantee the direct measurement of the flow velocity, providing that the flow is properly inseminated, namely that the velocity slip between the fluid and the tracking particles is considerably small. Such velocity measurements are thus considered to be direct, oppositely to those inferred from temperature and pressure measurements via the fluid thermodynamic model, here referred to as indirect velocity measurements.

Direct velocity measurements complement the flow characterization performed through pressure and temperature measurements and are highly valuable in non-ideal flows, especially for components whose operation is strictly related to the velocity field, such as turbomachines, heat exchangers and pipelines. Moreover, a comparison between velocity either measured or inferred from pressure and temperature data via the thermodynamic model allows to assess the attainable accuracy in providing the velocity field of directional pressure probes, which are still under development for non-ideal flows. These reasons motivate the opportunity of measuring the velocity field though techniques which do not require fluid-specific and/or thermodynamic condition-specific calibration.

The flow seeding is particularly critical due to the need of introducing particles in a high temperature, high pressure and possibly condensing flow; also, fluid contamination must be avoided. Solid particles, chemically compatible and insoluble in the working fluid, are the only viable option, due to the high temperature they need to withstand to without significantly altering their dimension and density.

To the authors’ knowledge, no experimental studies about direct velocity measurements in high speed non-ideal flows were found in the literature up to date. A recent research (Valori et al. 2019) applied PIV to an unconventional vapor in highly non-ideal state. However, the analyzed conditions were low temperature and high pressure, and the study dealt with natural convection, thus the flow field featured extremely low velocity. Flows to be studied in the campaign presented here feature at the same time high temperature, high pressure and high velocity, thus making LDV (or PIV) implementation not straightforward.

Though a pointwise technique, LDV features, with respect to PIV, an easier implementation (the laser emitter and the receiver may be placed in the same device) and a higher attainable spatial and temporal resolution. For these reasons, LDV technique (extensively described in Albrecht et al. 2003) was selected in this work to obtain direct velocity measurements capable of providing, with temperature and pressure data, a complete and high spatial resolution characterization of high temperature non-ideal vapor flows.

In this work, experiments were carried out on isentropic, non-ideal expanding flows of hexamethyldisiloxane vapor (MM), which target operating conditions relevant to typical ORC turbine applications. The flow is characterized at selected locations along the axis of planar nozzles at different Mach number and thermodynamic conditions, from subsonic to supersonic regimes. Flow description is obtained by measuring total pressure and temperature, static pressure, and velocity, both directly measured through LDV and inferred from measured pressure and temperature via a suitable thermodynamic model.

Three different flows are analyzed: a subsonic flow at \(M=0.7\), with almost zero velocity gradient, a supersonic flow at \(M=1.7\), in a region of near-zero velocity gradient, and a supersonic flow at \(M=1.4\), in a region of large velocity gradient. The experimental campaign was carried out, at the CREA Laboratory of Politecnico di Milano (Italy), on the Test Rig for Organic VApors (TROVA), a blow-down wind tunnel for organic vapors in non-ideal compressible flow conditions. The seeding particles selected for flow tracing are \(0.2\,\upmu \mathrm {m}\) diameter titanium dioxide (TiO\(_2\)) particles, and a specifically conceived seeding system was designed and implemented. A liquid suspension of the tracer particle in the fluid to be tested is injected by an atomizer into the main flow at high temperature and pressure. The injection point is located in a plenum ahead of the test section. Due to the high temperature, the liquid fraction evaporates and solid particles trace the flow, allowing to measure the flow velocity.

At the measuring points, the directly measured velocity was compared to both velocity inferred from pressure and temperature measurements and to the one extracted from CFD calculation. This allowed to assess the consistency of different data sets and thus verifying the reliability of the implemented seeding system and the applicability of LDV in highly compressible and non-ideal vapor flows occurring at high temperature.

The paper is organized as follows. The experimental facility instrumentation and measurement techniques for pressure and temperature are described in Sect. 2. Section 3 presents experiments and nozzle design, while Sect. 4 discusses the choice of the tracer particle, the design of the seeding system and its operation. The employed LDV system is also presented. Section 5 describes velocity signals processing, while results are presented and discussed in Sect. 6. Finally, conclusions are drawn in Sect. 7.

The experimental facility employed for this study is the Test Rig for Organic VApors, TROVA, a blow down wind tunnel designed to characterize non-ideal compressible flows of organic fluids of different nature; a detailed description of the facility can be found in Spinelli et al. (2013). To achieve the complete characterization of a fluid flow, the TROVA was conceived for the independent measurement of total pressure \(P_T\), total temperature \(T_T\), static pressure P and fluid velocity \(V_f\).

Tests at a maximum pressure of \({50}\,{\mathrm{bar}}\) and maximum temperature of \(400\,^{\circ }\mathrm{C}\) can be performed, providing that the working fluid does not undergo substantial decomposition at such temperature level. The fluid selected for the present campaign is siloxane MM, due to its widespread use in high-temperature ORC systems and the presence of a wide thermodynamic region where non-ideal effects are marked and thermal stability is preserved.

The TROVA implements a batch organic Rankine cycle, with a fixed test section replacing the turbine. The desired superheated or supercritical test conditions are reached by a isochoric heating of the fluid under scrutiny within a high pressure vessel, HPV. The tank pressure is higher than the test section stagnation pressure \(P_{T}\), which can be regulated through a control valve and measured, along with total temperature \(T_{T}\), in a plenum upstream of the test section. In all experimental runs, the test section is equipped with a planar nozzle. The fluid expands through the nozzle, where static pressure along the axis is measured; also, an optical access allows to perform LDV measurements and schlieren visualizations of the flow field. The exhausted vapor is discharged and condensed into a low pressure vessel, and liquid compression to the HPV through a metering pump closes the loop.

In the test section, the nozzle is obtained through a pair of steel contoured walls, shaped according to the test conditions. The frontal wall is provided by a planar quartz window, which represents the optical access, while in the rear steel plate a series of 0.3 mm pressure taps is machined. Both the test section and the plenum are electrically heated to avoid vapor condensation. Figure 1 reports a sketch of the two nozzle profiles employed as well as the location of pressure taps (diameter not in scale).

The nozzle characteristic time is more than two orders of magnitude lower than the HPV emptying process time (Spinelli et al. 2013; Pini et al. 2011), and thus the investigated flow can be regarded as a quasi-steady state one at each time instant. Therefore, several steady flow fields at different times and levels of non-ideality can be conveniently extracted from data acquired during the unsteady operation of the facility, since the employed instrumentation exhibits a sufficiently high-frequency response. As a practical measure for the level of non-ideality, the compressibility factor in stagnation conditions \(Z_T\) is used, where the compressibility factor is \(Z = Pv/RT\), with v specific volume and R gas constant. A more detailed description can be found in Spinelli et al. (2018).

Total temperature \(T_T\) and total pressure \(P_T\) are measured in the plenum and define the total condition of the nozzle flow under scrutiny. Two different thermocouples of type K and type J are employed, both located at the plenum axis.

Due to current unavailability of calibrated pressure probes for non-ideal flows, static pressure P is measured along the nozzle axis through wall taps, as well as total pressure \(P_T\), which is acquired at the plenum wall due to the negligible velocity of the local flow (\(\sim \,{1}\,{\mathrm{m}/\mathrm{s}}\)). Wall taps are connected to pressure sensors via pneumatic lines directly machined either at the plenum wall or within the test section rear plate. Miniaturized piezoresistive transducers with a sensing element diameter of 3.8 mm and maximum working temperature of \({454}\,{^{\circ }\mathrm{C}}\) are employed. The flow temperature significantly differs from the ambient one, and its variability is not negligible during a test; therefore, the considerable sensitivity to temperature of transducers required their calibration at different operating temperatures, obtaining a voltage \(V_P\), pressure P, temperature T calibration surface. An on-line zero procedure is carried out before each test run using a high precision barometer, to compensate for possible zero drift of the calibrating surface. All these uncertainty contributions are considered in the formulation of the combined pressure uncertainty.

To assess the accuracy of direct velocity measurements in a non-ideal vapor flow of a molecularly complex fluid, a test campaign with different nozzles and thermodynamic conditions was planned. A converging and a converging-diverging nozzle were considered, both with constant thickness (planar, two-dimensional nozzles). The feasibility of direct velocity measurements in the TROVA by means of LDV was first assessed in a subsonic flow, thus at relatively moderate velocity, and in a region of low or no velocity gradient. Then, a supersonic flow was characterized, both in regions of high and almost zero velocity gradient. Therefore, two different nozzles were employed, whose characteristics are reported in Table 1.

The first, called CM07, is a converging nozzle, where the flow is expanded and accelerated up to sonic conditions at the nozzle outlet, which corresponds to the geometrical throat. The nozzle profile is sketched in Fig. 1. It features a large portion at constant cross-sectional area. This region is designed to obtain Mach number \(M\approx 0.7\) with low velocity gradient. Indeed, the flow slightly expands due to a limited growth of the wall boundary layer.

The first converging portion profile is obtained through a 5th-order polynomial, while the second one is composed by a straight segment connected to the constant area region through a cubic spline. The design of both converging portions is made so to avoid possible separation bubbles or excessive gradients.

The outlet section was designed with sharp edge, so to set the geometrical throat. The outlet semi-height is \(H={17.1}\,{\mathrm{mm}}\), while the constant area semi-height is \(Y_c={19}\,{\mathrm{mm}}\), thus resulting in the area ratio \(H/Y_c=A_t/A_c=0.9\) needed to achieve Mach number \(M=0.7\) at design conditions for MM (see Table 1); \(A_t\) is the throat area and \(A_c\) the area of the zero gradient region. The design total conditions are \(P_T={5}\,{\mathrm{bar}}\) and \(T_T= {210}\,{^{\circ }\mathrm{C}}\) with siloxane MM. It is worth noticing that the nozzle operates in choked and under-expanded conditions; moreover, due to the significant change in total conditions it almost always operates off-design.

The second nozzle (Fig. 1) is of the converging-diverging type. It is called M16 and is used to perform measurements in the supersonic regime in both a region of almost zero velocity gradient and a region of high velocity gradient.

The diverging portion of the nozzle was designed using the method of characteristics implemented with state-of-the art thermodynamic models able to capture the non-ideal behavior of the flow (Guardone et al. 2013). A first portion of the diverging section is a circular arc, that provides an expansion to the desired outlet pressure value. Downstream, the flow is then deviated to obtain a uniform profile at the outlet in the so-called turning region, designed imposing mass conservation. A 5th-order polynomial was used for the nozzle profile of the converging part. The two portions of nozzle profiles are connected by imposing the same first and second derivatives at the nozzle throat.

M16 is designed for a uniform outlet Mach number \(M=1.6\) at \(P_T={21.4}\,{\mathrm{bar}}\) and \(T_T={254}\,{^{\circ }\mathrm{C}}\) with siloxane MM. The throat semi-height is \(H={8}\,{\mathrm{mm}}\), while the non-dimensional curvature is \(r_t/H=5\). Also in this case, the nozzle normally operates off-design, and thus the flow at outlet is not perfectly uniform and the delivered Mach number differs from \(M=1.6\), being typically higher at lower \(P_T\).

Tests span from strongly non-ideal conditions to almost ideal ones. A single velocity measurement point along the nozzle axis is taken for each run. Indeed, the blow-down operation of the facility entails time-dependent total conditions, as well as measured pressure ratio due to flow non-ideality, and velocity. Rather than traversing the nozzle, it was preferred to take measurements at one location only for the whole test duration, in order to observe, at one measurement point, the flow field evolution across the maximum span of non-ideality level. The condition at lowest total compressibility factor encountered in this test campaign features \(Z_T\approx 0.75\) at \(P_T\approx {8.5}\,{\mathrm{bar}}\) and \(T_T\approx {208}\,{^{\circ }\mathrm{C}}\). For tests at \(M\approx 0.7\) and almost zero velocity gradient, the measurement point is located in the second half of the constant area region, where both CFD calculations and preliminary tests showed an almost uniform flow. The measurement volume is located at the axial coordinate of tap 10 (Fig. 1) and at half depth of the channel. In supersonic tests (nozzle M16), the measurement point is placed just downstream the throat (at tap 8, see Fig. 1), in the expansion region of the nozzle, for the high gradient case, while it is placed in the uniform outlet region (tap 11) for the near-zero gradient case.

Tests were carried out acquiring pressure, temperature and velocity signals, which were synchronized through a trigger signal from the TROVA control computer. The LDV data set features six variables: x and y velocity components \(\tilde{V}_{x}\) and \(\tilde{V}_{y}\), particle arrival times \(\tilde{t}_{x}\) and \(\tilde{t}_{y}\) and transit times \(\tilde{tt}_{x}\) and \(\tilde{tt}_{y}\). The order of magnitude of data rate ranges from 10\(^2\) to \(10^3\,\mathrm{Hz}\).

Due to the blow-down operation of the TROVA, total conditions of the nozzle continuously change in time, resulting in a time-dependent velocity field. Thus, the statistical analysis is carried out on contiguous non-overlapping blocks of \({1}\,{\mathrm{s}}\) width. This provides a sufficient number of signals for the statistical analysis, while keeping the mean velocity variation in time within the interval approximately below 0.5%. Centered time intervals are adopted \(\left[ t-{0.5}\,{\mathrm{s}};t+{0.5}\,{\mathrm{s}}\right]\) to further reduce the approximation introduced by the batch operation.

To tackle the issue of the velocity bias introduced by the arithmetic mean, the transit time weighted mean is implemented, which is the recommended choice if transit time is available (Albrecht et al. 2003). Thus, mean velocity is computed according to Eq. (2), where \(\hat{V}_{x,i}\) and \(\hat{tt}_{x,i}\) are, respectively, the velocity observations and the transit time in the \(k-th\) time intervaland similarly for the y component.

Diverse options are available in the open literature to define the standard deviation of the weighted average. Some relations, each with some drawbacks, were proposed in literature and Gatz and Smith (1995) reported an assessment of their accuracy, using the bootstrap method as a benchmark. This method is a computer-based technique which permits to compute the accuracy of statistical estimates, without making parametric assumptions on the distribution underlying the data (see the book by Efron and Tibshirani 1994 for details). It is particularly useful in this case, where the distribution function of LDV samples is not known and a weighted average is being used. To avoid possible drawbacks of available relations for the standard deviation of the weighted mean and due to the very low computer power required to analyze the LDV data set, the bootstrap method was directly implemented.

The obtained standard deviation interval is taken as the contribution of the dispersion of data to the 95% confidence level uncertainty of the LDV measurement. This method was applied to both the x and y velocity component separately, to obtain the profile of average velocity and its dispersion as a function of test time.

Other uncertainty sources may arise from mean velocity gradients normal to the direction of the component being measured. It can be proved (Albrecht et al. 2003) that the mean x velocity averaged over the measurement volume \(\bar{V}_{x,MV}\) can be approximated aswhere \(\bar{V}_{x}\left( y\right)\) is the mean x velocity, \(y_c\) is the y coordinate of the center of the measurement volume, and \(b_d\) is the ellipsoid semi-axis in the y direction. Here, the error contribution was estimated from velocity profiles extracted at the nozzle axis from CFD calculations (which allowed to compute the second order derivative appearing in last term in Eq. (3)) and resulted of the order of \(1\times 10^{-4}\mathrm{m}/\mathrm{s}\) in the worst case; thus, it was considered negligible.

During a test, density undergoes a significant variation, and thus the refractive index n in the measurement point changes as a result of the Gladstone–Dale equation (Eq. (4))where k is the Gladstone–Dale coefficient. The angle \(\Theta\) between the two beams changes accordingly, thus causing a slight movement of the measurement volume in the direction of positive z, toward the rear plate. This displacement was estimated for all performed tests, obtaining a maximum value of 1.1% of the nozzle semi-depth, that is considered negligible, due to the planar nature of the flow.

The last considered source of error is the effect of a gradient of refractive index of the vapor. Indeed, due to a density gradient, laser beams can be deflected, leading to both a displacing and rotation of the measurement volume. In the laser Doppler velocimetry case, beams enter the perturbed region with a non-negligible angle in the plane of laser beams with respect to the density gradient.

Given that x, y, z are the directions of the nozzle axis, height and depth, respectively, the displacement of the measurement volume in the streamwise direction is given bywhere subscript 2 refers to the nozzle axis and L is the nozzle semi-depth.

The value of n to be used in previous calculations can be obtained from Eq. (4), if the Gladstone–Dale coefficient k for MM is known. Under the hypothesis that the molecular polarizability is a weak function of temperature and pressure, the refractive index of MM can be obtained from the Lorentz-Lorenz relation (Liu and Daum 2008) aswhere \(R_M=\frac{\left( n^2-1\right) M_m}{\left( n^2+2\right) \rho }\) is the molar refraction, \(\rho\) is the material density, and \(M_m\) is the molar mass. This equation, for low densities, is well approximated by a linear relation between n and \(\rho\). A Gladstone–Dale constant \(k_{\mathrm{MM}}={4.5\times 10^{-4}}\,{\mathrm{m}^3/\mathrm{kg}}\) was found by applying the least squares method to values obtained from Eq. (6) with \(R_M\) computed with a refractive index of liquid MM \(n_{\mathrm{MM},\mathrm{liq}}=1.3772\) (Wohlfarth 2017).

The sources of error arising from refractive index gradients were evaluated for the case of maximum gradient using data from CFD simulations (see 6.3), yielding a maximum angular deviation \(|\epsilon |={0.18}{^{\circ }}\) and beam displacement \(x_{2}-\left( x_{2}\right) _{\partial n/\partial x=0}={14}\,{\upmu \mathrm{m}}\). The error from the angular deviation of the velocity component being measured (the apparent velocity) with respect to the x axis is negligible, being approximately \(5\cdot 10^{-4}{\%}\). The displacement of the beam leads to an error which depends on the velocity gradient at the measurement point and was estimated to be \({0.07}\,{\mathrm{m}/\mathrm{s}}\) with respect to a mean velocity of \(\approx \,{180}\,{\mathrm{m}/\mathrm{s}}\). Further \(x_{2}-\left( x_{2}\right) _{\partial n/\partial x=0}\) is to be compared to the diameter of pressure taps, which is \({300}\,{\upmu \mathrm{m}}\), thus the displacement is lower than the resolution of pressure measurements. For the aforementioned reasons, the contribution of density gradient on measurement uncertainty was neglected.

Summarizing, the only considered contribution to velocity measurement uncertainty is the one originating from the scatter of velocity samples. In case of measurements in high axial velocity gradient regions, the contribution to the uncertainty related to the measurement volume positioning was also considered, referring to a position uncertainty of \({0.5}\,{\mathrm{mm}}\) along the nozzle axis. Contrarily, the contribution related to positioning in the plane normal to the nozzle axis was not considered due to negligible velocity gradients. The expanded uncertainty (95% confidence level) for the mean velocity, resulted of the order of 0.1–0.2% for measurement taken within almost uniform flows, while it reaches the order of 1–2% in the case of high velocity gradient regions.

In this section, results of the complete characterization of non-ideal compressible fluid flows by means of pressure, temperature and direct velocity measurements are reported and thoroughly discussed. The flow is characterized in a single measurement point on the nozzle axis. Three different cases are analyzed, a subsonic flow at \(M\approx 0.7\), with near-zero velocity gradient (Sect. 6.1), a supersonic flow at \(M\approx 1.7\), in a region at near-zero velocity gradient (Sect. 6.2), and a supersonic flow at \(M\approx 1.4\), in a region at high velocity gradient (Sect. 6.3).

The first-ever complete characterization of a point in a non-ideal compressible fluid flow was reported. To reach this goal, temperature and pressure measurement were complemented with the first-ever direct velocity measurements in such flows, carried out by means of laser Doppler velocimetry.

The hostile environment represented by the flow to be characterized required the employment of solid materials for flow tracing. Titanium dioxide particles of about \({200}\,\mathrm{nm}\) diameter were selected as the best compromise between dynamic and optical properties. Also, a novel seeding system, based on the atomization of a suspension of solid particles in the working fluid, was designed and implemented. The seeding system proved to be operational and suited for the purpose of accurately tracing non-ideal high temperature vapor flows. Nozzle flows of siloxane MM vapor were tested from non-ideal to almost ideal thermodynamic states, and measurements were taken at one single point along the expansion at three different conditions, corresponding to a subsonic (\(M\approx 0.7\)) non-accelerating flow, a supersonic (\(M\approx 1.7\)) non-accelerating flow, and a supersonic (\(M\approx 1.4\)) accelerating flow. The flow field was analyzed in terms of pressure ratio and velocity as a function of total conditions. Measured trends show a clear dependency on total conditions, thus proving the non-ideality of the analyzed flows. Also, experimental distributions of pressure ratio were compared to two-dimensional and viscous CFD simulations, while measured velocity was compared with both CFD results and values computed from pressure and temperature measurement using the isentropic expansion hypothesis.

Concerning measured pressure ratio, discrepancies from CFD are satisfactorily below 2.2% except for uniform flow at \(M\approx 1.7\) where a difference of the order of 10% is observed, with a peak of 14.3% at the most non-ideal conditions. Such difference is probably to be partially ascribed to the influence of side wall boundary layer, which is not accounted for in the 2D simulations, and partially to a possible small fraction of kinetic head recovered due to tap edge undesired chamfering.

Regarding directly measured velocity, the observed discrepancy of CFD results is always below 3% except for the supersonic (\(M\approx 1.4\)) accelerating flow test, where the maximum deviation is 6.6% at the test start. The presence of impurities or agglomerated particles during the initial portion of the test could motivate an underestimation of measured velocity for the accelerating case; however, this hypothesis needs to be further investigated. Instead, the difference between CFD data and the velocity inferred from pressure and temperature measurements ranges between 4 and 5.4%, while discrepancy between LDV measured and velocity inferred from \(P_T, T_T, P\) measurements is always below 4% except for the supersonic accelerating flow test.

The two different methods employed for measuring the velocity revealed to be congruent and also in accordance with the predictions of CFD simulations, thus proving the applicability of laser Doppler velocimetry for direct velocity measurement in high-temperature vapor flows with diverse levels of non-ideality, as well as the possibility of resorting to total and static pressure measurements to retrieve the flow velocity as it is required to develop directional pressure probes.