Pressure measurements

A probe rake comprising a pitot tube, a static pressure tube and a hot-wire probe for measuring turbulence spectra (discussion is not part of this paper) was traversed in the cylinder wake. The pitot pressure probes’ head had an outer diameter of 1.2 mm and an inner diameter of 0.8 mm. Traversing was performed from x=30 mm to 125 mm downstream of the cylinder and from y=−75 mm to +75 mm in the lateral expansion with a step size Δ from 3 to 15 mm (larger in the far wake). An areal probe traverse finally consisted of 450 measurement points. The irregular probe data spacing was interpolated to a regular grid which has identical resolution of the PIV grid, which is introduced in the next section.

The probe pressure data was acquired using differential pressure transducers in a rack-mount pressure scanner 98RK-1 from Pressure Systems Inc. offering a pressure range of ±5psi (or ±34.5 kPa) and a full-scale accuracy of 0.05% (i.e. 17.3 Pa). The pressure operating conditions such as ambient pressure or total pressure of the wind tunnel were measured separately by using a Mensor CPG pressure scanner with increased measurement accuracy option of 0.001% (7.5 Pa or 10 Pa absolute uncertainty for ambient and total pressure, respectively). All pressure data were measured for 10 s sampled at 10 Hz.

PIV measurements

The general plausibility of applying particle-seeded flow diagnostics to turbomachinery flows and in particular in the HGK environment which partly operates at the density limits of the Stokes-flow conditions, was investigated in e.g., Ruck (1990) and Engel (2007) or Bitter et al. (2016). The particles’ following behavior is widely sufficient for flows down to static pressure levels of pref>5 kPa. Below 5 kPa the particle trajectories may be dominated by inertial effects, especially in the vicinity of strong accelerations and stream-line curvature. This pressure regime is touched at the operating point ReD=10,000 at Maexit>0.7.

The PIV laser plane was introduced into the test section from the far wake behind the cylinder. An Innolas Spitlight 1000 Nd:YAG double-frame laser provided 480 mJ optical power per pulse to illuminate the particles in the field. A scientific CMOS camera with double-shutter option was installed at a distance of 500 mm away from the laser plane. The camera was equipped with a 35 mm Zeiss macro lens to capture a field of view which even slightly extends the dimensions from the probes’ field traverse. The field of view was calibrated with a planar calibration plate which was manually aligned with the flow field and the laser plane with an angular accuracy of better than 0.1 degree. The calibration scheme which is based on a second order polynomial as implemented in DaVis was applied for image reconstruction.

The flow was seeded with DEHS particles of 0.9μm RMS size using 2 PIVpart45 seeding generators from PivTec. In order to resolve the imaged particle displacement with a dynamic of around 15 px throughout the entire experimental range of operating points, the delay between both PIV frames was linearly adjusted from 2.2 to 6μs (lower for higher Mach numbers).

In order to ensure fully-converged Reynolds stress fields which are required for the application of the RANS equation in the pressure reconstruction scheme, 5,000 PIV images were captured and analyzed at each operating point. PIV data evaluation was performed with LaVisions’ latest release of the commercially available state-of-the-art PIV software Davis 10. An iterative window-based cross-correlation approach which is standard approach for PIV evaluation was applied. The vector field of each PIV snap-shot was individually calculated based on a reducing interrogation window size (IWS) from 96 px down to 48 px at 75 %. From the authors’ experience, this is a good compromise between spatial resolution and vector field calculation accuracy as a consequence of the particle density. The final PIV field had a spatial vector density of 0.46 mm/vector, which is about 8–10 times higher compared to a classical pressure probe traverse. Data post-processing and moderate spatial interpolation was applied in cases where required. Both, the data availability and vector validity were better than 99% for the cases of ReD>10,000. Especially, in the direct vicinity of the cylinder wake, these values suffered from low seeding density due to the dominance of inertial effects (in particular at Ma>0.7).

Cylinder in crossflow

The results presented in this section are based on the 5 steps introduced in the previous chapter. The raw pressure fields, as they were calculated with LaVision (2021) by applying the outlined procedure from the previous section, serve as the starting point for further discussion. Beyond this initial discussion, only the fitted/corrected pressure data is discussed in the context of the paper. First, the raw reconstructed pressure data from PIV will be compared to classic probe-based total pressure measurements for Mach numbers ranging from 0.3 to 0.85. The data comparison is shown for selected operating conditions in Figure 3. The correlation between the probe-based total pressure versus the corresponding total pressure reconstructed from PIV is plotted on the left-hand side. It should be noted that the plots for Ma = 0.3, 0.5 and 0.7 have been shifted in y-direction for a better display by an individual offset of Δpt/pt,1=0.3, 0.2 and, 0.1, respectively. Hence, all plots would collapse and end at pt/pt,1=1. The individual symbols distinguish between the Mach numbers. When evaluating Figure 3, two key features directly stand out: on the one hand, the non-linearity of the corresponding relationships increases with increasing Mach number and, on the other hand, the fluctuation range, which is characterized by the RMS for each individual plot, also increases in the same manner. This is a major consequence of the non-consideration of compressibility effects in the pressure reconstruction scheme in LaVision (2021) which is implemented to deal with incompressible flows. In the following discussion, such a fitting function is calculated for each operation point and inversely applied to the raw total pressure data from PIV.

Figure 3.

In-situ fits and their corresponding statistics for selected operating conditions (fit - black line, raw data - symbols). The plots on the left show the correlation between raw values reconstructed from PIV (x-axis) versus total pressures from probe measurements (y-axis) and prior to fitting. The plots on the right show the dependency of the relative deviations in pressure loss Δζint between pitot probe and PIV with respect to the number of reference points which were selected to create the in-situ fits.

The dependency of the fit results in relation to the number of reference points N is plotted on the right in Figure 3. The data points were selected from the min-max pressure range in the field. This range was sub-divided into N−1 equal steps to select the corresponding pressure data, whereas the placement of the reference positions was arbitrary. As indicated by the integral difference between the fitted total pressures from PIV and the pitot probe Δζint the fitting delivers stable results from already a few reference points from 10 to 20 until the fully available dataset. This means, that the number of provided reference data points is fairly low, if these probes are properly distributed to capture the full pressure range within the reconstruction field. For the generation of the polynomial fit for further discussion, a fairly large number of 100 reference data points from the pressure traverse and their corresponding counterparts from PIV have been used, and 350 points for statistical evaluation. The corresponding second-order polynomial fits and their statistics are included in the plot on the left.

Four selected planar total pressure loss fields in the wake of the cylinder are plotted in Figure 4 representing distributions from incompressible low-Re conditions (top left) to compressible high-Re in the bottom row. The top half of each sub-figure carries the probe data and compares it to the in-situ corrected pressure data from PIV on the bottom half. The local total pressure losses ζloc are calculated as introduced further above in Equation 1. In addition, the illustrations on the right-hand side contain a comparison in the form of a classical wake traverse, such as the one used to assess profile losses e.g., in a linear cascade. These traverses were extracted from both, probe and PIV plots at x/D=10. The authors are less intended to highlight aerodynamic effects in the individual plots rather than discussing the agreement in the local and integral pressure loss coefficient. To prevent the pressure sensors from range overload, probe measurements where not carried out in the nearest vicinity of the cylinder at some operating points (esp. high Ma/high Re combinations). Since data evaluation was automated and the results were aligned among each other, all pressure field plots representing pitot data start from x/D≈3.5.

Figure 4.

Comparison between total pressure losses in the wake of a cylinder measured with pitot probe in the top half of each frame (a)–(d) and reconstructed and fitted from Particle Image Velocimetry (bottom half) at different Mach and Reynolds numbers. The traverse line plots were extracted at x/D=10. (a) ReD=10,000;Ma=0.3. (b) ReD=50,000;Ma=0.3. (c) ReD=50,000;Ma=0.7 and (d) ReD=80,000;Ma=0.7.

In general, a very good local and quantitative agreement can be demonstrated. It is obvious, that the wake core close to the cylinder provides larger mismatch compared to the remaining flow field. As a consequence of strong but very thin shear gradients in this area, the pressure loss production is also significantly affected by the turbulent stresses. For a highly accurate measurement of turbulent quantities, both measurement techniques are not the perfect approach, but, at least PIV (if properly applied), can help to estimate their order of magnitude. It is also known from multi-hole probe measurements, that the geometric size of the probe head and the traversing step size may have significant impact on the measured pressure data in terms of spatial resolution, especially if the flow structures become finer and the gradients get sharper, cp. Vinnemeier et al. (1990). For this reason, conventional probe traverses for total pressure loss characterization shall be performed in the mixed-out wake, were the flow is not dominated by these sharp gradients.

A criterion which may separate turbulent wakes between non mixed-out and mixed-out regions was proposed by Proudian (1964) who related the dominant turbulent fluctuations to the local wake velocity deficit according to Ωz=u′2/ΔV2. In the mixed-out wake, the gradient dΩz/dx converges to a steady value. This does not necessarily mean, that the total pressure profile and the wake deficit is completely flattened out but rather distinguishes the axial position behind a test specimen which may not be dominated by relative turbulent fluctuations compared to the origin, e.g. directly at the trailing edge. Of course, turbulent fluctuations are present and can be higher compared to the undisturbed outer flow field. The plots in the upper rows of Figure 5 show the axial gradient of the span-wise integral of Ω along the wake length. It can be seen that the position, where the criterion converges, clearly varies between incompressible flow at Ma=0.3 and a compressible wake at Ma=0.7. The corresponding central and lower plots in the same figure display the axial slopes of the integral pressure loss coefficient (central plot) according to Equation 2 in comparison between PIV (solid) and probe measurements (dashed) and the relative local deviation between both experimental methods (lower plot), respectively. Again, it gets obvious that the approach performs slightly more accurate in the case of the incompressible low-speed flow. As discussed above in Figure 3, data spreading rises with increasing Mach numbers. This directly affects the inclination /offset of the in-situ fit. This may lead to a constant offset or a tilt in the reconstructed data. As also mentioned earlier, it is not the authors’ claim to expect a 100 percent alignment between both data. On the other hand, an integral agreement within less than 2 percent seems valuable for the authors and their proposed approach, knowing that both measurement techniques also have their own weaknesses.

Figure 5.

Spanwise-averaged wake properties showing: the axial gradient of the “mixed-out” criterion Ωz=u′2/ΔV2 (top); the axial total pressure losses from PIV and pitot tube (center) and the relative difference between PIV and pitot measurements (bottom).

In particular transonic (wake) flows can be of interest for the application of the proposed approach since the flow field may be complex and dominated by strong velocity and pressure gradients (shocks). By means of Figure 6, the dependency of the spatial shock position on the final interrogation window size from the PIV evaluation shall be highlighted for the operation point at Ma=0.85 and ReD=50,000. The PIV evaluation was repeated while the final IWS was changed ranging from 24 px as the finest to 96 px at the coarsest vector field, including 48 px as the standard resolution discussed throughout the entire paper. The left and right subfigures represent the axial and lateral velocity gradients for 48 px in colors, respectively. The contour line of Ma=1 in the vicinity of the cylinder is also included. Two corresponding pressure line plots were extracted along the dashed slopes highlighted in both subfigures. These plots represent both, a cut through the shock and through the strong gradient of a transonic wakes’ shear layer. In order to overcome spurious effects in the pressure reconstruction, which are linked to a bad spatial resolution of the final PIV vector field, DaVis seems to perform integration on a refined sub-grid. This approach widely eliminates the first-order dependency of the reconstructed pressure field on the final IWS. As a consequence of a reduced particle ensemble number with reduced IWS size and the corresponding raised uncertainty of the corresponding PIV vector field, the data with IWS 24 px exhibits a slightly different gradient in the wake compared to the other IWS. This effect is less pronounced in the outer flow field (cut through the shock) since the particle concentration is a-priori less in the wake compared to the outer flow. From this study it can be stated, that the proposed approach also returns reasonable results in terms of the reconstructed pressure pattern in the flow. But, as already discussed, the uncertainty of the reconstructed pressure field increases with increasing Mach number. This trend is finally highlighted by means of Figure 7. For this purpose, the integral total pressure losses of a traverse along the position x/D=10 were calculated in the same manner as above. The slopes show the corresponding pressure loss graphs for the probe measurements (filled symbols) and PIV (hollow symbols) as a primary function of the Mach number (left) and of the Reynolds number (right). The top row contains the absolute loss values. The pitot data were added up by the error bars of measurement uncertainty from the linear error propagation. The bottom row contains the mean relative deviations between the measurement techniques and the error bars were calculated from the corresponding standard deviations. (Note: By now, the PIV reconstruction tool does not provide any pressure uncertainty.)

Figure 6.

Shock position as a result of a PIV grid refinement study at Ma=0.85 and ReD=50,000. The velocity gradient du/dx (left) and dv/dy (right) around the cylinder top half are plotted color-coded. Vertical and horizontal cuts through the field show reconstructed pressure ratios for PIV interrogation window sizes of 96 px, 48 px (standard for all other figures) and 24 px.

Figure 7.

Comparison of integral total pressure losses between pitot probe and PIV at x/D=10; left: Mach number effect; right: Reynolds number effect; in absolute numbers (top row) and relative deviations (bottom row).

The raising trend of the integral losses with respected to the test Mach number is evident as a consequence of rising mixing losses. The negative trend of reduced total pressure losses with increased Reynolds number is also known for viscous wake flow. As a consequence of pronounced shear forces which lead to a strong shear gradient and a reduced shear layer width, the integral losses mix out more rapidly. Both trends are well prominent in wake flows, in particular for cascades flows, whose further investigation with the proposed approach is the authors’ main motivation. It is shown that both, pitot and fitted PIV pressures provide data which are in very good agreement and whose integral deviation is in the order of 1 percent or even less. The uncertainty is higher for highest Mach numbers and at lowest ambient and dynamic pressure conditions as a consequence of a) the uncertainty of the fit raises with increased Mach number as stated above, and b) the resolution of low (dynamic) pressure differences with pressure transducers of range ±34.5 kPa might not be the best choice but was required for the sake of keeping the test campaign in a proper time frame.

It can be stated in summary of all presented results so far, that the presented approach enables the reconstruction of reliable total pressure distributions from the PIV over a wide Reynolds and Mach number range when measured in the HGK facility. The relative deviations in the determination of the loss coefficient between the classic pressure measurement with a pressure probe (e.g., pitot or multi-hole probe) and the reconstructed distributions from PIV over the examined flow range are not larger than 2 % if the measurement uncertainty is considered, too. With respect to the simplicity of the presented approach, this high agreement is remarkably good, especially if the limitations (e.g. in the implementation of the pressure reconstruction scheme) are considered. It was also shown that the driving factor of growing uncertainty are increased compressibility effects due to the raised test Mach number, whereas the dependence on the Reynolds number appears to be comparatively small. With this knowledge, the approach appears to the authors to be valid enough to proof its performance at this level for a usage in cascade flows.

Low-pressure turbine cascade

Finally, the proposed approach is applied and cross-validated to the exit flow field downstream of a low-pressure turbine (LPT) cascade comprised of 7 blades. A detailed discussion of the cascade itself and the particular flow fields at Maexit>0.6 with an inflow turbulence level of 4% is outlined in Bitter and Niehuis (2019). The wake flow field of the central blade was investigated with PIV and 5HP measurements. Pressure traverses were measured at two axial positions downstream of the cascade trailing edges, i.e. xax/cax=[20;80]% %. The PIV fields’ height and width extend over 1.5 blade pitches P or 1.5 axial blade lengths cax, respectively. The LPT cascade was operated at static pressures ranging from about 5 kPa at the lowest chord-based Reynolds number of 40,000 up to about 15 kPa at the highest Reynolds number of 120,000. The total pressure loss distributions in the wake at highest Reynolds number are shown on the left in Figure 8. The pressure field also embodies the local comparison of the loss traverses at 20 % and 80 % relative axial chord length xax/cax downstream the blades’ trailing edge, whereas the PIV data is represented by the solid lines and the 5HP slopes by the dashed lines. The local loss values from PIV were in-situ corrected in the same manner presented above using all points from the 5HP traverses at both axial positions as in-situ fit reference data. The local values from PIV were normalized with the integral values from the 5HP measurements at position 80%. In accordance to the cylinder results from above, the agreement in the mixed-out region at 80% is remarkable. At 20%, the topology of the wake seems a bit slender for the 5HP results. This might be the consequence of the traverse position, since the wake seems not mixed out at 20% as indicated by the evaluation of the mixed-out criterion on the right in Figure 8. Anyhow, the integral losses calculated at 80% and compared between 5HP and PIV at all tested Reynolds numbers in Figure 9 agree remarkably well. The ζ values in the figure were normalized with the same reference value as above. The relative deviations are even covered by the uncertainty of the probe measurements, except for Rec=40,000. Towards the lowest Reynolds number, the deviations grow as a consequence of the low operating density in the wind tunnel which tends to approach the limits of the stokes-flow criteria for the particle following constraint. The trend of raising pressure losses with respect to reduced Reynolds numbers not just holds for the cylinder wake flow but is also typical behavior of (LPT) airfoils in general. At all, the results presented so far proof a valid concept for total pressure loss estimation from PIV supported by local multi-hole pressure measurements.

Figure 8.

Left: Total pressure loss coefficient field in the wake of an LPT cascade at Rec=120,000 reconstructed from planar PIV data at Maexit>0.6. The absolute values were normalized with the integral loss coefficients measured with a 5-hole probe at xax/cax=0.8. Right: Color-coded evolution of the “mixed-out” criterion in direction of the LPT wake.

Figure 9.

Comparison of integral total pressure losses in the wake of an LPT at xax/cax=0.8 measured with 5HP (filled/dashed) and reconstructed from PIV (hollow/solid) at Maexit>0.6.

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