8TH INTERNATIONAL SYMPOSIUM ON PARTICLE IMAGE VELOCIMETRY - PIV09 Melbourne, Victoria, Australia, August 25-28, 2009

Instantaneous Pressure Field Determination in a 3D Flow using Time-Resolved Thin Volume Tomographic-PIV

R. de Kat, B.W. van Oudheusden and F. Scarano

Faculty of Aerospace Engineering, Delft University of Technology, 2629 HS, The Netherlands r.dekat@tudelft.nl

ABSTRACT This paper describes the application of time-resolved tomographic particle image velocimetry to determine instantaneous planar pressure fields in the three-dimensional wake of a square-section cylinder, with the face normal to the flow, for ReD = 9,500, where D is the section dimension. The pressure fields are determined by solving an in-plane Poisson pressure formulation. From the resulting series of instantaneous planar pressure fields a time signal is extracted and compared to pressure signals from two pressure transducers located on either side of the measurement volume. An assessment of the influence of the out-of-plane velocity gradient components on the pressure field evaluation is performed. The results show that the agreement between the PIV-based pressure signal and the signal from the transducers at the base of the cylinder improves when using the full 3D information with respect to using a 2D sub-set. 1. INTRODUCTION The instantaneous pressure field around a body immersed in a fluid is a property of great practical importance in aerodynamic applications, in particular for its relevance to integral aerodynamic loads. Particle image velocimetry [10] (PIV) affords itself to the determination of the pressure fields by combining experimental velocity data with the momentum equations [1], [4], and [9]. Practical implementation of PIV-based approaches is confronted with particular challenges when applied to unsteady three-dimensional (3D) flows. Extension of pressure-determination strategies to unsteady flows can be achieved by using time-resolved PIV (TR-PIV) [6]. Pressure dominated flows with large separation-regions -like bluff-body wakes- contain complex unsteady 3D features, especially at moderate to high Reynolds numbers. These features make them particularly challenging test cases for procedures that rely on planar flow diagnostics (2C- or 3C-PIV). Previous experiments performed on a square-section cylinder, [5], have shown that the instantaneous pressure can be inferred along the side of the cylinder, where the flow is predominantly two-dimensional (2D). However, due to the strong 3D nature of the wake the instantaneous pressure at the base was poorly captured. This indicates the necessity of including the 3D terms in the instantaneous pressure-determination procedure. In the present investigation a thin-volume tomographic-PIV [3] (Tomo-PIV) approach is followed to determine all terms of the velocity gradient and assess their significance in pressure field evaluation in the 3D flow in the wake of the square-section cylinder. 2. PRESSURE DETERMINATION PROCEDURE The instantaneous velocity fields are acquired by TR-Tomo-PIV and are used to determine the in-plane pressure gradient components from the momentum equations given in equation 1. In the present study an in-plane Poisson

formulation for 3D flow is used to calculate a planar pressure field (equation 2), in contrast to [4] where a Poisson formulation for 2D flow was used. This formulation uses the in-plane divergence (∇xy⋅f = ∂fx/∂x + ∂fy/∂y) of the pressure gradient. The in-plane momentum equations are given by

p u u u uu v w

x t x y z

p v v v vu v w

y t x y z

ρ

ρ

∂ ∂ ∂ ∂ ∂= − + + + ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂= − + + + ∂ ∂ ∂ ∂ ∂

, (1)

where the viscous terms are neglected. The in-plane Poisson formulation is given by

( ) ( ) ( )2 2

2 2

A

xy xy

p p

x y tρ∂ ∂ ∂+ = − ∇ ⋅ + ⋅∇ ∇ ⋅ +∂ ∂ ∂

u u u�������

22

B2D

2u u v v u w v w

x y x y z x z y

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

������������������

, (2)

where part of the terms indicated with “A” and “B” completely contain out-of-plane velocity gradient components and “2D” is the complete right-hand-side for 2D flow. To assess the influence of the out-of-plane components on the pressure field evaluation the results for two different inputs are compared: (i) a fully 3D approach (3D input) and (ii) a planar sub-set (2D input). For the 2D input the parts containing out-of-plane velocity gradient components (part of “A” and “B” completely in equation 2) are omitted. The results with 2D input resemble the results that can be obtained with 2C- or 3C-PIV. 3. EXPERIMENTAL ARRANGEMENT Experiments were performed in a low-speed, open-jet wind-tunnel at the Aerodynamics laboratory at Delft University of Technology. The tunnel outlet has dimensions 40 by 40 cm2. The turbulence level in the test-section is 0.11 % [11]. The square-section cylinder with dimension 30 x 30 mm2 (D x D) and 34.5 cm in width was fitted with end plates and positioned in the middle of the jet (see figure 1). The geometric blockage was 6.5 %. In the present experiment the free-stream velocity V∞ was 4.7 m⋅s-1 (dynamic pressure, q∞ = 13.5 Pa) giving a Reynolds number ReD = (V∞⋅D) ⋅ν -1 = 9,500, where ν is the kinematic viscosity. The Strouhal number is derived from half the dominant frequency at the base, f = 40 Hz (cf. [5]) and estimated to be St = f⋅D⋅V∞

-1 = 0.13.

Figure 1. Experimental setup. The cylinder was instrumented with two flush mounted pressure transducers located in close proximity of the centre-line of the model (see figure 1) to provide reference values for the pressure signals extracted from the PIV data. The pressure transducers used are Endevco 8507-C1 with a range of +/- 1 Psi (+/- 6890 Pa) and a typical sensitivity of 175 mV/Psi (25 µV/Pa). Signal recording was performed using a National Instruments data acquisition system (comprising of: PCI-6250, SCXI-1001, SCXI-1520, and SCXI 1314) operating at 10 kHz. The resulting noise level was 4 µV RMS giving a resolution of 0.3 Pa (two times the RMS). The transducers 1 & 2, placed 8 mm apart, are shown in figure 1, right and left respectively. None of the results presented have been corrected for the effects of wind tunnel blockage. A high-repetition rate tomographic-PIV system was used to capture the flow in a thin volume around the symmetry plane of the model. Flow seeding was provided by a Safex smoke generator, which delivered droplets of about 1 µm in diameter. The volume was illuminated by a Quantronix Darwin-Duo laser system with an average output of 80 W at 3 kHz with wavelength of λ = 527 nm (equivalent pulse-energy at 2.7 kHz is 2 x 15 mJ). The laser-pulse separation was 90 µs. Images are acquired by four Photron Fascam SA1 cameras with a 1024 x 1024 pixels sensor (pixel-pitch = 20 µm), recording image-pairs at 2.7 kHz (cameras running at 5.4 kHz), equipped with Nikon lenses with focal length 60 mm and the f-number set at 2.8 for the bottom cameras and 5.6 for the top cameras. The particle concentration was approximately

Figure 2. Schematic representation of the integration domain. “A” indicates the Dirichlet boundary condition. “B” indicates the Neumann boundary condition.

0.05 particles per pixel, resulting in circa 4 particles per interrogation volume. The illuminated volume was 70 x 70 x 4 mm3 (2.3 D x 2.3 D x 0.1 D), with a digital resolution of 14.3 voxels mm-1. A total of 5400 image-pairs per camera (2 s) were recorded. For the results presented in this paper only 272 image-pairs were used (0.1 s). Particle images were processed using DaVis 7.4 software, including tomographic self-calibration [13], with a final interrogation volume-size of 16 x 16 x 16 pixels (0.04 D x 0.04 D x 0.04 D) and an overlap-factor of 50 % giving (after cropping) a vector grid of 104 x 109 x 8 vectors with a vector spacing of 0.6 mm (0.02 D). The vector fields were processed with a median test [12] combined with a multiple peak check. Spurious vectors were removed and replaced using linear interpolation. The number of spurious vectors in the region used in the evaluation of the right-hand-side of equation 2 was less than 5 %. The average signal-to-noise ratio and the normalized correlation coefficient were 2.82 and 0.63 respectively. The acceleration was determined using central differencing in time and the velocity gradient using central differencing in space. The in-plane pressure gradient was evaluated in the mid-plane of the volume. Subsequently the pressure field was computed (see figure 2) using a Dirichlet boundary condition (prescribed pressure values using Bernoulli’s equation) on the lower edge of the pressure evaluation domain, where the flow was quasi-steady and irrotational. Additional Neumann boundary conditions are enforced on the remaining edges of the integration domain. 4. RESULTS AND DISCUSSION 4.1 Velocity results and derived quantities An instantaneous 3D visualization of vorticity is given in figure 3. It shows that the wake of the square-section cylinder is composed of vortices of different orientations. The vortex located in the top of the volume (label A) is a clockwise rotating Kármán vortex shed from the upper side of the model. The vortex just below the middle on the left of the volume (label B) is a counter-clockwise rotating Kármán vortex forming from the bottom side. In between these two vortices, a distinct coherent secondary vortex (label C) is present obliquely aligned in the plane, similar to what is reported for circular cylinders at lower Reynolds numbers [14]. This secondary vortex is surrounded by smaller vortices with a variety of orientations. Figure 4 shows the pressure fields corresponding to the visualization in figure 3 for the 3D and 2D input, top and bottom respectively. The pressure coefficient is defined as cp = ( p - p∞ ) ⋅ q∞

-1. The pressure field from the 3D input shows an agreement in structure with the 3D visualization in figure 3. All labelled vortices show up as low-pressure zones. In the 2D input result the major difference is the lack of the low-pressure zone in the region where the secondary vortex is located. It only shows low-pressure zones where out-of-plane vorticity is present. 4.2 Comparison of pressure signals at the base Figure 5 shows the pressure signals from the pressure transducers and the pressure signals extracted from the PIV fields with 3D and 2D input, top and bottom respectively. The

Figure 3. 3D visualization of the flow at t = 0.078 s. Vectors show in-plane velocities in the mid-plane (1 out of 3 vectors is shown in X-direction). Isosurfaces show vorticity. Light gray: positive ωz. Black: negative ωz. Gray: sqrt(ωx

2+ωy2). (The velocity volume used for this

visualization was smoothed). results from the 3D input show better agreement with the transducer results than the results from the 2D input. Transducer 1 had a mean value of -1.56 q∞, transducer 2: -1.53 q∞, and transducers 1 & 2: -1.55 q∞. The results are consistent with literature, where a range of -1.5 to -1.6 is given [2], [7], and [8]. The PIV-based results are for the 3D input: -1.46 q∞, and 2D input -1.42 q∞, which are in agreement with the PIV-based results found in [5]. The PIV-based results differ 5.8 % and 8.4 %, with transducers 1 & 2, for the 3D and 2D input respectively (Table 1). The transducers differ 2 % with respect to each other. The improvement when changing from 2D to 3D input is 2.6 %. For both the methods, the PIV-based signal shows reasonable to good agreement with the transducer signals indicated by the correlation coefficients of 0.52 for the 2D input and 0.66 for the 3D input, when compared with both transducers (Table 1). The correlation between the transducer signals themselves is 0.70, which indicates considerable spanwise influences. The value of the correlation between the PIV-based signal and the transducer signal is close to the correlation value of the transducers suggesting that the major influence on the difference between the PIV-based signal and the transducer are the spanwise variations. The change from 2D to 3D input improves the correlation by 0.14. The amplitude of the fluctuations, σp, for the transducers was 0.31 q∞ for transducer 1, 0.24 q∞ for transducer 2, and 0.25 q∞

for 1 & 2. These values are slightly larger than found in [2] and [5], the reader should take into account that the current

Figure 4. Pressure fields from PIV-data at t = 0.078 s. Top: pressure field resulting from 3D input. Bottom: pressure field resulting from 2D input.

Figure 5. Pressure signals from PIV compared with the signals from the pressure transducers. Top: pressure signal for 3D input. Bottom: pressure signal for 2D input.

values are preliminary and based on only 0.1 s of data (approx 2 shedding cycles). The PIV-based results are 0.36 for the 3D input and 0.40 for the 2D input. The PIV-based results differ 44 % and 60 % with transducers 1 & 2, for the 3D and 2D input respectively (Table 1). This shows a large improvement over the differences reported in [5], which were larger than 100 %. The 3D input improves 16 % over the 2D input. To give an estimate of the accuracy of the PIV based results the RMS difference between the PIV and transducer signals is used. The RMS differences were 0.26 q∞ and 0.34 q∞ for the 3D and 2D inputs respectively (Table 1). The result for the 2D input case is consistent with [5], where a RMS difference of 0.34 q∞ was found. The change from 2D to 3D input gives an improvement of 0.08 q∞. The RMS difference between the transducer signals is 0.22 q∞. This also suggests that the difference between the PIV-based result and transducer signals may largely be caused by spanwise variation. Transducer 3D input 2D input

1 0.58 0.41 2 0.65 0.56

1 & 2 0.66 0.52

Correlation coefficient

1 6.4 % 9.0 % 2 4.6 % 7.2 %

1 & 2 5.8 % 8.4 %

(pmean(trans)−pmean(piv))⋅ ⋅pmean(trans)-1

1 16 % 29 % 2 50 % 66 %

1 & 2 44 % 60 %

(σp(trans)–σp(piv))⋅ ⋅σp(trans)-1

1 0.30 0.39 2 0.27 0.33

RMS of (ppiv -ptrans) ⋅ q∞-1

1 & 2 0.26 0.34 Table 1. Correlation and differences between pressure signals from PIV and the pressure signals from the transducers. Transducer 1 & 2 means 0.5 ⋅ (p1+p2). 5. CONCLUSIONS This paper shows that pressure can be derived from tomographic-PIV data and that pressure signals extracted at the base of the square-section cylinder from the PIV agree with the signals from two pressure transducers located at the base on either side of the measurement volume. Two different inputs were tested, one where the full 3D velocity information from the tomo-PIV experiment is used, including the out-of-plane velocity gradient component (3D input), and one where it has been omitted (2D input). The latter is similar to information obtained from planar PIV. The 3D input is an improvement over the 2D input, improving the correlation coefficient, the mean difference with the transducers, the difference in fluctuating pressures, and the RMS difference between the signals. The results for the 3D input and the pressure transducers suggest that the difference between the 3D input and the transducers are largely caused by spanwise variations. ACKNOWLEDGMENTS This work is sponsored by the Dutch Technology Foundation “STW”, grant 07645.

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