Instantaneous planar pressure from PIV: analytic and ...

15th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 05-08 July, 2010

- 1 -

Instantaneous planar pressure from PIV:

analytic and experimental test-cases

Roeland de Kat1, Bas W. van Oudheusden1

1: Faculty of Aerospace Engineering, Delft University of Technology, Delft, Netherlands.

1 corresponding author: [email protected] Abstract This paper makes a comparative assessment of different procedures to determine instantaneous planar pressure from particle image velocimetry measurement data. A description of the basic operation principles of the methods addressing the different procedures for obtaining the pressure gradient (Eulerian form or Lagrangian form) and for the subsequent spatial integration to obtain the pressure field itself (direct spatial integration of a Poisson formulation). First, an analytic test-case of a convecting Gaussian vortex is considered to give insight in the relevant parameters for a successful determination of pressure. In the second part of the paper an experimental test-case is considered, where the pressure was determined from stereoscopic and tomographic time-resolved PIV data around a stationary square-section cylinder. Pressure time series were recorded simultaneously with pressure transducers to validate the pressure obtained from the PIV-data. The results illustrate the potential of using PIV-data for pressure determination, as well as providing guidelines to assess the critical experimental parameters. 1. Introduction The instantaneous pressure field around a body immersed in a fluid is of great practical interest, amongst others for its practical relevance in the determination of integral aerodynamic loads. Techniques to measure surface pressure (taps) and integral loads (force balances) directly are commonly in practice. However, the cost, complexity, and intrusiveness associated to many aspects of these techniques motivated the search for alternative methods in order to nonintrusively obtain aerodynamic quantities.

With the development and success of nonintrusive flow field diagnostics, such as particle image velocimetry (PIV, Raffel et al. 2007) in particular, new potentials have emerged to determine instantaneous aerodynamic loads. PIV has already proven its capability in characterizing instantaneous velocity fields and derived quantities such as vorticity, whereas its use in determining the instantaneous pressure field remains relatively unexplored.

Gurka et al. (1999) derived from a steady velocity field the pressure distribution in a channel flow. Concurrently, Baur & Köngeter (1999) explored determination of instantaneous pressure from time-resolved velocity data, addressing the local pressure reduction in the vortices shed from a wall-mounted obstacle, using a two-dimensional (2D) approach. Hosokawa et al. (2003) used PIV-data to obtain the pressure distribution around single bubbles, while Fujisawa, Tanahashi & Srinivas (2005) derived pressure fields around and fluid forces on a circular cylinder from PIV-data. Liu & Katz (2006) show the application of pressure determination from PIV on a cavity flow. Fujisawa et al. (2006) apply the pressure reconstruction on a micro channel using micro-PIV data. Pressure evaluation from PIV-data has even found its extension into the compressible regime as demonstrated by van Oudheusden (2008).

As the velocity-data used as input for the pressure and load determination are commonly obtained from planar PIV, most of these studies are hampered by the restriction to 2D (average) flow or necessarily making 2D flow assumptions, where it is not obvious what the impact of this approximation can be. Also, no comprehensive analysis of the parameters that will determine the success of pressure evaluation from PIV has been reported yet.


- 2 -

The first objective of the present paper is to assess the importance of the main parameters on the pressure evaluation using an analytic test case, employing synthetic PIV velocity information generated from a theoretical flow case for which the pressure field is known (a convecting Gaussian vortex). We demonstrate how pressure may be derived from instantaneous velocity data and describe what the key parameters are that need to be taken into account. We test two different approaches to determine the local pressure gradient (Eulerian and Lagrangian) and test two different methods to evaluate the pressure from these pressure gradients (direct spatial integration and a Poisson formulation). As an experimental test case we consider the flow around a square-section cylinder. Pressure dominated flows around bluff bodies pose relevant and challenging test cases for pressure evaluation from planar-PIV, due to the complex time-evolving three-dimensional (3D) nature of the flow field, especially at moderate to high Reynolds numbers (see e.g. Williamson 1996). We use this flow case to show the viability of the pressure evaluation methods by using data from stereoscopic PIV (stereo-PIV) and tomographic PIV (tomo-PIV, Elsinga et al. 2006) to assess the impact of 3D flow effects, employing surface pressure data for validation. 2. Pressure evaluation from PIV Pressure evaluation from PIV velocity data involves two steps. First the pressure gradient is determined by locally applying the momentum equation in differential form. Subsequently, the pressure gradient is spatially integrated to obtain the pressure field. 2.1 Operation principle The incompressible momentum equations for 3D flow allow relating the pressure gradient to the velocity data as shown in (1). The two different expressions are equivalent and express the flow acceleration in either a Lagrangian or Eulerian form. These two different approaches have been investigated and were found to have different strengths and weaknesses. Jakobsen, Dewhirst & Greated (1997) compared the Eulerian and Lagrangian form to determine the material acceleration in waves and found that for their case the Lagrangian form out-performed the Eulerian, but was only working within a smaller time-separation range. Christensen & Adrian (2002) found that for their convecting turbulence experiment the material acceleration was about one order of magnitude smaller than the time change of the velocity at one point, which would promote the use of the Lagrangian approach. These findings motivate the comparison of the two forms for determining the pressure gradient and evaluate their performance.

( )2 2DD

pt t

ρ ν ρ ν∂⎧ ⎫ ⎧ ⎫∇ = − − ∇ = − + ⋅∇ − ∇⎨ ⎬ ⎨ ⎬∂⎩ ⎭ ⎩ ⎭u uu u u u (1)

In case of 2D flow, planar time-resolved PIV will suffice in determining the pressure gradients, but for 3D flow all components of the velocity and velocity-gradient are needed, which may be accomplished by a time-resolved tomo-PIV procedure (see Schröder et al. 2008), for example.

To evaluate the pressure in a plane (here defined as the x-y plane), only the in-plane components of the pressure gradient are needed. The reader should note however that these in-plane pressure gradient components contain in- and out-of-plane components of the velocity and velocity-gradient. To obtain the pressure field, the pressure gradient can be spatially integrated using either a direct spatial integration approach or a Poisson approach. In the latter approach the in-plane divergence of the pressure gradient is taken (2) and subsequently integrated by a Poisson solver.


- 3 -

2 2

2 2xy xyp pp f

x yρ∂ ∂

∇ ⋅∇ = + = −∂ ∂

(2)

Now, even for 3D flow, most of the terms that appear can be extracted from planar-PIV. In (3) the right-hand-side of (2) is given, with omission of the viscous terms. The additional 3D flow contributions contain the in-plane divergence of the velocity, which can also be derived from planar PIV-data, and only for the parts containing out-of-plane gradients 3D velocity information is needed.

( ) ( )( )22

Additional 3D flow terms2D flow

2xy xy xyu u v v w u w vfx y x y t x z y z

⎧ ⎫⎛ ⎞ ⎧ ⎫∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪⎛ ⎞= + + + ∇ ⋅ + ⋅∇ ∇ ⋅ + +⎨ ⎬ ⎨ ⎬⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎩ ⎭⎪ ⎪⎩ ⎭u u u

1444444444244444444431444442444443

(3)

2.2 Numerical implementation As expressed by the two alternative formulations in (1), the pressure gradient can be computed in two different ways: the Lagrangian form where all the quantities are evaluated with respect to an element moving with the flow and the Eulerian form where everything is taken relative to a fixed spatial location. For the latter approach we use a second-order central finite differences in space and time. For the Lagrangian approach we need to reconstruct the fluid-parcel trajectory. In the present study the fluid trajectory is reconstructed using a pseudo-tracking approach, which is derived from velocity fields instead of particle locations (see Liu & Katz 2006). A second-order fluid path is reconstructed using an iterative approach given in (4). For both the Eulerian and the Lagrangian approaches we switch to linear forward or backward schemes at domain edges.

1

2 2

D ( , ) ( , )D 2

D D( , ) ; ( , )D D

k

k k

t t t tt t

t t t t t tt t

+ + −

+ −

+ ∆ + ∆ − −∆ −∆=

∆

∆ = ∆ + ∆ ∆ = − ∆ + ∆

u u x x u x x

u ux u x x u x (4)

For the integration of the pressure gradient into a pressure field two different approaches have been evaluated. First a direct spatial integration approach that integrates the pressure gradient directly from reference points (where the pressure has been prescribed) using a spatial marching erosion scheme, first introduced by Baur & Köngeter (1999), details on the current implementation can be found in van Oudheusden (2008). One disadvantage of this method is its dependency on its integration path. Liu & Katz (2006) have investigated an approach that removes this dependency by an omnidirectional integration of the pressure gradient.

The Poisson approach solves the in-plane Poisson formulation of (2) using a standard 5-point discretization scheme (second-order central finite differences) and gaussian elimination. Next to the reference boundary condition, the Poisson approach uses Neumann conditions along the remaining boundaries of the domain, which is implemented using ghost points.

As a reference point for the pressure and boundary condition for the pressure evaluation a point or domain in the steady inviscid outer-flow can be used in combination with the Bernoulli equation. However, due to the limited measurement domain of PIV, the boundary conditions need to be enforced within the disturbed flow domain. For this, the reference pressure is computed with an extended version of the Bernoulli equation that holds for an irrotational unsteady flow with small mean velocity gradients as given in (5).


- 4 -

( )2 21 .2

p constρ+ + =u u' (5)

3. Analytic assessment Vortices are arguably the most relevant flow structure elements, likely to be encountered in many fluid dynamic studies where pressure is of interest (e.g. separated flows, bluff body flows). The convection of a Gaussian vortex is taken to serve as an analytic test case for the pressure evaluation procedures. Analytic solutions for the velocity field are used to generate synthetic PIV velocity fields and the analytic exact pressure field is used as a reference to validate the pressure field computed from the synthetic PIV velocity fields. In the simulated experiments the influence of resolution in space and time is considered, as well as the noise and spatial filtering caused by PIV algorithms, and the effects of 3D flow on planar determination of pressure. We look at two main settings, one that is typical for a planar or stereo-PIV experiment and one typical for a thin-volume tomo-PIV experiment. 3.1 Analytic flow field The flow field consists of a superposition of a Gaussian vortex and a uniform convective velocity in x-direction (see figure 1). The flow field relative to the vortex centre has Vθ as the tangential velocity in a cylindrical polar coordinate system moving with the vortex. The corresponding velocity and pressure distributions are given in (6) where the parameter cθ = rc

2/γ, with γ = 1.256431 to have the maximum tangential velocity at rc.

2 22

2 22

1 12

2 2 2 2

211 ;

2 4 2 2

r rr c cc

r rE Ec ce eV e p

r r r c r c

θ θθ θ θ

θθ θ

ρπ π

− −−

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎛ ⎞Γ Γ ⎝ ⎠ ⎝ ⎠⎜ ⎟⎜ ⎟= − = − − + + −⎜ ⎟⎜ ⎟

⎝ ⎠ ⎜ ⎟⎜ ⎟⎝ ⎠

(6)

For the simulation of PIV experiments the velocity is evaluated on a rectangular grid that cuts through the vortex (see figure 1). Two angles determine the orientation of the vortex with respect to the evaluation plane: α is the angle between the z-axis and the vortex axis and β is the angle between the x-axis and the vortex axis projection. The flow field is evaluated in the plane or in a thin-volume expansion of it, to simulate the stereo- and tomo-PIV data, respectively. 3.2 Parametric study The pressure is evaluated on a domain of 60 x 90 points (W x H), which is sufficiently large to represent a PIV result, the simulated thin volume had a total of 15 planes in z-direction to allow the Lagrangian approach to find points within the volume. For the pressure integration a Dirichlet condition is applied to the lower edge of the domain and for the Poisson approach additional Neumann conditions are enforced on the remaining boundaries (see figure 1). Three different locations are considered for the position of the vortex centre: one with the vortex in the centre of the domain and two near either the upper or the right boundary to simulate a vortex moving along a boundary and a vortex crossing a boundary respectively.


- 5 -

Figure 1: Analytic assessment configuration. Left: Vortex axis orientation with respect to the

evaluation domain, BC indicates the location of the Dirichlet condition. Right: tangential velocity distribution.

Eight different parameters were varied to investigate the influence on the pressure evaluation in

the different approaches. All parameters were normalized with respect to the the spatial and temporal steps sizes, h (= ∆x = ∆y) and ∆t respectively. The following parameters were evaluated: the convective velocity Uc, the peak tangential velocity Vθ,max, the vortex core radius rc, the angle between the z-axis and the vortex-axis α, the angle of the vortex-axis with respect to the to the Dirichlet boundary condition β, the simulated noise σsim added to the velocity field, the spatial filter from PIV (simulated by a moving average over the simulated window size), which is determined by the overlap-factor OF, and the distance to the edge of the pressure evaluation domain de. All the pressure results are scaled by the minimum analytic pressure pmin. Simulation of a viscous vortex and inclusion of viscous terms in the pressure determination was found to have no significant effect on the quality of the pressure evaluation and will therefore be discarded in the further discussion.

Table 1: List of different values for the parametric study. Parameter Values Parameter Values OF 0, 0.5, 0.75 Uc 0, 2, 4, 6, 8 h/∆t σsim 0%, 1%, 2%, 3%, 4% Umax α 0o, 20o, 40o, 60o, 80o Vθ,max 1, 2, 3, 4, 5 h/∆t β 0o, 30o, 60o, 90o rc 1, 2, 3, 5, 8 h de 1, 5, 9, 13, 17 h For each parameter the different values assessed in this study are listed in table 1. A total number of the order of 50,000 different parameter combinations were evaluated. To ensure that the influence of noise was captured correctly, each run with noise was repeated three times with a different noise distribution.

To illustrate the influences of the different parameters, a baseline case is defined, which has OF = 50%, σsim= 2% Umax, Vθ,max = 3 h/∆t, rc = 3 h, Uc = 4 h/∆t, α = 0o, β = 0o. Figure 2 shows the associated analytic velocity and pressure fields along with the simulated PIV velocity fields. The corresponding pressure fields for the four different approaches are shown in figure 3. It is clear that the pressure fields from the different methods are quite different and at this stage it is not clear yet, which step(s) in the pressure evaluation caused these differences. Because the results of the direct spatial integration approach are clearly poorer than the Poisson approach in both directional dependence and noise, they are excluded from further discussion.

To investigate the sources of the errors we break down the pressure evaluation for the Poisson approach in a number of subsequent steps that will indicate what part of the pressure evaluation is sensitive to a particular parameter. The break-down will be done in the following manner: first we assess the influence of the Poisson approach (P) itself by integrating the analytic pressure gradient


- 6 -

and evaluate the difference with respect to the analytic pressure field. Next we assess the influence of the pressure gradient determination (Eulerian, E. Lagrangian, L) approach by integrating the pressure gradient obtained with each approach and evaluate the differences with the previous step. Next, we assess the additional effects of the spatial PIV filter (F) and addition of noise (N). As indicated all the influences are evaluated in a cumulative way, i.e. with respect to the previous step.

Figure 2: Pressure and velocity fields for the base line case. Top: analytic results. Bottom: simulated

results. From left to right: pressure, u-component, and v-component.

Figure 3: Results from pressure evaluation. Top: pressure fields. Bottom: difference between the

calculated pressure field and the analytic pressure field (pana). From left to right: direct spatial integration approach combined with Eulerian form and with Lagrangian form, and Poisson

approach combined with Eulerian form and with Lagrangian form.

Figure 4 shows these influences and it can be seen that each approach has a different reaction to the different steps in the pressure evaluation process. The effects of the Poisson integration and the Eulerian determination of the pressure gradient are minimal. Surprisingly the effect of the Lagrangian approach shows a substancial difference of over 5% |pmin| around the vortex centre. This behaviour is observed for most cases. The effect of the simulated spatial PIV filter is as expected; it filters the structure and reduces the velocity around the peaks. The peak for the Lagrangian approach seems a bit wider and lower for this step. Finally, for both cases the noise affects the entire field, where the Lagrangian approach is less sensitive to it.

pana / |pmin| u ∆t / h v ∆t / h

p |pmin|

p-pana

|pmin|


- 7 -

Figure 4: Break-down of pressure evaluation results. Left: influence of the Poisson approach (P).

Top: Eulerian form (E). Bottom: Lagrangian form (L). From left to right: incremental influences of the pressure gradient determination, simulated spatial PIV filter (F), and simulated PIV noise (N).

3.3 Sensitivity study To provide an overall impression of how the parameters influence each pressure evaluation step a linear least-squares fit on the rms-errors around the base-line case is performed. In addition to the parameters already mentioned we apply the analysis to three different areas to identify in what region of the flow domain the error arises. This additional parameter (Cmp, compactness) shows whether the error introduced is located in the vortex core or is spread over the entire field; the higher the value the more the error is located in the vortex centre. Table 2 lists these linear sensitivity coefficients and the rms-error values for the baseline case. The rms-error values for the Poisson approach and Eulerian approach are small and they are biased to the centre, as was seen in figure 4. The Lagrangian approach shows a reasonable impact on the rms-error, which agrees with what is seen in figure 4. Most linear sensitivity coefficients are as expected. Larger values of σsim mean more rms-error, bigger structures (larger values for rc) lead to less rms-error (better resolved), and larger values of α mean more rms-error. The biggest contribution to the rms-error is the influence of noise σsim. There is one coefficient that is surprising though and that is the sensitivity coefficient of the Lagrangian approach to Vθ,max that is positive. This means a stronger vortex results in a higher rms-error value. This is a highly unwanted effect, since stronger vortices perturb the pressure field more and are therefore more important to fluid dynamics and force evaluation. This effect is most likely related to the second-order fluid path determination based on velocity fields, which consistently overestimates the material acceleration for the curved flow paths. To remedy this effect a much more elaborate data processing is needed, where the fluid flow paths should be based on the particle images and the points along the path determined by correlation rather than by a local flow expansion based on the velocity fields.

The total of all rms-error values and linear sensitivity coefficient are added and listed in the bottom of table 2. The Eulerain and Langrangian approach are fairly matched for the baseline case, even though they have errors occurring in different steps. The sensitivity of the methods to the size of the vortex rc is similar as is the sensitivity to 3D flow. The linear sensitivity coefficient of the pressure evaluation from planar-PIV to 3D flow is significant; adding for both approaches approximately a rms-error of circa 3 % |pmin| to the error per 10o of α, the inclination of the vortex axis with respect to the measurement plane. Only a small effect was found for the variation of β. The major differences between the methods are the sensitivities to σsim, Vθ and Uc. The Eulerian approach is sensitive to noise and convective velocity, but increasing the tangential velocity conversely improves the result of this approach. The Lagrangian approach is less sensitive to σsim and hardly affected by changing velocities.

∆p / |pmin|


- 8 -

Table 2: Results from the linear least-squares fit

Baseline Linear influences, ∆prms(~)/|pmin| prms/|pmin| OF σsim Vθ,max rc α β Uc Cmp. % %/OF %/(%Umax) %/(h/∆t) %/h %/10o %/10o %/(h/∆t) -

P 0.1 -0.2 0 0 -0.6 -0.1 0.1 0 1.9 E 0.1 1.4 0 -0.7 -0.6 3.2 -0.8 0.4 7.3 EF 0.4 7.1 0 -0.1 -1.0 -0.2 0.2 0.1 3.9 EFN 1.4 -0.4 2.0 -2.4 0.1 0.2 0.2 1.1 0.5 L 0.6 1.2 0 1.5 -1.0 2.6 -0.5 0 8.9 LF 0.3 5.0 0 -0.9 -0.5 -0.2 0.1 0 2.5 LFN 0.7 -0.2 0.7 -0.9 0 0.1 0.1 0.3 0.1 Total E 2.0 7.9 2.0 -3.2 -2.1 3.1 -0.3 1.6 -- Total L 1.7 5.8 0.7 -0.3 -2.1 2.4 -0.2 0.3 -- For the assessment of the edge effects we have performed a similar analysis as before. Instead of varying the angle at which the vortex-centre intersects the evaluation domain the distance to the edge of the domain was varied. The main effect was found in the pressure gradient determination step. For a vortex moving with Uc = 4 h/∆t parallel to a boundary an additional rms-error of 0.5-1.0 % |pmin| and 0.2-0.4 % |pmin| were found for the Eulerian and Lagrangian approaches, respectively. For a vortex moving with Uc = 4 h/∆t across an edge of the domain an additional rms-error of 1.0-1.5 % |pmin| and 0.5-1.0 % |pmin| were found for the Eulerian and Lagrangian approaches, respectively. The additional rms-errors increase with increasing Uc for both approaches.

To assess the improvement that can be made by capturing the complete 3D flow field the results from the simulated tomo-PIV were compared with the results of the simulated planar-PIV. The results showed that the influence of the angle with respect to the measurement plane is almost completely removed. The Lagrangian approach was found to be insensitive to the angles. The Eulerian approach showed to be insensitive to the angles for no convection. For a combination of convection of the vortex and misalignment of the vortex axis with one of the grid axis the approach suffers mildly from additional errors. No significant other changes were found in the behaviour of the pressure evaluation approaches.

In conclusion, the way to improve pressure determination from PIV based on these results is to reduce the noise, increase the temporal resolution (edge effects), and increase the spatial resolution. However, there are limits to what extent the accuracy can be increased. Once the time separation between velocity fields is the same as the laser-pulse time separation the increase of temporal resolution also results in an increase in noise (error of the cross-correlation with respect to the particle displacement). Increasing the OF increases the rms-error but also reduces the vector spacing h, therefore it has both negative and positive effects on the pressure evaluation that approximately cancel each other. Therefore an increase of resolution should preferably be done by increasing the sensor-size or by increasing the magnification. The increase of magnification is limited to still have a reference point (Dirichlet) in the domain where the pressure is known or where it is possible to obtain it via an equivalent of (5). The reader should be aware that these conclusions are based on a linear sensitivity study and a non-linear analysis needs to be made to assess the sensitivity at points significantly different from the baseline case. 4. Experimental assessment In previous experiments (de Kat, van Oudheusden & Scarano 2009a & 2009b) the flow around a square-section cylinder was observed to be predominantly 2D along the side of the cylinder and found to have considerable 3D structures in the wake. Using similar experimental PIV-data we will


- 9 -

describe the results of pressure evaluation and how they link with the analytic assessment. First we present the results from stereo-PIV to show what accuracy the pressure evaluation can achieve using planar PIV. Secondly, the tomo-PIV results are used to assess the influence of 3D flow effects and to what extent the pressure evaluation improves with inclusion of the 3D flow terms. Following the results of the analytical assessment, we restrict the assessment of the pressure integration to the application of the Poisson approach in combination with the Eulerian and the Lagrangian pressure gradient determination. 4.1 Experimental arrangement and procedure Experiments were performed in a low-speed, open-jet wind tunnel at the Aerodynamics laboratory at Delft University of Technology. The tunnel outlet had dimensions 400 x 400 mm. A square-section cylinder with dimensions 30 mm x 30 mm (D x D) and 345 mm in span was fitted with endplates and positioned in the middle of the free-stream. The geometric blockage was 6.5%. The nominal free-stream velocity U was 4.7 m s-1 (nominal dynamic pressure, q = 13.5 Pa), giving a Reynolds number ReD = UD/ν = 9,500. The main vortex shedding frequency was fs = 20 Hz, corresponding to a Strouhal number of St = fsD/U = 0.13. The free-stream turbulence intensity was approximately 0.1%.

The cylinder was instrumented with two flush mounted pressure transducers located in close proximity of the centreline/cross-sectional plane of the model to provide reference values for the pressure signals extracted from the pressure derived from the PIV data. For the stereo-PIV setup one transducer was located at the bottom of the cylinder and the other at the base (see figure 5). For the tomo-PIV setup both transducers were located at the base on either side of the measurement volume (see figure 5).

The pressure transducers, Endevco 8507-C1, have a range of 6895 Pa and a typical sensitivity of 25 µV Pa-1, and were unaffected by temperature under operating conditions (< 0.2% change). The transducers were calibrated, using a closed system with an U-tube, against a Mensor DPG 2001 (range 3447 Pa, uncertainty 0.010% FS). Signal recording was performed using a National Instruments data acquisition system operating at 10 kHz. The resulting noise level on the pressure measurements were smaller than 4µV giving a resolution of 0.3 Pa (twice the noise level). The zero-drift during each run was less than 2µV. Unless stated otherwise the pressure signals are unfiltered and not corrected for wind tunnel blockage effects. The pressure measurement on the side of the model during the stereo-PIV measurements suffered from laser-influences. The laser-influence was corrected by removing erroneous points; filling the gaps by interpolation and applying a low-pass filter in the form of a second-order fit (robust Loess) over 25 points. The power-spectrum was checked against a pressure measurement without laser-interference and was found to be unchanged for frequencies up to 400 Hz.

Figure 5: Experimental setup. Left: tomo-PIV setup (de Kat et al. 2009b). Right: pressure evaluation domains, red line indicates the location of the Dirichlet boundary condition.


- 10 -

A high-repetition-rate PIV-system was used to capture the flow. Flow seeding was provided by a Safex smoke generator, which delivered droplets of about 1 µm in diameter. The measurement plane was illuminated by a Quantronix Darwin-Duo laser system with an average output of 80 W at 3 kHz at a wavelength of 527 nm. The typical energy per pulse was 16 mJ at 2.7 kHz. The laser-pulse separation was 90 µs. Particle images were acquired by Photron Fastcam SA1 cameras (two cameras for stereo-PIV and four cameras for tomo-PIV) with a 1024 x 1024 pixel sensor (pixel-pitch 20 µm), recording image-pairs at 2.7 kHz, equipped with Nikon lenses with focal length 60 mm and aperture set at 2.8 (for tomo-PIV, the top cameras had an aperture set at 5.6). A total of 2728 image-pairs spanning just over 1s were captured for both PIV-setups.

For stereo-PIV the field-of-view (FOV) was 65 mm x 65 mm giving a digital resolution of 15.7 pixel mm-1. The final interrogation window-size of 16 x 16 pixels in combination with an OF of 50% gave a vector grid of 118 x 125 vectors (after cropping) with a vector-spacing of 0.51 mm.

For tomo-PIV the illuminated volume was 70 mm x 70 mm x 4 mm. It was captured with a digital resolution of 14.3 pixel mm-1. The final interrogation volume-size of 16 x 16 x 16 voxels with OF 50% gave a vector grid of 99 x 110 x 7 vectors (after cropping) with a vector-spacing of 0.56 mm. The vector fields were processed with a median test (Westerweel & Scarano 2005) combined with a multiple peak check. Remaining spurious vectors were removed and replaced using linear interpolation. The total number of spurious vectors was less that 2% for both data-sets.

The in-plane pressure gradient was determined using either the Lagrangian or the Eulerian approach. The pressure field was obtained by the Poisson approach with a Dirichlet condition on the lower edge of the domain and Neumann conditions on the remaining edges (see figure 5). 4.2 Estimating the uncertainty with the analytic test case results An estimate of the uncertainty of the pressure evaluation was made, based on the measurement uncertainty for the velocity field, the size of the observed structures, an estimate of the maximum tangential velocity field of the structures, and the convective velocity of these structures. Using a linear uncertainty-propagation we found the uncertainty associated with the instantaneous velocity (based on the free-stream displacement) is 1.5% U. An estimate of the normalized sizes, strength and convection of the vortices in the flow gives Vθ,max = 3 h/∆t, rc = 2-30 h, and Uc = 0-3 h/∆t. Using the results from the analytic test-case gives a rms-uncertainty estimate of 2-3% of pmin, which corresponds to a rms-uncertainty of about 10% q for both the Eulerian and the Lagrangian approach. Edge effects are estimated to be about 10% q. For the stereo-PIV experiment this results in an uncertainty of 10% q on the side pressure and 40% q for the base pressure, where the latter consists of uncertainty due to vortices moving, two edges close (see figure 5) to the base and about 10% (rough estimate) for the 3D effects. For the tomo-PIV there is only one edge close to the base so this results in an estimated rms-uncertainty of 30% for the 2D input and 20% for the 3D input. 4.3 Results Parts of the time series of the pressure signal from the transducer and of the pressure signal extracted from the pressure fields obtained from PIV for the stereo-PIV and tomo-PIV experiment are shown in figure 6. As expected from the results of the analytic test case, the Eulerian and the Lagrangian approach give almost indistinguishable results. On the left in figure 6 the results for the stereo-PIV experiment are shown. For the side pressure the time-series are in good agreement with each other. There is just a slight difference between the signal from the transducer and the signals obtained from PIV, possibly to errors in the reference pressure. The time-series for the base pressure show a reasonable overall agreement between the transducer signal and the signals obtained from PIV, but the latter look rather noisy. The results for the tomo-PIV experiment are shown on the right in figure 6. The time-series for the base from the 2D input (figure 6, top-right)


- 11 -

show a slight improvement with respect to the results from the stereo-PIV experiment, which is most likely due to the down-stream edge of the domain being farther away. When the full 3D input is used (figure 6, bottom-right) the pressure results obtained from PIV show an improved agreement with the signals from the transducers. The Lagrangian approach displays some occasional peaks. These peaks were found to coincide with images where the out-of-plane flow was that large that for some points the Lagrangian approach could not determine a material acceleration. Although these points were masked from the pressure evaluation they still affect the pressure field.

A statistical comparison of the signals is listed in table 3 for the stereo-PIV and tomo-PIV experiment. They confirm the observations from the time-series, showing a high correlation of and a small rms-difference between the PIV-based pressure and the transducer pressure for the side pressure, and an improvement of the correlation and the rms-difference for the base pressure when .

Figure 6: Time-series of pressure from PIV compared to surface pressure measurements. Top-left: stereo-PIV experiment, side pressure. Bottom-left: stereo-PIV, base pressure. Top-right: tomo-PIV,

base pressure, 2D input. Bottom-right: tomo-PIV, base pressure, 3D input. (1 out of 3 points marked for Lagrangian, 1 out of 4 for Eulerian)

Table 3: Statistical results for the pressure from the experimental test-cases

Cp Correlation coefficient ∆Cp, rms Stereo-PIV mean Rms Trans side Trans base Trans side Trans base

Transducer, side -1.576 0.778 -- -- -- -- Transducer, base -1.542 0.326 -- -- -- -- Side, Eulerian -1.613 0.657 0.981 -- 0.186 -- Side, Lagrangian -1.634 0.669 0.980 -- 0.180 -- Base, Eulerain -1.607 0.491 -- 0.573 -- 0.405 Base, Lagrangian -1.631 0.488 -- 0.578 -- 0.400

Cp Correlation coefficient ∆Cp, rms Tomo-PIV mean rms Trans 1 Trans 2 Trans 1 Trans 2

Transducer 1 -1.518 0.343 -- 0.816 -- 0.203 Transducer 2 -1.543 0.322 0.816 -- 0.203 -- Eulerain, 2D -1.488 0.429 0.639 0.639 0.337 0.333 Lagrangian, 2D -1.506 0.431 0.651 0.651 0.333 0.330 Eulerain, 3D -1.522 0.391 0.791 0.767 0.241 0.252 Lagrangian, 3D -1.437 0.417 0.682 0.666 0.310 0.314


- 12 -

3D flow information is added. Overall the results from the Eulerian and Lagrangian approach are very close for all cases, except the 3D Lagrangian, which suffered from the thin volume. For the stereo-PIV experiment the prediction based on the results from the analytic test case are reasonable for the uncertainty on the side pressure, where the unaccounted rms-uncertainty is probably due to the reference boundary condition. The prediction for the uncertainty on the base pressure is good. For the tomo-PIV results the predictions of the uncertainty are good, except for the 3D Lagragian. 5. Summary & conclusions Instantaneous planar pressure evaluation results are shown and evaluated for convecting Gaussian vortex and for a stereo- and tomo-PIV experiment on a square-section cylinder. The results from the analytic test case show that the main sources of error are the 3D flow effect, the temporal and spatial resolution, and measurement noise. Predictions based on the analytic results agree with the experimental results by comparing the difference between surface pressure signals obtained from PIV and measured by transducers. The results from the analytic test case show that improvements in accuracy of the pressure evaluation are possible. Acknowledgments This work was supported by the Dutch Technology Foundation STW, grant 07645. References Baur, T. & Köngeter, J. (1999), “PIV with high temporal resolution for the determination of local

pressure reductions from coherent turbulence phenomena”, 3rd International Workshop on Particle Image Velocimetry, Santa Barbara, 16-18 September.

Christensen, K. & Adrian, R. (2002), “Measurement of instantaneous eulerian acceleration fields by particle image accelerometry: methods and accuracy”, Exp. Fluids 33(6), 759-769.

Elsinga, G.E., Scarano, F, Wieneke, B. & van Oudheusden, B.W. (2006), “Tomographic particle image velocimetry”, Exp. Fluids 41(6), 933-947.

Fujisawa, N., Nakamura, Y., Matsuura, F. & Sato, Y. (2006), “Pressure field evaluation in microchannel junction flows through µPIV measurement”, Microfluid. Nanofluid. 2(5), 447-453.

Fujisawa, N., Tanahashi, S. & Srinivas, K. (2005), “Evaluation of pressure field and fluid forces”, Meas. Sci. Technol. 16, 989-996.

Gurka, R., Liberzon, A., Hefetz, D., Rubinstein, D. & Shavit, U. (1999), “Computation of pressure distribution using PIV velocity data”, 3rd International Workshop on Particle Image Velocimetry, Santa Barbara, 16-18 September.

Hosokawa, S., Moriyama, S., Tomiyama, A. & Takada, N. (2003), “PIV measurement of pressure distributions about single bubbles”, J. Nucl. Sci. Technol. 40(10), 754-762.

Jakobsen, M., Dewhirst, T. & Geated, C. (1997), “Particle image velocimetry for predictions of acceleration fielsd within fluid flows”, Meas. Sci. Technol. 8, 1502-1516.

de Kat, R., van Oudheusden, B.W. & Scarano, F. (2009a), “Instantaneous pressure field around a square-section cylinder using time-resolved stereo-PIV”, 39th AIAA Fluid Dynamics Conference, San Antonio, 22-25 June.

de Kat, R., van Oudheusden, B.W. & Scarano, F. (2009b), “Instantaneous pressure field determination in a 3D flow using time-resolved thin volume tomograhpic-PIV”, 8th International Symposium on Particle Image Velocimetry, Melbourne, 25-28 August.

Liu, X. & Katz, J. (2006), “Instantaneous pressure and material acceleration using a four-exposure


- 13 -

PIV system”, Exp. Fluids 41(2), 227-240. van Oudheusden, B.W. (2008), “Principles and application of velocimetry-based planar pressure

imaging in compressible flows with shocks”, Exp. Fluids 45(4), 657-674. Raffel, M., Willert, C.E., Wereley, S.T. & Komperhans, J. (2007), “Particle image velocimetry, a

practical guide”, Second edition, Springer-Verlag Berlin Heidelberg. Schröder, A., Geisler, R., Elsinga, G.E., Scarano, R. & Dierksheide, U. (2008), “Investigation of a

turbulent spot and a tripped turbulent boundary layer flow using time-resolved tomographic PIV”, Exp. Fluids 44(2), 305-316.

Westerweel, J. & Scarano, F. (2005), “Universal outlier detection for PIV data”, Exp. Fluids 39(6), 1096-1100.

Williamson, C.H.K. (1996), “Vortex dynamics in the cylinder wake”, Annu. Rev. Fluid Mech. 28(1), 477-539.

Instantaneous planar pressure from PIV: analytic and ...

Documents

Transcript of Instantaneous planar pressure from PIV: analytic and ...