Full-field pressure from 3D PIV snapshots in convective turbulent flow

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1 Full-field pressure from 3D PIV snapshots in convective turbulent flow Angeliki Laskari *, Roeland de Kat, Bharathram Ganapathisubramani Aerodynamics and Flight Mechanics Research Group, University of Southampton, Southampton, UK * correspondent author: A.Laskari@soton.ac.uk Abstract This paper presents full-field pressure determination from volumetric three-dimensional (3D) synthetic particle image velocimetry (PIV) velocity data. In view of the many difficulties involved in experimental direct pressure measurement techniques, indirect methods of acquiring pressure estimations are rapidly advancing. In the present work a technique that combines an Eulerian approach with Taylor s hypothesis for the computation of the flow acceleration is used (de Kat and Ganapathisubramani 2013). Independent time - frames from Direct Numerical Simulations (DNS) of a turbulent channel flow are used to create synthetic 3D PIV velocity volumes including the effect of noise and filtering inherent to PIV. Using the exact pressure field available from the DNS database, the method s accuracy is evaluated. The technique s dependence on grid resolution and noise level is also assessed, as well as the performance of different convection velocity approaches employed in the Taylor s Hypothesis formulation. The results indicate that the method can achieve a correlation of 0.75 with the exact pressure field, when using the mean streamwise velocity as the convection velocity. The proposed approach is sensitive to both noise and grid resolution, exhibiting a decrease in correlation of around 20% when the noise level is increased from 0% to 4% and also when the resolution is reduced by a factor of four. 1. Introduction Pressure estimation is of great interest in many engineering applications. It provides, together with velocity, a full description of the flow field. For bodies immersed in a fluid, pressure is the main factor contributing to the aerodynamic loads exerted on them. In aeroacoustics, estimation of pressure fluctuations is essential in noise quantification. Pressure is also responsible for cavitation phenomena in turbomachinery, marine propellers, valves and pumps. However, simultaneous direct measurement of full pressure and velocity fields is yet to be achieved experimentally. Surface pressure can be obtained using microphones or pressure transducers (whose performance is limited due to their intrusive nature), whereas for static pressure only point-wise measurements using probes are usually available, which are also intrusive and mainly provide mean flow quantities (van Oudheusden 2013). These difficulties or in some applications inability of existent techniques to directly measure pressure led to the development of indirect methods. More specifically, as indicated by the link between pressure and velocity in the Navier-Stokes equations, pressure can be obtained using velocity data. Therefore, these methods exploit the rapid development of particle image velocimetry (PIV) as a standard non-intrusive technique to acquire the velocity data, and based on the fluid flow equations, can provide a viable way to estimate pressure. In recent studies, Liu and Katz (2006) use a four-exposure PIV system to measure material accelerations, which are subsequently integrated employing an omni-directional virtual boundary integration scheme to obtain the pressure distribution. De Kat et al. (2008) obtain planar pressure fields around a squaresection cylinder using time resolved stereo PIV velocity data and compare the results with surface pressure measurements. Using artificial and real PIV velocity data, Charonko et al. (2010) present a detailed assessment of different Eulerian methods that calculate pressure, as well as their dependence on grid resolution, sampling rate, measurement error and off-axis recording. The material acceleration is an integral part of the Navier-Stokes equations and can also be determined with a Lagrangian approach. Using PIV on surface waves to predict flow accelerations and forces, Jakobsen et. al (1997) compare Eulerian and Lagrangian approaches and the results indicate that the former approach matches closely the analytical calculations while the latter - 1 -

2 exhibits a small bias, which leads to a systematic error in the estimation of flow acceleration. They also show that, the Lagrangian approach seems to be limited due to poor tracking or deformation of fluid volume, an observation that was also supported by de Kat and Oudheusden (2012), who performed stereo and Tomo-PIV measurements of a flow around a square section cylinder. They found that an Eulerian approach suffers more from measurement noise and is severely limited by advection of structures on the boundaries. Nonetheless, a Lagrangian approach, limited in time by the turnover time of the structures, exhibits more severe restrictions requiring thick measurement volumes to accurately reconstruct the fluid path (de Kat and Oudheusden 2012). In contrast to these findings, results from pressure field evaluation of rod-airfoil flow from time-resolved PIV (Violato et. al 2011) suggest that a Lagrangian approach manages a lower precision error with a larger timestep than an Eulerian one, which is again shown to suffer from measurement noise. In line with this, Ghaemi et al. (2012), using time-resolved, Tomo-PIV on turbulent boundary layer to estimate pressure, show that a Lagrangian formulation performs much better than an Eulerian one, when compared with direct surface pressure measurements. In view of these contrasting results, an alternative method for convective flows was proposed by de Kat and Ganapathisubramani (2013), which combines an Eulerian formulation but also avoids the aforementioned temporal limitations by using Taylor s Hypothesis for the estimation of flow acceleration. Implementation of the method on a synthetic flow field and on time-resolved stereo- PIV data from a turbulent jet indicate that it can accurately determine the pressure field, if an appropriate estimate of the convection velocity is chosen (de Kat and Ganapathisubramani 2013). In the present work, we apply the method proposed by de Kat and Ganapathisubramani (2013) to derive pressure from synthetic PIV velocity volumes obtained from DNS data from the John s Hopkins University Database (Graham et al. 2013, Li et al. 2008, Perlman et al. 2007). Different convection velocities are compared and the dependence of the method on noise level and grid resolution is also assessed. 2. Mathematic Formulation Throughout this paper, we use the coordinate system x, y, and z to denote the streamwise, wallnormal and spanwise directions respectively and u, v, w to denote the corresponding velocity components. The incompressible Navier Stokes equations can be rewritten, solving for the pressure gradient, as follows: where u is the velocity vector field, p is the pressure field, ρ is the fluid s density and ν the kinematic viscosity. Taking the divergence of the pressure gradient, the result is a Poisson equation (2.2), which can be solved by spatial integration using a Poisson solver (see de Kat and Ganapathisubramani 2013). The boundary conditions used are Neumann using eq. (2.1). The equation for the pressure gradient, besides convective and viscous terms that can be computed - 2 -

3 from the available velocity data, includes the acceleration term which is approximated using an Eulerian approach combined with Taylor s Hypothesis. Applying Reynolds decomposition, the velocity vector field u = (u, v, w, t) can be written as the sum of the temporal mean U = (U, V, W) and the fluctuations around it, u = (u, v, w, t): Taylor s hypothesis states that, if the mean velocity is significantly larger than the turbulent fluctuations, the turbulent eddies are frozen in time and are simply convected by the mean flow (Taylor 1938), with a convection velocity, U c = (U c, V c, W c): When only volumetric velocity data are available, the missing temporal information can be extracted from (2.3), given that an appropriate convection velocity is chosen: From this equation, it is clear that the choice of the convection velocity is fundamental in the proposed approach. Since the flow velocity is known, it is the choice of convection velocity that will determine the acceleration term and subsequently the pressure gradient. Using equation (2.4), the pressure gradient (eq. 2.1) can now be written as: Using the above equation, the pressure field can subsequently be estimated following equation Synthetic PIV volumes All the simulations presented throughout this paper are based on the DNS data from a turbulent channel flow, available from the John s Hopkins University Database (Graham et al. 2013, Li et al. 2008, Perlman et al. 2007). The main simulation parameters and flow statistics of the dataset are shown in Table 3.1 (nondimensional) and the grid spacing (in wall units) in Table 3.2: Friction velocity Reynolds number: Reτ x 10 2 Database timestep: δt Friction velocity: uτ x 10-2 Bulk velocity: Ub Viscosity: ν x 10-5 Domain Length: Lx x Ly x Lz 8πh x 2h x 3πh Grid: Nx x Ny x Nz 2048 x 512 x 1536 Table 3.1 Main simulation parameters and flow statistics x- direction: Δx y- direction- first point: Δy

4 y- direction- centreline: Δyc z- direction: Δz Table 3.2 Grid spacing From the full channel pressure and velocity datasets, an initial volume of 88x101x96 points in x, y, and z respectively, is chosen. The volume is located in the middle of the channel in the streamwise and spanwise direction and spreads from the viscous sublayer up to the wake region of the boundary layer (y + =2 up to y/δ=0.3) in the wall normal direction. In order to simulate the effect of synthetic PIV, the first step is to interpolate this data onto an equidistant grid in x, y, and z. Additionally, the data also needs to be filtered, since it has been shown that image cross correlation procedure of the PIV techniques have a similar effect as a moving average filter (Scarano and Riethmuller 2000). The interpolation onto an equidistant grid and the filtering are performed in a single step in order to avoid aliasing. For filtering a Hanning window is applied. Also, since the original data is non-uniformly spaced in the wall normal direction, the filtering in y is combined with a weighting function to counteract the bias induced due to the non-uniform spacing. The volume is subsequently cropped to avoid edge effects of filtering. The final volume, after interpolation, filtering and cropping is located entirely in the log region in the wall normal direction and in the middle of the channel width and length. The final size of the resulting FOV is 240x120x180 (in wall units) in x, y, and z respectively. In order to assess the dependence of the method in grid resolution, three different interrogation volume sizes are tested. Simulating an overlap factor (OF) of 75% in all three directions, the resulting number of vectors for each resolution is (Table 3.3): Interrogation volume size (l+) Grid spacing (Δh + =Δx + =Δy + =Δz + ) Resulting number of vectors (x,y,z) x41x x21x x11x16 Table 3.3 Vectors capturing the FOV, simulating an OF of 75% for three different resolutions. Besides the filtering effect mentioned above, PIV has been shown to create additional noise on the velocity field, on top of the physical noise originated from the particle motion (Foucaut et al 2004). To simulate this influence, separate random noise fields are generated, filtered and cropped, following the same procedure described above and subsequently added to the three velocity components. To evaluate the dependence of the proposed technique on the noise level, four different noise magnitudes are tested, each one scaled so that its root-mean-square (RMS) value, ε u, is given as a percentage of the maximum velocity occurring in the flow (de Kat and Ganapathisubramani 2013). In order to adequately assess the average performance of the technique, a sufficient number of different samples should be processed. Therefore, from the available time history 30 time frames are chosen, several timesteps away from each other so that they can be considered independent. All the performance measures that will follow are averaged over this set of samples. 4. Pressure Estimation - 4 -

5 Using the synthetic velocity field produced as discussed above and following the method outlined in the second paragraph, the pressure field can be estimated for the different grid resolutions and noise levels. As mentioned earlier, the performance of a Taylor s Hypothesis method mainly depends on the convection velocity chosen. In this study, apart from the grid and noise variations, three different approaches for the convection velocity are tested with respect to the accuracy of the pressure field estimation: Mean velocity (one dimension) For convective flows, one of the widely used approaches is to choose the mean velocity in the dominant direction as the convection velocity. Here, we are using the mean value of the streamwise velocity, U c=u(y), constant in the x-z plane and varying only in the wall normal direction, y. Filtered streamwise velocity Following de Kat and Ganapathisubramani (2013), a locally changing convection velocity improves the accuracy of the pressure estimation in the case of time-resolved stereo PIV measurements. To test this approach for volumetric PIV, in the present study the streamwise velocity component is filtered in the three dimensions: U c=u f, using a moving average with kernel l f, chosen such that the accuracy of the pressure estimation is maximised. Mean velocity (three dimensions) Even though in most convective flows there is a dominant direction, using only one component for the convection velocity in this direction might be too restrictive. Therefore, a third method is implemented as an extension to the first one, where the convection velocity is represented by a three dimensional vector. Each component is the mean of the respective flow velocity component: U c=u(y), V(y), W(y) constant in the x-z plane, in line with the process followed in the first approach. For the method s performance assessment, three different measures are computed: The normalised variance of the difference between the exact and estimated pressure field, S Pi : The correlation coefficient between the exact and estimated pressure fields, r Pi : The mean squared error between the probability density functions (pdfs) of the exact and estimated pressure fields, mse i: Where the brackets denote volumetric averaging, the subscript i denotes the case studied and n denotes the discrete values where the probability density function is computed. It should also be noted here that for the presentation of the results, unless indicated otherwise: U c=u(y), l + =12, ε u=1% 4.1 Noise Dependence As already mentioned, four different noise levels were filtered and added to the velocity field, - 5 -

6 with the Root-Mean-Squared (rms) value of the final noise fields, ε u, being a percentage of the maximum velocity present in the flow. The correlation coefficient, normalized variance and meansquared error of the pdf for the different convection velocities, are presented on Figures Figure Correlation coefficient for different noise levels, ε u (l + =12) Figure Normalised variance for different noise levels, ε u (l + =12) - 6 -

7 Figure Mean squared error for different noise levels, ε u (l + =12) As expected, as the level of noise on the velocity field increases, the estimated pressure fields are more weakly correlated to the exact one (Fig ). For all convection velocities, a similar decreasing trend can be observed for the correlation coefficient, which from around 0.75 for zero noise drops to 0.6 for the largest noise level (4% of the maximum velocity in the flow). From Fig it is also clear that the use of the streamwise mean as the convection velocity slightly outperforms the other two approaches, in terms of pressure correlation. The normalized variance of the difference between the exact and estimated pressure fields (Fig ) shows that increasing noise level results to an increase in variance for all the convection velocities. It is worth noting that, even though for zero noise level the use of a mean streamwise velocity results in a slightly lower variance, as the noise increases the filtered streamwise velocity approach becomes consistently better than the other two. This effect can be attributed to the smoother convection velocity field used for this approach (filtering of the noisy velocity field in the dominant direction of the flow), which partially attenuates the noise influence on the estimated pressure field. Finally, the mean squared error of the estimated pdf doesn t seem to follow a clear trend with increasing noise level (Fig ). Since the noise added is random, low noise levels might shift the probability density function of the estimated pressure field in a way that it is closer to the original one for some time frames. However when the noise reaches high magnitudes (4 and 5% of U max) the average error increases, denoting a significant deformation of the estimated pressure s pdf. The only exception to this trend is the case of using the filtered streamwise velocity as the convection velocity. For this approach the error decreases consistently for increasing noise and as also noted above, even though for zero noise level it underperforms with the respect to the other two approaches, the accuracy gets better for the highest noise levels. This can again be explained when taking into account the effect of the filtering on the noisy velocity fields, which becomes more pronounced as the noise increases. A clear representation of the effect noise has on the estimated pressure field is given on Figure 4.1.4, where pressure contours are plotted for the first time frame (U c=u(y), l + =12), for all the different noise levels tested. It is obvious that, for low noise levels the structures (Fig a, 4.1.4b) on the pressure field are still retained even though they are much noisier. However, as the noise level increases, only the largest structures can be identified, while the rest are severely deformed, explaining the correlation coefficient drop noted above

8 Figure Pressure contours of time frame No 1 (U c=u(y), l + =12), for different noise levels. (a) ε u/u max=0. (b) ε u/u max=1%. (c) ε u/u max=2%. (d) ε u/u max=3%. (e) ε u/u max=4%. 4.2 Resolution Dependence The three performance measures of the proposed technique are presented below (Fig ) for the different grid resolutions tested (Table 3.3) and for the three convection velocity approaches used. Figure Correlation coefficient for different resolutions, l + (ε u/u max=1%) - 8 -

9 Figure Normalised variance for different resolutions, l + (ε u/u max=1%) Figure Mean squared error for different resolutions, l + (ε u/u max=1%) The figures show that the decrease of grid resolution leads to poorer performance of the method, for all the convection velocities used and for all the measures computed. Similar to the influence of noise (Fig ), lower grid resolution results in a decrease of the pressure correlation coefficient from about 0.72 for the best resolution to about 0.6 for the lowest one (Fig ) Also in line with the previous results, the use of the streamwise mean as the convection velocity is slightly better than the other two approaches in terms of the correlation coefficient. Considering the normalised variance, the three approaches for the convection velocity perform almost identically, decreasing in accuracy with lower resolution (Fig ). However, in terms of the mean-squared error of the pressure s pdf, there is a notable difference in their performance, with the use of the mean streamwise velocity as the convection velocity providing more accurate results, whereas the use of a filtered streamwise velocity, although performing better with increasing noise (Fig ), clearly suffers in the case of low resolution (Fig ). This can be attributed to the fact that the kernel size used for the filtering was not grid dependent

10 Figure Pressure contours of time frame No 1 (U c=u(y), ε u/u max=1%), for different resolutions. (a) l + =12. (b) l + =24. (c) l + =48. In Figure 4.2.4, pressure contours for a constant wall normal location are plotted for all three different resolutions. It is apparent that, with lower resolution, even though the form of the structures is similar the distortion is quite significant, especially for l + =48, which reflects the much lower correlation coefficient and the higher variance and mean-squared error observed in the previous figures. 4.3 Convection velocity Dependence Using the three different convection velocities outlined above, the results for the average correlation coefficient, normalized variance and mean squared error are presented on Table U c=u(y) U c=u f U c=u(y), V(y), W(y) r Pi S pi mse i Table Average correlation coefficient and normalised variance for the 3 different convection velocities (l =12, ε + u/u max=1%) The results show that for this resolution and noise level, using the streamwise mean velocity as the convection velocity is the most accurate approach in terms of all three measures, in line with the previous findings. The average correlation coefficient for all the snapshots tested is around 0.75, and about 6% larger than for the other two approaches. With respect to the normalized variance and mean-squared error of the pressure s probability density function it is also clear that using the first approach for the convection velocity provides more accurate results than the other two. In figure 4.3.1, pressure contours of the first time frame are presented (l + =12, ε u/u max=1%) that show the exact and estimated pressure fields using the three different convection velocity approaches. The results support the previous findings, since the pressure field estimated using the mean streamwise velocity as the convection velocity, figure 4.3.1(a), exhibits the most accurate representation of the exact field

11 Figure Pressure contours of time frame No 1 (l + =12, ε u/u max=1%). (a) DNS field. (b) U c=u(y). (c) U c=u f. (d) U c=u(y), V(y), W(y). 5. Conclusions Taylor s hypothesis combined with an Eulerian approach is used to estimate pressure from 3D synthetic PIV data. Based on DNS snapshots from a turbulent channel flow, a synthetic PIV dataset is created by spatial filtering and noise adding on the velocity field. Using the pressure field from DNS, the performance of the method is assessed in terms of grid resolution, noise level and convection velocity used. The results indicate that in terms of the convection velocity, the most accurate approach is to use the mean streamwise velocity which attains, for zero noise and the best resolution, a pressure correlation coefficient of about By increasing the noise level, the method becomes less accurate and for the maximum noise level in the best resolution the correlation coefficient drops to 0.6. A similar loss of accuracy is observed with decreasing resolution. Future analysis of the full time history of the flow will enable the implementation of a purely Eulerian technique for the estimation of acceleration as well as a Lagrangian technique in order to compare their performance with the accuracy provided by the proposed method. Acknowledgements The research leading to these results has received funding from the European Research Council under the European Union's seventh framework program (FP7/ ) / ERC grant agreement no (WBT project) and grant agreement no (NIOPLEX project). References

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