10TH INTERNATIONAL SYMPOSIUM ON PARTICLE IMAGE VELOCIMETRY – PIV13 Delft, The Netherlands, July 1-3, 2013

Vortex-in-Cell method for time-supersampling of PIV data

Jan Schneiders1, Richard P. Dwight1 and Fulvio Scarano1

1 Aerospace Engineering, Delft University of Technology, Delft, The Netherlands janschneiders@gmail.com, r.p.dwight@tudelft.nl

ABSTRACT The present work investigates the use of a vortex-in-cell (VIC) inviscid, incompressible Navier-Stokes solver [1,2] to increase temporal resolution of time-resolved 3D fluid-velocity data obtained from tomographic PIV experiments. The measurement rate in such experiments is limited by available laser power, and the requirements of laser pulse energy and quality of particle image digital recordings. The principle of time-supersampling is that by using an interpolation based on a largely complete description of the flow-physics, high spatial-resolution can be leveraged to increase low temporal-resolution. The numerical solver simulating the fluid is applied on a domain corresponding to the 3D measurement volume, and time-integration is performed between each pair of consecutive measurements. Initial conditions are taken from the first measurement field, and time-resolved boundary conditions can be approximated either by linear interpolation or advection-model-based interpolation between the two fields. To obtain a continuous representation of the velocity in time, a weighted-average of forward- and backward-time integration is made. To make the backward integration problem well-posed, viscosity is neglected. This is justified given the short time-interval between measurements. The supersampling method is applied to a turbulent wake, and a cylindrical jet, and compared to reference high-frequency experimental data, as well as an advection-model-based supersampling approach. In both test cases temporal resolution substantially above the Nyquist limit is achieved. INTRODUCTION The measurement rate for time resolved 3d tomographic PIV is at the moment insufficient for aerodynamic problems where the flow velocity may approach 100 m/s. The present work investigates the use of a vortex-in-cell (VIC) inviscid, incompressible Navier-Stokes solver [1,2] to increase of temporal resolution of time-resolved 3D velocity data obtained from tomographic PIV experiments. We name this procedure time-supersampling. The objective is to achieve a temporal resolution not available to the original measurement system. The unsteady Navier-Stokes equations are solved using VIC, on the experimental measurement domain, starting from an initial condition given by one measurement field. Appropriate boundary conditions must be specified, and smooth matching of the solution at the measurement times is necessary. As a side-effect the unsteady pressure field is also computed. The work follows a previous study [4] that investigated the use of the advection equation for the same purpose. There a significant increase the temporal resolution of PIV time-series was demonstrated for certain classes of flows. For example the method showed generally good accuracy in flows with strong uniform background advection, where the turbulence was approximately frozen. Less accurate results were obtained in a jet into quiescent fluid, where the free shear layer is dominated by strong vortices that roll-up under the effects of the Kelvin-Helmholtz instability. This is a direct consequence of the advection model not including the non-linear physics. In this study the advection model is compared against VIC. Other related work has involved filling spatial gaps in PIV data using a N-S solver [3]. The VIC method only applies to substantially divergence-free velocity fields. Therefore the study does not treat the case of 2d N-S and planar PIV measurements, where the divergence-free condition is almost never met. Instead, the work concentrates on 3d datasets produced from tomographic PIV, where experimental velocities are available everywhere in the domain in which the governing equations are solved. The paper first introduces the VIC method. Then two cases are considered: the first one is the fully developed turbulent flow at the trailing edge of a NACA airfoil [5]; the second case is a transitional jet in water [6] with a vortex-dominated flow field. In both cases tomo-PIV measurements well-resolved in time are available. This data is first under-sampled by some factor and then the VIC method is applied to this reduced data-set. The removed samples act as reference solutions for evaluating the accuracy of the reconstruction. The method is also compared to linear interpolation (no physics) and the advection model (linear physics). In both cases the VIC method is able to accurately reconstruct the velocity time series. Notably in the jet case, this remains true even when the sampling rate is reduced below the shedding frequency of the vortices.

VORTEX-IN-CELL METHOD The VIC method first appeared in Christiansen [1] in 1973. The method is based on a rewriting of the inviscid, incompressible Navier-Stokes equations in terms of vorticity. The vorticity transport equation is: 𝜕𝝃

𝜕𝑡+ 𝒖 ⋅ 𝛁 𝝃 = 𝝃 ⋅ 𝛁 𝒖, (1)

where the velocity field is related to the 3d vorticity distribution by a Poisson equation: Δ𝒖 = −𝛁  ×  𝛏 (2) Initial conditions on vorticity are computed from experimental data at time 𝑡!, time-dependent boundary conditions are based on interpolation of experimental data. This is proposed as a model for the fluid in the measurement domain in our experiment, and the effectiveness of the supersampling will depend upon the quality of this model. For example we rely on an almost incompressible (typically also isothermal) fluid. If the model is 2d then we require that - in the measurement domain - the flow is almost 2d. This is rarely the case to sufficient accuracy in practice. One consequence is that 2d PIV data has non-zero divergence, making it unsuitable as an initial condition for the divergence-free field 𝑢. Therefore to eliminate this source of error we proceed directly to the 3d case using tomo-PIV data. In the applications presented later, the main error in this model is due to neglecting viscosity. This approximation is made to preserve the time-reversibility of the governing equations, and this is necessary for our method of matching solutions at final times. This is a negligible error under the circumstances of very short integration times (between two consecutive measurements) and convection dominated flow, as we expect to occur in our applications. NUMERICAL TREATMENT A full discussion of the VIC method is given in [2]. The basic principle is to discretize the vorticity distribution by a limited number of point vortices, 𝝃 𝒙, 𝑡 = 𝜞!

!

𝛿[𝒙 − 𝒙! 𝑡 ] (3)

where 𝜞! is an estimate of the circulation in a small region Vp around the vortex particle at location xp.

𝜞! = 𝝃  𝑑𝒙!!

(4)

In case of 3d PIV data produced from tomographic PIV measurements, it is convenient to initialize a vortex particle at each grid node making use of the measured velocity field, i.e. use the uniform Cartesian grid defined by the experimental data-set. The vorticity field is evaluated using central differences. The distribution of point vortices is integrated in time using the following procedure:

1. Advect the vortex particles: 𝑑𝒙!

𝑑𝑡= 𝒖(𝒙!, 𝑡)

(5)

2. Update the particle circulation to account for vortex stretching: 𝑑𝜞!

𝑑𝑡= 𝜞! ⋅ 𝛁𝒖(𝒙!, 𝑡) (6)

3. Evaluate the updated vorticity field at grid nodes:

𝝃 𝒙!, 𝑡 =1ℎ!

𝜞!𝜑𝒙! − 𝒙!(𝑡)

ℎ!

(7)

where h is the mesh cell width and 𝜑 𝒙 = 𝜑 𝑥  𝜑 𝑦  𝜑(𝑧) with

𝜑 𝑥 =

0 𝑥 > 2122 − 𝑥 !(1 − 𝑥 ) 1 ≤ |𝑥| ≤ 2

12(2 − 5 𝑥 ! + 3 𝑥 ! |𝑥| ≤ 1

radially symmetric interpolation function, introduced by Monaghan [7]. The function has been used successfully for this purpose in general VIC codes—see for example in [2,8,9]. Using this function, each particle can influence 4×4×4 neighbouring particles. To allow the assignment procedure to be vectorized, vortex particles are ordered such that they are at least five grid points apart and assign their strength to different subsets of the mesh, as suggested in [10].

4. Calculate the velocity field on the grid using (2).

Solving the Poisson equation in the last step is done using a Fast Poisson solver based on the Fourier transform. The time integration is performed time-marching from a given PIV snapshot towards the subsequent one. As in previous work based on the advection equation [4], the time-marching procedure is also performed backwards (i.e. from the subsequent snapshot to the previous one), and the forward and backward velocity fields are averaged with a weight corresponding to the distance in time from their respective initial conditions. This operation ensures that the time-history remains continuous also at the measurements times, although it will no longer satisfy exactly the N-S equations. The necessity of backward integration means the governing equations must be time-reversible, hence viscosity is not allowed. INITIAL CONDITIONS AND BOUNDARY CONDITIONS Initial conditions are taken by calculating the vorticity from the first measured velocity field. Time-resolved velocity boundary conditions required for solving the Poisson equation can be imposed either with a linear interpolation or better, making use of an advection-based interpolation as proposed in [4]. The vorticity values on the domain boundary need to be corrected for the lack of vortex particles just outside of the boundary. This can be done using an iterative approach as suggested in [2], however, to allow for cheaper computation an estimate of the vorticity field can be calculated from the advection-based interpolation between the measurements and overlaid over the boundaries. EXPERIMENTAL ASSESSMENT The VIC method is utilized to supersample PIV time-resolved sequences and its accuracy is scrutinized considering two test cases that are detailed in literature and where the datasets are available to the authors. The method is compared with two other reconstruction techniques: the first one is a point-wise linear interpolation; the second one makes use of the advection equation as described in [4]. NACA AIRFOIL TRAILING EDGE FLOW The first case is the fully developed turbulent boundary layer leaving the trailing edge of a NACA 0012 airfoil [5]. This case has been chosen in order to verify whether the VIC method is able to reconstruct the

time-resolved velocity fields in conditions close to “frozen turbulence”. Here the advection-based model has been demonstrated to yield results with good accuracy and is therefore considered as a reference technique.

Free stream velocity 14 m/s Repetition rate 2700 Hz Measurement field 47 × 47 × 8 mm Interrogation volume (IV) 32 × 32 × 32 voxel (1.47 × 1.47 × 1.47 mm3) Vectors per field 128 × 128 × 22

Table 1 – Experimental parameters for the turbulent wake flow as done by Ghaemi [5]

The flow in the near wake of the airfoil features the typical pattern of fully developed boundary layers with elongated regions at alternating low and high-speed streaks (Figure 1). Table 1 lists the relevant experimental parameters and a full description of the flow field and experiment is presented in Ghaemi [5].

Figure 1 – Tomographic experiments at the trailing edge of a NACA airfoil. Schematic layout of experiment (left) and sample instantaneous flow field (right). Organization of the low-speed (blue/dark grey) and high speed (green/light grey) streaks visualized by iso-surfaces of u/U∞ = ±0.1. Axes are scaled to momentum thickness of the boundary layer at the airfoil edge (repr. from Ghaemi et al. [5]). The assessment is performed by initially sub-sampling the available time series (open circles in Figure 2) to a coarser set of time samples (filled circles). Subsequently, the time-supersampling techniques are applied and the resulting velocity fields are compared to the original time series. The Sub-Sampling Factor (SSF) defines the extent to which the data is sub-sampled.

SSF =𝑓!"#$%&"!"'(𝑓!"#!!"#$%&'

(1)

The relative error is considered as the Euclidean norm for all velocity components normalized by a reference velocity. A statistical estimator is obtained by averaging in time at each grid point. In the present case of the turbulent wake, the reference speed is chosen to be the free stream velocity of 14 m/s.

ϵ =1V!"#

   1𝑁   𝒖!"",! − 𝒖!"#$,!

!  !

!!!

!/!

(1)

The time history of the velocity at a point chosen in the center of the measurement domain is shown in Figure 2 for the case of SSF = 6. In these conditions the measurements have been sub-sampled from 2700 Hz to 450 Hz. As expected, the linear interpolation fails to reconstruct the temporal velocity fluctuations caused by the convection of turbulent fluctuations. In contrast, both the advection and VIC method are quite adequate in this case as they yield a velocity field reconstruction close to that of the fully sampled data series. The spatial distribution of the statistical error along a plane parallel to the airfoil is reported in Figure 3. It can be seen that the error obtained by the VIC method is in the same order of magnitude as the advection model error. Furthermore, especially for the larger SSF = 8 it can be seen that the VIC error is slightly lower. Near the boundaries the advection and vortex methods yield essentially equivalent results, because the advection-based interpolation has been used to set boundary conditions for the VIC method. Further away from the boundaries, the VIC method improves upon the advection model. In Figure 3 it can be seen that as expected for a larger SSF the relative error is larger. The effect of increasing the time separation between subsequent measurements used in the calculation (i.e. virtually decreasing the measurement rate) is monitored by following the average relative error, evaluated over the whole domain and over a long sequence of 50 snapshots. Figure 4 confirms that indeed the error increases with increasing SSF. Furthermore, the error of the linear point-wise interpolation increases rapidly and as also observed in Figure 2, the linear-interpolation is rapidly inadequate. Already at very low SSF, the error made by the linear-interpolation is in the order of the RMS velocity fluctuations. Figure 4 shows a substantially better reconstruction is obtained by both the advection and vortex models. The models perform similar, with the VIC method achieving slightly smaller errors as also seen in Figure 3.

Figure 2 - Time history of two velocity components in the center of the measurement domain, as calculated from velocity fields sub-sampled with SSF = 6.

Figure 3 - Contour plot of the relative error in a plane parallel to the airfoil for SSF = 4 and SSF = 8

Figure 4 – Average relative error calculated over the whole domain for both the wake and jet cases

TRANSITIONAL CIRCULAR JET The second case is that of a transitional circular jet measured in a water tank. For this flow, less accurate results were obtained using the advection model and therefore it is chosen as a second test case for the VIC method. The free shear layer is dominated by strong vortices that roll-up under the effect of Kelvin-Helmholtz instability and the assumption of frozen turbulence is not valid in this region. The relevant experimental parameters are listed in Table 2 and full details of the experiment can be found in [6]. For reference the flow is visualized in Figure 5, which shows an instantaneous velocity vector slice in the axial plane. Figure 6 shows the time history of the radial and axial velocity components in a point in the shear layer. As can be seen both the linear interpolation and the advection model fail to give a correct representation of the velocity fluctuations. This is expected, as the sampling frequency has been reduced from 1000 Hz to 25 Hz, which is below the frequency of approximately 30 Hz at which vortex are shed. The VIC method however is able to accurately reconstruct the velocity fluctuations.

Jet velocity 0.5 m/s Repetition rate 1000 Hz Measurement field Cylinder 30 mm (d) × 50 mm (h) Interrogation window (IO) 40 × 40 × 40 voxel (2 × 2 × 2 mm) Vectors per field 61 × 102 × 61

Table 2 – Experimental parameters for the transitional jet as done by Violato and Scarano [6]

Figure 5 – Instantaneous velocity and vorticity field in a transitional jet. Velocity vectors on an axial plane. Cyan iso-surface for azimuthal vorticity. Red iso-surface for positive axial velocity fluctuations. Yellow/green iso-surfaces for

positive/negative streamwise vorticity (repr. from Violato and Scarano [6]).

2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

SSF

Aver

age

rela

tive

erro

r

Average Relative Error for Turbulent Wake case

Linear InterpolationAdvection ModelVortex Model

5 10 15 20 25 30 35 400

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Aver

age

rela

tive

erro

r

SSF

Average Relative Error for Transitional Jet case

Linear InterpolationAdvection ModelVortex Model

Figure 6 – Time history of the radial and axial velocity in the jet, calculated using a linear interpolation, the advection

model and the proposed vortex model from measurements sub-sampled from 1000 Hz to 25 Hz.

Figure 7 – Contour plot of the relative error in the axial plane of the transitional jet at SSF = 24 and SSF = 40 The relative error is plotted in the axial plane in Figure 7. For the jet case the error has been normalized with the jet velocity of 0.5 m/s. As can be seen the vortex model gives a substantial improvement over the advection model and reduces the reconstruction error to acceptable values. This implies that using the vortex model, the experiment can be performed with a repetition rate of 25 Hz, which is far below the repetition rate dictated by the Nyquist frequency based on vortex shedding at 30 Hz. Figure 3 shows also for the jet case the average relative error in the domain. One can see the dramatic decrease in effectiveness of the advection model in situations where the frozen turbulence assumption is not valid. The vortex method on the other hand allows for accurate reconstruction of the velocity field, even at high sub-sampling factors.

CONCLUSIONS A Vortex-in-Cell (VIC) Navier-Stokes solver is proposed to increase the temporal resolution of time-resolved tomographic PIV experiments. The numerical solver is applied on a domain corresponding to the 3D measurement volume, and time-integration is performed between each pair of consecutive measurements to achieve a temporal resolution not available to the original measurement system. The measurements serve as initial conditions for the VIC method simulating the fluid and in this way the spatial resolution available by the measurements is projected into time. The VIC method is compared to a linear interpolation (no physics) and an advection based approach (linear physics) to increase the temporal resolution of measured time series in two experimental test cases. First, in the case of the turbulent wake, where Taylor’s hypothesis of frozen turbulence holds to a large extent, both the advection model and VIC method are able to accurately reconstruct the velocity fluctuations. Second, in the case of a transitional jet, where the free shear layer is dominated by strong vortices that roll-up under the effect of the Kelvin-Helmholtz instability, the advection model yields less accurate results. At high sub-sampling factors, the advection model fails to represent the velocity fluctuations. The VIC method on the other hand is able to accurately reconstruct the velocity time series, even when the sampling rate is reduced below the Nyquist frequency.

0 10 20 30 40 50 60 70 80−3

−2

−1

0

1

2

3u velocity component (radial) − R/D = 0.6, Y/D = 3, Z/D = 0

radi

al v

eloc

ity [v

oxel

s/m

s]

time [ms]0 10 20 30 40 50 60 70 80

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5v velocity component (axial) − R/D = 0.6, Y/D = 3, Z/D = 0

axia

l vel

ocity

[vox

els/

ms]

time [ms]

Reference Measurements (1000 Hz)Sub−Sampled Measurements (25 Hz)Linear InterpolationSuper−Sampled with Advection ModelSuper−Sampled with Vortex Model

0 0.05 0.1 0.15 0.2 0.25 0.3

R/D

SSF = 40

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

0 0.05 0.1 0.15 0.2 0.25 0.3

R/D

SSF = 24

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

0 0.05 0.1 0.15 0.2 0.25 0.3

R/D

SSF = 40

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

0 0.05 0.1 0.15 0.2 0.25 0.3

R/D

SSF = 24

Y/D

Advection Model Vortex Model

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

The study demonstrates that the sampling rate requirements for tomographic PIV can be strongly reduced when the data is time super-sampled using the Vortex-in-Cell method. As anticipated, this can allow for larger measurement domains, higher spatial resolution and new flow cases to be analyzed using tomographic PIV, at lower measurement rates than dictated by principles such as the Nyquist frequency. REFERENCES

[1] Christiansen JP, “Numerical Simulation of Hydrodynamics by the Method of Point Vortices” J. of Comput. Phys. 13 (1973) 363 [2] Cottet GH and Koumoutsakos P, “Vortex Methods – Theory and Practice” New York: Cambridge University Press (2000) [3] Sciacchitano A, Dwight RP, Scarano F, “Navier-Stokes simulations in gappy PIV data” Exp. Fluids 53 (2012) 1421 [4] Scarano F and Moore PD, “An advection-based model to increase the temporal resolution of PIV time series” Exp. Fluids 52 (2012) 919 [5] Ghaemi S and Scarano F, “Counter-hairpin vortices in the turbulent wake of a sharp trailing edge” J. Fluid Mech. 689 (2011) 317 [6] Violato D and Scarano F, “Three-dimensional evolution of flow structures in transitional circular and chevron jets” Phys. Fluids 23

(2011) 124104 [7] Monaghan J, “Extrapolating B splines for interpolation” Journal of Computational Physics 60 (1985) 253 [8] Cottet GH and Poncet P, “Advances in direct numerical simulations of 3D wall-bounded flows by Vortex-in-Cell methods” Journal of

Computational Physics 193 (2004) 136 [9] Kosior A and Kudela H, “Parallel computations on GPU in 3D using the vortex particle method” Computers & Fluids 80 (2013) 423 [10] Walther J and Koumoutsakos P, “Three- Dimensional Vortex Methods for Particle- Laden Flows with Two-Way Coupling” Journal of

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