PIV13-VICSupersampling Schneiders Richard

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    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: + = , (1) where the velocity field is related to the 3d vorticity distribution by a Poisson equation: = (2) Initial conditions on vorticity are computed from experimental data at time !, 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 . 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, , = !! [ ! ] (3) where ! is an estimate of the circulation in a small region Vp around the vortex particle at location xp. ! = !! (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: ! = (!, )


    2. Update the particle circulation to account for vortex stretching: ! = ! (!, ) (6)

  • 3. Evaluate the updated vorticity field at grid nodes:

    !, = 1! ! ! !()! (7)

    where h is the mesh cell width and = () with

    = 0 > 212 2 !(1 ) 1 || 212 (2 5 ! + 3 ! || 1

    radially symmetric interpolation function, introduced by Monaghan [7]. The function has been used successfully for this purpose in general VIC codessee for example in [2,8,9]. Using this function, each particle can influence 444 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 scr