On optimal interpolation schemes for particle tracking in turbulence

An important aspect in numerical simulations of particle-laden turbulent ¿ows is the interpolation of the ¿ow ¿eld needed for the computation of the Lagrangian trajectories. The accuracy of the interpolation method has direct consequences for the acceleration spectrum of the ¿uid particles and is therefore also important for the correct evaluation of the hydrodynamic forces for almost neutrally buoyant particles, common in many environmental applications. In order to systematically choose the optimal tradeo¿ between interpolation accuracy and computational cost we focuss on comparing errors: the interpolation error is compared with the discretisation error of the ¿ow ¿eld. In this way one can prevent unnecessary computations and still retain the accuracy of the turbulent ¿ow simulation. From the analysis a practical method is proposed that enables direct estimation of the interpolation and discretization error from the energy spectrum. The theory is validated by means of Direct Numerical Simulations (DNS) of homogeneous, isotropic turbulence using a spectral code, where the trajectories of ¿uid tracers are computed using several interpolation methods. We show that B-spline interpolation has the best accuracy given the computational cost. Finally, the optimal interpolation order for the di¿erent methods is shown as a function of the resolution of the DNS simulation