Development of a Second-Order Corrector for an Accurate Laplacian Operator in SPH

A new numerical method, called Smoothed Particle Hydrodynamics (SPH) is applied in the field of aeroengines. In the last decade, SPH has shown promising advantages such as the natural advection of the phase interface without numerical diffusion, in case of multiphase flows. This makes it a serious alternative to traditional grid-based methods, leading to a numerous number of industrial applications. However, this method suffers from numerical weaknesses. For instance, it is observed that the nominal Laplacian operator, involved in viscosity models, fails to retrieve first order consistency. ... mehrTo circumvent this problem, a sophisticated mathematical correction is required to obtain high-fidelity results.
In this paper, a high order corrector for the Laplacian operator is implemented into the SPH code developed at the Institute for Thermal Turbomachines (KIT-ITS). The corrector is validated against a generic quadratic field and the results are compared to common approaches to estimate the Laplacian. The validation confirms the high accuracy of the corrected Laplacian operator, even for disordered particle lattices. In a second step, the corrector is adapted to describe non-Newtonian multiphase flows. Based on this adaptation, a hybrid viscosity model is developed. It combines advantages of the traditional approach and the corrected Laplacian operator. The new viscosity model is applied to air assisted atomization in the context of biomass conversion to synthetic fuels. The simulation consists of a 2D configuration of a co-axial liquid/gas nozzle. The results were compared to (i) the traditional approach for the viscous term and (ii) to experimental results provided by the Institute of Technical Chemistry (KIT-ITC). In terms of spray morphology, the hybrid model shows more similarities to the experimental data compared to a simulation that only used the traditional viscosity model. Moreover, the frequency of the primary Kelvin-Helmholtz instability can be predicted with a deviation of only 16% compared to the experimental data whereas the traditional approach shows a deviation of 60%.

A new numerical method, called Smoothed Particle Hydrodynamics (SPH) is applied in the field of aero-
engines. In the last decade, SPH has shown promising advantages such as the natural advection of the
phase interface without numerical diffusion, in case of multiphase flows. This makes it a serious alterna-
tive to traditional grid-based methods, leading to a numerous number of industrial applications. However,
this method suffers from numerical weaknesses. For instance, it is observed that the nominal Laplacian
operator, involved in viscosity models, fails to retrieve first order consistency. ... mehrTo circumvent this problem, a
sophisticated mathematical correction is required to obtain high-fidelity results.
In this paper, a high order corrector for the Laplacian operator is implemented into the SPH code developed
at the Institute for Thermal Turbomachines (KIT-ITS). The corrector is validated against a generic quadratic
field and the results are compared to common approaches to estimate the Laplacian. The validation
confirms the high accuracy of the corrected Laplacian operator, even for disordered particle lattices. In
a second step, the corrector is adapted to describe non-Newtonian multiphase flows. Based on this
adaptation, a hybrid viscosity model is developed. It combines advantages of the traditional approach
and the corrected Laplacian operator. The new viscosity model is applied to air assisted atomization in
the context of biomass conversion to synthetic fuels. The simulation consists of a 2D configuration of a
co-axial liquid/gas nozzle. The results were compared to (i) the traditional approach for the viscous term
and (ii) to experimental results provided by the Institute of Technical Chemistry (KIT-ITC). In terms of spray
morphology, the hybrid model shows more similarities to the experimental data compared to a simulation
that only used the traditional viscosity model. Moreover, the frequency of the primary Kelvin-Helmholtz
instability can be predicted with a deviation of only 16% compared to the experimental data whereas the
traditional approach shows a deviation of 60%.