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Theme : "Mechanical Engineering Congress, 2024 Japan (MECJ-24)" Preface Hideo MORI, Tetsuya KANAGAWA Modeling of the Reynolds stress transport equation – use of generative AI Shinnosuke OBI (Keio University) Principled thinking: Numerical simulation of fluid flow and turbulence analysis Yohei MORINISHI (Nagoya Institute of Technology) Performance Evaluation of Thrust Increasing Guide for Drone Atsushi KASE (University of Toyama) Flow measurements on wind turbine research Yasunari KAMADA (Mie University) PIV Measurement of Mini Centrifugal Pump Toru SHIGEMITSU (Tokushima University)

 

Principled thinking: Numerical simulation of fluid flow and turbulence analysis

Yohei MORINISHI Nagoya Institute of Technology

Abstract

In this keynote lecture, fully conservative convection schemes for incompressible flows ( Morinishi (1995), Morinishi et al. (1998) ), for compressible flows ( Morinishi (2010) ), and for flows on moving grid ( Morinishi and Koga (2014) ) were introduced.

The fully conservative convection schemes are the finite difference schemes for the convection term which preserve primary and secondary conservation quantities simultaneously. Therefore, they are now recognized as useful tools for unsteady turbulence simulations like direct numerical simulation (DNS) and large eddy simulation (LES). The key principles of the scheme construction are; (i) analytical relations for the flow equations (commutability among the convection forms with the aid of the continuity and conservation properties), (ii) discrete analogue of the Leibniz rule, and (iii) analogy between incompressible and compressible flow equations. The divergence and advection forms are commutable with the aid of the continuity, and the skew-symmetric form is defined as the average of the divergence and advection forms. In addition, the divergence form is (primary) conservative a priori, and the skew-symmetric forms is secondary conservative a priori. In order to construct the convection schemes, some discrete operators (finite difference, interpolation, and permanent product) were defined. Some identities among the discrete operators were also specified, where some of them were a discrete analogue of the Leibniz rule.

Based on the discrete operators and identities among the operators, the author proposed the proper sets of 4th and higher order fully conservative convection schemes for incompressible flows ( Morinishi (1995), Morinishi et al. (1998) ).

Then, the fully conservative convection schemes for compressible flows were proposed in 2010 ( Morinishi (2010) ) by using the analogy between incompressible and compressible flow equations. The commutability among the convection forms with the aid of the continuity is satisfied including time derivative terms for compressible flows. The proper skew-symmetric form which is secondary conservative is also defined as the average of the divergence and advection forms for compressible flows. A couple of time derivative terms in the skew-symmetric form can be transformed into a single form. For fully discretized and fully conservative convection schemes for compressible flows, a special temporal interpolation is required for the commutability and the conservation properties for compressible flow equations (density weighted equations).

The fully conservative convection schemes for flows on moving grid ( Morinishi and Koga (2014) ) were proposed in 2014. An ALE type moving grid is considered, and the governing equations in a physical space are transformed into a computational space. Here, the transformed equations are Jacobian and density weighted, while the original equations are density weighted. Therefore, the analogy between different weights (density for compressible flows, Jacobian and density for flows on moving grid) is also available for the scheme construction.

Key words

Fully conservative convection schemes, incompressible flows, compressible flows, flows on moving grid

References

[1]	Morinishi, Y., Conservative properties of finite difference schemes for incompressible flow, CTR Annual Research Briefs 1995, Center for Turbulence Research, Stanford Univ./NASA Ames Research Center, (1996), pp.121-132.
[2]	Morinishi, Y., Lund, T. S., Vasilyev, O.V., Moin, P., Fully conservative higher order finite difference schemes for incompressible flow, J. Comput. Phys., Vol.143, (1998), pp.90-124.
[3]	Morinishi, Y., Skew-symmetric form of convective terms and fully conservative finite difference schemes for variable density low-Mach number flows, J. Comput. Phys., Vol. 229, (2010), pp.276-300.
[4]	Morinishi, Y., Koga, K., Skew-symmetric convection form and secondary conservative finite difference methods for moving grids, J. Comput. Phys., Vol. 257, (2014), pp.1081-1112.