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How can we engage in direct modeling for computational fluid dynamics?

World Scientific


IMAGE: This is the cover for Direct Modeling for Computational Fluid Dynamics. view more

Credit: World Scientific, 2014

Direct modeling for computational fluid dynamics provides an effective methodology to develop multi-scale numerical algorithms for flow simulation in all regimes from rarefied to continuum ones, which will help to improve the conventional CFD methods which are based on numerical partial differential equations.

The study of fluid dynamics needs go beyond the traditional numerical partial differential equations (PDE). The emerging engineering applications, such as air-vehicle design for near space flight and flow and heat transfer in micro-devices, do require further expansion of the concept of gas dynamic equations to a larger domain of physical reality, rather than the traditional distinguishable governing equations, such as the Navier-Stokes and the Boltzmann equations.

At the current stage, the non-equilibrium flow physics, especially in the transition regime, has not been well explored and clearly understood due to the lack of appropriate governing equation and the corresponding numerical tools. Under the current numerical PDE approach, it is hard to develop such a meaningful tool due to the absence of valid PDEs. Theoretically, the flow dynamics should have a continuous spectrum from the kinetic Boltzmann equation to the hydrodynamic Navier-Stokes ones. In order to develop efficient multi-scale and multi-physics numerical methods to cover such a continuous flow dynamics, similar to the modeling process of constructing the governing equations, the numerical algorithm is better based on the first principle of physical modeling.

In Professor Kun Xu's book published by World Scientific, instead of following traditional numerical PDE path, a direct modeling as a principle for the computational fluid dynamics algorithm development is proposed. Since all computations are conducted in a discretized space with limited resolution, the flow physics to be modeled has to be done in the mesh size and time step scales. Here, the computational fluid dynamics is to directly construct a discrete numerical evolution equation, where the mesh size and time step themselves will play dynamic roles in its modeling process.

With the variation of the ratio between mesh size and local particle mean free path, the unified scheme will capture flow physics from the kinetic particle transport and collision to the hydrodynamic wave propagation. Based on the direct modeling, a continuous dynamics of flow motion will be simulated in the unified gas-kinetic scheme. This scheme can be faithfully used to study the unexplored non-equilibrium flows in the transition regime.


The book retails for US$118 / £78 at all major bookstores. More information on the book can be found at:

About World Scientific Publishing

World Scientific Publishing is a leading independent publisher of books and journals for the scholarly, research and professional communities. The company publishes about 500 books annually and more than 120 journals in various fields. World Scientific collaborates with prestigious organisations like the Nobel Foundation, US National Academies Press, as well as its subsidiary, the Imperial College Press, amongst others, to bring high quality academic and professional content to researchers and academics worldwide. To find out more about World Scientific, please visit

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