Multi-scale Method (Block-Spectral Model)

For many problems in aerospace and power generation industries, there are very different length scales. The fine (micro) scale features of a huge number of geometrically largely similar regions collectively interact with globally large (macro) scale phenomena.  Examples include flows through porous medium, effusion film-cooling through many holes of a tiny size, acoustical liners for duct noise reduction, and surface mini-scale treatment for flow control (e.g. dimples etc).  The traditional treatment of this kind of problems would be to use empirically based models to count for the effects of the small scale features and to solve an up-scaled problem on a coarse mesh. The generality of the solutions following this kind of approaches are of course limited by the very empirical nature of these fine-scale models.

Here we are interested in developing a new multi-scale methodology, called ‘Block-Spectral Method’. The main intended attribute of the new approach is that the same numerical discretization scheme and integration method are used for both the coarse (macro) and fine (micro) scales, so that the numerical resolution is consistently and completely dictated by the mesh scales. A blocking of the fine resolution domain is introduced to facilitate the two basic but competing requirements:

     a) high resolution for fine scale flow features;

     b) avoidance of  having to have fine meshes for a large domain

The block spectral approach can be simply illustrated by comparing a direct solution and a block spectral solution, as shown in Fig.1.  The method has been shown to lead to a significant gain in solving micro-scale problems (up to 102 reduction of degrees of freedom). An important perspective is that the methodology would enable to resolve the kind of the micro-scale problems currently intractable.  


Dimpled Surface


Unsteady Effusion Cooling (~1000 micro holes) 


Intake Duct with Acoustical Liner (micro cavities)




-    He L, “Block-spectral Approach to Film-cooling Modelling”,   Journal of Turbomachinery, Vol.134, No.2, March, 2012.

-    He L, “Fourier Spectral Modelling for Multi-scale Aero-thermal Analysis’ , International Journal of Computational Fluid Dynamics, Vol.27, No.2, 2013.

-    He L, “Block-spectral Mapping for Multi-scale Solution”, Journal of Computational Physics, Vol.250, pp13-26, 2013.