In this paper, based on simplified Boltzmann equation, we explore the inverse-design of mesoscopic models for compressible flow using the Chapman-Enskog analysis. Starting from the single-relaxation-time Boltzmann equation with an additional source term, two model Boltzmann equations for two reduced distribution functions are obtained, each then also having an additional undetermined source term. Under this general framework and using Navier-Stokes-Fourier (NSF) equations as constraints, the structures of the distribution functions are obtained by the leading-order Chapman-Enskog analysis. Next, five basic constraints for the design of the two source terms are obtained in order to recover the NSF system in the continuum limit. These constraints allow for adjustable bulk-to-shear viscosity ratio, Prandtl number as well as a thermal energy source. The specific forms of the two source terms can be determined through proper physical considerations and numerical implementation requirements. By employing the truncated Hermite expansion, one design for the two source terms is proposed. Moreover, three well-known mesoscopic models in the literature are shown to be compatible with these five constraints. In addition, the consistent implementation of boundary conditions is also explored by using the Chapman-Enskog expansion at the NSF order. Finally, based on the higher-order Chapman-Enskog expansion of the distribution functions, we derive the complete analytical expressions for the viscous stress tensor and the heat flux. Some underlying physics can be further explored using the DNS simulation data based on the proposed model.
- NSF-PAR ID:
- 10322686
- Date Published:
- Journal Name:
- ArXivorg
- ISSN:
- 2331-8422
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Active learning—
doing —supports learning.Collaborative learning—doing
together —supports learning.Classroom discourse—focused, relevant
discussion , not just listening—supports learning.Clear articulation of one's thinking, even just to oneself, helps develop that thinking.
What this paper adds
The common languages we use for classroom mathematics—natural language for conveying the meaning and context of mathematical situations and for explaining our reasoning; and the formal (written) language of conventional mathematical notation, the symbols we use in mathematical expressions and equations—are both essential but each presents hurdles that necessitate the other. Yet, even together, they are insufficient especially for young learners.
Programming, appropriately designed and used, can be the third language that both reduces barriers and provides the missing expressive and creative capabilities children need.
Appropriate design for use in regular mathematics classrooms requires making key mathematical content obvious, strong and the ‘driver’ of the activities, and requires reducing tech ‘overhead’ to near zero.
Continued usefulness across the grades requires developing children's sophistication and knowledge with the language; the powerful ways that children rapidly acquire facility with (natural) language provides guidance for ways they can learn a formal language as well.
Implications for policy and/or practice
Mathematics teaching can take advantage of the ways children learn through experimentation and attention to the results, and of the ways children use their language brain even for mathematics.
In particular, programming—in microworlds driven by the mathematical content, designed to minimise distraction and overhead, open to exploration and discovery
en route to focused aims, and in which childrenself ‐evaluate—can allow clear articulation of thought, experimentation with immediate feedback.As it aids the mathematics, it also builds computational thinking and satisfies schools' increasing concerns to broaden access to ideas of computer science.