SUMMARYThe method of manufactured solutions is used to verify the order of accuracy of two ÿnite-volume Euler and Navier-Stokes codes. The Premo code employs a node-centred approach using unstructured meshes, while the Wind code employs a similar scheme on structured meshes. Both codes use Roe's upwind method with MUSCL extrapolation for the convective terms and central di erences for the di usion terms, thus yielding a numerical scheme that is formally second-order accurate. The method of manufactured solutions is employed to generate exact solutions to the governing Euler and NavierStokes equations in two dimensions along with additional source terms. These exact solutions are then used to accurately evaluate the discretization error in the numerical solutions. Through global discretization error analyses, the spatial order of accuracy is observed to be second order for both codes, thus giving a high degree of conÿdence that the two codes are free from coding mistakes in the options exercised. Examples of coding mistakes discovered using the method are also given.

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This paper presents a collection of fluid mechanics problems with exact solutions which can be used to verify the numerical accuracy of solutions obtained by CFD codes. This document is a product of the AIAA Committee On Standards (COS). It is intended to serve as the start of a catalog of exact solutions for fluid mechanics problems, and as a complement to the Verification and Validation Guide prepared by the AIAA. While the solutions presented in this paper do not necessarily test all aspects of a code or all terms of the governing equations, they constitute fundamentals tests for identifying coding errors. Hence, documenting these solutions is an important step towards consolidating significant verification cases.

Various personal dimensions of students—particularly motivation, self-efficacy beliefs, and epistemic beliefs—can change in response to teaching, affect student learning, and be conceptualized as learning dispositions. We propose that these learning dispositions serve as learning outcomes in their own right; that patterns of interrelationships among these specific learning dispositions are likely; and that differing constellations (or learning disposition profiles) may have meaningful implications for instructional practices. In this observational study, we examine changes in these learning dispositions in the context of six courses at four institutions designed to scaffold undergraduate thesis writing and promote students’ scientific reasoning in writing in science, technology, engineering, and mathematics. We explore the utility of cluster analysis for generating meaningful learning disposition profiles and building a more sophisticated understanding of students as complex, multidimensional learners. For example, while students’ self-efficacy beliefs about writing and science increased across capstone writing courses on average, there was considerable variability at the level of individual students. When responses on all of the personal dimensions were analyzed jointly using cluster analysis, several distinct and meaningful learning disposition profiles emerged. We explore these profiles in this work and discuss the implications of this framework for describing developmental trajectories of students’ scientific identities.

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