Shock wave structure in highly rarefied flows

The Boltzmann equation is written in terms of two functions associated with the gain and loss of a certain type of molecule due to collisions. Its integral form is then applied to the problem of normal shock structure, and an iteration technique is used to determine the solution. The first approximation to the velocity distribution function of the Chapman-Enskog sequence, which leads to the Navier-Stokes equations, is used to initiate the iteration scheme. Expressions for the distribution function and the flow parameters pertinent to the first iteration are derived and show that the B-G-K model results can be obtained as a special case. This model is found to be valid in the continuum regime only, and is consequently limited to the study of strong shocks. In the present treatment the iteration is carried out on the distribution function and the analysis indicates that the method is equally valid for variations in both Mach and Knudsen numbers. Finally, the results of the first approximation are simplified, and expressed in a form suitable for numerical computation, and the range of their validity is discussed. The method should be equally suitable for other flow problems of linear or nonlinear nature

On the instability of Goertler vortices to nonlinear travelling waves

Recent theoretical work by Hall and Seddougui (1989) has shown that strongly nonlinear, high wavenumber Goertler vortices developing within a boundary layer flow are susceptible to a secondary instability which takes the form of travelling waves confined to a thin region centered at the outer edge of the vortex. The case is considered in which the secondary mode could be satisfactorily described by a linear stability theory and herein the objective is to extend this investigation of Hall and Seddougui (1989) into the nonlinear regime. It was found that at this stage not only does the secondary mode become nonlinear but it also interacts with itself so as to modify the governing equations for the primary Goertler vortex. In this case then, the vortex and the travelling wave drive each other and, indeed, the whole flow structure is described by an infinite set of coupled, nonlinear differential equations. A Stuart-Watson type of weakly nonlinear analysis of these equations is undertaken and concluded, in particular, that on this basis there exist stable flow configurations in which the travelling mode is of finite amplitude. Implications of the findings for practical situations are discussed and it is shown that the theoretical conclusions drawn here are in good qualitative agreement with available experimental observations

Numerical methodologies for the simulation of liquid metal flows

The fluid-dynamic modelling for the simulation of the Lead-Bismuth flow in the EADF was reviewed. The general form of the non-dimensional governing equation was derived, and the analysis of the orders of magnitude of the different terms in the case of a the liquid metal flows in the EADF was performed, through a flow-Mach number asymptotic analysis. It was found that the resulting form of the equations is the one commonly used in commercial CFD codes for the simulation of liquid flows, which can then be used for our applications. The most common numerical methods for flow-Mach number applications were also presented. These methods are general and can be applied to liquid metal flows without any modification. The peculiarity of the numerical simulation of liquid metal flows lies in the modelling of the turbulent heat transfer, due to the flow Prandtl number of this type of fluids. This subject is discussed in [21]

Homogenization of the Equations Governing the Flow Between a Slider and a Rough Spinning Disk

We have analyzed the behavior of the flow between a slider bearing and a hard-drive magnetic disk under two types of surface roughness. For both cases the length scale of the roughness along the surface is small as compared to the scale of the slider, so that a homogenization of the governing equations was performed. For the case of longitudinal roughness, we derived a one-dimensional lubrication-type equation for the leading behavior of the pressure in the direction parallel to the velocity of the disk. The coefficients of the equation are determined by solving linear elliptic equations on a domain bounded by the gap height in the vertical direction and the period of the roughness in the span-wise direction. For the case of transverse roughness the unsteady lubrication equations were reduced, following a multiple scale homogenization analysis, to a steady equation for the leading behavior of the pressure in the gap. The reduced equation involves certain averages of the gap height, but retains the same form of the usual steady, compressible lubrication equations. Numerical calculations were performed for both cases, and the solution for the case of transverse roughness was shown be in excellent agreement with a corresponding numerical calculation of the original unsteady equations

A Relational Logic for Higher-Order Programs

Relational program verification is a variant of program verification where one can reason about two programs and as a special case about two executions of a single program on different inputs. Relational program verification can be used for reasoning about a broad range of properties, including equivalence and refinement, and specialized notions such as continuity, information flow security or relative cost. In a higher-order setting, relational program verification can be achieved using relational refinement type systems, a form of refinement types where assertions have a relational interpretation. Relational refinement type systems excel at relating structurally equivalent terms but provide limited support for relating terms with very different structures. We present a logic, called Relational Higher Order Logic (RHOL), for proving relational properties of a simply typed $\lambda$-calculus with inductive types and recursive definitions. RHOL retains the type-directed flavour of relational refinement type systems but achieves greater expressivity through rules which simultaneously reason about the two terms as well as rules which only contemplate one of the two terms. We show that RHOL has strong foundations, by proving an equivalence with higher-order logic (HOL), and leverage this equivalence to derive key meta-theoretical properties: subject reduction, admissibility of a transitivity rule and set-theoretical soundness. Moreover, we define sound embeddings for several existing relational type systems such as relational refinement types and type systems for dependency analysis and relative cost, and we verify examples that were out of reach of prior work.Comment: Submitted to ICFP 201

Probabilistic pointer analysis for multithreaded programs

The use of pointers and data-structures based on pointers results in circular memory references that are interpreted by a vital compiler analysis, namely pointer analysis. For a pair of memory references at a program point, a typical pointer analysis specifies if the points-to relation between them may exist, definitely does not exist, or definitely exists. The "may be" case, which describes the points-to relation for most of the pairs, cannot be dealt with by most compiler optimizations. This is so to guarantee the soundness of these optimizations. However, the "may be" case can be capitalized by the modern class of speculative optimizations if the probability that two memory references alias can be measured. Focusing on multithreading, a prevailing technique of programming, this paper presents a new flow-sensitive technique for probabilistic pointer analysis of multithreaded programs. The proposed technique has the form of a type system and calculates the probability of every points-to relation at each program point. The key to our approach is to calculate the points-to information via a post-type derivation. The use of type systems has the advantage of associating each analysis results with a justification (proof) for the correctness of the results. This justification has the form of a type derivation and is very much required in applications like certified code.Comment: 12 page

Mean curvature flow in an extended Ricci flow background

In this paper, we consider functionals related to mean curvature flow in an ambient space which evolves by an extended Ricci flow from the perspective introduced by Lott when studying a mean curvature flow in a Ricci flow background. One of them is a weighted extended version of the Gibbons-Hawking-York action on Riemannian metrics in compact manifolds with boundary. We compute its variational properties from which naturally arise boundary conditions to the analysis of its time-derivative under Perelman's modified extended Ricci flow. For instance, the boundary integrand term provides an extension of Hamilton's differential Harnack expression for mean curvature flows in Euclidean space. We also derive the evolution equations for both the second fundamental form and the mean curvature under mean curvature flow in an extended Ricci flow background. In the special case of gradient solitons to the extended Ricci flow, we discuss mean curvature solitons and establish a Huisken's monotonicity-type formula. We show how to construct a family of mean curvature solitons and establish a characterization of such a family. Also, we show how for constructing examples of mean curvature solitons in an extended Ricci flow background.Comment: 24 pages. Suggestions and comments are welcom

The Application of Ansys-Fluent Software for Aerodynamic Analysis on Rectangular and Moderate Swept Wing Planform

ANSYS-Fluent software represents a CFD software having the capability for solving various engineering flow problems. Besides offering a variety of flow solvers, this software also offers various type of turbulence model can be used in the flow analysis. The present work focuses on the use of this software applied to two type wing models, a moderately swept wing and (2) a rectangular wing planform. The moderately swept wing geometry and experimental data were obtained from AGARD AR-138, whereas the rectangular wing planform was obtained from RTO-TR-026. The first model evaluated by using five different turbulent models, namely (1) Spalart-Allmaras, (2) k-ε Standard, (3) k-ε Realizable, (4) k-ω Standard and (5) k-ω SST turbulence models. Comparisons result with AGARD shows that all turbulent models are able to provide in a good agreement. However, Spalart-Allmaras and k-ω SST turbulence models give less CPU time than the others. These two turbulent models then applied to the case of a rectangular wing plan form. The result from the second test case, the k-ω SST turbulence models, give a more accurate result compared to the Spalart-Allmaras turbulence models. It gives a better result compared with the Spalart-Allmaras turbulence models. The k-ω SST turbulence model makes the ANSYS-Fluent result just differ 11.5% from the experiment result while for the Spalart-Allmaras turbulence model differs 14.05%.  Here it can be concluded that k-ω SST turbulence mode may represent a suitable turbulence model for solving flow over a rectangular wing to the moderate swept wing plan form