Engineering topics

McHardy, James David

Publications and source records attributed to McHardy, James David.

4 records

DOE OSTI2019

Group-invariant solutions for one dimensional, inviscid hydrodynamics

In this paper, the results of the Lie group method carried out by Ovsiannikov are utilized to study the one-dimensional hydrodynamic equations governing compressible, inviscid fluid flow in the absence of heat conduction. One-parameter subgroups of the admissible R-parameter Lie group of point transformations of the system are applied to reduce the first-order, non-linear system of partial differential equations (PDE)s to a first-order system of ordinary differential equations (ODE)s. Closed-form solutions to the reduced ODE systems are subsequently determined using a linear velocity profile ansatz. These solutions are valid in one-dimensional (1D) planar, cylindrical and spherical geometries and are connected to solutions of the governing system of PDEs through an inverse map. The linear velocity type solutions were first considered by Sedov in 1953 and constitute a subclass of all possible similarity solutions of the compressible hydrodynamics equations. They further serve as illustrative examples of using solutions obtained for the reduced ODEs to find solutions of the associated system of PDEs. Consequently, the reduced systems of ODEs are provided in their entirety inviting additional solutions to be determined via alternative explicit or numerical solution techniques.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2019

Application of symmetries to differential equations: symmetry reduction and solution transformation examples

In the following lecture notes, we provide an outline of what we mean by symmetries of differential equations and illustrate their utility in simplification of differential equations which may lead to solutions. Loosely, a symmetry of a differential equation is a local group of transformations acting on the space of independent and dependent variables that transform solutions of a differential equation to new solutions, i.e., if Δ(x,u,p)=0 denotes a differential equation with a solution prescribed by [x,u,p], then knowlege of a symmetry facilitates the attainment of a new solution, [$\tilde{x}$, $\tilde{u}$, $\tilde{p}$], such that Δ(x,u,p)=Δ($\tilde{x}$, $\tilde{u}$, $\tilde{p}$)=0. In Equations (1) and (2), x, u and p denote the independent variables, dependent variables and partial derivatives, respectively and the tildes denote the transformations. Equation (2) will subsequently be referred to as the symmetry criterion. We can apply the knowlede of the existence of a symmetry in two ways: 1) If not solution is known apriori, the symmetry can be applied to simplify the differential equation through the method of symmetry reduction. Often, the resultant simplification may enable a solution to be determined; 2) If we already possess a solution to a different equation, knowledge of an admissible symmetry enables us to generate new solutions.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Lie groups of variable cross-section channel flow

This paper considers Lie groups of scaling transformations for the system of equations governing inviscid, compressible, quasi-one dimensional (quasi-1D) fluid flow in channels with variable cross-section. We determine the coupling between the admissibility of the group transformations and the equation of state (EOS) model used to close the system as well as the spatial and temporal dependence of the channel cross-section. The results presented extend prior group analysis performed in Ovsiannikov and Boyd et al. on the equations of gas dynamics for 1D flow. In the context of channel flow, the analysis in Ovsiannikov and Boyd et al. is only applicable for a very limited scope of channels, namely for channels possessing planar, cylindrical and spherical symmetry where the flow is truly 1D. To illustrate the extension achieved using the quasi-1D model, we consider the classical Noh problem set up in a channel with a variable area cross-section. First, a new set of canonical variables are derived using the admissible scaling transformations. Then, by re-expressing the governing equations of motion in terms of the new variables, a solvable system of ordinary differential equations (ODEs) is acquired. The resulting explicit solutions of the problem extend the previous results of Ramsey et al. to solutions of the Noh problem for quasi-1D flow.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2018

Symmetry Analysis Using Symbolic Computation

The symmetry group of a system of differential equations reveals basic invariant properties of the underlying model and its solutions. In many cases symmetry groups have simple geometric interpretations that can be applied to simplify the model and determine certain classes of solutions explicitly. The general procedure to compute the symmetry group of a given system of differential equations follows a well-known algorithm. However, this procedure involves a large number of laborious and usually error-prone calculations, especially in the case of higher-order systems. The purpose of this report is explain how to use an existing software program designed to automate the major steps of this algorithm in order to calculate the symmetry group of any system of differential equations more efficiently.

97 MATHEMATICS AND COMPUTING↗