Patient-Relativity in Morality

It is common to distinguish moral rules, reasons, or values that are agent-relative from those that are agent-neutral. One can also distinguish moral rules, reasons, or values that are moment-relative from those that are moment-neutral. In this article, I introduce a third distinction that stands alongside these two distinctions—the distinction between moral rules, reasons, or values that are patient-relative and those that are patient-neutral. I then show how patient-relativity plays an important role in several moral theories, gives us a better understanding of agent-relativity and moment-relativity, and provides a novel objection to Derek Parfit’s “appeal to full relativity” argument

Boundary-layer receptivity for a parabolic leading edge. Part 2. The small-Strouhal-number limit

In Hammerton & Kerschen (1996), the effect of the nose radius of a body on boundary-layer receptivity was analysed for the case of a symmetric mean flow past a two-dimensional body with a parabolic leading edge. A low-Mach-number two-dimensional flow was considered. The radius of curvature of the leading edge, rn, enters the theory through a Strouhal number, S=?rn/U, where ? is the frequency of the unsteady free-stream disturbance and U is the mean flow speed. Numerical results revealed that the variation of receptivity for small S was very different for free-stream acoustic waves propagating parallel to the mean flow and those free-stream waves propagating at an angle to the mean flow. In this paper the small-S asymptotic theory is presented. For free-stream acoustic waves propagating parallel to the symmetric mean flow, the receptivity is found to vary linearly with S, giving a small increase in the amplitude of the receptivity coefficient for small S compared to the flat-plate value. In contrast, for oblique free-stream acoustic waves, the receptivity varies with S1/2, leading to a sharp decrease in the amplitude of the receptivity coefficient relative to the flat-plate value. Comparison of the asymptotic theory with numerical results obtained in the earlier paper confirms the asymptotic results but reveals that the numerical results diverge from the asymptotic result for unexpectedly small values of S

Boundary-layer receptivity for a parabolic leading edge

The effect of the nose radius of a body on boundary-layer receptivity is analysed for the case of a symmetric mean flow past a body with a parabolic leading edge. Asymptotic methods based on large Reynolds number are used, supplemented by numerical results. The Mach number is assumed small, and acoustic free-stream disturbances are considered. The case of free-stream acoustic waves, propagating obliquely to the symmetric mean flow is considered. The body nose radius, rn, enters the theory through a Strouhal number, S = ?rn/U, where ? is the frequency of the acoustic wave and U is the mean flow speed. The finite nose radius dramatically reduces the receptivity level compared to that for a flat plate, the amplitude of the instability waves in the boundary layer being decreased by an order of magnitude when S = 0.3. Oblique acoustic waves produce much higher receptivity levels than acoustic waves propagating parallel to the body chord

Effect of molecular relaxation on the propagation of sonic booms through a stratified atmosphere

Nonlinear acoustic wave propagation through a stratified atmosphere is considered. The initial signal is taken to be an isolated N-wave, which is the disturbance that is generated some distance away from a supersonic body in horizontal flight. The effect of cylindrical spreading and exponential density stratification on the propagation of the disturbance is considered, with the shock structure controlled by molecular relaxation mechanisms and by thermoviscous diffusion. An augmented Burgers equation is obtained and asymptotic solutions are derived based on the limit of small dissipation and dispersion. For a single relaxation mode, the solution depends on whether relaxation alone can support the shock or whether a sub-shock arises controlled by other mechanisms. The resulting shock structures are known as fully dispersed and partly dispersed shocks, respectively. In this paper, the spatial location of the transition between fully dispersed and partly dispersed shocks is identified for shocks propagating above and below the horizontal. This phenomenon is important in understanding the character of sonic booms since the transition to a partly dispersed shock structure leads to the appearance of a shorter scale in the shock rise-time, associated with the embedded sub-shock

Leading-edge receptivity for bodies with mean aerodynamic loading

Boundary-layer receptivity in the leading-edge region of a cambered thin airfoil is analysed for the case of a low-Mach-number flow. Acoustic free-stream disturbances are considered. Asymptotic results based on large Reynolds number ($U^2 / \omega \nu \,{\gg}\, 1$) are presented, supplemented by numerical solutions. The influence of mean aerodynamic loading enters the theory through a parameter $\mu$, which provides a measure of the flow speed variations in the leading-edge region, due to flow around the leading edge from the lower surface to the upper. A Strouhal number based on airfoil nose radius, $S\,{=}\,\omega r_n/U$, also enters the theory. The variation of the receptivity level as a function of $\mu$ and $S$ is analysed. Modest levels of aerodynamic loading are found to decrease the receptivity level for the upper surface of the airfoil, while the receptivity is increased for the lower surface. For larger angles of attack close to the critical angle for boundary layer separation, a local rise in the receptivity occurs for the upper surface, while on the lower surface the receptivity decreases. These effects are more pronounced at larger values of $S$. While the Tollmien–Schlichting wave does not emerge until a downstream distance of $O((U^2 / \omega \nu)^{1/3} U / \omega)$, the amplitude of the Tollmien–Schlichting wave is influenced by the acoustic free-stream disturbances only in a relatively small region near the leading edge, of length approximately $4 U/\omega$. The numerical receptivity coefficients calculated, together with the asymptotic eigenfunctions presented, provide all the necessary information for transition analysis from the interaction of acoustic disturbances with leading-edge geometry

Analysis of the unstable Tollmien--Schlichting mode on bodies with a rounded leading edge using the parabolized stability equation

The interaction between free-stream disturbances and the boundary layer on a body with a rounded leading edge is considered in this paper. A method which incorporates calculations using the parabolized stability equation in the Orr-Sommerfeld region, along with an upstream boundary condition derived from asymptotic theory in the vicinity of the leading edge, is generalized to bodies with an inviscid slip velocity which tends to a constant far downstream. We present results for the position of the lower branch neutral stability point and the magnitude of the unstable Tollmien-Schlichting (T-S) mode at this point for both a parabolic body and the Rankine body. For the Rankine body, which has an adverse pressure gradient along its surface far from the nose, we find a double maximum in the T-S wave amplitude for sufficiently large Reynolds numbers

Numerical and asymptotic solutions of generalised Burgers’ equation

The generalised Burgers’ equation has been subject to a considerable amount of research on how the equation should behave according to asymptotic analysis, however there has been limited research verifying the asymptotic analysis. In order to verify the asymptotic analysis, this paper aims to run long time and detailed numerical simulations of Burgers’ equation by employing suitable rescalings of Burgers’ equation. It is hoped that this technique will make it possible to notice subtle changes in the shock structure which would otherwise be impossible to observe. The main aim of this paper is to validate the numerical methods used in order to allow further research into shock evolution where further relaxation effects will be included