Hydrodynamic linear stability of the two-dimensional bluff-body wake through modal analysis and initial-value problem formulation

In this paper a new Newton-Raphson (NR) method and a new implicit Euler method are developed for numerically solving the HNC and related equations for simple fluids. The new methods are efficient, stable and easy to implement. The implicit Euler method is derived through a reformulation of the usual HNC problem into the form of an initial value problem (IVP). The IVP formulation allows application of specialized numerical techniques developed for the solution of stiff systems of differential equations. The implicit Euler method is shown to have excellent stability and rate of convergence. The relationship between the implicit Euler method and the NR method is elucidated, and it is shown that in the limit of infinite step-size, the implicit Euler method reduces to the NR method.

We propose a physically sensible formulation of initial value problem for black hole perturbations in higher-order scalar–tensor theories. As a first application, we study monopole perturbations around stealth Schwarzschild solutions in a shift- and reflection-symmetric subclass of degenerate higher-order scalar–tensor (DHOST) theories. In particular, we investigate the time evolution of the monopole perturbations by solving a two-dimensional wave equation and analyze the Vishveshwara’s classical scattering experiment, i.e., the time evolution of a Gaussian wave packet. As a result, we confirm that stealth Schwarzschild solutions in the DHOST theory are dynamically stable against the monopole perturbations with the wavelength comparable or shorter than the size of the black hole horizon. We also find that the damped oscillations at the late time do not show up unlike the ringdown phase in the standard case of general relativity. Moreover, we investigate the characteristic curves of the monopole perturbations as well as a static spherically symmetric solution with monopole hair.

The paper presents a nonparallel stability analysis of the intermediate region of the two-dimensional wake behind a bluff body. In particular, it analyzes the convective instabilities using a Wentzel-Kramers-Brillouin-Jeffreys method on a basic flow previously derived from intermediate asymptotics [D. Tordella and M. Belan, Phys. Fluids 15, 1897 (2003)]. The multiscaling is carried out to explicitly account for the effects associated to the lateral momentum dynamics at a given Reynolds number. These effects are an important feature of the base flow and are included in the perturbative equation as well as in the associated modulation equation. At the first order in the multiscaling, the disturbance is locally tuned to the property of the instability, as can be seen in the zero-order theory (near-parallel parametric Orr-Sommerfeld treatment). This leads to a synthetic analysis of the nonparallel correction of the instability characteristics. The system is, in fact, considered to be locally perturbed by waves with a wave number that varies along the intermediate wake and which is equal to the wave number of the dominant saddle point of the zero order dispersion relation, taken at different Reynolds numbers. In this study, the Reynolds number is thus the only parameter. It is shown that the corrections to the frequency, and to the temporal and spatial growth rates are remarkable in the first part of the intermediate wake and lead to absolute instability in regions that extend to about ten body scales. The correction increases with the Reynolds number and agrees with data from laboratory and numerical experiments in literature. An eigenfunction and eigenvalue asymptotic analysis for the far wake is included, which is in excellent agreement with the complete problem.

An approximated Navier–Stokes steady solution is here presented for the two dimensional bluff body wake region that is intermediate between the field on the body scale LD, which includes the two symmetric counter-rotating eddies, and the ultimate far wake. The nonparallelism of the streamlines in the intermediate wake cannot yet be considered negligible. The R is of the order of the critical value for the onset of the first instability and the limiting behavior for large R is not considered. The solution is obtained by matching an inner solution—a Navier–Stokes expansion in powers of the inverse of the longitudinal coordinate—and an outer solution, which is a Navier–Stokes asymptotic expansion in powers of the inverse of the distance from the body. The matching is built on the criteria that, where the two solutions meet, the longitunal pressure gradients and the vorticities must be equal and the flow toward the inner layer must be equal to the outflow from the external stream. At high orders in the inner expansion solution, the lateral decay turns out to be algebraic. This approximate solution is here examined in relation to the class of asymptotic solutions that, in the past, were obtained by adopting the rapid decay principle, which implies an irrotational outer flow. The theme running through this paper is the necessity of the addition of this criterion to the equations of motion to build a solution that describes the intermediate wake. The present solution has been obtained by relaxing the imposition of the rapid decay principle. It can be concluded that, at Reynolds numbers as low as the first critical value and where the nonparallelism of the streamlines is not yet negligible, the division of the field into two basic parts—an inner vortical boundary layer flow and an outer potential flow—is spontaneously shown up to the second order of accuracy: at higher orders in the expansion solution the vorticity is first convected and then diffused in the outer field. If exploited to represent the basic flow of bluff body wakes, the analytical simplicity of this asymptotic expansion could be useful for the nonparallel analysis of the instability of two-dimensional wakes.

An expansion theorem is derived for Rossby normal modes in a closed rectangular basin and the set of Rossby normal modes is proved to be complete. This theorem provides a general linear solution to the initial value problem as well as to the response problem. In particular, the Green's function is obtained for the instantaneous localized torque anywhere in the basin. Weakly nonlinear versions are solved also by the combination of the general linear solution with the asymptotic expansion in terms of small amplitude. Further, an application is suggested to the spectral method of numerical simulation based on Rossby normal modes relevant to the more nonlinear evolution equation on aβ-plane, instead ofsin functions or Chebyshev polynomials, which have been employed conventionally for this purpose.

On the construction of the solution of an equation describing an axially moving string

Frequency analysis of carbon and silicon nanosheet with surface effects

In a recent publication [Galley, Phys. Rev. Lett. 110, 174301 (2013)], Galley proposed an initial value problem formulation of Hamilton's principle that enables consideration of dissipative systems. Here we explore this formulation at the level of field theories with infinite degrees of freedom. In particular, we illustrate that it affords a previously unwarranted and appealing as well as broadly relevant possibility, namely, to generalize the popular collective coordinate or variational method to open systems, i.e., nonconservative ones. To showcase the relevance or validity of the method we explore two case examples from the timely area of $\mathcal{PT}$-symmetric variants of field theories, in this case for a sine-Gordon and for a ${\ensuremath{\phi}}^{4}$ model.

In this paper, an approximate analytical solution for Frank-Kamenetskii equation modeling a thermal explosion in a sphere, is obtained. The approximate solution is obtained by perturbation methods in terms of small and large distance parameter. The approximate solution is compared with the numerical solution obtained from an initial value problem formulation to the original boundary value problem. The approximate solution obtained is valid for all values of the distance parameter. For the original boundary value problem and for a given Frank-Kamenetskii parameter, a nonlinear algebraic equation needs to be solved to be able to apply the approximate solution.

: We present a new method for stability and modal analysis of shear flows and their acoustic radiation. The Euler equations are modified and solved as a spatial initial value problem in which initial perturbations are specified at the ow inlet and propagated downstream by integration of the equations. The modified equations, which we call one-way Euler equations, differ from the usual Euler equations in that they do not support upstream acoustic waves. It is necessary to remove these modes from the Euler operator because, if retained, they cause instability in the spatial marching procedure. These modes are removed using a two-step process. First, the upstream modes are partially decoupled from the down- stream modes using a linear similarity transformation. Second, the error in the first step is eliminated using a convergent recursive filtering technique. A previous spatial marching method called the parabolized stability equations uses numerical damping to stabilize the march, but this has the unintended consequence of heavily damping the downstream acoustic waves. Therefore, the one-way Euler equation could be used to obtain improved accuracy over the parabolized stability equations as a low-order model for noise simulation of mixing layers and jets.

A new very high-order finite-difference method for linear stability analysis and bi-orthogonal decomposition of hypersonic boundary layer flow

Abstract

We revisit singularities of two distinct kinds in the Cauchy problem of general scalar-tensor theories of gravity (previously discussed in the literature), and of metric and Palatini f(R) gravity, in both their Jordan and Einstein frame representations. Examples and toy models are used to shed light onto the problem and it is shown that, contrary to common lore, the two conformal frames are equivalent with respect to the initial value problem.