On Approximate Aesthetic Curves

Symmetry plays an essential role for generating aesthetic forms. The curve is the basic element used by designers to obtain aesthetic forms. A curve with a linear logarithmic curvature graph gradient is called aesthetic curve. The aesthetic value of a curve increases when its gradient is close to a straight line. We introduce the notions of approximate aesthetic curves and approximate neutral curves and obtain estimations between the curvature of an approximate aesthetic/neutral curve and an aesthetic curve.

Effect of porous layer on the Faraday instability in viscous liquid

A linear stability analysis of a viscous liquid on a vertically oscillating porous plane is performed for infinitesimal disturbances of arbitrary wavenumbers. A time-dependent boundary value problem is derived and solved based on the Floquet theory along with the complex Fourier series expansion. Numerical results show that the Faraday instability is dominated by the subharmonic solution at high forcing frequency, but it responds harmonically at low forcing frequency. The unstable regions corresponding to both subharmonic and harmonic solutions enhance with the increasing value of permeability and yields a destabilizing effect on the Faraday instability. Further, the presence of porous layer makes faster the transition process from subharmonic instability to harmonic instability in the wavenumber regime. In addition, the first harmonic solution shrinks gradually and becomes an unstable island, and ultimately disappears from the neutral curve if the porous layer thickness is increased. In contrast, the first and second subharmonic solutions coalesce, and the onset of Faraday instability is dominated by the subharmonic solution. In a special case, the study of Faraday instability of a viscous liquid on a porous substrate can be replaced by a study of Faraday instability of a viscous liquid on a slippery substrate when the permeability of the porous substrate is very low. Further, the Faraday instability can be destabilized by introducing a slip effect at the bottom plane.

Exploration of Coriolis Force on the Linear Stability of Couple Stress Fluid Flow Induced by Double Diffusive Convection

In this paper, the effect of Coriolis force is explored on convective instability of a doubly diffusive incompressible couple stress fluid layer with gravity acting downward. A linear stability analysis is used to obtain the conditions for the onset of stationary and oscillatory convection in the closed form. Being a multiparameter instability problem, results for some isolated cases have been presented to illustrate interesting corners of parameter space. It is found that the neutral curve for oscillatory onset forms a closed-loop which is separate from the neutral curve for stationary onset indicating the requirement of three critical thermal Rayleigh numbers to specify the linear instability criteria instead of the usual single value. Besides, the simultaneous presence of rotation and the addition of heavy solute to the bottom of the layer exhibit an intriguing possibility of destabilizing the system under certain conditions, in contrast to their stabilizing effect when they are present in isolation. The implication of couple stresses on each of the aforementioned anomalies is clearly brought out. The spatial wavelength of convective cells at the onset is also discussed.

Neutral stability curves of compressible Görtler flow generated by low-frequency free-stream vortical disturbances

Pre-transitional compressible boundary layers perturbed by low-frequency free-stream vortical disturbances and flowing over plates with streamwise-concave curvature are studied via matched asymptotic expansions and numerically. The Mach number, the Görtler number and the frequency of the free-stream disturbance are varied to obtain the neutral stability curves, i.e. curves in the space of the parameters that distinguish spatially growing from spatially decaying perturbations. The receptivity approach is used to calculate the evolution of Klebanoff modes, highly oblique Tollmien–Schlichting waves influenced by the concave curvature of the wall, and Görtler vortices. The Klebanoff modes always evolve from the leading edge, the Görtler vortices dominate when the influence of the curvature becomes significant and the Tollmien–Schlichting waves may precede the Görtler vortices for moderate Görtler numbers. For relatively high frequencies the triple-deck formalism allows us to confirm the numerical result of the negligible influence of the curvature on the Tollmien–Schlichting waves when the Görtler number is an order-one quantity. Experimental data for compressible Görtler flows are mapped onto our neutral-curve graphs and earlier theoretical results are compared with our predictions.

Analytic description of flame intrinsic instability in one-dimensional model of open–open combustors with ideal and non-ideal end boundaries

This paper is concerned with the theoretical study of thermo-acoustic instabilities in combustors and focuses upon recently discovered flame intrinsic modes. Here, a complete analytical description of the salient properties of intrinsic modes is provided for a linearized one-dimensional model of open–open combustors with temperature and cross-section jump across the flame taken into account. The standard [Formula: see text] model of heat release is adopted, where n is the interaction index and τ is the time lag. We build upon the recent key finding that for a closed–lopen combustor, on the neutral curve, the intrinsic mode frequencies become completely decoupled from the combustor parameters like cross-section jump, temperature jump and flame location. Here, we show that this remarkable decoupling phenomenon holds not only for closed–open combustors but also for all combustors with the ideal boundary conditions (i.e. closed–open, open–open and closed–closed). Making use of this decoupling phenomenon for the open–open combustors, we derive explicit analytic expressions for the neutral curve of intrinsic mode instability on the [Formula: see text] plane as well as for the linear growth/decay rate near the neutral curve taking into account temperature and cross-section jumps. The instability domain on the [Formula: see text] plane is shown to be qualitatively different from that of the closed–open combustor; in open–open combustors it is not confined for large τ. To find the instability domain and growth rate characteristics for non-ideal open–open boundaries the combustor end boundaries are perturbed and explicit analytical formulae derived and verified by numerics.

A simple non-equilibrium bedload transport equation for the formation of dune in a shallowwater flow over an erodible bed

In this work, we consider the long-standing problem of capturing dune formation in an erodible-bed channel at subcritical speed by using a reduced order model of depth-averaged equations. The pioneering study by Reynolds [1] showed that the standard Saint-Venant-Exner equations are unconditionally stable at subcritical Froude number. Hence, the use of depthaveraged flow equations, which are commonly used by the hydraulic community, prevents the formation of bedforms as dunes. Recently, Cañada-Pereira & Bohorquez [2] have proposed a simple sediment transport formulation able to capture the formation of dune when coupled with the Saint-Venant equations. We replace the standard Exner equation with a non-equilibrium sediment transport equation that includes the following necessary ingredients: first, a phase shift in the particle entrainment rate; second, a particle diffusivity and an eddy viscosity. Subsequently, we solve the linear stability problem of an erodiblebed channel and show that the neutral curve properly captures the bed instability both in subcritical regime (i.e. dune) and supercritical flow (i.e. antidune and roll wave). Finally, we corroborate the capabilities of the model by means of non-linear numerical simulations which reproduce the growth of dune and antidune in agreement with experiments.

Instabilities in flexible channel flow with large external pressure

We examine the stability of laminar high-Reynolds-number flow through an asymmetric flexible-walled channel driven by a fixed upstream flux and subject to a (large) uniform external pressure. We construct a long-wavelength, spatially one-dimensional model using a flow profile assumption, modelling the flexible wall as a thin tensioned membrane subject to a large axial pre-stress. We numerically construct the non-uniform static shape of the flexible wall and consider its stability using both a global eigensolver and numerical simulation of the nonlinear governing equations. The system admits multiple static solutions, including a highly collapsed steady state where the membrane has a single constriction which increases with increasing external pressure. We demonstrate that the non-uniform static state is unstable to two distinct (infinite) families of normal modes which we characterise in the limit of large external pressure. In particular, there is a family of low-frequency oscillatory modes which each persist to low membrane tensions, where the most unstable mode has an oscillating membrane profile which is outwardly bulged at the centre of the domain with a narrow constriction at the downstream end. In addition, there is a family of high-frequency oscillatory modes which are each unstable beyond a critical value of the tension within a two-branch neutral curve. Unstable modes along the lower branch of the neutral curve are sustained by a leading-order balance between unsteady inertia and the restoring force of membrane tension along the channel. In addition, we elucidate the mechanism of energy transfer to sustain the self-excited oscillations: oscillations decrease the mean maximal constriction of the channel over a period, which reduces the overall dissipation of the mean flow and releases energy to sustain the instability. Fully nonlinear simulations indicate that as the Reynolds number increases these unstable normal modes can grow supercritically into sustained large-amplitude ‘slamming’ oscillations, where the membrane is periodically drawn very close to the opposite rigid wall before recovering.

Linear stability of a viscoelastic liquid flow on an oscillating plane

Linear stability of a viscoelastic liquid on an oscillating plane is studied for disturbances of arbitrary wavenumbers. The main aim is to extend the earlier study of Dandapat & Gupta (J. Fluid Mech., vol. 72, 1975, pp. 425–432) to the finite wavenumber regime, which has not been attempted so far in the literature. The Orr–Sommerfeld boundary value problem is formulated for an unsteady base flow, and it is resolved numerically based on the Chebyshev spectral collocation method along with the Floquet theory. The analytical solution predicts that U-shaped unstable regions appear in the separated bandwidths of the imposed frequency, and the dominant mode of the long-wave instability intensifies in the presence of the viscoelastic parameter. The numerical solution shows that oblique neutral curves come out from the branch points of the U-shaped neutral curves at finite wavenumber and continue with the imposed frequency until the curves cross the next U-shaped neutral curve. As a consequence, in the finite wavenumber regime, no stable bandwidth of the imposed frequency is predicted by the long-wavelength analysis. Further, in some frequency ranges, the finite wavenumber instability is more dangerous than the long-wave instability.