Nonlinear Inviscid Research Articles

Unveiling the intricacies: Analytical insights into time and space fractional order inviscid burger's equations using adomian decomposition method

In this study, we analyze complex partial differential equations, specifically focusing on nonlinear fractional-order Inviscid Burger's equations. By employing the Adomian Decomposition Method, we develop analytical approximations in the form of convergent series to solve these equations with time and space fractional derivatives in the Caputo sense. Our approach provides explicit, efficient, and highly accurate solutions, which we verify against exact solutions, eliminating the need for closure approximations and perturbation theory. We present our findings through detailed tables and graphs, offering valuable insights into the behavior of the solutions. This work advances analytical techniques and establishes a foundation for new applications in various scientific and engineering fields.

Nonlinear inviscid damping for a class of monotone shear flows in a finite channel

Nonlinear inviscid damping for a class of monotone shear flows in a finite channel

Elastic-viscoplastic behaviors of polymer-blend geocell sheets: Numerical and experimental investigations

Polymer-blend geocell sheets (PBGS) have been developed as substitute materials for manufacturing geocells. Various attempts have been made to test and predict the behaviors of commonly used geogrids, geotextiles, geomembranes, and geocells. However, the elastic-viscoplastic behaviors of novel-developed geocell sheets are still poorly understood. Therefore, this paper investigates the elastic-viscoplastic behaviors of PBGS to gain a comprehensive understanding of their mechanical properties. Furthermore, the tensile load-strain history under various loading conditions is simulated by numerical calculation for widespread utilization. To achieve this goal, monotonic loading tests, short-term creep and stress relaxation tests, and multi-load-path tests (also known as arbitrary loading history tests) are performed using a universal testing machine. The results are simulated using the nonlinear three-component (NLTC) model, which consists of three nonlinear components, i.e. a hypo-elastic component, a nonlinear inviscid component, and a nonlinear viscid component. The experimental and numerical results demonstrate that PBGS exhibit significant elastic-viscoplastic behavior that can be accurately predicted by the NLTC model. Moreover, the tensile strain rates significantly influence the tensile load, with higher strain rates resulting in increased tensile loads and more linear load-strain curves. Also, parametric analysis of the rheological characteristics reveals that the initial tensile strain rates have negligible impact on the results. The rate-sensitivity coefficient of PBGS is approximately 0.163, which falls within the typical range observed in most geosynthetics.

Dynamics near Couette flow for the β-plane equation

In this paper, we study stationary structures near the planar Couette flow in Sobolev spaces on a channel T×[−1,1], and asymptotic behavior of Couette flow in Gevrey spaces on T×R for the β-plane equation. Let T>0 be the horizontal period of the channel and α=2πT be the wave number. We obtain a sharp region O in the whole (α,β) half-plane such that non-shear traveling waves do not exist for (α,β)∈O and such traveling waves indeed exist for (α,β) in the remaining regions, near Couette flow for H≥5 velocity perturbation. The borderlines between the region O and its remaining are determined by two curves of the principal eigenvalues of singular Rayleigh-Kuo operators. Our results reveal that there exists β⁎>0 such that if |β|≤β⁎, then non-shear traveling waves do not exist for any T>0, while if |β|>β⁎, then there exists a critical period Tβ>0 so that such traveling waves exist for T∈[Tβ,∞) and do not exist for T∈(0,Tβ), near Couette flow for H≥5 velocity perturbation. This contrasting dynamics plays an important role in studying the long time dynamics near Couette flow with Coriolis effects. Moreover, for any β≠0 and T>0, we prove that there exist no non-shear traveling waves with traveling speeds converging in (−1,1) near Couette flow for H≥5 velocity perturbation, in contrast to this, we construct non-shear stationary solutions near Couette flow for H<52 velocity perturbation, which is a generalization of Theorem 1 in [22] but the construction is more difficult due to the Coriolis effects. Finally, we prove nonlinear inviscid damping for Couette flow in some Gevrey spaces by extending the method of [4] to the β-plane equation on T×R.

On zonal steady solutions to the 2D Euler equations on the rotating unit sphere

The present paper studies the structure of the set of stationary solutions to the incompressible Euler equations on the rotating unit sphere that are near two basic zonal flows: the zonal Rossby–Haurwitz solution of degree 2 and the zonal rigid rotation along the polar axis. We construct a new family of non-zonal steady solutions arbitrarily close in analytic regularity to the second degree zonal Rossby–Haurwitz stream function, for any given rotation of the sphere. This shows that any non-linear inviscid damping to a zonal flow cannot be expected for solutions near this Rossby–Haurwitz solution. On the other hand, we prove that, under suitable conditions on the rotation of the sphere, any stationary solution close enough to the rigid rotation zonal flow must itself be zonal, witnessing some sort of rigidity inherited from the equation, the geometry of the sphere and the base flow. Nevertheless, when the conditions on the rotation of the sphere fail, the set of solutions is much richer and we are able to prove the existence of both explicit stationary and travelling wave non-zonal solutions bifurcating from , in the same spirit as those emanating from the zonal Rossby–Haurwitz solution of degree 2.

Nonlinear inviscid damping and shear‐buoyancy instability in the two‐dimensional Boussinesq equations

AbstractWe investigate the long‐time properties of the two‐dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. Under the classical Miles‐Howard stability condition on the Richardson number, we prove that the system experiences a shear‐buoyancy instability: the density variation and velocity undergo an inviscid damping while the vorticity and density gradient grow as . The result holds at least until the natural, nonlinear timescale . Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles‐Howard spectral stability condition; (B) a variation of the Fourier time‐dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.

Nonlinear inviscid aerodynamics of a wind turbine rotor in surge, sway, and yaw motions using a free-wake panel method

Abstract. We investigate the aerodynamics of a surging, heaving, and yawing wind turbine with numerical simulations based on a free-wake panel method. We focus on the UNAFLOW (UNsteady Aerodynamics of FLOating Wind turbines) case: a surging wind turbine which was modeled experimentally and with various numerical methods. Good agreement with experimental data is observed for amplitude and phase of the thrust with surge motion. We achieve numerical results of a wind turbine wake that accurately reproduce experimentally verified effects of surging motion. We then extend our simulations beyond the frequency range of the UNAFLOW experiments and reach results that do not follow a quasi-steady response for surge. Finally, simulations are done with the turbine in yaw and heave motion, and the impact of the wake motion on the blade thrust is examined. Our work seeks to contribute a different method to the pool of results for the UNAFLOW case while extending the analysis to conditions that have not been simulated before and providing insights into nonlinear aerodynamic effects of wind turbine motion.

Nonlinear inviscid damping near monotonic shear flows

Nonlinear inviscid damping near monotonic shear flows

Large amplitude non-spherical bubbles

SummaryWe consider the long-term evolution of an axisymmetric bubble and explore the ways in which it may develop. Linearised inviscid analysis is used to predict the stability of the bubble with a small disturbance while a nonlinear inviscid extension shows that the growth of unstable modes is ultimately limited by the formation of axisymmetric curvature singularities. The addition of surface tension is shown to delay, but not entirely prevent, these singularities. Our results are found to agree well with a viscous Boussinesq theory at least to early times. The inclusion of viscosity means that the development of the bubble structure is not limited by the creation of singularities, and the bubble may ultimately adopt one of a wide range of possible large-scale deformations. Among these, perhaps the most exotic are jet-like structures which can pinch off and break into several distinct parts. Spectral methods are employed to solve the inviscid and Boussinesq models while the linearised inviscid model admits a closed-form series solution.

Stability analysis of nonlinear inviscid microscopic and macroscopic traffic flow models of bidirectional cruise-controlled vehicles

Abstract The paper introduces a new bidirectional microscopic inviscid Adaptive Cruise Control (ACC) model that uses only spacing information from the preceding and following vehicles in order to select the proper control action to avoid collisions and maintain a desired speed. $KL$ estimates that guarantee uniform convergence of the ACC model to the set of equilibria are provided. Moreover, the corresponding macroscopic model is derived, consisting of a conservation equation and a momentum equation that contains a nonlinear relaxation term. It is shown that, if the density is sufficiently small, then the macroscopic model has a solution that approaches exponentially the equilibrium speed (in the sup norm) while the density converges exponentially to a traveling wave. Numerical simulations are also provided, illustrating the properties of the microscopic and macroscopic inviscid ACC models.

Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation

In everyday life human faces shock waves and rarefaction largely in their surroundings. Hence it’s necessary to know the behavior of these waves to protect destructive effects. The aim of this work to observe the propagation of shock and rarefaction waves in various dynamics due to solve non-linear hyperbolic inviscid Burgers’ equation numerically. The models adopted here two numerical schemes which enable us to solve non-linear hyperbolic Burgers’ equation numerically. The first order explicit upwind scheme (EUDS) and second order Lax-Wendroff schemes are used to solve this equation to improve our understanding of the numerical diffusion (smearing) and oscillations that can be present when using such schemes. In order to understand the behavior of the solution we use method of characteristics to find the exact solution of inviscid Burgers’ equation. Numerical solutions are studied for different initial conditions and the shock and rarefaction waves are investigated for Riemann problem. We present stability analysis of the schemes and establish stability condition which leads to determine time step selection in terms of spatial step size with maximum initial value. Numerical result for these schemes are compared with an exact solution of inviscid Burgers’ equation in terms of accuracy by error estimation. The numerical features of the rate of convergence are presented graphically. This analysis helps us to understand a wide range of physical phenomenon of the properties of wave as well as saves in several aspects in real life.

Modeling Asymmetric Interactions in Economy

We consider a general nonlinear kinetic type equation that can describe the time evolution of a variable related to an economical state of an individual agent of the system. We assume asymmetric interactions between the agents. We show that in a corresponding limit, it is asymptotically equivalent to a nonlinear inviscid Burgers type equation.

Vibrational control of mechanical systems with piecewise linear damping and high-frequency inputs

This paper discusses averaging and vibrational control of mechanical control-affine systems with piecewise linear damping and high-frequency inputs. The results are used for dynamic analysis and vibrational control of a three-degree-of-freedom, planar, horizontal pendulum. The system has non-symmetric, linear viscous damping, and the pivot point of the pendulum moves along a prescribed elliptical path with “high” frequency. The dynamic analysis shows that the presence of non-symmetric damping results in change of periodic orbits or equilibria of the system or instability of an otherwise stable system. This paper presents the stability conditions of the system in terms of the physical parameters of the pendulum and the prescribed path. Using the results of dynamic analysis, the paper also discusses vibrational control of the system of horizontal pendulum. The analysis can be extended to mechanical control systems with nonlinear viscous forces, dry friction, or inviscid flow forces.

On the Forced Euler and Navier–Stokes Equations: Linear Damping and Modified Scattering

We study the asymptotic behavior of the forced linear Euler and nonlinear Navier–Stokes equations close to Couette flow on $$\mathbb {T}\times I$$ T×I. As our main result we show that for smooth time-periodic forcing linear inviscid damping persists, i.e. the velocity field (weakly) asymptotically converges. However, stability and scattering to the transport problem fail in $$H^{s}, s>-1$$ Hs,s>-1. We further show that this behavior is consistent with the nonlinear Euler equations and that a similar result also holds for the nonlinear Navier–Stokes equations. Hence, these results provide an indication that nonlinear inviscid damping may still hold in Sobolev regularity in the above sense despite the Gevrey regularity instability results of Deng and Masmoudi (Long time instability of the Couette flow in low Gevrey spaces, 2018. arXiv:1803.01246).

Linear Inviscid Damping in Gevrey Spaces

We prove linear inviscid damping near a general class of monotone shear flows in a finite channel, in Gevrey spaces. It is an essential step towards proving nonlinear inviscid damping for general shear flows that are not close to the Couette flow, which is a major open problem in 2d Euler equations.

The correction to the nonlinear inviscid model of the Rayleigh–Taylor instability suppressed by a phase transition

The Rayleigh–Taylor instability, complicated by heat and mass transfer, is studied for a thin vapor film covering the horizontal surface of a flat heater, which is immersed in a subcooled volume of liquid. A correction to the nonlinear inviscid model of the instability suppression by a phase transition is proposed. It is based on the fact that, at finite Prandtl numbers, the strength of the phase transition effect can have the limitation described before for the linear problem. It is shown that the nonlinearity alone, without taking the viscosity into account, is able to stabilize the thin vapor film, but only for the situations when the convective heat exchange in the liquid obeys the Newton–Richman law.

Nonlinear pressure–velocity waveforms in large arteries, shock waves and wave separation

Nonlinear inviscid 1D blood flow equations were studied analytically using the method of characteristics. The initial–boundary value problem for a class of the initial–boundary conditions (including a triangle-shaped profile) at the aortic outlet was considered. The nonlinear pressure–velocity profile and shock conditions were derived as closed analytical expressions with second order accuracy on the relative luminal area change. Finally, the fully nonlinear wave separation expressions were obtained using the Riemann invariants. The results show good correspondence with the data from the literature.

Characterization of a 3DOF aeroelastic system with freeplay and aerodynamic nonlinearities – Part I: Higher-order spectra

Abstract The identification of nonlinear systems in aeroelasticity poses a significant challenge for practitioners, often hampered by the complex nature of aeroelastic response data which may contain multiple forms of nonlinearity. Characterizing and quantifying nonlinearities is further hampered when the response is obtained at a location which is away from the nonlinear source and/or the response is contaminated by noise. In the present paper, a three-degree-of-freedom airfoil with a freeplay nonlinearity located in the control surface and exposed to transonic flow is investigated. In this Part I paper the main form of analysis is via higher-order spectra techniques to unveil features of the nonlinear mechanism which result from i) structural nonlinearities (freeplay) in isolation and ii) freeplay with Euler derived nonlinear inviscid aerodynamic phenomena (transition between Tijdeman Type-A and Type-B shock motion). It is shown that the control surface structural freeplay nonlinearity is characterized by strong cubic phase-coupling between linear and nonlinear modes. On the other hand, nonlinear inviscid flow phenomena are shown to be characterized by quadratic phase-coupling between linear and nonlinear modular modes, the strength of which is related to the strength of the aerodynamic nonlinearity (amplitude of the shock motion). The nonlinear inviscid flow phenomena do not appear to affect the identification of the freeplay nonlinearity. Conjectures are made which address the transition between aperiodic, quasi-periodic and periodic behavior (pre-flutter), further physical support towards these conjectures is provided in Part II [1] . The limitations of the higher-order spectra approach are assessed, in particular, the analysis demonstrates the difficulty in extracting natural frequencies with this approach.

A general nonlocal variable coefficient KdV equation with shifted parity and delayed time reversal

A general nonlocal time-dependent variable coefficient KdV (VCKdV) equation with shifted parity and delayed time reversal is derived from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a $$\beta $$ -plane. A special transformation is established to change it into a nonlocal constant coefficient KdV (CCKdV) equation with shifted parity and delayed time reversal. Making advantage of this transformation, exact solutions of the nonlocal CCKdV equation can be utilized to construct exact solutions of the nonlocal VCKdV equation. Two kinds of nonlinear wave excitations are presented explicitly and graphically. Though they possess very simple wave profiles, they can move in abundant ways due to the arbitrary time-dependent functions in their exact solutions, and can be used to model various blocking events in climate disasters. It is demonstrated that a special approximate solution of the original stream functions can capture a kind of two correlated dipole blocking events with a lifetime.

A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal

A general nonlocal nonlinear Schrodinger equation with shifted parity, charge-conjugate and delayed time reversal is derived from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a $$\beta $$ -plane. The modulational instability (MI) of the obtained system is studied, which reveals a number of possibilities for the MI regions due to the generalized dispersion relation that relates the frequency and wavenumber of the modulating perturbations. Exact periodic solutions in terms of Jacobi elliptic functions are obtained, which, in the limit of the modulus approaches unity, reduce to soliton, kink solutions and their linear superpositions. Representative profiles of different nonlinear wave excitations are displayed graphically. These solutions can be used to model different blocking events in climate disasters. As an illustration, a special approximate solution is given to describe a kind of two correlated dipole blocking events.