Dynamic Boundary Conditions Research Articles

Nonlinear longitudinal–bending–twisting vibrations of extensible slowly rotating beam with tip mass

Nonlinear longitudinal–bending–twisting vibrations of a slowly rotating extensible beam are studied in the paper. The beam model is based on extended Euler–Bernoulli theory with shear deformation neglected. Geometrical nonlinearities, caused by large oscillations, couple longitudinal vibrations, bending in two directions and twist. The beam is fixed to a rigid hub at a given preset angle and carries mass attached at the beam’s tip. The governing partial differential equations and dynamical boundary conditions are derived from the extended Hamilton principle. The model is solved analytically up to the third order perturbation by the multiple time scale method applied directly to the partial differential equations. The influence of angular velocity of the hub, tip mass and other structural beam parameters on nonlinear vibrations around selected resonance zones is presented.

Large Eddy Simulation of the Layout Effects on Wind Farm Performance Coupling With Wind Turbine Control Strategies

Abstract Large eddy simulation (LES) coupling with wind turbine control strategies is newly developed to quantitatively study the layout effects on wind farm performance. The turbine rotor is parameterized with an actuator line method (ALM), and the five-region generator-torque control and the proportional-integral (PI) pitch control are newly introduced to regulate the operation of the wind turbine. First, a dynamic inflow boundary condition is designed to validate the current simulation framework. The validation results show that the simulated power curve agrees well with the real power curve of the wind turbine, and the maximum power error of the simulation only accounts for 5% of the rated power. Then, to study the layout effects, four kinds of wind farm arrangements are designed by varying the alignment method and the turbine spacing. The results show that the staggered arrangement and increasing the stream-wise spacing are beneficial to reduce the velocity deficit. The power comparison results show that the staggered arrangement has obvious advantages among the four cases, and it increases the capacity factor (CF) by 25% and improves the wind farm efficiency by about 50% compared with the aligned arrangement. The present simulation framework can be used to optimize the turbine layout for the potential wind farms.

Exponential stability for wave equation with delay and dynamic boundary conditions

In this paper, we consider a multi-dimensional wave equation with delay and dynamic boundary conditions, related to the Kelvin–Voigt damping. By using the Faedo–Galerkin approximations together with some priori estimates, we prove the local existence of solution. Since the damping may stabilize the system while the delay may destabilize it, we discuss the interaction between the damping and the delay, and obtain that the system is uniformly stable when the effect of damping is greater than that of time delay. Exponential stability result of system is also established by constructing suitable Lyapunov functionals.

Changes in the ozone chemical regime over the contiguous United States inferred by the inversion of NOx and VOC emissions using satellite observation

To investigate changes in the ozone (O3) chemical production regime over the contiguous United States (CONUS) with accurate knowledge of concentrations of its precursors, we applied an inverse modeling technique with Ozone Monitoring Instrument (OMI) tropospheric nitrogen dioxide (NO2) and total formaldehyde (HCHO) retrieval products in the summers of 2011, 2014, and 2017, years in which United States National Emission Inventory were based. The inclusion of dynamic chemical lateral boundary conditions and lightning-induced nitric oxide emissions significantly account for the contribution of background sources in the free troposphere. Satellite-constrained nitrogen oxide (NOx) and non-methane volatile organic compounds (NMVOCs) emissions mitigate the discrepancy between satellite and modeled columns: the inversion suggested 2.33–2.84 (1.07–1.34) times higher NOx over the CONUS (over urban regions) and 0.28–0.81 times fewer NMVOCs emissions over the southeastern United States. The model-derived HCHO/NO2 column ratio shows gradual spatial changes in the O3 production regime near urban cores relative to previously defined threshold values representing NOx and VOC sensitive conditions. We also found apparent shifts from the NOx-saturated regime to the transition regime (or the transition regime to the NOx-limited regime) over the major cities in the western United States. In contrast, rural areas, especially in the east-southeastern United States, exhibit a decreased HCHO/NO2 column ratio by −1.30 ± 1.71 with a reduction in HCHO column primarily driven by meteorology, becoming sensitive to VOC emissions. Results show that incorporating satellite observations into numerical modeling could help policymakers implement appropriate emission control policies for O3 pollution.

Elastic membrane equation with dynamic boundary conditions and infinite memory

In this paper, we study the elastic membrane equation with dynamic boundary conditions, source term and a nonlinear weak damping localized on a part of the boundary and past history. Under some appropriate assumptions on the relaxation function the general decay for the energy have been established using the perturbed Lyapunov functionals and some properties of convex functions.

Polynomial Decay for the Timoshenko System with Dynamical Boundary Conditions

Polynomial Decay for the Timoshenko System with Dynamical Boundary Conditions

Modeling and Numerical Simulation of the Thermal Interaction between Vegetation Cover and Soil

In this work, we propose a mathematical model representing the thermal interaction between vegetation cover and the soil underneath it. This model consists of a one-dimensional reaction–diffusion equation describing the evolution of the temperature in the vegetation cover coupled with a two-dimensional reaction–diffusion equation to represent the evolution of the temperature in the soil. The thermal interaction between the vegetation cover and the soil is studied and the distribution of temperatures in the soil with depth is also obtained. The vegetation cover acts in this model as a dynamic and diffusive boundary condition for the soil. The developed model takes into account the latent heat of fusion, which appears when the transformation of ice into liquid water or vice versa occurs inside the soil. The numerical approach for the solution of the mathematical model conducted in this work is based on the finite volume method with Weighted Essentially Non-Oscillatory technique for spatial reconstruction and the third-order Runge–Kutta Total Variation Diminishing numerical scheme is used for time integration, which is very efficient to obtain the numerical solution of this type of model. Some numerical examples are solved to obtain the distribution of temperature both in the vegetation cover and the soil.

A review on the Cahn–Hilliard equation: classical results and recent advances in dynamic boundary conditions

<abstract><p>The Cahn–Hilliard equation is a fundamental model that describes the phase separation process in multi-component mixtures. It has been successfully extended to different contexts in various scientific fields. In this survey article, we briefly review the derivation, structure as well as some analytical issues for the Cahn–Hilliard equation and its variants. Our focus will be placed on the well-posedness as well as long-time behavior of global solutions for the Cahn–Hilliard equation in the classical setting and recent progresses on the dynamic boundary conditions that describe non-trivial boundary effects.</p></abstract>

A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition

<p style='text-indent:20px;'>We propose a structure-preserving finite difference scheme for the Cahn–Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM) proposed by Furihata and Matsuo [<xref ref-type="bibr" rid="b14">14</xref>]. In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme proposed by Fukao–Yoshikawa–Wada [<xref ref-type="bibr" rid="b13">13</xref>] is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for our proposed scheme. Computation examples demonstrate the effectiveness of our proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.</p>

Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

<p style='text-indent:20px;'>This paper deals with the hierarchical control of the anisotropic heat equation with dynamic boundary conditions and drift terms. We use the Stackelberg-Nash strategy with one leader and two followers. To each fixed leader, we find a Nash equilibrium corresponding to a bi-objective optimal control problem for the followers. Then, by some new Carleman estimates, we prove a null controllability result.</p>

Null controllability for semilinear heat equation with dynamic boundary conditions

<p style='text-indent:20px;'>This paper deals with the null controllability of the semilinear heat equation with dynamic boundary conditions of surface diffusion type, with nonlinearities involving drift terms. First, we prove a negative result for some function <inline-formula><tex-math id="M1">\begin{document}$ F $\end{document}</tex-math></inline-formula> that behaves at infinity like <inline-formula><tex-math id="M2">\begin{document}$ |s| \ln ^{p}(1+|s|), $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ p > 2 $\end{document}</tex-math></inline-formula>. Then, by a careful analysis of the linearized system and a fixed point method, a null controllability result is proved for nonlinearties <inline-formula><tex-math id="M4">\begin{document}$ F(s, \xi) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ G(s, \xi) $\end{document}</tex-math></inline-formula> growing slower than <inline-formula><tex-math id="M6">\begin{document}$ |s| \ln ^{3 / 2}(1+|s|+\|\xi\|)+\|\xi\| \ln^{1 / 2}(1+|s|+\|\xi\|) $\end{document}</tex-math></inline-formula> at infinity.</p>

Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control

<p style='text-indent:20px;'>This article deals with the dynamic stability of a flexible cable attached at its top end to a cart and a load mass at its bottom end. The model is governed by a system of one partial differential equation coupled with two ordinary differential equations. Assuming that a time-dependent delay occurs in one boundary, the main concern of this paper is to stabilize the dynamics of the cable as well as the dynamical terms related to the cart and the load mass. To do so, we first prove that the problem is well-posed in the sense of semigroups theory provided that some conditions on the delay are satisfied. Thereafter, an appropriate Lyapunov function is put forward, which leads to the exponential decay of the energy as well as an estimate of the decay rate.</p>

Numerical study on a thermoacoustic refrigerator with continuous and staggered arrangements

Thermoacoustic devices require heat exchangers with oscillating flow, but there is currently no viable design approach for them. A heat exchanger with a staggered structure can efficiently improve the velocity disturbance and promote heat transfer in steady flow. The flow and heat transfer characteristics of a standing-wave thermoacoustic refrigerator and an ambient heat exchanger with staggered parallel plates under the oscillating flow condition are investigated in this study, primarily focusing on the geometric influences and differences between staggered and non-staggered (continuous) arrangements. The CFD simulation is a mainstream tool for the numerical simulation of complex thermoacoustic phe?nomena. The flow field around the stack and heat exchanger plate is simulated by introducing the dynamic mesh boundary conditions. Through numerical simulation, the flow field characteristics of non-linear vortices generation around the heat exchanger are presented. By changing the staggered column number in the ambient heat exchanger, it is observed that the larger the column number of staggered parallel plates, the more significant the refrigeration effect through the thermoacoustic effect.

On the stabilization of a flexible structure via a nonlinear delayed boundary control

<p style='text-indent:20px;'>The main concern of this article is to investigate the stabilization problem of a flexible structure with one dynamical boundary condition subject to two nonlinearities in the proposed boundary control. Specifically, the first nonlinearity is related to the velocity term, while the second one arises from the delayed term. Despite such a situation, it is shown that the system has a global solution by means of semigroups theory of nonlinear operators. More importantly, we prove that the system energy decays. An estimate of the decay rate of the energy is explicitly provided depending on the type of the nonlinearities.</p>

An Improved Dynamic Boundary Condition in SPH Method

Dynamic boundary condition (DBC) has been widely used in SPH method. However, in certain situations, it was found that a few fluid particles could break through the boundary or were not reflected specularly. Of course, these phenomena are unphysical. To improve the performance of DBC, an improved dynamic boundary condition (IDBC) was presented in this paper. To prevent fluid particles from breaking through the boundary, the repulsive force of boundary particles was enhanced by expanding the equation of state into a higher order. To deal with the asymmetry of DBC, a rectangular support domain attached to boundary particles and a corresponding correction factor are proposed. The results of three test cases showed that the performance of IDBC was satisfied.

On the Solvability of the Burgers Equation with Dynamic Boundary Conditions in a Degenerating Domain

On the Solvability of the Burgers Equation with Dynamic Boundary Conditions in a Degenerating Domain

Liquid walls and interfaces in arbitrary directions stabilized by vibrations

Gravity shapes liquids and plays a crucial role in their internal balance. Creating new equilibrium configurations irrespective of the presence of a gravitational field is challenging with applications on Earth as well as in zero-gravity environments. Vibrations are known to alter the shape of liquid interfaces and also to change internal dynamics and stability in depth. Here, we show that vibrations can also create an "artificial gravity" in any direction. We demonstrate that a liquid can maintain an inclined interface when shaken in an arbitrary direction. A necessary condition for the equilibrium to occur is the existence of a velocity gradient determined by dynamical boundary conditions. However, the no-slip boundary condition and incompressibility can perturb the required velocity profile, leading to a destabilization of the equilibrium. We show that liquid layers provide a solution, and liquid walls of several centimeters in height can thus be stabilized. We show that the buoyancy equilibrium is not affected by the forcing.

Well-posedness of a nonlinear second-order anisotropic reaction-diffusion problem with nonlinear and inhomogeneous dynamic boundary conditions

The paper is concerned with a qualitative analysis for a nonlinear second-order boundary value problem, endowed with nonlinear and inhomogeneous dynamic boundary conditions, extending the types of bounday conditions already studied. Under certain assumptions on the input data: $f_{_1}(t,x)$, $w(t,x)$ and $u_0(x)$, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $W^{1,2}_p(Q)$. This extends previous works concerned with nonlinear dynamic boundary conditions, allowing to the present mathematical model to better approximate the real physical phenomena (the anisotropy effects, phase change in $\Omega$ and at the boundary $\partial\Omega$, etc.).

Improving predictability of high-ozone episodes through dynamic boundary conditions, emission refresh and chemical data assimilation during the Long Island Sound Tropospheric Ozone Study (LISTOS) field campaign

Abstract. Although air quality in the United States has improved remarkably in the past decades, ground-level ozone (O3) often rises in exceedance of the national ambient air quality standard in nonattainment areas, including the Long Island Sound (LIS) and its surrounding areas. Accurate prediction of high-ozone episodes is needed to assist government agencies and the public in mitigating harmful effects of air pollution. In this study, we have developed a suite of potential forecast improvements, including dynamic boundary conditions, rapid emission refresh and chemical data assimilation, in a 3 km resolution Community Multiscale Air Quality (CMAQ) modeling system. The purpose is to evaluate and assess the effectiveness of these forecasting techniques, individually or in combination, in improving forecast guidance for two major air pollutants: surface O3 and nitrogen dioxide (NO2). Experiments were conducted for a high-O3 episode (28–29 August 2018) during the Long Island Sound Tropospheric Ozone Study (LISTOS) field campaign, which provides abundant observations for evaluating model performance. The results show that these forecast system updates are useful in enhancing the capability of this 3 km forecasting model with varying effectiveness for different pollutants. For O3 prediction, the most significant improvement comes from the dynamic boundary conditions derived from the NOAA operational forecast system, National Air Quality Forecast Capability (NAQFC), which increases the correlation coefficient (R) from 0.81 to 0.93 and reduces the root mean square error (RMSE) from 14.97 to 8.22 ppbv, compared to that with the static boundary conditions (BCs). The NO2 from all high-resolution simulations outperforms that from the operational 12 km NAQFC simulation, regardless of the BCs used, highlighting the importance of spatially resolved emission and meteorology inputs for the prediction of short-lived pollutants. The effectiveness of improved initial concentrations through optimal interpolation (OI) is shown to be high in urban areas with high emission density. The influence of OI adjustment, however, is maintained for a longer period in rural areas, where emissions and chemical transformation make a smaller contribution to the O3 budget than that in high-emission areas. Following the assessment of individual updates, the forecasting system is configured with dynamic boundary conditions, optimal interpolation of initial concentrations and emission adjustment, to simulate a high-ozone episode during the 2018 LISTOS field campaign. The newly developed forecasting system significantly reduces the bias of surface NO2 prediction. When compared with the NASA Langley GeoCAPE Airborne Simulator (GCAS) vertical column density (VCD), this system is able to reproduce the NO2 VCD with a higher correlation (0.74), lower normalized mean bias (40 %) and normalized mean error (61 %) than NAQFC (0.57, 45 % and 76 %, respectively). The 3 km system captures magnitude and timing of surface O3 peaks and valleys better. In comparison with lidar, O3 profile variability of the vertical O3 is captured better by the new system (correlation coefficient of 0.71) than by NAQFC (correlation coefficient of 0.54). Although the experiments are limited to one pollution episode over the Long Island Sound, this study demonstrates feasible approaches to improve the predictability of high-O3 episodes in contemporary urban environments.

Cavity Detachment from a Wedge with Rounded Edges and the Surface Tension Effect

Cavity flow around a wedge with rounded edges was studied, taking into account the surface tension effect and the Brillouin–Villat criterion of cavity detachment. The liquid compressibility and viscosity were ignored. An analytical solution was obtained in parametric form by applying the integral hodograph method. This method gives the possibility of deriving analytical expressions for complex velocity and for potential, both defined in a parameter plane. An expression for the curvature of the cavity boundary was obtained analytically. By using the dynamic boundary condition on the cavity boundary, an integral equation in the velocity modulus was derived. The particular case of zero surface tension is a special case of the solution. The surface tension effect was computed over a wide range of the Weber number for various degrees of cavitation development. Numerical results are presented for the flow configuration, the drag force coefficient, and the position of cavity detachment. It was found that for each radius of the edges, there exists a critical Weber number, below which the iterative solution process fails to converge, so a steady flow solution cannot be computed. This critical Weber number increases as the radius of the edge decreases. As the edge radius tends to zero, the critical Weber number tends to infinity, or a steady cavity flow cannot be computed at any finite Weber number in the case of sharp wedge edges. This shows some limitations of the model based on the Brillouin–Villat criterion of cavity detachment.