Viscous Wave Research Articles

Stability of the composite wave of two planar viscous shock waves for the compressible Navier–Stokes system

We are concerned with the nonlinear stability of the composite wave consisting of two planar viscous shock waves to the three-dimensional compressible Navier–Stokes system. It is shown that if the shock strengths are suitably small but not necessarily of the same order of magnitude, and the initial perturbations are suitably small without the zero-mass conditions, then there exists a unique, globally strong solution in time to the compressible Navier–Stokes system, which asymptotically approaches the corresponding composite wave up to time-dependent shifts in L^{\infty} norm. The proof employs the weighted relative entropy method, an L^{2} -contraction technique with time-dependent shifts to the shocks developed by Kang and Vasseur [J. Eur. Math. Soc. (JEMS) 23 (2021), 585–638; Invent. Math. 224 (2021), 55–146]. We perform the stability analysis within the original H^{2} -perturbation framework instead of using the anti-derivative technique. Compared with the previous work of the author and Wang [J. Eur. Math. Soc. (JEMS) (2023), DOI 10.4171/JEMS/1486] for the single shock case, a major difficulty is the construction of shifts to ensure that the two shock waves are well separated.

Two-dimensional viscous study of coupled nonlinear fluid resonances in two narrow gaps

The wave-induced hydrodynamics of coupled nonlinear piston-mode fluid resonances within two narrow gaps between three barges are numerically investigated using a two-dimensional viscous wave flume. This study aims to explore the time-dependent nonlinear interactions between fluid oscillations in the two gaps. The coupled synchronous dynamic behaviors of fluid oscillations during the transient evolution stage are first examined in terms of amplitude and frequency modulation. It is shown that phase dynamics, including phase slipping, trapping, and locking, play significant roles in establishing the coupled synchronous dynamic evolutions of fluid oscillations in the two gaps. The quasi-steady state of the amplitude- and phase-frequency responses of fluid oscillations within the gaps, along with the reflection and transmission waves in front of and behind the three-barge system, are further analyzed. This analysis clarifies the significance of viscous damping energy dissipation and radiation damping energy transfer involved in gap resonance problems. This clarification also explains the performance of fully nonlinear potential flow solvers in predicting fluid resonances in the two narrow gaps. Finally, the nonlinear dynamic features of fluid oscillations are examined. The effects of incident wave nonlinearity, i.e., wave steepness, on resonant frequencies, response amplitudes, energy dissipation, and reflection and transmission coefficients are investigated. Harmonic analysis via Fourier transformation reveals the contributions of first-, second-, and third-order harmonics to the overall response amplitudes. The physical insights gained from this study provide a deeper understanding of the coupled nonlinear dynamics of piston-mode fluid resonances in multiple narrow gaps.

Nonlinear hydrodynamic assessment of a floating solar double-hull substructure using viscous numerical wave tank

This paper assesses a viscous numerical solver for hydrodynamic simulations of floating double-hull substructures supporting solar panels, critical for advancing floating solar technologies. Using the OpenFOAM repository, the study develops a model based on the Finite Volume method and Volume of Fluid approach, focusing on a range of wave frequencies, from weakly to strongly nonlinear, at intermediate depths. A free decay test calibrated the mesh morphology of the control volume, enabling comparative analysis of single and double-cylinder structures exposed to Stokes second-order nonlinear waves, alongside nonlinear potential models and experimental data. Spectral analysis of these solutions quantifies the nonlinearities' influence on structural responses and high-order dynamic prediction accuracy. The key findings demonstrate that the viscous model accurately predicted heave and surge responses, outperforming the potential model under steep waves. The gap distance between the cylinder in the double-cylinder platforms significantly affects fluid flow and stability, especially under shorter waves. Accurate wave-body simulations require appropriate mesh resolution and tailored wavemaker functions for nonlinear waves. These findings highlight the need to incorporate viscous effects in floating solar platform design to enhance stability and performance in real-world environments.

Lattice Boltzmann Model for a Class of Viscous Wave Equation

Lattice Boltzmann Model for a Class of Viscous Wave Equation

Stability of viscous contact wave for the full compressible Navier–Stokes–Korteweg equations with large perturbation

This study focuses on the stability of the viscous contact wave for the one-dimensional full Navier–Stokes–Korteweg equations with density-temperature dependent transport coefficients and large perturbation. Our findings demonstrate a Nishida–Smoller type result, indicating that the solution remains stable under large perturbation as long as γ−1 is sufficiently small. Notably, the smallness of the capillary coefficient is unnecessary. We then employ the initial layer analysis technique to investigate the asymptotic behaviour in the Wk,p norm. We show that the capillary term has a smoothing effect, which implies that the strong solution is indeed a smooth one. Our results represent an improvement over those previously reported in Chen and Sheng (2019 Nonlinearity 32 395–444). Furthermore, by applying the method in this study to the isothermal case, we can achieve a better outcome than Germain and LeFloch (2016 Commun. Pure Appl. Math. 69 3–61).

Nonlinear stability of planar viscous shock wave to three-dimensional compressible Navier–Stokes equations

We prove the nonlinear stability of the planar viscous shock up to a time-dependent shift for the three-dimensional (3D) compressible Navier–Stokes equations under the generic perturbations, in particular, without zero mass conditions. Moreover, the time-dependent shift function keeps the shock profile shape time-asymptotically. Our stability result is unconditional for the weak planar Navier–Stokes shock. Our proof is motivated by the a -contraction method (a kind of weighted L^{2} -relative entropy method) with time-dependent shift for the stability of viscous shock in the one-dimensional (1D) case. Instead of the classical anti-derivative techniques, we perform the stability analysis of the planar Navier–Stokes shock in the original H^{2} -perturbation framework and therefore zero mass conditions are not necessarily needed, which, in turn, brings out the essential difficulties due to the compressibility of viscous shock. Furthermore, compared with 1D case, there are additional difficulties coming from the wave propagations along the multi-dimensional transverse directions and their interactions with the viscous shock. To overcome these difficulties, a multi-dimensional version of the sharp weighted Poincaré inequality, a -contraction techniques with time-dependent shift, and some essential physical structures of the multi-dimen­sional Navier–Stokes system are fully used.

Validation of predictive performance models for supersonic gas-jet nozzles at the Laboratory for Laser Energetics.

We present results characterizing the neutral-density distributions produced by the supersonic nozzles used in experiments on the OMEGA-60 and OMEGA-EP laser systems at the University of Rochester's Laboratory for Laser Energetics (LLE). Axisymmetric Fluent® simulations using LLE nozzle specifications capture the viscous effects, gas expansion, and shock waves that complicate flow predictions for offsets above the nozzle exit. These simulations show good agreement with neutral-density measurements obtained using a four-wave shearing interferometer. An analytical form is given for the plateau length. Fits to simulation data for boundary layer thickness, mean plateau density, and density ramps are given as functions of nozzle offset and nozzle backing pressure for a number of nozzles and gases.

Convergence rate toward shock wave under periodic perturbation for generalized Korteweg–de Vries–Burgers equation

In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg–de Vries–Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.

On the decay of viscous surface waves in 3D

We consider the incompressible viscous surface wave problem in the setting that the fluid domain is a horizontal infinite layer in 3D. The fluid dynamics is governed by the gravity-driven incompressible Navier–Stokes equations, and the effect of surface tension is ignored on the upper free surface. We prove the optimal time-decay rate of the low-order energy of the solution with minimal derivative count 3, which implies that the Lipschitz norm of the velocity decays at the rate (1+t)−1. This together with a time-weighted estimate for the highest order spatial derivatives of the free surface function leads to the boundedness of the high-order energy, which improves the result of Wang [9].

The waveform comparison of three fractional viscous acoustic wave equations

The waveform comparison of three fractional viscous acoustic wave equations

The hydrodynamic study of the interaction between waves and objects in permeable reef environment

This study employed a self-developed viscous numerical wave tank based on the piston type wave maker and user-defined functions of Fluent software to explore the wave shadowing effect of the transmission of waves near the underwater object and the reef. In the study, various factors such as the permeability of the reef, changes in topography, and objects motion were considered, and detailed simulation analyses were conducted for five scenarios: a solitary reef, a fixed underwater object, a forced moving underwater object, and a free moving underwater object. The results showed that this method can effectively simulate the complex flow field changes around the underwater object and the reef under different conditions, while also fully considering the strong nonlinearity and fluid viscosity effects between the wave surface and the object. Furthermore, it was found that between the free-moving object and the reef, there would be significant swirls and streamline distortion. This indicates that a portion of the incident wave energy is transformed into kinetic energy of swirls, resulting in a smoother and more uniform flow field and reducing wave forces.

Акустический пограничный слой твёрдой абсолютно теплопроводной поверхности

The paper presents the analysis results of formation theoretical descriptions of an acoustic boundary layer near solid absolutely thermally conductive surface, obtained by G. Kirchhoff and L.D. Landau. In both cases, the acoustic boundary layer is formed by inhomogeneous viscous and thermal waves in the wall layer of a liquid medium in contact with the surface of a solid body, from which a plane traveling sound wave is reflected. Based on the analysis, conclusions can be drawn: the analyzed problem solutions are physically sound, independent and complementary to each other. During the formation of an acoustic boundary layer, viscous and thermal waves are excited synchronously in pairs. Inside the acoustic boundary layer, each pair of inhomogeneous waves propagates towards each other. Inhomogeneous waves originate on parallel surfaces that limit the volume of the acoustic boundary layer. The analysis of the process of transformation of heat waves into additional one-dimensional inhomogeneous waves, the appearance of which in the boundary layer was predicted by G. Kirchhoff. It is shown that when interacting with the surface of the body of a traveling sound wave in the sound frequency range, these waves do not affect the formation of the boundary layer. The expressions allowing for a numerical estimation of the heat dissipation power density in the boundary layer are refined. A formula has been obtained that allows us to determine the proportion of the energy of the sound wave that is absorbed in the acoustic boundary layer. In practice, the results obtained in the article can be used, for example, in aeroacoustics to assess the dissipative properties of solid surfaces.

Transient gap resonance between two closely-spaced boxes triggered by nonlinear focused wave groups

A two-dimensional (2D) viscous numerical wave flume is developed based on OpenFOAM® to investigate the transient gap resonance problem triggered by nonlinear focused wave groups. The narrow gap is formed by two closely-spaced boxes, where the weather-side box is allowed to heave freely under wave actions, but the rear one is firmly fixed. The overset mesh method is introduced in the numerical model to capture the weather-side box's large-amplitude motion accurately. This work mainly focuses on the coupled effects of free-surface nonlinearity and heave-freedom of the weather-side box on the fluid oscillations in the gap and heave motions of the weather-side box. The heave-freedom of the weather-side box is demonstrated to play an important effect on the period in which the maximum value of non-dimensional amplitude of fluid oscillation in the gap occurs. It is revealed that the free-surface nonlinearity significantly affects the nonlinear fluid oscillations in the gap but has a limited effect on the heave motions of the weather-side box. The four-phase combination method is further employed to conduct the higher harmonic component analysis to clarify the effects of higher harmonic components on the nonlinear response of fluid and box.

Norm inflation for the viscous nonlinear wave equation

Norm inflation for the viscous nonlinear wave equation

Hydrodynamic performance optimization of a cost-effective WEC-type floating breakwater with half-airfoil bottom

Aiming to design and develop an affordable integrated floating breakwater and wave energy converter (WEC) system that has high performance in power absorption and wave attenuation, a novel integrated floater with a half-airfoil bottom is proposed. The genetic algorithm combined with the boundary element method is employed to optimize the half-airfoil bottom surface represented by the class-shape function transformation parameterization method, for maximizing the power density over the operating bandwidth. Then the operating performance of the optimal half-airfoil bottom floater is further studied in a validated two-dimensional viscous numerical wave tank. The result shows that the optimal half-airfoil bottom floater is a more effective and affordable solution than square and triangular bottom ones. The effective ratio of the floater breadth to the wavelength can be reduced to about a tenth, far below that of the conventional floating breakwater, denoting its excellent wave attenuation capability. The increase in floater breadth induces more intense vortex dynamic behaviors and thus higher dissipation coefficient. A shallow front-wall draft design can improve the maximum conversion efficiency to more than 74%, slightly broaden the resonance bandwidth, and reduce energy dissipation while minimizing costs. The nonlinear PTO damping force contributes to a more stable response. The research results denote the feasibility of bio-inspired WEC-type floating breakwaters and show their superiority in energy extraction, wave attenuation, as well as cost reduction.

Finite-difference frequency-domain method with QR-decomposition-based complex-valued adaptive coefficients for 3D diffusive viscous wave modelling

Abstract The diffusive viscous (DV) model is a useful tool for interpreting low-frequency seismic attenuation and the influence of fluid saturation on frequency-dependent reflections. Among present methods for the numerical solution of the corresponding DV wave equation, the finite-difference frequency-domain (FDFD) method with complex-valued adaptive coefficients (CVAC) has the advantage of efficiently suppressing both numerical dispersion and numerical attenuation. In this research, the FDFD method with CVAC is first generalized to a 3D DV equation. In addition, the current calculation of CVAC involves the numerical integration of propagation angles, conjugate gradient (CG) iterative optimization, and the sequential selection of initial values, which is difficult and inefficient for implementation. An improved method is developed for calculating CVAC, in which a complex-valued least-squares problem is constructed by substituting the 3D complex-valued plane-wave solutions into the FDFD scheme. The QR-decomposition method is used to efficiently solve the least-squares problem. Numerical dispersion and attenuation analyses reveal that the FDFD method with CVAC requires ∼2.5 spatial points in a wavelength within a dispersion deviation of 1% and an attenuation deviation of 10% for the 3D DV equation. An analytic solution for 3D DV wave equation in homogeneous media is proposed to verify the effectiveness of the proposed method. Numerical examples also demonstrate that the FDFD method with CVAC can obtain accurate wavefield modelling results for 3D DV models with a limited number of spatial points in a wavelength, and the FDFD method with QR-based CVAC requires less computational time than the FDFD method with CG-based CVAC.

Surface gravity wave-induced drift of floating objects in the diffraction regime

Floating objects will drift due to the action of surface gravity waves. This drift will depart from that of a perfect Lagrangian tracer due to both viscous effects (non-potential flow) and wave–body interaction (potential flow). We examine the drift of freely floating objects in regular (non-breaking) deep-water wave fields for object sizes that are large enough to cause significant diffraction. Systematic numerical simulations are performed using a hybrid numerical solver, qaleFOAM, which deals with both viscosity and wave–body interaction. For very small objects, the model predicts a wave-induced drift equal to the Stokes drift. For larger objects, the drift is generally greater and increases with object size (we examine object sizes up to $10\,\%$ of the wavelength). The effects of different shapes, sizes and submergence depths and steepnesses are examined. Furthermore, we derive a ‘diffraction-modified Stokes drift’ akin to Stokes (Trans. Camb. Phil. Soc., vol. 8, 1847, pp. 411–455), but based on the combination of incident, diffracted and radiated wave fields, which are based on potential-flow theory and obtained using the boundary element method. This diffraction-modified Stokes drift explains both qualitatively and quantitatively the increase in drift. Generally, round objects do not diffract the wave field significantly and do not experience a significant drift enhancement as a result. For box-shape objects, drift enhancement is greater for larger objects with greater submergence depths (we report an increase of $92\,\%$ for simulations without viscosity and $113\,\%$ with viscosity for a round-cornered box whose size is $10\,\%$ of the wavelength). We identify the specific standing wave pattern that arises near the object because of diffraction as the main cause of the enhanced drift. Viscosity plays a small positive role in the enhanced drift behaviour of large objects, increasing the drift further by approximately $20\,\%$ .

Stability of planar shock wave for the 3-dimensional compressible Navier-Stokes-Poisson equations

This paper is concerned with the stability of planar viscous shock wave for the 3-dimensional compressible Navier-Stokes-Poisson (NSP) system in the domain Ω:=R×T2 with T2=(R/Z)2. The stability problem of viscous shock under small 1-dimensional perturbations was solved in Duan-Liu-Zhang [7]. In this paper, we prove the viscous shock is still stable under small 3-d perturbations. Firstly, we decompose the perturbation into the zero mode and non-zero mode. Then we can show that both the perturbation and zero-mode time-asymptotically tend to zero by the anti-derivative technique and crucial estimates on the zero-mode. Moreover, we can further prove that the non-zero mode tends to zero with exponential decay rate. The key point is to estimate the non-zero mode of nonlinear terms involving electronic potential, see Lemma 6.1 below.

Global stability of viscous contact wave for compressible Navier–Stokes equations with temperature‐dependent transport coefficients and large data

We are concerned with the large‐time behavior of the viscous contact wave for a one‐dimensional compressible Navier–Stokes equations whose transport coefficients depend on the temperature. It is shown that if the adiabatic exponent satisfies a suitable inequality and strength is small enough, the unique solution global in time to ideal polytropic gas exists and asymptotically tends toward the viscous contact wave under large initial perturbation for any . Subsequently, we also prove that the large‐time asymptotic stability of the contact wave has a uniform convergence rate under initial data. Our results improve ‐rate in [F. Huang, Z. Xin, and T. Yang, Adv. Math. 219 (2008), 1246–1297] which studied the constant coefficients Navier–Stokes system. The proofs are given by the elementary energy estimate and anti‐derivative method.

Stability and decay rate of viscous contact wave to one-dimensional compressible Navier-Stokes equations

Stability and decay rate of viscous contact wave to one-dimensional compressible Navier-Stokes equations