Understanding discrete capillary-wave turbulence using a quasi-resonant kinetic equation

2017 ◽  
Vol 816 ◽  
Author(s):  
Yulin Pan ◽  
Dick K. P. Yue
Keyword(s):  
Kinetic Equation ◽  
Euler Equations ◽  
Power Law ◽  
Capillary Wave ◽  
Quantitative Measure ◽  
Energy Cascade ◽  
Turbulence Theory ◽  
Finite Domain ◽  
Spectral Slope ◽  
Infinite Domain

Experimental and numerical studies have shown that, with sufficient nonlinearity, the theoretical capillary-wave power-law spectrum derived from the kinetic equation (KE) of weak turbulence theory can be realized. This is despite the fact that the KE is derived assuming an infinite domain with continuous wavenumber, while experiments and numerical simulations are conducted in realistic finite domains with discrete wavenumbers for which the KE theoretically allows no energy transfer. To understand this, we first analyse results from direct simulations of the primitive Euler equations to elucidate the role of nonlinear resonance broadening (NRB) in discrete turbulence. We define a quantitative measure of the NRB, explaining its dependence on the nonlinearity level and its effect on the properties of the obtained stationary power-law spectra. This inspires us to develop a new quasi-resonant kinetic equation (QKE) for discrete turbulence, which incorporates the mechanism of NRB, governed by a single parameter $\unicode[STIX]{x1D705}$ expressing the ratio of NRB and wavenumber discreteness. At $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{0}\approx 0.02$, the QKE recovers simultaneously the spectral slope $\unicode[STIX]{x1D6FC}_{0}=-17/4$ and the Kolmogorov constant $C_{0}=6.97$ (corrected from the original derivation) of the theoretical continuous spectrum, which physically represents the upper bound of energy cascade capacity for the discrete turbulence. For $\unicode[STIX]{x1D705}<\unicode[STIX]{x1D705}_{0}$, the obtained spectra represent those corresponding to a finite domain with insufficient nonlinearity, resulting in a steeper spectral slope $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{0}$ and reduced capacity of energy cascade $C>C_{0}$. The physical insights from the QKE are corroborated by direct simulation results of the Euler equations.

Decaying capillary wave turbulence under broad-scale dissipation

2015 ◽  
Vol 780 ◽  
Author(s):  
Yulin Pan ◽  
Dick K. P. Yue
Keyword(s):  
Energy Transfer ◽  
Energy Dissipation ◽  
Energy Flux ◽  
Power Law ◽  
Time Moment ◽  
Capillary Waves ◽  
Inertial Range ◽  
Turbulence Theory ◽  
Weak Turbulence ◽  
Spectral Slope

We study the freely decaying weak turbulence of capillary waves by direct numerical solution of the primitive Euler equations. By introducing a small amount of wave dissipation, measured by the viscosity magnitude ${\it\gamma}_{0}$, we are able to recover phenomena observed in experiments that are not described by weak-turbulence theory (WTT), including the exponential modal decay and time variation of the width and power-law spectral slope ${\it\alpha}$ of the inertial range. In contrast to WTT, this problem also involves non-constant inter-modal energy transfer across the inertial range, which imposes a difficulty in quantifying and measuring the energy flux $P$ associated with a certain power-law spectrum. We propose an effective and novel way to evaluate $P$ in such cases by physically considering the unsteady effects of the spectrum and variation of the inter-modal energy transfer. Our results show the fundamental difference between the energy flux $P$ and the total energy dissipation rate ${\it\Gamma}$, which is due to significant energy dissipation within the inertial range. This settles the previous debate on the measurement of $P$ which assumes the equivalence of the two. Based on our numerical data, we obtain a general form of the time-evolving inertial-range spectrum, where the parameters involved are functions of ${\it\gamma}_{0}$ only. The value of the spectral slope ${\it\alpha}$ at each time moment in the decay, however, is found to be uniquely related to the spectral magnitude at that time and irrespective of ${\it\gamma}_{0}$, in the range we consider. This physically reveals the dominant effect of nonlinear wave interaction in forming the power-law spectrum within the inertial range. The evolutions of the inertial-range energy are shown to be predicted by analytical integration of the evolving spectra for different values of ${\it\gamma}_{0}$.

scholarly journals Oceanic Internal-Wave Field: Theory of Scale-Invariant Spectra

2010 ◽  
Vol 40 (12) ◽  
pp. 2605-2623 ◽  
Author(s):  
Yuri V. Lvov ◽  
Kurt L. Polzin ◽  
Esteban G. Tabak ◽  
Naoto Yokoyama
Keyword(s):  
Kinetic Equation ◽  
Power Law ◽  
Spectral Power ◽  
Internal Gravity Waves ◽  
Collision Integral ◽  
Turbulence Theory ◽  
Coriolis Parameter ◽  
Scale Invariant ◽  
Weakly Nonlinear ◽  
Resonant Interactions

Abstract Steady scale-invariant solutions of a kinetic equation describing the statistics of oceanic internal gravity waves based on wave turbulence theory are investigated. It is shown in the nonrotating scale-invariant limit that the collision integral in the kinetic equation diverges for almost all spectral power-law exponents. These divergences come from resonant interactions with the smallest horizontal wavenumbers and/or the largest horizontal wavenumbers with extreme scale separations. A small domain is identified in which the scale-invariant collision integral converges and numerically find a convergent power-law solution. This numerical solution is close to the Garrett–Munk spectrum. Power-law exponents that potentially permit a balance between the infrared and ultraviolet divergences are investigated. The balanced exponents are generalizations of an exact solution of the scale-invariant kinetic equation, the Pelinovsky–Raevsky spectrum. A small but finite Coriolis parameter representing the effects of rotation is introduced into the kinetic equation to determine solutions over the divergent part of the domain using rigorous asymptotic arguments. This gives rise to the induced diffusion regime. The derivation of the kinetic equation is based on an assumption of weak nonlinearity. Dominance of the nonlocal interactions puts the self-consistency of the kinetic equation at risk. However, these weakly nonlinear stationary states are consistent with much of the observational evidence.

scholarly journals Turbulence of capillary waves forced by steep gravity waves

2018 ◽  
Vol 850 ◽  
pp. 803-843 ◽  
Author(s):  
M. Berhanu ◽  
E. Falcon ◽  
L. Deike
Keyword(s):  
Nonlinear Wave ◽  
Power Law ◽  
Gravity Waves ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Turbulence Theory ◽  
Wave Turbulence ◽  
Small Scale ◽  
Turbulent Regime ◽  
Strong Nonlinearity

We study experimentally the dynamics and statistics of capillary waves forced by random steep gravity waves mechanically generated in the laboratory. Capillary waves are produced here by gravity waves from nonlinear wave interactions. Using a spatio-temporal measurement of the free surface, we characterize statistically the random regimes of capillary waves in the spatial and temporal Fourier spaces. For a significant wave steepness (0.2–0.3), power-law spectra are observed both in space and time, defining a turbulent regime of capillary waves transferring energy from the large scale to the small scale. Analysis of temporal fluctuations of the spatial spectrum demonstrates that the capillary power-law spectra result from the temporal averaging over intermittent and strong nonlinear events transferring energy to the small scale in a fast time scale, when capillary wave trains are generated in a way similar to the parasitic capillary wave generation mechanism. The frequency and wavenumber power-law exponents of the wave spectra are found to be in agreement with those of the weakly nonlinear wave turbulence theory. However, the energy flux is not constant through the scales and the wave spectrum scaling with this flux is not in good agreement with wave turbulence theory. These results suggest that theoretical developments beyond the classic wave turbulence theory are necessary to describe the dynamics and statistics of capillary waves in a natural environment. In particular, in the presence of broad-scale viscous dissipation and strong nonlinearity, the role of non-local and non-resonant interactions should be reconsidered.

A new Reliable Numerical Algorithm Based on the First Kind of Bessel Functions to Solve Prandtl–Blasius Laminar Viscous Flow over a Semi-Infinite Flat Plate

2012 ◽  
Vol 67 (12) ◽  
pp. 665-673 ◽  
Author(s):  
Kourosh Parand ◽  
Mehran Nikarya ◽  
Jamal Amani Rad ◽  
Fatemeh Baharifard
Keyword(s):  
Viscous Flow ◽  
Flat Plate ◽  
Numerical Algorithm ◽  
Bessel Functions ◽  
Nonlinear Ordinary Differential Equation ◽  
Finite Domain ◽  
Algebraic Equations ◽  
Infinite Domain ◽  
Third Order ◽  
Domain Truncation

In this paper, a new numerical algorithm is introduced to solve the Blasius equation, which is a third-order nonlinear ordinary differential equation arising in the problem of two-dimensional steady state laminar viscous flow over a semi-infinite flat plate. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, defined on ℝ and is convergent for any x ∊ℝ. In this work, we solve the problem on semi-infinite domain without any domain truncation, variable transformation basis functions or transformation of the domain of the problem to a finite domain. This method reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show the applicability and efficiency of our method.

Swirling Flow Over an Oscillatory Stretchable Disk

2014 ◽  
Vol 30 (4) ◽  
pp. 339-347 ◽  
Author(s):  
S. Munawar ◽  
A. Ali ◽  
N. Saleem ◽  
A. Naqeeb
Keyword(s):  
Oscillatory Flow ◽  
Swirling Flow ◽  
Three Dimensional ◽  
Finite Domain ◽  
Governing Equations ◽  
Infinite Domain ◽  
Similarity Transformations ◽  
Sinusoidal Oscillations ◽  
Dimensional Boundary ◽  
Layer Flow

AbstractIn this work a numerical investigation has been conducted to study the unsteady oscillatory flow of a viscous fluid induced by a swirling disk. The disk stretches radially with the time-based sinusoidal oscillations. The governing equations for the three-dimensional boundary layer-flow are normalized using a suitable set of similarity transformations. The normalized partial differential equations are then solved numerically using a finite difference scheme by altering the semi-infinite domain to a finite domain. The effects of various imperative parameters on the oscillatory flow are discussed with graphs and tables.

Numerical Investigation of Mechanisms Underlying Oceanic Internal Gravity Wave Power-Law Spectra

2020 ◽  
Vol 50 (9) ◽  
pp. 2713-2733
Author(s):  
Yulin Pan ◽  
Brian K. Arbic ◽  
Arin D. Nelson ◽  
Dimitris Menemenlis ◽  
W. R. Peltier ◽  
...  
Keyword(s):  
Power Law ◽  
Gravity Waves ◽  
High Frequency ◽  
Internal Gravity Wave ◽  
Numerical Data ◽  
Field Measurements ◽  
Internal Gravity Waves ◽  
Collision Integral ◽  
Turbulence Theory ◽  
Nonlinear Interactions

AbstractWe consider the power-law spectra of internal gravity waves in a rotating and stratified ocean. Field measurements have shown considerable variability of spectral slopes compared to the high-wavenumber, high-frequency portion of the Garrett–Munk (GM) spectrum. Theoretical explanations have been developed through wave turbulence theory (WTT), where different power-law solutions of the kinetic equation can be found depending on the mechanisms underlying the nonlinear interactions. Mathematically, these are reflected by the convergence properties of the so-called collision integral (CL) at low- and high-frequency limits. In this work, we study the mechanisms in the formation of the power-law spectra of internal gravity waves, utilizing numerical data from the high-resolution modeling of internal waves (HRMIW) in a region northwest of Hawaii. The model captures the power-law spectra in broad ranges of space and time scales, with scalings ω−2.05±0.2 in frequency and m−2.58±0.4 in vertical wavenumber. The latter clearly deviates from the GM76 spectrum but is closer to a family of induced-diffusion-dominated solutions predicted by WTT. Our analysis of nonlinear interactions is performed directly on these model outputs, which is fundamentally different from previous work assuming a GM76 spectrum. By applying a bicoherence analysis and evaluations of modal energy transfer, we show that the CL is dominated by nonlocal interactions between modes in the power-law range and low-frequency inertial motions. We further identify induced diffusion and the near-resonances at its spectral vicinity as dominating the formation of power-law spectrum.

Infinite Boundary Element Formulation for the Analysis of CP System for Submersible Pump

2013 ◽  
Vol 471 ◽  
pp. 313-318 ◽  
Author(s):  
M. Safuadi ◽  
S. Fonna ◽  
M. Ridha ◽  
Zebua ◽  
Ahmad K. Ariffin ◽  
...  
Keyword(s):  
Boundary Element ◽  
Cathodic Protection ◽  
Domain Boundary ◽  
Protection System ◽  
Finite Domain ◽  
Element Formulation ◽  
Submersible Pump ◽  
Infinite Domain ◽  
Cathodic Protection System ◽  
Early Damage

The effectiveness of the cathodic protection system is very important to be maintained for the submersible pump structure. Early damage of the infrastructure can be caused by improper design of the protection system. However, nowadays the effectiveness of the cathodic protection system could not be evaluated before the system applied in the field. This study is conducted on development of 3D infinite domain boundary element method (BEM) to evaluate the cathodic protection system for submersible pump structure using aluminum sacrificial anode. In this study, the potential in the domain was modeled using Laplace equation. The equation was solved by applying BEM, hence the potential distribution and current density on the metal surface and at any location in the domain can be obtained. The numerical analysis result shows that the 3D infinite domain BEM can be used to simulate the cathodic protection system. Moreover, the execution time for infinite domain BEM is less than the finite domain for the evaluated case.

scholarly journals Numerical study on the energy cascade of pulsatile Newtonian and power-law flow models in an ICA bifurcation

PLoS ONE ◽  
2021 ◽  
Vol 16 (1) ◽  
pp. e0245775
Author(s):  
Samar A. Mahrous ◽  
Nor Azwadi Che Sidik ◽  
Khalid M. Saqr
Keyword(s):  
Reynolds Number ◽  
Power Law ◽  
Numerical Study ◽  
Carotid Bifurcation ◽  
Shear Thinning ◽  
Energy Cascade ◽  
Hemodynamic Parameters ◽  
Vortex Identification ◽  
Newtonian Viscosity ◽  
The Impact

The complex physics and biology underlying intracranial hemodynamics are yet to be fully revealed. A fully resolved direct numerical simulation (DNS) study has been performed to identify the intrinsic flow dynamics in an idealized carotid bifurcation model. To shed the light on the significance of considering blood shear-thinning properties, the power-law model is compared to the commonly used Newtonian viscosity hypothesis. We scrutinize the kinetic energy cascade (KEC) rates in the Fourier domain and the vortex structure of both fluid models and examine the impact of the power-law viscosity model. The flow intrinsically contains coherent structures which has frequencies corresponding to the boundary frequency, which could be associated with the regulation of endothelial cells. From the proposed comparative study, it is found that KEC rates and the vortex-identification are significantly influenced by the shear-thinning blood properties. Conclusively, from the obtained results, it is found that neglecting the non-Newtonian behavior could lead to underestimation of the hemodynamic parameters at low Reynolds number and overestimation of the hemodynamic parameters by increasing the Reynolds number. In addition, we provide physical insight and discussion onto the hemodynamics associated with endothelial dysfunction which plays significant role in the pathogenesis of intracranial aneurysms.

Sign in / Sign up

Export Citation Format

Share Document