Second-order Quantities Research Articles

Displacement norm in the presence of an inverse-square perturbing acceleration in the reference frame associated with the radius vector

The problem of motion of a zero-mass point under the influence of attraction to the central body and a small perturbing acceleration P′ = P/r 2 is considered, where r is the distance to the attracting center, components of the vector P are assumed to be constant in a reference system with axes directed along the radius vector, the transversal and the angular momentum vector. Previously, for this problem, we found equations of motion in the mean elements and formulas for the transition from the osculating elements to the mean elements in the first order of smallness; we neglected second-order quantities. In this work, the Euclidean (root–mean–square over the mean anomaly) displacement norm ||dr||2 is obtained, where dr represents the difference between the position vectors on the osculating and mean orbit. It turned out that ||dr||2 depends only on the components of the vector P (positive definite quadratic form), the semi–major axis (proportional to the second power) and the eccentricity of the osculating ellipse. The norm ||dr||2 is obtained in the form of series in powers of the β=e/1+1-e2and in powers of the eccentricity e. The results are applied to the problem of the motion of asteroids under the influence of a perturbing acceleration inversely proportional to the square of the heliocentric distance, in particular, under the influence of the Yarkovsky effect.

The Fourier–Malliavin Volatility (FMVol) MATLAB® library

This paper presents the Fourier–Malliavin Volatility (FMVol) estimation library for MATLAB®. This library includes functions that implement Fourier–Malliavin estimators (see Malliavin and Mancino (2002, 2009)) of the volatility and co-volatility of continuous stochastic volatility processes and second-order quantities, like the quarticity (the squared volatility), the volatility of volatility and the leverage (the covariance between changes in the process and changes in its volatility). The Fourier–Malliavin method is fully non-parametric, does not require equally-spaced observations and is robust to measurement errors, or noise, without any preliminary bias correction or pre-treatment of the observations. Furthermore, in its multivariate version, it is intrinsically robust to irregular and asynchronous sampling. Although originally introduced for a specific application in financial econometrics, namely the estimation of asset volatilities, the Fourier–Malliavin method is a general method that can be applied whenever one is interested in reconstructing the latent volatility and second-order quantities of a continuous stochastic volatility process from discrete observations.

Predicting pressure fields from incomplete velocity fields based on deep convolutional neural network

Acquisition of an instantaneous pressure field in a flow field is essential for experimental fluid dynamics research because pressure is associated with various flow phenomena. Machine learning has performed well in various fluid mechanics studies in recent years due to its nonlinearity. In this paper, we use the machine learning method to predict the pressure fields from incomplete velocity fields while reconstructing the missing velocity information, as exemplified by the stationary circular cylinder flow and circular cylinder undergoing forced oscillating problems, in which the incompleteness of the velocity field is caused by the optical path occlusion in experiments. We propose a network model based on convolutional neural networks. The experimental results show that the error of machine learning methods is around 4%, and even if the resolution of the training set is reduced, it does not affect the prediction accuracy of the network. For statistical averaging of prediction results, the network has high accuracy in predicting first-order quantities, but its ability to predict second-order quantities is limited. We also explore the effects of the amount of noise on the performance of networks. When applying the model to the forced oscillating cylinder problem, we use transfer learning to train the network given the similarity of the two problems, and the training error yields faster convergence and more minor convergence results. In addition to this, we use flow field data with a single vibration frequency to construct a single dataset, and flow field data with multiple vibration frequencies to construct a mixed dataset, and study the effect of different datasets on the prediction accuracy, which shows that it is difficult for neural networks to achieve both accuracy and generalization.

On the second-order neutral Volterra integro-differential equation and its numerical solution

In this paper, we consider an initial-value problem for a second-order neutral Volterra integro-differential equation. First, we give the stability inequality indicating the stability of the problem with respect to the right-side and initial conditions. Further, we develop a finite difference method that uses for differential part second difference derivative, for the integral part appropriate composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points. Error estimate for the approximate solution is established, which shows the second-order accuracy. Finally, the numerical experiments are presented confirming the accuracy of the proposed scheme.

Flutter stability analysis of aeroelastic systems with consideration of hybrid uncertain parameters

Uncertainties are usually ubiquitous and of different sources, and especially hybrid uncertainties widely exist in the aeroelastic system including external loads and material properties. In this context, a novel numerical method for flutter stability analysis of aeroelastic systems considering hybrid uncertain parameters is proposed. Uncertain parameters in aeroelastic systems are characterized by stochastic and interval model, and the flutter analysis model with hybrid uncertain parameters is established. The interval-based probability density evolution method is then developed to capture the probability density function of the lower and upper bound of generalized eigenvalues. By introducing the virtual time parameter, the probability density evolution equation can be transformed into a standard form. Using the finite difference method and the total variation diminishing (TVD) scheme, the probability density function as well as the second-order statistical quantities of the lower and upper bound of the maximum real part of generalized eigenvalues are predicted efficiently. Thus, the stability of aeroelastic systems with hybrid uncertain parameters can be analyzed based on the probability distribution of the interval bound of the maximum real part of generalized eigenvalues. Two numerical examples demonstrate that the proposed method yields results consistent with Monte-Carlo simulation method and improve the computational efficiency.

PyWolf: A PyOpenCL implementation for simulating the propagation of partially coherent light

We present PyWolf, an open-source software capable of performing numerical simulations of partially coherent light propagation from two-dimensional light sources. PyWolf computes the evolution of a user-defined cross-spectral density function in the Fresnel and far field approximations, which enables the retrieval of second-order optical quantities of interest such as the spectral degree of coherence and spectral density for a given frequency. The open-source tool kit PyOpenCL is used to increase the computation speed. We present examples of propagation of different source models and optical systems to validate our implementation. Performance results for the computation speed when using parallel computation through PyOpenCL is shown. Source models and propagation systems can be easily added to PyWolf, which has a graphical user interface built with PyQt5. This software can be of great utility for partially coherent light simulation problems that are difficult to treat analytically. Program summaryProgram Title: PyWolfCPC Library link to program files:https://doi.org/10.17632/frjscxypkd.1Developer's repository link:https://github.com/tiagoecmagalhaes/PyWolfLicensing provisions: GPLv3Programming language: PythonExternal routines: NumPy, SciPy, Matplotlib, PyOpenCL, PyQt5Nature of problem: Numerical simulations of the propagation of partially coherent light from planar sources in the Fresnel or far field approximations. Computation time can be improved by using optimized versions of the Fast Fourier Transform (FFT) algorithm, but other calculations can only be increased through parallel computation.Solution method: We use the open-source toolkit PyOpenCL to perform parallel computation on time-consuming calculations. The user can easily modify and add more features to the software, such as source models and propagation methods. A graphical user interface is also built using PyQt5 which enables the user to set input parameters, including the user's custom models, view the simulation results, export data, and save and load sessions.

기준 좌표계에 따른 탄성체의 일반화 파랑 하중 및 응답에 대한 연구

In this paper, the generalized hydrodynamic force and response of a flexible body are calculated at different reference coordinate systems. We generalize the equation of motion for a flexible body by using the conservation of momentum (Mei et al., 2005). To obtain the equations in the generalized mode, two different reference coordinates are adopted. The first is the body-fixed coordinate system by a rigid body motion. The other is the inertial coordinate system which has been adopted for the analysis. Using the perturbation scheme in the weakly-nonlinear assumption, the equations of motion are expanded up to second-order quantities and several second-order forces are obtained. Numerical tests are conducted for the flexible barge model in head waves and the vertical bending is only considered in the hydroelastic responses. The results show that the linear response does not have the difference between the two formulations. On the other hand, second-order quantities have different values for which the rigid body motion is relatively large. However, the total summation of second-order quantities has not shown a large difference at each reference coordinate system.

Turbulent kinetic dissipation analysis for residual-based large eddy simulation of incompressible turbulent flow by variational multiscale method

The underlying physical mechanism of the residual-based large eddy simulation (LES) based on the variational multiscale (VMS) method is clarified. Resolved large-scale energy transportation equation is mathematically derived for turbulent kinetic energy budget analysis. Firstly, statistical results of benchmark turbulent channel flow at Reτ=180 obtained using a coarse mesh are compared with the results obtained by the classical LES with the Smagorinsky and dynamic subgrid stress (SGS) model. The present LES shows an advantage in predicting the statistical results of the incompressible turbulent flows. Secondly, the contributions of the unresolved small-scale presentation terms (Term I-IV in Eq. (10)) to the turbulent kinetic dissipation are analysed for the VMS method. The results show that the turbulent kinetic dissipation provided by the numerical diffusion in the VMS method is smaller in the inner layer, larger in the outer layer of the channel flow than those by the Smagorinsky and dynamic SGS model. The turbulent kinetic dissipation in the VMS method is mainly given by the numerical diffusion provided by one of the “cross-stress” terms (Term I, same as the stabilization term in the SUPG method) and LSIC term (Term IV). The other one of the “cross-stress” terms (Term II) gives rise to the positive turbulent kinetic energy budget, and does not dissipate the turbulent kinetic energy. The so-called “Reynolds stress” term (Term III) dissipates the turbulent energy but provides a very small numerical diffusion. Finally, on the basis of the turbulent kinetic energy dissipation analysis, a new residual-based stabilized finite element formulation is proposed by modifying the large-scale equation in the VMS method. Numerical experiments of 2D lid-driven cavity flow and 3D incompressible turbulent channel flow are tested to validate the proposed formulation. It is shown that all the stabilization terms in the proposed formulation produce additional numerical diffusions and physically increase the total turbulent kinetic dissipation. Consequently, an apparent improvement in both the first-order and second-order statistical quantities are pursued by the new stabilized finite element formulation.

Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

Poisson processes in the space of (d-1)-dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature -1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all dge 2, it is shown that in case (ii) the central limit theorem holds for din {2,3} and fails if dge 4 and k=d-1 or if dge 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

Flutter stability analysis of stochastic aeroelastic systems via the generalized eigenvalue-based probability density evolution method

This paper proposes a generalized eigenvalue-based probability density evolution method for aeroelastic systems considering stochastic uncertainties in structural and aerodynamic parameters. Uncertain parameters in aeroelastic systems are characterized by stochastic model, and the flutter analysis model with stochastic parameters is established. The generalized eigenvalue-based probability density evolution method is then developed to capture the probability density function of the maximum real part of generalized eigenvalues. By introducing the virtual time parameter, the probability density evolution equation can be transformed into a standard form. Using the finite difference method and the total variation diminishing (TVD) scheme, the probability density function as well as the second-order statistical quantities of the maximum real part of generalized eigenvalues are predicted efficiently. Thus, the flutter stability of aeroelastic systems with stochastic parameters can be analyzed based on the probability distribution of the maximum real part of generalized eigenvalues. Two numerical examples demonstrate that the proposed method yields results consistent with Monte-Carlo simulation method and improves the computational efficiency.

Numerical study on the second-order hydrodynamic force and response of an elastic body – In bichromatic waves

The second-order hydrodynamic force and response of an elastic body in bichromatic waves are studied by using higher-order boundary element method (HOBEM). To solve the boundary value problem, the free-surface wave Green function is adopted in the boundary integral equation and the discretization is conducted by quadratic shape function. In the boundary condition, the normal vector variation on the body surface is re-defined to consider both rigid and elastic body motions. It is invoked especially for the derivation of second-order body boundary condition and several generalized forces in zero forward speed condition. In the second-order quantities, the second-order velocity potential force is obtained by using indirect formulation in the generalized mode and the contribution of each component is investigated. The validation for the second-order forces is first conducted by comparing with published data. The excitation force on the rigid body motion of a hemisphere is calculated and showed a good agreement with other semi-analytic and numerical results. Using simplified structural model, several vertical bending modes of an elastic hemisphere are considered as a numerical test. Second-order hydrodynamic force and response for the two-node vertical bending are obtained and properties of second-order quantities are confirmed.

Are all KdV-type shallow water wave equations the same with uniform solutions?

ABSTRACT Cnoidal wave and its extreme case, the solitary wave, can be described by the KdV equation, which was first derived by Korteweg and de Vires with the first-order accuracy. Subsequently, different authors proposed their derivations and claimed that their equations, sharing similar expressions but different corresponding coefficients, were the same as the original one. After introducing a unified dimensionless frame, this study re-derived the KdV equation with respect to seven existing methods and confirmed that KdV equation indeed refers to a type of first-order equations, rather than a specified one. Differences in equations come from the influence of the second-order quantities associated with the derivation process. Regarding the cnoidal wave and solitary wave, the KdV-type equations obtained using different methods present the same first-order solution for wave profile. Nevertheless, in their directly derived results, different wave celerities and water particle velocities are presented due to the influence of second-order quantities. Additionally, comparing with the second-order solutions, all directly derived wave celerity solutions predict well for the Ursell number between 20 and 100. As for the first-order solution of the water particle velocity, all methods present the same result except Dean’s expression which contains a different coefficient.

Assessment of RANS and LES turbulence models for natural convection in a differentially heated square cavity

Development of computationally efficient modeling techniques for thermally driven buoyant flows remains an ongoing challenge for the computational fluid dynamics (CFD) community due to the complex interactions of buoyancy, heat transfer, and turbulence. Although several “best practice” guides are available for certain scenarios, comprehensive validation studies against benchmark-quality data must occur to ensure the accuracy and suitability of these computational models. To this end, the present study provides a robust assessment of 16 different turbulence treatments − 13 Reynolds-averaged Navier-Stokes (RANS) formulations and 3 large-eddy simulation (LES) sub-grid scale models – and their ability to predict various first- and second-order system response quantities (SRQs) in a differentially heated enclosure at a Rayleigh number of 1.58 × 109. Current ASME standards are used to quantify the latent discretization errors in the RANS predictions while a sub-grid activity parameter is used to justify the spatial resolution of the LES models. In general, most RANS models suitably replicate surface heat transfer and first-order SRQs; however, certain low-Reynolds-number formulations markedly mischaracterize the same parameters. Following a thorough comparison of turbulent statistics and turbulence kinetic energy budgets, these modeling errors are traced back to a misprediction of turbulent viscosity and direct production of turbulence from buoyancy. The LES predictions from all three sub-grid scale models are in good agreement with the corresponding experimental measurements with only minor disparities in the horizontal and vertical components of the turbulent heat flux.

Predicting stochastic characteristics of generalized eigenvalues via a novel sensitivity-based probability density evolution method

This paper proposes a novel numerical method for predicting the probability density function of generalized eigenvalues in the mechanical vibration system with consideration of uncertainties in structural parameters. The eigenproblem of structural vibration is presented by first and the sensitivity of generalized eigenvalues with respect to structural parameters can be derived. The probability density evolution method is then developed to capture the probability density function of generalized eigenvalues considering uncertain material properties. Within the proposed method, the probability density evolution equation for the generalized eigenvalue problem is established accounting for the sensitivity of generalized eigenvalues with respect to structural parameters. A new variable which connects generalized eigenvalues to structural parameters is then introduced to simplify the original probability density evolution equation. Next, the simplified probability density evolution equation is solved by using the finite difference method with total variation diminishing schemes. Finally, the probability density function as well as the second-order statistical quantities of generalized eigenvalues can be predicted. Numerical examples demonstrate that the proposed method yields results consistent with Monte-Carlo simulation method within significantly less computation time and the coefficients of variation of uncertain parameters as well as the total number of them have remarkable effects on stochastic characteristics of generalized eigenvalues.

Stirring anisotropic turbulence with an active grid

We study spectra and high-order structure functions in anisotropic wind tunnel turbulence, which is generated using an active grid. In the first experiment, we impose homogeneous shear turbulence with a constant gradient of the mean flow and (approximately) homogeneous turbulent fluctuations. We measure mixed structure functions of order 2, 3, 4, 6, and 10 using an array of two-component hotwires. These structure functions, which vanish for isotropic turbulence, display scaling with scaling exponents that highlight intermittency: the return to isotropy at small scales of large fluctuations is much slower than expected on the basis of a simple Kolmogorov-like scaling argument [J. L. Lumley, “Similarity and the turbulent energy spectrum,” Phys. Fluids 10, 855 (1967)]. In the second experiment, we impose anisotropy in otherwise homogeneous turbulence through the time modulation of the active grid. This is done by driving the grid using signals from a turbulence (shell) model, which acts as a convenient turbulent random signal generator. In this way, different statistical properties of different velocity components could be imposed. Similar to the first experiment, our interest is in the return to isotropy of the small-scale turbulent fluctuations, which is quantified using second-order quantities such as spectra and correlation functions. Also, in this case, the strongly anisotropic correlations induced by the forcing at large scales tend to return to isotropy at small, inertial-range scales, but with the imprint of large-scale anisotropy retained.

Vector acoustic study of ship noise during the 2017 Sediment Characterization Experiment off New England

The Intensity Vector Autonomous Recorder (IVAR) is a bottom deployed system measuring particle velocity and pressure. Results using IVAR in the Sediment Characterization Experiment (SBCEX) conducted off New England (spring 2017), involving active sources have been presented [P. H. Dahl and D. R. Dall’Osto, IEEE J. Ocean. Eng. (2019)]. Here, passive ship noise is studied from a 200-m length cargo vessel that is tracked over a 10 km course a 15 knots for which the closest point of approach (CPA) to the IVAR deployment location (1.25 m above the seafloor) was 500 m, as confirmed by Automatic Identification System (AIS) data. The time-frequency interference pattern of the noise as the vessel closes and opens in range is studied by way of vector acoustic field indicators. These are non-dimensional ratios of second-order quantities, e.g., kinetic energy over potential energy, which preserve magnitude over the track. Inferences on the seabed made from study of the field indicators are compared with geoacoustic models emerging from studies related to SBCEX. The aspect-dependent source level of the vessel is also estimated, with method tested on a known, omni-directional source towed past IVAR from the research vessel R/V Endeavor. [Study supported by Office of Naval Research.]

PIV measurements and CFD simulations of the particle-scale flow distribution in a packed bed

A comparison of the particle-scale PIV measurements with the corresponding particle-resolved CFD simulations in the turbulent flow regime using SST k-ω model is performed. A cylindrical packed bed containing spherical particles with the tube-to-particle diameter ratio of ∼4.3 operating at Rebed in the range of 1100–6600 was considered. The measured and predicted distribution of the first-order (Vx and Vy) and second-order (vorticity and strain rate) mean velocity quantities showed a reasonable agreement, which improved with the increase in the Rebed. The observed deviations were caused by the differences in the geometry (particle position and upstream packing condition) used in the CFD model compared to the measurements. On the other hand, the turbulent quantities (k and ε) were under-predicted in the simulations. However, for the ε, the agreement with the measurements was found to be better at higher Rebed compared to that of the k illustrating the influence of the turbulence model on the predictions. From the results, it can be inferred that the SST k-ω turbulence model appears to be more suitable for the high-Re flows. The present work helps to establish a methodology to validate the particle-resolved CFD simulations in the turbulent flow regime.

Quadratic quantities in acoustics: Scattering cross-section and radiation force

Time-harmonic acoustic waves interact with a scatterer: this interaction is calculated using linear, first-order theory. However, there are quadratic, second-order quantities that are of interest. These include the scattering cross-section and the steady radiation force; these quantities can be expressed as integrals of products of first-order quantities over a sphere. These integrals are evaluated exactly. The results are infinite series of products of the coefficients in the spherical multipole expansions of the incident and scattered fields; they do not depend on the radius of the spherical integration surface. For a specific scattering problem, the coefficients can be connected by an appropriate T-matrix. In most previous work, the spherical surface is moved to infinity so that far-field quantities can be introduced: it is shown that this process is not straightforward and it may introduce spurious difficulties.

Simulation of an Evolving Convective Boundary Layer Using a Scale-Dependent Dynamic Smagorinsky Model at Near-Gray-Zone Resolutions

AbstractA scale-dependent Lagrangian-averaged dynamic Smagorinsky subgrid scheme with stratification effects is used to simulate the evolving convective boundary layer of the Wangara (Australia) case study in the gray-zone regime (specifically, for grid lengths from 25 to 400 m). The dynamic Smagorinsky and standard Smagorinsky approaches are assessed for first- and second-order quantities in comparison with results derived from coarse-grained large-eddy simulation (LES) fields. In the LES regime, the subgrid schemes produce very similar results, albeit with some modest differences near the surface. At coarser resolutions, the use of the standard Smagorinsky approach significantly delays the onset of resolved turbulence, with the delay increasing with coarsening resolution. In contrast, the dynamic Smagorinsky scheme much improves the spinup and so is also able to maintain consistency with the LES temperature profiles at the coarser resolutions. Moreover, the resolved part of the turbulence reproduces well the turbulence profiles obtained from the coarse-grained fields, especially in the near gray zone. The dynamic scheme does become somewhat overenergetic with further coarsening of the resolution, especially near the surface. The dynamic scheme reaches its limit in the current configuration when the test filter starts to sample at the unresolved scales, returning very small Smagorinsky coefficients. Sensitivity tests reveal that the dynamic model can adapt to changes in the imposed numerical or subgrid diffusion by adjusting the Smagorinsky constant to the changing flow field and minimizing the dissipation effects on the resolved turbulence structures.

DNS of turbulent mixed convection over a vertical backward-facing step for lead-bismuth eutectic

Studies of turbulent mixed convection over a vertical backward-facing step for liquid metals are indispensable to research the buoyancy effects in flow separation and reattachment scenarios that occurs in innovative nuclear and solar power facilities. Direct numerical simulations of turbulent buoyancy-aided flow at two moderate Richardson numbers (Ri = 0.1, 0.2) and buoyancy-opposed flow at Ri = −0.04 for lead–bismuth eutectic (LBE, Prandtl number Pr = 0.025) are performed. The forced convection and air mixed convection (Pr = 0.7, Ri = 0.1) are also simulated for comparison. In all the cases Reynolds number is 4805 and the expansion ratio is 1.5. Mean statistics like velocity components, temperature and heat fluxes are given. Second-order quantities such as Reynolds stresses, temperature fluctuations, and turbulent heat fluxes are also discussed. The results show that buoyancy has substantially altered the flow field for LBE flow compared to air flow at moderate Ri. For buoyancy-aided LBE flow, the recirculation zone is reduced in size and the reattachment length is shortened. With Ri increasing, the second vortex gradually prevails over the main vortex and finally pushes it detach from the wall, leading to positive skin-friction coefficient. Buoyancy weakens the turbulent quantities such as Reynolds stresses, temperature fluctuations and turbulent heat fluxes. Both skin-friction coefficient and Nusselt number increase due to the reduced reversed flow resulting from buoyancy acceleration near the wall. However, it is opposite for the case of buoyancy-opposed flow. The skin-friction coefficient and Nusselt number decrease due to the increased reversed flow though the turbulence is enhanced in buoyancy-opposed flow. These high-resolved data could promote the understanding of this scenario and improve turbulent modelling for flow separation and reattachment of low Prandtl fluids.