Approximate Velocity Research Articles

Layer-averaged approximation of Navier–Stokes system with complex rheologies

In this work, we present a family of layer-averaged models for the Navier–Stokes equations. For its derivation, we consider a layerwise linear vertical profile for the horizontal velocity component. As a particular case, we also obtain layer-averaged models with the common layerwise constant approximation of the horizontal velocity. The approximation of the derivatives of the velocity components is set by following the theory of distributions to account for the discontinuities at the internal interfaces. Several models has been proposed, depending on the order of approximation of an asymptotic analysis respect to the shallowness parameter. Then, we obtain a hydrostatic model with vertical viscous effects, a hydrostatic model where the pressure depends on the stress tensor, and fully non-hydrostatic models, with a complex rheology. It is remarkable that the proposed models generalize plenty of previous models in the literature. Furthermore, all of them satisfy an exact dissipative energy balance. We also propose a model that is second-order accurate in the vertical direction thanks to a correction of the shear stress approximation. Finally, we show how effective the layerwise linear approach is to notably improve, with respect to the layerwise constant method, the approximation of the velocity profile for some geophysical flows. Namely, a Newtonian fluid and some complex viscoplastic (dry granular and Herschel–Bulkley) materials are considered.

Time-lapse global inversion for surface-waves: A differential approach using a linear approximation of the Rayleigh wave phase velocity

Surface-wave methods have a wide range of possible applications in the monitoring of near-surface media. In classical monitoring methods, data measured at different times are inverted separately and independently, and the difference in the inversion results reflects the temporal changes in the medium. However, the results obtained in this way can be affected by the reproducibility of experimental measurements, measurement and inversion uncertainties, etc. In this study, we introduce Differential Inversion in surface-wave methods, which uses the difference between measured data as inversion input data, instead of the measured data themselves. More precisely, a linear approximation of Rayleigh wave phase velocity associated with its sensitivity kernel is used to relate the data difference with the model variations. This differential inversion approach is firstly tested with numerical data generated in a series of two-layer models in order to estimate the limitations of the linear approximation with a global inversion approach. It is shown that when the shear-wave velocity variation is more than 5%, the linear assumption is no longer relevant. With respect to the 5% of variation, we apply the differential inversion approach on laboratory data, obtained from three reduced-scale epoxy-resin two-layer models. The results show the feasibility of the differential inversion to estimate the shear-wave velocity differences between epoxy-resin models. The short calculation time is one of its advantages. However, a good estimation on the baseline is required in order to calculate the sensitivity kernel. Finally, the robustness of the proposed approach is verified by numerical data, using the Spectral Element Method to simulate wave propagation in the presence of an under-surface cavity.

Mixed Isogeometric Analysis of the Brinkman Equation

This study focuses on numerical solution to the Brinkman equation with mixed Dirichlet–Neumann boundary conditions utilizing isogeometric analysis (IGA) based on non-uniform rational B-splines (NURBS) within the Galerkin method framework. The authors suggest using different choices of compatible NURBS spaces, which may be considered a generalization of traditional finite element spaces for velocity and pressure approximation. In order to investigate the numerical properties of the suggested elements, two numerical experiments based on a square and a quarter of an annulus are discussed. The preliminary results for the Stokes problem are presented in References.

POD-ROMs for Incompressible Flows Including Snapshots of the Temporal Derivative of the Full Order Solution

In this paper we study the influence of including snapshots that approach the velocity time derivative in the numerical approximation of the incompressible Navier–Stokes equations by means of proper orthogonal decomposition (POD) methods. Our set of snapshots includes the velocity approximation at the initial time from a full order mixed finite element method (FOM) together with approximations to the time derivative at different times. The approximation at the initial velocity can be replaced by the mean value of the velocities at the different times so that when implementing the method to the fluctuations, as done mostly in practice, only approximations to the time derivatives are included in the set of snapshots. For the POD method we study the differences between projecting onto and . In both cases pointwise in time error bounds can be proved. Including grad-div stabilization in both the FOM and the POD methods, error bounds with constants independent of inverse powers of the viscosity can be obtained.

Analytical approximations for elastic limit angular velocities of rotating annular disks with hyperbolic thickness

Analytical approximations for elastic limit angular velocities of rotating annular disks with hyperbolic thickness

Improved composite implicit time integration method for dynamic analysis of viscoelastic damping systems

In this work, a composite implicit time integration method is generalized for viscoelastic damping (VED) system. The interpolation quadrature formula and velocity approximation function scheme are employed for the convolution integration of VED system. This method is a generic one for any VED models. For comparison, three classical implicit methods are also extended to VED system. The representative numerical examples illustrate that the proposed method can achieved high efficiency and accuracy for VED system.

Divergence-Conforming Velocity and Vorticity Approximations for Incompressible Fluids Obtained with Minimal Facet Coupling

We introduce two new lowest order methods, a mixed method, and a hybrid discontinuous Galerkin method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi–Douglas–Marini space for approximating the velocity and the lowest order Raviart–Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.

Pressure‐robust and conforming discretization of the Stokes equations on anisotropic meshes

AbstractPressure‐robust discretizations for incompressible flows have been in the focus of research for the past years. Many publications construct exactly divergence‐free methods or use a reconstruction approach [13] for existing methods like the Crouzeix–Raviart element in order to achieve pressure‐robustness. To the best of our knowledge, except for our recent publications [3, 4], all those articles impose a condition on the shape‐regularity of the mesh, and the two mentioned papers that allow for anisotropic elements use a non‐conforming velocity approximation. Based on the classical Bernardi–Raugel element we provide a conforming pressure‐robust discretization using the reconstruction approach on anisotropic meshes. Numerical examples support the theory.

Improved Operator-Splitting Method for Dynamic Real-Time Substructure Testing of Multiple-Degree-of-Freedom Structure

The operator-splitting (OS) method for real-time substructure testing (RTST) provides explicit and unconditionally stable solutions for structures with softening type stiffness and damping has been widely used for precisely disclosing the seismic performance of structures with velocity-dependent specimens. However, the OS method for RTST provides only explicit target displacement and velocity but not an explicit target acceleration, so that it is essentially an implicit method for Dynamic real-time substructure testing (DRTST) with dynamic system specimens. For precisely disclosing the seismic performance of multiple-degree-of-freedom structures with dynamic system specimens, one improved OS method is proposed for DRTST in this paper. The methodology and stability of the proposed OS method are investigated, and the effectiveness of the proposed OS method is validated by numerical simulations and experimental tests. By using the predictor displacement, velocity, and acceleration approximations, the OS method is an explicit algorithm for DRTST. The stability analysis shows that the proposed OS method is conditionally stable for DRTST, and its stability limit decreases with the increase of the mass ratio and the stiffness ratio, while it increases first and then decreases with the increase of the damping ratio. The numerical and experimental results confirm the stability and accuracy of the OS method. The OS method for DRTST provides a practical and precise investigation strategy for structures with dynamic system specimens and may have broad application prospects in civil engineering.

Error estimate of a consistent splitting GSAV scheme for the Navier-Stokes equations

We carry out a rigorous error analysis of the first-order semi-discrete (in time) consistent splitting scheme coupled with a generalized scalar auxiliary variable (GSAV) approach for the Navier-Stokes equations with no-slip boundary conditions. The scheme is linear, unconditionally stable, and only requires solving a sequence of Poisson type equations at each time step. By using the build-in unconditional stability of the GSAV approach, we derive optimal global (resp. local) in time error estimates in the two (resp. three) dimensional case for the velocity and pressure approximations.

Approximation of seismic velocities from the spectrum of weighted graphs

Extracting seismic velocities from recorded seismic data requires converting the shot gathers to mid-point gathers, calculating the velocity spectrum, and picking the velocity values. In this paper, we propose to use graph theory to extract the seismic velocity values directly from the mid-point gathers. We use spectral data of a weighted graph model for approximating the seismic velocity. We develop a regression model to predict the seismic velocity from the largest eigenvalue of the graph representing the physical system. The approach is tested on a synthetic seismic data that represent a typical near-surface geological situation. The method was able to predict the seismic velocity of the second layer with 99.4% accuracy.

Higher-Order Accurate Explicit Time Schemes with Improved Dissipation Properties

In this paper, a novel family of fourth-order accurate explicit time integration schemes is developed by combining the new time approximations and the explicit fourth-order Runge–Kutta (RK4) method. Inaccurate predictions due to the presence of excessive numerical dissipation are often observed in practical analyses of structural dynamics when the RK4 is employed for both displacement and velocity approximations. To remedy this, novel time approximations with adjustable algorithmic parameters are employed for the displacement vectors while the velocity vectors are approximated by using the RK4. For the complete elimination of numerical dissipation, algorithmic parameters are unconventionally determined by taking the determinant of the amplification matrix as unity. A set of algorithmic parameters obtained from this process makes the new schemes completely non-dissipative while keeping the computational cost the same as the RK4. Due to this improvement, the new schemes have enhanced total energy-conserving capabilities for nonlinear systems and give noticeably more accurate predictions in practical analyses. Until now, controllable numerical dissipation and fourth-order accuracy are not attained simultaneously in a unified set of time schemes. In the new schemes, however, a systematic way to adjust the level of numerical dissipation is also presented while attaining fourth-order accuracy. The numerical results of various test problems show that the new time schemes can provide more accurate predictions for nonlinear conservative dynamic problems than the existing time schemes.

Dynamics of dust grains in turbulent molecular clouds

Context. Dust grain dynamics in molecular clouds is regulated by its interplay with supersonic turbulent gas motions. The conditions under which interstellar dust grains decouple from the dynamics of gas in molecular clouds remain poorly constrained. Aims. We first aim to investigate the critical dust grain size for dynamical decoupling, using both analytical predictions and numerical experiments. Second, we aim to set the range of validity of two fundamentally different numerical implementations for the evolution of dust and gas mixtures in turbulent molecular clouds. Methods. We carried out a suite of numerical experiments using two different schemes to integrate the dust grain equation of motion within the same framework. First, we used a monofluid formalism (or often referred to as single fluid) in the terminal velocity approximation. This scheme follows the evolution of the barycentre of mass between the gas and the dust on a Eulerian grid. Second, we used a two-fluid scheme, in which the dust dynamics is handled with Lagrangian super-particles, and the gas dynamics on a Eulerian grid. Results. The monofluid results are in good agreement with the theoretical critical size for decoupling. We report dust dynamics decoupling for Stokes number St > 0.1, that is, dust grains of s > 4 μm in size. We find that the terminal velocity approximation is well suited for grain sizes of 10 μm in molecular clouds, in particular in the densest regions. However, the maximum dust enrichment measured in the low-density material - where St > 1 - is questionable. In the Lagrangian dust experiments, we show that the results are affected by the numerics for all dust grain sizes. At St ≪ 1, the dust dynamics is largely affected by artificial trapping in the high-density regions, leading to spurious variations of the dust concentration. At St > 1 , the maximum dust enrichment is regulated by the grid resolution used for the gas dynamics. Conclusions. Dust enrichment of submicron dust grains is unlikely to occur in the densest parts of molecular clouds. Two fluid implementations using a mixture of Eulerian and Lagrangian descriptions for the dust and gas mixture dynamics lead to spurious dust concentration variations in the strongly and weakly coupled regimes. Conversely, the monofluid implementation using the terminalvelocity approximation does not accurately capture dust dynamics in the low-density regions, that is, where St > 1 . The results of previous similar numerical work should therefore be revisited with respect to the limitations we highlight in this study.

Heat and mass transfer characteristics in MHD Casson fluid flow over a cylinder in a wavy channel: Higher-order FEM computations

In this paper, an extensive analysis of heat and mass transfer characteristics of a Casson fluid flow past a cylinder in a wavy channel has been performed. Due to the non-availability of analytical solution the model represented by coupled non-linear partial differential equations has been simulated by implementing higher-order Finite Element Method (FEM). For the approximation of velocity, temperature, and concentration profiles, a finite element space involving the cubic polynomial (P3) is selected whereas the pressure estimation is accomplished through a space of quadratic polynomial (P2). Such a choice gives rise to 10° of freedom for each of the velocity components, temperature, and concentration profile while 6° of freedom for the pressure. The Newton's method is used to linearize the systems of equations. The linearized inner system of equations are solved with a direct solver PARDISO. Computational results are demonstrated in the form of velocity, isotherms, isoconcentrations, and for several quantities of interest including the drag and lift coefficients. It has been observed that both drag and lift coefficients show a decreasing trend with Magnetic parater M. In addition, for all values of Bn, the drag coefficient has a linear growth profile as a result of fluid forces dominating at higher values of Bn. Furthermore, the concentration is high in the center of the channel and has a decreasing trend as we move towards the walls.

Dissipation factor and phase velocity approximate formulas for P-waves in fluid-saturated porous media

Actual sedimentary rocks are generally homogeneous porous media saturated by fluid, which are described by poroelastic media. Wave velocity and dissipation factor in poroelastic media provide essential information about rock and fluid properties and are widely used in many fields. The exact wave velocity and dissipation factor can be obtained from wavenumbers determined by dispersion equation through complex mathematic operations. Reported approximate formulas are most given under the assumption that the characteristic angular frequency (ωc ) is far larger than the angular frequency (ω) and becomes invalid for high frequency. Thus, this paper derived phase velocity and dissipation factor approximation of the two P-waves in poroelastic media for the whole low-frequency range. Based on the Geertsma–Smit approximation, we modified the infinite-frequency limiting velocity for fast P-wave approximation and removed the ω/ωc <<1 assumption for slow P-wave approximation. Numerical analysis shows that the proposed approximate formulas for P-waves in porous media can reveal the exact variations of phase velocity and dissipation factor in the low-frequency range, which is from zero frequency for isotropic media to the characteristic frequency. Phase velocity approximations of the two P-waves in poroelastic media are very close to exact values, with maximum relative errors being far less than 1%.

Pressure data-driven variational multiscale reduced order models

In this paper, we develop data-driven closure/correction terms to increase the pressure and velocity accuracy of reduced order models (ROMs) for fluid flows. Specifically, we propose the first pressure-based data-driven variational multiscale ROM, in which we use the available data to construct closure/correction terms for both the momentum equation and the continuity equation. Our numerical investigation of the two-dimensional flow past a circular cylinder at Re=50,000 in the marginally-resolved regime shows that the novel pressure data-driven variational multiscale ROM yields significantly more accurate velocity and pressure approximations than the standard ROM and, more importantly, than the original data-driven variational multiscale ROM (i.e., without pressure components). In particular, our numerical results show that adding the closure/correction term in the momentum equation significantly improves both the velocity and the pressure approximations, whereas adding the closure/correction term in the continuity equation improves only the pressure approximation.

A Linearized Alternating Direction Method of Multipliers Algorithm for Prestack Seismic Inversion in VTI Media Using the Quadratic PP-Reflectivity Approximation

Periodic horizontal thin interlayers, or formations developing horizontally layered fractures in a homogeneous background, are commonly studied as a transverse isotropy with a vertical axis of symmetry (VTI) media. Anisotropy needs to be considered in predicting such special reservoirs, and can be inferred from the amplitude variation with offset (AVO) by the exact or linear reflectivity equations. However, the linear approximation is usually derived based on the assumptions of weak anisotropy and weak contrast, the accuracy suffers when the anisotropy or the contrast of two adjacent media increases. In addition, due to the high nonlinearity of the Graebner equation, the stability and efficiency of the exact inversion are limited. To address these issues, we propose a robust pre-stack seismic inversion for VTI media using quadratic PP-reflectivity approximation. Firstly, by deriving quadratic approximations for phase velocity and polarization, we analyze the advantage of the second-order terms at a higher-contrast interface. Then, based on perturbation theory, we derive a quadratic PP-reflectivity approximation from the exact equation. It consists of first- and second-order terms with isotropic and anisotropic components, and the accuracy is significantly improved even with enhanced anisotropy. Finally, we introduce the Hadamard product operator to linearly characterize the objective function, and develop an improved Alternating Direction Method of Multipliers (ADMM) algorithm to handle the nonlinearity of the quadratic equations. The feasibility and stability of the proposed method are tested on two datasets, that is, the logging curves used as numerical experiments and a field dataset from a shale reservoir.

Instability in axion inflation with strong backreaction from gauge modes

We perform an analytical study of the stability of the background solution [1] of the model in which an inflaton, through an axionic coupling to a U(1) gauge field, causes an amplification of the gauge field modes that strongly backreact on its dynamics. To this goal, we study the evolution of the gauge field modes coupled to the inflaton zero mode, treating perturbatively the deviation of the inflaton velocity from its mean-field value. As long as the system is in the strong backreaction regime we find that the inflaton velocity performs oscillations of increasing amplitude about the value it would have in the approximation of constant velocity, confirming an instability that has been observed in numerical studies.

Error Analysis of Nonconforming Virtual Element Method for Stokes Problem with Low Regularity

In this paper, the nonconforming virtual element method is used to solve the Stokes problem where the velocity and pressure are allowed to have the low regularity. With the help of an enriching operator, the consistency error is estimated under the low regularity condition. Then the optimal error estimates are obtained for the velocity and pressure approximations, which implies that the nonconforming virtual element method has the good convergence even for the Stokes problem with the low regularity.

Groundwater flow in saturated riparian buffers and implications for nitrate removal.

A saturated riparian buffer (SRB) is an edge-of-field conservation practice that intercepts tile drainage and reduces nitrate flux to nearby streams by redistributing the flow as shallow groundwater. In this study, a three-dimensional, finite-difference groundwater model representative of SRBs in central Iowa was developed to assess the flow of groundwater and implications for nitrate removal during spring conditions, when flow to the SRB is highest. The model reproduces field observations of water level with Nash-Sutcliffe efficiency of 0.68, which is deemed acceptable for hydrologic models. The modeling shows that groundwater flow is three-dimensional near the distribution pipe and the stream and primarily one-dimensional in the rest of the buffer. The path the water takes in flowing toward the stream depends on where it exits the distribution pipe. When nitrate is not limiting, the potential for nitrate removal depends on the length of the path-and thus travel time-and depth because denitrification potential varies with depth. Travel time Tt can be estimated well with slight modifications to a one-dimensional approximation: Tt = 1.11Lx /vx , where Lx is the buffer width and vx is a one-dimensional approximation of the average linear velocity of groundwater. Refining knowledge of SRB function is an important step toward enhancing design for improving water quality.