Viscous Problems Research Articles

Use of Pulsed Potentials for Electropolishing TA6V Parts Elaborated By Additive Manufacturing in Deep Eutectic Solvent

The Ti-6Al-4V alloy's high mechanical properties, low density, excellent biocompatibility, and corrosion resistance make it an ideal material for sectors such as aeronautics, astronautics and medical technology. The demand for increasingly complex part designs is speeding up the development of Additive Manufacturing (AM) technology, enabling the creation of shapes unachievable with traditional machining. However, the parts manufactured present significant surface roughness, thus reducing resistance to corrosion and mechanical stress. To mitigate these risks, homogeneous polishing of the entire shape is required. Electrochemical polishing is an all-round part treatment technique ideal for addressing AM challenges. Deep Eutectic Solvent (DES) electrolytes are well-suited for this application. The aim of this study is to develop electropolishing of Ti-6Al-4V parts in a DES solvent of ethylene glycol and choline chloride. The challenges posed by viscous layer thickness problems, as well as two methods to overcome them, are described, including agitation control and pulsed potentials.

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].

Comparison of Libration- and Precession-Driven Flows: From Linear Responses to Broadband Dynamics

Libration and precession are different body forces that are ubiquitous in many rapidly rotating systems, particularly in geophysical and astrophysical flows. Libration is a modulation of the background rotation magnitude, whereas precession is a modulation of the background rotation direction. Assessing the consequences of these body forces in large-scale flows is challenging. The Ekman number, the ratio of the rotation time scale to the viscous time scale quantifying the rotation speed, is extremely small, leading to extremely thin and intense shear layers in the flows even when the amplitudes of the body forces are very small. We consider the consequences of libration and precession numerically in a geometrically simple container, a cube, which lends itself to very efficient, accurate, and robust numerical treatment, with the axis of rotation passing through opposite vertices, so that all walls of the cube are at oblique angles to the rotation axis. This results in the geometric focusing of inertial wavebeams reflecting off the walls, whereby the energy density of the wavebeams increases along with the magnitude of their wavevector. The nature of this focusing depends on the forcing frequency but not on the body force. In the inviscid setting, wavebeams form infinitesimally thin vortex sheets, and their energy density becomes unbounded upon focusing. We present linear inviscid ray tracing to set the scene for the focusing of wavebeams and then consider viscous problems at an Ekman number that is typical of current state-of-the-art laboratory experiments. We begin by considering the linear responses, which are comprised of focusing viscous shear layers, of which their details are mostly captured via ray tracing, and particular solutions accounting for the body forces. These have complicated spatio-temporal structures, which differ for libration and precession. Increasing the forcing amplitude from zero introduces nonlinear interactions, enhances the focusing effects via vortex tilting and stretching when the shear layers reflect at the walls, and also introduces temporal superharmonics and a mean flow. When the magnitude of the mean flow is within a few percent of the magnitude of the instantaneous flow, instabilities breaking the spatio-temporal symmetries set in. These are localized in the oscillatory boundary layers where the reflections are concentrated and introduce broadband dynamics in the boundary layers, with additional inertial wavebeams emitted into the interior. The details again depend on the specifics of the body forces.

Application of lattice Boltzmann method to solution of viscous incompressible fluid dynamics problems

Application of lattice Boltzmann method to solution of viscous incompressible fluid dynamics problems

On the investigation of shock wave/boundary layer interaction with a high-order scheme based on lattice Boltzmann flux solver

Shock wave/boundary layer interaction (SWBLI) continues to pose a significant challenge in the field of aerospace engineering. This paper aims to address this issue by proposing a novel approach for predicting aerodynamic coefficients and heat transfer in viscous supersonic and hypersonic flows using a high-order flux reconstruction technique. Currently, finite volume methods are extensively employed for the computation of skin aerodynamic coefficients and heat transfer. Nevertheless, these numerical methods exhibit considerable susceptibility to a range of factors, including the inviscid flux function and the computational mesh. The application of high-order flux reconstruction techniques offers promising potential in alleviating these challenges. In contrast to other high-order methods, the flux reconstruction is combined with the lattice Boltzmann flux solver in this study. The current method evaluates the common inviscid flux at the cell interface by locally reconstructing the lattice Boltzmann equation solution from macroscopic flow variables at solution points. Consequently, this framework performs a positivity-preserving, entropy-based adaptive filtering method for shock capturing. The present approach is validated by simulating the double Mach reflection, and then simulating some typical viscous problems. The results demonstrate that the current method accurately predicts aerodynamic coefficients and heat transfer, providing valuable insights into the application of high-order methods for shock wave/boundary layer interaction.

A Viscous Ergodic Problem with Unbounded and Measurable Ingredients, Part 1: HJB Equation

A Viscous Ergodic Problem with Unbounded and Measurable Ingredients, Part 1: HJB Equation

Inertial wave super-attractor in a truncated elliptic cone

We consider inertial waves propagating in a fluid contained in a non-axisymmetric three-dimensional rotating cavity. We focus on the particular case of a fluid enclosed inside a truncated cone or frustum, which is the volume that lies between two horizontal parallel planes cutting an upright cone. While this geometry has been studied in the past, we generalise it by breaking its axisymmetry and consider the case of a truncated elliptic cone for which the horizontal sections are elliptic instead of circular. The problem is first tackled using ray tracing, where local wave packets are geometrically propagated and reflected within the closed volume without attenuation. We complement these results with a local asymptotic analysis and numerical simulations of the original linear viscous problem. We show that the attractors, well known in two dimensional or axisymmetric domains, can be trapped in a particular plane in three dimensions provided that the axisymmetry of the domain is broken. Contrary to previous examples of attractors in three-dimensional domains, all rays converge towards the same limit cycle regardless of initial conditions, and it is localised in the bulk of the fluid.

Analytical and numerical analyses of a viscous strain gradient problem involving type Ⅱ thermoelasticity

<abstract><p>In this paper, a thermoelastic problem involving a viscous strain gradient beam is considered from the analytical and numerical points of view. The so-called type Ⅱ thermal law is used to model the heat conduction and two possible dissipation mechanisms are introduced in the mechanical part, which is considered for the first time within strain gradient theory. An existence and uniqueness result is proved by using semigroup arguments, and the exponential energy decay is obtained. The lack of differentiability for the semigroup of contractions is also shown. Then, fully discrete approximations are introduced by using the finite element method and the backward time scheme, for which a discrete stability property and a priori error estimates are proved. The linear convergence is derived under suitable additional regularity conditions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the discrete energy decay.</p></abstract>

Visualization of Calculations of the Discontinuous Particle Method in Problems with Viscosity

In previous studies, it was shown that the discontinuous particle method performs well in computational hydrodynamics problems with strong gradients, exemplified by the formation of an oblique stress jump. This article explores the application of the discontinuous particle method to problems involving viscosity. The investigation includes a one-dimensional Burgers' equation with an initial condition in the form of a smoothed wave and a two-dimensional Blasius problem. Numerical experiments showed agreement between the obtained solution and the analytical one. However, in the two-dimensional case, the algorithm's performance significantly decreases due to the need to determine particle neighbors. It is concluded that the discontinuous particle method can handle viscosity problems in one dimension, but modifications to the existing algorithm are required for higher-dimensional cases. The study of applying the discontinuous particle method to viscous problems was conducted as part of a comprehensive research effort comparing the relative accuracy of numerical methods on benchmark solutions.

Role of interfacial rheology on fingering instabilities in lifting Hele-Shaw flows.

The lifting Hele-Shaw cell setup is a popular modification of the classic, fixed-gap, radial viscous fingering problem. In the lifting cell configuration, the upper cell plate is lifted such that a more viscous inner fluid is invaded by an inward-moving outer fluid. As the fluid-fluid interface contracts, one observes the rising of distinctive patterns in which penetrating fingers having rounded tips compete among themselves, reaching different lengths. Despite the scholarly and practical relevance of this confined lifting flow problem, the impact of interfacial rheology effects on its pattern-forming dynamics has been overlooked. Authors of recent studies on the traditional injection-induced radial Hele-Shaw flow and its centrifugally driven variant have shown that, if the fluid-fluid interface is structured (i.e., laden with surfactants, particles, proteins, or other surface-active entities), surface rheological stresses start to act, influencing the development of the viscous fingering patterns. In this paper, we investigate how interfacial rheology affects the stability as well as the shape of the emerging fingered structures in lifting Hele-Shaw flows, at linear and early nonlinear dynamic stages. We tackle the problem by utilizing the Boussinesq-Scriven model to describe the interface and by employing a perturbative mode-coupling scheme. Our linear stability results show that interfacial rheology effects destabilize the interface. Furthermore, our second-order findings indicate that interfacial rheology significantly alters intrinsically nonlinear morphological features of the shrinking interface, inducing the formation of narrow sharp-tip penetrating fingers and favoring enhanced competition among them.

Aerodynamic optimization with large shape and topology changes using a differentiable embedded boundary method

Embedded (or immersed) boundary methods (EBMs) for CFD and fluid-structure interaction (FSI) are attractive for aerodynamic optimization problems characterized by large shape deformations and surface topology changes. At each iteration, they eliminate the need for explicitly remeshing a computational fluid domain and avoid the pitfalls of transferring information from one CFD mesh to another. However, they are vulnerable to discrete events that compromise the smoothness of the aerodynamic objective and/or constraint functions they may compute. For this reason, the realization of EBMs for gradient-based aerodynamic shape optimization requires first their endowment with sufficient smoothness guarantees. Drawing on a recently developed EBM for the solution of compressible viscous fluid and FSI problems with such guarantees, this paper demonstrates the potential of shape-differentiable EBMs for discovering optimal aerodynamic designs featuring significant shape deformations and surface topology changes. Specifically, it highlights their advantages in terms of reliability with respect to large shape adjustments and surface topology changes, efficiency, simplicity, and reduced need for user intervention. To this end, the paper showcases the application of an advanced EBM with smoothness guarantees to wing shape, wing section, and nacelle-pylon placement optimization problems formulated for a NASA Common Research Model (CRM).

Three-Dimensional Numerical Modeling of Lava Dynamics Using the Smoothed Particle Hydrodynamics Method

Lava domes and lava flows are major manifestations of effusive volcanic eruptions. Less viscous lava tends to flow long distances, depending on the volcanic slope topography, the eruption rate, and the viscosity of the erupted magma. When magma is highly viscous, its eruption to the surface leads to the formation of lava domes and their growth. The meshless smoothed particle hydrodynamics (SPH) method is used in this paper to simulate lava dynamics. We describe the SPH method and present a numerical algorithm to compute lava dynamics models. The numerical method is verified by solving a model of cylindrical dam-break fluid flow, and the modelled results are compared to the analytical solution of the axisymmetric thin-layer viscous current problem. The SPH method is applied to study three models of lava advancement along the volcanic slope, when the lava viscosity is constant, depends on time and on the volume fraction of crystals in the lava. Simulation results show characteristic features of lava flows, such as lava channel and tube formation, and lava domes, such as the formation of a highly viscous carapace versus a less viscous dome core. Finally, the simulation results and their dependence on a particle size in the SPH method are discussed.

Liouville-type results for the time-dependent three-dimensional (inviscid and viscous) water wave problem with an interface

Under consideration here are time-dependent three-dimensional stratified water flows of finite depth with a free surface and an interface (playing the role of an internal wave and separating two layers of constant and different densities). The main outcome of the paper is that, under the assumption that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other, we prove that bounded solutions to the three-dimensional equations are essentially two-dimensional. More precisely, the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion. In addition, one of the horizontal components of the velocity field is constant throughout the flow and the vorticity also points in the direction orthogonal to the wave propagation direction. The rigid lid case is analyzed from the perspective of relaxed assumptions concerning the vorticity vectors. We also consider the viscous water wave problem case with normal and tangential stress conditions at the free surface and find that under similar conditions on the vorticity vectors, the conclusion about the two-dimensionality remains in place. Along the way, we also provide an explicit solution for the time-dependent three-dimensional inviscid water wave problem with an interface.

A study on the effect of driver section length on the flow field inside a two-dimensional closed-ended viscous shock tube: A numerical investigation

The interaction of a reflected shock wave with the boundary layer inside a closed-ended shock tube is studied by many researchers using different numerical schemes. However, the interaction of a reflected shock wave having a decaying pressure profile ahead of the reflected shock is lacking in the literature. It is essential to quantify the deviation in flow characteristics experienced in tests performed inside the shock tube compared to field tests while studying blast wave attenuation and mitigation. In the present study, the effect of transient upstream pressure on the shock wave reflecting from a solid wall and the associated flow separation inside the boundary layer is examined in detail through numerical simulations using the viscous shock tube problem for Re = 1000. Simulations were performed using an in-house 13th-order high-resolution hybrid solver with 50 million grid points. The transient pressure profile behind the incident shock wave is achieved by varying the non-dimensional driver section length from 0.5 to 0.02 while keeping the overall shock tube length as 1. The flow field is scrutinized through numerical schlieren and vorticity plots. It is observed that the decaying pressure profiles behind the incident shock wave have an immense impact on the dynamics of shockwave-boundary layer interaction and the vortices emanating from the shear layer due to Kelvin–Helmholtz Instability. The secondary vortices inside the boundary layer, vortices at the shear layer, and compression waves in the separated boundary layer region reduce with a decrease in driver section length. The triple point height increases as driver section length decreases due to the weakening of upstream velocity. The Kelvin–Helmholtz vortices originating from the triple point moves back into the separated boundary layer rather than interacting with the end wall. It suggests that the decaying pressure behind the incident shock wave minimizes the secondary effects on the solid wall during shock wave impingement.

A Bernstein–Bézier Lagrange–Galerkin method for three-dimensional advection-dominated problems

We present a high-order Bernstein–Bézier finite element discretization to accurately solve three-dimensional advection-dominated problems on unstructured tetrahedral meshes. The key idea consists of implementing a modified method of characteristics to discretize the advection terms in a Bernstein–Bésier finite element framework. The proposed Bernstein–Bézier Lagrange–Galerkin method has been designed so that the Courant–Friedrichs–Lewy condition is strongly relaxed using semi-Lagrangian time discretization. A low complexity procedures in building finite element matrices and load vectors is also achieved in the present work by both the analytical rule and the sum factorization method using the tensorial feature of Bernstein polynomials. Several numerical examples including advection–diffusion equations with known analytical solutions and the viscous Burgers problem are considered to illustrate the accuracy, robustness and performance of the proposed approach. The computed results support our expectations for a stable and highly accurate Bernstein–Bézier Lagrange–Galerkin finite element method for three-dimensional advection-dominated problems.

Dynamic stabilisation of Rayleigh–Plateau modes on a liquid cylinder

We demonstrate dynamic stabilisation of axisymmetric Fourier modes susceptible to the classical Rayleigh–Plateau (RP) instability on a liquid cylinder by subjecting it to a radial oscillatory body force. Viscosity is found to play a crucial role in this stabilisation. Linear stability predictions are obtained via Floquet analysis demonstrating that RP unstable modes can be stabilised using radial forcing. We also solve the linearised, viscous initial-value problem for free-surface deformation obtaining an equation governing the amplitude of a three-dimensional Fourier mode. This equation generalizes the Mathieu equation governing Faraday waves on a cylinder derived earlier in Patankaret al.(J. Fluid Mech., vol. 857, 2018, pp. 80–110), is non-local in time and represents the cylindrical analogue of its Cartesian counterpart (Beyer & Friedrich,Phys. Rev.E, vol. 51, issue 2, 1995, p. 1162). The memory term in this equation is physically interpreted and it is shown that, for highly viscous fluids, its contribution can be sizeable. Predictions from the numerical solution to this equation demonstrate the predicted RP mode stabilisation and are in excellent agreement with simulations of the incompressible Navier–Stokes equations (up to the simulation time of several hundred forcing cycles).

Bernstein-Bézier Galerkin-Characteristics Finite Element Method for Convection-Diffusion Problems

A class of Bernstein-Bézier basis based high-order finite element methods is developed for the Galerkin-characteristics solution of convection-diffusion problems. The Galerkin-characteristics formulation is derived using a semi-Lagrangian discretization of the total derivative in the considered problems. The spatial discretization is performed using the finite element method on unstructured meshes. The Lagrangian interpretation in this approach greatly reduces the time truncation errors in the Eulerian methods. To achieve high-order accuracy in the Galerkin-characteristics solver, the semi-Lagrangian method requires high-order interpolating procedures. In the present work, this step is carried out using the Bernstein-Bézier basis functions to evaluate the solution at the departure points. Triangular Bernstein-Bézier patches are constructed in a simple and inherent manner over finite elements along the characteristics. An efficient preconditioned conjugate gradient solver is used for the linear systems of algebraic equations. Several numerical examples including advection-diffusion equations with known analytical solutions and the viscous Burgers problem are considered to illustrate the accuracy, robustness and performance of the proposed approach. The computed results support our expectations for a stable and highly accurate Bernstein-Bézier Galerkin-characteristics finite element method for convection-diffusion problems.

An Arbitrarily High-order Spectral Difference Method with Divergence Cleaning (SDDC) for Compressible Magnetohydrodynamic Simulations on Unstructured Grids

This paper reports a recent development of the high-order spectral difference method with divergence cleaning (SDDC) for accurate simulations of both ideal and resistive magnetohydrodynamics (MHD) on curved unstructured grids consisting of high-order isoparametric quadrilateral elements. The divergence cleaning approach is based on the improved generalized Lagrange multiplier, which is thermodynamically consistent. The SDDC method can achieve an arbitrarily high order of accuracy in spatial discretization, as demonstrated in the test problems with smooth solutions. The high-order SDDC method combined with the artificial dissipation method can sharply capture shock interfaces with the oscillation-free property and resolve small-scale vortex structures and density fluctuations on relatively sparse grids. The robustness of the codes is demonstrated through long time simulations of ideal MHD problems with progressively interacting shock structures, resistive MHD problems with high Lundquist numbers, and viscous resistive MHD problems on complex curved domains.

Numerical approximation of viscous contact problems applied to glacial sliding

Viscous contact problems describe the time evolution of fluid flows in contact with a surface from which they can detach and reattach. These problems are of particular importance in glaciology, where they arise in the study of grounding lines and subglacial cavities. In this work, we propose a novel numerical method for solving viscous contact problems based on a mixed formulation with Lagrange multipliers of a variational inequality involving the Stokes equations. The advection equation for evolving the geometry of the domain occupied by the fluid is then solved via a specially-built upwinding scheme, leading to a robust and accurate algorithm for viscous contact problems. We first verify the method by comparing the numerical results to analytical results obtained by a linearised method. Then we use this numerical scheme to reconstruct friction laws for glacial sliding with cavitation. Finally, we compute the evolution of cavities from a steady state under oscillating water pressures. The results depend strongly on the location of the initial steady state along the friction law. In particular, we find that if the steady state is located on the downsloping or rate-weakening part of the friction law, then the cavity evolves towards the upsloping section, indicating that the downsloping part is unstable.

On the application of physics informed neural networks (PINN) to solve boundary layer thermal-fluid problems

Deep neural network is a powerful technique in discovering the hidden physics behind the transport phenomena through big-data training. In this study, the application of physic-informed neural networks is extended to solve viscous and thermal boundary layer problems. Three benchmark problems including Blasius-Pohlhausen, Falkner-Skan, and Natural convection are selected to investigate the effects of nonlinearity of the equations and unbounded boundary conditions on adjusting the network structure's width and depth, leading to reasonable solutions. TensorFlow is used to build and train the models, and the resulted predictions are compared with those obtained by finite difference technique with Richardson extrapolation. The results reveal that the Prandtl number in the heat equation is a key factor which its value drastically changes the required number of neurons and layers to achieve the desired solutions. Also, setting the unbounded boundary at a higher distance from the origin demands an adequate number of layers and correspondingly neurons to deal with the infinity boundary condition. Finally, trained models are successfully applied to unseen data to evaluate the boundary layer thicknesses.