Linear dynamics of dilute hard-sphere suspensions

We study the linear dynamics of a dilute dispersion of solid spherical particles in an incompressible viscous fluid. We start from a rest situation, in which the spheres are assumed to be distributed at random. The rest situation is perturbed linearly by small oscillatory applied forces and torques acting on the solute particles, and by an oscillatory force density acting on the fluid. By averaging over an ensemble of rest configurations we obtain linear average equations of motion for the two-phase system with well-defined transport coefficients. We evaluate the transport coefficients as a function of wavenumber and frequency to first order in the volume fraction.     

Deforming composite grids for solving fluid structure problems

We describe a mixed Eulerian–Lagrangian approach for solving fluid–structure interaction (FSI) problems. The technique, which uses deforming composite grids (DCG), is applied to FSI problems that couple high speed compressible flow with elastic solids. The fluid and solid domains are discretized with composite overlapping grids. Curvilinear grids are aligned with each interface and these grids deform as the interface evolves. The majority of grid points in the fluid domain generally belong to background Cartesian grids which do not move during a simulation. The FSI-DCG approach allows large displacements of the interfaces while retaining high quality grids. Efficiency is obtained through the use of structured grids and Cartesian grids. The governing equations in the fluid and solid domains are evolved in a partitioned approach. We solve the compressible Euler equations in the fluid domains using a high-order Godunov finite-volume scheme. We solve the linear elastodynamic equations in the solid domains using a second-order upwind scheme. We develop interface approximations based on the solution of a fluid–solid Riemann problem that results in a stable scheme even for the difficult case of light solids coupled to heavy fluids. The FSI-DCG approach is verified for three problems with known solutions, an elastic-piston problem, the superseismic shock problem and a deforming diffuser. In addition, a self convergence study is performed for an elastic shock hitting a fluid filled cavity. The overall FSI-DCG scheme is shown to be second-order accurate in the max-norm for smooth solutions, and robust and stable for problems with discontinuous solutions for a wide range of constitutive parameters.     

The Modified Ghost Fluid Method Applied to Fluid-Elastic Structure Interaction

In this work, the modified ghost fluid method is developed to deal with 2D compressible fluid interacting with elastic solid in an Euler-Lagrange coupled system. In applying the modified Ghost Fluid Method to treat the fluid-elastic solid coupling, the Navier equations for elastic solid are cast into a system similar to the Euler equations but in Lagrangian coordinates. Furthermore, to take into account the influence of material deformation and nonlinear wave interaction at the interface, an Euler-Lagrange Riemann problem is constructed and solved approximately along the normal direction of the interface to predict the interfacial status and then define the ghost fluid and ghost solid states. Numerical tests are presented to verify the resultant method.     

The flexural vibration of a line-stiffened plate with fluid loading,part I: Analysis

This paper considers the vibration of a symmetrical system consisting of an infinite fluid loaded plate bearing a finite number of parallel stiffeners. The system is driven at the central stiffener by a travelling wave line force. Formal solutions for the equations of motion are found in terms of cosine transforms. Manipulation of the equations allows the problem to be reduced to the solution of a set of linear algebraic equations in the vibration amplitudes at the stiffeners. The coefficients in these equations depend in a simple way upon the stiffener parameters, and upon particular values of the cosine transform of a function which depends only on the plate and fluid parameters, and the stiffener positions.     

Aeroelastic flutter of a disk rotating in an unbounded acoustic medium

The linear aeroelastic stability of an unbaffled flexible disk rotating in an unbounded fluid is investigated by modeling the disk-fluid system as a rotating Kirchhoff plate coupled to the irrotational motions of a compressible inviscid fluid. A perturbed eigenvalue formulation is used to compute systematically the coupled system eigenvalues. Both a semi-analytical and a numerical method are employed to solve the fluid boundary value problem. The semi-analytical approach involves a perturbation series solution of the dual integral equations arising from the fluid boundary value problem. The numerical approach is a boundary element method based on the Hadamard finite part. Unlike previous works, it is found that a disk with zero material damping destabilizes immediately beyond its lowest critical speed. Upon the inclusion of small disk material damping, the flutter speeds become supercritical and increase with decreasing fluid density. The competing effects of radiation damping into the surrounding fluid and disk material damping control the onset of flutter at supercritical speed. The results are expected to be relevant for the design of rotating disk systems in data storage, turbomachinery and manufacturing applications.     

Approximation of the inductionless MHD problem using a stabilized finite element method

In this work, we present a stabilized formulation to solve the inductionless magnetohydrodynamic (MHD) problem using the finite element (FE) method. The MHD problem couples the Navier–Stokes equations and a Darcy-type system for the electric potential via Lorentz’s force in the momentum equation of the Navier–Stokes equations and the currents generated by the moving fluid in Ohm’s law. The key feature of the FE formulation resides in the design of the stabilization terms, which serve several purposes. First, the formulation is suitable for convection dominated flows. Second, there is no need to use interpolation spaces constrained to a compatibility condition in both sub-problems and therefore, equal-order interpolation spaces can be used for all the unknowns. Finally, this formulation leads to a coupled linear system; this monolithic approach is effective, since the coupling can be dealt by effective preconditioning and iterative solvers that allows to deal with high Hartmann numbers.     

Natural vibrations and stability of shells of revolution interacting with an internal fluid flow

A finite-element algorithm is proposed to investigate the dynamic behavior of elastic shells of revolution containing a quiescent or a flowing inviscid fluid in the framework of linear theory. The fluid behavior is described using the perturbed velocity potential. The shell behavior is treated in the framework of the classical shell theory and variational principle of virtual displacements incorporating a linearized Bernoulli equation for calculation of hydrodynamic pressure acting on the shell. The problem reduces to evaluation and analysis of the eigenvalues in the connected system of equations obtained by coupling the equations for velocity perturbations with the equations for shell displacements. For cylindrical shells, the results of numerical simulations are compared with recently published experimental, analytical and numerical data. The paper also reports the results of studying the dynamic behavior of shells under various boundary conditions for the perturbed velocity potential. The investigation made for conical shells has shown that under certain conditions an increase in the cone angle can change a divergent type of instability to a flutter type.     

Rotational oscillations of solid cylinder and viscous fluid

A new exact solution of a problem for motion equations of a solid and ambient viscous fluid is found.     

Viscous scattering of a pressure wave: calculation of the fluid tractions on a biomimetic acoustic velocity sensor

In the paper we give a method for calculating the tractions (local forces) of the fluid motion determined by an incoming plane pressure wave on an artificial hair cell transducer structure. The sensing element of the transducer is a standing high aspect ratio cilium in the shape of a narrow thin curved beam (tape-like), which can be easily fabricated in micro-/nanotechnology. The method is based on considering the system of partial differential equations describing the motion of the compressible viscous fluid in an acoustic linearized approximation, and representation of the velocity field as a viscous acoustic single-layer potential. The boundary conditions, stating the cancellation of the velocity components on the solid beam, yield a two-dimensional (2-D) system of three integral equations over the beam's surface for the traction components. In the case of a narrow cilium, the system of integral equations furnishes a system of two 1-D integral equations over the symmetry curve of the structure for obtaining the tangential and normal components of the traction. This system is solved numerically by a finite (boundary) element method. The numerical code written for solving the problem was applied to some particular structures. The last structure is similar to the trichobothrium of a spider Cupiennius salei. The results obtained show that the curvature of the hair is enhancing sensitivity to flows directed normal to the main shaft of the hair confirming the assertion of Barth et al. [Philos. Trans. R. Soc. London, Ser. B 340, 445-461 (1993)].     

Plane waves in noncommutative fluids

We study the dynamics of the noncommutative fluid in the Snyder space perturbatively at the first order in powers of the noncommutative parameter. The linearized noncommutative fluid dynamics is described by a system of coupled linear partial differential equations in which the variables are the fluid density and the fluid potentials. We show that these equations admit a set of solutions that are monochromatic plane waves for the fluid density and two of the potentials and a linear function for the third potential. The energy–momentum tensor of the plane waves is calculated.     

The analysis of the impact response of a thin plate via fractional derivative standard linear solid model

The impact of a rigid body upon an infinite isotropic plate is investigated for the case when the viscoelastic features of the plate represent themselves only in the place of contact and are governed by the standard linear solid model with fractional derivatives. Thus, the problem concerns the shock interaction of the dropping mass and the target, wherein instead of the Hertz contact law the generalized fractional derivative standard linear solid law is employed as a law of interaction. The part of the plate beyond the contact domain is assumed to be elastic, and its behaviour is described by the equations of motion which take rotary inertia and shear deformations into account. It is assumed that transient waves generate in the plate at the moment of impact, the influence of which on the contact domain is considered using the theory of discontinuities. To determine the desired values behind the transverse shear wave front, one-term ray expansions are used, as well as the equations of motion of the falling mass and the contact region. As a result, we are led to a set of two linear differential equations, the solution of which is found analytically by the Laplace transform and by the Euler substitution method. This allows the contact force to be determined as a function of time.     

A Kato Type Theorem for the Inviscid Limit of the Navier-Stokes Equations with a Moving Rigid Body

The issue of the inviscid limit for the incompressible Navier-Stokes equations when a no-slip condition is prescribed on the boundary is a famous open problem. A result by Kato (Math Sci Res Inst Publ 2:85?C98, 1984) says that convergence to the Euler equations holds true in the energy space if and only if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes. Of course, if one considers the motion of a solid body in an incompressible fluid, with a no-slip condition at the interface, the issue of the inviscid limit is as least as difficult. However it is not clear if the additional difficulties linked to the body??s dynamic make this issue more difficult or not. In this paper we consider the motion of a rigid body in an incompressible fluid occupying the complementary set in the space and we prove that a Kato type condition implies the convergence of the fluid velocity and of the body velocity as well, which seems to indicate that an answer in the case of a fixed boundary could also bring an answer to the case where there is a moving body in the fluid.     

Hermite-DG methods for pdf equations modelling particle transport and deposition in turbulent boundary layers

A novel methodology is presented for the numerical treatment of multi-dimensional pdf (probability density function) models used to study particle transport in turbulent boundary layers. A system of coupled Fokker–Planck type equations is constructed to describe the transport of phase-space conditioned moments of particle and fluid velocities, both streamwise and wall-normal. This system, unlike conventional moment-based transport equations, allows for an exact treatment of particle deposition at the flow boundary and provides an efficient way to handle the 5-dimensional phase-space domain. Moreover, the equations in the system are linear and can be solved in a sequential fashion; there is no closure problem to address.A hybrid Hermite-Discontinuous Galerkin scheme is developed to treat the system. The choice of Hermite basis functions in combination with an iterative scaling approach permits the efficient computation of solutions to high accuracy. Results demonstrate the effectiveness of the methodology in resolving the extreme gradients characteristic of distributions near an absorbing boundary.     

A finite difference method for a coupled model of wave propagation in poroelastic materials

A computational method for time-domain multi-physics simulation of wave propagation in a poroelastic medium is presented. The medium is composed of an elastic matrix saturated with a Newtonian fluid, and the method operates on a digital representation of the medium where a distinct material phase and properties are specified at each volume cell. The dynamic response to an acoustic excitation is modeled mathematically with a coupled system of equations: elastic wave equation in the solid matrix and linearized Navier-Stokes equation in the fluid. Implementation of the solution is simplified by introducing a common numerical form for both solid and fluid cells and using a rotated-staggered-grid which allows stable solutions without explicitly handling the fluid-solid boundary conditions. A stability analysis is presented which can be used to select gridding and time step size as a function of material properties. The numerical results are shown to agree with the analytical solution for an idealized porous medium of periodically alternating solid and fluid layers.     

基于有限立体角测量的谱色测温法

辐射测温是通过测量物体表面发射的辐射来反演温度. 本文结合线性发射率模型从辐射测温方程封闭求解的角度, 解释了谱色测温通常需采用微元立体角测量或针对漫发射体的有限立体角辐射测量的原因, 并推导出了有限立体角辐射测量条件下, 具有非漫发射性质物体表面温度测量的辐射测温方程, 该方程具有测量普适性. 以此方程为基础, 推导了具有测量普适性的谱色测温方程组, 发现不同的辐射测量条件下, 发射率标尺的取值范围相同, 但物理意义发生了明显变化.     

Numerical solution of an extended White–Metzner model for eccentric Taylor–Couette flow

In this study, we have developed a new numerical approach to solve differential-type viscoelastic fluid models for a commonly used benchmark problem, namely, the steady Taylor—Couette flow between eccentric cylinders. The proposed numerical approach is special in that the nonlinear system of discretized algebraic flow equations is solved iteratively using a Newton–Krylov method along with an inverse-based incomplete lower-upper preconditioner. The numerical approach has been validated by solving the benchmark problem for the upper-convected Maxwell model at a large Deborah number. Excellent agreement with the numerical data reported in the literature has been found. In addition, a parameter study was performed for an extended White–Metzner model. A large eccentricity ratio was chosen for the cylinder system in order to allow flow recirculation to occur. We detected several interesting phenomena caused by the large eccentricity ratio of the cylinder system and by the viscoelastic nature of the fluid. Encouraged by the results of this study, we intend to investigate other polymeric fluids having a more complex microstructure in an eccentric annular flow field.     

A Schur complement formulation for solving free-boundary,Stefan problems of phase change

A Schur complement formulation that utilizes a linear iterative solver is derived to solve a free-boundary, Stefan problem describing steady-state phase change via the Isotherm–Newton approach, which employs Newton’s method to simultaneously and efficiently solve for both interface and field equations. This formulation is tested alongside more traditional solution strategies that employ direct or iterative linear solvers on the entire Jacobian matrix for a two-dimensional sample problem that discretizes the field equations using a Galerkin finite-element method and employs a deforming-grid approach to represent the melt–solid interface. All methods demonstrate quadratic convergence for sufficiently accurate Newton solves, but the two approaches utilizing linear iterative solvers show better scaling of computational effort with problem size. Of these two approaches, the Schur formulation proves to be more robust, converging with significantly smaller Krylov subspaces than those required to solve the global system of equations. Further improvement of performance are made through approximations and preconditioning of the Schur complement problem. Hence, the new Schur formulation shows promise as an affordable, robust, and scalable method to solve free-boundary, Stefan problems. Such models are employed to study a wide array of applications, including casting, welding, glass forming, planetary mantle and glacier dynamics, thermal energy storage, food processing, cryosurgery, metallurgical solidification, and crystal growth.     

MHD mixed convection stagnation point flow of a hybrid nanofluid past a vertical flat plate with convective boundary condition

The steady magnetohydrodynamics (MHD) mixed convection stagnation point flow of a hybrid nanofluid past a vertical flat plate with convective boundary condition is investigated in this study. By choosing appropriate similarity variables, the partial differential equations are transformed into a system of linear equations which are solved afterwards by using the boundary value problem solver, namely bvp4c in Matlab software. The solutions of the ordinary differential equations are proven to have two branches which are upper and lower solutions in a certain range of the mixed convection parameter. This leads to the execution of a stability analysis that verifies the achievable solution of the upper branch which is considered stable while the lower branch is unstable. The findings exposed that the fluid velocity increases with the increasing value of the magnetic parameter while the fluid temperature is decreased contradictory.     

A class of Cartesian grid embedded boundary algorithms for incompressible flow with time-varying complex geometries

We present a class of numerical algorithms for simulating viscous fluid problems of incompressible flow interacting with moving rigid structures. The proposed Cartesian grid embedded boundary algorithms employ a slightly different idea from the traditional direct-forcing immersed boundary methods: the proposed algorithms calculate and apply the force density in the extended solid domain to uphold the solid velocity and hence the boundary condition at the rigid-body surface. The principle of the embedded boundary algorithm allows us to solve the fluid equations on a Cartesian grid with a set of external forces spread onto the grid points occupied by the rigid structure. The proposed algorithms use the MAC (marker and cell) algorithm to solve the incompressible Navier-Stokes equations. Unlike projection methods, the MAC scheme incorporates the gradient of the force density in solving the pressure Poisson equation, so that the dipole force, due to the jump of pressure across the solid-fluid interface, is directly balanced by the gradient of the force density. We validate the proposed algorithms via the classical benchmark problem of flow past a cylinder. Our numerical experiments show that numerical solutions of the velocity field obtained by using the proposed algorithms are smooth across the solid-fluid interface. Finally, we consider the problem of a cylinder moving between two parallel plane walls. Numerical solutions of this problem obtained by using the proposed algorithms are compared with the classical asymptotic solutions. We show that the two solutions are in good agreement.     

AEROELASTIC FLUTTER OF CIRCULAR ROTATING DISKS: A SIMPLE PREDICTIVE MODEL

This paper is an attempt to predict aeroelastic flutter of a rotating disk in an unbounded fluid. In the first part of the paper, the linear vibration of a rotating, potential fluid driven by transverse, harmonic motion of a rotating disk is solved. We extend the existing solution for a rigid disk to include flexible disks and compare alternative numerical evaluation schemes. Our principal interest in this problem is the identification of possible physical mechanisms for aeroelastic flutter. In the forced vibration problem considered here, fluid rotation renders the governing equations hyperbolic for low-frequency oscillation. As a result, the fluid motion may be discontinuous along the two characteristics that emanate from the rim of the disk. These discontinuities suggest the presence of previously unrecognized boundary layers near the rim of the disk that may be important for aeroelastic flutter. This idea is used to develop a simple mathematical model for predicting aeroelastic flutter. The model and its dependence on the dimensionless parameters describing the system are derived from first principles except for the compressible boundary layer, which is described by a simple function whose magnitude is empirically determined by fitting experimental data. Although the model is simple, its predictions are quantitatively similar to the experimental evidence and gives analytic predictions of aeroelastic flutter that are within an order of magnitude of the experimental values.