Incompressible Case Research Articles

Acoustic/steady streaming from a motionless boundary and related phenomena: generalized treatment of the inner streaming and examples

As originally realized by Nyborg (J. Acoust. Soc. Am., vol. 30, 1958, p. 329), the problem of the inner acoustic/steady streaming can be analysed in quite general terms. The inner streaming is the one that develops in the high-frequency limit in a thin Stokes (shear-wave) layer at a boundary, in contrast to the outer streaming in the main bulk of the fluid. The analysis provides relevant inner-streaming characteristics through a given distribution of the acoustic amplitude along the boundary. Here such a generalized treatment is revisited for a motionless boundary. By working in terms of surface vectors, though in elementary notations, new compact and easy-to-use expressions are obtained. The most important ones are those for the effective (apparent) slip velocity at the boundary as seen from a perspective of the main bulk of the fluid, which is often the sole driving factor behind the outer streaming, and for the induced (acoustic) steady tangential stress on the boundary. As another novel development, non-adiabatic effects in the Stokes layer are taken into account, which become apparent through the fluctuating density and viscosity perturbations, and whose contribution into the streaming is often ignored in the literature. Some important particular cases, such as the axisymmetric case and the incompressible case, are emphasized. As far as the application of the derived general inner-streaming expressions is concerned, a few examples provided here involve a plane acoustic standing wave, which either grazes a wall parallel to its direction (convenient for the estimation of the non-adiabatic effects), or into which a small (compared to the acoustic wavelength) rigid sphere is placed. If there are simultaneously two such waves, out-of-phase and, say, in mutually orthogonal directions, a disk placed coplanarly with them will undergo a steady torque, which is calculated here as another example. Two further examples deal with translational high-frequency harmonic vibrations of particles relative to an incompressible fluid medium, viz. of a rigid oblate spheroid (along its axis) and of a sphere (arbitrary three dimensional). The latter can be a fixed rigid sphere, one free to rotate or even a (viscous) spherical drop, for which the outer streaming and the internal circulation are also considered.

Study on Mode I Quasi-Static Growing Crack Field in the Pressure-Sensitive Dilatant Material

A mechanical model of the pressure-sensitive dilatant material is established in order to investigate the viscous effect in mode I quasi-static growing crack-tip field. The constitutive equations on the pressure-sensitive dilatant material are deducted. Through asymptotic analysis, it is shown that in the stable creep growing stage, the elastic-deformation and the visco-deformation are equally dominant in the near-tip field, as . The asymptotic solutions of separative variable in the crack-tip field of plane stress mode I quasi-static are aslo obtained. According to numerical calculation, the curves of stress, strain and displacement in terms of various parameters are given. The asymptotic solutions of quasi-static growing crack-tip field gained here can conveniently degenerate the incompressible case, when the Poisson ratio , named as HR field. The conclusions can provide the references for further studying the dynamic growing crack-tip field in the pressure-sensitive dilatant material.

Forcing for statistically stationary compressible isotropic turbulence

Linear forcing has been proposed as a useful method for forced isotropic turbulence simulations because it is a physically realistic forcing method with a straightforward implementation in physical-space numerical codes [T. S. Lundgren, “Linearly forced isotropic turbulence,” Annual Research Briefs (Center for Turbulence Research, Stanford, CA, 2003), p. 461; C. Rosales and C. Meneveau, “Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties,” Phys. Fluids 17, 095106 (2005)]. Here, extensions to the compressible case are discussed. It is shown that, unlike the incompressible case, separate solenoidal and dilatational parts for the forcing term are necessary for controlling the stationary state of the compressible case. In addition, the forcing coefficients can be cast in a form that allows the control of the stationary state values of the total dissipation (and thus the Kolmogorov microscale) and the ratio of dilatational to solenoidal dissipation. Linear full spectrum forcing is also compared to its low wavenumber restriction. Low wavenumber forcing achieves much larger Taylor Reynolds number at the same resolution. Thus, high Reynolds number asymptotics can be more readily probed with low wavenumber forced simulations. Since, in both cases, a solenoidal/dilatational decomposition of the velocity field is required, the simplicity of the full spectrum linear forcing implementation in physical-space numerical codes is lost. Nevertheless, low wavenumber forcing can be implemented without using a full Fourier transform, and so is computationally less demanding.

Bubble growth in visco-elastic magma: implications to magma fragmentation and bubble nucleation

We present a visco-elastic bubble growth model, accounting for viscous and elastic deformations and for volatile mass transfer between bubbles and melt. We define the borders between previous bubble growth models accounting for incompressible viscous melt, and our new model accounting also for elastic deformation; this is done by a set of end-member analytical solutions and numerical simulations. Elastic deformation is most prominent for magma of small vesicularity, where the growth regime depends on the shear modulus. For high shear modulus, bubble growth is slow and follows an exponential law in a viscous growth regime, while for low shear modulus bubbles quickly follow a square-root diffusive solution. Our model provides all the elastic components (stresses, strains and strain rates) required for defining criteria for failure and magma fragmentation. We suggest two failure criteria, a stress related one based on the internal friction and the Mohr-Coulomb failure theory, and a strain related one based on fibre elongation experiments. We argue that both criteria are equivalent if we consider their shear modulus dependency and its effect on magma rheology. Last, we apply our model to the process of bubble nucleation. In the incompressible case, following nucleation, growth is slow and leads to long incubation times during which bubbles may be dissolved back into the melt. The elastic response in magmas with low shear modulus results in a short incubation time, increasing the probability of survival. The above effects emphasize the significance of visco-elasticity for the dynamic processes occurring in magmas during volcanic activity.

Existence and Stability of Viscoelastic Shock Profiles

We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic–parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in our investigations of numerical Evans function computations determining stability of large-amplitude and nonclassical type shock profiles.

Magnetic reconnection with asymmetry in the outflow direction

Magnetic reconnection with asymmetry in the outflow direction occurs in the Earth's magnetotail, coronal mass ejections, flux cancellation events, astrophysical disks, spheromak merging experiments, and elsewhere in nature and the laboratory. A control volume analysis is performed for the case of steady antiparallel magnetic reconnection with asymmetric downstream pressure to derive scaling relations for the outflow velocity from each side of the current sheet and the reconnection rate. Simple relationships for outflow velocity are presented for the incompressible case and the case of symmetric downstream pressure but asymmetric downstream density. Asymmetry alone is not found to greatly affect the reconnection rate. The flow stagnation point and magnetic field null do not coincide in a steady state unless the pressure gradient is negligible at the flow stagnation point. The predictions of the model are compared with resistive MHD simulations of driven reconnection with asymmetry in the outflow direction.

Visualization of compressibility effects on large-scale structures in compressible mixing layers

Vorticity, x; velocity divergence, r u ~; and the magnitude of pressure gradient, rp j j are visualized in a same flow domain, to reveal large-scale, dilatation and shocklet structures in a compressible mixing layer from direct numerical simulation results. We simulated 2-dimensional (2-D) and 3-dimensional (3-D) compressible mixing layers with a third-order Discontinuous Galerkin method (Cockburn and Shu 1998) and a third order explicit Runge–Kutta time advancement scheme. The grid numbers are 5:04 10 for 2-D cases, and 4:6 10 for 3-D case. The resulting variants are normalized with the density q; the speed of sound a and qa of the free streams. A perturbation signal with a broadband spectrum and a peak at the most unstable frequency in incompressible case, in conjunction with a hyperbolic tangent mean velocity profile are imposed Fig. 1 Vorticity, x; velocity divergence, r u ~; and the magnitude of pressure gradient, rp j j; from DNS of 2-D compressible mixing layers. The mixing layer convects from left to right. The higher speed stream is at the upper portion of the flow field, and the lower speed stream is at the lower side of the flow field. x is plotted by contour, the colors of which represent velocity divergence. Pressure waves are identified by rp j j with blue contours. a Mc = 0.4. The contour levels for x range from -1.24 to 0.01 with 30 levels. The color levels for r u ~range from -0.001 to 0.001. b Mc = 0.8. The contour levels of x range from -1.24 to 0.01 with 30 levels. The color levels for r u ~ range from -0.005 to 0.005. The boundary for rp j j is determined by 0.1

Compressible laminar flow in a channel

Laminar flow of a compressible Newtonian fluid in a channel is analyzed. An analytic solution to the vorticity-stream function form of the hydrodynamics equations is found for weakly compressible flow using a regular perturbation method. In contrast with previous studies, the present analysis does not invoke the lubrication approximation and, consequently, predicts both a nonzero transverse velocity and a nonzero transverse pressure gradient. Predicted velocity and pressure fields from the perturbation solution are compared with previously published analytical and numerical solutions. Expressions for pressure drop are also given for compressible laminar flow in a channel that display significant deviations from the incompressible case. In addition, experimental data from the literature for the flow of gases in microchannels are analyzed and compared with predictions from the analytical solution. We find that a commonly used method for analyzing microchannel flow experiments obscures a rather simple dependence of pressure drop on the same dimensionless parameter used in the perturbation solution.

Three-phase compressible flow in porous media: Total Differential Compatible interpolation of relative permeabilities

We describe the construction of Total Differential (TD) three-phase data for the implementation of the exact global pressure formulation for the modeling of three-phase compressible flow in porous media. This global formulation is preferred since it reduces the coupling between the pressure and saturation equations, compared to phase or weighted formulations. It simplifies the numerical analysis of the problem and boosts its computational efficiency. However, this global pressure approach exists only for three-phase data (relative permeabilities, capillary pressures) which satisfy a TD condition. Such TD three-phase data are determined by the choice of a global capillary pressure function and a global mobility function, which take both saturations and global pressure level as argument. Boundary conditions for global capillary pressure and global mobility are given such that the corresponding three-phase data are consistent with a given set of three two-phase data. The numerical construction of global capillary pressure and global mobility functions by C1 and C0 finite element is then performed using bi-Laplacian and Laplacian interpolation. Examples of the corresponding TD three-phase data are given for a compressible and an incompressible case.

Response to “Shear Impossibility—Comments on ‘Void Growth by Dislocation Emission’ and ‘Void Growth in Metals’”

We thank the authors of Ref. [1] for their comments and interest in our work [2–6]. We agree that the emission of shear loops on one single shear plane cannot lead to void growth. However, our simulations suggest that coordinated shear on non-parallel planes can lead to void growth/shrinkage [3–5]. Detailed descriptions of this process are still being worked out and represent an exciting area for future research. Thus, there seem to be two separate dislocation mechanisms operating either sequentially or simultaneously: shear and prismatic loop emission. Whereas prismatic loops are well known and have been extensively documented, the postulation of “special” shear loops is novel. The authors [1] have correctly pointed out that special restrictions need to be applied to these shear loops, something not mentioned in Refs. [2–6]. On the subject of shear impossibility, the authors [1] bring some good points in their comments of our papers. Indeed, the closed loop emitted from the surface of the void does not by itself expand the void. This is obvious from our papers, in which we considered a growth of a cylindrical void by emission of an edge dislocation [2], and a collapse of the void by an emission of an opposite-signed edge dislocation [5]. Thus, if both (positive and negative) dislocations were emitted (which is a 2-D counterpart of a dislocation loop), no net effect on the void growth would take place. Void growth takes place by the outward transfer of the material, which is possible even in an incompressible case due to the outward movement of the remote boundary of the body (the plastically deformed region around the void is surrounded by an elastically deformed region). Eq. (1) of the Comments on our papers [1] nicely states that the net amount of added material associated with the crea-

Preferential concentration versus clustering in inertial particle transport by random velocity fields

The concept of preferential concentration in the transport of inertial particles by random velocity fields is extended to account for the possibility of zero correlation time and compressibility of the velocity field. It is shown that, in the case of an uncorrelated in time random velocity field, preferential concentration takes the form of a condition on the field history leading to the current particle positions. This generalized form of preferential concentration appears to be a necessary condition for clustering in the uncorrelated in time case. The standard interpretation of preferential concentration is recovered considering local time averages of the velocity field. In the compressible case, preferential concentration occurs in regions of negative divergence of the field. In the incompressible case, it occurs in regions of simultaneously high strain and negative field skewness.

Relative shear modulus reconstruction for visualization with no geometrical artifact

We previously reported several useful shear modulus reconstruction methods obtained from motion or static equations and reference shear moduli that yield an absolute reconstruction of shear modulus, density or inertia using the finite difference method (FDM) or finite element method (FEM) with strain tensor measurement (i.e., methods referred to as Methods A to C and F). Methods A to C are particularly useful for three-dimensional (3D) shear modulus reconstruction in an incompressible case or in the case of an inhomogeneous Poisson’s ratio by using the mean normal stress as an unknown, whereas Method F using a typical Poisson’s ratio is particularly useful when the reconstruction dimensionality is low or when the Poisson’s ratio is homogeneous. For instance, when using FDM, although Method B permits the absolute reconstruction of a mean normal stress by using a reference mean normal stress, Methods A and C yield biased mean normal stress reconstructions using no reference value and a quasi-reference value (e.g., zero), respectively. In this report, we describe our newly developed Methods D and E obtained from Methods A to C and F, which yield a unique relative shear modulus reconstruction with no geometrical artifact. In Method D, no reference shear modulus is used with Methods A to C and F, whereas in Method E, an arbitrary finite value (e.g., unity) is used with Methods A to C and F as a quasi-reference shear modulus at one or more arbitrary points (i.e., a quasi-reference point) or in an arbitrary region (i.e., a quasi-reference region) in the region of interest. That is, both Method D and Method E permit the visualization of the shear modulus distribution without using an absolute reference value (e.g., no reference material or typical value), which is particularly useful when dealing with deeply located tissues such as the liver and heart tissues. Methods D and E use iterative methods such as the conjugate gradient method to solve algebraic equations. As confirmed from simulated phantom deformation data, each version of Method E results in a higher convergence speed and reconstruction accuracy than the corresponding Method D. Although several versions of Method D using FEM are practically useful, Method E obtainable should be used for such relative reconstructions rather than the corresponding version of Method D, in combination with Methods A to C and F.

Sound modes in holographic superfluids

Superfluids support many different types of sound waves. We investigate the relation between the sound waves in a relativistic and a non-relativistic superfluid by using hydrodynamics to calculate the various sound speeds. Then, using a particular holographic scalar gravity realization of a strongly interacting superfluid, we compute first, second and fourth sound speeds as a function of the temperature. The relativistic low temperature results for second sound differ from Landau's well known prediction for the non-relativistic, incompressible case.

Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity

AbstractOverlapping Schwarz methods are considered for mixed finite element approximations of linear elasticity, with discontinuous pressure spaces, as well as for compressible elasticity approximated by standard conforming finite elements. The coarse components of the preconditioners are based on spaces, with a number of degrees of freedom per subdomain which are uniformly bounded, which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type, i.e. methods based on nonoverlapping decompositions of the domain. The local components of the new preconditioners are based on solvers on a set of overlapping subdomains.In the current study, the dimension of the coarse spaces is smaller than in recently developed algorithms; in the compressible case all independent face degrees of freedom have been eliminated while in the almost incompressible case five out of six are not needed. In many cases, this will result in a reduction of the dimension of the coarse space by about one half compared with that of the algorithm previously considered. In addition, in spite of using overlapping subdomains to define the local components of the preconditioner, values of the residual and the approximate solution need only to be retained on the interface between the subdomains in the iteration of the new hybrid Schwarz algorithm. The use of discontinuous pressures makes it possible to work exclusively with symmetric, positive‐definite problems and the standard preconditioned conjugate gradient method.Bounds are established for the condition number of the preconditioned operators. The bound for the almost incompressible case grows in proportion to the square of the logarithm of the number of degrees of freedom of individual subdomains and the third power of the relative overlap between the overlapping subdomains, and it is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. Numerical results illustrate the findings. Copyright © 2009 John Wiley & Sons, Ltd.

Study on Stress and Strain Field of Mode II Quasi-Static Growing Crack in Pressure-Sensitive Dilatant Material

A mechanical model of the pressure-sensitive dilatant material is established in order to investigate the viscous effect in quasi-static growing crack-tip field. The constitutive equations on the pressure-sensitive dilatant material are deducted. Through asymptotic analysis, it is shown that in the stable creep growing stage, the elastic-deformation and the visco-deformation are equally dominant in the near-tip field, as . The asymptotic solutions of separative variable in the crack-tip field of plane stress mode II quasi-static are aslo obtained. According to numerical calculation, the curves of stress, strain and displacement in terms of various parameters are given. The asymptotic solutions of quasi-static growing crack-tip field gained here can conveniently degenerate the incompressible case, when the Poisson ratio , named as HR field. The conclusions can provide the references for further studying the dynamic growing crack-tip field in the pressure-sensitive dilatant material.

From the Hu–Washizu formulation to the average nodal strain formulation

We present a stabilized finite element method for the Hu–Washizu formulation of linear elasticity based on simplicial meshes leading to the stabilized nodal strain formulation or node-based uniform strain elements. We show that the finite element approximation converges uniformly to the exact solution for the nearly incompressible case.

Remarks on The Use and Misuse of The Semi-Inverse Method in The Nonlinear Theory of Elasticity

We show, by considering some specific examples, that in the nonlinear theory of elasticity the application of the semi-inverse method has to be carried out very carefully. Usually, the strategy followed in applying the semi-inverse method while dealing with complex models is to generalize forms of solutions already known within the framework of a simpler theory. This is a smart strategy from a mathematical point of view, but from the physical point of view, it may be imprudent. For example, in the theory of nonlinear elasticity, we are at times motivated by the results obtained in the incompressible case to arrive at an understanding of what happens in the compressible case, and by doing so, many important exact solutions for special classes of compressible elastic materials have been obtained successfully. Sometimes, the admissibility of a given deformation field is considered to delineate special classes of constitutive laws. We wish to point out that the classes of constitutive equations thus identified from the standpoint that it may admit a type of deformation may lead to models that exhibit physically unacceptable mechanical behavior. To illustrate the dangers inherent to merely turning the mathematical crank to determine classes of constitutive equations where a certain class of deformations are possible, we consider the torsion of a cylindrical shaft and the propagation of transverse waves in a compressible nonlinear elastic material and show that care has to be exercised in appealing to the semi-inverse method.

Compressibility Effects on Maximum Range Cruise at Constant Altitude

I NA recent paper, Pargett and Ardema [1] analyze the problem of range maximization in cruise at constant altitude as a singular optimal control problem. The same problem is also analyzed using different approaches by Miele [2] and Torenbeek [3], who consider the case of quasi-steady flight. In [1], the singular arc that defines the optimal path is studied; in this study, a simple aircraft model defined by a parabolic drag polar of constant coefficients is considered and only one value of cruise altitude. Many other analyses of cruise optimization can be found in the literature, with different flight constraints and performance indices; see, for instance, the study of Menon [4], in which both speed and altitude are allowed to vary, and the many references therein. In this work, we generalize the analysis of [1] considering a general drag polar, so that compressibility effects can be taken into account. The singular arc in this general case is obtained and is particularized for both asymmetric and symmetric compressible parabolic drag polars. The results show that compressibility effects are very important; the differences with the incompressible case are shown to be not only quantitative, but also qualitative. We also analyze the influence of cruise altitude on the optimal paths, showing that it can be important from a qualitative point of view. Themaximum range is calculated as a function of cruise altitude; in particular, the best altitude that leads to the largest maximum range is obtained. Results are presented for a model of a Boeing 767-300ER.

A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity

In this note we propose a finite element method for incompressible (or compressible) elasticity problems with discontinuous modulus of elasticity (or, if compressible, Poisson’s ratio). The problem is written on mixed form using P 1 -continuous displacements and elementwise P 0 pressures, leading to the possibility of eliminating the pressure beforehand in the compressible case. In the incompressible case, the method is augmented by a stabilization term, penalizing the pressure jumps. We show a priori error estimates under certain regularity hypothesis. In particular we prove that if the exact solution is sufficiently smooth in each subdomain then the convergence order is optimal.

Dynamics of Isotropic Homogeneous Turbulence with Linear Forcing Using a Lattice Boltzmann Method

The Lattice Boltzmann Method (LBM) has proved to be a promising approach to solve the Navier–Stokes equations, especially for incompressible and isothermal cases. For turbulent flows, the quality of the predictions has been previously studied considering standard spectral forced (ten Cate et al., Comput Fluids 35:1239–1251, 2006) statistically homogeneous isotropic turbulence. In the present contribution, a recently proposed linear forcing scheme working in physical space (Lundgren 2003; Rosales and Meneveau, Phys Fluids 17(9):095106–1,8, 2005) has been integrated in a three-dimensional fifteen-velocity LBM formulation. Results have been analyzed, with special attention to the dynamics of the flow through the invariants of the velocity tensor. This topic had not been studied yet for the linear forcing, regardless of the nature (spectral or LBM) of the numerical method. Results fully agree with standard pseudo-spectral direct numerical simulations, results proving the validity of the LBM with linear forcing in real space to study this kind of turbulent flows.