Incompressible Case Research Articles

Studies of poisson's ratio variation for solid propellant grains under ignition pressure loading

For structural integrity evaluation of solid propellant grains, an accurate estimation of the stress and strain response is essential, because of the viscoelastic nature of the polymer material. The results of an investigation on solid propellant grains considering the effect of Poisson's ratio υ variation under ignition pressurization are presented. Traditionally the υ value of solid propellant grains is assumed to be a constant (e.g. 0.4999) to simplify the experimental task, but the real υ value is from 0.47 to 0.4999 depending on the chemical design of the solid propellant grains. In order to simulate the time–temperature-dependent behaviour of viscoelastic polymer materials for a range of Poisson's ratio values, the concepts of a time–temperature shift principle and reduced integration were used. In addition, eight different Poisson's ratio values from 0.47 to 0.4999 were assumed and compared using the finite element method. Results show that the Poisson's ratio variation effect is very important for structural integrity of solid propellant grains, and the stress and strain responses versus Poisson's ratio variation are nonlinear because the type of polymer material is changed from incompressible to compressible. Under ignition pressure loading, the stress and strain responses in the compressible case ( υ≠0.5) are much higher than those of the incompressible case ( υ≒0.5). Therefore unlike metallic structures, an exact value of Poisson's ratio for polymer materials is very important, and an improper assumption ( υ≒0.5) may cause the structural integrity of a missile system subjected to pressure loading to be wrongly evaluated.

Planar velocity measurements in a weakly compressible mixing layer

High-vector-density planar velocity fields were obtained for a weakly compressible mixing layer using particle image velocimetry (PIV). The velocity ratio of the mixing layer was 0.53, the density ratio was 0.67, and the convective Mach number was 0.38. At the location where the PIV images were obtained, . The instantaneous planar velocity fields fall into three regimes characterized by the size and number of large-scale structures present. The large-scale rollers are either circular or elliptical, with the elliptical rollers having, in general, horizontal major axes. The transverse velocity fluctuations and Reynolds shear stress are suppressed for the weakly compressible mixing-layer as compared to the incompressible case. The spatial correlations of velocity fluctuations also occupy a smaller fraction of the mixing-layer thickness than for an incompressible mixing layer. The linear stochastic estimate of a roller structure is elliptical with the major axis oriented in the streamwise direction and with an eccentricity greater than for the incompressible case. The linear stochastic estimate of a braid suggests that the braids are vertically oriented, as opposed to the oblique orientation seen in incompressible mixing layers. In addition, the braids in the weakly compressible case have a vertically oriented stagnation line, as opposed to the braids in the incompressible mixing layer where stagnation occurs at a point.

Alfven resonances, forced magnetic reconnection and model of solar flares

This paper briefly reviews the surface-wave-induced-magnetic reconnection (SWIMR) model based on the Alfven resonance theory near the neutral point. Extending this model to magnetosonic waves it is seen that it not only has features of formation of current sheets and plasmoids as in the incompressible case but in addition energy considerations show the triggering of solar flares by the catastrophic loss of equilibrium. The similarity of SWIMR model to the emerging flux and plasmoid induced magnetic reconnection model for solar flares is discussed.This article was scheduled to appear in issue 5 of Plasma Phys. Control. Fusion. To access this Special issue please follow this link: http://www.iop.org/EJ/toc/0741-3335/45/5

The Fully Compressible Semi-Geostrophic System from Meteorology

The semi-geostrophic equations are an approximation to the 3-D Euler equations for an atmosphere where the effects of rotation dominate. They are used by meteorologists to model the formation of fronts. Mathematically rigorous results were obtained by Benamou and Brenier and by Cullen and Gangbo in the incompressible case. In this paper we extend the results of Benamou and Brenier to the fully compressible case and show the existence of weak solutions to a reformulation of these equations in so-called dual variables. The Monge-Kantorovich theory appears as a material tool in our study.

Interchange Method in Compressible Magnetized Couette Flow: Magnetorotational and Magnetoconvective Instabilities

We obtain the general forms of the axisymmetric stability criteria in a magnetized compressible Couette flow using an energy variational principle, the so-called interchange or Chandrasekhar s met hod, which we applied successfully in the incompressible case. This formulation accounts for the simultaneous presence of gravity, rotation, a toroidal magnetic field, a weak axial magnetic field, entropy gradients, and density gradients in the initial equilibrium state. The power of the method lies in its simplicity which allows us to derive extremely compact and physically clear expressions for the relevant stability criteria despite the inclusion of so many physical effects. In the implementation of the method, all the applicable conservation laws are explicitly taken into account during the variations of a quantity with dimensions of energy which we call the free energy function. As in the incompressible case, the presence of an axial field invalidates the conservation laws of angular momentum and azimuthal magnetic flux and introduces instead isorotation and axial current conservation along field lines. Our results are therefore markedly different depending on whether an axial magnetic field is present, and generalize in two simple expressions all previously known, partial stability criteria for the appearance of magnetorotational instability. Furthermore, the coupling between magnetic tension and buoyancy and its influence to the dynamics of nonhomoentropic magnetized flows becomes quite clear from our results. In the limits of plane-parallel atmospheres and homoentropic flows, our formulation easily recovers the stability criteria for suppression of convective and Parker instabilities, as well as some related special cases studied over 40 years ago by Newcomb and Tserkovnikov via laborious variational techniques.

Asymptotic bifurcation results for the eversion of elastic shells

We apply the WKB method to the bifurcation analysis of everted cylindrical and spherical shells composed of Varga material. Both compressible and incompressible cases are considered. The method is degenerate but we obtain explicit bifurcation criteria and compare with previous numerical approximations. Different strategies are compared.

A mixed finite element method for acoustic wave propagation in moving fluids based on an Eulerian-Lagrangian description.

A nonstandard wave equation, established by Galbrun in 1931, is used to study sound propagation in nonuniform flows. Galbrun's equation describes exactly the same physical phenomenon as the linearized Euler's equations (LEE) but is derived from an Eulerian-Lagrangian description and written only in term of the Lagrangian perturbation of the displacement. This equation has interesting properties and may be a good alternative to the LEE: only acoustic displacement is involved (even in nonhomentropic cases), it provides exact expressions of acoustic intensity and energy, and boundary conditions are easily expressed because acoustic displacement whose normal component is continuous appears explicitly. In this paper, Galbrun's equation is solved using a finite element method in the axisymmetric case. With standard finite elements, the direct displacement-based variational formulation gives some corrupted results. Instead, a mixed finite element satisfying the inf-sup condition is proposed to avoid this problem. A first set of results is compared with semianalytical solutions for a straight duct containing a sheared flow (obtained from Pridmore-Brown's equation). A second set of results concerns a more complex duct geometry with a potential flow and is compared to results obtained from a multiple-scale method (which is an adaptation for the incompressible case of Rienstra's recent work).

Discrete Velocity Fields with Explicitly Computable Lagrangian Law

We introduce a class of random velocity fields on the periodic lattice and in discrete time having a certain hidden Markov structure. The generalized Lagrangian velocity (the velocity field as viewed from the location of a single moving particle) has similar hidden Markov structure, and its law is found explicitly. Its rate of convergence to equilibrium is studied in small numerical examples and in rigorous results giving absolute and relative bounds on the size of the second–largest eigenvalue modulus. The effect of molecular diffusion on the rate of convergence is also investigated; in some cases it slows convergence to equilibrium. After repeating the velocity field periodically throughout the integer lattice, it is shown that, with the usual diffusive rescaling, the single–particle motion converges to Brownian motion in both compressible and incompressible cases. An exact formula for the effective diffusivity is given and numerical examples are shown.

A mixed least squares method for solving problems in linear elasticity: theoretical study

In a previous paper we proposed a mixed least squares method for solving problems in linear elasticity. The solution to the equations of linear elasticity was obtained via minimization of a least squares functional depending on displacements and stresses. The performance of the method was tested numerically for low order elements for classical examples with well known analytical solutions. In this paper we derive a condition for the existence and uniqueness of the solution of the discrete problem for both compressible and incompressible cases, and verify the uniqueness of the solution analytically for two low order piece-wise polynomial FEM spaces.

Measures of intermittency in driven supersonic flows.

Scaling exponents for structure functions of the velocity, density, and entropy are computed for driven supersonic flows for rms Mach numbers of order unity, with numerical simulations using the piecewise parabolic method algorithm on grids of up to 512(3) points. The driving is made up of either one or three orthogonal shear waves. In all cases studied, the compressible component of the velocity in the statistically steady regime is weaker than its solenoidal counterpart by roughly a factor of 6. Exponents for the longitudinal component of the velocity are comparable to what is found in the incompressible case and appear insensitive to the presence of numerous shocks. Scaling exponents of the transverse components of the velocity are comparable to those for the longitudinal component. Density and entropy structure functions display strong departures from linear scaling. Finally, the scaling of structure functions of the energy transfer is also given and compared with the Kolmogorov refined similarity hypothesis.

Axisymmetric modal analysis of liquid‐storage tanks considering compressibility effects

AbstractIn this paper, we address analytical and numerical studies on the free vibration of fluid–structure interaction problems considering the fluid compressibility. According to the separation of variables together with the boundary condition enforcement, we first derive a compressible‐fluid velocity potential function. Next, we split the structure region into the wet and dry parts, for which we apply the Novozhilov thin shell theory. Combining two dynamic displacement fields, for two split structure parts, using the displacement compatibility conditions, we finally obtain a simultaneous equation system for computing natural frequencies and modes. According to the derived analytical formulae, we compute natural frequencies and modes, stress resultants, together with the comparison with the FEM analysis and the incompressible case. Numerical results show, compared to the incompressible case, that the compressible case produces lower natural frequencies and, furthermore, the relative difference is influenced by the slenderness of tanks and the relative liquid fill height. Copyright © 2002 John Wiley & Sons, Ltd.

Post‐Newtonian Treatment of Bar Mode Instability in Rigidly Rotating Equilibrium Configurations for Polytropic Stars

In this paper we determine the onset point of secular instability for the nonaxisymmetric bar mode in rigidly rotating equilibrium configurations in the post-Newtonian approximation, in order to apply it to neutron stars. The treatment is based on a precedent Newtonian analytic energy variational method that we have extended to the post-Newtonian case. This method, based on Landau's theory of second-order phase transitions, provides the critical value of the ellipsoid polar eccentricity e at the onset of instability, i.e., at the bifurcation point from the axisymmetric Maclaurin to the triaxial Jacobi ellipsoids, and it is valid for any equation of state. The extension of this method to post-Newtonian fluid configurations has been accomplished by combining two earlier orthogonal works, specialized, respectively, to slow rotating configurations but with arbitrary density profile and to constant mass density but arbitrarily fast rotating ellipsoids. We also determine the explicit expressions for the density functionals that allow the generalization of the physical quantities involved in our treatment from the constant mass density to an arbitrary density profile form. We find that, considering homogeneous ellipsoids, the value of the critical eccentricity increases as the stars become more relativistic, in qualitative agreement with previous investigations but with a less marked amount of such an increase. Then we have studied the dependence of this critical value on the configuration equation of state. Considering polytropic matter distributions, we find that the increase in the eccentricity at the onset of instability with the star compactness is confirmed for softer equations of state (with respect to the incompressible case). The amount of this stabilizing effect is nearly independent of the polytropic index.

A multi-temperature multiphase flow model

In this paper we formulate a multiphase model with nonequilibrated temperatures but with equal velocities and pressures for each species. Turbulent mixing is driven by diffusion in these equations. The closure equations are defined in part by reference to a more exact chunk mix model developed by the authors and coworkers which has separate pressures, temperatures, and velocities for each species. There are two main results in this paper. The first is to identify a thermodynamic constraint, in the form of a process dependence, for pressure equilibrated models. The second is to determine one of the diffusion coefficients needed for the closure of the equilibrated pressure multiphase flow equations, in the incompressible case. The diffusion coefficients depend on entrainment times derived from the chunk mix model. These entrainment times are determined here first via general formulas and then explicitly for Rayleigh-Taylor and Richtmyer-Meshkov large time asymptotic flows. We also determine volume fractions for these flows, using the chunk mix model.

Functional approach to the problem of self-gravitating systems: Conditions of integrability

Using a functional method based on the introduction of a velocity potential to solve the Euler, continuity and Poisson equations, a new analytic study of the equilibrium of self-gravitating rotating systems with a polytropic equation of state has permitted the formulation of the conditions of integrability. For the polytropic index $n=1$ in the incompressible case $(\ensuremath{\nabla}\ensuremath{\cdot}\stackrel{\ensuremath{\rightarrow}}{v}=0),$ we are able to find the conditions for solving the problem of the equilibrium of polytropic self-gravitating systems that rotate and have nonuniform vorticity. This work contains the conditions which give analytic and quasi-analytic solutions for the equilibrium of polytropic stars and galactic systems in Newtonian gravity. In special cases, explicit analytic solutions are presented.

New statistical mechanical treatment of systems near surfaces. V. Incompressible blend of interacting polydisperse linear polymers

We consider a lattice model of an incompressible blend of interacting (repulsive, attractive, or neutral) polydisperse polymers of two species, A and B. The blend is next to an infinite plane surface whose interaction with A can be attractive, repulsive, or neutral. This is the only parameter required to completely specify the effect of the surface on both components of the blend. We numerically study various density profiles and surface functions, as we move away from the surface, by using the method of Chhajer and Gujrati that has already been successfully applied to study a polymer solution next to a surface. The resulting density profiles show the oscillations that are seen in Monte Carlo simulations (but with magnitude enhanced and range diminished due to the presence of free volume in simulations), and the enrichment of the smaller species at a neutral surface. The method is computationally ultrafast and can be carried out on a PC, even in the incompressible case, when Monte Carlo simulations are not feasible. The calculations usually take a few seconds to a minute.

The compressible evolution of the super-Alfvénic magnetized wake

The effects of compressibility on the linear and nonlinear properties of the magnetized wake are examined, with an emphasis on the high speed flow situation. It is found that compressibility can modify properties of this system previously identified for the incompressible case. Of particular interest is an investigation of how the properties of the magnetized wake vary with the sonic Mach number. It is found that, in general, the growth rates of the unstable sinuous and varicose modes decrease with increasing Mach number and with increasing Alfvén number. However, at high sonic Mach numbers the varicose modes can have a growth rate which increases as the spanwise wave number increases, a significant difference from the incompressible case. The linear compressible equations are solved by a Chebyshev collocation technique. Nonlinear computations based on a finite volume method are also presented. Growth rates computed by both codes in the linear regime are in excellent agreement. At long times the system relaminarizes to an overall accelerated and broadened wake channel. It is found that variations in the Mach and Alfvén numbers have a strong affect on the evolution of the magnetized wake, e.g., for high M fast magnetosonic shocks are observed to develop.

Anomalous scaling in two models of passive scalar advection: effects of anisotropy and compressibility.

The problem of the effects of compressibility and large-scale anisotropy on anomalous scaling behavior is considered for two models describing passive advection of scalar density and tracer fields. The advecting velocity field is Gaussian, delta correlated in time, and scales with a positive exponent epsilon. Explicit inertial-range expressions for the scalar correlation functions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal anomalous exponents (dependent only on epsilon and alpha, the compressibility parameter). The complete set of anomalous exponents for the pair correlation functions is found nonperturbatively, in any space dimension d, using the zero-mode technique. For higher-order correlation functions, the anomalous exponents are calculated to O(epsilon(2)) using the renormalization group. As in the incompressible case, the exponents exhibit a hierarchy related to the degree of anisotropy: the leading contributions to the even correlation functions are given by the exponents from the isotropic shell, in agreement with the idea of restored small-scale isotropy. As the degree of compressibility increases, the corrections become closer to the leading terms. The small-scale anisotropy reveals itself in the odd ratios of correlation functions: the skewness factor slowly decreases going down to small scales for the incompressible case, but starts to increase if alpha is large enough. The higher odd dimensionless ratios (hyperskewness, etc.) increase, thus signaling persistent small-scale anisotropy; this effect becomes more pronounced for larger values of alpha.

A generalized orthotropic hyperelastic material model with application to incompressible shells

AbstractIn the present paper a new orthotropic hyperelastic constitutive model is proposed which can be applied to the numerical simulation of a wide range of anisotropic materials and particularly biological soft tissues. The model represents a non‐linear extension of the orthotropic St. Venant–Kirchhoff material and is described in each principal material direction by an arbitrary isotropic tensor function coupled with the corresponding structural tensor. In the special case of isotropy this constitutive formulation reduces to the Valanis–Landel hypothesis and may therefore be considered as its generalization to the case of orthotropy. Constitutive relations and tangent moduli of the model are expressed in terms of eigenvalue bases of the right Cauchy–Green tensor C and obtained for the case of distinct and coinciding eigenvalues as well. For the analysis of shells the model is then coupled with a six (five in incompressible case) parametric shell kinematics able to deal with large strains as well as finite rotations. The application of the developed finite shell element is finally illustrated by a number of numerical examples. Copyright © 2001 John Wiley & Sons, Ltd.

1302 直接数値シミュレーションによる圧縮性溝乱流場の構造解析(GS-5 数値流体力学と乱流(1))

To investigate the effect of comprssibelity on near-wall turblent structures , direct numerical simulation (DNS) is conducted for a supersonic wall-bounded channel flow. Mach number defined as bulk mean velocity and sound speed at the wall, is 1.5. In the nearwall region, typical turbulence structure, such as streaky structure and coherent vortices are observed similar to the incompressible turbulent flow. However, quantitatively the number or length scale of vortices is distinct from incompressible case. A conditional sampling technique is applied to study the flow field surrounding the strong down-wash events. The diltation dissipation denoting compressiblity effect is fairly small around vortices, the solenoidal dissipation due to the vortex itself is crucial even in this compressible flow.

Covariant vortex in superconducting-superfluid-normal fluid mixtures with a stiff equation of state

Using a covariant description, we obtain the integrals of motion for a cylindrically symmetric, stationary vortex configuration in a mixture of interacting superconductors, superfluids and normal fluids. We then integrate the stress-energy density and find a very simple, closed expression for the energy per unit length and the relevant stress coefficients of the vortex with respect to a vortex-free reference state. This result is found assuming a ``stiff'' equation of state for the fluid mixture, which is the least compressible but still causal equation of state (contrary to the incompressible case). As an illustration for these general results, we discuss some applications to ``real'' superfluid-superconducting systems that are contained as special cases. These include the two-fluid model for He-II, uncharged binary superfluid mixtures, conventional superconductors and the superfluid neutron-proton-electron plasma in the outer core of neutron stars.