Hamilton principle for an inviscid compressible fluid model in Eulerian coordinates

It is shown that the well-known variational principles for the ideal compressible fluid model in Eulerian coordinates have the following deficiencies:

1. They are not related to the corresponding variational principles in Lagrangian coordinates;
2. The variation procedure in these variational problems does not lead to the equations of motion themselves in the Euler form; rather it leads to relations which correspond to definite classes of solutions of the Euler equations. Here allowance for the equations of the constraints imposed by the adiabaticity and continuity conditions limits the region of application of these variational principles to only potential flows;
3. More general results, involving flows other than potential, are achieved by artificial selection of certain additional constraint conditions imposed on the quantities being varied, and in this case additional clarification is required to ascertain whether any inviscid compressible fluid flow is the extremum of the corresponding variational problem.

A new formulation of the Hamilton principle for the inviscid compressible fluid in Eulerian coordinates is suggested which is free from these deficiencies.     

Boundary Element Analysis of Vibro-Acoustic Interaction between Vibrating Membrane and Thin Fluid Layer

This paper presents a variational formulation for the study of the acoustic propagation and radiation of a vibrating membrane coupled to a cavity filled with a visco-thermal fluid. This formulation is obtained by combining a variational formulation by integral equations of both internal and external fluids, which takes account of acoustic and entropic waves coupling, with a classical variational formulation of the membrane. Numerical results obtained by this new formulation are compared to those obtained when the viscous and thermal effects are not considered. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical Simulation of the Fall of a Body with a Corrugated Bottom on Water

A numerical model is proposed for the potential flow of an ideal incompressible fluid produced by impact of a body with concave bottom on water. Compression of the entrapped air is taken into account. The algorithm is based on joint solution of the equations of motion for the body and the fluid by the finite difference method with approximation in time. At each time, the boundaryvalue problem for the Laplace equation is solved by the boundaryelement method. Calculation results are given. The effects of the air layer, dimensions and shape of the corrugations, initial velocity, and other parameters on the impact process are shown.     

Inelastic Analysis by Symmetrization of the Constitutive Law

This paper deals with the transformation of the inelastic continuum problem from the differential to the variational form. Due to the lack of symmetry of the constitutive law, the variational formulation of the problem has been, above all, obtained using techniques of symmetrization of the overall problem operator (as with Tontis techniques). In the present paper, instead, the adopted method is based on the symmetrization of the constitutive law only. On this basis, taking into account the consequent transformations to be done on the equilibrium and on the compatibility boundary condition operators, a symmetric global operator for the original problem is achieved. This leads to a new enlarged formulation and then, by the choice of a suitable operator, to a new wide class of variational principles and, by specialization, to classical and recent variational principles.     

Models of Elastic Shells in Contact with a Rigid Foundation: An Asymptotic Approach

We consider a family of linearly elastic shells with thickness \(2\varepsilon\) (where \(\varepsilon\) is a small parameter). The shells are clamped along a portion of their lateral face, all having the same middle surface \(S\), and may enter in contact with a rigid foundation along the bottom face.We are interested in studying the limit behavior of both the three-dimensional problems, given in curvilinear coordinates, and their solutions (displacements \(\boldsymbol{u}^{\varepsilon}\) of covariant components \(u_{i}^{\varepsilon}\)) when \(\varepsilon\) tends to zero. To do that, we use asymptotic analysis methods. On one hand, we find that if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), a suitable approximation of the variational formulation of the contact problem is a two-dimensional variational inequality which can be identified as the variational formulation of the obstacle problem for an elastic membrane. On the other hand, if the applied body force density is \(O(\varepsilon^{2})\) and surface tractions density is \(O(\varepsilon^{3})\), the corresponding approximation is a different two-dimensional inequality which can be identified as the variational formulation of the obstacle problem for an elastic flexural shell. We finally discuss the existence and uniqueness of solution for the limit two-dimensional variational problems found.     

First- and second-order sensitivities of beams with respect to cross-sectional cracks

This paper deals with a variational formulation for the sensitivity problem of beam systems in the context of deformable solids with cracks. Natural frequencies are defined as state variables involved in the energy functional of the system, while the cracks length and position are treated as system parameters. The hierarchical equation system is formed and solved for the first and second derivatives of the natural frequency functions with respect to the cracks length and position. An analytical procedure for calculations of the second-order sensitivities of natural frequencies is proposed for the non-symmetrical equation system operator. Numerical algorithms are worked out and implemented computationally. Analytical and numerical aspects of the problem are discussed in detail through a number of illustrative results.The support of this work by the State Committee for Scientific Research (KBN) under Grant No. 4-050-0148/17-98-00 is gratefully acknowledged.     

GURTIN VARIATIONAL PRINCIPLE AND FINITE ELEMENT SIMULATION FOR DYNAMICAL PROBLEMS OF FLUID-SATURATED POROUS MEDIA

Based on the theory of porous media,a general Gurtin variational principle for theinitial boundary value problem of dynamical response of fluid-saturated elastic porous media isdeveloped by assuming infinitesimal deformation and incompressible constituents of the solid andfluid phase.The finite element formulation based on this variational principle is also derived.Asthe functional of the variational principle is a spatial integral of the convolution formulation,thegeneral finite element discretization in space results in symmetrical differential-integral equationsin the time domain.In some situations,the differential-integral equations can be reduced to sym-metrical differential equations and,as a numerical example,it is employed to analyze the reflectionof one-dimensional longitudinal wave in a fluid-saturated porous solid.The numerical results canprovide further understanding of the wave propagation in porous media.     

Analytical analysis of the static interaction of fluid and cylindrical membrane structures

The static behavior of an inflated cylindrical membrane is theoretically investigated under different conditions of internal pressures, upstream and downstream fluid parameters. The membrane is attached to a horizontal base along two generators and can be inflated with a compressible fluid (air), an incompressible fluid (water), or a combination of them. The base width, curved perimeter, internal pressure, upstream and downstream fluid properties are given. Large deformation of the membrane due to the internal and external pressures makes the governing equation of the problem to be non-linear. In the present study, an analytical approach for the non-linear analysis of the static interaction of the fluid and the cylindrical membrane with different load distributions and boundary conditions is developed. Both geometric and equilibrium relations of the membrane element are used to obtain the membrane profile in explicit closed form. The validity of the present analytical approach is confirmed by comparing the results with experimental and numerical results obtained from the literature. It is shown that the present formulation is an appropriate method and a new approach to predict the static non-linear interaction of the fluid and the membrane structures with a good accuracy and less numerical effort.     

Entrance pressure losses for power-law fluid at small Reynolds number

Finite-element results are presented for the extra pressure loss for a power-law fluid due to an abrupt contraction of ratio 2, 4, 8 and for both the axisymmetric and planar cases. Contrary to earlier results in the literature, it is found that the variational result of Duda and Vrentas forms a very good upper bound for the axisymmetric case, even for small values of the power-law index. This variational method has now been extended to the planar case, where it is again found to form a very good upper bound.     

Optimal numerical flux of power-law fluids in some partially full pipes

Consider the steady state pressure driven flow of a power-law fluid in a partially filled straight pipe. It is known that an increase in flux can be achieved for a fixed pressure by partially filling the pipe and having the remaining volume either void or filled with a less viscous, lubricating fluid. If the pipe has circular cross section, the fluid level which maximizes flux is the level which avoids contact with exactly 25% of the boundary. This result can be proved analytically for Newtonian fluids and has been verified numerically for certain non-Newtonian models.

This paper provides a generalization of this work numerically to pipes with non-circular cross sections which are partially full with a power-law fluid. A simple and physically plausible geometric condition is presented which can be used to approximate the fluid level that maximizes flux in a wide range of pipe geometries. Additional increases in flux for a given pressure can be obtained by changing the shape of the pipe but leaving the perimeter fixed. This computational analysis of flux as a function of both fluid level and pipe geometry has not been considered to our knowledge.

Fluxes are computed using a special discretization scheme, designed to uncover general properties which are only dependent on fluid level and/or pipe cross-sectional geometry. Computations use finite elements and take advantage of the variational structure inherent in the power-law model. A minimization technique for approximating the critical points of the associated non-linear energy functional is used. In particular, the numerical scheme for the non-linear partial differential equation has been proved to be convergent with known error estimates. The numerical results obtained in this work can be useful for designing pipes and canals for transportation of non-Newtonian fluids, such as those in chemical engineering and food processing engineering.     

Variational principle of thermoelectroelasticity and its application to the problem of vibrations of a thin‐wall member

The dynamic behavior of thinwall members manufactured from materials with the pyroelectric effect was studied. A variational formulation of the problem is used, and a variational principle is formulated that differs from the wellknown one. Correct boundaryvalue problems describing the tension, compression, and bending of a thinwall pyroelectric member are constructed using the variational principle and a number of hypotheses on the distribution of the components of physical fields along the width of the member.     

Numerical solution of plane and axially symmetric jet flow problems based on the variational inequality formulation

The numerical analysis of plane and axially symmetric jet flows of an incompressible inviscid fluid is treated. A new formulation of the variational inequality type is developed from the variational principle associated with jet problems. A successive approximation method is formulated by the combined use of variational inequality and the finite element method. Numerical examples based on the iterative method are presented. The results obtained agree well with those by other methods.     

Elasto-plastic analysis of Reissner-Mindlin plates by the use of hybrid stress FEM with layered model

Based on a _mR -type variational formulation featuring a cross-depth layered model in conjunction with a mechanical sub-element for simulating the material constitution, the cross-depth plasticity development of the Reissner-Mindlin plate is investigated by following the loading process. A 4-node quadrilateral hybrid-stressc ⁰-continuous plate-bending element HPT-9 is formulated. Numerical examinations demonstrate its remarkable characteristic behavior in being free from spurious kinematic mode, capable of alleviating locking difficulties as the thin plate limit is approached and providing numerical results with remarkable accuracy and computational efficiency over Spilker's counterpart LH₄. An elasto-plastic analysis of Reissner-Mindlin plates has justified the validity and effectiveness of the present scheme in depicting the cross-depth plasticity development following the loading process.The Project is Supported by National Natural Science Foundation of China.     

Variational principles of time-dependent fluid flows through porous media with discontinuous reservoir properties

The use of variational principles as the initial basis for constructing continuum models was investigated by Sedov and his disciples. In this study the variational formalism is developed for calculating time-dependent fluid flows through porous and fractured-porous media with inhomogeneous, discontinuous, and, in particular, piecewise-constant properties. It is proved that, in the case of a medium with discontinuous properties, from the basic variational relation W = 0 there follows not only the differential equations of the flow models but also the conditions on the surfaces of discontinuity of the reservoir properties. This clears the way for the generalization and effective use of direct variational methods for calculating flow fields in complex-structure reservoirs. The methods proposed are illustrated by particular examples.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 115–123.Original Russian Text Copyright © 2004 by Volnitskaya.     

A geometrically exact Cosserat shell-model including size effects,avoiding degeneracy in the thin shell limit. Part I: Formal dimensional reduction for elastic plates and existence of minimizers for positive Cosserat couple modulus

This contribution is concerned with a consistent formal dimensional reduction of a previously introduced finite-strain three-dimensional Cosserat micropolar elasticity model to the two-dimensional situation of thin plates and shells. Contrary to the direct modelling of a shell as a Cosserat surface with additional directors, we obtain the shell model from the Cosserat bulk model which already includes a triad of rigid directors. The reduction is achieved by assumed kinematics, quadratic through the thickness. The three-dimensional transverse boundary conditions can be evaluated analytically in terms of the assumed kinematics and determines exactly two appearing coefficients in the chosen ansatz. Further simplifications with subsequent analytical integration through the thickness determine the reduced model in a variational setting. The resulting membrane energy turns out to be a quadratic, elliptic, first order, non degenerate energy in contrast to classical approaches. The bending contribution is augmented by a curvature term representing an additional stiffness of the Cosserat model and the corresponding system of balance equations remains of second order. The lateral boundary conditions for simple support are non-standard. The model includes size-effects, transverse shear resistance, drilling degrees of freedom and accounts implicitly for thickness extension and asymmetric shift of the midsurface. The formal thin shell membrane limit without classical h ³-bending term is non-degenerate due to the additional Cosserat curvature stiffness and control of drill rotations. In our formulation, the drill-rotations are strictly related to the size-effects of the bulk model and not introduced artificially for numerical convenience. Upon linearization with zero Cosserat couple modulus we recover the well known infinitesimal-displacement Reissner-Mindlin model without size-effects and without drill-rotations. It is shown that the dimensionally reduced Cosserat formulation is well-posed for positive Cosserat couple modulus by means of the direct methods of variations along the same line of argument which showed the well-posedness of the three-dimensional Cosserat bulk model [72].Received: 16 April 2004, Accepted: 3 May 2004, Published online: 17 September 2004     

Analytical and numerical study of natural convection heat transfer in an inclined composite enclosure

Laminar natural convection heat transfer in inclined fluid layers divided by a partition with finite thickness and conductivity is studied analytically and numerically. The governing equations for the fluid layers are solved analytically in the limit of a thin layered system with constant flux boundary conditions. The study covers of the range of Ra from 10³ to 10⁷, from 0° to 180° and the thermal conductivity ratio of partition to fluid ratioK from 10^–2 to 10⁶. The Prandtl number was 0.72 (for air). Results are obtained in terms of an overall Nusselt number as a function of Rayleigh number, angle of inclination of the system, mid layer thickness, and mid layer thermal conductivity. The critical Rayleigh number for the onset of convection in a bottom-heated horizontal system is predicted. The results are compared with the numerical results obtained by solving the complete system of governing equations, using SIMPLER method, as well as with the limiting cases in the literature.     

流体饱和多孔介质的动力学Gurtin型变分原理和有限元模拟

基于多孔介质理论。在两相不可压和小变形的假设下，建立了流体饱和弹性多孔介质的动力学Gurtin型变分原理，并导出了以此变分原理为基础的有限元离散公式，由于Gurtin型变分原理是卷积型的空间积分泛函，空间的有限元离散导致一个关于时间的对称微分—积分方程组，在一般条件下。该积分—微分方程组可转化为对称的微分方程组，这组方程有别于标准Galerkin有限元的非对称离散方程组，作为数值例子，分析了流体饱和弹性多孔介质中一维纵向波的传播和反射，其结果进一步揭示了饱和多孔介质中波的传播特性。     

Optimum lifting body shapes in hypersonic flow at high angles of attack

A variational procedure for the determination of lifting body configurations having a maximum lift-to-drag ratio K _max in hypersonic flight at high angles of attack , is proposed. It is based on an analytical solution to the problem for three-dimensional hypersonic flow over small aspect ratio wings using thin shock-layer theory. This reduces the variational problem of finding K _max, and the corresponding optimized wing shape, to the minimization of a linear functional subject to various constraints. The contributions of nonequilibrium thermochemical effects and laminar or turbulent viscous drag effects are also included in the problem formulation. The solution shows that optimized wings have an unbent forward part and a concave lower surface. Due to bifurcation in the optimization process, the planform may have either a sharp apex or a straight nose cut. Corresponding values of K _max() significantly exceed the limiting value K _N=cot for a flat wing. Real thermochemical effects and air viscosity are shown to cause a decrease in K _max and sometimes to influence the optimized wing geometry; however, the relative increment of K _max to K _N is still retained.     

Hamiltonian Formulation for the Motion of a Two-Fluid System with Free Surface

In this work, a theoretical investigation is performed on modeling interfacial and surface waves in a layered fluid system. The physical system consists of two immiscible liquid layers of different densities ₁ > ₂ with an interfacial surface and a free surface, inside a prismatic-section tank. On the basis of the potential formulation of the fluid motion, we derive a nonlinear system of partial differential equations using the Hamiltonian formulation for irrotational flow of the two fluids of different density subject to conservative force. As a consequence of the assumption of potential velocity, the dynamics of the system can be described in terms of variables evaluated only at the boundary of the fluid system, namely the separation surface and the free surface. This Hamiltonian formulation enables one to define the evolution equations of the system in a canonical form by using the functional derivatives.