Incompressible Case Research Articles

Investigation of seismic behavior of fluid–rectangular tank–soil/foundation systems in frequency domain

Main purpose of this study is to evaluate the dynamic behavior of fluid–rectangular tank–soil/foundation system with a simple and fast seismic analysis procedure. In this procedure, interaction effects are presented by Housner's two mass approximations for fluid and the cone model for soil/foundation system. This approach can determine; displacement at the height of the impulsive mass, the sloshing displacement and base forces for the soil/foundation system conditions including embedment and incompressible soil cases. Models and equations for proposed method were briefly explained for different tank–soil/foundation system combinations. By means of changing soil/foundation conditions, some comparisons are made on base forces and sloshing responses for the cases of embedment and no embedment. The results showed that the displacements and base shear forces generally decreased, with decreasing soil stiffness. However, embedment, wall flexibility, and soil–structure interaction (SSI) did not considerably affect the sloshing displacement.

On volumetric locking‐free behaviour of p‐version finite elements under finite deformations

AbstractWe demonstrate the locking‐free properties of the displacement formulation of p‐finite elements when applied to nearly incompressible Neo‐Hookean material under finite deformations. For an axisymmetric model problem we provide semi‐analytical solutions for a nearly incompressible Neo‐Hookean material exploited to investigate the robustness of p‐FEM with respect to volumetric locking. An analytical solution for the incompressible case is also derived to demonstrate the convergence of the compressible numerical solution towards the incompressible case when the compression modulus is increased. Copyright © 2007 John Wiley & Sons, Ltd.

Locking-free finite element methods for linear and nonlinear elasticity in 2D and 3D

The uniform convergence of finite element approximations based on a modified Hu–Washizu formulation for the nearly incompressible linear elasticity is analyzed. We show the optimal and robust convergence of the displacement-based discrete formulation in the nearly incompressible case with the choice of approximations based on quadrilateral and hexahedral elements. These choices include bases that are well known, as well as newly constructed bases. Starting from a suitable three-field problem, we extend our α-dependent three-field formulation to geometrically nonlinear elasticity with Saint-Venant Kirchhoff law. Additionally, an α-dependent three-field formulation for a general hyperelastic material model is proposed. A range of numerical examples using different material laws for small and large strain elasticity is presented.

Homogenization of the unstationary incompressible Reynolds equation

This paper is devoted to the effects of surface roughness during hydrodynamic lubrication. In the numerical analysis a very fine mesh is needed to resolve the surface roughness, suggesting some type of averaging. A rigorous way to do this is to use the general theory of homogenization. In most works about the influence of surface roughness, it is assumed that only the stationary surface is rough. This means that the governing Reynolds type equation does not involve time. However, recently, homogenization was successfully applied to analyze a situation where both surfaces are rough and the lubricant is assumed to have constant bulk modulus. In this paper we will consider a case where both surfaces are assumed to be rough, but the lubricant is incompressible. It is also clearly demonstrated, in this case that homogenization is an efficient approach. Moreover, several numerical results are presented and compared with those corresponding to where a constant bulk modulus is assumed to govern the lubricant compressibility. In particular, the result shows a significant difference in the asymptotic behavior between the incompressible case and that with constant bulk modulus.

On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy

This paper discusses the multiplicative decomposition of the deformation gradient into its volumetric and isochoric parts and its implications in the case of anisotropy. An analysis is carried out showing that the volumetric-isochoric split of the stored energy function can be justified and systematically derived on the basis of the physical assumption that the spherical part of the stress depends on the determinant of the deformation gradient without ad hoc introduction of the multiplicative split. The analysis shows that care must be exercised in the case of anisotropic material description in order not to violate certain physical requirements. Additive splits of the energy can be justified on the basis of certain physical observations and independent of the multiplicative decomposition of the deformation gradient. Specifically, it is shown that a spherical state of stress will cause even in the incompressible case, a change of shape. In fibre reinforced materials, the split of the stored energy function into a part related to the matrix and a part related to the fibre is considered, showing that the volumetric-isochoric split should be applied to the matrix part only.

Stability study for the MHD problem in perforated and parallel walls

We have studied the stability of a conducting fluid when it is continuously injected or ejected through a pair of parallel porous walls and escapes in both directions along the channel. The flow forms a stagnation point at the center and the effluence is restricted by a magnetic field perpendicular to the walls. A theoretical analysis of the steady state solutions of the MHD equations in the incompressible case is given as a function of three parameters R e , R m and M A ( R e : Reynolds number, R m : magnetic Reynolds number, M A : Alfvenic Mach number), for some asymptotic limits. In the case of suction, critical values of the parameters are found for which there are bifurcations in the system. Such bifurcations are pitchfork type. In the case of injection, the flux results always stable.

Fully explicit Agmon's condition for general states of a special incompressible elastic material

Agmon's condition arises as a necessary condition at the boundary for minimizers in compressible and incompressible elasticity. It is commonly formulated as a statement concerning the solution set of a family of ODEs with constant coefficients. As such, it is algebraic “in principle”. In both the compressible and incompressible cases, Agmon's condition may be recast in a more overtly algebraic form, namely the requirement that a certain family of algebraic Riccati equations (parametrized over the tangent plane) should possess positive solutions. In order to reduce Agmon's condition to a fully explicit set of inequalities involving the components of the incremental elasticity tensor, one must be able to solve the algebraic Riccati equation explicitly. Known situations where this can be done tend to involve highly symmetric states of isotropic materials. It is therefore noteworthy that Agmon's condition may be rendered explicit for any boundary-point of an arbitrarily deformed incompressible neo-Hookean body.

Compressible turbulent flows: Aspects of prediction and analysis

AbstractCompressible turbulent flows are an important element of high‐speed flight. Boundary layers developing along fuselage and wings of an aircraft and along engine compressor and turbine blades are compressible and mostly turbulent. The high‐speed flow around rockets and through rocket nozzles involves compressible turbulence and flow separation. Turbulent mixing and combustion in scramjet engines is another example where compressibility dominates the flow physics. Although compressible turbulent flows have attracted researchers since the fifties of the last century, they are not completely understood. Especially interactions between compressible turbulence and combustion lead to challenging, yet unsolved problems. Direct numerical simulation (DNS) and large‐eddy simulation (LES) represent modern powerful research tools which allow to mimic such flows in great detail and to analyze underlying physical mechanisms, even those which cannot be accessed by the experiment. The present lecture provides a short description of these tools and some of their numerical characteristics. It then describes DNS and LES results of fully‐developed channel and pipe flow and highlights effects of compressibility on the turbulence structure. The analysis of pressure fluctuations in such flows with isothermal cooled walls leads to the conclusion that the pressure‐strain correlation tensor decreases in the wall layer and that the turbulence anisotropy increases, since the mean density falls off relative to the incompressible flow case. Similar increases in turbulence anisotropy due to compressibility are observed in inert and reacting temporal mixing layers. The nature of the pressure fluctuations is however two‐facetted. While inert compressible mixing layers reveal wave‐propagation effects in the pressure and density fluctuations, compressible reacting mixing layers seem to generate pressure fluctuations that are controlled by the time‐rate of change of heat release and mean density variation as sources in the pressure Poisson equation rather than wave propagation phenomena. This opens new challenges to turbulence modellers and underlines the need to treat compressibility effects via the full Reynolds stress equations.

New Model for Compressible Vortices

A new analytical solution for self-similar compressible vortices is derived in this paper. Based on the previous incompressible formulation of intense vortices, we derived a theoretical model that includes density and temperature variations. The governing equations are simplified assuming strong vortex conditions. Part of the hydrodynamic problem (mass and momentum) is shown to be analogous to the incompressible kind and as such the velocity is obtained through a straightforward variable transformation. Since all the velocity components are bounded in the radial direction, the density and pressure are then determined by standard numerical integration without the usual stringent simplification for the radial velocity. While compressibility is shown not to affect the tangential velocity, it influences only the meridional flow (radial and axial velocities). The temperature, pressure, and density are found to decrease along the converging flow direction. The traditional homentropic flow hypothesis, often employed in vortex stability and optical studies, is shown to undervalue the density and greatly overestimate the temperature. Comparable to vorticity diffusion balance for the incompressible case, the incoming flow carries the required energy to offset the contributions of conduction, viscous dissipation, and material expansion, thus keeping the temperature steady. This model is general and can be used to obtain a compressible version for all classical previous incompressible analysis from the literature such as Rankine, Burgers, Taylor, and Sullivan vortices.

An approach for obtaining approximate formulas for the Rayleigh wave velocity

In this paper, we introduce an approach for finding analytical approximate formulas for the Rayleigh wave velocity for isotropic elastic solids and anisotropic elastic media as well. The approach is based on the least-square principle. To demonstrate its application, we applied it in order to obtain an explanation for Bergmann’s approximation, the earliest known approximation of the Rayleigh wave velocity for isotropic elastic solids, and used it to establish a new approximation. By employing this approach, the best approximate polynomials of the second order of the cubic power and the quartic power in the interval [0, 1] were found. By using the best approximate polynomial of the second order of the cubic power, we derived an approximate formula for the Rayleigh wave speed in isotropic elastic solids which is slightly better than the one given recently by Rahman and Michelitsch by employing Lanczos’s approximation. Also by using this second order polynomial, analytical approximate expressions for orthotropic, incompressible and compressible elastic solids were found. For incompressible case, it is shown that the approximation is comparable with Rahman and Michelitsch’s approximation, while for the compressible case, it is shown that our approximate formulas are more accurate than Mozhaev’s ones. Remarkably, by using the best approximate polynomials of the second order of the cubic power and the quartic power in the interval [0, 1], we derived an approximate formula of the Rayleigh wave velocity in incompressible monoclinic materials, where the explicit exact formulas of the Rayleigh wave velocity so far are not available.

Evaluation of integral forces and pressure fields from planar velocimetry data for incompressible and compressible flows

The approach to determine pressure fields and integral loads from planar velocimetry data is discussed, in relation to the implementation for incompressible and compressible flows around two-dimensional objects. The method relies upon the application of control-volume approaches in combination with the deduction of the pressure field from the experimental data, by making use of the flow constitutive equations. In this paper the implementation for two specific application areas is addressed. The first is time-mean pressure field and force evaluation from velocity ensemble statistics, as obtained from time-uncorrelated PIV acquisition, for incompressible flow. Two test cases are considered for this flow regime: the unsteady vortical flow around a square section cylinder at incidence, as well as the force characterization of a low-speed airfoil. The second topic considers the extension of the method to steady compressible flow, with the supersonic flow around a bi-convex airfoil as experimental test case. As in this flow regime the density appears as an extra unknown in the momentum equation, additional flow equations need to be invoked. A convenient approach for this was found, using the gas law and the adiabatic flow condition, with which the pressure-integration procedure becomes essentially the same as for the incompressible case.

Theory of residual stresses with application to an arterial geometry

This paper presents a theory of residual stresses, with applications to biomechanics, especially to arteries. For a hyperelastic material, we use an initial local deformation tensor K as a descriptor of residual strain. This tensor, in general, is not the gradient of a global deformation, and a stress-free reference configuration, denoted B - , therefore, becomes incompatible. Any compatible reference configuration B 0 will, in general, be residually stressed. However, when a certain curvature tensor vanishes, there actually exists a compatible and stress-free configuration, and we show that the traditional treatment of residual stresses in arteries, using the opening–angle method, relates to such a situation. Boundary value problems of nonlinear elasticity are preferably formulated on a fixed integration domain. For residually stressed bodies, three such formulations naturally appear: (i) a formulation relating to B 0 with a non-Euclidean metric structure; (ii) a formulation relating to B 0 with a Euclidean metric structure; and (iii) a formulation relating to the incompatible configuration B - . We state these formulations, show that (i) and (ii) coincide in the incompressible case, and that an extra term appears in a formulation on B - , due to the incompatibility.

Gaseous slip flow analysis of a micromachined flow sensor for ultra small flow applications

The velocity slip of a fluid at a wall is one of the most typical phenomena in microscale gas flows. This paper presents a flow analysis considering the velocity slip in a capacitive micro gas flow sensor based on pressure difference measurements along a microchannel. The tangential momentum accommodation coefficient (TMAC) measurements of a particular channel wall in planar microchannels will be presented while the previous micro gas flow studies have been based on the same TMACs on both walls. The sensors consist of a pair of capacitive pressure sensors, inlet/outlet and a microchannel. The main microchannel is 128.0 µm wide, 4.64 µm deep and 5680 µm long, and operated under nearly atmospheric conditions where the outlet Knudsen number is 0.0137. The sensor was fabricated using silicon wet etching, ultrasonic drilling, deep reactive ion etching (DRIE) and anodic bonding. The capacitance change of the sensor and the mass flow rate of nitrogen were measured as the inlet-to-outlet pressure ratio was varied from 1.00 to 1.24. The measured maximum mass flow rate was 3.86 × 10−10 kg s−1 (0.019 sccm) at the highest pressure ratio tested. As the pressure difference increased, both the capacitance of the differential pressure sensor and the flow rate through the main microchannel increased. The laminar friction constant f ⋅ Re, an important consideration in sensor design, varied from the incompressible no-slip case and the mass sensitivity and resolution of this sensor were discussed. Using the current slip flow formulae, a microchannel with much smaller mass flow rates can be designed at the same pressure ratios.

Non-Riemannian geometry of twisted flux tubes

New examples of the theory recently proposed by Ricca [PRA(1991)] on the generalization of Da Rios-Betchov intrinsic equations on curvature and torsion of classical non-Riemannian vortex higher-dimensional string are given. In particular we consider applications to 3-dimesional fluid dynamics, including the case of a twisted flux tube and the fluid rotation. In this case use is made of Da Rios equation to constrain the fluid. Integrals on the Cartan connection are shown to be related to the integrals which represent the total Frenet torsion and total curvature. By analogy with the blue phases twisted tubes in liquid crystals, non-Riemannian geometrical formulation of the twisted flux tube in fluid dynamics is obtained. A theorem by Ricca and Moffatt on invariant integrals for the Frenet curvature is used to place limits on the Cartan integrals. The stationary incompressible flow case is also addressed in the non-Riemannian case where Cartan torsion scalars are shown to correspond to abnormalities of the congruence. Geodesic motion is shown to be torsionless. Vorticity is shown to be expressed in terms of abnormalities of the congruence, which is analogous to the result recenly obtained [Garcia de Andrade,PRD(2004)], where the vorticity of the superfluid plays the role of Cartan contortion vector in the context of analog gravity.

A limit problem in natural convection

We discuss a model limit problem which arises as a first step in the mathematical justification of our Boussinesq-type approximation (4), which takes into account dissipative heating in natural convection. We treat a simplified highly non linear system depending on a (perturbation) para- meter e. The main difficulty is that for e = 0 the velocity is not solenoidal. First we prove that our system has weak solutions for each fixed e. More- over, while the chosen perturbation parameter e tends to zero we show, that we arrive at the usual incompressible case and the standard Boussinesq approximation.

Macroscopic theory of multicomponent flows: Irreversibility and well-posed equations

A theory of the coupling of N systems of Navier–Stokes equations is presented. A set of coupled Navier–Stokes systems contains the macroscopic equations of motion for flows of N different material components or phases. Six postulates are formulated from which three Navier–Stokes sets are derived for arbitrary N . The main postulate concerns the second law of thermodynamics, which the present sets satisfy for arbitrary N . Appropriate averages of velocities and pressures distribute the entropy production of each irreversible interaction process among the interacting components. Increase of entropy is guaranteed if the interaction terms are suitably constructed from positive definite operators. A strictly irreversible mass exchange law, expressed in the difference of Gibbs free energies of the exchanging components, is proposed. Furthermore, the characteristic velocities of each Navier–Stokes set are investigated, since their realness is required for well-posedness of the equations. For both Navier–Stokes sets with multiple pressures, the characteristic velocities are proven to be real for arbitrary N . The canonical problem of complex characteristics of incompressible multiphase flows with a single pressure is revisited. A solution to this problem is found by generalizing the standard algebraic Stokes drag law to a positive definite operator acting on the velocity difference. In this way, effects of forces due to nonuniform velocity fields can be incorporated. A generalized drag law is specified by the inclusion of m -th order spatial derivatives of the velocity differences between two components ( m ≥ 2 ). The characteristic polynomial is derived for an arbitrary number of components and its roots, the characteristic velocities, appear to be real. This is rigorously proven for two- and three-phase flows, and conjectured to be true for flows with four or more components. The linear stability of the incompressible one-dimensional case is also investigated. Ill-posedness of the standard case leads to an infinite growth rate for infinitely short waves. In the case of generalized drag, this growth rate becomes finite for even and neutral for odd m .

Reynolds-Averaged Navier-Stokes/Large-Eddy Simulations of Supersonic Base Flow

Several zonal and nonzonal hybrid Reynolds-averaged Navier-Stokes/large-eddy simulation approaches have been assessed to handle a high Reynolds number supersonic base flow. The results obtained on 5 and 13.5 x 10 6 points grids are compared to the available experimental data (Herrin, J. L., and Dutton, J. C., Supersonic Base Flow Experiments in the Near Wake of a Cylindrical Afterbody,AIAA Journal, Vol.32, No. 77,1994), and the capabilities of these different methodologies to predict supersonic flows are discussed. The highly compressible separated shear layer proved to be a challenging issue for hybrid methods due to an alteration in the instability process (as compared to the incompressible case), leading to three-dimensional coherent structures. Numerous numerical parameters relevant to hybrid methods have been assessed. The incoming boundary layer thickness needs to be properly set, while a weak influence of the subgrid scale model is observed when small-scale structures in the separating mixing layer are resolved. Another finding of the present study is the dramatic influence displayed by the numerical dissipation on the flowfield. Sensitivity to the C DES model constant is observed even with the finest grid, leading to a delay in the generation of instabilities in the shear layer.

Self-similar solutions to the isothermal compressible Navier–Stokes equations

We investigate the self-similar solutions to the isothermal compressible Navier-Stokes equations. The aim of this paper is to show that there exist neither forward nor backward self-similar solutions with finite total energy. This generalizes the results for the incompressible case in Necas, J., Ruzicka, M. & Sverak, V. (1996, On Leray's self-similar solutions of the Navier-Stokes equations. Acta. Math., 176, 283-294), and is consistent with the (unproved) existence of regular solutions globally in time for the compressible Navier-Stokes equations.

Elasticity solution for an axially compressed and encased cylindrical specimen

We present the solution of the linear elasticity equations governing the deformation of an elastic cylinder encased in a tube and subjected to uniform compression on the flat ends. The solutions for the stresses, strains, and displacements in the encased and compressed cylinder are all systematically determined from the basic solution of Lamé's classical elasticity problem of the long tube subjected to internal and external pressures. We first derive the general elastostatic analysis for an encased hollow cylinder, stress-free at the cavity, and later particularize the solution to a solid cylindrical specimen. The effective modulus E eff of the encased sample is found to be a function of the bulk modulus k and Poisson's ratio ν of the material. E eff differs from k except for nearly incompressible materials, where E eff approaches the bulk modulus value. In the incompressible case, we also show how a load applied on the cylinder's flat ends is equivalent to, and can be replaced by, the same load acting on the curved surface. For compressible materials, a more general expression for E eff is found that also accounts for the case deformation. These results explain the deformation of an axially compressed and encased cylindrical specimen tested in compressibility measuring devices such as those described by Matsuoka and Maxwell [Response of linear high polymers to hydrostatic pressure. Journal of Polymer Science 1958; 32:131–59]. The present analysis thus contributes to a better understanding of how this device works and to the interpretation of measurements taken with it.

Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at M=2.25

The interaction of a spatially developing adiabatic boundary layer flow at M∞=2.25 and Reθ=3725 with an impinging oblique shock wave (β=33.2°) is analyzed by means of direct numerical simulation of the compressible Navier-Stokes equations. Under the selected flow conditions the incoming boundary layer undergoes mild separation due to the adverse pressure gradient. Coherent structures are shed near the average separation point and the flow field exhibits large-scale low-frequency unsteadiness. The formation of the mixing layer is primarily responsible for the amplification of turbulence, which relaxes to an equilibrium state past the interaction. Complete equilibrium is attained in the inner part of the boundary layer, while in the outer region the relaxation process is incomplete. Far from the interaction zone, turbulence exhibits a universal behavior and it shows similarities with the incompressible case. The interaction of the coherent structures with the incident shock produces acoustic waves that propagate upstream, thus inducing an oscillatory motion of the separation bubble and a subsequent flapping motion of the reflected shock. The simulation indicates the occurrence of low-frequency tones in the interaction zone associated with peaks in the pressure spectra at discrete frequencies. We propose that such large-scale low-frequency unsteadiness is due to a resonance mechanism that establishes in the interaction region, and which has close similarities with those responsible for the generation of tones in cavity flows and screeching jets. In order to support our claim, we develop a simplified model for the acoustic resonance that is capable to predict the characteristic frequencies of the tones.