Nonlinear waves in incompressible viscoelastic Maxwell medium

In this paper we study two-dimensional flows of incompressible viscoelastic Maxwell media with Jaumann corotational derivative in the rheological constitutive law. In the general case, due to the incompressibility condition, the equations of motion have both real and complex characteristics. Group properties of this system are studied. On this basis, two submodels of the Maxwell model are selected, which can be reduced to hyperbolic ones. More precisely, we consider plane shear flow between two parallel planes and Couette type flow caused by the inertial cylinder rotation. As a result, we obtain the closed systems of three equations of mixed type, which describe nonlinear transverse waves in an incompressible Maxwell fluid. It is demonstrated that discontinuities can develop in elastic media even from smooth initial data. Stability of shocks in the Maxwell fluid with and without retardation time is discussed.     

On shock-capturing schemes using artificial wind

We introduce and apply a new way for the construction of efficient non-oscillatory shock-capturing schemes for fluid dynamic and magneto-hydro-dynamic simulations. The basic idea is to solve the governing equations in different steadily moving frames of reference chosen in such a way that the flow would be supersonic there resulting in simple upwind formulas for fluxes across control volume faces. An extra velocity (artificial wind) is added to the velocity of the flow under simulation when the system of coordinates is changed. The approach allows to simplify existing schemes and to get new modifications. Test problems demonstrate that the derived schemes provide accurate results while being simple and efficient. Received: 21 May 1999 / Accepted 10 September 1999     

Numerical studies of slow viscous rotating flow past a sphere. III

The flow of steady incompressible viscous fluid rotating about the z-axis with angular velocity ω and moving with velocity u past a sphere of radius a which is kept fixed at the origin is investigated by means of a numerical method for small values of the Reynolds number Re_ω. The Navier–Stokes equations governing the axisymmetric flow can be written as three coupled non-linear partial differential equations for the streamfunction, vorticity and rotational velocity component. Central differences are applied to the partial differential equations for solution by the Peaceman–Rachford ADI method, and the resulting algebraic equations are solved by the ‘method of sweeps’. The results obtained by solving the non-linear partial differential equations are compared with the results obtained by linearizing the equations for very small values of Re_ω. Streamlines are plotted for Ψ = 0·05, 0·2, 0·5 for both linear and non-linear cases. The magnitude of the vorticity vector near the body, i.e. at z = 0·2, is plotted for Re_ω = 0·05, 0·24, 0·5. The correction to the Stokes drag as a result of rotation of the fluid is calculated.     

Decay of vortex velocity and diffusion of temperature in a generalized second grade fluid

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid was introduced. The velocity and temperature fields of the vortex flow of a generalized second fluid with fractional derivative model were described by fractional partial differential equations. Exact analytical solutions of these differential equations were obtained by using the discrete Laplace transform of the sequential fractional derivatives and generalized Mittag-Leffier function. The influence of fractional coefficient on the decay of vortex velocity and diffusion of temperature was also analyzed.     

Assessment of a high-order accurate Discontinuous Galerkin method for turbomachinery flows

In this work the capabilities of a high-order Discontinuous Galerkin (DG) method applied to the computation of turbomachinery flows are investigated. The Reynolds averaged Navier–Stokes equations coupled with the two equations k-ω turbulence model are solved to predict the flow features, either in a fixed or rotating reference frame, to simulate the fluid flow around bodies that operate under an imposed steady rotation. To ensure, by design, the positivity of all thermodynamic variables at a discrete level, a set of primitive variables based on pressure and temperature logarithms is used. The flow fields through the MTU T106A low-pressure turbine cascade and the NASA Rotor 37 axial compressor have been computed up to fourth-order of accuracy and compared to the experimental and numerical data available in the literature.     

Flow of power-law fluids over a moving wedge surface with wall mass injection

The boundary layer problem of a power-law fluid flow with fluid injection on a wedge whose surface is moving with a constant velocity in the opposite direction to that of the uniform mainstream is analyzed. The free stream velocity, the injection velocity at the surface, moving velocity of the wedge surface, the wedge angle and the power law index of non-Newtonian fluid are assumed variables. The fourth order Runge–Kutta method modified by Gill is used to solve the non-dimensional boundary layer equations for non-Newtonian flow field. Without fluid injection, for every angle of wedge β, a limiting value for velocity ratio λ _cr (velocity of the wedge surface/velocity of the uniform flow) is found for each power-law index n. The value of λ _cr increases with the increasing wedge angle β. The value of wedge angle also restricts the physical characteristics of the fluid to be used. The effects of the different parameters on velocity profile and on skin friction are studied and the drag reduction is discussed. In case of C = 2.5 and velocity ratio λ = 0.2 for wedge angle β = 0.5 with the fluid with power law-index n = 0.5, 48.8% drag reduction is obtained.     

关于时空标架变换和物质标架无关性原理

建立了欧几里德标架无关性的一般表述与某一参考标架\phi下作相对刚体运动的一般曲线坐标、分量表述之间的关系,从而使欧几里德标架无关性的张量表述有相应的分量表述形式.建立这种联系的基础是, 参考标架\phi^\ast 下描述的运动在参考标架\phi 中某坐标系下的分量结果, 对应于\phi下另一相对该坐标系作相同刚体运动的坐标系下描述\phi观测的运动的分量结果. 通过例子说明了Truesdell和Noll用等价动力过程,关于物质标架无关性原理的陈述,比Svendsen和Bertram及Liu的本构关系在参考标架变换下的形式不变性更合理且方便,尤其是对微分方程形式的本构关系更是如此.      

An implicit algorithm for hypoelasto-plastic and hypoelasto-viscoplastic endochronic theory in finite strain isotropic–kinematic-hardening model

This paper is concerned with objective stress update algorithm for elasto-plastic and elasto-viscoplastic endochronic theory within the framework of additive plasticity. The elastic response is stated in terms of hypoelastic model and endochronic constitutive equations are stated in unrotated frame of reference. A trivially incrementally objective integration scheme for rate constitutive equations is established. Algorithmic modulus consistent with numerical integration algorithm of constitutive equations is extracted. The implementation is validated by means of a set of simple deformation paths (simple shear, extension and rotation), two benchmark test in nonlinear mechanics (the necking of a circular bar and expansion of a thick-walled cylinder), a test which demonstrates the capabilities of the proposed model in simulation of cyclic loading and ratcheting in finite strain case (cyclically loaded notched bar) and finally, the analysis of a tensile test, which presents a shear band with a finite thickness independent of the finite element mesh using endochronic viscoplastic constitutive model.     

An Analysis of Journal Orbits for Nonlinear Dynamic Bearing Systems

An analysis of journal centre orbits is presented in this paper based on a non-isothermal non-Newtonian fluid model for dynamically loaded bearing systems. A spectral element approach is used to solve a full set of coupled equations (kinematics and constitutive) governing the flow of the lubricant, and an operator-splitting spectral element technique is used to evaluate the dynamic energy equation. The motion of the journal is calculated on the basis of Newtonian mechanics incorporated with a simple cavitation model. The stability of the journal orbits is investigated under a wide range of the rotation speeds of journal. The unstable orbits arise as a sub-harmonic motion when the journal rotation speed is increased beyond a critical value. The influences of the oscillation speeds of the applied loads on the journal orbits are examined. The numerical simulations demonstrate that both the rotation speed of the journal and the oscillation speed of the applied load play an important role in determining the pattern of the journal orbits. The effects of square-wave and rotating applied loads on the journal orbits are also investigated. Received 22 April 1998 and accepted 26 May 1999     

A non-ordinary state-based peridynamic method to model solid material deformation and fracture

In this paper, we develop a new non-ordinary state-based peridynamic method to solve transient dynamic solid mechanics problems. This new peridynamic method has advantages over the previously developed bond-based and ordinary state-based peridynamic methods in that its bonds are not restricted to central forces, nor is it restricted to a Poisson’s ratio of 1/4 as with the bond-based method. First, we obtain non-local nodal deformation gradients that are used to define nodal strain tensors. The deformation gradient tensors are used with the nodal strain tensors to obtain rate of deformation tensors in the deformed configuration. The polar decomposition of the deformation gradient tensors are then used to obtain the nodal rotation tensors which are used to rotate the rate of deformation tensors and previous Cauchy stress tensors into an unrotated configuration. These are then used with conventional Cauchy stress constitutive models in the unrotated state where the unrotated Cauchy stress rate is objective. We then obtain the unrotated Cauchy nodal stress tensors and rotate them back into the deformed configuration where they are used to define the forces in the nodal connecting bonds. As a first example we quasi-statically stretch a bar, hold it, and then rotate it ninety degrees to illustrate the methods finite rotation capabilities. Next, we verify our new method by comparing small strain results from a bar fixed at one end and subjected to an initial velocity gradient with results obtained from the corresponding one-dimensional small strain analytical solution. As a last example, we show the fracture capabilities of the method using both a notched and un-notched bar.     

A new particle tracking algorithm based on deterministic annealing and alternative distance measures

We describe a new particle tracking algorithm for the interrogation of double frame single exposure data, which is obtained with particle image velocimetry. The new procedure is based on an algorithm which has recently been proposed by Gold et al. (Gold et al., 1998) for solving point matching problems in statistical pattern recognition. For a given interrogation window, the algorithm simultaneously extracts: (i) the correct correspondences between particles in both frames and (ii) an estimate of the local flow-field parameters. Contrary to previous methods, the algorithm determines not only the local velocity, but other local components of the flow field, for example rotation and shear. This makes the new interrogation method superior to standard methods in particular in regions with high velocity gradients (e.g. vortices or shear flows). We perform benchmarks with three standard particle image velocimetry (PIV) and particle tracking velocimetry (PTV) methods: cross-correlation, nearest neighbour search, and image relaxation. We show that the new algorithm requires less particles per interrogation window than cross-correlation and allows for much higher particle densities than the other PTV methods. Consequently, one may obtain the velocity field at high spatial resolution even in regions of very fast flows. Finally, we find that the new algorithm is more robust against out-of-plane noise than previously proposed methods. Received: 1 March 1999 / Accepted: 29 July 1999     

On the Mathematical Modelling of a Compressible Viscoelastic Fluid

Thermodynamical considerations have largely been avoided in the modelling of complex fluids by invoking the assumption of incompressibility. This approximation allows pressure to be defined as a Lagrange multiplier, and therefore its natural connection with other thermodynamic variables such as density and temperature is irretrievably lost. Relaxing this condition to allow more realistic modelling involves much more than prescribing an equation of state. Even for a simple isothermal viscoelastic model, as explored in this paper, the transition to a compressible model is non-trivial. This paper shows that pressure enters the governing equations in a non-intuitive way. Furthermore, a fluid volume element, which is no longer constant, radically changes the way the basic element of the constitutive equations is viewed—stress is no longer the fundamental constitutive link between the momentum equations and velocity. The importance of geometry in fluid modelling is emphasised through the use of the Lie derivative, which is of a more fundamental character than the “upper” and “lower” convected derivatives prevalent in the literature and which are found to be almost redundant for a compressible fluid. There is now a strong body of non-equilibrium thermodynamics theory for flowing systems, which proves indispensible for this development. These fundamental principles are described herein using methodology and examples, that are sometimes conflicting, from the literature. The main conflict arises from the relationship between thermodynamic pressure and the trace of Cauchy stress, where the current preferred choice is (up to a constant) to set them equal—this is shown to be incorrect. Other issues such as the dependence of viscosity on density, bulk viscosity, integral modelling, the principle of objectivity and convected derivatives, are also clarified and resolved.     

Transonic Euler computation in streamfunction co-ordinates

Numerical simulation by a finite element method is used to examine the problem of the rotating flow of a viscoelastic fluid in a cylindrical vessel agitated with a paddle impeller. The mathematical model consists of a viscoelastic constitutive equation of Oldroyd B type coupled to the hydrodynamic equations expressed in a rotating frame. This system is solved by using an unsteady approach for velocity, pressure and stress fields. For Reynolds numbers in the range 0.1–10, viscoelastic effects are taken into account up to a Deborah number De of 1.33 and viscoelasticity and inertia cross-effects are studied. Examining the velocity and stress fields as well as the power consumption, it is found that their evolutions are significantly different for low and moderate inertia. These results confirm the trends of experimental studies and show the specific contribution of elasticity without interference of the pseudoplastic character found in actual fluids.     

Modelling of two‐dimensional laminar flow using finite element method

In this paper, a Galerkin weighted residual finite element numerical solution method, with velocity material time derivative discretisation, is applied to solve for a classical fluid mechanics system of partial differential equations modelling two‐dimensional stationary incompressible Newtonian fluid flow. Classical examples of driven cavity laminar flow and laminar flow past a cylinder are presented. Numerical results are compared with data found in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite element computation of torsional plastic waves in a thin-walled tube

Summary  A finite element technique is presented for the analysis of one-dimensional torsional plastic waves in a thin-walled tube. Three different nonlinear consitutive relations deduced from elementary mechanical models are used to describe the shear stress–strain characteristics of the tube material at high rates of strain. The resulting incremental equations of torsional motion for the tube are solved by applying a direct numerical integration technique in conjunction with the constitutive relations. The finite element solutions for torsional plastic waves in a long copper tube subjected to an imposed angular velocity at one end are given, and a comparison with available experimental results to assess the accuracy of the constitutive relations considered is conducted. It is demonstrated that the strain-rate dependent solutions show a better agreement with the experimental results than the strain-rate independent solutions. The limitations of the constitutive equations are discussed, and some modifications are suggested. Received 9 February 1999; accepted for publication 28 March 2000     

Extended thermodynamics of molecular ideal gases

The objective of extended thermodynamics of molecular ideal gases is the determination of the 17 fields ofmass density, velocity, energy density, pressure deviator, heat flux, intrinsic energy density and intrinsic heat flux. The intrinsic energy represents the rotational or the vibrational energy of the molecules. The necessary field equations are based upon balance laws and the system of equations is closed by constitutive relations which are characteristic for the gas under consideration. The generality of the constitutive relations is restricted by theprinciple of material frame indifference, and by the entropy principle. These principles reduce the constitutive coefficients of all fluxes to the thermal and caloric equation of state of the gas and provide inequalities for the transport coefficients. The transport coefficients can be related to the shear viscosity, the heat conductivity, and the coefficients of self-diffusion and attenuation of sound waves, so that the field equations become quite specific. The theory is in perfect agreement with the kinetic theory of molecular gases. It is shown that in non-equilibrium the temperature is discontinuous at thermometric walls. The dynamic pressure and the volume viscosity, are discussed and it is shown how extended thermodynamics and ordinary thermodynamics are related.