Numerical simulation of three-dimensional duct flows of incompressible fluids by using the stream-tube method

Numerical simulation of three-dimensional flows generally involves solving large-scale problems. In this paper we consider the stream-tube method in three-dimensional duct flows. The analysis uses the concept of stream tubes in a mapped computational domain of the physical domain where the streamlines are parallel and straight. The incompressibility equation is automatically verified and the primary unknowns of the problem are, in addition to the pressure, the transformation functions between the two domains. It is also shown that the flow may be computed by considering successive subdomains (the stream tubes). This results in a reduction of computing time and storage area. Incompressible viscous and elastic liquids involving memory-integral equations may be considered in the flow simulations. This part of the paper examines three-dimensional flows of Newtonian fluids. The method is applied to the flow in a duct involving a threefold rotational symmetry, where the discretized relevant equations are solved by using the Levenberg-Marquardt algorithm.     

Particle and bubble dynamics in a creeping flow

Dynamics of a solid particle and non-deformable gaseous bubble in viscous fluid are studied analytically and numerically within the framework of creeping flow regime (flow at vanishingly small Reynolds numbers). Equations of motion for the particle and bubble include the consideration of the buoyancy force, Stokes drag force and memory-integral drag force. Exact analytical solutions are obtained and categorised in terms of inclusion (particle or bubble) density with respect to the density of a surrounding fluid. Through the analytical and numerical solutions, the dynamics of solid particle and air bubble in water have been found to behave differently especially at the early stages of motion, whereas some qualitative similarities exist in the long-term asymptotic.     

Modelization of extensional experiments under constant force using memory-integral constitutive equations and stream-tube analysis

In this paper, we propose a numerical simulation of axisymmetric extensional experiments on a viscoelastic polydimethylsiloxane (PDMS) material, using a falling-weight extensional rheometer. The polymer behaviour is represented by a K-BKZ memory-integral constitutive equation, involving a damping function of the Wagner type. Under the assumption of a homogeneous flow zone in the sample, a numerical model is set up, using the stream-tube method and approximating functions. The governing equations of the problem, associated to a limited number of unknowns, are solved by means of the Levenberg-Marquardt optimization algorithm. The numerical results are found to be consistent with the experimental data and reveal the importance of the non-homogeneous flow zone, in relation to the estimation of the extensional strain rate. The calculations involve the sensitivity of the model on the fluid parameters and those concerning the size of the initial column of fluid. The limited computing (CPU) time of the code is also to be underlined.     

Correcting the initialization of models with fractional derivatives via history-dependent conditions

Fractional differential equations are more and more used in modeling memory(history-dependent,nonlocal,or hereditary) phenomena.Conventional initial values of fractional differential equations are define at a point,while recent works defin initial conditions over histories.We prove that the conventional initialization of fractional differential equations with a Riemann–Liouville derivative is wrong with a simple counter-example.The initial values were assumed to be arbitrarily given for a typical fractional differential equation,but we fin one of these values can only be zero.We show that fractional differential equations are of infinit dimensions,and the initial conditions,initial histories,are define as functions over intervals.We obtain the equivalent integral equation for Caputo case.With a simple fractional model of materials,we illustrate that the recovery behavior is correct with the initial creep history,but is wrong with initial values at the starting point of the recovery.We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.     

Small Reynolds number instabilities in two-layer Couette flow

Instabilities in two-layer Couette flow are investigated from a small Reynolds number expansion of the eigenvalue problem governing linear stability. The wave velocity and growth rate are given explicitly, and previous results for long waves and short waves are retrieved as special cases. In addition to the inertial instability due to viscous stratification, the flow may be subject to the Rayleigh–Taylor instability. As a result of the competition of these two instabilities, inertia may completely stabilise a gravity-unstable flow above some finite critical Froude number, or conversely, for a gravity-stable flow, inertia may give rise to finite wavenumber instability above some finite critical Weber number. General conditions for these phenomena are given, as well as exact or approximate values of the critical numbers. The validity domain of the many asymptotic expansions is then investigated from comparison with the numerical solution. It appears that the small-Re expansion gives good results beyond Re = 1, with an error less that 1%. For Reynolds numbers of a few hundred, we show from the eigenfunctions and the energy equation that the nature of the instability changes: instability still arises from the interfacial mode (there is no mode crossing), but this mode takes all the features of a shear mode. The other modes correspond to the stable eigenmodes of the single-layer Couette flow, which are recovered when one fluid is rigidified by increasing its viscosity or surface tension.     

Validation of adaptive unstructured hexahedral mesh computations of flow around a wind turbine airfoil

In this paper we investigate local adaptive refinement of unstructured hexahedral meshes for computations of the flow around the DU91 wind turbine airfoil. This is a 25% thick airfoil, found at the mid‐span section of a wind turbine blade. Wind turbine applications typically involve unsteady flows due to changes in the angle of attack and to unsteady flow separation at high angles of attack. In order to obtain reasonably accurate results for all these conditions one should use a mesh which is refined in many regions, which is not computationally efficient. Our solution is to apply an automated mesh adaptation technique. In this paper we test an adaptive refinement strategy developed for unstructured hexahedral meshes for steady flow conditions. The automated mesh adaptation is based on local flow sensors for pressure, velocity, density or a combination of these flow variables. This way the mesh is refined only in those regions necessary for high accuracy, retaining computational efficiency. A validation study is performed for two cases: attached flow at an angle of 6° and separated flow at 12°. The results obtained using our adaptive mesh strategy are compared with experimental data and with results obtained with an equally sized non‐adapted mesh. From these computations it can be concluded that for a given computing time, adapted meshes result in solutions closer to the experimental data compared to non‐adapted meshes for attached flow. Finally, we show results for unsteady computations. Copyright © 2005 John Wiley & Sons, Ltd.     

Three-dimensional extrudate swell experimental and numerical study of a polyethylene melt obeying a memory-integral equation

The present work deals with the experimental and numerical features of the flow of a linear low-density polyethylene melt (LLDPE) at 160°C at the exit of a die of square cross-section. The rheological properties of the fluid are fitted by a Wagner's memory-integral constitutive equation. The characteristics of the extrudate jet are determined by optical means at different flow rates. The stream-tube analysis, already applied to two-dimensional extrudate swell problems involving rate and integral constitutive equations, is considered to simulate the flow field. The method avoids particle tracking problems related to integral models and allows computation of the unknown free surface by considering only a `peripheral stream tube' limited by the wall and the jet surface and an inner stream surface. Those boundary surfaces are determined by considering the conservation equations together with boundary condition equations, solved by the Levenberg–Marquardt optimization algorithm. The method leads to a considerable reduction in the number of degrees of freedom and the storage area. The numerical results are found to be generally consistent with the experimental data and highlight the growing importance of stress peaks due to the singularity at the exit when the flow rate increases.     

A comparative study of poiseuille flow of a polar fluid under various boundary conditions with applications to blood flow

In this paper, Poiseuille flow of a polar fluid (model of a red blood cell suspension) under various boundary conditions at the wall, viz., slip or no-slip in the axial velocity and couple stresses zero or non-zero at the boundary, is considered from the point of view of its applications to blood flow. Analytic expressions for axial and rotational velocities, flow rate, effective viscosity and stresses are obtained. The magnitudes of the length ratioL and the coupling number N are determined in accordance with concentration and tube radius (in the existing literature, values ofL andN are chosen arbitrarily). Velocity profiles (both axial and rotational) and the variation of the effective viscosity with concentration, tube radius and for various values of the boundary condition parameters are shown graphically. The analytic results obtained are compared with experimental results (for blood flow). It is found that they are in a reasonably good agreement. The effective viscosity exhibits the Inverse Fahraeus-Lindquist Effect in all the cases (including the slip or no-slip in the velocity fields). A method is given for determining the non-zero couple stress boundary condition for a given concentration. Applications of this theory to blood flow are briefly discussed.     

Approximate subsonic gas flows under assigned boundary conditions

Summary The solution to compressible flow problems under fully assigned boundary conditions is discussed. It is shown that Schwarz's results on minimal surfaces can be immediately applied for two-dimensional flow, and several special cases and examples are given. Extensions of these results provide certain particular types of three-dimensional flow.     

Necessary and sufficient conditions for the stability of flows of incompressible viscous fluids

Summary We perturb a steady flow of an incompressible viscous fluid and derive a necessary and sufficient condition for the marginal cases for monotonie energy stability and stability against small (infinitesimal) disturbances to coincide. Evaluation of this condition in two examples singles out, in terms of the parameters of the problem, the cases where necessary and sufficient conditions for stability coincide and thus the steady flow first becomes unstable, together with the class of perturbations responsible for the instability. The analysis is done within the range of strict solutions of each underlying problem; the precise regularity and existence classes are given in Sec. 0. The examples we treat are plane parallel shear flow with a non-symmetric profile in an infinite rotating layer and the effect of rotation on convection.     

Dynamic non-shakedown theorem for elastic perfectly-plastic continua

Sufficient and necessary conditions of a kinematic nature are established for alternating plasticity or incremental collapse (i.e. inadaptation or non-shakedown) of elastic perfectly-plastic media subjected to given histories of loads and thermal strains, in the presence of significant inertia and viscous damping forces. The classical second shakedown theorem, due to W.T. Koiter, is thus extended to the dynamic range.     

A representation of the solution of the Gromeko problem for viscous and elastico-viscous fluids

The solution of the Gromeko problem [1] on unsteady flow of a viscous fluid in a long circular pipe is among the few exact solutions of the Navier-Stokes equations. Its effective solution is obtained only when the longitudinal pressure gradient is given as an arbitrary time function. However, in practice we encounter cases when the flow rate is a known time function. This sort of problem arises, in particular, in rheological experiments using viscometers with a given flow rate. In this case the determination of the pressure gradient from the given flow rate leads in the general case to a very unwieldy expression. Below we present an effective solution of this problem for viscous and elasticoviscous media using the method of solving the inlet flow problem for a steady flow of a viscous fluid in a semi-infinite pipe. It is shown that for the case of a viscous fluid these two problems are actually equivalent.     

A semi-implicit lagrangian scheme for 3D shallow water flow with a two-layer turbulence model

A semi-implicit Lagrangian finite difference scheme for 3D shallow water flow has been developed to include an eddy viscosity model for turbulent mixing in the vertical direction. The α-co-ordinate system for the vertical direction has been introduced to give accurate definition of bed and surface boundary conditions. The simple two-layer mixing length model for rough surfaces is used with the standard assumption that the shear stress across the wall region at a given horizontal location is constant. The bed condition is thus defined only by its roughness height (avoiding the need for a friction formula relating to depth-averaged flow, e.g. Chezy, used previously). The method is shown to be efficient and stable with an explicit Lagrangian formulation for convective terms and terms for surface elevation and vertical mixing handled implicitly. The method is applied to current flow around a circular island with gently sloping sides which produce periodic recirculation zones (vortex shedding). Comparisons are made with experimental measurements of velocity using laser Doppler anemometry (time histories at specific points) and surface particle-tracking velocimetry.     

Constitutive equations for polymer solutions derived from the bead/spring model of Rouse and Zimm

Rheologica Acta - The theory of dilute polymer solutions based on the bead/spring model in the form given by Zimm is reformulated for arbitrary homogeneous flow histories. It is shown that the...     

Weak Local Minimizers in Finite Elasticity

This paper deals with necessary conditions and sufficient conditions for a weak local minimum of the energy of a hyperelastic body. We consider anisotropic bodies of arbitrary shape, subject to prescribed displacements on a given portion of the boundary. As an example, we consider the uniaxial stretching of a cylinder, in the two cases of compressible and incompressible material. In both cases we find that there is a continuous path across the natural state, made of local energy minimizers. For the Blatz-Ko compressible material and for the Mooney-Rivlin incompressible material, explicit estimates of the minimizing path are given and compared with those available in the literature. Dedicated to the memory of Victor J. Mizel.     

Some uses of Green's theorem in solving the Navier–Stokes equations

This paper gives a review of methods where Green's theorem may be employed in solving numerically the Navier–Stokes equations for incompressible fluid motion. They are based on the concept of using the theorem to transform local boundary conditions given on the boundary of a closed region in the solution domain into global, or integral, conditions taken over it. Two formulations of the Navier–Stokes equations are considered: that in terms of the streamfunction and vorticity for two-dimensional motion and that in terms of the primitive variables of the velocity components and the pressure. In the first formulation overspecification of conditions for the streamfunction is utilized to obtain conditions of integral type for the vorticity and in the second formulation integral conditions for the pressure are found. Some illustrations of the principle of the method are given in one space dimension, including some derived from two-dimensional flows using the series truncation method. In particular, an illustration is given of the calculation of surface vorticity for two-dimensional flow normal to a flat plate. An account is also given of the implementation of these methods for general two-dimensional flows in both of the mentioned formulations and a numerical illustration is given.     

A generalized Brinkman number for non-Newtonian duct flows

When viscous dissipation effects are important in duct flows the Brinkman number is widely used to quantify the relationship between the heat generated by dissipation and the heat exchanged at the wall. For Newtonian laminar fully developed pipe flow the use of the classical expression for this dimensionless group is appropriate, but under different conditions it can lead to misleading conclusions, such as when comparing flows through different cross-section ducts, flow regimes and mainly non-Newtonian flows. In this work a generalized Brinkman number is proposed, based on an energy balance for the power dissipated by friction, that allows proper quantification of viscous heating effects and reduces to the classical definition in laminar Newtonian pipe flow. The advantages of the new definition are shown and expressions are given for generalized Brinkman numbers in the most common cases.     

A hypersonic chemically nonequilibrium viscous shock layer oh wings with a catalytic surface

Within the framework of the theory of a hypersonic viscous shock layer a study is made of flow round wings of infinite span with blunt leading edges at various angles of attack and slip. Account is taken of multicomponent diffusion, and homogeneous chemical reactions, including dissociation-recombination reactions and exchange reactions. On the shock wave the generalized Rankine-Hugoniot conditions are given, and on the surface of the body conditions which allow for heterogeneous catalytic reactions of the first order with reaction rate constants depending [1] or not depending [2] on the temperature. The cases of an ideally catalytic and a noncatalytic surface are also considered. The surface of the body is assumed to be heatinsulated. A numerical study was made of the problem in a broad range of variation in the angles of attack and slip for different cases of prescribed constants representing the rates of the heterogeneous reactions. The conditions of the flow corresponded to the motion of a body which possess a lifting force along the trajectory of entry into the Earth's atmosphere [3]. The dependences are given of the equilibrium temperature of the surface along the stagnation line of the wing on the height of the flight and the distribution of this temperature along the surface of wings with parabolic and hyperbolic contours. It is shown that for flow regimes with a relatively high degree of dissociation in cases when the proportion of atoms recombined on the surface of the body is small, the dependences of the heat flow and the temperature of the surface on the angle of slip are of a nonmonotonic nature.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza., No. 6, pp. 127–135, November–December, 1984.     

Heat transfer effects on particle motion under rarefied conditions

Gas–solid flows occurring on very small spatial scales (of the order of micro and nanometres) are of great relevance in a number of industrial applications. It is currently not well established how particle motion and filtration are affected by non-isothermal conditions at such scales. Furthermore, when the particle size is comparable to the mean free path of the gas, rarefaction effects become important. In the present work we investigate the effects of heat transfer and non-isothermal conditions on the motion of small particles in rarefied flow. For that purpose, a suitable framework is developed here as a generic multiphase DNS method for rarefied flows. The resulting model is valid for low particle Reynolds number flows, irrespective of the Biot number, and for particle Knudsen numbers up to unity in unbounded flow. Using this model, we show that there is different settling behaviour of particles with an internal heat source in rarefied and continuum cases of the carrier gas respectively. It is shown that the chances for thermal levitation and/or lifting up of a particle due to buoyancy effects are significantly reduced under rarefied conditions.