Showing papers in "European Journal of Mechanics B-fluids in 2003"

Physical Mechanisms of the Rogue Wave Phenomenon

Abstract: A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrodinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.

Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. II. Slip and jump coefficients

Abstract: The Cercignani–Lampis scattering kernel of the gas–surface interaction is applied to numerical calculations of the viscous slip coefficient, the thermal slip coefficient and the temperature jump coefficient. The S model of the Boltzmann equation is numerically solved by the discrete velocity method. The calculations have been carried out in the wide ranges of the accommodation coefficients of momentum and energy. Comparing the present results with experimental data on the viscous slip coefficient the values of the accommodation coefficients are calculated for some gases and glass surface.

Mixed convection boundary-layer flow over a vertical surface embedded in a porous medium

Abstract: In this paper we have numerically investigated the existence and uniqueness of a vertically flowing fluid passed a model of a thin vertical fin in a saturated porous media. We have assumed the two-dimensional mixed convection from a fin, which is modelled as a fixed, semi-infinite vertical surface, embedded in a fluid-saturated porous media under the boundary-layer approximation. We have taken the temperature, in excess of the constant temperature in the ambient fluid on the fin, to vary as x λ , where x is measured from the leading edge of the plate and λ is a fixed constant. The Rayleigh number is assumed to be large so that the boundary-layer approximation may be made and the fluid velocity at the edge of the boundary-layer is assumed to vary as x λ . The problem then depends on two parameters, namely λ and e, the ratio of the Rayleigh to Peclet numbers. It is found that when λ>0 ( there are (is) dual (unique) solution(s) when e is grater than some negative values of e (which depends on λ). When λ 0 and λ

Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. III. Poiseuille flow and thermal creep through a long tube

Abstract: The Cercignani–Lampis scattering kernel of the gas–surface interaction is applied to numerical calculations of the Poiseuille flow and thermal creep through a long tube. The S model of the Boltzmann equation was numerically solved by the discrete velocity method. The calculations have been carried out in the wide ranges of the rarefaction parameter and of the accommodation coefficients. Comparing the present results with experimental data the values of the accommodation coefficients have been calculated.

Buoyant flow in long vertical enclosures in the presence of a strong horizontal magnetic field. Part 2. Finite enclosures

Abstract: Numerical computations and experiments were carried out for a buoyant flow of liquid metal (mercury in the experiments) in a long vertical enclosure of square cross-section, in the presence of a uniform horizontal magnetic field. A strong emphasis is put on the case of a magnetic field perpendicular to the applied temperature gradient for two reasons: (1) the MHD damping is smaller than with any other orientation, and (2) the quasi-two-dimensionality of the flow in this case yields a quite efficient velocity measurement technique. The enclosure is heated by a thermally controlled flow of water from one of the vertical walls and cooled by a similar technique from the facing wall. Those two walls are good thermal conductors (thick copper plates in the experiments), whereas the four other walls are thermally insulating. All walls are electrically insulated from the fluid. In this paper, as well as in the companion paper by Tagawa et al. (Eur. J. Mech. B Fluids 21 (4) (2002) 383–398), we model analytically the Hartmann layers present along the walls perpendicular to the magnetic field. This modeling, which yields boundary conditions for the core flow without any meshing of the thin layers, is quite accurate when Hartmann layers are stable. The numerical results are in fairly good agreement with the experimental data. They namely reveal how the heat flux and the fluid flow organization depend on the magnetic field.

Axisymmetric stagnation flow obliquely impinging on a circular cylinder

Abstract: Laminar stagnation flow, axisymmetrically yet obliquely impinging on the generators of a circular cylinder, is formulated as an exact solution of the Navier–Stokes equations. The outer stream is composed of a rotational axial flow superposed onto irrotational radial stagnation flow normal to the cylinder. The relative importance of these two flows is measured by a parameter γ . The viscous problem is reduced to a coupled pair of ordinary differential equations governed by a Reynolds number R introduced by Wang (1974). Two-term asymptotic formulae valid for large R are derived for the wall shear stress and for the position and slope of streamline attachment. These results agree well with exact numerical calculations for R >30. In checking the consistency of our solution in the planar limit R →∞ we uncover and correct an error in the work of Dorrepaal (2000) for the position of viscous streamline attachment.

Miscible displacements between silicone oils in capillary tubes

Abstract: An experiment in which one viscous silicone oil displaces another more viscous silicone oil in capillary tubes, leading to a version of so-called fingering displacement, is reported here. The fractional volume of the more viscous fluid left on the tube wall after the finger front had passed was measured. This fraction, m, defined as m=1−Um/Utip, is a critical diagnostic parameter used to identify finger patterns. Here, Um is the mean velocity of the Poiseuille flow ahead the finger front and Utip the propagation speed of the finger tip. In the present case, the fraction m is a function of the Peclet number, Pe, a viscous Atwood number, At, and a gravitational parameter, F. In this experiment, m was obtained by measuring the finger propagation speed, Utip, for a range of injection rates, tube diameters and orientations and fluid viscosities to cover a range of Pe, At and F. For large Pe, the results show the fraction m reached a constant value that depended only on Atwood number. For small Pe, gravitational effects were substantial as measured by the magnitude of F, which is the ratio of gravitational to viscous effects. In this case m is a function of all three parameters and it becomes possible to interpolate the results to obtain the value of m when gravity is absent.

Flow of magnetorheological fluid through porous media

Abstract: Flow of a magneto-rheological (MR) fluid through different types of porous media (bundle of cylinders, packed beds of magnetic and non-magnetic spheres and cylinders) is considered, both theoretically and experimentally. The theory is based on averaging the magnetic and rheological properties of MR fluid in tortuous channels making different angles between local field and local velocity. A comparison of the pressure drop through porous beds and spiral channels is analyzed and practical recommendations are developed. It is shown that the mean yield stress of Bingham MR fluid (as well as the pressure drop, ΔP) depends on the mutual orientation of the external magnetic field and the main axis of the flow. This theory is tested against our experimental results and is shown to well predict the pressure drop obtained in different porous media.

3D numerical simulation of regular structure formation in a locally heated falling film

Abstract: A thin-film flow on a locally heated vertical plate is studied numerically by solving a full 3D nonlinear time-dependent problem. The method of particles for incompressible fluid has been adapted to take into account viscous and surface tension forces. The effect of periodic rivulet-like structure formation, observed experimentally at moderate heating, has been simulated and investigated. The interesting flow picture including the spots of a strong reverse and spanwise thermocapillary surface flows is revealed. Some qualitative and quantitative comparisons with the experimental results are presented.

Heat transfer and evaporation/condensation problems based on the linearized Boltzmann equation

Abstract: A polynomial expansion procedure and an analytical discrete-ordinates method are used to solve four basic problems, all based on the linearized Boltzmann equation for rigid-sphere interactions, that describe heat transfer and/or evaporation–condensation between two parallel surfaces or for the case of a semi-infinite half space. Relevant to the case of two surfaces, the basic problem of heat transfer driven by a temperature difference at two confining walls described by a general Maxwell gas–surface interaction law (a mixture of specular and diffuse reflection) is solved for the case where different accommodation coefficients can be used for each of the two bounding surfaces. In addition, the classical problem of “reverse temperature gradient” in the theory of evaporation and condensation is also solved for the case of two parallel liquid–vapor interfaces kept at different temperatures. In regard to half-space applications, an evaporation/condensation problem based on a presumed known interface condition and a heat-conduction problem (with no net flow) driven by energy flow from a bounding surface with know properties are each solved with what is considered a high degree of accuracy.

A new Finite Element Method for magneto-dynamical problems: two-dimensional results

Abstract: A mixed Lagrange finite element technique is used to solve the Maxwell equations in the magneto-hydrodynamic (MHD) limit in an hybrid domain composed of vacuum and conducting regions. The originality of the approach is that no artificial boundary condition is enforced at the interface between the conducting and the insulating regions and the non-conducting medium is not approximated by a weakly conducting medium as is frequently done in the literature. As a first evaluation of the performance of the method, we study two-dimensional (2D) configurations, where the flow streamlines of the conducting fluid are planar, i.e., invariant in one direction, and either the magnetic field (“magnetic scalar” case) or the electric field (“electric scalar” case) is parallel to the invariant direction. Induction heating, eddy current generation, and magnetic field stretching are investigated showing the usefulness of finite element methods to solve magneto-dynamical problems with complex insulating boundaries.

Optimal linear growth in the asymptotic suction boundary layer

Abstract: A variational technique in the temporal framework is used to study initial configurations of disturbance velocity which maximize perturbation kinetic energy in the asymptotic suction boundary layer ...

Hydrodynamic stability of the Ekman boundary layer including interaction with a compliant surface: a numerical framework

Abstract: A model is developed for the interaction of the Ekman boundary layer with a compliant two-dimensional surface. To study the hydrodynamic instability of this interaction a new accurate numerical framework extending the compound matrix method is introduced. Preliminary results are presented on the implications of the compliant surface on the stability of the Ekman layer which show that the compliant surface has negligible effect on the critical Reynolds number.

Flow structures in zero pressure-gradient turbulent boundary layers at high Reynolds numbers

Abstract: The near-wall region of zero-pressure gradient turbulent boundary layers was studied through correlation- and other two-point measurements over a wide range of Reynolds numbers. The requirements of high spatial resolution were met by use of a MEMS-type of hot-film sensor array together with a small, in-house built hot-wire probe. Streak-spacing and characteristics of buffer region shear-layer events were studied. At high Reynolds numbers the motions that are of substantially larger scale than the streaks have a significant influence on the near-wall dynamics. By removing such scales through high-pass filtering a streak spacing was recovered that is close to that found in low Reynolds number flows. The frequency of occurrence of shear-layer events was found to scale with a mixed time scale, in analogy with earlier findings in channel flow, again indicating the increasing relative influence of large scales with increasing Reynolds number.

Weakly nonlinear stability of Hartmann boundary layers

Abstract: By means of a weakly nonlinear stability analysis it is shown that the Hartmann boundary layer presents subcritical instability in the proximity of the minimum linear critical Reynolds number. This gives further support to earlier speculations that finite amplitude effects account for the discrepancies between the results of the linear stability analysis and experimental evidence on laminarisation.

Vapor flows with evaporation and condensation in the continuum limit: effect of a trace of noncondensable gas

Abstract: Steady flows of a vapor with evaporation and condensation on the boundary consisting of the condensed phase of the vapor are considered in the following situation: (i) the boundary is of arbitrary smooth shape; (ii) the Knudsen number Kn, the ratio of the typical mean free path of the vapor molecules to the characteristic length of the system, is small; (iii) a small amount of a noncondensable gas is contained in the system; more specifically, the amount is such that the average concentration of the noncondensable gas is of the order of Kn in the case of a closed domain (the case of an infinite domain is also discussed). The steady behavior of the vapor and the noncondensable gas, in particular, that in the continuum limit where Kn vanishes, is investigated by means of a systematic asymptotic analysis based on kinetic theory. In this situation, the average concentration of the noncondensable gas becomes infinitely small in the continuum limit in the case of a closed domain. However, it is shown that the noncondensable gas accumulates in the infinitely thin Knudsen layer on the boundary where condensation is taking place and has a significant effect on the global vapor flow in the continuum limit. An example demonstrating such an effect is also given.

On the breaking of long, axisymmetric liquid bridges between unequal supporting disks at minimum volume stability limit

Abstract: The stability of axisymmetric liquid bridges held between non-equal circular supporting disks, and subjected to an axial acceleration, has been analyzed both theoretically and experimentally. Some characteristics of the breaking process which takes place when the stability limit of minimum volume is reached (mainly the dependence with the disks separation of the volume of the liquid drops resulting after the liquid bridge breakage) have been theoretically studied by using standard asymptotic expansion techniques. From the analysis of the nature of the unstable equilibrium shapes of minimum volume at the stability limit it is concluded that the relative volume of the main drops resulting from the liquid bridge rupture drastically change as the disks separation grows. Theoretical predictions have been experimentally checked by working with very small size liquid bridges (supporting disks being some 1 millimeter in diameter), the agreement between theoretical predictions and experimental results being remarkable.

Linear dynamics of axisymmetric liquid bridges

Abstract: The dynamical response of an axisymmetric liquid bridge to small-magnitude perturbations is studied in the framework of the Cosserat model. A numerical procedure to deal with this problem is proposed. The method is found to provide very accurate results from a comparison with the analytical predictions for the cylindrical configuration. The frequencies and damping rates characterizing the oscillation modes are obtained numerically for arbitrary axisymmetric liquid bridge shapes, considering the combined effects of residual gravity, the liquid bridge rotation, the inequality of the disks, and the liquid bridge volume. The results are compared with the predictions obtained from the three-dimensional model for inviscid liquid bridges.

Forced convection past a heated cylinder in a porous medium using a thermal nonequilibrium model: Boundary layer analysis

Abstract: We study the forced convective heat transfer from a uniform temperature cylinder placed perpendicular to an otherwise uniform fluid stream in a porous medium. Attention is focussed on how the absence of local thermal equilibrium between the solid and fluid phases affects the rate of heat transfer from the cylinder when the Peclet number is very large. It is found in all cases that the surface rate of heat transfer for the fluid is always greater than that of the solid matrix. Detailed numerical results are given for a wide range of parameter values, and these are supplemented by asymptotic analyses for both small and large values of the inter-phase heat transfer coefficient, H. When this coefficient is small the thermal field corresponding to the solid phase occupies a much greater region than does the thermal field of the fluid phase.

The effect of fine structure on the stability of planar vortices

Abstract: This study considers the linear, inviscid response to an external strain field of classes of planar vortices. The case of a Gaussian vortex has been considered elsewhere, and an enstrophy rebound phenomenon was noted: after the vortex is disturbed enstrophy feeds from the non-axisymmetric to mean flow. At the same time an irreversible spiral wind-up of vorticity fluctuations takes place. A top-hat or Rankine vortex, on the other hand, can support a non-decaying normal mode. In vortex dynamics processes such as stripping and collisions generate vortices with sharp edges and often with bands or rings of fine scale vorticity at their periphery, rather than smooth profiles. This paper considers the stability and response of a family of vortices that vary from a broad profile to a top-hat vortex. As the edge of the vortex becomes sharper, a quasi-mode emerges and vorticity winds up in a critical layer, at the radius where the angular velocity of the fluid matches that of a normal mode on a top-hat vortex. The decay rate of these quasi-modes is proportional to the vorticity gradient at the critical layer, in agreement with theory. As the vortex edge becomes sharper it is found that the rebound of enstrophy becomes stronger but slower. The stability and linear behaviour of coherent vortices is then studied for distributions which exhibit additional fine structure within the critical layer. In particular we consider vorticity profiles with ‘bumps’, ‘troughs’ or ‘steps’ as this fine structure. The modified evolution equation that governs the critical layer is studied using numerical simulations and asymptotic analysis. It is shown that depending on the form of the short-scale vorticity distribution, this can stabilise or destabilise quasi-modes, and it may also lead to oscillatory behaviour.

On the relationship between the pressure and the projection function in the numerical computation of viscous incompressible flow

Abstract: The relationship between the pressure p and the projection function φ employed in the numerical computation of viscous incompressible flow using the fractional-step method is discussed. To leading order, the difference p−φ is proportional to the divergence of the intermediate velocity, u ∗ , computed by integrating the equation of motion in the absence of the pressure gradient. Previous authors have shown that the intermediate rate of expansion, α ∗ ≡∇· u ∗ is supported by numerical boundary-layers of thickness δ≃(νΔt)1/2, where Δt is the time step and ν is the kinematic viscosity. We demonstrate that, in the absence of singularities due to discontinuous boundary velocity, the magnitude of α ∗ changes by an amount of order δ across the boundary layers and of higher order in the bulk of the flow, and argue that adding a computable correction to the projection function allows us to recover the pressure with temporal accuracy whose order matches that of the method used for carrying out the convection–diffusion step. When the boundary velocity is discontinuous, the normal derivative of the pressure exhibits strong singularities, and the computation of the pressure using finite-difference methods on non-staggered grids is notably sensitive to the numerical implementation. In contrast, in spite of the singular behavior of the intermediate rate of expansion, the solution of the Poisson equation for the projection function subject to the homogeneous Neumann boundary condition is less sensitive to the numerical method. Computing the projection function thus emerges as a preferred venue of approximating the pressure on non-staggered grids even under demanding conditions.

Experiments on control of streamwise streaks

Abstract: Control of streamwise velocity streaks are studied experimentally in a plane channel flow. High and low-velocity streaks are created by suction through streamwise slots and, further downstream, the ...

Nonlinear oscillations in three-armed tubes

Abstract: We consider nonlinear gravitational oscillations of inviscid liquid in arrangements of three tubes joined at their bases, for which the dynamical system is four-dimensional and conservative. Though the problem of two joined tubes was solved in 1738, that of three tubes appears to have remained unstudied. We consider both weakly-nonlinear theory, which gives rise to coupled amplitude evolution equations; and also full numerical solutions. In this way, an understanding is reached of the strengths and limitations of the weakly-nonlinear approximation. Both the weakly-nonlinear approximation and the full system display amplitude modulations on a slow timescale; but only the full system captures a narrow region of chaos.

Modal growth and non-modal growth in a stretched shear layer

Abstract: The evolution of two-dimensional linear perturbations in a uniform shear layer stretched along the streamwise direction is considered in this work. The velocity field of the basic flow is assumed to be given by the following exact solution of Navier–Stokes equations U=(γx+(1/S(t))erf(y/a(t)),−γy,0) where erf is the error function, a(t) and S(t) are time-varying functions. The solution is governed by two parameters: the Reynolds number Re and the stretching rate γ (non-dimensionalized by the initial maximum vorticity) which is assumed to be a positive constant. Using a direct-adjoint technique, perturbations which maximize the energy gain during a time interval (0,tf) are computed for various tf, γ and Re. For each case, the results are compared with those obtained by considering a single local normal mode (WKBJ approach). For small tf (tf 20), instability takes over transients: the WKBJ approximation is shown to provide a good estimate of the maximum gain whatever the Reynolds number (>10) and the stretching rate (<0.025). However, differences concerning the most amplified wavenumbers remain visible and increase with γ. For very large times, stretching moves the local wavenumber toward zero. A non-viscous asymptotic study performed for small k shows that although the perturbation energy ultimately diminishes, it decreases less rapidly than the basic flow energy density. Stretching therefore never stabilizes the shear layer for large Reynolds numbers. The results obtained in the WKBJ framework are also extended to more general configurations including three-dimensional perturbations and triaxial stretching fields.

Comparison of a lattice Boltzmann simulation of steep internal waves and laboratory measurements using particle image velocimetry

Abstract: Progressive interfacial waves are simulated between two viscous, immiscible fluids using a lattice Boltzmann model and compared to waves generated in a laboratory tank, where their velocities are measured using a film-based particle image velocimetry system. The numerical and experimental velocities are compared for two similar, steep waves and the results are found to be in good agreement. The results are also compared with a first order wave theory, in both cases they predict velocities up to 40% greater than the theory.

Wall functions for numerical modeling of laminar MHD flows

Abstract: A general wall function treatment is presented for the numerical modeling of laminar magnetohydrodynamic (MHD) flows. The wall function expressions are derived analytically from the steady-state momentum and electric potential equations, making use only of local variables of the numerical solution. No assumptions are made regarding the orientation of the magnetic field relative to the wall, nor of the magnitude of the Hartmann number, or the wall conductivity. The wall functions are used for defining implicit boundary conditions for velocity and electric potential, and for computing mass flow and electrical currents in near wall-cells. The wall function treatment was validated in a finite volume formulation, and compared with an analytic solution for a fully developed channel flow in a transverse magnetic field. For the case with insulating walls, a uniform 20×20 grid, and Hartmann numbers Ha={10,30,100}, the accuracy of pressure drop and wall shear stress predictions was {1.1%,1.6%,0.5%}, respectively. Comparable results were obtained also with conducting Hartmann walls. The accuracy of predicted pressure drop and wall shear stress was essentially independent of the resolution of the Hartmann layers. When applied also to the parallel walls, the wall functions reduced the errors by a factor two to three. The wall functions can be implemented in any general flow solver, to allow accurate predictions at reasonable cost even for complex geometries and nonuniform magnetic fields.