Oseen Approximation Research Articles

Dynamics of laminar flows with coaxial oppositely-rotating layers

Introduction. The work relates to the scientific foundations of hydraulic and energy construction and is devoted to the study of laminar flows with coaxial oppositely-rotating layers. In the literature, such flows are called counter-vortex. In the turbulent range, counter-vortex flows are characterized by intensive mixing of the medium, which is widely used in the technologies of mixing non-natural and multi-phase media in thermal and atomic energy, in systems of mass- and heat transfer, in chemistry and microbiology, ecology, engine and rocket production. The aim of the theoretical study is to study the physical laws of the hydrodynamics of counter-vortex flows. Research methods. The theoretical Navier-Stokes equations and continuity equation are the basis of the theoretical model of the laminar counter-vortex flow. Results. Assuming the radial velocities are much less than the azimuthal and axial velocities and taking the Oseen approximation, the solution of the Navier - Stokes equations is obtained as Fourier - Bessel series or products of Fourier - Bessel series. In particular, the following were obtained: formulas for calculating the radial-longitudinal distributions of the normalized azimuthal, axial and radial velocities in the flow under study, the velocities are presented graphically in the form of radial profiles; equations for the calculation of current lines and viscous vortex fields, which are also presented in the form of graphs, were obtained. The two-layer and four-layer counter-vortex flows are considered. The analysis of the obtained theoretical results is performed. Conclusions. On the axis at the beginning of the active zone, the formation of a return flow with significant negative velocities is characteristic. This leads to the formation of a recirculation region, the mass exchange between which and the external flow is absent. Cascades of concentric vortexes of such high intensity that are not found in streams of a different nature are generated in the active zone. Calculation formulas include exp (-λ2x/Re) exponent multiplied by Reynolds number in degree b = 0 or b = -1, therefore increasing Reynolds number when b = 0 leads to proportional transfer of arbitrary characteristic counter-vortex flow down the pipe; and at b = -1, the bias of characteristic is accompanied by a proportional decrease in its scale.

Hydrodynamic interaction of trapped active Janus particles in two dimensions.

The dynamics of a pair of identical artificial microswimmers bound inside two harmonic traps, in a thin sheared fluid film, is numerically investigated. In a two-dimensional Oseen approximation, the hydrodynamic pair coupling is long-ranged and proportional to the particle radius to film thickness ratio. On increasing such ratio above a certain threshold, a transition occurs between a free regime, where each swimmer orbits in its own trap with random phase, and a strong synchronization regime, where the two swimmers strongly repel each other to an average distance larger than both the trap distance and their free orbit diameter. Moreover, the swimmers tend to synchronize their positions opposite the center of the system.

Formulae for predicting non-acoustical parameters of deformed fibrous porous materials.

Formulae to predict non-acoustical parameters (i.e., flow resistivity, tortuosity, and viscous and thermal characteristic lengths) of deformed fibrous porous materials are proposed provided that the original values of these parameters are known in advance. These formulae are developed using numerical fluid analyses. The flow resistivity was calculated by using the finite element method for a two-dimensional incompressible viscous fluid approximated by Oseen flow. The tortuosity and characteristic lengths were calculated by using the complex variable boundary element method for a two-dimensional potential flow. These calculations showed that the flow resistivity was inversely proportional to the porosity multiplied by the three-halves power of the compression ratio, that the tortuosity can be represented by a linear expression of the porosity, and that both characteristic lengths changed in the same manner with respect to the porosity. These tendencies agreed well with measurements of real glass wools of various bulk densities. The proposed prediction formulae for the parameters were then derived from the tendencies obtained from the numerical analyses. The predicted parameter values were compared with the calculated parameters and good agreement was obtained, confirming the validity of the proposed formulae.

Flow of a viscous fluid past a heterogeneous porous sphere at low Reynolds numbers

A flow past a heterogeneous porous sphere is investigated by using the perturbation theory. The flow through the sphere is divided into two zones, which are fully saturated with the viscous fluid, and the flow in these zones is governed by the Brinkman equation. The space outside the sphere, where a clear fluid flows, is also divided into two zones: the Navier–Stokes zone and the Oseen flow zone. The solutions on the interface inside the sphere are matched with the condition proposed by Merrikh and Mohammad. The stream function in the Navier–Stokes zone is matched with that on the sphere surface by the condition proposed by Ochoa-Tapia and Whitaker. It is found that the drag on the spherical shell decreases as the permeability toward the sphere boundary increases.

Stability of momentumless wakes

The caudal fin of swimming animals can be modelled as a thrust-producing flapping foil. When considered alone, such a foil produces on average a jet wake with a positive net momentum. It has been argued that the instability characteristics of these averaged wakes are linked to the propulsion efficiency of swimming animals. Here, we reconsider this question by taking into account both the thrust and the drag exerted on a self-propelled swimming body. To do so, we study the stability of a family of momentumless wakes, constructed as the Oseen approximation of a force doublet moving at constant velocity. By performing a local stability analysis, we first show that these wakes undergo a transition from absolute to convective instability. Then, using the time stepper approach by integrating the linearised Navier–Stokes system, we investigate the global stability and reveal the influence of a non-parallel base flow as well as the role of the locally absolutely unstable upstream region in the wake. Finally, to complete the global scenario, we address the nonlinear evolution of the wake disturbance. These results are then discussed in the context of aquatic locomotion. According to the present stability results, and assuming the Oseen approximation whose validity has been assessed only for moderate Reynolds number, the momentumless wake of aquatic animals is generally stable, whereas the corresponding thrust part is unstable. It is therefore essential to consider all forces exerted on a self-propelled animal when discussing its wake stability and its propulsion efficiency.

Lift force due to odd Hall viscosity

We study the problem of flow of a neutral gas past an infinite cylinder at right angle to its axis at low Reynolds number when the fluid is characterized by broken time-reversal invariance, and hence by odd viscosity in addition to the normal even one. We solve the Oseen approximation to Navier-Stokes equation and calculate the lift force which appears due to the odd viscosity.

Аналитическое решение задачи об обтекании параболического дендрита наклонным потоком вязкой жидкости в приближении Осеена

Аналитическое решение задачи об обтекании параболического дендрита наклонным потоком вязкой жидкости в приближении Осеена

The application of adjoint method for shape optimization in Stokes–Oseen flow

This paper presents a numerical method for shape optimization of a body immersed in an incompressible viscous flow governed by Stokes–Oseen equations. The purpose of this work is to optimize the shape that minimizes a given cost functional. Based on the continuous adjoint method, the shape gradient of the cost functional is derived by involving a Lagrangian functional with the function space parametrization technique. Then, a gradient‐type algorithm is applied to the shape optimization problem. The numerical examples indicate the proposed algorithm is feasible and effective in low Reynolds number flow. Copyright © 2016 John Wiley & Sons, Ltd.

Fluid flow over and through a regular bundle of rigid fibres

The interaction between a fluid flow and a transversely isotropic porous medium is described. A homogenized model is used to treat the flow field in the porous region, and different interface conditions, needed to match solutions at the boundary between the pure fluid and the porous regions, are evaluated. Two problems in different flow regimes (laminar and turbulent) are considered to validate the system, which includes inertia in the leading-order equations for the permeability tensor through a Oseen approximation. The components of the permeability, which characterize microscopically the porous medium and determine the flow field at the macroscopic scale, are reasonably well estimated by the theory, both in the laminar and the turbulent case. This is demonstrated by comparing the model’s results to both experimental measurements and direct numerical simulations of the Navier–Stokes equations which resolve the flow also through the pores of the medium.

求解Oseen流的交替线松弛多重网格方法

The 1st-order upwind discretization form of the Oseen flow was obtained through the Godunov-type flux-difference splitting approach based on the Riemann solver. The convergence analysis of 2 kinds of cycling algorithms, i.e., the V-cycle and the W-cycle in the multigrid method for the solution of the discretized equations, was performed. Furthermore, the smooth properties of the collective symmetrical alternating-line Gauss-Seidel relaxation was investigated by means of the local Fourier analysis. The numerical results show that the collective symmetrical alternating-line Gauss-Seidel relaxation has sound smooth properties, and the convergence of the W-cycle algorithm is better than that of the V-cycle one in the multigrid method for the solution of the Oseen flow with different Reynolds numbers.

A General Approach to Time Periodic Incompressible Viscous Fluid Flow Problems

This article develops a general approach to time periodic incompressible fluid flow problems and semilinear evolution equations. It yields, on the one hand, a unified approach to various classical problems in incompressible fluid flow and, on the other hand, gives new results for periodic solutions to the Navier–Stokes–Oseen flow, the Navier–Stokes flow past rotating obstacles, and, in the geophysical setting, for Ornstein–Uhlenbeck and various diffusion equations with rough coefficients. The method is based on a combination of interpolation and topological arguments, as well as on the smoothing properties of the linearized equation.

Simulation of aerosol fibrous filters at Reynolds numbers of the order of unity

The deposition of aerosol particles from a flow in a model filter composed of parallel rows of parallel fibers oriented normal to the flow direction has been studied at Reynolds numbers of the order of unity, Re ∼ 1, which are inherent in analytical filtration. The Oseen flow field in a row has been determined by the method of fundamental solutions. It has been shown that the drag forces of the fibers to the flow and the efficiencies of inertial particle collection on the fibers calculated based on the obtained field insubstantially differ from those calculated by the Navier-Stokes equations. The higher the filter packing density, the larger the Re numbers at which the drag forces begin to differ. In the case of inertial particle deposition, the difference between the collection efficiencies increases with the interception parameter. For rows with a/h > 0.2 (a is the fiber radius, and 2h is the interfiber distance in a row), including rows with a nonuniform arrangement of fibers, the calculations performed by both the linearized and complete Navier-Stokes equations yield almost equal results up to Re ∼ 1.

Computation of steady incompressible flows in unbounded domains

In this study we revisit the problem of computing steady Navier–Stokes flows in two-dimensional unbounded domains. Precise quantitative characterization of such flows in the high-Reynolds number limit remains an open problem of theoretical fluid dynamics. Following a review of key mathematical properties of such solutions related to the slow decay of the velocity field at large distances from the obstacle, we develop and carefully validate a spectrally-accurate computational approach which ensures the correct behavior of the solution at infinity. In the proposed method the numerical solution is defined on the entire unbounded domain without the need to truncate this domain to a finite box with some artificial boundary conditions prescribed at its boundaries. Since our approach relies on the streamfunction–vorticity formulation, the main complication is the presence of a discontinuity in the streamfunction field at infinity which is related to the slow decay of this field. We demonstrate how this difficulty can be overcome by reformulating the problem using a suitable background “skeleton” field expressed in terms of the corresponding Oseen flow combined with spectral filtering. The method is thoroughly validated for Reynolds numbers spanning two orders of magnitude with the results comparing favorably against known theoretical predictions and the data available in the literature.

A Numerical Method for Shape Optimal Design in the Oseen Flow with Heat Transfer

This paper is concerned with the optimal design of an obstacle located in the viscous and incompressible fluid which is driven by the steady-state Oseen equations with thermal effects. The structure of shape gradient of the cost functional is derived by applying the differentiability of a minimax formulation involving a Lagrange functional with a space parametrization technique. A gradient type algorithm is employed to the shape optimization problem. Numerical examples indicate that our theory is useful for practical purpose and the proposed algorithm is feasible.

Scaling laws of free dendritic growth in a forced Oseen flow

We use the method presented in M von Kurnatowski et al (2013 Phys. Rev. E 87 042405) to solve the nonlinear problem of free dendritic growth in an Oseen flow. The growth process is assumed to be limited by thermal transport via diffusion and convection. A singular perturbation expansion is treated to lowest nontrivial order in the framework of asymptotic decomposition. The resulting complex integro-differential equation is solved using an elaborate numerical method. The approximate scaling laws and for the growth velocity and the tip radius of curvature of the dendrite, respectively, are found as a function of the forced flow velocity. The results are compared to those by Pelcé and Bouissou, constituting the only other attempt so far to treat the problem analytically.

Experimental validation of a correlation capturing the boundary layer effect on spread rate in the kinetic regime of opposed-flow flame spread

In opposed-flow flame spread over thermally thin solid fuels, an increase in flow velocity eventually leads to flame extinguishment. While the chemical time is independent of the flow velocity, the residence time of the oxidizer at the flame leading edge is inversely proportional to the flow velocity. A competition between the two leads to a situation where finite-rate kinetics dominates the flame spread behavior, leading to blow-off extinguishment. The ratio of the two competing, the Damkohler number, captures this finite-rate effect and has been used to correlate the non-dimensional spread rate with opposing flow velocity and ambient oxygen level. Although qualitatively successful, these correlations show considerable spread in the data despite the use of several variations of the definition of the Damkohler number. One possible reason could be the neglect of the developing boundary layer in front of the flame by the experimentalists. In this work we present new flame spread data over ashless filter paper acquired in an eight meter tall vertical steel chamber, in which a sample is moved at desired velocities to create a relative opposing flow. The developing boundary layer over the fuel sample is shown to have a significant effect on the measured spread rate and flame shape that does not correlate with opposing flow velocity if an Oseen flow approximation is used. Once the data is adjusted for the developing boundary layer using an effective flow velocity formula obtained from scaling arguments and refined through computational results, the correlation improves drastically, establishing the importance of the boundary layer in the kinetic regime. Much of the thermally thin fuel spread rate data available in literature does not mention the development length and must be cautiously used.

Uniqueness and nonuniqueness of the Stokes and Oseen flows

The uniqueness class for the solutions to the Cauchy problem for flows modeled by the time-dependent Stokes and Oseen systems of equations is determined as in the growth class C exp(α|x|2). An example of a type considered by Tychonoff [Mat. Sb. 42 (1935), 199–216] is given that establishes the lack of uniqueness for such Stokes and Oseen systems. Even for the incompressible Navier–Stokes system, an example shows that rapidly growing nonphysical mathematical solutions exist.

Evaluation on Leading Order Vortex Decay behind an Aircraft Using an Analytical Model

A theoretical model in this article which is used to evaluate leading order vortex decay behind an aircraft, is presented. This model is then applied to two important problems. The first evaluates safety separation distances for the super class of aircraft exemplified by the Airbus A380. The second evaluates the swirl parameter which is shown to decay to a sufficiently lowlevel that short-wave instabilities are likely to be instigated. The model uses the Oseen approximation,which assumes that in the far-field the ratio of the perturbed velocity to the aircraft velocity is small, and developed from recent publications by the first author. The atmospheric turbulence is incorporated into the model by using effective viscosity.

Pointwise spatial decay of time-dependent Oseen flows: The case of data with noncompact support

The article deals with the time-dependent Oseen system ina 3D exterior domain.It is shown that the velocity part of a weak solution to that systemdecays as $\bigl(\, |x| \cdot (1+|x|-x_1) \,\bigr) ^{-1}$,and its spatial gradient as $\bigl(\, |x| \cdot (1+|x|-x_1) \,\bigr) ^{-3/2}$, for$|x|\to \infty $. This result is obtained for data that need not have compact support.

Cauchy integral formula for generalized analytic functions in hydrodynamics

It is shown that for several classes of generalized analytic functions arising in linearized equations of hydrodynamics and magnetohydrodynamics, the Cauchy integral formulae follow from the one for generalized holomorphic vectors in a uniform fashion. If hydrodynamic fields (velocity, pressure and vorticity) admit representations in terms of corresponding generalized analytic functions, those representations and the Cauchy integral formulae form two essential parts of the generalized analytic function approach, which readily yields either closed-form solutions or boundary integral equations. This approach is demonstrated for problems of axisymmetric and asymmetric Stokes flows, two-phase axisymmetric Stokes flows, two-dimensional and axisymmetric Oseen flows.