Integral equations for the generalized stokes operator: Applications to high reynolds number flows

This paper deals with a boundary integral equation for the generalized Stokes problem and its approximation by simpler integral equations when the Reynolds number tends to infinity. The two-dimensional case has been treated in [1]. This paper addresses the three-dimensional case. © 1994 John Wiley & Sons, Inc.     

Critical reynolds numbers for flows containing particles

Zusammenfassung Die Bedingungen, unter denen in der Strömung vorhandene Teilchen die kritische Reynoldszahl beeinflussen, wurden untersucht. Die Teilchen werden in laminarer Strömung instabil, wenn der vertikale Geschwindigkeitsgradient eine bestimmte Relation zum Teilchenradius bekommt. Dies ist dann der Fall, wenn der Teilchenradius ungefähr die Grössenordnung des Prandtlschen Mischungsweges oder der kleinsten Turbulenzelemente annimmt. Dies gilt für den Fall, dass Flüssigkeit und Teilchen gleiche Dichte haben. Bei ungleicher Dichte werden die Ergebnisse entsprechend modifiziert.     

Problems with discontinuous boundary conditions describing laminar flows at high reynolds numbers

The asymptotic structure of perturbed laminar flows, which are described by problems with unsteady, discontinuous boundary conditions, is investigated. As a result of an asymptotic analysis, the structure of the flow domains is found and appropriate boundary conditions are formulated. Numerical solutions are obtained for one of the processes, which is described by an inhomogeneous Burgers equation and corresponds to discontinuities in the vertical longitudinal velocities, as well as to the blow out of a tangential jet.     

A partitioned second‐order method for magnetohydrodynamic flows at small magnetic reynolds numbers

This article aims to study the partitioned method for magnetohydrodynamic flows at small magnetic Reynolds numbers. We design a partitioned second‐order method and show that this method is stable under a time step () restrict condition. Our method can decouple the magnetohydrodynamic equations so that we can solve two relatively simple subproblems separately at each time step, which is computationally economic. A complete theoretical analysis of error estimates is also given. Finally, we present numerical experiments to support our theory.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1966–1986, 2017     

An efficient,partitioned ensemble algorithm for simulating ensembles of evolutionary MHD flows at low magnetic Reynolds number

Studying the propagation of uncertainties in a nonlinear dynamical system usually involves generating a set of samples in the stochastic parameter space and then repeated simulations with different sampled parameters. The main difficulty faced in the process is the excessive computational cost. In this paper, we present an efficient, partitioned ensemble algorithm to determine multiple realizations of a reduced Magnetohydrodynamics (MHD) system, which models MHD flows at low magnetic Reynolds number. The algorithm decouples the fully coupled problem into two smaller subphysics problems, which reduces the size of the linear systems that to be solved and allows the use of optimized codes for each subphysics problem. Moreover, the resulting coefficient matrices are the same for all realizations at each time step, which allows faster computation of all realizations and significant savings in computational cost. We prove this algorithm is first order accurate and long time stable under a time step condition. Numerical examples are provided to verify the theoretical results and demonstrate the efficiency of the algorithm.     

Solution method for MHD-problems in plasma with unconstrained magnetic reynolds number

Translated from Chislennye Metody Resheniya Obratnykh Zadach Matematicheskoi Fiziki, pp. 171–179, 1988.     

Sequential and parallel algorithms for minimum flows

First, we present two classes of sequential algorithms for minimum flow problem: decreasing path algorithms and preflow algorithms. Then we describe another approach of the minimum flow problem, that consists of applying any maximum flow algorithm in a modified network. In section 5 we present several parallel preflow algorithms that solve the minimum flow problem. Finally, we present an application of the minimum flow problem.     

Inessential Combinations and Colorings of Models

We define the operations of an inessential combination and an almost inessential combination of models and theories. We establish basedness for an (almost) inessential combination of theories. We also establish that the properties of smallness and -stability are preserved upon passing to (almost) inessential combinations of theories. We define the notions of coloring of a model, colored model, and colored theory, and transfer the assertions about combinations to the case of colorings. We characterize the inessential colorings of a polygonometry.     

Defect correction method for viscoelastic fluid flows at high Weissenberg number

We study a defect correction method for the approximation of viscoelastic fluid flow. In the defect step, the constitutive equation is computed with an artificially reduced Weissenberg parameter for stability, and the resulting residual is corrected in the correction step. We prove the convergence of the defect correction method and derive an error estimate for the Oseen‐viscoelastic model problem. The derived theoretical results are supported by numerical tests for both the Oseen‐viscoelastic problem and the Johnson‐Segalman model problem. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A high accuracy minimally invasive regularization technique for navier–stokes equations at high reynolds number

A method is presented, that combines the defect and deferred correction approaches to approximate solutions of Navier–Stokes equations at high Reynolds number. The method is of high accuracy in both space and time, and it allows for the usage of legacy codes a frequent requirement in the simulation of turbulent flows in complex geometries. The two‐step method is considered here; to obtain a regularization that is second order accurate in space and time, the method computes a low‐order accurate, stable, and computationally inexpensive approximation (Backward Euler with artificial viscosity) twice. The results are readily extendable to the higher order accuracy cases by adding more correction steps. Both the theoretical results and the numerical tests provided demonstrate that the computed solution is stable and the accuracy in both space and time is improved after the correction step. We also perform a qualitative test to demonstrate that the method is capable of capturing qualitative features of a turbulent flow, even on a very coarse mesh. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 814–839, 2017     

On the low Mach number limit of compressible flows in exterior moving domains

We study the incompressible limit of solutions to the compressible barotropic Navier–Stokes system in the exterior of a bounded domain undergoing a simple translation. The problem is reformulated using a change of coordinates to fixed exterior domain. Using the spectral analysis of the wave propagator, the dispersion of acoustic waves is proved by means of the RAGE theorem. The solution to the incompressible Navier–Stokes equations is identified as a limit.     

Nonlinear characteristics of vortex-induced vibration at low Reynolds number

Numerical simulations of vortex-induced vibration of a two-dimensional elastic circular cylinder under the uniform flow are calculated when Reynolds number is 200. In order to achieve the vortex-induced vibration, two-dimensional incompressible Navier–Stokes equations are solved with the space–time finite element method, the equations of the cylinder motion are solved with the new explicit integral method and the emeshing is achieved by the spring analogy technology. Considering vortex-induced vibration with the low reduced damping parameters, the variety trends of the lift coefficient, the drag coefficient, the displacement of cylinder are analyzed under different oscillating frequencies of cylinder. The nonlinear phenomena of locked-in, beat and phaseswith are captured successfully. The limit cycle and bifurcation of lift coefficient and displacement are analyzed. Besides, the Poincare sections of the lift coefficient are used for discussing the bifurcation of periodic solution. There are some differences in nonlinear characteristics between the results of the one degree of freedom cylinder model and those of the two degrees of freedom cylinder model. The streamwise vibration has a certain effect on the lateral vibration.     

Simplified radiative models for low-Mach number reactive flows

We introduce a class of numerical models for the simulation of radiative effects in low-Mach number reactive flows. These models are based on simplified P_N approximations for radiative heat transfer, low-Mach asymptotic in the compressible flow, and reduced four-step chemical reaction for reacting species. The models presented here remove on one hand, acoustic wave propagation while retaining the compressibilty effects resulting from combustion and on the other hand, simplify the integro-differential equation for radiative transfer to a set of differential equations independent of the angle variable and compatible to those used for modelling flow and combustion. We briefly discuss the basic discretization methodology for the combined equations and its implementation in a modified projection method. We present validation computations for a two-dimensional methane/air flame in which the computational results are compared for nongray participating media.     

On inertial wave and flow structure at low Rossby number

Flow disturbances produced by the slow relative motion of an impermeable body in a large, rapidly rotating vessel are studied as an asymptotic theory for an inviscid, incompressible fluid at a small Rossby number (u _c /L1). The axial distance between the vessel wallsH is assumed to be so much larger than the body scaleL that the reduced height H/L is of unit order or greater. This flow admits a columnar structure near the body and an outer nonlinear structure of the inertial-wave type far above the forcing region, at distances along the rotation-axis (z=0(L/)). The inner boundary condition for the outer problem is provided by transferring the impermeable surface condition through the columnar structure; the outer solution in turn determines the vorticity and the solution of the columnar inner region.For thin obstacles or shallow ground topography (1), the outer solution is governed by an equation comparable to a linear one for inertial waves. The linear solutions obtained for steady transverse motion in an infinite domain () shows that, in this case, surface (perturbation) velocities are orthogonal to those in an irrotational (non-rotating) flow over the same ground topography. In the far field, i.e., ( z/L l) disturbances are confined mainly behind a wedge-shaped caustic front downstream of the rotation axis , wherein their amplitudes are comparable to , in general accord with Lighthill's result from group-velocity consideration. The field behind the caustic supports, however, lee waves extending far downstream with diminishing strength. Their wave lengths belong to orderLz/x, and therefore these waves should appear to be densely packed in the wedge-shaped region. The question on tilting of the Taylor column is delineated; the structure of the caustic zone and its upstream flow behavior are also analyzed.

---

Zusammenfassung Die Strömung um einen Körper, der sich langsam in einem grossen, rotierenden Behälter bewegt, wird untersucht mit einer asymptotischen Theorie für eine reibungsfreie, inkompressible Flüssigkeit bei einer kleinen Rossby-Zahl, d.h. u _c /L1. Der axiale AbstandH zwischen den Gefässwänden wird als soviel grösser als die Körperabmessung angenommen, dass für die reduzierte Höhe =H/L 0(1) gilt. Diese Strömung erlaubt eine säulenähnliche Struktur (Taylor column) in Körpernähe und eine äussere, nichtlineare Struktur vom Wellentyp für Höhenz=0(L/) über dem Köper. Die innere Randbedingung für das äussere Problem wird erhalten, indem die Bedingung an der Wand durch die Säule verschoben wird. Die äussere Lösung bestimmt ihrerseits die Rotation und damit die Lösung im inneren, säulenartigen Bereich.Wenn Körper oder Bodenform flach sind (1), so wird die Aussenlösung durch eine Gleichung bestimmt, welche vergleichbar ist mit einer linearen Gleichung für Trägheitswellen. Lineare Lösungen werden für gleichförmige Bewegung von der Achse weg in einem unbegrenzten Bereich ) gegeben. Sie zeigen, dass die Oberflächenstromlinien orthogonal zu den Stromlinien sind, die bei rotationsfreier (nichtdrehender) Strömung über der gleichen Bodenform entstehen. Im Fernfeld ( z/L l) sind Störungen hauptsächlich auf ein keilförmiges Gebiet stromabwärts von der Rotationsachse ( ) begrenzt. In diesem Gebiet geht ihre Amplitude mit , in Uebereinstimmung mit Lighthill's Resultat, das mit der Gruppengeschwindigkeit hergeleitet wurde. Im keiförmigen Gebiet bestehen drei Familien von Lee-Wellen, welche sich mit unveränderter Stärke weit stromabwärts erstrecken. Ihre Wellenlängen gehören zur OrdnungLx/z, weshalb die Wellen im keilförmigen Bereich dicht gepackt erscheinen. Die Frage der Neigung der Taylor Säule behandelt und die Struktur des Randes vom keilförmigen Bereich wird analysiert.

---

---

This work is dedicated to Professor Nicholas Rott on the occasion of his sixtieth birthday.     

Structure of the linearized problem for compressible parallel fluid flows

We consider the linearized compressible Navier-Stokes equation near a parallel flow in a cylindrical domain restricting our study to perturbations periodic in the generatrix direction. For any parameter values, we show that the initial value linear evolution problem is solved by the direct sum of a (strictly) contraction semi-group and an analytic semi-group. Any unbounded in time solution of this linear problem comes from isolated eigenvalues with finite multiplicities, which have non negative real part, and whose imaginary part is bounded. In addition, we precise the structure of the spectrum of the generator of the semi-group, locating the essential spectrum stricly on the left side of the complex plane.

---

Sunto Si linearizzano le equazioni di Navier-Stokes comprimibili attorno ad un moto lineare in un cilindro. Senza imporre restrizioni sul moto di base, si studia il problema lineare per perturbazioni periodiche nella direzione della generatrice del cilindro. Si prova che il problema di evoluzione lineare è dato dalla somma diretta di un semigruppo di stretta contrazione ed un semigruppo analitico. Ogni soluzione temporalmente non limitata deriva da autovalori isolati di molteplicità finita, aventi parte reale positiva e parte immaginaria limitata. Si studia anche la struttura dello spettro del generatore del semigruppo, costituito nel lato destro del piano complesso da autovalori di molteplicità finita, e nel lato sinistro di autovalori e da uno spettro essenziale situato su di una retta parallela all’asse immaginario.

     

A coin vibrational motor swimming at low Reynolds number

Low-cost coin vibrational motors, used in haptic feedback, exhibit rotational internal motion inside a rigid case. Because the motor case motion exhibits rotational symmetry, when placed into a fluid such as glycerin, the motor does not swim even though its oscillatory motions induce steady streaming in the fluid. However, a piece of rubber foam stuck to the curved case and giving the motor neutral buoyancy also breaks the rotational symmetry allowing it to swim. We measured a 1 cm diameter coin vibrational motor swimming in glycerin at a speed of a body length in 3 seconds or at 3 mm/s. The swim speed puts the vibrational motor in a low Reynolds number regime similar to bacterial motility, but because of the oscillations of the motor it is not analogous to biological organisms. Rather the swimming vibrational motor may inspire small inexpensive robotic swimmers that are robust as they contain no external moving parts. A time dependent Stokes equation planar sheet model suggests that the swim speed depends on a steady streaming velocity V _stream ~ Re _s ^1/2 U ₀ where U ₀ is the velocity of surface oscillations, and streaming Reynolds number Re _s = U ₀ ² /(ων) for motor angular frequency ω and fluid kinematic viscosity ν.