Three-dimensional divergent waves on a model viscoelastic coating in a potential incompressible fluid flow

Generation of three-dimensional nonlinear waves on a model viscoelastic coating in a potential flow of an incompressible fluid is studied. Periodic nonlinear waves enhanced by the development of quasi-static instability (wave divergence) are considered. The coating is modeled by a flexible flat plate supported by a distributed nonlinearly-elastic spring foundation. Plate flexure is described on the basis of the Karman equations of the theory of thin plates. Perturbations of surface pressure in the potential flow are found in the small slope approximation to an accuracy to terms of the second order of smallness. Numerical simulation reveals a jump-like transition from two-dimensional nonlinear waves to three-dimensional wave structures, which are also observed in experiments.     

Well-Posedness of Hydrodynamics on the Moving Elastic Surface

The dynamics of a membrane are that of a coupled system comprising a moving elastic surface and an incompressible membrane fluid. We will consider a reduced elastic surface model, which involves the evolution equations of the moving surface, the dynamic equations of the two-dimensional fluid, and the incompressible equation, all of which operate within a curved geometry. In this paper, we prove the local existence and uniqueness of the solution to the reduced elastic surface model by reformulating the model into a new system in the isothermal coordinates. One major difficulty is that of constructing an appropriate iterative scheme such that the limit system is consistent with the original system.     

A model of hydraulic transport of a permeable elastic body in a pipe

A steady-state axisymmetric model of transport and transition into a plastic state of a permeable elastic body carried by an incompressible power-law fluid in a vertical pipe flow is constructed. Possible flow regimes are found and the threshold values of the parameters responsible for the change of a regime are determined. Semi-analytical expressions for the fluid velocity distribution in the annulus, the pressure difference, and the steady-state velocity of the body are obtained. The conditions of the transition of the body into a plastic state are studied depending on the governing parameters of the problem.     

Boundary layer development from transition provoking devices

The results of an experimental investigation of four different, incompressible, transitional boundary layer situations are presented. The experiments were carried out in zero pressure gradient conditions and transition was initiated from two- and from three-dimensional provoking agents.

The measurements of transitional intermittency from two-dimensional tripping agents showed a trend consistent with that reported elsewhere in the literature, with the development of mean and fluctuating component velocity profiles and local skin friction coefficient exhibiting approximate similarity through the transition region.

Disturbance frequency and spread angles for turbulent wedge growth behind isolated roughness elements were similar to those reported by others.

Computer predictions using a transition model based on the present correlations show reasonable agreement with the data.     

A mixture differential quadrature method for solving two-dimensional incompressible Navier-Stokes equations

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Symmetry breaking and preliminary results about a Hopf bifurcation for incompressible viscous flow in an expansion channel

This computational study shows, for the first time, a clear transition to two-dimensional Hopf bifurcation for laminar incompressible flows in symmetric plane expansion channels. Due to the well-known extreme sensitivity of this study on computational mesh, the critical Reynolds numbers for both the known symmetry-breaking (pitchfork) bifurcation and Hopf bifurcation were investigated for several layers of mesh refinement. It is found that under-refined meshes lead to an overestimation of the critical Reynolds number for the symmetry breaking and an underestimation of the critical Reynolds number for the Hopf bifurcation.     

Continuous Mode Transition in High-speed Boundary-layers

Mode interaction is studied via direct numerical simulations of a Mach 4.5 boundary layer with discrete and continuous modes imposed at the inflow. An approximate decoupling procedure is developed to create separate vortical, acoustic and entropic continuous mode components. Oblique horizontal vorticity modes induce boundary layer disturbances that grow with downstream distance, similarly to their incompressible counterpart. One salient difference is that a low frequency vorticity mode, alone, is found to induce transition by spawning two-dimensional, unstable discrete modes. The discrete modes are non-linearly excited at high harmonics of the inlet perturbation. Adding a Mack second mode, in addition to the vorticity mode, causes even earlier transition, suggesting that, in supersonic flow, unstable discrete modes play a crucial role in breakdown of boundary-layer streaks.     

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.     

The inviscid compressible Görtler problem in three-dimensional boundary layers

In this paper we investigate numerical solutions for the growth rates of Görtler vortices in a compressible three-dimensional flow in the inviscid limit of a large Görtler number. We look at a range of Mach numbers and find that there are three different types of behaviour for the mode growth-rate, corresponding to whether the flow is incompressible, has a Mach number small enough so that temperature-adjustment-layer modes do not appear in the two-dimensional case, or has a Mach number large enough so that they do. We find that it takes a considerably greater crossflow to destroy the Görtler vortices for moderate Mach numbers than it did in the incompressible case looked at by Bassom and Hall (1991). From this we believe that Görtler vortices may well still be a cause of transition for many practical compressible inviscid three-dimensional flows.Support for the author from SERC is acknowledged.     

Formation fronts of a nonlinear elastic medium from a medium without shear stresses

The fronts of phase transition of a medium without shear stresses to a nonlinear incompressible anisotropic elastic medium are considered. The mass flux through unit area of a front is assumed to be known. The variation of the tangential components of the medium’s velocity and the variation of the arising shear stresses are studied. An explicit form of boundary conditions is found using the existence condition of a discontinuity front structure. The Kelvin–Voight viscoelastic model is adopted for this structure.     

On asymptotic theory of separated flows past bodies

Symmetric two-dimensional steady flow past a body in a homogeneous incompressible fluid stream at high Reynolds numbers is considered. A slow motion in the reverse flow zone is investigated and the solution for the flow in the external region is obtained in the second approximation. Additional considerations of the fact that the flow in the closure region of the separation zone and in the wake behind this zone is turbulent are presented. The laminar-turbulent transition in the mixing layer is analyzed and an analogy between this process and the propagation of perturbations upstream of the boundary layer interaction regions is revealed.     

Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face

This note examines the two-dimensional unsteady isothermal free surface flow of an incompressible fluid in a non-deformable, homogeneous, isotropic, and saturated porous medium (with zero recharge and neglecting capillary effects). Coupling a Boussinesq-type model for nonlinear water waves with Darcy's law, the two-dimensional flow problem is solved using one-dimensional model equations including vertical effects and seepage face. In order to take into account the seepage face development, the system equations (given by the continuity and momentum equations) are completed by an integral relation (deduced from the Cauchy theorem). After testing the model against data sets available in the literature, some numerical simulations, concerning the unsteady flow through a rectangular dam (with an impermeable horizontal bottom), are presented and discussed.     

On the origins of vortex shedding in two-dimensional incompressible flows

An exegesis of a novel mechanism leading to vortex splitting and subsequent shedding that is valid for two-dimensional incompressible, inviscid or viscous, and external or internal or wall-bounded flows, is detailed in this research. The mechanism, termed the vortex shedding mechanism (VSM) is simple and intuitive, requiring only two coincident conditions in the flow: (1) the existence of a location with zero momentum and (2) the presence of a net force having a positive divergence. Numerical solutions of several model problems illustrate causality of the VSM. Moreover, the VSM criteria is proved to be a necessary and sufficient condition for a vortex splitting event in any two-dimensional, incompressible flow. The VSM is shown to exist in several canonical problems including the external flow past a circular cylinder. Suppression of the von Kármán vortex street is demonstrated for Reynolds numbers of 100 and 400 by mitigating the VSM.     

Limit cycle oscillations of two-dimensional panels in low subsonic flow

Limit cycle oscillations of a two-dimensional panel in low subsonic flow have been studied theoretically and experimentally. The panel is clamped at its leading edge and free at its trailing edge. A structural non-linearity arises in both the bending stiffness and the mass inertia. Two-dimensional incompressible (linear) vortex lattice aerodynamic theory and a corresponding reduced order aerodynamic model were used to calculate the linear flutter boundary and also the limit cycle oscillations (that occur beyond the linear flutter boundary).     

离心叶轮正冲角流场计算的边界层方法

本文使用stanitz快速分析法计算主流,用积分法计算带旋转和曲率的二维不可压湍流边界层.通过流量方程将两者紧密联系起来、同时求解,能够连续计算至分离区.可方便快速地用于估算离心叶轮在正冲角工况下的流场.     

Statistically equilibrium two-dimensional vortices

Equilibrium statistical mechanics is used for describing two-dimensional vortices in an unbounded incompressible ideal fluid. Both the energy and angular momentum integrals and a set of invariants are taken into account. The latter follows from the condition that any vorticity distribution can be obtained from an initial distribution by a differentiable areas-preserving transformation. The equations for the statistically equilibrium vorticity and passive admixture distributions are derived. It is argued that taking subsidiary invariants into account weakens the arbitrariness associated with the choice of a finite-dimensional approximation of the flow. The case in which the vorticity cloud behaves like a thermodynamic system undergoing an ordering phase transition is discussed.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 47–55, September–October, 1995.     

Obtaining the exact solutions of the Navier-Stokes equations

The Navier-Stokes equations of two-dimensional, steady, incompressible flow are studied using the unknown streamlines as one of the coordinates. The solution of the equations for in-plane motion is obtained, which corresponds to the well known solution of Couette flow, radial flow, spiral flow and the flow between two concentric cylinders. Possibilities of obtaining other solutions by this approach are also discussed.     

Direct and large-eddy simulations of transition in the compressible boundary layer

The transition to turbulence in the three-dimensional compressible boundary layer over a semi-infinite insulated flat plate is studied by means of direct and large-eddy simulations. Results are presented in the quasi-incompressible (Mach number equal to 0.5) and high supersonic (Mach number equal to 5) cases, both in temporal and spatial configurations. Simulations of controlled transition, in which a two-dimensional wave corresponding to the primary instability is introduced at the initial stage, allows us to study the secondary instability of the flow. The latter is triggered with the aid of a three-dimensional white-noise perturbation of small amplitude superposed upon the wave. At a low Mach number, a direct-numerical simulation shows that the fundamental mode is selected, leading to the peak-valley structure found by Klebanoff et al. (1962). The complete transition process is then studied, with emphasis on vortex-filament dynamics. It is shown that the development to turbulence is well simulated, at least for the prediction of average quantities of the flow. In the high Mach number case, no direct-numerical simulation is possible, and we use a subgrid-scale model, the structure-function model, in order to perform a large-eddy simulation of the transition. In this case, the subharmonic mode appears, giving rise to a staggered pattern of vortices. These vortices, which affect the whole thickness of the boundary layer, are more elongated than in the incompressible case.This work was supported by CNES-Avions Marcel Dassault in the frame of the Hermès program (Contract No. RDMF3/86), by DRET (Contract Nos. 87/808/11 and 88/150), and by CNRS (GDR Mécanique des Fluides Numérique and GDR Hypersonique).Unité associée CNRS.     

Point vortex dynamics: recent results and open problems

he concept of point vortex motion, a classical model in the theory of two-dimensional, incompressible fluid mechanics, was introduced by Helmholtz in 1858. Exploration of the solutions to these equations has made fitful progress since that time as the point vortex model has been brought to bear on various physical situations: atomic structure, large scale weather patterns, “vortex street” wakes, vortex lattices in superfluids and superconductors, etc. The point vortex equations also provide an interesting example of transition to chaotic behavior. We give a brief historical introduction to these topics and develop two of them in particular to the point of current understanding: (i) Steadily moving configurations of point vortices; and (ii) Collision dynamics of vortex pairs.     

Modes of vortex formation and transition to three-dimensionality in the wake of a freely vibrating cylinder

The present work is aimed to give some insight into the relation between vortex shedding modes and transition to three-dimensionality in the wake of a freely vibrating cylinder by establishing a numerical model and analyzing the relevant results of two- and three-dimensional simulations. The compressible flow past an elastically-mounted cylinder is solved by using the immersed boundary method (IB method). The cylinder is free to vibrate in the transverse direction with zero structure damping. The response of displacement amplitude is studied with the variation of reduced velocity. Whether P+S mode exists in three-dimensional flow and the occurrence of 2P mode is caused by flow transition from two-dimensional to three-dimensional are problems of concern. Both 2P and P+S wake modes are observed in two- and three-dimensional simulations. The numerical results indicate that the flow transition from two-dimensional to three-dimensional is coupled with the cylinder vibration in the synchronization/lock-in regime. The wake formation given by three-dimensional simulations suggests that the P+S mode might exist in reality when the flow is reverted to two-dimensional by vortex induced vibration (VIV) at Re=300–350. When Reynolds number increases to 425, the wake formation undergoes transition to three-dimensionality and 2P mode is observed. The effect of mass ratio on the flow transition to three-dimensionality is studied. The relationship between wake modes and aerodynamic forces is discussed.