Supersonic Laminar-Turbulent Gas Flow Past a Circular Cylinder

The flow past a circular cylinder with an isothermal surface at Mach numbers M=2 and 5 and Reynolds numbers ranging from 10⁴ to 10⁸ is investigated on the basis of the Reynolds equations using a differential two-equation turbulence model. The effect of flow turbulization on the separation point displacement, the separation zone length, the maximum velocity reduction in the separation zone, and the increase in the heat flux at the rear stagnation point is determined.     

Turbulent Transition in the Inlet Region of a Circular Pipe

Calculated and experimental data on turbulent transition in a circular pipe are analyzed. The calculations were performed using the three-parameter turbulence model. The dependence of the distance from the inlet to the point of minimum friction during transition on the Reynolds number for fixed inlet conditions and the distribution of the turbulence parameters over the pipe length and radius are obtained. The dependence of the maximum (critical) Reynolds number, Re^*, for which there is no transition in the pipe, on the inlet intensity and scale of turbulence is found. It is suggested that Re^* depends on the inlet perturbation parameters up to Re^* = 1000, where the friction coefficients for laminar and turbulent flows coincide.     

Laminar-Turbulent Transition in Spherical Couette Flow for Counter-Rotating Boundaries

The results of an experimental investigation of the transition to stochasticity are given for a thick spherical layer = (r ₂ – r ₁ = 1.006( r ₁ and r ₂ are the radii of the inner and outer spheres). The three-dimensional structure and the velocity fluctuation spectra of the supercritical flow regimes are studied over the range of variation of the Reynolds numbers, calculated on the basis of the parameters of the inner and outer spheres, -950 < Re₂ = ₂ r ₂ ² /v < -700 and 250 < Re₂ = ₂ r ₂ ² /v < 450, where ₂ and ₁ are the angular velocities of rotation of the inner and outer spheres, is the kinematic viscosity of the fluid in the layer, and negative values of Re₂ correspond to rotation of the layer boundaries in opposite directions.     

Mach Wave Effect on Laminar-Turbulent Transition in Supersonic Flow over a Flat Plate

The effect of a Mach wave (N wave) on laminar-turbulent transition induced by the first instability mode (Tollmien–Schlichting wave) in the flat-plate boundary layer is investigated on the basis of the numerical solution of Navier–Stokes equations at the freestream Mach number of 2.5. In accordance with the experiment, the N wave is generated by a two-dimensional roughness at the computation domain boundary corresponding to the side wall of the test section of a wind tunnel. It is shown that the disturbance induced by the backward front of the N wave in the boundary layer has no effect on the beginning of transition but displaces downstream the nonlinear stage of the first mode development. The disturbance induced by the forward front of the N wave displaces the beginning of transition upstream.     

Periodic Perturbations of a Class of Functional Differential Equations

Journal of Dynamics and Differential Equations - We study the existence of a connected “branch” of periodic solutions of T-periodic perturbations of a particular class of functional...     

Nonlinear Instability in the Region of Laminar-Turbulent Transition in Supersonic Three-Dimensional Flow over a Flat Plate

Direct numerical simulation is applied to obtain laminar-turbulent transition in supersonic flow over a flat plate. It is shown that, due to the nonlinear instability, Tollmien–Schlichting waves generated in the boundary layer lead to the formation of oblique disturbances in the flow. These represent a combination of compression and expansion waves, whose intensities can be two orders higher than that of external harmonic disturbances. The patterns of the three-dimensional flow over the plate are presented and the structures of the turbulent flat-plate boundary layers are described for the freestream Mach numbers M = 2 and 4.     

On Methods of Experimental Investigation of the Laminar-Turbulent Transition at Transonic Velocities

The results of an experimental investigation of the laminar-turbulent transition on a flat plate at transonic velocities, performed in a transonic wind tunnel, are presented. A comparative analysis of the transition data obtained by various methods is given.     

Periodic Solutions of Symmetric Perturbations of Gravitational Potentials

It is shown that for certain symmetric perturbations of gravitational potentials in the space, which admit two first integrals of motion, a circular solution of the unperturbed system with inclination different from 0 and π gives rise to a periodic solution of the reduced dynamics which is defined in the quotient space of the action by the subgroup that fixes the symmetry axis. In the planar case, if we assume that the system admits a first integral of motion which is also symmetric with respect to the origin, then it is shown that each circular solution of the unperturbed problem gives rise to a periodic solution of the perturbed system.     

An Updated Portrait of Transition to Turbulence in Laminar Pipe Flows with Periodic Time Dependence (A Correlation Study)

Transition to turbulence in axially symmetrical laminar pipe flows with periodic time dependence classified as pure oscillating and pulsatile (pulsating) ones is the concern of the paper. The current state of art on the transitional characteristics of pulsatile and oscillating pipe flows is introduced with a particular attention to the utilized terminology and methodology. Transition from laminar to turbulent regime is usually described by the presence of the disturbed flow with small amplitude perturbations followed by the growth of turbulent bursts. The visual treatment of velocity waveforms is therefore a preferred inspection method. The observation of turbulent bursts first in the decelerating phase and covering the whole cycle of oscillation are used to define the critical states of the start and end of transition, respectively. A correlation study referring to the available experimental data of the literature particularly at the start of transition are presented in terms of the governing periodic flow parameters. In this respect critical oscillating and time averaged Reynolds numbers at the start of transition; Re _os,crit and Re _ta,crit are expressed as a major function of Womersley number, $\sqrt {\omega ^\prime } $ defined as dimensionless frequency of oscillation, f. The correlation study indicates that in oscillating flows, an increase in Re _os,crit with increasing magnitudes of $\sqrt {\omega ^\prime } $ is observed in the covered range of $1<\sqrt {\omega ^\prime } <72$ . The proposed equation (Eq. 7), ${\rm{Re}}_{os,crit} ={\rm{Re}}_{os,crit} \left( {\sqrt {\omega ^\prime } } \right)$ , can be utilized to estimate the critical magnitude of $\sqrt {\omega ^\prime }$ at the start of transition with an accuracy of ±12?% in the range of $\sqrt {\omega ^\prime } <41$ . However in pulsatile flows, the influence of $\sqrt {\omega ^\prime }$ on Re _ta,crit seems to be different in the ranges of $\sqrt {\omega ^\prime } <8$ and $\sqrt {\omega ^\prime } >8$ . Furthermore there is rather insufficient experimental data in pulsatile flows considering interactive influences of $\sqrt {\omega ^\prime } $ and velocity amplitude ratio, A ₁. For the purpose, the measurements conducted at the start of transition of a laminar sinusoidal pulsatile pipe flow test case covering the range of 0.21<?A ₁?<0.95 with $\sqrt {\omega ^\prime } <8$ are evaluated. In conformity with the literature, the start of transition corresponds to the observation of first turbulent bursts in the decelerating phase of oscillation. The measured data indicate that increase in $\sqrt {\omega ^\prime } $ is associated with an increase in Re _ta,crit up to $\sqrt {\omega ^\prime } =3.85$ while a decrease in Re _ta,crit is observed with an increase in $\sqrt {\omega ^\prime } $ for $\sqrt {{\omega }'} >3.85$ . Eventually updated portrait is pointing out the need for further measurements on i) the end of transition both in oscillating and pulsatile flows with the ranges of $\sqrt {\omega ^\prime } <8$ and $\sqrt {\omega ^\prime } >8$ , and ii) the interactive influences of $\sqrt {\omega ^\prime } $ and A ₁ on Re _ta,crit in pulsatile flows with the range of $\sqrt {\omega ^\prime } >8$ .     

Numerical Simulation of the Flow Around an Infinitely Long Circular Cylinder in the Transition Regime

The development of three-dimensional structures and the succeeding transition to turbulence occurs in the wake of a circular cylinder at Reynolds numbers 190≤Re≤330. This regime is investigated numerically by means of a spectral element method. Earlier numerical works aimed mainly at reproducing characteristic wake patterns observed in experiments. Small sizes of computational domains and short integration times were chosen to save computational resources. Consequently, the quantitative results show a considerable scatter. Within this work, a step by step approach to highly accurate direct numerical simulations is described. Thorough studies of the effect of resolution and blockage are performed in the laminar, two-dimensional regime, resulting in Reynolds number relationships that exactly reproduce experimental data. Based on these results, a stability analysis is performed to obtain wavelengths that are unstable against spanwise perturbations and the critical Reynolds number for the onset of three-dimensionality. The most unstable wavelengths of the “mode A” and “mode B” instabilities and its multiples are used as periodicity length for direct numerical simulations. Effects of integration time, resolution in streamwise as well as spanwise directions, and periodicity length on the flow quantities are studied. Numerically obtained Reynolds number relationships of Strouhal number and base-pressure coefficients that fit accurately within measured results are given for the first time. Curves for drag and lift coefficients are provided and compared with previous numerical studies. Furthermore, physical interpretations of the wake transition are discussed. Since the separation of physical features and effects of experimental arrangements are frequently an open question, our numerical results are able to supply a contribution to the understanding of the physics of cylinder flow. Received 12 September 2000 and accepted 26 June 2001     

On Periodic Perturbations of Uniform Motion of Maxwell's Planetary Ring

We examine the case when equally sized small moons arrange themselves on the vertices of a regular n-gon for n 7. For n 4, there are at least 3 pure imaginary characteristic exponents, each of which has multiplicity = 1, a surprising result that makes it possible to apply the Lyapunov center theorem to verify the existence of some periodic perturbations. For sufficiently large n, when the regular n-gon is the unique central configuration, the number of families of periodic perturbations is at least equal to 2n – (n + 1)/4, where x is the greatest integer less than or equal to x.     

Numerical Study of the Effect of Periodic Velocity Excitation on Aerodynamic Characteristics of an Oscillating Circular Cylinder

A finite element method based on ALE formulation has been adopted in order to examine the effect of periodic velocity excitation on the aerodynamic characteristics of an oscillating circular cylinder. Periodic excitation, which was placed on the cylinder surface, stimulated the separated shear layers around the cylinder, and numerical results showed that some excitation can reduce negative damping, which is caused by unsteady lift force, and thereby stabilize the aerodynamics of the cylinder. Furthermore, the change of lift phase caused by periodic excitation seems to be important in stabilizing the aerodynamics of the cylinder. The simulation also confirmed that periodic excitation can suppress the vortex-induced vibration of the cylinder.     

管内湍流旋流的数值计算

本文用有限差分法对直管内的湍流旋流进行了数值模拟。计算中采用Ｂｏｕｓｓｉｎｅｓｑ湍流涡粘性假设的基本思想和Ｋ－ε双方程模型来求解雷诺应力各分量。为了反映旋流中湍流转输的非均匀性和各向异性特征，对雷诺应力各分量及与之相主尖的各湍流粘性系数分别进行计算。计算结果表明该模型能较好地反映直管内湍流旋流的流动结构。     

Stability Analysis of Continuous Systems with Structural Perturbations

The paper is a survey of the development of the direct Lyapunov method based on matrix-valued functions for systems with structural perturbations. Sufficient conditions for stability in the large, uniform asymptotic stability in the large, exponential stability in the large, and instability of nonlinear continuous systems under structural perturbations are presented.     

On the Lower Critical Reynolds Number for Flow in a Circular Pipe

Flow in a circular pipe is investigated experimentally at Reynolds numbers higher than that at which the resistance coefficients calculated from the Blasius formula for laminar flow and from the Prandtl formula for turbulent flow are equal. The corresponding Reynolds number based on the mean-flow velocity and the pipe diameter is about 1000. The experiments were performed at a high level of inlet pulsations produced by feeding gas into the pipe through a hole with a diameter several times smaller than the pipe diameter. In our experiments the critical Reynolds number was determined as the value, independent of the distance from the inlet, at which the ratio of the axial to the mean-flow velocity as a function of the Reynolds number deviated from 2. At the maximum ratio of the pipe cross-sectional area to the area of the hole through which the gas entered the pipe, equal to 26, the critical Reynolds number was about 2300. After a fivefold increase in the hole area the critical Reynolds number increased by approximately 4%.At Reynolds numbers below 2000, after at a high level of the inlet pulsations an almost laminar flow had developed in the pipe, a perturbation was introduced by inserting a diametrically oriented cylindrical rod with a diameter 10–20 times smaller than the pipe diameter. In these experiments, at Reynolds numbers higher than 1000, at a distance from the rod equal to 50 pipe diameters the axial to mean-flow velocity ratio was less than 2, approaching this value again at large distances from the rod. The insertion of the rod led to a decrease in the critical Reynolds number by approximately 12%.     

On the Possibility of Delaying Laminar-Turbulent Transition at High Reynolds Numbers by the Optimal Choice of Body Forces

The possibility of delaying laminar-turbulent transition on a flat plate in a longitudinal viscous incompressible flow by the optimal choice of the body force distribution in the boundary layer is discussed. It is shown that even for very high Reynolds numbers, Re 10¹⁰, a body force distribution can be found such that the corresponding boundary layer flow is absolutely stable, while the total drag of the body is less than that in the absence of body force action on the flow.     

Analysis of Linear Viscous Flow with a Free Surface in a Circular Cylinder Subjected to Small-Amplitude Circular Oscillations

A fluid flow with a free surface inside a circular cylinder subjected to horizontal, circular oscillation was analyzed theoretically and numerically under the assumption of small-amplitude oscillation and high Reynolds number. It was shown that the nature of the oscillatory flow is of a standing-wave type when projected onto the axial plane and of a progressive-wave type when projected onto the azimuthal plane. The Stokes drift motion in the azimuthal direction and the steady streaming velocity at the edge of the bottom- and side-wall boundary layers are then used in the numerical computation for the steady axisymmetric recirculatory flow outside the boundary layers. We have found that the solutions can be well predicted by asymptotic analysis for the full Navier–Stokes equations in the low streaming-Reynolds-number limit. A simple experiment on flow visualization revealed a good agreement in the surface flow pattern on the bottom wall. It also provided steady recirculatory flows that were not much different from the numerical results on the whole. Received 16 August 1999 and accepted 14 January 2000     

Numerical Simulation of the Unsteady Laminar Flow Past a Circular Cylinder with a Perforated Sheath

The unsteady two-dimensional laminar flow past a circular cylinder encased in a perforated sheath is numerically simulated. On the basis of the calculated results a technique for controlling the wake flow by diverting a portion of the flow from the forward stagnation point through internal ducts to orifices in the sheath located in the separation zone, is analyzed.     

多圆孔圆板问题的数值解

提出了一种求解多圆孔圆板问题的新方法。首先引入了基本特解，它由主要部分和附加部分组成。主要部分为带奇点无限平板的一个特殊弹性力学解，奇点取在内圆孔的中心处。附加部分为实心圆饭的一个特解。整个基本特解满足外圆周界为自由条件。文中把待求解取为特解系的形式，其中待定系数可用变分原理得出。最后给出了算例。