Mixed Convection Flow Past a Horizontal Plate: The Trailing Edge

The mixed convection flow past a horizontal plate which is aligned under a small angle of attack to a uniform free stream will be considered in the limit of large Reynolds number and small Richardson number close to the trailing edge. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transient Flow of a Radiating Gas Between Concentric Spheres

The transient flow of a radiating gas between concentric rotatingspheres is discussed under the optically thin gas approximation.The temperatures of the spheres are large but the differencein the temperatures is assumed small, so that the Grashof numberis correspondingly small. When time t'<0, the spheres arestationary and a steadystate nonrotating flow exists betweenthe spheres as a result of the impressed temperature gradient.At t'=0, the spheres are rotated impulsively from rest and thesubsequent flow pattern is studied by perturbation series expansionfor small Grashof number. The flow characteristic is typifiedby an exponentially decaying component superimposed on a steadyflow. The conclusion is that the flow is stable in the givencircumstances. The temperature distribution shows a marked differencewith that in the absence of radiation and this is discussed.     

Decomposition of the Flow Polynomial

The flow polynomials denote the number of nowhere-zero flows on graphs, and are related to the well-known Tutte polynomials and chromatic polynomials. We will show the decomposition of the flow polynomials by edge-cuts and vertex-cuts of size 2 or 3. Moreover by using this decomposition, we will consider what kind of graphs have the same flow polynomials. Another application of the decomposition results is that if a bridgeless graph G does not admit a nowhere-zero k-flow and G has a small vertex- or edge-cut, then a proper bridgeless subgraph of G (a graph minor) does not admit a nowhere-zero k-flow either.     

Direct Resonance of Nonaxisymmetric Disturbances in Pipe Flow

A mechanism which may lead to algebraic growth, followed by exponential decay, of small nonaxisymmetric disturbances in pipe flow is considered. The mechanism is interpreted as a direct resonance between the perturbations of the pressure and the streamwise velocity. The eigenvalue problems for the pressure and the velocity modes have been solved numerically for complex streamwise wave number, and 36 resonances have been investigated. A plot of the propagation speed versus the damping rate shows that the resonances follow certain sequences as the azimuthal wave number increases. The largest propagation speed is found to be ≈ 0.69 times the centerline velocity. No lowest speed is obtained, and as the azimuthal wave number increases the propagation speed decreases. The effects of changing the Reynolds number have also been investigated. It is found that the streamwise wave number and the damping rate are proportional to 1/R as R → ∞. The complex phase speed and the propagation speed become independent of R in the same limit.     

On Flow Past Bodies in Aligned Magnetic Fields

In inviscid, incompressible flow past a body in a strong alignedmagnetic field at very low magnetic Reynolds number, the flowfar from the body may be rectilinear but non-uniform. The paperdiscusses the evolution of small perturbations towards or awayfrom such a rectilinear flow and concludes that such evolutioncan only occur gradually (i.e. over many diameters) ahead ofthe body. Downstream the decay must occur close to the body.Two dimensional and axisymmetric flows are considered.     

Formation of Coherent Structures in Kolmogorov Flow with Stratification and Drag

We study a weakly stratified Kolmogorov flow under the effect of a small linear drag. We perform a linear stability analysis of the basic state. We construct the finite dimensional dynamical system deriving from the truncated Fourier mode approximation. Using the Reynolds number as bifurcation parameter we build the corresponding diagram up to Re=100. We observe the coexistence of three coherent structures.     

微通道周期流动电位势及电粘性效应

求解了双电层的Poisson-Boltzmann方程和流体运动的Navier-Stokes方程,得到在周期压差作用下,二维微通道的周期流动电位势,流动诱导电场和液体流动速度的解析解．量纲分析表明,流体电粘性力与以下3个参数有关:1) 电粘性数,它表示定常流动时,通道最大电粘性力与压力梯度的比;2) 形状函数,它表示电粘性力在通道横截面的分布形态; 3) 耦合系数,它表示电粘性力的振幅衰减特征和相位差．分析结果表明,微通道周期流动诱导电场、流动速度与频率Reynolds数有关．在频率Reynolds数小于1时,流动诱导电场随频率Reynolds数变化很慢．在频率Reynolds数大于1时,流动诱导电场随频率Reynolds数的增加快速衰减．在通道宽度与双电层厚度比值较小情况下,电粘性效应对周期流动速度和流动诱导电场有重要影响．     

Initial Algebraic Growth of Small Angular Dependent Disturbances in Pipe Poiseuille Flow

The evolution of small, angular dependent velocity disturbances in laminar pipe flow is studied. In particular, streamwise independent perturbations are considered. To fully describe the flow field, two equations are required, one for the radial and the other for the streamwise velocity perturbation. Whereas the former is homogeneous, the latter has the radial velocity component as a forcing term. First, the normal modes of the system are determined and analytical solutions for eigenfunctions, damping rates, and phase velocities are calculated. As the azimuthal wave number (n) increases, the damping rate increases and the phase velocities decrease. Particularly interesting are results showing that the phase velocities associated with the streamwise eigenfunctions are independent of the radial mode index when n = 1, and when n = 5 the same is obtained for phase velocities associated with the eigenfunctions of the radial component. Then, the initial value problem is treated and the time development of the disturbances is determined. The radial and the azimuthal velocity components always decay but, owing to the forcing, the streamwise component shows an initial algebraic growth, followed by a decay. The kinetic energy density is used to characterize the induced streamwise disturbance. Its dependence on the Reynolds number, the radial mode, and the azimuthal wave number is investigated. With a normalized initial disturbance, n = 1 gives the largest amplification, followed by n = 2 etc. However, for small times, higher values of n are associated with the largest energy density. As n increases, the distribution of the streamwise velocity perturbation becomes more concentrated to the region near the pipe wall.     

A Hybrid Method for Low Reynolds Number Flow Past an Asymmetric Cylindrical Body

Low Reynolds number fluid flow past a cylindrical body of arbitrary shape in an unbounded, two-dimensional domain is a singular perturbation problem involving an infinite logarithmic expansion in the small parameter ε, representing the Reynolds number. We apply a hybrid asymptotic–numerical method to compute the drag coefficient, C _D and lift coefficient C _L to within all logarithmic terms. The hybrid method solution involves a matrix M , depending only on the shape of the body, which we compute using a boundary integral method. We illustrate the hybrid method results on an elliptic object and on a more complicated profile.     

DNS of a Turbulent Channel Flow with Streamwise Rotation - Study of the Reverse Effect of the Cross Flow

Modeling of rotating turbulent flows is a major issue in engineering applications. In this work a turbulent channel flow rotating about the streamwise direction is presented. The theory is based on the investigations of [3] employing Lie group analysis. It was found that a cross flow in spanwise direction is induced. A series of direct numerical simulations (DNS) has been conducted for both different rotation rates and different Reynolds numbers to validate the cross flow. In addition some new interesting effects were observed. The averaged profile ū₃ of the cross flow is formed like a ‘S’ that means it exhibits a triple zero-crossing which denotes regions of reverse flow. Alaso a reverse effect is seen which means that for small rotation rates up to Ro=10 the spanwise mean velocity profiles increase and at rotation number Ro=14 this effect appears to reverse. Both effects were observed at two different Reynolds numbers Re = 180 and Re = 270. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Aerodynamic Interaction of Systems of Bodies in a Supersonic Flow

Use of numerical experiment methods gives broad opportunities for studying complex flows by rather simple and inexpensive means. This work is devoted to researches on aerodynamic interference of bodies in supersonic flows. Interest to similar problems is shown in a number of areas of modern mechanics: meteoritics, movement of multiphase media, applied fields. The head shock wave before a configuration of spheres with growth of distance h monotonously breaks up to individual shock waves before spheres of forward lines. The basic conclusion of this research is the following: distortion of supersonic stream around systems of bodies as though is transferred downstream, i.e. the influence of back bodies on the flow nearby forward bodies is small. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flow of a conducting dusty gas past a flat plate

An exact solution is derived by Laplace-transform technique for the problem of the flow of a conducting dusty gas occupying a semi-infinite space in the presence of a transverse magnet field. It is assumed that the flow is independent of the distance parallel to the plate and that the mass concentration of dust is small. Formulas are derived in terms of a constant external impulsive velocity field for the velocity profiles of both the dust and the conducting gas only for values of Hartmann number greater than or equal to unity. For these values of the Hartmann number the skin friction is also obtained.     

Nonlinear Spatial Evolution of Small Disturbances from Boundary Conditions in Flat Inclined-Channel Flow

The asymptotic behavior of small disturbances as they evolve spatially from boundary conditions in a flat inclined channel is determined. These small disturbances develop into traveling waves called roll waves, first discussed by Dressler in 1949. Roll waves exist if the Froude number F exceeds 2, which consist of a periodic pattern of bores, or discontinuities. After confirming the instability condition for   F > 2  for the linearized equations in the boundary value case, the nonlinear boundary value problem for the weakly unstable region of F slightly larger than 2 is studied. Multiple scales and the Fredholm alternative theorem are applied to determine the evolution of the solution in space. It is found that the solution is dominated by the evolution of the disturbance along one characteristic. The shock conditions governing the asymptotic solution are determined and these conditions are used to determine the approximate shape of the resulting traveling wave from the solution. Both asymptotic and numerical results for periodic disturbances are presented.     

Compressible Modes of the Rotating-Disk Boundary-Layer Flow Leading to Absolute Instability

This work is devoted to the clarification of the viscous compressible modes particularly leading to absolute instability of the three-dimensional generalized Von Karman's boundary-layer flow due to a rotating disk. The infinitesimally small perturbations are superimposed onto the basic Von Karman's flow to achieve linearized viscous compressible stability equations. A numerical treatment of these equations is then undertaken to search for the modes causing absolute instability within the principle of Briggs–Bers pinching. Having verified the earlier incompressible and inviscid compressible results of [ 1–3 ], and also confirming the correct match of the viscous modes onto the inviscid ones in the large Reynolds number limit, the influences of the compressibility on the subject matter are investigated taking into consideration both the wall insulation and heat transfer. Results clearly demonstrate that compressibility, as the Mach number increases, acts in favor of stabilizing the boundary-layer flow, especially in the inviscid limit, as far as the absolute instability is concerned, although wall heating and insulation greatly enhances the viscous absolutely unstable modes (even more dramatic in the case of wall insulation) by lowering down the critical Reynolds number for the onset of instability, unlike the wall cooling.     

A Suggested Mechanism for Nonlinear Wall Roughness Effects on High Reynolds Number Flow Stability

The interactions between an uneven wall and free stream unsteadiness and their resultant nonlinear influence on flow stability are considered by means of a related model problem concerning the nonlinear stability of streaming flow past a moving wavy wall. The particular streaming flows studied are plane Poiseuille flow and attached boundary-layer flow, and the theory is presented for the high Reynolds number regime in each case. That regime can permit inter alia much more analytical and physical understanding to be obtained than the finite Reynolds number regime; this may be at the expense of some loss of real application, but not necessarily so, as the present study shows. The fundamental differences found between the forced nonlinear stability properties of the two cases are influenced to a large extent by the surprising contrasts existing even in the unforced situations. For the high Reynolds number effects of nonlinearity alone are destabilizing for plane Poiseuille flow, in contrast with both the initial suggestion of earlier numerical work (our prediction is shown to be consistent with these results nevertheless) and the corresponding high Reynolds number effects in boundary-layer stability. A small amplitude of unevenness at the wall can still have a significant impact on the bifurcation of disturbances to finite-amplitude periodic solutions, however, producing a destabilizing influence on plane Poiseuille flow but a stabilizing influence on boundary-layer flow.     

The Trailing Edge Problem For Mixed Convection Flow Past a Horizontal Plate

The influence of buoyancy onto the boundary‐layer flow past a horizontal plate aligned parallel to a uniform free stream is characterized by the buoyancy parameter K = Gr/Re^5/2 where Gr and Re are the Grashof and Reynolds number, respectively. An asymptotiy analysis of the complete flow field including potential flow, boundary layer, wake and interaction region is given for small buoyancy parameters and large Reynolds numbers in the distinguished limit KRe^1/4 = O(1). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Steady Viscous Flow of Two Differentially Rotating Immiscible Fluids

We have computed the steady, axisymmetric viscous boundary layers on either side of an interface between two immiscible, incompressible fluids that are in rigid body rotation far from the interface. The internal rotational Froude number is assumed small so that the interface may be considered horizontal. An application of our results to the spinup from rest of two immiscible slightly viscous fluids in a vertically mounted cylinder is discussed.     

Optimal Disturbance Growth in Watertable Flow

The linear stability of a liquid flow down an inclined plane is investigated. The equations governing the evolution of the disturbance are written in vector form where the dependent variables are the normal velocity and the normal vorticity. Similar to other shear flows, it is shown that there can be transient growth in the energy of a disturbance followed by an exponential decay although all eigenvalues predict decay only. Parameter studies reveal that the maximum amplification occur for waves with no streamwise dependence and with a spanwise wavenumber of (1). The mechanism involved in this growth is analyzed. A free surface parameter (S) can be identified that is related to the extent gravity and surface tension influence the free surface. A scaling of the equations is studied which revealed that the maximum transient growth scales with the Reynolds number as Re² if k2 S Re² is kept constant, where k is the absolute value of the wavenumber vector. For small values of S exponential growth of free-surface modes also exists. In general, however, we have found that for moderate times the transient growth dominates over the exponential growth and that its characteristics are similar to the transient growth found in other shear flows, e.g., plane Poiseuille flow.     

变截面圆形管道粘性流动的伽辽金-摄动杂交解

采用伽辽金-摄动杂交法来研究壁面是正弦形状的变截面圆形管道的粘性流动,从而避免了摄动小参数的局限性和单纯伽辽金法基函数选取的任意性的困难．讨论了边界和雷诺数对流动的影响,获得流动分离点和附着点的位置,还分析了壁面剪应力和摩擦系数沿轴向的变化情况．在小参数的情况下,计算所获得的结果与摄动解吻合良好．     

Stability of Kähler-Ricci Flow

We prove the convergence of Kähler-Ricci flow with some small initial curvature conditions. As applications, we discuss the convergence of Kähler-Ricci flow when the complex structure varies on a Kähler-Einstein manifold.