Energy transient growth in curved channel flow

Transient growth of perturbations due to non-normality for the curved channel Poiseuille flow (CCPF) is presented. The study covers a wide range of the radii ratios ?? as well as axial and azimuthal modes, with the purpose of complementing the results of linear stability for this flow with a study of the optimal linear growth possible in the linearly stable parameter regions. For the wide-gap case, the transient growth of streamwise-azimuthal modes that grow most is of low level and suppressed by curvature. It is also found that as curved channel flow approaches the flow in a straight channel enough, both the normal and non-normal stability characteristics are almost identical to that of plane Poiseuille flow. The modulation of the basic circular Poiseuille flow by the presence of azimuthal streaks resulting from the significant growth of initial perturbations can be clearly visualized. For the transition region between the wide-gap case and the narrow-gap case, the sensitivity of eigenvalues is shown to be closely related to the magnitude of transient growth, which is tuned by curvature in a smooth way.     

Critical threshold in pipe flow transition

This study provides a numerical characterization of the basin of attraction of the laminar Hagen-Poiseuille flow by measuring the minimal amplitude of a perturbation required to trigger transition. For pressure-driven pipe flow, the analysis presented here covers autonomous and impulsive scenarios where either the flow is perturbed with an initial disturbance with a well-defined norm or perturbed by means of local impulsive forcing that mimics injections through the pipe wall. In both the cases, the exploration is carried out for a wide range of Reynolds numbers by means of a computational method that numerically resolves the transitional dynamics. For , the present work provides critical amplitudes that decay as Re(-3/2) and Re(-1) for the autonomous and impulsive scenarios, respectively. For Re=2875, accurate threshold amplitudes are found for constant mass-flux pipe by means of a shooting method that provides critical trajectories that never relaminarize or trigger transition. These transient states are used as initial guesses in a damped Newton-Krylov method formulated to find periodic travelling wave solutions that either travel downstream or exhibit a helicoidal advection.     

Transition to Turbulence in Pipe Flow

We review the results of recent experimental investigations into transition to turbulence in fluid flow through a circular straight pipe, at room temperature. The stability of Hagen–Poiseuille flow was investigated using impulsive perturbations by either injecting or sucking small amounts of fluid through holes in the wall of the pipe. The evolution of the induced patches of disturbed flow were observed using flow visualization and laser Doppler velocimetry. The principle result obtained was a finite amplitude stability curve where the critical amplitude of the disturbance required to cause transition is found to be inversely proportional to the Reynolds number. Estimates for the lower threshold value of Reynolds number which is required to sustain turbulence were also measured.     

圆管流动的二次转捩

利用直接数值模拟方法求解N-S方程来研究空间发展的圆管转捩,计算中雷诺数Re选定为3000。在局部壁面引入的周期性吹吸(PSB)扰动作用下,圆管中首先出现了塞流结构,并向下游迁移,同时有稳定的流向条带结构形成;在塞流结构离开计算域后,随着扰动的不断发展,流向条带结构逐渐破裂失稳,圆管流动出现了第二次转捩,这是一种新的转捩形式,我们称之为“二次转捩”。     

Capture of convective vortices by an unsteady-state flow in a Hele-Shaw cell

The specific transient regime of thermal convection in a Hele-Shaw cell has been studied experimentally and theoretically. It is a kind of unsteady-state, four-vortex flow, symmetric about the vertical axis, in the form of periodic pulsations of the lower convective cells, with the latter being partially adsorbed by the upper vortices. The transient pulsational regimes in a Hele-Shaw cell were revealed experimentally in studying convective motions in a transformer oil. These pulsational flows are unstable and with time they are always rearranged into a steady four-vortex vibrational regime with reconnection of corner vortices. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 80, No. 1, pp. 100–106, January–February, 2007.     

An optimal path to transition in a duct

This paper is concerned with the transition of the laminar flow in a duct of square cross section. As in the similar case of pipe flow, the motion is linearly stable for all Reynolds numbers, rendering this flow a suitable candidate for a study of the 'bypass' path to turbulence. It has already been shown that the classical linear optimal perturbation problem, yielding optimal disturbances in the form of longitudinal vortices, fails to provide an 'optimal' path to turbulence, i.e. optimal perturbations do not elicit a significant nonlinear response from the flow. Previous simulations have also indicated that a pair of travelling waves generates immediately, by nonlinear quadratic interactions, an unstable mean flow distortion, responsible for rapid breakdown. By the use of functions quantifying the sensitivity of the motion to deviations in the base flow, the optimal travelling wave associated with its specific defect is found by a variational approach. This optimal solution is then integrated in time and shown to display a qualitative similarity to the so-called 'minimal defect', for the same parameters. Finally, numerical simulations of an 'edge state' are conducted, to identify an unstable solution that mediates laminar-turbulent transition and relate it to results of the optimization procedure.     

SPH Simulation of Low Reynolds Number Planar Shear Flow and Heat Convection

The present study reports on a set of computer programmable SPH formulations, which are used to simulate transient planar shear flows, and in particular Poiseuille flow and Couette flow with different types of body forces. The flows examined have Reynolds numbers within the range 0.05∼50. SPH results agree well with analytical solutions for those situations amenable to an analytical treatment, with the largest deviation being less than 2.0 %. The accuracy of a SPH formulation for heat convection, with particular emphasis in the viscous heat dissipation, is also tested via a steady convective heat transfer case for a combined Poiseuille and Couette flow.     

驻波场中圆柱形换热管外声流特性的数值研究

基于非线性Navier-Stokes方程推导了一般性的声流控制方程,并利用有限元软件COMSOL对平面驻波声场中单根圆柱形换热管和双圆柱形换热管外包含二阶流场信息的Navier-Stokes方程进行全域求解,得到了一阶和二阶流场信息。研究了雷诺数Re和斯特劳哈尔数Sr对换热管外声流结构的影响规律。研究发现:随Re和Sr的增大,换热管边界层内的涡流区域变小,直至消失,而边界层外的涡流区域逐渐增大;且单换热管外涡流个数由8个减少到4个,双换热管外涡流个数从12个减少到8个。此外,边界层内和边界层外的涡流结构呈反向旋转;边界层外流场沿着振荡方向远离换热管,而在垂直方向流向换热管。     

Symmetric and asymmetric Taylor vortex flow in spherical gaps

Summary The rotationally symmetric flow between two concentric rotating spheres is investigated both theoretically and experimentally. The non-uniqueness of the supercritical flow exhibits three different modes with zero, one and two Taylor vortices in each hemisphere. These modes are realized by different accelerations of the inner sphere from the state at rest. A initial value code, based on a finite difference method, is used for the numerical simulation. The existence regions of the different supercritical flows are connected with symmetric and asymmetric transitions. It is found, that a steady state can exist asymmetric with respect to the equator. The flow is analyzed by plotting the size of the Taylor vortices, the depending variables , ,V, the velocity distributions and the torque. A comparison between theory and experiments for the observed modes of flow is given.     

Control of vortex shedding on two- and three-dimensional aerofoils

We review and expand on the control of separated flows over flat plates and aerofoils at low Reynolds numbers associated with micro air vehicles. Experimental observations of the steady-state and transient lift response to actuation, and its dependence on the actuator, aerofoil geometry and flow conditions, are discussed and an attempt is made to unify them in terms of their excitation of periodic and transient vortex shedding. We also examine strategies for closed-loop flow and flight control using actuation of leading-edge vortices.     

Exact solutions for Couette and Poiseuille flows for fourth grade fluids

Summary Exact solutions for four types of flows between two parallel plates are presented, viz. Couette flow, plug flow, Poiseuille flow and generalized Couette flow. The nonlinear second-order ordinary differential equation for the velocity field is solved exactly in each case. These solutions are compared to those found by perturbation and homotopy analysis methods by Siddiqui et al. [1].     

Some indications from instability results about the effectiveness of wall heating as a control option for channel flow

This paper is a review of some of our recent work on the effect of wall heating on the stability of laminar flow in a channel. The summary of our results, some of them unexpected, is as follows. Viscosity stratification has very little effect on transient growth, whereas it results in linear mode stabilising or destabilising by an order of magnitude. It has hitherto been accepted that heat diffusivity does not affect stability. This is however true only for linear instability, transient growth is affected by an order of magnitude. Unusually, the growth is spanwise-independent and not in the form of streamwise vortices. It is also shown that flow is destabilised by secondary modes as the viscosity ratio increases. However, the viscosity ratio has no role in the selection of the pattern of Λ vortices.     

Transient behaviour and stability points of the Poiseuille flow of a KBKZ-fluid

The Poiseuille flow of a KBKZ-fluid, being a nonlinear viscoelastic model for a polymeric fluid, is studied. The flow starts from rest and especially the transient phase of the flow is considered. It is shown that under certain conditions the steady flow equation has three different equilibrium points. The stability of these points is investigated. It is proved that two points are stable, whereas the remaining one is unstable, leading to several peculiar phenomena such as discontinuities in the velocity gradient near the wall of the pipe (spurt) and hysteresis. Our theoretical results are confirmed by numerical calculationsof the velocity gradient.     

Primary crossflow vortices,secondary absolute instabilities and their control in the rotating-disk boundary layer

The three-dimensional boundary layer produced by a disk rotating in otherwise still fluid is analytically investigated and its stability properties are systematically established. Using a local parallel flow approximation, finite-amplitude primary travelling vortices governed by a nonlinear dispersion relation are obtained. A secondary stability analysis yields the secondary linear dispersion relation and the secondary absolute growth rate, which determines the long-term stability of the primary nonlinear vortex-trains. By using these local characteristics, spatially developing global patterns of crossflow vortices are derived by employing asymptotic techniques. This approach accounts for both the self-sustained behaviour, exhibiting a sharp transition from laminar to turbulent flow, and the spatial response to external harmonic forcing, for which onset of nonlinearity and transition both depend on the forcing parameters. Based on these results, an open-loop control method is described in detail. Its aim is not to suppress the primary fluctuations but rather to enhance them and to tune them to externally imposed frequency and modenumber, and thereby to delay onset of secondary absolute instability and transition. It is shown that transition can be delayed by more than 100 boundary-layer units.     

The flow structure of a puff

From time-resolved stereoscopic particle image velocimetry measurements over the entire circular cross section of a pipe, a first-of-its-kind quasi-instantaneous three-dimensional velocity field of a turbulent puff at a low Reynolds number is reconstructed. At the trailing edge of the puff, where the laminar flow undergoes transition to turbulence, pairs of counterrotating streamwise vortices are observed that form the legs of large hairpin vortices. At the upstream end of the puff, a quasi-periodic regeneration of streamwise vortices takes place. Initially, the vortex structure resembles a travelling wave solution, but as the vortices propagate into the turbulent region of the puff, they continue to develop into strong hairpin vortices. These hairpin vortices extract so much energy from the mean flow that they cannot be sustained. This structure provides a possible explanation for the intermittent character of the puffs in pipe flow at low Reynolds numbers.     

Study on boundary layer transition of a rotating disk

Summary Behaviour of spiral vortices being generated in transition regime of a disk rotating in otherwise undisturbed fluid is experimentally studied in detail. Through visualizations of the transition regime by using close-up camera, new striped flow patterns originating along the axis of spital vortices are found to be ring-like vortices which occur on the surfaces of each spiral vortices. Mechanism of the spiral vortex is clarified by cutting the vortices by strobo slit light. It is also found out experimentally that the phase velocity of the vortices is zero.     

BEM approaches and simplified kinetic models for the analysis of damping in deformable MEMS

Microflows occurring in MEMS are addressed by applying a continuum BEM code for quasi-static Stokes flow accounting for slip boundary condition. Working conditions typical of transition and rarefied flows are treated in a simplified manner by employing a viscosity corrected according to semi-analytical solutions for the linearized BGK model of Couette and Poiseuille flows. Numerical results are compared with experiments showing excellent agreement     

Homotopy analysis of Couette and Poiseuille flows for fourth grade fluids

Summary The steady flow of a fluid, called a fourth grade fluid, between two parallel plates is considered. Depending upon the relative motion of the plates we analyze four types of flows: Couette flow, plug flow, Poiseuille flow and generalized Couette flow. In each case, the nonlinear differential equation describing the velocity field is solved using perturbation technique and homotopy analysis method. The pressure distribution is also found. It is observed that the homotopy analysis method is more efficient and flexible than the perturbation technique.     

Stability of concentrated suspensions under Couette and Poiseuille flow

The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. A linear stability analysis of the two-phase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Poiseuille flow gives rise to base state solutions that exhibit a jammed and unyielded region, due to shear-induced migration, as the maximum packing fraction is approached. The stability characteristics of the resulting Bingham-type flow is investigated, and the connections to the stability problem for the related classical Bingham flow problem are discussed.     

Turbulent flow in smooth and rough pipes

Recent experiments at Princeton University have revealed aspects of smooth pipe flow behaviour that suggest a more complex scaling than previously noted. In particular, the pressure gradient results yield a new friction factor relationship for smooth pipes, and the velocity profiles indicate the presence of a power-law region near the wall and, for Reynolds numbers greater than about 400x103 (R+>9x103), a logarithmic region further out. New experiments on a rough pipe with a honed surface finish with krms/D=19.4x10-6, over a Reynolds number range of 57x103-21x106, show that in the transitionally rough regime this surface follows an inflectional friction factor relationship rather than the monotonic relationship given in the Moody diagram. Outer-layer scaling of the mean velocity data and streamwise turbulence intensities for the rough pipe show excellent collapse and provide strong support for Townsend's outer-layer similarity hypothesis for rough-walled flows. The streamwise rough-wall spectra also agree well with the corresponding smooth-wall data. The pipe exhibited smooth behaviour for ks+ < or =3.5, which supports the suggestion that the original smooth pipe was indeed hydraulically smooth for ReD< or =24x106. The relationship between the velocity shift, DeltaU/utau, and the roughness Reynolds number, ks+, has been used to generalize the form of the transition from smooth to fully rough flow for an arbitrary relative roughness krms/D. These predictions apply for honed pipes when the separation of pipe diameter to roughness height is large, and they differ significantly from the traditional Moody curves.