On the stability of plane parallel viscoelastic shear flows in the limit of infinite Weissenberg and Reynolds numbers

Elastic effects on the hydrodynamic instability of inviscid parallel shear flows are investigated through a linear stability analysis. We focus on the upper convected Maxwell model in the limit of infinite Weissenberg and Reynolds numbers. We study the effects of elasticity on the instability of a few classes of simple parallel flows, specifically plane Poiseuille and Couette flows, the hyperbolic-tangent shear layer and the Bickley jet.The equation for stability is derived and solved numerically using the spectral Chebyshev collocation method. This algorithm is computationally efficient and accurate in reproducing the eigenvalues. We consider flows bounded by walls as well as flows bounded by free surfaces. In the inviscid, nonelastic case all the flows we study are unstable for free surfaces. In the case of wall bounded flow, there are instabilities in the shear layer and Bickley jet flows. In all cases, the effect of elasticity is to reduce and ultimately suppress the inviscid instability.     

Velocity and spanwise vorticity measurements in an excited mixing layer of a plane jet

The mixing layer of a plane jet was subjected to periodic weak excitation at two different frequencies corresponding to shear layer mode (St =0.012) and preferred mode (St _D =0.36). The nozzle exit boundary layer was identical for the unexcited and excited flows. Measurements of mean velocity, longitudinal and lateral velocity fluctuations, Reynolds shear stress and spanwise component of fluctuating vorticity were made over a longitudinal distance x/D of 6 for both the unexcited and the excited flows. Even weak excitation was observed to influence the development of the mixing layer. Under shear layer mode of excitation, the width of the layer and longitudinal turbulence level decrease compared to the naturally developing (unexcited) flow whereas preferred mode of excitation results in increase in the width and turbulence levels. The rms spanwise vorticity showed an increase for shear layer mode of excitation whereas the preferred mode of excitation resulted in a decrease compared to the values in an unexcited flow. Spectra of velocity and vorticity fluctuations exhibited subharmonic peaks, suggesting the possible occurrence of vortex pairing in both unexpected and excited flows. The influence of excitation is found to decrease as x/D increases and is not significant at x/D=6.This study was partly supported by a grant from the Research Grant Council, Hong Kong. The support and hospitality of the Department of Mechanical Engineering, University of Hong Kong are gratefully acknowledged by SR. The authors are grateful the referees for valuable comments.     

Pulsed-wire measurements in the near-wall layer in a reattaching separated flow

Pulsed-wire velocity measurements have been made in the near-wall layer, including the viscous sublayer, beneath a separated flow. A method for correcting the error caused by fluctuations in velocity gradient is given, extending the work of Schober et al. (1998). The measurements show that the r.m.s. of the streamwise velocity fluctuations scale closely in accordance with an inner-layer scaling, where the velocity scale, , is based on the r.m.s. of the wall shear stress fluctuations (measured by means of a pulsed-wire shear stress probe), rather than the mean wall shear stress. The effects of velocity gradient are only significant beneath of 10 or less.List of symbols C Calibration constant - f Function representing mean velocity - h_f Height of fence above splitter plate surface - L Length scale of outer-layer structures - s Distance between pulsed and sensor wires - u r.m.s. of U - Velocity scale based on r.m.s of wall shear stress fluctuation - U Instantaneous velocity in x-direction - U_m Instantaneous measured velocity in x-direction - U_r Free-stream reference velocity - x Streamwise direction from separation point - y Distance from splitter plate surface, in normal direction - X_r Length of separation bubble - ₀ Thickness scale in oscillating layer - Blasius laminar boundary layer parameter - Density - Wall shear stress - r.m.s. of wall shear stress fluctuation - Frequency of oscillating layer - Kinematic viscosity - Overbar denotes time average     

Interactions between pairs of oblique waves in a Bickley jet

We consider the weakly nonlinear spatial evolution of a pair of varicose oblique waves and a pair of sinuous oblique waves superimposed on an inviscid Bickley jet, with each wave being slightly amplified on a linear basis. The two pairs are assumed to both be inclined at the same angle to the plane of the jet. A nonlinear critical layer analysis is employed to derive equations governing the evolution of the instability wave amplitudes, which contain a coupling between the modes. These equations are discussed and solved numerically, and it is shown that, as in related work for other flows, these equations may develop a singularity at a finite distance downstream.     

Artificial eigenmodes in truncated flow domains

Whenever linear eigenmodes of open flows are computed on a numerical domain that is truncated in the streamwise direction, artificial boundary conditions may give rise to spurious pressure signals that are capable of providing unwanted perturbation feedback to upstream locations. The manifestation of such feedback in the eigenmode spectrum is analysed here for two simple configurations. First, explicitly prescribed feedback in a Ginzburg–Landau model is shown to produce a spurious eigenmode branch, named the ‘arc branch’, that strongly resembles a characteristic family of eigenmodes typically present in open shear flow calculations. Second, corresponding mode branches in the global spectrum of an incompressible parallel jet in a truncated domain are examined. It is demonstrated that these eigenmodes of the numerical model depend on the presence of spurious forcing of a local \(k^+\) instability wave at the inflow, caused by pressure signals that appear to be generated at the outflow. Multiple local \(k^+\) branches result in multiple global eigenmode branches. For the particular boundary treatment chosen here, the strength of the pressure feedback from the outflow towards the inflow boundary is found to decay with the cube of the numerical domain length. It is concluded that arc branch eigenmodes are artefacts of domain truncation, with limited value for physical analysis. It is demonstrated, for the example of a non-parallel jet, how spurious feedback may be reduced by an absorbing layer near the outflow boundary.     

Singularly perturbed phenomena of semilinear second order systems

In this paper we consider singular perturbed phenomena of semilinear second ordersystems, under appropriate assumptions, the existence and asymptotic behavior asε→0⁺ of solution of vector boundary value problem are proved by constructing specialinvariant regions in which solutions display so-called boundary layer phenomena andangular layer phenomena.     

High‐Frequency Changes in the Orientation of Nematic Liquid Crystals in Radio Frequency Crossed Electric Fields

The orientational dynamics of the director of a nematic liquid crystal located in radio frequency crossed electric fields were studied by numerical calculations and experimentally. This system is shown to be a physical object of nonlinear dynamics. Depending on the parameters of the problem, the following types of states of the director were observed: stationary (an analog of the nonthreshold Freedericksz transition), periodic, quasiperiodic (multimode), and stochastic of the strange attractor type. In the calculations, all states were obtained by solving a deterministic system of two timedependent nonlinear differential equations of the first order with no electrohydrodynamic terms. All types of solutions obtained, including stochastic ones, were observed experimentally.     

Adjoint optimization of natural convection problems: differentially heated cavity

Optimization of natural convection-driven flows may provide significant improvements to the performance of cooling devices, but a theoretical investigation of such flows has been rarely done. The present paper illustrates an efficient gradient-based optimization method for analyzing such systems. We consider numerically the natural convection-driven flow in a differentially heated cavity with three Prandtl numbers (\(Pr=0.15{-}7\)) at super-critical conditions. All results and implementations were done with the spectral element code Nek5000. The flow is analyzed using linear direct and adjoint computations about a nonlinear base flow, extracting in particular optimal initial conditions using power iteration and the solution of the full adjoint direct eigenproblem. The cost function for both temperature and velocity is based on the kinetic energy and the concept of entransy, which yields a quadratic functional. Results are presented as a function of Prandtl number, time horizons and weights between kinetic energy and entransy. In particular, it is shown that the maximum transient growth is achieved at time horizons on the order of 5 time units for all cases, whereas for larger time horizons the adjoint mode is recovered as optimal initial condition. For smaller time horizons, the influence of the weights leads either to a concentric temperature distribution or to an initial condition pattern that opposes the mean shear and grows according to the Orr mechanism. For specific cases, it could also been shown that the computation of optimal initial conditions leads to a degenerate problem, with a potential loss of symmetry. In these situations, it turns out that any initial condition lying in a specific span of the eigenfunctions will yield exactly the same transient amplification. As a consequence, the power iteration converges very slowly and fails to extract all possible optimal initial conditions. According to the authors’ knowledge, this behavior is illustrated here for the first time.     

Mode transition and oscillation suppression in supersonic cavity flow

Supersonic flows past two-dimensional cavities with/without control are investigated by the direct numerical simulation(DNS). For an uncontrolled cavity, as the thickness of the boundary layer declines, transition of the dominant mode from the steady mode to the Rossiter Ⅱ mode and then to the Rossiter Ⅲ mode is observed due to the change of vortex-corner interactions. Meanwhile, a low frequency mode appears. However, the wake mode observed in a subsonic cavity flow is absent in the current simulation.The oscillation frequencies obtained from a global dynamic mode decomposition(DMD)approach are consistent with the local power spectral density(PSD) analysis. The dominant mode transition is clearly shown by the dynamic modes obtained from the DMD. A passive control technique of substituting the cavity trailing edge with a quarter-circle is studied. As the effective cavity length increases, the dominant mode transition from the Rossiter Ⅱ mode to the Rossiter Ⅲ mode occurs. With the control, the pressure oscillations are reduced significantly. The interaction of the shear layer and the recirculation zone is greatly weakened, combined with weaker shear layer instability, responsible for the suppression of pressure oscillations. Moreover, active control using steady subsonic mass injection upstream of a cavity leading edge can stabilize the flow.     

Flicker noise and self‐organized criticality in crisis boiling regimes

The paper describes an experimental study of thermal fluctuations during transition from bubble to film boiling of water on a wire heater and fluctuations of the shape of a superheated liquid jet discharged from a highpressure vessel. It is found that for a heattransfer crisis on the wire heater and for intense volume boiling of the superheated liquid jet, the fluctuation power spectrum has a lowfrequency component (flicker noise) that diverges under the law 1/f. This effect is due to nonequilibrium phase transitions in the system: the heattransfer crisis during transition from bubble to film boiling and a flow crisis during boiling of the superheated liquid jet.     

Semi-similar solutions to the three-dimensional laminar boundary layer

The theory of semi-similar solutions is developed for and applied to the problem of three-dimensional laminar boundary layer flow. A number of specific examples are calculated. Particular attention is given to certain flows in which separation is approached and the nature of three-dimensional laminar boundary layer separation is inferred from the behavior of these solutions close to separation. Two types of separation are observed: singular separation characterized by the vanishing of the total shear along the line of separation and ordinary separation characterized by limiting streamlines which become parallel to the line of separation.     

Prediction of stagnation point heat transfer for a single round jet impinging on a concave hemispherical surface

This paper deals with a systematic procedure for assessment of fluid flow and heat transfer parameters for a single round jet impinging on a concave hemispherical surface. Based on Scholkemeier's modifications of the Karman-Pohlhausen integral method, expressions are derived for evaluation of the momentum thickness, boundary layer thickness and the displacement thickness at the stagnation point. This is followed by the estimation of thermal boundary layer thickness and local heat transfer coefficients. A correlation is presented for the Nusselt number at the stagnation point as a function of the Reynolds number for different non-dimensional distances from the exit plane of the jet to the impingement surface.

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Bestimmung des Staupunktes bei der Wärmeübertragung für einen einzelnen Strahl, der auf eine konkave halbkugelige Oberfläche trifft

Zusammenfassung Diese Arbeit beschäftigt sich mit dem systematischen Verfahren der Bewertung von Fluidströmungen und Wärmeübertragungsparametern für einen einzelnen runden Strahl, der auf eine konkave halbkugelförmige Oberfläche trifft. Das Verfahren beruht auf Scholkemeiers Modifikation des Karman-Pohlhausen Integrationsverfahrens. Ausdrücke sind für die Berechnung der Impuls-Dicke, der Grenzschichtdicke und der Verschiebungsdicke am Staupunkt hergeleitet worden. Dies ist aus der Berechnung der thermischen Grenzschichtdicke und des lokalen Wärmeübertragungskoeffizienten abgeleitet worden. Es wird eine Gleichung für die Nusselt-Zahl am Staupunkt als Funktion der Reynolds-Zahl für verschiedene dimensionslose Abstände vom Strahlaustrittspunkt bis zum Auftreffpunkt auf die Oberfläche vorgestellt.

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Nomenclature c _p specific heat at constant pressure - d diameter of single round nozzle - h ₀ heat transfer coefficient at the stagnation point - H distance from the exit plane of the jet to the impingement surface - k thermal conductivity - Nu _0.5 Nusselt number based on impinging jet quantities=h _0.50/k - Nu _{0.5, 0} stagnation point Nusselt number=h _{0 0,50}/k - p pressure - p _a ambient pressure - p ₀ maximum pressure or stagnation pressure - p(x) static pressure at a distancex from the stagnation point - R radius of curvature of the hemisphere - Re _J jet Reynolds number=U _Jd/ - Re _0.5 Reynolds number based on impinging jet quantities=u _{m0 0.50}/ - T temperature - T _a room temperature - T _J jet temperature - T _W wall temperature - u velocity component inx andx directions (Fig. 1) - u _m jet centerline (or maximum) free jet velocity: external (or maximum) boundary layer velocity aty= _m - u _m0 arrival velocity defined as the maximum velocity the free jet would have at the plane of impingement if the plane were not there - U _J jet exit velocity - x* non-dimensional coordinate starting at the stagnation point=x/2 _0.50 - x, y rectangular Cartesian coordinates - y coordinate normal to the wall starting at the wall - ratio of thermal to velocity boundary layer thickness= _T/_m - ₀ ratio of thermal to velocity boundary layer thickness at the stagnation point - * inner layer displacement thickness - _0.50 jet half width at the plane of impingement if the plate were not there - _m inner boundary layer thickness atu=u _m - Pohlhausen's form parameter - dynamic viscosity - kinematic viscosity=/ - fluid density - momentum thickness - ₀ momentum thickness at the stagnation point     

Drag reduction and heat transfer in surfactant solutions with excess counterion

We present the results of a study of turbulent drag reduction in a small circulating loop using surfactant solutions with excess counterion. In addition, these solutions were used in measurements of heat transfer, both in pipe flow and in an impinging jet. Both frictional drag and heat transfer were reduced in the pipe flow experiments. Measurements of heat transfer in the impinging jet revealed a dependence on the molar concentration ratio of the counterion. When the counterion was added at a molar concentration 30 times higher than that of the surfactant, the resulting surfactant solution did not reduce the rate of heat transfer in the impinging jet. By using this surfactant system in an impinging jet, we show both a reduction in pipe friction and normal heat transfer potential in a circulating heat exchange system. In order to investigate this difference in heat transfer between pipe flows and impinging jet flows, a comparison was made of the wall shear stress between these two flow regimes. The estimated wall shear stress was of the same order in both flows, and thus was not considered to be the primary cause of the difference in heat transfer. It is instead suggested that the micellar structure of the surfactant is influenced by a compressive deformation of the impinging flow in a manner that is different from the shear deformation observed in pipe flow.     

Natural convection flow from an isothermal horizontal circular cylinder with temperature dependent viscosity

Free convection flow from an isothermal horizontal circular cylinder immersed in a fluid with viscosity proportional to an inverse linear function of temperature is studied. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations is reduced to local non-similarity equations which are solved numerically by a very efficient implicit finite difference method together with Keller box scheme. Numerical results are presented by velocity and viscosity profiles of the fluid as well as heat transfer characteristics, namely the local heat transfer rate and the local skin-friction coefficients for a wide range of viscosity parameter (= 0.0, 0.5, 1.0, 2.0, 3.0,4.0) and the Prandtl number Pr (= 1.0, 7.0, 10.0, 15.0, 20.0, 30.0).     

Three dimensional simulations of penetrative convection in a porous medium with internal heat sources

The problem of penetrative convection in a fluid saturated porous medium heated internally is analysed. The linear instability theory and nonlinear energy theory are derived and then tested using three dimensions simulation.Critical Rayleigh numbers are obtained numerically for the case of a uniform heat source in a layer with two fixed surfaces. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using a three dimensional simulation. Our results show that the linear threshold accurately predicts the onset of instability in the basic steady state. However, the required time to arrive at the basic steady state increases significantly as the Rayleigh number tends to the linear threshold.     

Space–time correlation measurements of high-speed axisymmetric jets using optical deflectometry

An optical deflectometry system is used to provide unique space–time correlation measurements at two positions separated by varying axial distances within a high-speed jet shear layer. The measurements were made for both pure air and for helium/air mixture jets at Mach numbers M=0.9 and M=1.5. The jets issue from round nozzles and the sensing volumes at the two measurement positions consist of small light filaments along spanwise lines that are tangential to the annular jet shear layer. Applying this technique to obtain measurements detailing the level of correlation, spectral content, and convection velocity for jet flows in these flow regimes near the end of the potential core is particularly important in the understanding and prediction of jet noise. Measurements near the end of the potential core along the jet lip line exhibit distinct cross-correlation curves for the pure air jet cases. However, helium/air mixture jets display much lower levels of correlation and little evidence of large-scale structure in the measured spectra. It is believed that the thick visual density gradients dominated by smaller scales throughout the shear layer of the helium/air mixture jets effectively mask the large-scale structure, thus, reflecting a limitation of this optical deflectometer. Finally, a decrease in normalized convection velocity with helium addition is observed.     

开放剪切流空间全局动力学与空间相位斑图间的关系

开放流动空间动力学可基于两类全局能量关系式进行研究;而空间相位斑图则可通过互谱空间演化加以测定。全局能量关系式以时间Fourier系数的形式建立流场任意两点问速度脉动能量间的关系,籍此可定义全局意义上的线性、非线性和线性一非线性机制。基于轴对称剪切流、变密度轴对称圆射流以及平面对称剪切流的实验发现：轴对称旋涡结构的配对由线性、线性一非线性机制表征,对应有序空间相位斑图;并且能量可通过线性一非线性机制在具有相同相速度的扰动间传递。螺旋结构由线性机制表征,对应有序相位斑图。全局自激励振荡由非线性的能量共振表征,对应无序相位斑图。籍此,有序空间相位斑图对应线性和线性一非线性机制;而混沌相位斑图则对应非线性机制。     

Experimental investigation of planar offset attaching jets with small offset distances

An experimental investigation was performed to characterize the development of planar jets initially issuing parallel to an adjacent wall with offset distances of up to 1 jet height and Reynolds number of 44,000. The results showed that the initial development of the mean flow field in the planar offset jets could be divided into five regions; three associated with the jet attaching to the wall similar to other reattaching shear layer flows and two associated with the resulting planar wall jet flow. The transition from the reattaching flow to the wall jet flow was also characterized by a significant change in the characteristic frequency, size, and convection velocity of the large-scale structures in the flows. The research was funded by the Natural Sciences and Engineering Research Council of Canada.