Effect of streamwise and spanwise electric fields on transient growth in a two-fluid channel flow

A linear model of a two-fluid channel flow under streamwise/spanwise electric field is built. Both the fluids are assumed to be incompressible, viscous and perfectly dielectric. The effect of the streamwise and spanwise electric fields on transient behavior of small three-dimensional disturbances is studied. The numerical result shows that the streamwise electric field suppresses transient growth of the disturbance with spanwise uniform wave number. The spanwise electric field diminishes transient growth of the disturbance with streamwise uniform wave number. Two peaks of optimal growth are detected in the wave number plane. The peak at relatively large spanwise wave number is dominated by the lift-up mechanism and little influenced by electric field. Differently, the peak at relatively small wave number is associated with the characteristic of the interface and possibly influenced by electric field. The effect of the Weber number, the Reynolds number and the relative electrical permittivity on optimal growth is studied as well. A scaling law is obtained for relatively small Weber numbers and relatively large Reynolds numbers.     

Flow instability of nanofuilds in jet

The flow instability of nanofluids in a jet is studied numerically under various shape factors of the velocity profile, Reynolds numbers, nanoparticle mass loadings,Knudsen numbers, and Stokes numbers. The numerical results are compared with the available theoretical results for validation. The results show that the presence of nanoparticles enhances the flow stability, and there exists a critical particle mass loading beyond which the flow is stable. As the shape factor of the velocity profile and the Reynolds number increase, the flow becomes more unstable. However, the flow becomes more stable with the increase of the particle mass loading. The wavenumber corresponding to the maximum of wave amplification becomes large with the increase of the shape factor of the velocity profile, and with the decrease of the particle mass loading and the Reynolds number. The variations of wave amplification with the Stokes number and the Knudsen number are not monotonic increasing or decreasing, and there exists a critical Stokes number and a Knudsen number with which the flow is relatively stable and most unstable,respectively, when other parameters remain unchanged. The perturbation with the first azimuthal mode makes the flow unstable more easily than that with the axisymmetric azimuthal mode. The wavenumbers corresponding to the maximum of wave amplification are more concentrated for the perturbation with the axisymmetric azimuthal mode.     

Perturbation Structure and Spectra in Turbulent Channel Flow

The strong mean shear in the vicinity of the boundaries in turbulent boundary layer flows preferentially amplifies a particular class of perturbations resulting in the appearance of coherent structures and in characteristic associated spatial and temporal velocity spectra. This enhanced response to certain perturbations can be traced to the nonnormality of the linearized dynamical operator through which transient growth arising in dynamical systems with asymptotically stable operators is expressed. This dynamical amplification process can be comprehensively probed by forcing the linearized operator associated with the boundary layer flow stochastically to obtain the statistically stationary response. In this work the spatial wave-number/temporal frequency spectra obtained by stochastically forcing the linearized model boundary layer operator associated with wall-bounded shear flow at large Reynolds number are compared with observations of boundary layer turbulence. The verisimilitude of the stochastically excited synthetic turbulence supports the identification of the underlying dynamics maintaining the turbulence with nonnormal perturbation growth. Received 30 January 1997 and accepted 27 March 1998     

Algorithm for transient growth of perturbations in channel Poiseuille flow

This study develops a direct optimal growth algorithm for three-dimensional transient growth analysis of perturbations in channel flows which are globally stable but locally unstable. Different from traditional non-modal methods based on the OrrSommerfeld and Squire(OSS) equations that assume simple base flows, this algorithm can be applied to arbitrarily complex base flows. In the proposed algorithm, a reorthogonalization Arnoldi method is used to improve orthogonality of the orthogonal basis of the Krylov subspace generated by solving the linearized forward and adjoint Navier-Stokes(N-S) equations. The linearized adjoint N-S equations with the specific boundary conditions for the channel are derived, and a new convergence criterion is proposed. The algorithm is then applied to a one-dimensional base flow(the plane Poiseuille flow) and a two-dimensional base flow(the plane Poiseuille flow with a low-speed streak)in a channel. For one-dimensional cases, the effects of the spanwise width of the channel and the Reynolds number on the transient growth of perturbations are studied. For two-dimensional cases, the effect of strength of initial low-speed streak is discussed. The presence of the streak in the plane Poiseuille flow leads to a larger and quicker growth of the perturbations than that in the one-dimensional case. For both cases, the results show that an optimal flow field leading to the largest growth of perturbations is characterized by high-and low-speed streaks and the corresponding streamwise vortical structures.The lift-up mechanism that induces the transient growth of perturbations is discussed.The performance of the re-orthogonalization Arnoldi technique in the algorithm for both one-and two-dimensional base flows is demonstrated, and the algorithm is validated by comparing the results with those obtained from the OSS equations method and the crosscheck method.     

Transient growth and bypass transition in stenotic flow with a physiological waveform

A direct analysis method is applied to compute optimal transient growth initial conditions for physiologically relevant pulsatile flows in a smooth axisymmetric stenosis with 75% occlusion. The flow waveform employed represents phase-average measurements obtained in the human common carotid artery. Floquet analysis shows that the periodic flow is stable to infinitesimal eigenmodal-type perturbations that would grow from one cycle to the next at the Reynolds numbers considered. However, the same flows display explosive transient growth of optimal disturbances, with our analysis predicting disturbance energy growths of order 10²⁵ within half a pulse period at a mean bulk flow Reynolds number Re = 300, which is significantly lower than the physiological value of Re = 450 at this location. Direct numerical simulation at Re = 300 shows that when the base flow is perturbed a small amount with the optimal growth initial condition, the disturbance grows rapidly in time in agreement with the linear analysis, and saturates to provide a locally turbulent state within half a pulse period. This transition resulting from non-normal growth mechanisms shows the flow exhibits bypass transition to turbulence. Our analysis suggests that this route to localized turbulent states could be relatively common in human arterial flows.     

Numerical study of the turbulent channel flow under space-dependent electromagnetic force control at different Reynolds numbers

In this paper, the control of turbulent channel flow by space-dependent electromagnetic force and the mechanism of drag reduction are investigated with the direct numerical simulation(DNS) methods for different Reynolds numbers. A formulation is derived to express the relation between the drag and the Reynolds shear stress. With the application of optimal electromagnetic force, the in-depth relations among characteristic structures in the flow field, mean Reynolds shear stress, and the effect of drag reduction for different Reynolds numbers are discussed. The results indicate that the maximum drag reductions can be obtained with an optimal combination of parameters for each case of different Reynolds numbers. The regular quasi-streamwise vortex structures, which appear in the flow field, have the same period with that of the electromagnetic force.These structures suppress the random velocity fluctuations, which leads to the absolute value of mean Reynolds shear stress decreasing and the distribution of that moving away from the wall. Moreover, the wave number of optimal electromagnetic force increases,and the scale of the regular quasi-streamwise vortex structures decreases as the Reynolds number increases. Therefore, the rate of drag reduction decreases with the increase in the Reynolds number since the scale of the regular quasi-streamwise vortex structures decreases.     

Linear stability of viscous incompressible flow in a circular viscoelastic tube

Flow stability in rigid tubes has been the subject of much research [1]. The overwhelming majority of authors of both theoretical and experimental studies now conclude that Poiseuille flow in a circular rigid tube is linearly stable. However, real tubes all possess elastic properties, the influence of which has not been investigated in such detail. For certain selected values of the parameters characterizing an elastic tube it has been shown that with respect to infinitesimal axisymmetric perturbations Poiseuille flow in the tube can be unstable [2]. In this case boundary conditions that did not take into account the fairly large velocity gradient of the undisturbed flow near the tube wall were used. The present paper reports the results of a numerical investigation of the linear stability of Poiseuille flow in a circular elastic tube with respect to three-dimensional perturbations in the form of traveling waves propagated along the system (azimuthal perturbation modes with numbers 0, 1, 2, 3, 4, and 5 are considered). It is shown that the elastic properties of the tube can have an important influence on the linear stability spectrum. In the case of axisymmetric perturbations it is possible to detect an instability which, at Reynolds numbers of more than 200, exists only for tubes whose modulus of elasticity is substantially less than that of materials in common use. The instability to perturbations of the second azimuthal mode is different in character, inasmuch as at Reynolds numbers greater than unity it occurs in stiffer tubes. Moreover, as the Reynolds number increases it can also occur in tubes of greater stiffness. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 126–134, November–December, 1986.     

Circular Couette flow with pressure-driven axial flow and a porous inner cylinder

In a rotating filter separator a suspension is introduced at one end of the annulus between a rotating porous inner cylinder and a fixed impermeable outer cylinder. The filtrate is removed through the inner cylinder and the concentrate is removed from the opposite end of the annulus from which the suspension entered. The flow in a rotating filter separator is circular Couette flow with a pressure-driven axial flow and a suction boundary condition at the inner cylinder. Flow visualization was used to determine the effect of the Taylor number, axial Reynolds number, and radial Reynolds number on the types of flows present in the annulus. A rich variety of secondary vortical flows appear, depending upon the flow parameters. The radial inflow at the inner cylinder delays the appearance of supercritical circular Couette flow and prevents the appearance of certain flow regimes that have a helical vortex structure. Nevertheless, the average azimuthal velocity measured using laser Doppler velocimetry indicates that the velocity profile is nearly the same for all supercritical flow regimes.This work was supported by a grant from The Whitaker Foundation     

Direct numerical simulation of supersonic pipe flow at moderate Reynolds number

We study compressible turbulent flow in a circular pipe at computationally high Reynolds number. Classical related issues are addressed and discussed in light of the DNS data, including validity of compressibility transformations, velocity/temperature relations, passive scalar statistics, and size of turbulent eddies. Regarding velocity statistics, we find that Huang’s transformation yields excellent universality of the scaled Reynolds stresses distributions, whereas the transformation proposed by Trettel and Larsson (2016) yields better representation of the effects of strong variation of density and viscosity occurring in the buffer layer on the mean velocity distribution. A clear logarithmic layer is recovered in terms of transformed velocity and wall distance coordinates at the higher Reynolds number under scrutiny (Re_τ ≈ 1000), whereas the core part of the flow is found to be characterized by a universal parabolic velocity profile. Based on formal similarity between the streamwise velocity and the passive scalar transport equations, we further propose an extension of the above compressibility transformations to also achieve universality of passive scalar statistics. Analysis of the velocity/temperature relationship provides evidence for quadratic dependence which is very well approximated by the thermal analogy proposed by Zhang et al. (2014). The azimuthal velocity and scalar spectra show an organization very similar to canonical incompressible flow, with a bump-shaped distribution across the flow scales, whose peak increases with the wall distance. We find that the size growth effect is well accounted for through an effective length scale accounting for the local friction velocity and for the local mean shear.     

Vortex dynamics of turbulence–coherent structure interaction

We study the interaction between a coherent structure (CS) and imposed external turbulence by employing direct numerical simulations (DNS) designed for unbounded flows with compact vorticity distribution. Flow evolution comprises (i) the reorganization of turbulence into finer-scale spiral filaments, (ii) the growth of wave-like perturbations within the vortex core, and (iii) the eventual arrest of production, leading to the decay of ambient turbulence. The filaments, preferentially aligned in the azimuthal direction, undergo two types of interactions: parallel filaments pair to form higher-circulation “threads”, and anti-parallel threads form dipoles that self-advect radially outwards. The consequent radial transport of angular momentum manifests as an overshoot of the mean circulation profile—a theoretically known consequence of faster-than-viscous vortex decay. It is found that while the resulting centrifugal instability can enhance turbulence production, vortex decay is arrested by the dampening of the instability due to the “turbulent mixing” caused by instability-generated threads. Ensemble-averaged turbulence statistics show strong fluctuations within the core; these are triggered by the external turbulence, and grow even as the turbulence decays. This surprising growth on a normal-mode-stable vortex results from algebraic amplification through “linear transient growth”. Transient growth is examined by initializing DNS with the “optimal” modes obtained from linear analysis. The simulations show that the growth of transient modes reproduces the prominent dynamics of CS-turbulence interaction: formation of thread-dipoles, growth of core fluctuations, and appearance of bending waves on the column’s core. At the larger Reynolds numbers prevailing in practical flows, transient growth may enable accelerated vortex decay through vortex column breakdown.     

Stability analysis in spatial mode for channel flow of fiber suspensions

Different from previous temporal evolution assumption, the spatially growing mode was employed to analyze the linear stability for the channel flow of fiber suspensions. The stability equation applicable to fiber suspensions was established and solutions for a wide range of Reynolds number and angular frequency were given numerically . The results show that, the flow instability is governed by a parameter H which represents a ratio between the axial stretching resistance of fiber and the inertial force of the fluid. An increase of H leads to a raise of the critical Reynolds number, a decrease of corresponding wave number, a slowdown of the decreasing of phase velocity , a growth of the spatial attenuation rate and a diminishment of the peak value of disturbance velocity. Although the unstable region is reduced on the whole, long wave disturbances are susceptible to fibers.     

Transient growth mechanisms of low Reynolds number flow over a low-pressure turbine blade

A direct transient growth analysis for three-dimensional perturbations to flow past a periodic array of T-106/300 low-pressure turbine fan blades is presented. The methodology is based on a singular value decomposition of the flow evolution operator, linearised about a steady or periodic base flow. This analysis yields the optimal growth modes. Previous work on global mode stability analysis of this flow geometry showed the flow is asymptotically stable, indicating a non-modal explanation of transition may be more appropriate. The present work extends previous investigations into the transient growth around a steady base flow, to higher Reynolds numbers and periodic base flows. It is found that the notable transient growth of the optimal modes suggests a plausible route to transition in comparison to modal growth for this configuration. The spatial extent and localisation of the optimal modes is examined and possible physical triggering mechanisms are discussed. It is found that for longer times and longer spanwise wavelengths, a separation in the shear layer excites the wake mode. For shorter times and spanwise wavelengths, smaller growth associated with excitation of the near wake are observed.     

Surface instabilities of thin liquid film flow on a rotating disk

 Steady flow of a liquid jet from a nozzle onto the centre of a rotating disk is studied with a streak line method to determine the superficial velocity of the spreading liquid film. Good agreement is found with an asymptotic analysis of the unperturbed flow field. Experimentally, the liquid surface is always perturbed by surface waves which appear as regular spirals, steady in the laboratory system in the low Reynolds number range. It could be shown that wave formation is very sensitive to entrance conditions. Therefore, it is assumed that wave generation is an entrance effect which acts as periodic forcing on the forming liquid film. Wave velocities outside the entrance region are measured and proved to be in good agreement with the prediction of a linear stability theory, as long as the flow rate and entrance perturbations are small. At higher flow rates or stronger disturbances, the radial development of the wave velocities takes on the characteristics predicted by nonlinear stability theories and is in qualitative agreement with experiments performed on an inclined plane. Received: 15 January 1998/Accepted: 8 June 1998     

Axially symmetric vortical flow of a nonviscous liquid in the semiinfinite gap between two coaxial circular cylinders

In the framework of the Hromek-Lamb equations we investigate the axially symmetric vortical flow of a nonviscous incompressible liquid in both semiinfinite and infinite gaps between two coaxial circular cylinders. The investigation is carried out for two circulation and flow functions and two different Bernoulli constants which are chosen in the form of a third-order polynomial in the flow function. This makes it possible to determine the effect of the azimuthal velocity component on the flow in an axial plane with radial and axial components of the velocity. It is shown that under certain circumstances wave oscillations in the flow are possible, in agreement with the results of [1–3] which investigated the flow in an infinite tube [1], in a semiinfinite tube with simpler circulation functions and Bernoulli constants [2], and in the two-dimensional case [3]. We determine the dependence of the formation of wave perturbations on the third term of the Bernoulli constant and on the azimuthal velocity component. The results of this work agree with investigations by other authors [1–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 38–45, September–October, 1977.The author thanks Yu. P. Gupalo and Yu. S. Ryazantsev for suggesting this problem and for their interest in the work. Thanks are also due to G. Yu. Stepanov for discussions and valuable comments.     

Correlation of swirl number for a radial-type swirl generator

An experimental investigation was undertaken to derive a new correlation for the swirl number of a radial-type swirl generator under various Reynolds numbers and various vane angle conditions. A radial-type swirl generator with 16 rotary guide vanes was used to generate an annular swirling jet flow. The Reynolds numbers ranged from 60 to 6000, and the vane angles from 0° to 56°. Quantitative measurements for the velocities were made by using an optical method of laser-Doppler anemometry (LDA). Three-component velocity profiles of axial, radial, and azimuthal components at the swirling jet exit were measured for various flow conditions. A flow visualization method using smoke-wire and still photography was also applied to observe the flow patterns of the recirculation region behind the circular bluff body. Under low Reynolds number conditions, the swirl strength was found to be strongly dependent on the Reynolds number as well as on the guide vane angle. Based on the experimental results, a modified swirl number S is derived to characterize the swirling flow, which is useful for the design of a swirl generator.     

Large-eddy simulation of counter-rotating Taylor–Couette flow: The effects of angular velocity and eccentricity

In the present work, turbulent flow in the annulus of a counter-rotating Taylor-Couette (CRTC) system is studied using large-eddy simulation. The numerical methodology employed is validated, for both the mean and second-order statistics, with the direct numerical simulation (DNS) data available in the literature, for a range of Reynolds numbers from 500 to 4000. Thereafter, turbulent flow occurring in this system at Reynolds numbers of 8000 and 16000 are studied, and the results obtained are analyzed using mean and second-order statistics, vortical structures, velocity vector plots and power energy spectra. Further, the spatio-temporal variation of azimuthal velocity, extracted near the inner cylinder, shows the existence of herringbone like patterns similar to that observed in the previous studies. The effect of eccentricity of the inner cylinder with respect to the outer cylinder is studied, on the turbulent flow in the CRTC system, for two different eccentricity ratios of 0.2 and 0.5 and for two different Reynolds numbers of 1500 and 4000. The results of the eccentric CRTC are analyzed using contours of pressure, mean and second-order statistics, velocity vectors, vortical structures, and turbulence anisotropy maps. It is observed from the eccentric CRTC simulations that the smaller-gap region seems to contain higher amplitude fluctuations and more vortical structures when compared with the larger-gap region. The mean turbulent kinetic energy contours do not change qualitatively with the Reynolds number, however, quantitatively a higher turbulent kinetic energy is observed in the higher Reynolds number case of 4000.     

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.     

Effect of fluid temperature and uniformly accelerated rotation of the inner sphere on the selection of the flow pattern in a spherical layer

The process of the selection of one of the two flow patterns possible in the hysteresis region, when the Reynolds number is varied in different directions, and differing with respect to the azimuthal wavenumber, 3 or 4, is experimentally investigated. The flow pattern selection proceeds under the influence of an increase in the rotation velocity of the inner sphere at a constant acceleration, the post-acceleration velocity remaining constant. The spherical layer thickness is equal to the inner sphere radius and the outer boundary is fixed. It is established that there is a time lag between the beginning (end) of the sphere acceleration and the beginning (end) of the variation in the measured azimuthal velocity component. It is found that the acceleration necessary for one flow pattern to be replaced by the other significantly depends not only on the Reynolds numbers at which the acceleration begins and ends but also on the fluid temperature in the layer. It is shown that the temperature dependence can be attributed to the variation in the Reynolds number corresponding to the position of the hysteresis boundary when the working fluid viscosity is varied in the layer.     

Calculation of autooscillations of the type of azimuthal waves occurring at loss of stability of flow of a viscous fluid between concentric cylinders rotating in opposite directions

Starting with the Navier-Stokes equation we use the Lyapunov-Schmidt method to investigate the nature of the loss of stability of Couette flow between cylinders as the Reynolds number passes through its critical value. We consider the rotation of the cylinders in opposite directions with the ratio of the angular velocities such that the role of the most dangerous disturbances passes over from rotationally symmetric to nonrotationally symmetric disturbances. Branching nonstationary secondary flows (autooscillations) are found in the form of azimuthal waves; the longitudinal wave number and the azimuthal wave number m are assumed given. The amplitude of autooscillations and the wave velocity are calculated for m = 1, and it is shown that depending on the value of both weak excitation of stable and strong excitation of unstable autooscillations are possible and the wave number for which the critical Reynolds number is a minimum corresponds to a stable wave regime in the supercritical region. The linear problem of the stability of the circular flow of a viscous fluid with respect to nonrotationally symmetric disturbances is discussed in [1–3]. Di Prima [1] solved the problem numerically by the Galerkin method when the gap is small and the cylinders rotate in the same direction. Di Prima's analysis is extended in [2] to cylinders rotating in opposite directions, and in [3] it is extended to gaps which are not small. The nonlinear stability problem is treated in [4], where for fixed = 3 and cylinders rotating in opposite directions the axisymmetric stationary secondary flow the Taylor vortex is calculated. The formation of azimuthal waves in the fluid between the cylinders was studied experimentally in detail by Coles [5].Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskoi Fiziki, No. 2, pp. 68–75, March–April, 1976.     

Numerical prediction of vortex shedding behind a square cylinder

It is common knowledge that flow around bluff bodies exhibits oscillatory behaviour. The aim of the present study is to compute the steady two-dimensional flow around a square cylinder at different Reynolds numbers and to determine the onset of unsteadiness through a linear stability analysis of the steady flow. Stability of the steady flow to small two-dimensional perturbations is analysed by computing the evolution of these perturbations. An analysis of various time-stepping techniques is carried out to select the most appropriate technique for predicting the growth of the perturbations and hence the stability of the flow. The critical Reynolds number is determined from the growth rate of the perturbations. Computations are then made for periodic unsteady flow at a Reynolds number above the critical value. The predicted Strouhal number agrees well with experimental data. Heat transfer from the cylinder is also studied for the unsteady laminar flow.