Showing papers in "Journal of Fluid Mechanics in 1993"

Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence

Abstract: The average settling velocity in homogeneous turbulence of a small rigid spherical particle, subject to a Stokes drag force, has been shown to differ from that in still fluid owing to a bias from the particle inertia (Maxey 1987). Previous numerical results for particles in a random flow field, where the flow dynamics were not considered, showed an increase in the average settling velocity. Direct numerical simulations of the motion of heavy particles in isotropic homogeneous turbulence have been performed where the flow dynamics are included. These show that a significant increase in the average settling velocity can occur for particles with inertial response time and still-fluid terminal velocity comparable to the Kolmogorov scales of the turbulence. This increase may be as much as 50% of the terminal velocity, which is much larger than was previously found. The concentration field of the heavy particles, obtained from direct numerical simulations, shows the importance of the inertial bias with particles tending to collect in elongated sheets on the peripheries of local vortical structures. This is coupled then to a preferential sweeping of the particles in downward moving fluid. Again the importance of Kolmogorov scaling to these processes is demonstrated. Finally, some consideration is given to larger particles that are subject to a nonlinear drag force where it is found that the nonlinearity reduces the net increase in settling velocity.

The structure of intense vorticity in isotropic turbulence

Abstract: The structure of the intense-vorticity regions is studied in numerically simulated homogeneous, isotropic, equilibrium turbulent flow fields at four different Reynolds numbers, in the range Re, = 35-170. In accordance with previous investigators this vorticity is found to be organized in coherent, cylindrical or ribbon-like, vortices (‘worms’). A statistical study suggests that they are simply especially intense features of the background, O(o’), vorticity. Their radii scale with the Kolmogorov microscale and their lengths with the integral scale of the flow. An interesting observation is that the Reynolds number y/v, based on the circulation of the intense vortices, increases monotonically with ReA, raising the question of the stability of the structures in the limit of Re, --z co. Conversely, the average rate of stretching of these vortices increases only slowly with their peak vorticity, suggesting that self-stretching is not important in their evolution. One- and two-dimensional statistics of vorticity and strain are presented; they are non-Gaussian and the behaviour of their tails depends strongly on the Reynolds number. There is no evidence of convergence to a limiting distribution in this range of Re,, even though the energy spectra and the energy dissipation rate show good asymptotic properties in the higher-Reynolds-number cases. Evidence is presented to show that worms are natural features of the flow and that they do not depend on the particular forcing scheme.

On Explicit Algebraic Stress Models for Complex Turbulent Flows

Abstract: Explicit algebraic stress models that are valid for three-dimensional turbulent flows in noninertial frames are systematically derived from a hierarchy of second-order closure models. This represents a generalization of the model derived by Pope who based his analysis on the Launder, Reece, and Rodi model restricted to two-dimensional turbulent flows in an inertial frame. The relationship between the new models and traditional algebraic stress models -- as well as anistropic eddy visosity models -- is theoretically established. The need for regularization is demonstrated in an effort to explain why traditional algebraic stress models have failed in complex flows. It is also shown that these explicit algebraic stress models can shed new light on what second-order closure models predict for the equilibrium states of homogeneous turbulent flows and can serve as a useful alternative in practical computations.

Energy growth in viscous channel flows

Abstract: In recent work it has been shown that there can be substantial transient growth in the energy of small perturbations to plane Poiseuille and Couette flows if the Reynolds number is below the critical value predicted by linear stability analysis. This growth, which may be as large as O( lOOO), occurs in the absence of nonlinear effects and can be explained by the non-normality of the governing linear operator - that is, the nonorthogonality of the associated eigenfunctions. In this paper we study various aspects of this energy growth for two- and three-dimensional Poiseuille and Couette flows using energy methods, linear stability analysis, and a direct numerical procedure for computing the transient growth. We examine conditions for no energy growth, the dependence of the growth on the streamwise and spanwise wavenumbers, the time dependence of the growth, and the effects of degenerate eigenvalues. We show that the maximum transient growth behaves like O(R2), where R is the Reynolds number. We derive conditions for no energy growth by applying the Hille-Yosida theorem to the governing linear operator and show that these conditions yield the same results as those derived by energy methods, which can be applied to perturbations of arbitrary amplitude. These results emphasize the fact that subcritical transition can occur for Poiseuille and Couette flows because the governing linear operator is non-normal.

Turbulence measurements in axisymmetric jets of air and helium. I - Air jet. II - Helium jet

Abstract: Results are presented of measurements on turbulent round jets of air and of helium of the same nozzle momentum efflux, using, for the air jets, x-wire hot-wire probes mounted on a moving shuttle and, for He jets, a composite probe consisting of an interference probe of the Way-Libby type and an x-probe. Current models for scalar triple moments were evaluated. It was found that the performance of the model termed the Full model, which includes all terms except advection, was very good for both the air and the He jets.

Direct numerical simulation of turbulent flow over riblets

Abstract: Direct numerical simulations of turbulent flows over riblet-mounted surfaces are performed to educe the mechanism of drag reduction by riblets. The computed drag on the riblet surfaces is in good agreement with the existing experimental data. The mean-velocity profiles show upward and downward shifts in the log–law for drag-decreasing and drag-increasing cases, respectively. Turbulence statistics above the riblets are computed and compared with those above a flat plate. Differences in the mean-velocity profile and turbulence quantities are found to be limited to the inner region of the boundary layer. Velocity and vorticity fluctuations as well as the Reynolds shear stresses above the riblets are reduced in drag-reducing configurations. Quadrant analysis indicates that riblets mitigate the positive Reynolds-shear-stress-producing events in drag-reducing configurations. From examination of the instantaneous flow fields, a drag reduction mechanism by riblets is proposed: riblets with small spacings reduce viscous drag by restricting the location of the streamwise vortices above the wetted surface such that only a limited area of the riblets is exposed to the downwash of high-speed fluid that the vortices induce.

Water entry of two-dimensional bodies

Abstract: A numerical method for studying water entry of a two-dimensional body of arbitrary cross-section is presented. It is a nonlinear boundary element method with a jet flow approximation. The method has been verified by comparisons with new similarity solution results for wedges with deadrise angles varying from 4° to 81°. A simple asymptotic solution for small deadrise angles α based on Wagner (1932) agrees with the similarity solution for small α.

Dynamics of bubble growth and detachment from a needle

Abstract: Several aspects of the growth and departure of bubbles from a submerged needle are considered. A simple model shows the existence of two different growth regimes according to whether the gas flow rate into the bubble is smaller or greater than a critical value. These conclusions are refined by means of a boundary-integral potential-flow calculation that gives results in remarkable agreement with experiment. It is shown that bubbles growing in a liquid flowing parallel to the needle may detach with a considerably smaller radius than in a quiescent liquid. The study also demonstrates the critical role played by the gas flow resistance in the needle. A considerable control on the rate and size of bubble production can be achieved by a careful consideration of this parameter. The effect is particularly noticeable in the case of small bubbles, which are the most difficult ones to produce in practice.

A Reynolds stress model for near-wall turbulence

Abstract: A tensorially consistent near-wall second-order closure model is formulated. Redistributive terms in the Reynolds stress equations are modelled by an elliptic relaxation equation in order to represent strongly non-homogeneous effects produced by the presence of walls; this replaces the quasi-homogeneous algebraic models that are usually employed, and avoids the need for ad hoc damping functions. A quasi-homogeneous model appears as the source term in the elliptic relaxation equation-here we use the simple Rotta return to isotropy and isotropization of production formulae. The formulation of the model equations enables appropriate boundary conditions to be satisfied. The model is solved for channel flow and boundary layers with zero and adverse pressure gradients. Good predictions of Reynolds stress components, mean flow, skin friction and displacement thickness are obtained in various comparisons to experimental and direct numerical simulation data. The model is also applied to a boundary layer flowing along a wall with a 90°, constant-radius, convex bend. Because the model is of a general, tensorially invariant form, special modifications for curvature effects are not needed; the equations are simply transformed to curvilinear coordinates. The model predicts many important features of this flow. These include: the abrupt drop of skin friction and Stanton number at the start of the curve, and their more gradual recovery after the bend; the suppression of turbulent intensity in the outer part of the boundary layer; a region of negative (counter-gradient) Reynolds shear stress; and recovery from curvature in the form of a Reynolds stress ‘bore’ propagating out from the surface. A shortcoming of the present model is that it overpredicts the rate of this recovery. A heat flux model is developed. It is shown that curvature effects on heat transfer can also be accounted for automatically by a tensorially invariant formulation.

Experimental and numerical study of a turbulent boundary layer with pressure gradients

Abstract: Results are presented of an experimental and numerical study of a turbulent boundary layer with pressure gradients conducted using the recent 'fringe method' with its numerical advantages and good inflow quality. After an inflow transient good agreement is observed; the differences, of up to 13 percent, are discussed. Moderate deviations from the law of the wall are found in the velocity profiles of the simulation. They are fully correlated with the pressure gradient, are in fair quantitative agreement with the experimental results of Nagano et al. (1992), and are roughly the opposite of uncorrected mixing-length-model predictions. Large deviations from the wall scaling are observed for other quantities, notably for the turbulence dissipation rate. The a(1) structure parameter drops mildly in the upper layer with adverse pressure gradient.

Direct simulations of low-Reynolds-number turbulent flow in a rotating channel

Abstract: Direct numerical simulations of fully developed pressure-driven turbulent flow in a rotating channel have been performed The unsteady Navier–Stokes equations were written for flow in a constantly rotating frame of reference and solved numerically by means of a finite-difference technique on a 128 × 128 × 128 computational mesh The Reynolds number, based on the bulk mean velocity Um and the channel half-width h, was about 2900, while the rotation number Ro = 2|Ω|h/Um varied from 0 to 05 Without system rotation, results of the simulation were in good agreement with the accurate reference simulation of Kim, Moin & Moser (1987) and available experimental data The simulated flow fields subject to rotation revealed fascinating effects exerted by the Coriolis force on channel flow turbulence With weak rotation (Ro = 001) the turbulence statistics across the channel varied only slightly compared with the nonrotating case, and opposite effects were observed near the pressure and suction sides of the channel With increasing rotation the augmentation and damping of the turbulence along the pressure and suction sides, respectively, became more significant, resulting in highly asymmetric profiles of mean velocity and turbulent Reynolds stresses In accordance with the experimental observations of Johnston, Halleen & Lezius (1972), the mean velocity profile exhibited an appreciable region with slope 2Ω At Ro = 050 the Reynolds stresses vanished in the vicinity of the stabilized side, and the nearly complete suppression of the turbulent agitation was confirmed by marker particle trackings and two-point velocity correlations Rotational-induced Taylor-Gortler-like counter-rotating streamwise vortices have been identified, and the simulations suggest that the vortices are shifted slightly towards the pressure side with increasing rotation rates, and the number of vortex pairs therefore tend to increase with Ro

Vortex-induced vibrations of a long flexible circular cylinder

Abstract: In an experimental study of the vortex-induced oscillations of a long flexible circular cylinder, the observed stationary amplitudes describe an hysteresis loop partially different from earlier studies. Each branch of the loop is associated with a vortex shedding mode and, as a jump from one branch to the other occurs, the phase difference between the cylinder displacement and the vortex shedding undergoes an abrupt change. The critical flow velocities at which the jump occurs concur with the flow visualization observations of Williamson & Roshko (1988) on the vortex shedding modes near the fundamental synchronization region. Impulsive regimes, obtained at a given flow velocity with the cylinder initially at rest or pre-excited, and progressive regimes resulting from a variation of the flow velocity, are examined. The occurrence of bifurcations is detected for a flow velocity range in the case of the impulsive regimes. The coordinates of the bifurcations define a boundary between two vortex shedding modes, a boundary that verifies the critical curve obtained by Williamson & Roshko (1988). The experimental set-up of this study simulates half the wavelength of a vibrating cable, eliminates the end effects present in oscillating rigid cylinder set-up and has one of the lowest damping ratios reported for the study of this phenomenon.

Transient high-Rayleigh-number thermal convection with large viscosity variations

Abstract: The characteristics of thermal convection in a fluid whose viscosity varies strongly with temperature are studied in the laboratory. At the start of an experiment, the upper boundary of an isothermal layer of Golden Syrup is cooled rapidly and maintained at a fixed temperature. The fluid layer is insulated at the bottom and cools continuously. Rayleigh numbers calculated with the viscosity of the well-mixed interior are between 106 and 108 and viscosity contrasts are up to 106. Thermal convection develops only in the lower part of the thermal boundary layer, and the upper part remains stagnant. Vertical profiles of temperature are measured with precision, allowing deduction of the thickness of the stagnant lid and the convective heat flux. At the onset of convection, the viscosity contrast across the unstable boundary layer has a value of about 3. In fully developed convection, this viscosity contrast is higher, with a typical value of 10. The heat flux through the top of the layer depends solely on local conditions in the unstable boundary layer and may be written \[Q_{\rm s} = - CK_{\rm m} (\alpha g/\kappa u_{\rm m})^{\frac{1}{3}} \Delta T^{\frac{4}{3}}_{\rm v}\], where km and νm are thermal conductivity and kinematic viscosity at the temperature of the well-mixed interior, κ thermal diffusivity, α the coefficient of thermal expansion, g the acceleration due to gravity. ΔTv, is the ‘viscous’ temperature scale defined by \[\Delta T_{\rm v} = - \frac{\mu (T_{\rm m})}{({\rm d}\mu /{\rm d}T)(T_{\rm m})}\] where μ(T) is the fluid viscosity and Tm the temperature of the well-mixed interior. Constant C takes a value of 0.47 ± 0.03. Using these relations, the magnitude of temperature fluctuations and the thickness of the stagnant lid are calculated to be in excellent agreement with the experimental data. One condition for the existence of a stagnant lid is that the applied temperature difference exceeds a threshold value equal to (2ΔTv).

Particle-driven gravity currents.

Abstract: Gravity currents created by the release of a fixed volume of a suspension into a lighter ambient fluid are studied theoretically and experimentally. The greater density of the current and the buoyancy force driving its motion arise primarily from dense particles suspended in the interstitial fluid of the current. The dynamics of the current are assumed to be dominated by a balance between inertial and buoyancy forces; viscous forces are assumed negligible. The currents considered are two-dimensional and flow over a rigid horizontal surface. The flow is modelled by either the single- or the twolayer shallow-water equations, the two-layer equations being necessary to include the effects of the overlying fluid, which are important when the depth of the current is comparable to the depth of the overlying fluid. Because the local density of the gravity current depends on the concentration of particles, the buoyancy contribution to the momentum balance depends on the variation of the particle concentration. A transport equation for the particle concentration is derived by assuming that the particles are vertically well-mixed by the turbulence in the current, are advected by the mean flow and settle out through the viscous sublayer at the bottom of the current. The boundary condition at the moving front of the current relates the velocity and the pressure head at that point. The resulting equations are solved numerically, which reveals that two types of shock can occur in the current. In the late stages of all particle-driven gravity currents, an internal bore develops that separates a particle-free jet-like flow in the rear from a dense gravity-current flow near the front. The second type of bore occurs if the initial height of the current is comparable to the depth of the ambient fluid. This bore develops during the early lock-exchange flow between the two fluids and strongly changes the structure of the current and its transport of particles from those of a current in very deep surroundings. To test the theory, several experiments were performed to measure the length of particle-driven gravity currents as a function of time and their deposition patterns for a variety of particle sizes and initial masses of sediment. The comparison between the theoretical predictions, which have no adjustable parameters, and the experimental results are very good.

Scaling laws for fully developed turbulent shear flows. Part 1. Basic hypotheses and analysis

Abstract: The present work consists of two parts. Here in Part 1, a scaling law (incomplete similarity with respect to local Reynolds number based on distance from the wall) is proposed for the mean velocity distribution in developed turbulent shear flow. The proposed scaling law involves a special dependence of the power exponent and multiplicative factor on the flow Reynolds number. It emerges that the universal logarithmic law is closely related to the envelope of a family of power-type curves, each corresponding to a fixed Reynolds number. A skin-friction law, corresponding to the proposed scaling law for the mean velocity distribution, is derived.In Part 2 (Barenblatt & Prostokishin 1993), both the scaling law for the velocity distribution and the corresponding friction law are compared with experimental data.

The instability of the steady flow past spheres and disks

Abstract: We consider the instability of the steady, axisymmetric base flow past a sphere, and a circular disk (oriented broadside-on to the incoming flow). Finite-element methods are used to compute the steady axisymmetric base flows, and to examine their linear instability to three-dimensional modal perturbations. The numerical results show that for the sphere and the circular disk, the first instability of the base flow is through a regular bifurcation, and the critical Reynolds number (based on the body radius) is 105 for the sphere, and 58.25 for the circular disk. In both cases, the unstable mode is non-axisymmetric with azimuthal wavenumber m = 1. These computational results are consistent with previous experimental observations (Magarvey & Bishop 1961 a, b; Nakamura 1976; Willmarth, Hawk & Harvey 1964).

Direct numerical simulation of turbulent flow in a square duct

Abstract: A direct numerical simulation of a fully developed, low-Reynolds-number turbulent flow in a square duct is presented. The numerical scheme employs a time-splitting method to integrate the three-dimensional, incompressible Navier-Stokes equations using spectral/high-order finite-difference discretization on a staggered mesh ; the nonlinear terms are represented by fifth-order upwind-biased finite differences. The unsteady flow field was simulated at a Reynolds number of 600 based on the mean friction velocity and the duct width, using 96 x 101 x 101 grid points. Turbulence statistics from the fully developed turbulent field are compared with existing experimental and numerical square duct data, providing good qualitative agreement. Results from the present study furnish the details of the corner effects and near-wall effects in this complex turbulent flow field; also included is a detailed description of the terms in the Reynolds-averaged streamwise momentum and vorticity equations. Mechanisms responsible for the generation of the stress-driven secondary flow are studied by quadrant analysis and by analysing the instantaneous turbulence structures. It is demonstrated that the mean secondary flow pattern, the distorted isotachs and the anisotropic Reynolds stress distribution can be explained by the preferred location of an ejection structure near the corner and the interaction between bursts from the two intersecting walls. Corner effects are also manifested in the behaviour of the pressure-strain and velocity-pressure gradient correlations.

The three-dimensional evolution of a plane mixing layer: pairing and transition to turbulence

Abstract: The evolution of three-dimensional temporally evolving plane mixing layers through as many as three pairings has been simulated numerically. All simulations were begun from a few low-wavenumber disturbances, usually derived from linear stability theory, in addition to the mean velocity. Three-dimensional perturbations were used with amplitudes ranging from infinitesimal to large enough to trigger a rapid transition to turbulence. Pairing is found to inhibit the growth of infinitesimal three-dimensional disturbances, and to trigger the transition to turbulence in highly three-dimensional flows. The mechanisms responsible for the growth of three-dimensionality and onset of transition to turbulence are described. The transition to turbulence is accompanied by the formation of thin sheets of spanwise vorticity, which undergo secondary rollups. The post-transitional simulated flow fields exhibit many properties characteristic of turbulent flows.

Measurements of the primary instabilities of film flows

Abstract: We present novel measurements of the primary instabilities of thin liquid films flowing down an incline. A fluorescence imaging method allows accurate measurements of film thickness h(x, y, t) in real time with a sensitivity of several microns, and laser beam deflection yields local measurements with a sensitivity of less than one micron. We locate the instability with good accuracy despite the fact that it occurs (asymptotically) at zero wavenumber, and determine the critical Reynolds number R, for the onset of waves as a function of angle ,8. The measurements of R,(/3) are found to be in good agreement with calculations, as are the growth rates and wave velocities. We show experimentally that the initial instability is convective and that the waves are noisesustained. This means that the waveform and its amplitude are strongly affected by external noise at the source. We investigate the role of noise by varying the level of periodic external forcing. The nonlinear evolution of the waves depends strongly on the initial wavenumber (or the frequency f). A new phase boundary e(R) is measured, which separates the regimes of saturated finite amplitude waves (at high f) from multipeaked solitary waves (at low f). This boundary probably corresponds approximately to the sign reversal of the third Landau coefficient in weakly nonlinear theory. Finally, we show that periodic waves are unstable over a wide frequency band with respect to a convective subharmonic instability. This instability leads to disordered two-dimensional waves.

Gas bubbles bursting at a free surface

Abstract: When a small air bubble bursts from an equilibrium position at an air/water interface, a complex motion ensues resulting in the production of a high-speed liquid jet. This free-surface motion following the burst is modelled numerically using a boundary integral method. Jet formation and liquid entrainment rates from jet breakup into drops are calculated and compared with existing experimental evidence. In order to investigate viscous effects, a boundary layer is included in the calculations by employing a time-stepping technique which allows the boundary mesh to remain orthogonal to the surface. This allows an approximation of the vorticity development in the region of boundary-layer separation during jet formation. Calculated values of pressure and energy dissipation rates in the fluid indicate a violent motion, particularly for smaller bubbles. This has important implications for the biological industry where animal cells in bioreactors have been found to be killed by the presence of small bubbles.

Direct numerical simulation of isotropic turbulence interacting with a weak shock wave

Abstract: Direct numerical simulations are used to investigate the interaction of isotropic quasi-incompressible turbulence with a weak shock wave. A linear analysis of the interaction is conducted for comparison with the simulations. Both the simulations and the analysis show that turbulence is enhanced during the interaction. Turbulent kinetic energy and transverse vorticity components are amplified, and turbulent lengthscales are decreased. It is suggested that the amplification mechanism is primarily linear. Simulations also showed a rapid evolution of turbulent kinetic energy just downstream of the shock, a behavior not reproduced by the linear analysis. Analysis of the budget of the turbulent kinetic energy transport equation shows that this behavior can be attributed to the pressure transport term. Multiple compression peaks were found along the mean streamlines at locations where the local shock thickness had increased significantly.

Feedback control of vortex shedding at low Reynolds numbers

Abstract: This paper describes experiments undertaken to study in detail the control of vortex shedding from circular cylinders at low Reynolds numbers by using feedback to stabilize the wake instability. Experiments have been performed both in a wind tunnel and in an open water channel with flow visualization. It has been found that feedback control is able to delay the onset of the wake instability, rendering the wake stable at Reynolds numbers about 20% higher than otherwise. At higher flow rates, however, it was not possible to use single-channel feedback to stabilize the wake - although, deceptively, it was possible to reduce the unsteadiness recorded by a near-wake sensor. When control is applied to a long span only the region near the control sensor is controlled. The results presented in this paper generally support the analytical results of other researchers.

Saltating particles over flat beds

Abstract: Measurements of ejection and impact velocities, trajectory lengths and maximum rise heights of sand grains (median diameters 118 and 188 μm) in saltation over a flat sand bed in a wind tunnel have been obtained from the digitization of multiple-image photo graphs. The mean angle of ejection is found to be about 30 o from the horizontal (rather than 90 o ) with mean vertical ejection velocity of about 2u * , where u * is the friction velocity. Trajectories of saltating grains have been computed, using the measurements of the initial ejection velocities and the mean velocity profile of the air flow.

Flow in deformable porous media. Part 1 Simple analysis

Abstract: Many processes in the Earth, such as magma migration, can be described by the flow of a low-viscosity fluid in a viscously deformable, permeable matrix. The purpose of this and a companion paper is to develop a better physical understanding of the equations governing these two-phase flows. This paper presents a series of analytic approximate solutions to the governing equations to show that the equations describe two different modes of matrix deformation. Shear deformation of the matrix is governed by Stokes equation and can lead to porosity-driven convection. Volume changes of the matrix are governed by a nonlinear dispersive wave equation for porosity. Porosity waves exist because the fluid flux is an increasing function of porosity and the matrix can expand or compact in response to variations in the fluid flux. The speed and behaviour of the waves depend on the functional relationship between permeability and porosity. If the partial derivative of the permeability with respect to porosity, ∂kϕ/∂ϕ, is also an increasing function of porosity, then the waves travel faster than the fluid in the pores and can steepen into porosity shocks. The propagation of porosity waves, however, is resisted by the viscous resistance of the matrix to volume changes. Linear analysis shows that viscous stresses cause plane waves to disperse and provide additional pressure gradients that deflect the flow of fluid around obstacles. When viscous resistance is neglected in the nonlinear equations, porosity shock waves form from obstructions in the fluid flux. Using the method of characteristics, we quantify the specific criteria for shocks to develop in one and two dimensions. A companion paper uses numerical schemes to show that in the full equations, viscous resistance to volume changes causes simple shocks to disperse into trains of nonlinear solitary waves.

Turbulent shear flow over slowly moving waves

Abstract: We investigate the changes to a fully developed turbulent boundary layer caused by the presence of a two-dimensional moving wave of wavelength L = 2π/k and amplitude a. Attention is focused on small slopes, ak, and small wave speeds, c, so that the linear perturbations are calculated as asymptotic sequences in the limit (u* + c)/UB(L) → 0 (u* is the unperturbed friction velocity and UB(L) is the approach-flow mean velocity at height L). The perturbations can then be described by an extension of the four-layer asymptotic structure developed by Hunt, Leibovich & Richards (1988) to calculate the changes to a boundary layer passing over a low hill.When (u* + c)/UB(L) is small, the matched height, zm (the height where UB equals c), lies within an inner surface layer, where the perturbation Reynolds shear stress varies only slowly. Solutions across the matched height are then constructed by considering an equation for the shear stress. The importance of the shear-stress perturbation at the matched height implies that the inviscid theory of Miles (1957) is inappropriate in this parameter range. The perturbations above the inner surface layer are not directly influenced by the matched height and the region of reversed flow below zm: they are similar to the perturbations due to a static undulation, but the ‘effective roughness length’ that determines the shape of the unperturbed velocity profile is modified to zm = z0 exp (kc/u*).The solutions for the perturbations to the boundary layer are used to calculate the growth rate of waves, which is determined at leading order by the asymmetric pressure perturbation induced by the thickening of the perturbed boundary layer on the leeside of the wave crest. At first order in (u* + c)/UB(L), however, there are three new effects which, numerically, contribute significantly to the growth rate, namely: the asymmetries in both the normal and shear Reynolds stresses associated with the leeside thickening of the boundary layer, and asymmetric perturbations induced by the varying surface velocity associated with the fluid motion in the wave; further asymmetries induced by the variation in the surface roughness along the wave may also be important.

Structure of bubble plumes in linearly stratified environments

Abstract: Bubble plumes in a linearly stratified ambient fluid are studied. Four well-defined flow regions were observed: an upward-moving bubble core, an inner plume consisting of a mixture of bubbles and relatively dense fluid, an annular downdraught and beyond that a horizontal intrusion flow. Depending on the gas flow rate with respect to the stratification, three types of intrusions were documented. At large gas flow rates a single intrusion was observed. As the gas flow rate was decreased, the buoyancy flux was insufficient to carry the lower fluid to the surface and a stack of intrusions were formed. At very low gas flow rates the intrusions became unsteady. The transition between these three regimes was observed to occur at critical values of the parameters N3H4/(QBg), QBg/ (4πα2u3sH), and H/HT, where N is the buoyancy frequency, H is the water depth, HT is equal to H + HA, HA being the atmospheric pressure head, QB is the gas flow rate at the bottom, g the acceleration due to gravity, α the entrainment coefficient and us the differential between the bubble and the average water velocity commonly called the slip velocity. The height between intrusions was found to scale with the Ozmidov length (QBg/N3)¼, the plunge point entrainment with the inner plume volume flux ( and the radial distance to the plunge point with (Q0g/N3)¾, where Q0 is the gas flow rate at the free surface.These results were used to construct a double annular plume model which was used to investigate the efficiency of conversion of the input bubble energy to potential energy of the stratification; the efficiency was found to first increase, reach a maximum, then decrease with decreasing gas flow rate. This agreed well with the results from the laboratory experiments.

A mechanism for bypass transition from localized disturbances in wall-bounded shear flows

Abstract: The linear, nonlinear and breakdown stages in the transition of localized disturbances in plane Poiseuille flow is studied by direct numerical simulations and analysis of the linearized Navier–Stokes equations. Three-dimensionality plays a key role and allows for algebraic growth of the normal vorticity through the linear lift-up mechanism. This growth primarily generates elongated structures in the streamwise direction since it is largest at low streamwise wavenumbers. For finite-amplitude disturbances such structures will be generated essentially independent of the details of the initial disturbance, since the preferred nonlinear interactions transfer energy to low streamwise wavenumbers. The nonlinear interactions also give a decrease in the spanwise scales. For the stronger initial disturbances the streamwise vorticity associated with the slightly inclined streaks was found to roll up into distinct streamwise vortices in the vicinity of which breakdown occurred. The breakdown starts with a local rapid growth of the normal velocity bringing low-speed fluid out from the wall. This phenomenon is similar to the low-velocity spikes previously observed in transition experiments. Soon thereafter a small turbulent spot is formed. This scenario represents a bypass of the regular Tollmien–Schlichting, secondary instability process. The simulations have been carried out with a sufficient spatial resolution to ensure an accurate description of all stages of the breakdown and spot formation processes. The generality of the observed processes is substantiated by use of different types of initial disturbances and by Blasius boundary-layer simulations. The present results point in the direction of universality of the observed transition mechanisms for localized disturbances in wall-bounded shear flows.

Manipulation of free shear flows using piezoelectric actuators

Abstract: An air jet emanating from a square conduit having an equivalent diameter of 4.34 cm and a centreline velocity of 4 m/s is forced using four resonantly driven piezoelectric actuators placed along the sides of the square exit. Excitation is effected via amplitude modulation of the resonant carrier waveform. The flow is normally receptive to time–harmonic excitation at the modulating frequency but not at the resonant frequency of the actuators. When the excitation amplitude is high enough, the excitation waveform is demodulated by a nonlinear process that is connected with the formation and coalescence of nominally spanwise vortices in the forced segments of the jet shear layer. As a result, the modulating and carrier wave trains undergo spatial amplification and attenuation, respectively, downstream of the exit plane. Strong instabilities of the jet column are excited when the jet is forced at phase relationships between actuators that correspond (to lowest order) to the azimuthal modes m = 0, ±1, ±2, and −1 of an axisymmetric flow. The streamwise velocity component is measured phase locked to the modulating signal in planes normal to the mean flow. Resonantly driving the actuators with different carrier amplitudes results in a distorted mean flow having a featureless spectrum that can be tailored to provide favourable conditions for the introduction and propagation of desirable low-frequency disturbances.

Low Reynolds number k —ε modelling with the aid of direct simulation data

Abstract: The constant C sub mu and the near-wall damping function f sub mu in the eddy-viscosity relation of the k-epsilon model are evaluated from direct numerical simulation (DNS) data for developed channel and boundary layer flow at two Reynolds numbers each. Various existing f sub mu model functions are compared with the DNS data, and a new function is fitted to the high-Reynolds-number channel flow data. The epsilon-budget is computed for the fully developed channel flow. The relative magnitude of the terms in the epsilon-equation is analyzed with the aid of scaling arguments, and the parameter governing this magnitude is established. Models for the sum of all source and sink terms in the epsilon-equation are tested against the DNS data, and an improved model is proposed.

Damping and reflection coefficient measurements for an open pipe at low Mach and low Helmholtz numbers

Abstract: The propagation of plane acoustic waves in smooth pipes and their reflection at open pipe terminations have been studied experimentally. The accuracy of the measurements is determined by comparison of experimental data with results of linear theory for the propagation of acoustic waves in a pipe with a quiescent fluid. The damping and the reflection at an unflanged pipe termination are compared.In the presence of a fully developed turbulent mean flow the measurements of the damping confirm the results of Ronneberger & Ahrens (1977). In the high-frequency limit the quasi-laminar theory of Ronneberger (1975) predicts accurately the convective effects on the damping of acoustic waves. For low frequencies a simple theory combining the rigid-plate model of Ronneberger & Ahrens (1977) with the theoretical approach of Howe (1984) yields a fair prediction of the influence of turbulence on the shear stress. The finite response time of the turbulence near the wall to the acoustic perturbations has to be taken into account in order to explain the experimental data. The model yields a quasi-stationary limit of the damping which does not take into account the fundamental difference between the viscous and thermal dissipation observed for low frequencies.Measurements of the nonlinear behaviour of the reflection properties for unflanged pipe terminations with thin and thick walls in the absence of a mean flow confirm the theory of Disselhorst & van Wijngaarden (1980), for the low-frequency limit. It appears however that a two-dimensional theory such as proposed by Disselhorst & van Wijngaarden (1980) for the high-frequency limit underestimates the acoustical energy absorption by vortex shedding by a factor 2.5.The measured influence of wall thickness on the reflection properties of an open pipe end confirms the linear theory of Ando (1969). In the presence of a mean flow the end correction δ of an unflanged pipe end varies from the value at the high-Strouhal-number limit of δ/a = 0.61, with a the pipe radius, which is close to the value in the absence of a mean flow given by Levine & Schwinger (1948) of δ/a = 0.6133, to a value of δ/a = 0.19 in the low-Strouhal-number limit which is close to the value predicted by Rienstra (1983) of δ/a = 0.26.The pressure reflection coefficient is found to agree with the theoretical predictions by Munt (1977, 1990) and Cargill (1982b) in which a full Kutta condition is included. The accuracy of the theory is fascinating in view of the dramatic simplifications introduced in the theory. For a thick-walled pipe end and a pipe terminated by a horn the end correction behaviour is similar. It is surprising that the nonlinear behaviour at low frequencies and high acoustic amplitudes in the absence of mean flow does not influence the end correction significantly.The aero-acoustic behaviour of the pipe end is dramatially influenced by the presence of a horn. In the presence of a mean flow the horn is a source of sound for a critical range of the Strouhal number.The high accuracy of the experimental data suggests that acoustic measurements can be used for a systematic study of turbulence in unsteady flow and of unsteady flow separation.