Research Article
Analysis and modelling of turbulent flow in an axially rotating pipe
 C. G. SPEZIALE, B. A. YOUNIS, S. A. BERGER

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 25 March 2000, pp. 126

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The analysis and modelling of the structure of turbulent flow in a circular pipe subjected to an axial rotation is presented. Particular attention is paid to determining the terms in various turbulence closures that generate the two main physical features that characterize this flow: a rotationally dependent axial mean velocity and a rotationally dependent mean azimuthal or swirl velocity relative to the rotating pipe. It is shown that the first feature is well represented by twodimensional explicit algebraic stress models but is irreproducible by traditional twoequation models. On the other hand, threedimensional framedependent models are needed to predict the presence of a mean swirl velocity. The latter is argued to be a secondary effect which arises from a cubic nonlinearity in standard algebraic models with conventional nearwall treatments. Secondorder closures are shown to give a more complete description of this flow and can describe both of these features fairly well. In this regard, quadratic pressure–strain models perform the best overall when extensive comparisons are made with the results of physical and numerical experiments. The physical significance of this problem and the implications for future research in turbulence are discussed in detail.
Scaling in thermal convection: a unifying theory
 SIEGFRIED GROSSMANN, DETLEF LOHSE

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 25 March 2000, pp. 2756

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A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh–Bénard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent largescale convection roll (‘wind of turbulence’) and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number Ra versus Prandtl number Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the kinetic boundary layer is thicker. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra [lsim ] 1011) the leading terms are Nu ∼ Ra1/4Pr1/8, Re ∼ Ra1/2Pr−3/4 for Pr [lsim ] 1 and Nu ∼ Ra1/4Pr−1/12, Re ∼ Ra1/2Pr−5/6 for Pr [gsim ] 1. In most measurements these laws are modified by additive corrections from the neighbouring regimes so that the impression of a slightly larger (effective) Nu vs. Ra scaling exponent can arise. The most important of the neighbouring regimes towards large Ra are a regime with scaling Nu ∼ Ra1/2Pr1/2, Re ∼ Ra1/2Pr−1/2 for medium Pr (‘Kraichnan regime’), a regime with scaling Nu ∼ Ra1/5Pr1/5, Re ∼ Ra2/5Pr−3/5 for small Pr, a regime with Nu ∼ Ra1/3, Re ∼ Ra4/9Pr−2/3 for larger Pr, and a regime with scaling Nu ∼ Ra3/7Pr−1/7, Re ∼ Ra4/7Pr−6/7 for even larger Pr. In particular, a linear combination of the ¼ and the 1/3 power laws for Nu with Ra, Nu = 0.27Ra1/4 + 0.038Ra1/3 (the prefactors follow from experiment), mimics a 2/7 powerlaw exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through the nonnormalnonlinear transition to turbulence and another regime emerges. The theory presented is best summarized in the phase diagram figure 2 and in table 2.
Turbulent thermal convection in a cell with ordered rough boundaries
 Y.B. DU, P. TONG

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 25 March 2000, pp. 5784

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A novel convection experiment is conducted in a cell with rough upper and lower surfaces. The measured heat transport in the rough cell is found to be increased by more than 76%. Flow visualization and nearwall temperature measurements reveal new dynamics for the emission of thermal plumes. The experiment shows that the interaction between the horizontal shear flow due to the largescale circulation and the ordered rough surface creates a secondary flow (eddies) in the groove region. The secondary flow together with the largescale circulation enhance the detachment of the thermal boundary layer from the tip of the rough elements. These extra thermal plumes are responsible for the enhanced heat transport in the rough cell. The discovery of the enhanced heat transport has important applications in engineering for more efficient heat transfer.
Threedimensional effects in wind tunnel studies of shock wave reflection
 B. W. SKEWS

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 25 March 2000, pp. 85104

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This paper concentrates on establishing the threedimensional flow geometry associated with studies of shock wave reflection between two symmetrical wedges in supersonic flow. It considers the issue of hysteresis in such flows, and draws a distinction between three different aspects of hysteresis, associated with: ideal twodimensional flow, flow with noise, and threedimensional effects. The threedimensional nature of the flow field is elucidated by the use of oblique shadowgraph photography where the optical axis of the shadowgraph system passes at an oblique angle, of as much as 55°, through the test section. The traces of the wave system reflecting off the tunnel window are identified and are used to assist in identification of wave profiles. The nature of the approach of the peripheral Mach reflections collapsing towards the centre of the flow becomes evident, as does the mechanism of transition from Mach reflection to regular reflection. Distinct evidence of the effects of flow perturbations at the mechanical equilibrium transition point are presented, as are changes in the rate of growth of the Mach stem near this point.
It is shown that threedimensional effects can have a major effect on the wedge angle for transition. In the present tests, at Mach 3.1 and a wedge aspect ratio of 0.5, this occurs at a wedge angle of about 5° higher than the theoretical maximum for the corresponding twodimensional flow, where the dual solution domain extends over only two degrees.
Enhanced dissipation for quasigeostrophic motion over smallscale topography
 JACQUES VANNESTE

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 25 March 2000, pp. 105122

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The effect of a smallscale topography on largescale, smallamplitude oceanic motion is analysed using a twodimensional quasigeostrophic model that includes freesurface and β effects, Ekman friction and viscous (or turbulent) dissipation. The topography is twodimensional and periodic; its slope is assumed to be much larger than the ratio of the ocean depth to the Earth's radius. An averaged equation of motion is derived for flows with spatial scales that are much larger than the scale of the topography and either (i) much larger than or (ii) comparable to the radius of deformation. Compared to the standard quasigeostrophic equation, this averaged equation contains an additional dissipative term that results from the interaction between topography and dissipation. In case (i) this term simply represents an additional Ekman friction, whereas in case (ii) it is given by an integral over the history of the largescale flow. The properties of the additional term are studied in detail. For case (i) in particular, numerical calculations are employed to analyse the dependence of the additional Ekman friction on the structure of the topography and on the strength of the original dissipation mechanisms.
The flow induced by a rotationally oscillating and translating circular cylinder
 S. C. R. DENNIS, P. NGUYEN, SERPIL KOCABIYIK

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 25 March 2000, pp. 123144

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The temporal development of twodimensional viscous incompressible flow induced by an impulsively started circular cylinder which performs timedependent rotational oscillations about its axis and translates at right angles to this axis is investigated. The investigation is based on the solutions of the unsteady Navier–Stokes equations. A series expansion for small times is developed. The Navier–Stokes equations are also integrated by a spectral–finite difference method for moderate values of time for both moderate and high Reynolds numbers. The numerical method is checked with the results of the analytical solution. The effects of the Reynolds number and of the forcing Strouhal number S on the laminar asymmetric flow structure in the nearwake region are studied. The lift and drag coefficients are also extracted from numerical results. An interesting phenomenon has been observed both in the flow patterns and in the behaviour of drag coefficients for S = π/2 at Reynolds number R = 500 and is discussed. For comparison purposes the startup flow is determined numerically at a low Reynolds number and is found to be in good agreement with previous experimental predictions.
Turbulent diffusion near a free surface
 LIAN SHEN, GEORGE S. TRIANTAFYLLOU, DICK K. P. YUE

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 25 March 2000, pp. 145166

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We study numerically and analytically the turbulent diffusion characteristics in a lowFroudenumber turbulent shear flow beneath a free surface. In the numerical study, the Navier–Stokes equations are solved directly subject to viscous boundary conditions at the free surface. From an ensemble of such simulations, we find that a boundary layer develops at the free surface characterized by a fast reduction in the value of the eddy viscosity. As the free surface is approached, the magnitude of the mean shear initially increases over the boundary (outer) layer, reaches a maximum and then drops to zero inside a much thinner inner layer. To understand and model this behaviour, we derive an analytical similarity solution for the mean flow. This solution predicts well the shape and the timescaling behaviour of the mean flow obtained in the direct simulations. The theoretical solution is then used to derive scaling relations for the thickness of the inner and outer layers. Based on this similarity solution, we propose a freesurface function model for largeeddy simulations of freesurface turbulence. This new model correctly accounts for the variations of the Smagorinsky coefficient over the freesurface boundary layer and is validated in both a priori and a posteriori tests.
Structure, diffusion and rheology of Brownian suspensions by Stokesian Dynamics simulation
 DAVID R. FOSS, JOHN F. BRADY

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 25 March 2000, pp. 167200

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The nonequilibrium behaviour of concentrated colloidal dispersions is studied using Stokesian Dynamics, a moleculardynamicslike simulation technique for analysing suspensions of particles immersed in a Newtonian fluid. The simulations are of a monodisperse suspension of Brownian hard spheres in simple shear flow as a function of the Péclet number, Pe, which measures the relative importance of hydrodynamic and Brownian forces, over a range of volume fraction 0.316 [les ] ϕ [les ] 0.49. For Pe < 10, Brownian motion dominates the behaviour, the suspension remains welldispersed, and the viscosity shear thins. The first normal stress difference is positive and the second negative. At higher Pe, hydrodynamics dominate resulting in an increase in the longtime selfdiffusivity and the viscosity. The first normal stress difference changes sign when hydrodynamics dominate. Simulation results are shown to agree well with both theory and experiment.
Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth
 ODD M. FALTINSEN, OLAV F. ROGNEBAKKE, IVAN A. LUKOVSKY, ALEXANDER N. TIMOKHA

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 25 March 2000, pp. 201234

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The discrete infinitedimensional modal system describing nonlinear sloshing of an incompressible fluid with irrotational flow partially occupying a tank performing an arbitrary threedimensional motion is derived in general form. The tank has vertical walls near the free surface and overturning waves are excluded. The derivation is based on the Bateman–Luke variational principle. The free surface motion and velocity potential are expanded in generalized Fourier series. The derived infinitedimensional modal system couples generalized timedependent coordinates of free surface elevation and the velocity potential. The procedure is not restricted by any order of smallness. The general multidimensional structure of the equations is approximated to analyse sloshing in a rectangular tank with finite water depth. The amplitude–frequency response is consistent with the fifthorder steadystate solutions by Waterhouse (1994). The theory is validated by new experimental results. It is shown that transients and associated nonlinear beating are important. An initial variation of excitation periods is more important than initial conditions. The theory is invalid when either the water depth is small or water impacts heavily on the tank ceiling. Alternative expressions for hydrodynamic loads are presented. The procedure facilitates simulations of a coupled vehicle–fluid system.
On the vorticity transport due to dissipating or breaking waves in shallowwater flow
 OLIVER BÜHLER

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 25 March 2000, pp. 235263

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Theoretical and numerical results are presented on the transport of vorticity (or potential vorticity) due to dissipating gravity waves in a shallowwater system with background rotation and bottom topography. The results are obtained under the assumption that the flow can be decomposed into smallscale gravity waves and a largescale mean flow. The particlefollowing formalism of ‘generalized Lagrangianmean’ theory is then used to derive an ‘effective mean force’ that captures the vorticity transport due to the dissipating waves. This can be achieved without neglecting other, nondissipative, effects which is an important practical consideration. It is then shown that the effective mean force obeys the socalled ‘pseudomomentum rule’, i.e. the force is approximately equal to minus the local dissipation rate of the wave's pseudomomentum. However, it is also shown that this holds only if the underlying dissipation mechanism is momentumconserving. This requirement has important implications for numerical simulations, and these are discussed.
The novelty of the results presented here is that they have been derived within a uniform theoretical framework, that they are not restricted to small wave amplitude, raytracing or JWKBtype approximations, and that they also include wave dissipation by breaking, or shock formation. The theory is tested carefully against shockcapturing nonlinear numerical simulations, which includes the detailed study of a wavetrain subject to slowly varying bottom topography. The theory is also crosschecked in the appropriate asymptotic limit against recently formulated weakly nonlinear theories. In addition to the general finiteamplitude theory, detailed smallamplitude expressions for the main results are provided in which the explicit appearance of Lagrangian fields can be avoided. The motivation for this work stems partly from an ongoing study of highaltitude breaking of internal gravity waves in the atmosphere, and some preliminary remarks on atmospheric applications and on threedimensional stratified versions of these results are given.
The role of dissipation and mixing in exchange flow through a contracting channel
 KRAIG B. WINTERS, HARVEY E. SEIM

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 25 March 2000, pp. 265290

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We investigate the transport of mass and momentum between layers in idealized exchange flow through a contracting channel. Lockexchange initial value problems are run to approximately steady state using a threedimensional, nonhydrostatic numerical model. The numerical model resolves the largescale exchange flow and shear instabilities that form at the interface, parameterizing the effects of subgridscale turbulence. The closure scheme is based on an assumed steady, local balance of turbulent production and dissipation in a densitystratified fluid.
The simulated flows are analysed using a twolayer decomposition and compared with predictions from twolayer hydraulic theory. Interlayer transport leads to a systematic deviation of the simulated maximal exchange flows from predictions. Relative to predictions, the observed flows exhibit lower Froude numbers, larger transports and wider regions of subcritical flow in the contraction. To describe entrainment and mixing between layers, the computed solutions are decomposed into a threelayer structure, with two bounding layers separated by an interfacial layer of finite thickness and variable properties. Both bounding layers lose fluid to the interfacial layer which carries a significant fraction of the horizontal transport. Entrainment is greatest from the faster moving layer, occurring preferentially downstream of the contraction.
Bottom friction exerts a drag on the lower layer, fundamentally altering the overall dynamics of the exchange. An example where bed friction leads to a submaximal exchange is discussed. The external forcing required to sustain a net transport is significantly less than predicted in the absence of bottom stresses.
Stability of fluid flow in a flexible tube to nonaxisymmetric disturbances
 V. SHANKAR, V. KUMARAN

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 25 March 2000, pp. 291314

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The stability of fluid flow in a flexible tube to nonaxisymmetric perturbations is analysed in this paper. In the first part of the paper, the equivalents of classical theorems of hydrodynamic stability are derived for inviscid flow in a flexible tube subjected to arbitrary nonaxisymmetric disturbances. Perturbations of the form vi = v˜i exp [ik(x − ct) + inθ] are imposed on a steady axisymmetric mean flow U(r) in a flexible tube, and the stability of mean flow velocity profiles and bounds for the phase velocity of the unstable modes are determined for arbitrary values of azimuthal wavenumber n. Here r, θ and x are respectively the radial, azimuthal and axial coordinates, and k and c are the axial wavenumber and phase velocity of disturbances. The flexible wall is represented by a standard constitutive relation which contains inertial, elastic and dissipative terms. The general results indicate that the fluid flow in a flexible tube is stable in the inviscid limit if the quantity Ud[Gscr ]/dr [ges ] 0, and could be unstable for Ud[Gscr ]/dr < 0, where [Gscr ] ≡ rU′/(n2 + k2r2). For the case of Hagen–Poiseuille flow, the general result implies that the flow is stable to axisymmetric disturbances (n = 0), but could be unstable to nonaxisymmetric disturbances with any nonzero azimuthal wavenumber (n ≠ 0). This is in marked contrast to plane parallel flows where twodimensional disturbances are always more unstable than threedimensional ones (Squire theorem). Some new bounds are derived which place restrictions on the real and imaginary parts of the phase velocity for arbitrary nonaxisymmetric disturbances.
In the second part of this paper, the stability of the Hagen–Poiseuille flow in a flexible tube to nonaxisymmetric disturbances is analysed in the high Reynolds number regime. An asymptotic analysis reveals that the Hagen–Poiseuille flow in a flexible tube is unstable to nonaxisymmetric disturbances even in the inviscid limit, and this agrees with the general results derived in this paper. The asymptotic results are extended numerically to the moderate Reynolds number regime. The numerical results reveal that the critical Reynolds number obtained for inviscid instability to nonaxisymmetric disturbances is much lower than the critical Reynolds numbers obtained in the previous studies for viscous instability to axisymmetric disturbances when the dimensionless parameter Σ = ρGR2/η2 is large. Here G is the shear modulus of the elastic medium, ρ is the density of the fluid, R is the radius of the tube and η is the viscosity of the fluid. The viscosity of the wall medium is found to have a stabilizing effect on this instability.
A scalar subgrid model with flow structure for largeeddy simulations of scalar variances
 P. FLOHR, J. C. VASSILICOS

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 25 March 2000, pp. 315349

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A new model to simulate passive scalar fields in largeeddy simulations of turbulence is presented. The scalar field is described by clouds of tracer particles and the subgrid contribution of the tracer displacement is modelled by a kinematic model which obeys Kolmogorov's inertialrange scaling, is incompressible and incorporates turbulentlike flow structure of the turbulent small scales. This makes it possible to study the scalar variance field with inertialrange effects explicitly resolved by the kinematic subgrid field while the LES determines the value of the Lagrangian integral time scale TL. In this way, the modelling approach does not rely on unknown Lagrangian input parameters which determine the absolute value of the scalar variance.
The mean separation of particle pairs displays a welldefined Richardson scaling in the inertial range, and we find that the Richardson constant GΔ ≈ 0.07 which is small compared to the value obtained from stochastic models with the same TL. The probability density function of the separation of particle pairs is found to be highly nonGaussian in the inertial range of times and for long times becomes Gaussian. We compute the scalar variance field for an instantaneous line source and find good agreement with experimental data.