Papers
Skin-friction measurements in high-enthalpy hypersonic boundary layers
- C. P. GOYNE, R. J. STALKER, A. PAULL
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- 24 June 2003, pp. 1-32
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Skin-friction measurements are reported for high-enthalpy and high-Mach-number laminar, transitional and turbulent boundary layers. The measurements were performed in a free-piston shock tunnel with air-flow Mach number, stagnation enthalpy and Reynolds numbers in the ranges of 4.4–6.7, 3–13 MJ kg$^{-1}$ and $0.16\times 10^{6}$–$21\times 10^{6}$, respectively. Wall temperatures were near 300 K and this resulted in ratios of wall enthalpy to flow-stagnation enthalpy in the range of 0.1–0.02. The experiments were performed using rectangular ducts. The measurements were accomplished using a new skin-friction gauge that was developed for impulse facility testing. The gauge was an acceleration compensated piezoelectric transducer and had a lowest natural frequency near 40 kHz. Turbulent skin-friction levels were measured to within a typical uncertainty of ± 7%. The systematic uncertainty in measured skin-friction coefficient was high for the tested laminar conditions; however, to within experimental uncertainty, the skin-friction and heat-transfer measurements were in agreement with the laminar theory of van Driest (1952). For predicting turbulent skin-friction coefficient, it was established that, for the range of Mach numbers and Reynolds numbers of the experiments, with cold walls and boundary layers approaching the turbulent equilibrium state, the Spalding & Chi (1964) method was the most suitable of the theories tested. It was also established that if the heat transfer rate to the wall is to be predicted, then the Spalding & Chi (1964) method should be used in conjunction with a Reynolds analogy factor near unity. If more accurate results are required, then an experimentally observed relationship between the Reynolds analogy factor and the skin-friction coefficient may be applied.
Space–time development of the onset of a shallow-water vortex
- J.-C. LIN, M. OZGOREN, D. ROCKWELL
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- 24 June 2003, pp. 33-66
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An impulsively started jet in shallow water gives rise to vortices having a characteristic diameter larger than the water depth. A technique of high-image-density particle image velocimetry allows characterization of the space–time development of the instantaneous flow patterns along planes representing the quasi-two-dimensional and three-dimensional vortex structure. The quasi-two-dimensional patterns exhibit different categories of vortex development and interaction, depending upon the depth of the shallow water layer. Despite these distinctions, the variations of normalized vortex position, diameter, and circulation, as well as peak vorticity within the vortex, are very similar for sufficiently small water depth.
These quasi-two-dimensional patterns are, in turn, related to specific forms of three-dimensional flow structure, which is highly ordered. A prevalent feature is a vortex orthogonal to, and just ahead of, the primary, quasi-two-dimensional vortex. Its streamline topology, on a plane parallel to the axis of the quasi-two-dimensional vortex, exhibits a separation bubble with a well-defined separatix at the bottom (bed) surface. Moreover, its vorticity can exceed that of the quasi-two-dimensional pattern by a factor of two. This feature is consistent for all values of water depth. When the depth becomes sufficiently large, however, the three-dimensional vortex pattern involves an array of vorticity concentrations, which extends across the entire depth of the water.
On a plane very close to the bottom surface (bed), global instantaneous distributions of velocity and vorticity exhibit large gradients; they are associated with small-scale vorticity concentrations characteristic of rapid transition. The corresponding streamline topology of the averaged flow close to the bed, however, exhibits a stable focus and is a direct indicator of the topology well above the bed.
Fluid displacement by Stokes flow past a spherical droplet
- I. EAMES, D. GOBBY, S. B. DALZIEL
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- 24 June 2003, pp. 67-85
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The concept of ‘drift’, which has been exploited in many high Reynolds number and inviscid flow problems, is here applied to examine transport by a spherical viscous droplet (of radius $a$) moving in a Stokes flow.
In an unbounded flow, the velocity in the direction of translation of a spherical droplet is positive everywhere because streamlines, in the fluid frame of reference, ‘close’ at infinity. Fluid particles are displaced a positive distance, $X$, forward, which is expressed in terms of the initial distance from the stagnation streamline $\rho_0$. Asymptotic expressions are developed for $X$ in the limits of $\rho_0/a\,{\ll}\,1$ and $\gg 1$. The nature of the singularity of the centreline displacement changes from $O(-a\log(\rho_0/a))$ to $O(a^2/\rho_0)$ as the viscosity of the droplet, compared to the ambient fluid, increases. By employing a mass-conservation argument, asymptotic expressions are calculated for the partial drift volume, $D_p$, associated with a circular material surface of radius $\rho_m$ which starts far in front of a droplet that translates a finite distance. Since the velocity perturbation decays slowly with distance from the droplet, $D_p$ tends to become unbounded as $\rho_m$ increases, in contrast to inviscid flows.
The presence of a rigid wall ensures that the velocity perturbation decays sufficiently rapidly that fluid particles, which do not lie on the stagnation streamline, are displaced a finite distance away from the wall. The distortion of a material surface lying a distance $h_L$ above a wall, by the droplet, starting a distance $h_S$ from the wall and moving away, is studied. The volume transported away from the wall, calculated using a multipolar flow approximation, is $D_p = \pi h_L^2 a(3\lambda+2)/(\lambda+1)$, and is weakly dependent on the starting position of the droplet, in accordance with numerical results. When the material surface is close to the wall ($h_L/a \ll 1$), the volume transported away from a wall is significantly smaller than for inviscid flows because the no-slip condition on the rigid wall tends to inhibit ‘scouring’. When the material surface is far from the wall ($h_L/a\gg 1$), the viscously dominated flow transports a larger volume of fluid away from the wall because the flow decays slowly with distance from the droplet.
These results can be generalized to arbitrarily shaped bodies, since the transport processes are dominated by the strength of the Stokeslet. The effect of boundaries and inertia on fluid transport processes is briefly discussed.
Dynamics of large-scale structures in turbulent flow over a wavy wall
- NILS KRUSE, AXEL GÜNTHER, PHILIPP RUDOLF VON ROHR
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- 24 June 2003, pp. 87-96
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We describe the dynamics of large-scale structures in a developed turbulent flow between a train of waves and a flat wall. A water channel facility, for which the wavelength, $\Lambda$, of the bottom wall equals the channel height and the wave amplitude is ten times smaller, is used. The channel is sufficiently wide so that structures of spanwise scale O$\{1.5\Lambda\}$ meander laterally. The paper dicusses the temporal behaviour and the meandering motion at a Reynolds number of 4500, defined with the half channel height and the bulk velocity. Digital particle image velocimetry is performed in a horizontal plane with a field of view of $2.6\Lambda\times 2.7\Lambda$. Ten ensembles of 90 consecutive image pairs are acquired at a rate of 15 Hz, a temporal resolution sufficient to assess how the largest flow scales evolve in time. The streamwise velocity $u(x,z,t)$ is filtered using the dominant eigenfunctions that are obtained by a proper orthogonal decomposition analysis. The very large temporal scales of the meandering motion of the O$\{1.5\Lambda\}$ structures could be followed over measurement times of up to 6 s, during which they are convected downstream by distance of 65 wavelengths. The observed coherent lengths in the streamwise direction are significantly larger than the streamwise domain extent of all large-eddy simulation and direct numerical simulation reported so far.
Thermocapillary suppression of the Plateau–Rayleigh instability: a model for long encapsulated liquid zones
- Y.-J. CHEN, R. ABBASCHIAN, P. H. STEEN
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- 24 June 2003, pp. 97-113
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A cylindrical liquid bridge is unstable when its length is longer than its circumference, the Plateau–Rayleigh limit. This capillary instability is modified by fluid motions adjacent to the interface, which can be induced by thermocapillary stress, among other means. A simple flow model with symmetry that mimics the situation in encapsulated floating zones is analysed. The interfacial balance equation is formulated as a bifurcation problem, appropriate when the flows are nearly rectilinear. This balance captures the competition between capillary stress and the flow-induced pressure. The fluid motions are shown to have a stabilizing effect; bridges much longer than the classical limit are stabilized. Numerical branch-tracing and the Lyapunov–Schmidt reduction methods provide the bifurcation structures of branching solutions. A normal-form analysis predicts standing-wave patterns due to mode–mode interaction. The model is proposed as an explanation for the extra long float zones observed in various spacelab experiments.
Small inertial effects on a spherical bubble, drop or particle moving near a wall in a time-dependent linear flow
- JACQUES MAGNAUDET
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- 24 June 2003, pp. 115-142
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The problem of a spherical drop of arbitrary density and viscosity moving near a wall under the effect of a body force is analysed theoretically in the limit where the wall lies in the inner region of the flow disturbance, the distance between the drop and the wall being large compared to the drop radius. The drop may move in an arbitrary direction with respect to the wall, and the undisturbed flow field is assumed to comprise a steady uniform shear or solid-body rotation and a time-dependent uniform stream, the variations of which take place over time scales large compared to the viscous diffusion time. An exact force balance with no limitation on the magnitude of inertial effects is obtained by using the reciprocal theorem. Explicit expressions for the contributions of temporal acceleration, slip and shear or rotation to the total hydrodynamic force are derived in the limit of small-but-finite inertial effects. The connection between these near-wall results and inertial lift and drag corrections in an unbounded flow is discussed. Situations of particular interest in which the lift force results from a combination of contributions due to unsteadiness and advection, like the case of a particle moving near the bottom wall of a centrifuge, are also examined.
Entrainment and detrainment from a model boundary layer
- MEIHONG SUN, SETH LICHTER
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- 24 June 2003, pp. 143-159
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A two-dimensional inviscid flow with piecewise-uniform regions of vorticity is studied as a model of the high-Reynolds-number mixing between a boundary layer and an outer layer. It is found that an initial disturbance to the boundary-layer thickness breaks down into a wave field plus, if the initial disturbance is steep enough, a volume of entrained fluid. The entrained fluid is drawn from the outer layer and then folded into a crevice. The crevice stretches, and eventually pinches off, becoming completely enveloped within the boundary layer. Though the entrained fluid is slender in shape, its volume is significant. Very steep disturbances result in detrainment, in which a small parcel of fluid detaches from the boundary layer and curls into the outer layer. The v-velocity field agrees with many features of Kovasznay et al.'s (1970) measurements in the turbulent boundary layer. This correspondence with fully turbulent flow, plus the characteristics of folding and stretching large volumes of fluid, make the process presented here a candidate for a mechanism by which high-Reynolds-number boundary layers mix with outer-layer fluid.
Experimental observation of resonances in modulated turbulence
- OLIVIER CADOT, JEAN HUGUES TITON, DANIEL BONN
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- 24 June 2003, pp. 161-170
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The response of turbulence to a periodic forcing of the Reynolds number is studied. The turbulent flow is produced in a closed geometry between two counter-rotating disks. The time series of both the global injected power and the local velocity are measured. It is found that the injected power fluctuations exhibit resonances for well-determined values of the modulation frequency, making it possible to estimate a turbulent cascade time. For modulation periods larger than the measured cascade time, the response of the velocity fluctuations is simply proportional to the modulation of the forcing. For high-frequency modulations, the velocity fluctuations are strongly damped: their amplitude decreases as the inverse of the modulation frequency.
A note on tidally generated sand waves
- G. BESIO, P. BLONDEAUX, P. FRISINA
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- 24 June 2003, pp. 171-190
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The process leading to the formation of sand waves in tide dominated coastal areas is investigated by means of the linear stability analysis of a flat sandy bottom subject to oscillatory tidal currents. The conditions for the decay or amplification of small bottom perturbations are determined for arbitrary values of the parameters of the problem. According to field observations, the initial growth of sand waves requires a minimum amplitude of the tidal current, even when the critical bed shear stress for the initial motion of sediment is set equal to zero. Moreover the minimum amplitude depends on sediment characteristics. In particular, the analysis shows that sand waves appear only for a sandy bottom and their growth does not take place when a coarse sediment covers the sea bed. The solution procedure extends the truncation method which is often used to describe the flow generated by the interaction of bottom perturbations with the oscillatory tidal current. The obtained results show that the truncation method describes the mechanism inducing the growth of sand waves, but values of the parameters exist for which its results are not quantitatively accurate. Finally, the asymptotic approach for large values of both r, which is the ratio between the amplitude of the horizontal tidal excursion and the wavelength of the bottom perturbations, and of the stress parameter s is modified in the bottom boundary layer to describe cases characterized by values of s of order one, which is the order of magnitude suggested by an analysis of field data.
Destabilization of a creeping flow by interfacial surfactant: linear theory extended to all wavenumbers
- DAVID HALPERN, ALEXANDER L. FRENKEL
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- 24 June 2003, pp. 191-220
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Creeping flow of a two-layer system with a monolayer of an insoluble surfactant on the interface is considered. The linear-stability theory of plane Couette–Poiseuille flow is developed in the Stokes approximation. To isolate the Marangoni effect, gravity is excluded. The shear-flow instability due to the interfacial surfactant, uncovered earlier for long waves only (Frenkel & Halpern 2002), is studied with inclusion of all wavelengths, and over the entire parameter space of the Marangoni number $M$, the viscosity ratio $m$, the interfacial velocity shear $s$, and the thickness ratio $n$ (${\ge}\,1$). The complex wave speed of normal modes solves a quadratic equation, and the growth rate function is continuous at all wavenumbers and all parameter values. If $M\,{>}\,0$, $s\,{\ne}\,0$, $m\,{<}\,n^2$, and $n\,{>}\,1$, the small disturbances grow provided they are sufficiently long wave. However, the instability is not long wave in the following sense: the unstable waves are not necessarily much longer than the smaller of the two layer thicknesses. On the other hand, there are parametric regimes for which the instability has a mid-wave character, the flow being stable at both sufficiently large and small wavelengths and unstable in between. The critical (instability-onset) manifold in the parameter space is investigated. Also, it is shown that for certain parametric limits the convergence of the dispersion function is non-uniform with respect to the wavenumber. This is used to explain the parametric discontinuities of the long-wave growth-rate exponents found earlier.
On the convectively unstable nature of optimal streaks in boundary layers
- LUCA BRANDT, CARLO COSSU, JEAN-MARC CHOMAZ, PATRICK HUERRE, DAN S. HENNINGSON
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- 24 June 2003, pp. 221-242
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The objective of the study is to determine the absolute/convective nature of the secondary instability experienced by finite-amplitude streaks in the flat-plate boundary layer. A family of parallel streaky base flows is defined by extracting velocity profiles from direct numerical simulations of nonlinearly saturated optimal streaks. The computed impulse response of the streaky base flows is then determined as a function of streak amplitude and streamwise station. Both the temporal and spatio-temporal instability properties are directly retrieved from the impulse response wave packet, without solving the dispersion relation or applying the pinching point criterion in the complex wavenumber plane. The instability of optimal streaks is found to be unambiguously convective for all streak amplitudes and streamwise stations. It is more convective than the Blasius boundary layer in the absence of streaks; the trailing edge-velocity of a Tollmien–Schlichting wave packet in the Blasius boundary layer is around 35% of the free-stream velocity, while that of the wave packet riding on the streaky base flow is around 70%. This is because the streak instability is primarily induced by the spanwise shear and the associated Reynolds stress production term is located further away from the wall, in a larger velocity region, than for the Tollmien–Schlichting instability. The streak impulse response consists of the sinuous mode of instability triggered by the spanwise wake-like profile, as confirmed by comparing the numerical results with the absolute/convective instability properties of the family of two-dimensional wakes introduced by Monkewitz (1988). The convective nature of the secondary streak instability implies that the type of bypass transition studied here involves streaks that behave as amplifiers of external noise.
Experimental study of the Richtmyer–Meshkov instability of incompressible fluids
- C. E. NIEDERHAUS, J. W. JACOBS
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- 24 June 2003, pp. 243-277
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The Richtmyer–Meshkov instability of a low-Atwood-number miscible two-liquid system is investigated experimentally. The initially stratified fluids are contained within a rectangular tank mounted on a sled that rides on a vertical set of rails. The instability is generated by dropping the sled onto a coil spring, producing a nearly impulsive upward acceleration. The subsequent free-fall that occurs as the container travels upward and then downward on the rails allows the instability to evolve in the absence of gravity. The interface separating the two liquids initially has a well-defined sinusoidal perturbation that quickly inverts and then grows in amplitude after undergoing the impulsive acceleration. Disturbance amplitudes are measured and compared to theoretical predictions. Linear stability theory gives excellent agreement with the measured initial growth rate, $\dot{a}_0$, for single-mode perturbations with the predicted amplitudes differing by less than 10% from experimental measurements up to a non-dimensional time $k\dot {a}_0 t = 0.7$, where $k$ is the wavenumber. Linear stability theory also provides excellent agreement for the individual mode amplitudes of multi-mode initial perturbations until the interface becomes multi-valued. Comparison with previously published weakly nonlinear single-mode models shows good agreement up to $k\dot{a}_0 t = 3$, whereas published nonlinear single-mode models provide good agreement up to $k\dot{a}_0 t = 30$. The effects of Reynolds number on the vortex core evolution and overall growth rate of the interface are also investigated. Measurements of the overall amplitude are found to be unaffected by the Reynolds number for the range of values studied here. However, experiments carried out at lower values of Reynolds numbers were found to have decreased vortex core rotation rates. In addition, an instability in the vortex cores is observed. The time of appearance of this instability was found to increase when the Reynolds number is decreased.
Nonlinear steady convection in rotating mushy layers
- D. N. RIAHI
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- 24 June 2003, pp. 279-306
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We consider the problem of nonlinear steady convection in a horizontal mushy layer rotating about a vertical axis. We analyse the stationary modes of convection in the form of two-dimensional oblique rolls and three-dimensional distorted patterns. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the two- and three-dimensional solutions to the weakly nonlinear problem by using a perturbation technique, and the stability of these solutions is investigated with respect to arbitrary three-dimensional disturbances. The results of the analyses in a particular range of values of the amplitude of convection indicate in particular that, over most of the range of values of the parameters, subcritical convection in the form of down-hexagons with down-flow at the cell centres and up-flow at the cell boundaries can be preferred over up-hexagonal convection, where the convective flow is upward at the cell centres and downward at the cell boundaries. For zero or very small values of ${\cal T}$ (${\cal T}\,{\ll}\,1$), which is the square root of a Taylor number, rolls are preferred over supercritical rectangles, while supercritical rectangles, which are characterized by an angle $\gamma$ of about $60^\circ$, are stable and preferred over the rolls for T above some value. Here, $\gamma$ or $180^\circ-\gamma$ are the angles between any two adjacent wavenumber vectors of a rectangular cell. For increasing values of T, these rectangles become subcritically unstable and are replaced by new supercritical rectangles of higher $\gamma$ values, and $\gamma$ increases with T until supercritical squares ($\gamma\,{=}\,90^\circ$) become stable. The significance and realizability of down-hexagons, rectangles and squares are found to be due to the interactions between the local solid fraction and the flow associated with the Coriolis term in the momentum–Darcy equation that are fully taken into account in the present study.
The absorption of axial acoustic waves by a perforated liner with bias flow
- JEFF D. ELDREDGE, ANN P. DOWLING
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- 24 June 2003, pp. 307-335
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The effectiveness of a cylindrical perforated liner with mean bias flow in its absorption of planar acoustic waves in a duct is investigated. The liner converts acoustic energy into flow energy through the excitation of vorticity fluctuations at the rims of the liner apertures. A one-dimensional model that embodies this absorption mechanism is developed. It utilizes a homogeneous liner compliance adapted from the Rayleigh conductivity of a single aperture with mean flow. The model is evaluated by comparing with experimental results, with excellent agreement. We show that such a system can absorb a large fraction of incoming energy, and can prevent all of the energy produced by an upstream source in certain frequency ranges from reflecting back. Moreover, the bandwidth of this strong absorption can be increased by appropriate placement of the liner system in the duct. An analysis of the acoustic energy flux is performed, revealing that local differences in fluctuating stagnation enthalpy, distributed over a finite length of duct, are responsible for absorption, and that both liners in a double-liner system are absorbant. A reduction of the model equations in the limit of long wavelength compared to liner length reveals an important parameter grouping, enabling the optimal design of liner systems.
Numerical study of pulsatile flow in a constricted channel
- R. MITTAL, S. P. SIMMONS, F. NAJJAR
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- 24 June 2003, pp. 337-378
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Pulsatile flow in a planar channel with a one-sided semicircular constriction has been simulated using direct numerical simulation and large-eddy simulation. This configuration is intended as a simple model for studying blood flow in a constricted artery. Simulations have been carried out over a range of Reynolds numbers (based on channel height and peak bulk velocity) from 750 to 2000 and a fixed non-dimensional pulsation frequency of 0.024. The results indicate that despite the simplicity of the chosen geometry, the simulated flow exhibits a number of features that have been observed in previous experiments carried out in more realistic configurations. It is found that over the entire Reynolds number range studied here, the flow downstream of the constriction is dominated by the complex dynamics associated with two shear-layers, one of which separates from the lip of the constriction and other from the opposite wall. Computed statistics indicate that for Reynolds numbers higher than about 1000, the flow transitions to turbulence downstream of the region where the separated shear layers first reattach to the channel walls. Large fluctuations in wall pressure and shear stress have also been associated with this reattachment phenomenon. Frequency spectra corresponding to velocity and pressure fluctuations have been analysed in detail and these indicate the presence of a characteristic shear-layer frequency which increases monotonically with Reynolds number. For Reynolds numbers greater than 1000, this frequency is found to be associated with the periodic formation of vortex structures in the shear-layers and the impact of this characteristic shear-layer frequency on the dynamics of the flow is described in detail.
BOOK REVIEWS
A Physical Introduction to Fluid Mechanics. By A. J. SMITS. Wiley, 2002. 527 pp. ISBN 0 471 2534 99. £41.95
- T. NICKELS
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- 24 June 2003, p. 379
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SHORT NOTICE
Annual Review of Fluid Mechanics, vol. 35. Edited by J. L. LUMLEY, S. H. DAVIS & P. MOIN. Annual Reviews, 2003. 565 pp. ISBN 0-8243-0735-6. Institutions: $160 (print or online only) or $195 (print and online); Individuals $70 (print and online).
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- 24 June 2003, p. 380
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