Papers
Consistent large-eddy simulation of a temporal mixing layer laden with evaporating drops. Part 1. Direct numerical simulation, formulation and a priori analysis
- NORA A. OKONG'O, JOSETTE BELLAN
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- Published online by Cambridge University Press:
- 30 January 2004, pp. 1-47
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Large-eddy simulation (LES) models are presented and evaluated on a database obtained from direct numerical simulation (DNS) of a three-dimensional temporal mixing layer with evaporating drops. The gas-phase equations are written in an Eulerian frame for two perfect gas species (carrier gas and vapour emanating from the drops), while the liquid-phase equations are written in a Lagrangian frame. The effect of drop evaporation on the gas phase is considered through mass, momentum and energy source terms. The DNS database consists of transitional states attained by layers with different initial Reynolds numbers and initial liquid-phase mass loadings. Budgets of the LES equations at the transitional states show that, for the mass loadings considered, the filtered source terms (FSTs) are smaller than the resolved inviscid terms and some subgrid scale (SGS) terms, but larger than the resolved viscous stress, heat flux and mass flux terms. The irreversible entropy production (i.e. the dissipation) expression for a two-phase flow with phase change is derived, showing that the dissipation contains contributions due to viscous stresses, heat and species-mass fluxes, and source terms. For both the DNS and filtered flow fields at transition, the two leading contributions are found to be the dissipation due to the energy source term and that due to the chemical potential of the mass source. Therefore, the modelling effort is focused on both the SGS fluxes and the FSTs in the LES equations. The FST models considered are applicable to LES in which the grid is coarser than the DNS grid and, consistently, ‘computational’ drops represent the DNS physical drops. Because the unfiltered flow field is required for the computation of the source terms, but would not be available in LES, it was approximated using the filtered flow field or the filtered flow field augmented by corrections based on the SGS variances. All of the FST models were found to overestimate DNS-field FSTs, with the relative error of modelling the unfiltered flow field compared to the error of using computational drops showing a complex dependence on filter width and number of computational drops. For modelling the SGS fluxes and (where possible) SGS variances, constant-coefficient Smagorinsky, gradient and scale-similarity models were assessed on the DNS database, and calibrated coefficients were statistically equivalent when computed on single-phase or two-phase flows. The gradient and scale-similarity models showed excellent correlation with the SGS quantities. An a posteriori study is proposed to evaluate the impact of the studied models on the flow-field development, so as to definitively assess their suitability for LES with evaporating drops.
Alternating bar instabilities in unsteady channel flows over erodible beds
- PHILIP HALL
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- 27 January 2004, pp. 49-73
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Channel flows over erodible beds are susceptible to instabilities of the coupled fluid and sediment flow equations. The most dangerous mode usually takes the form of a migrating alternating bar instability propagating in the flow direction. Previous theories have assumed that the underlying flow is steady and here the theory is extended to the unsteady situation. Stability characteristics are calculated for large-amplitude oscillations superimposed on a mean flow. In addition it is found that the basic instability is convective and we address the receptivity problem for bars induced by flow oscillations interacting with spatial variations associated with seepage, channel width variations etc. The effect of unsteadiness in the weakly nonlinear situation is also discussed. A mechanism which allows flow oscillations to interact with migrating bars to produce a sinusoidal structure fixed in space which might be relevant to meander formation is discussed.
Large interface deformation in two-layer thermal convection of miscible viscous fluids
- MICHAEL LE BARS, ANNE DAVAILLE
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- 27 January 2004, pp. 75-110
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Laboratory experiments have been performed to study two-layer thermal convection with large interface deformations. The two fluids have different densities and viscosities but there is neither surface tension nor chemical diffusion at the interface. The initial density stratification is stable, but can be reversed by thermal effects. Two different mechanisms of interface deformation are described: (i) purely thermal features due to convection inside each layer independently can locally and partially deform the interface, leading to dynamic topography; (ii) when the effective buoyancy number (the ratio of the stabilizing chemical density anomaly to the destabilizing thermal density anomaly) reaches a critical value, a whole-layer regime takes place, where the system is fully destabilized and one of the two layers invades the other one in the form of large domes. Several successive pulsations can be observed provided the viscosity ratio is large enough (i.e. ${>}\,5$). Typical scales (time, length, temperature excess, velocity) and behaviours (direction of spouting, shapes) are determined for each case. Both features are only transient states: because of stirring, the system systematically evolves towards one-layer Rayleigh–Bénard convection. However, this transient state can persist for a very long time compared to the characteristic time scale of thermal convection.
Scattering of incident disturbances by an annular cascade in a swirling flow
- H. M. ATASSI, A. A. ALI, O. V. ATASSI, I. V. VINOGRADOV
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- 27 January 2004, pp. 111-138
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Analytical and numerical analyses are developed for the interaction and scattering of incident acoustic and vortical disturbances by an unloaded annular cascade in a swirling flow. The mathematical formulation uses the Euler equations linearized about an axial and swirling mean flow. The incident disturbances are decomposed into nearly sonic and nearly convected disturbances using the results of a normal-mode analysis, namely the unsteady pressure is predominantly associated with the former. Exact non-reflecting inflow/outflow conditions are derived in terms of the normal modes using the group velocity to segregate the modes propagating downstream and upstream. An inflow condition is also derived for the nearly convected disturbances. An explicit primitive-variable scheme is implemented and validated by comparison with the uniform flow and narrow annulus limits. Acoustic and aerodynamic results are presented to examine how swirl modifies the scattering from that of the uniform flow and narrow annulus limits and to determine the conditions leading to strong scattering. The results indicate that the swirl changes the physics of the scattering in three major ways: (i) it modifies the number of acoustic modes in the duct, (ii) it changes their duct radial profile, and (iii) it causes significant amplitude and radial phase variations of the incident disturbance. The results also show that when the radial phase of the incident disturbance is different from that of the duct modes, weak scattering into the duct acoustic modes occurs. These results suggest that analysis of the radial variation of the incident disturbance and duct modes can provide an indication of the efficiency of the scattering process.
Impact jetting by a solid sphere
- S. T. THORODDSEN, T. G. ETOH, K. TAKEHARA, Y. TAKANO
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- 27 January 2004, pp. 139-148
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We use a novel ultra-high-speed video camera to study the initial stage of the impact of a solid sphere onto a liquid surface, finding a high-speed horizontal jet which emerges immediately following the intial contact. For ${\hbox{\it Re}} > 2 \times 10^4$ the jet emerges when the horizontal contact between the sphere and the liquid is only 12% of its diameter. For the largest Reynolds numbers this jet can travel at more than 30 times the impact velocity of the sphere. This jetting occurs sooner and at much higher normalized velocities than has been observed previously. The breakup of the jet into a spray of droplets sometimes occurs through formation of pockets in the liquid sheet. Early in the impact, the energy transferred to the jet and the subsequent spray sheet is estimated to be much larger than the energy associated with the added mass inside the liquid pool. The jetting will therefore greatly increase the initial impact force on the sphere.
Dynamics of cooling viscoplastic domes
- N. J. BALMFORTH, R. V. CRASTER, R. SASSI
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- 27 January 2004, pp. 149-182
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A variety of problems in engineering and geology involve spreading cooling non-Newtonian fluids. If the fluid is relatively shallow and spreads slowly, lubrication-style asymptotic approximations can be used to build reduced models for the spreading dynamics. The centrepiece of such models is a nonlinear diffusion equation for the local fluid thickness, and ideally this should become coupled to a correspondingly simple equation determining the local temperature field. However, when heat diffuses relatively slowly as the fluid flows, we cannot usefully reduce the temperature equation, and the asymptotic reduction couples the local thickness equation to an advection diffusion equation that crucially involves diffusion in the vertical. We present an efficient computational algorithm for numerically solving this more complicated type of lubrication model, and describe a suite of solutions that illustrate the dynamics captured by the model in the case of an expanding Bingham fluid with a temperature-dependent viscosity. Based on these solutions, we evaluate two simpler models that further approximate the temperature equation: a vertically isothermal theory, and a ‘skin theory’. The latter is based on the integral-balance method of heat-transfer theory, and demands that the vertical structure of the temperature field has the form of an advancing boundary layer, or skin. The vertically isothermal model performs well when the thermal conductivity is relatively large. The skin theory reproduces the full dynamics qualitatively, if not quantitatively, for all thermal conductivities. The main errors in both models arise near the fluid edge, where the numerical solutions show that chilled fluid is overridden as the fluid expands, creating an underlying collar of cold material. Encouraged by the success of the skin model, we extend the theory by incorporating extensional stresses in the skin, which emerge when cooling induces an extreme rheological change in the material, such as an exponential rise in the viscosity. The model predicts that when skin stresses are sufficiently strong, the skin is brought to rest, whilst hotter fluid expands underneath.
A ‘win–win’ mechanism for low-drag transients in controlled two-dimensional channel flow and its implications for sustained drag reduction
- THOMAS R. BEWLEY, OLE MORTEN AAMO
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- 27 January 2004, pp. 183-196
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A simple pressure-based feedback control strategy for wall-transpiration control of incompressible unsteady two-dimensional channel flow was recently investigated by Aamo, Krstic & Bewley (2003). Nonlinear two-dimensional channel flow simulations which implemented this control strategy resulted in flow transients with instantaneous drag far lower than that of the corresponding laminar flow. The present article examines the physical mechanism by which this very low level of instantaneous drag was attained. It then explores the possibility of achieving sustained drag reductions to below the laminar level by initiating such low-drag transients on a periodic basis. All attempts at sustaining the mean flow drag below the laminar level fail, perhaps providing indirect evidence in favour of the conjecture that the laminar state might provide a fundamental ‘performance limitation’ in such flows. Mathematical analysis of two-dimensional and three-dimensional channel-flow systems establishes a direct link between the average drag increase due to flow-field unsteadiness and a weighted space/time average of the Reynolds stress. Phenomenological justification of the conjecture is provided by a Reynolds analogy between convective momentum transport and convective heat transport. Proof of the conjecture remains an open problem.
The structure and dynamics of overturns in stably stratified homogeneous turbulence
- PETER J. DIAMESSIS, KEIKO K. NOMURA
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- 27 January 2004, pp. 197-229
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Direct numerical simulations of stably stratified homogeneous turbulence, with and without mean shear, are used to investigate the three-dimensional structure, evolution and energetic significance of density overturns. Although the flow conditions are idealized, examination of the full-field simulation data provides insight into flow energetics and mixing which may assist in the interpretation of physical measurements, typically limited to one-dimensional vertical profiles. Overturns, defined here through the density field as contiguous regions of non-zero Thorpe displacement, are initially generated by the stirring action of coherent vortex structures present in the flow and further develop through merging with adjacent overturns. During this growth phase, overturns exhibit irregular spatial structure in unsheared flow and elongated structure with distinct orientation in shear flow. Although most of the available potential energy (APE) and buoyancy flux are associated with stable (non-overturning) regions in the flow, young overturns actively contribute to the flow energetics. In particular, overturn peripheries are sites of high levels of APE, buoyancy flux and diapycnal mixing. A collapse phase may follow the growth phase in the absence of adequately strong mean shear. During this phase, buoyancy gradually assumes control of the overturns and their vertical scale steadily decreases. The energetic significance of the overturns diminishes, although high APE and diapycnal mixing continue to occur near their boundaries. In the final phase of their evolution, overturns contribute negligibly to the energetics. The remaining overturns are characterized by a viscous–buoyant balance which maintains their vertical scale. The overturns eventually vanish due to homogenization of their internal density distribution by diffusion. Activity diagrams, sampled at different points of flow evolution, show significant variation in overturn Reynolds and Froude numbers which may have implications for vertical sampling of a turbulent event.
Turbulent diffusion in rapidly rotating flows with and without stable stratification
- C. CAMBON, F. S. GODEFERD, F. C. G. A. NICOLLEAU, J. C. VASSILICOS
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- 27 January 2004, pp. 231-255
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In this work, three different approaches are used for evaluating some Lagrangian properties of homogeneous turbulence containing anisotropy due to the application of a stable stratification and a solid-body rotation. The two external frequencies are the magnitude of the system vorticity $2\Omega$, chosen vertical here, and the Brunt–Väisälä frequency $N$, which gives the strength of the vertical stratification. Analytical results are derived using linear theory for the Eulerian velocity correlations (single-point, two-time) in the vertical and the horizontal directions, and Lagrangian ones are assumed to be equivalent, in agreement with an additional Corrsin assumption used by Kaneda (2000). They are compared with results from the kinematic simulation model (KS) by Nicolleau & Vassilicos (2000), which also incorporates the wave–vortex dynamics inherited from linear theory, and directly yields Lagrangian correlations as well as Eulerian ones. Finally, results from direct numerical simulations (DNS) are obtained and compared for the rotation-dominant case $B\,{=}\,2\Omega/N\,{=}\,10$, the stratification-dominant case $B\,{=}\,1/10$, the non-dispersive case $B\,{=}\,1$, and pure stratification $B\,{=}\,0$ and pure rotation $N\,{=}\,0$. The last situation is shown to be singular with respect to the mixed stratified/rotating ones. We address the question of the validity of Corrsin's simplified hypothesis, which states the equivalence between Eulerian and Lagrangian correlations. Vertical correlations are found to follow this postulate, but not the horizontal ones. Consequences for the vertical and horizontal one-particle dispersion are examined. In the analytical model, the squared excursion lengths are calculated by time integrating the Lagrangian (equal to the Eulerian) two-time correlations, according to Taylor's procedure. These quantities are directly computed from fluctuating trajectories by both KS and DNS. In the case of pure rotation, the analytical procedure allows us to relate Brownian $t$-asymptotic laws of dispersion in both the horizontal and vertical directions to the angular phase-mixing properties of the inertial waves. If stratification is present, the inertia–gravity wave dynamics, which affects the vertical motion, yields a suppressed vertical diffusivity, but not a suppressed horizontal diffusivity, since part of the horizontal velocity field escapes wavy motion.
Continuously stratified exchange flow through a contraction in a channel
- ANDREW McC. HOGG, PETER D. KILLWORTH
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- 27 January 2004, pp. 257-276
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Existing solutions for exchange flow through straits rely upon the decomposition of the flow into a finite number of layers which have constant density. In this paper we provide a solution to inviscid steady exchange flow between continuously stratified reservoirs, where it is assumed that the flow in each direction is independently self-similar. The solution requires knowledge only of the two reservoir stratifications and an imposed net barotropic throughflow. The solution includes regions of stagnant fluid which separate two counter-flowing, stably stratified layers, with the provision that the two active layers may touch at no more than one point. Comparison of the theoretical solution with numerical simulations indicates that the assumption of self-similarity is reasonable, and that the disparity between the theoretical and simulated flows can be attributed to the inclusion of diffusion and viscosity in the numerical model.
On finite-amplitude subcritical instability in narrow-gap spherical Couette flow
- ANDREW P. BASSOM, ANDREW M. SOWARD
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- Published online by Cambridge University Press:
- 27 January 2004, pp. 277-314
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We consider the finite-amplitude instability of incompressible spherical Couette flow between two concentric spheres of radii $R_1$ and $R_2$$({>}R_1)$ in the narrow-gap limit, $\varepsilon\,{\equiv}\,(R_2-R_1)/R_1\,{\ll}\,1$, caused by rotating them both about a common axis with distinct angular velocities $\Omega_1$ and $\Omega_2$ respectively. In this limit it is well-known that the onset of (global) linear instability is manifested by Taylor vortices of roughly square cross-section close to the equator. According to linear theory this occurs at a critical Taylor number $T_{\rm crit}$ which, remarkably, exceeds the local value $T_c$ obtained by approximating the spheres as cylinders in the vicinity of the equator even as $\varepsilon\,{\downarrow}\,0$. Previous theoretical work on this problem has concentrated on the case of almost co-rotation with $\delta\,{\approx}\,(\Omega_1\,{-}\,\Omega_2)/\Omega_1\,{=}\,\OR(\varepsilon^{1/2})$ for which $T_{\rm crit}\,{=}\,T_c\,{+}\,\OR(\delta^2)\,{+}\,\OR(\varepsilon)$. In this limit the amplitude equation that governs the spatio-temporal modulation of the vortices on the latitudinal extent $\OR(\varepsilon^{1/2}R_1)$ gives rise to an interesting bifurcation sequence. In particular, the appearance of global bifurcations heralds the onset of complicated subcritical time-dependent finite-amplitude solutions.
Here we switch attention to the case when $\varepsilon^{1/2}\,{\ll}\,\delta\,{\le}\, 1$. We show that for Taylor numbers $T\,{=}\,T_c+\OR((\delta\varepsilon)^{2/3})$ there exists a locally unstable region of width $\OR((\delta\varepsilon)^{1/3}R_1)$ within which the amplitude equation admits solutions in the form of pulse-trains. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and consists of a wave propagating towards the equator under an envelope. The pulse drifts at a slow speed (relative to the wave velocity) proportional to its distance (and away) from the equator. Both the wavelength and the envelope width possess the same relatively short length scale $\OR((\varepsilon^{2}/\delta)^{1/3}R_1)$. The appropriate theory of spatially periodic pulse-trains is developed and numerical solutions found. Significantly, these solutions are strongly subcritical and have the property that $T\to T_c$ as $\varepsilon\,{\downarrow}\,0$.
Two particular limits of our theory are examined. In the first, $\varepsilon^{1/2}\,{\ll}\,\delta\,{\ll}\,1$, the spheres almost co-rotate and the pulse drift velocity is negligible. A comparison is made of the pulse-train predictions with previously obtained numerical results pertaining to large (but finite) values of $\delta/\varepsilon^{1/2}$. The agreement is excellent, despite the complicated long-time behaviour caused by inhomogeneity across the relatively wide unstable region.
Our second special case $\delta\,{=}\,1$ relates to the situation when the outer sphere is at rest. Now the poleward drift of the pulses leads to a slow but exponential increase of their separation with time. This systematic pulse movement, over and above the spatial inhomogeneity just mentioned, necessarily leads to complicated and presumably chaotic spatio-temporal behaviour across the wide unstable region of width $\OR(\varepsilon^{1/3}R_1)$ on its associated time scale, which is $\OR(\varepsilon^{-1/3})$ longer than the wave period. In view of the several length and time scales involved only qualitative comparison with experimental results is feasible. Nevertheless, the pulse-train structure is robust and likely to provide the building block of the ensuing complex dynamics.
A note on the barotropic instability of the Bickley jet
- L. ENGEVIK
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- 27 January 2004, pp. 315-326
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The linear stability of the Bickley jet in the framework of the beta-plane approximation is considered, with the objective of presenting analytical calculations which add to previous numerical computations. It is well-known that the equation governing the neutral solutions which are analytic at the critical layer can be transformed into an associated Legendre equation. It turns out that this particular equation has simple closed-form solutions other than those known already, which are the Legendre polynomial of degree two, and two associated Legendre functions of the first kind, respectively. This observation makes it possible to find analytic neutral modes and corresponding neutral curves in the $(\beta,k)$-plane not known previously, both for the bounded and the unbounded Bickley jet. Here $\beta$ denotes the beta-parameter and $k$ the wavenumber. These neutral curves comprise parts of the stability boundary. It is shown that the line segment ($\beta=-2$, $0<k<\sqrt{2}$) is part of the stability boundary for the unbounded Bickley jet, so the region for the unstable radiating modes is larger than the one obtained previously. Also, analytic sinuous and varicose modes and corresponding neutral curves are found in the case of the bounded flow where only numerical calculations have previously been presented. Furthermore, local stability analysis reveals weakly amplified modes with wave speed outside the range of the velocity profile for the Bickley jet. This is rather rare, although Pedlosky's theorem allows for it, and there are only a few examples of flows in which such modes occur. Here these modes are sinuous modes and occur when the flow is both bounded and unbounded.
The interaction of the piston vortex with a piston-generated vortex ring
- JOHN E. CATER, JULIO SORIA, T. T. LIM
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- 27 January 2004, pp. 327-343
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This paper presents the results of an experimental investigation of the effects of a piston vortex on the vorticity evolution of a vortex ring. The rings are produced by the roll-up of a shear layer at a circular orifice in a plane wall and have a Reynolds number of 2000 based on the ejection velocity and orifice diameter. The generation mechanism is a piston moving inside a cylinder with a stroke length of two piston diameters. The experimental apparatus is similar to that used by Glezer & Coles (1990) where the piston finishes flush with the orifice, with the result that a piston vortex produced by the apparatus interacts with the vortex ring. Instantaneous velocity field measurements using cross-correlation digital particle image velocimetry reveal that the piston vortex not only increases the circulation of the ring but also creates an asymmetric vorticity distribution of the vortex core. It is found that ‘imperfect’ merging of the piston vortex with the primary vortex ring promotes the growth of an instability which leads to early transition to turbulence of initially laminar vortex rings.
Steady streaming from an oblate spheroid due to vibrations along its axis
- A. Y. REDNIKOV, S. S. SADHAL
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- 27 January 2004, pp. 345-380
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Steady (acoustic) streaming around a rigid oblate spheroid is studied in the incompressible limit when the fluid medium and the particle are in a small-amplitude high-frequency relative oscillatory motion along the symmetry axis of the spheroid. The inner (inside the thin Stokes shear-wave layer) and outer streaming patterns are analysed. A solution for the outer streaming is obtained analytically for small Reynolds numbers. At large Reynolds numbers, a boundary-layer consideration is carried out. Steady streaming in the disk limit, as approached within the family of oblate spheroids, is systematically investigated in the large Reynolds number case, and qualitative implications for shapes other than oblate spheroidal are discussed.