The direct initiation of gaseous detonation is investigated experimentally in the cylindrical geometry. By using a long source of energy deposition along a line (i.e. pentaerythritoltetranitrate (PETN) detonating cord), undesirable charge initiation and confinement effects are eliminated. This permitted the different flow fields of direct initiation of detonation to be studied unambiguously. Although the detonation velocity in the detonating cord is finite, it was sufficiently large compared to the acoustic velocity in the surrounding gas to permit the different flow fields to be investigated within the hypersonic analogy framework, by which the detonating cord synchronizes a continuous series of cylindrical initiation events along its length. The hypersonic approximation was validated in experiments conducted in a non-reactive medium (air). In the supercritical regime of initiation in combustible gas, stable oblique detonations were observed, confirming their existence and stability. In the critical regime, the onset of detonation was observed to occur consistently from stochastic detonative centres. These centres appeared during the initial decay of the blast wave to sub-Chapman–Jouguet (CJ) velocities. The photographic evidence revealed the three-dimensional details of the detonation kernels' amplification. The present results in the cylindrical geometry are further used to discuss criteria for direct initiation of detonations. In conjunction with previous experiments in the spherical and planar geometries, a criterion for direct initiation is found to involve a critical decay rate of the reacting blast wave. In light of the experimental evidence of the inherent three-dimensional effects during the initiation phase, the strict one-dimensionality of current theoretical models is discussed.

The linear dynamics of rotating Rayleigh–Bénard convection with rigid stress-free boundaries has been thoroughly investigated by Chandrasekhar (1961) who determined the marginal stability boundary and critical horizontal wavenumbers for the onset of convection and overstability as a function of the Taylor number $T$. No closed-form formulae appeared to exist and the results were tabulated numerically. However, by taking the Rayleigh number $R$ as independent variable we have found remarkably simple expressions. When the Prandtl number $P \geq P_{\!\!c}=0.67659$, the marginal stability boundary is described by the curve $T(R)= R[(R/R_c)^{1/2} -1]$ where $R_c=\frac{27}{4}\pi^4$ is Rayleigh's famous critical value for the onset of stationary convection in a non-rotating system ($T=0$). For $P < P_{\!\!c}$ the marginal stability boundary is determined by this curve until it is intersected by the curve \[ T(R,P)= R{\Bigg[\bigg(\frac{1+P}{2^3 P^4}\bigg)^{1/2}(R/R_c)^{1/2} -\frac{1+P}{2P^2}\Bigg]}. \]A simple expression for the intersection point is derived and also for the critical horizontal wavenumbers for which, along the marginal stability boundary, instability sets in either as stationary convection or in an oscillatory fashion. A simple formula is derived for the frequency of the oscillations. Further, we have analytically determined critical points on the marginal stability boundary above which an increase of either viscosity or diffusivity is destabilizing. Finally, we show that if the fluid has zero viscosity the system is always unstable, in contradiction to Chandrasekhar's conclusion.

This paper describes a photon-counting study of the cavitation luminescence produced by flow over a hydrofoil. This has previously been identified in water saturated with xenon. The four objectives of this study are: to determine whether luminescence can be obtained using air-saturated water; to quantify this emission, if it is present, as a function of flow parameters; to determine whether the photon arrivals occur with random timing, or in ‘bursts’; to put limits on the rates associated with any bursts. The flow experiments were performed in a cavitation tunnel capable of achieving flow velocities of up to about 50 m s$^{-1}$ in the test section. The experimental hydrofoil was a NACA 009 blade. Parameters varied were the flow velocity, the incident angle of the hydrofoil and the cavitation index. The results show that significant photon counts are recorded when leading-edge cavitation takes place and U-shaped vortices (cavities) are shed from the main cavity. The photon count increases dramatically as the flow velocity increases or the cavitation index is reduced. Departures from a Poisson distribution in the arrival times of photons at the detector suggest the presence of ‘bursts’. These may be related to the way vortices are shed from the main cavity. Limits are inferred on the detection rates associated with bursts.

Flow of a viscous incompressible fluid through a conduit with rigid walls is considered, with the flux given at the entrance and exit. The velocity distribution which minimizes the rate of energy dissipation is characterized. It is a Stokes flow with the surface stress equal to a constant pressure at the entrance and another constant pressure at the exit.

A general stability condition for vortices in a two-dimensional incompressible inviscid flow field is presented. This condition is first applied to analyse the stability of symmetric vortices behind elliptic cylinders and circular cylinders with a splitter plate at the rear stagnation point. The effect of the size of the splitter plate on the stability of the vortices is studied. It is also shown that no stable symmetric vortices exist behind two-dimensional bodies based on the stability condition. The two-dimensional stability condition is then extended to analyse the absolute (temporal) stability of a symmetric vortex pair over three-dimensional slender conical bodies. The three-dimensional problem is reduced to a vortex stability problem for a pair of vortices in two dimensions by using the conical flow assumption, classical slender-body theory, and postulated separation positions. The bodies considered include circular cones and highly swept flat-plate wings with and without vertical fins, and elliptic cones of various eccentricities. There exists an intermediate cone with a finite thickness ratio between the circular cone and the flat-plate delta wing for which the symmetric vortices change from being unstable to being stable at a given angle of attack. The effects of the fin height and the separation position on the stability of the vortices are studied. Results agree well with known experimental observations.

We consider the averaged flow properties of a suspension in which the Reynolds number based on the particle diameter is finite so that the inertia of the fluid phase is important. When the inertia of the particles is sufficiently large, their trajectories, between successive particle collisions, are only weakly affected by the interstitial fluid. If the particle collisions are nearly elastic the particle velocity distribution is close to an isotropic Maxwellian. The rheological properties of the suspension can then be determined using kinetic theory, provided that one knows the granular temperature (energy contained in the particle velocity fluctuations). This energy results from a balance of the shear work with the loss due to the viscous dissipation in the interstitial fluid and the dissipation due to inelastic collisions. We use lattice-Boltzmann simulations to calculate the viscous dissipation as a function of particle volume fraction and Reynolds number (based on the particle diameter and granular temperature). The Reynolds stress induced in the interstitial fluid by the random motion of the particles is also determined. We also consider the case where the interstitial fluid is moving relative to the particles, as would occur if the particles experienced an external body force. Owing to the nonlinearity of the equations of motion for the interstitial fluid, there is a coupling between the viscous dissipation caused by the fluctuating motion of the particles and the drag associated with a mean relative motion of the two phases, and this coupling is explored by computing the dissipation and mean drag for a range of values of the Reynolds numbers based on the mean relative velocity and the granular temperature.

A tubeless siphon was created by sucking a 1% aqueous PolyoxPolyox is a registered trademark of Union Carbide. solution laden with particles from a beaker into a cylinder by moving a piston. The piston speed and particle concentration were varied. At very high rates of withdrawal, all the liquid could be removed before the siphon broke. In this case, the beaker was completely cleaned without a trace of liquid. The addition of small concentrations of small, nearly neutrally buoyant particles greatly enhanced the pulling power of the liquid, reducing the threshold speed of withdrawal at which the beaker was completely cleaned. At speeds of withdrawal smaller than the threshold not all of the fluid–particle mixture is pulled out of the beaker. The amount pulled out first increases, then decreases as the particle concentration is increased.

Measurements of nearly isotropic turbulence downstream of an active grid are performed as a high-Reynolds-number ($Re_{\lambda} \approx 720$) update of the Comte-Bellot & Corrsin (1971) data set. Energy spectra at four downstream distances from the grid, ranging from $x/M=20$ to $x/M=48$, are measured and documented for subsequent initialization of, and comparison with, large-eddy simulations (LES). Data are recorded using an array of four X-wire probes which enables measurement of filtered velocities, filtered in the streamwise (using Taylor's hypothesis) and cross-stream directions. Different filter sizes are considered by varying the separation between the four probes. Higher-order statistics of filtered velocity are quantified by measuring probability density functions, and hyper-flatness and skewness coefficients of two-point velocity increments. The data can be used to study the ability of LES to reproduce both spectral and higher-order statistics of the resolved velocity field. In this study, the Smagorinsky, dynamic Smagorinsky, and dynamic mixed nonlinear models are considered. They are implemented in simulations of decaying isotropic turbulence using a pseudospectral code with initial conditions that match the measured energy spectra at $x/M=20$. Overall, it is found that the various LES models predict accurate low-order statistics of resolved scales in isotropic turbulence during the decay. For the spectral cutoff filter, the dynamic Smagorinsky model simulates the energy spectrum more closely at smaller wavenumber, and the dynamic mixed nonlinear model has closer agreement at large wavenumber. For the graded physical-space (Gaussian) filter, the dynamic mixed nonlinear model provides the best spectral results. The three models considered here underpredict the intermittency of longitudinal velocity increments at small distances. For transverse velocity increments, the models' predictions are closer to the measured values, but differ among themselves, with the mixed nonlinear model predicting reduced intermittency. Comparisons of probability density functions of subgrid-scale dissipation and stresses from simulations and experimental data reveal pronounced differences among the different models.

This study considers the convective instability of a water-filled shallow wedge with an absorptive bottom subject to solar radiation. Previous studies have revealed that a thermal boundary layer develops along the sloping bottom due to the absorption of penetrative radiation there, and this boundary layer is potentially unstable to the Rayleigh–Bénard instability. The stability properties of the thermal boundary layer are determined in the present study by perturbing the three-dimensional numerical solution. The result of the direct stability analysis has confirmed previous scaling with respect to the convective instability. Additional features of the thermal layer instability have also been revealed from the direct stability analysis.

Buoyant convection induced between infinite horizontal walls by a horizontal temperature gradient is characterized by simple monodimensional parallel flows. In a layer of low-Prandtl-number fluid, these flows can involve two types of instabilities: two-dimensional stationary transverse instabilities and three-dimensional oscillatory longitudinal instabilities. The stabilization of such flows by a constant magnetic field (vertical, or horizontal with a direction transverse or longitudinal to the flow) is investigated in this paper through a linear stability analysis and energy considerations. The vertical magnetic field stabilizes the instabilities more quickly than the horizontal fields, but the stabilization is only obtained up to moderate values of Hartmann number $Ha$ (before disappearance of the instabilities). Characteristic laws, given by the critical Grashof number $\Gr_c$ as a function of $Ha$ (proportional to the intensity of the magnetic field), have been found for the initial stabilization at small $Ha$. They are $\Gr_c \sim \Gr_{c_0} \exp(Ha^2)$ for the two-dimensional instabilities and $\Gr_c - \Gr_{c_0} \sim Ha^2$ for the three-dimensional instabilities (where $\Gr_{c_0}$ is the critical Grashof number at $Ha=0$), indicating that the three-dimensional instabilities, less stabilized, will prevail in a vertical magnetic field. It has been shown by an energy analysis that the strong stabilization of the two-dimensional instabilities is connected to the strong diminution of the destabilizing shear energy term when the velocity profiles are modified by the vertical magnetic field, and affected little by the Lorentz energy term. For the horizontal magnetic fields, the stabilization is very weak at small $Ha$, but then reaches an asymptotic behaviour corresponding to $\Gr_c \sim Ha$. This asymptotic stabilization is connected to the decrease of the destabilizing shear energy term due to the increase of the marginal cell length in the horizontal magnetic field. In fact, this stabilization only concerns the two-dimensional modes in the longitudinal field and the three-dimensional modes in the transverse field.

New experiments have been conducted using a combined free-surface gradient detector (FSGD) and digital particle image velocimetry (DPIV) technique to study the interaction between a vertical shear layer, created by a surface-piercing splitter plate, and a free surface. The emphasis of this study is on understanding aspects of the interaction between the free-surface deformation (FSD) and the near-surface turbulence through the correlations between the elevation and the vorticity fields, and the spectral behaviour of the near-surface pressure. The Reynolds number of the present study, based on visual thickness and the velocity average of the two streams, is 12 100. Mean results for the velocity and vorticity fields show that self-similarity is achieved. Instantaneous data sets show that at the free surface, vortex tubes within the main rollers connect normally with the free surface as is evidenced by strong vorticity as well as the strong deformations at the free surface. The instantaneous data sets also show that the streamwise vortices near the braid regions, while weaker than those seen in the main rollers, also reconnect with the free surface. Statistical analyses show that the FSD is strongly correlated with the near-surface vorticity field, as the correlation coefficients are quite high ($\sim 0.7--0.8$). The pressure spectrum slope within the shear layer near the surface is found to be $-10/3$, compared with the analytically derived value of $-11/3$ (George et al. 1984) found for a shear layer in unbounded flow.

We study laminar thin film flows with large distortions of the free surface, using the method of averaging across the flow. Two specific problems are studied: the circular hydraulic jump and the flow down an inclined plane. For the circular hydraulic jump our method is able to handle an internal eddy and separated flow. Assuming a variable radial velocity profile as in Kármán–Pohlhausen's method, we obtain a system of two ordinary differential equations for stationary states that can smoothly go through the jump. Solutions of the system are in good agreement with experiments. For the flow down an inclined plane we take a similar approach and derive a simple model in which the velocity profile is not restricted to a parabolic or self-similar form. Two types of solutions with large surface distortions are found: solitary, kink-like propagating fronts, obtained when the flow rate is suddenly changed, and stationary jumps, obtained, for instance, behind a sluice gate. We then include time dependence in the model to study the stability of these waves. This allows us to distinguish between sub- and supercritical flows by calculating dispersion relations for wavelengths of the order of the width of the layer.

Viscous fingering experiments are performed in a radial Hele-Shaw cell for a liquid–liquid system in the presence of a well-characterized interfacial reaction capable of changing the surface tension on the time scale of the experiments. The reaction is a neutralization of a fatty acid by an alkaline material to form a surfactant, which exhibits first-order kinetics for the surface tension as a function of time. The experiments are carried out for capillary numbers, $Ca$, high enough for the fingering to always be in the fractal regime, and for a wide range of Damköhler numbers, $Da$. The fingers are typically wider in the presence of the chemical reaction than the non-reactive case. We observe two different-behaviours of the reactive fingering patterns. For intermediate values of $Da, 0.5 < Da < 4$, the fractal dimension $d_{f}$ is higher than the classical value measured for the non-reactive fingering patterns and reaches a maximum of about 1.9. For both small ($Da < 0.5$) and high $Da (Da > 4)$, the reactive fingering patterns are similar to the fingers with no reaction: the fractal dimension is found to be the same for both systems. These effects are consistent with the hypothesis that Marangoni stresses are present and produce wider fingers.

A non-parallel analysis of time-oscillatory instability of conical jets reveals important features not found in prior studies. Flow deceleration significantly enhances the shear-layer instability for both swirl-free and swirling jets. In swirl-free jets, flow deceleration causes the axisymmetric instability (absent in the parallel approximation). The critical Reynolds number $\hbox{\it Re}_{a}$ for this instability is an order of magnitude smaller than the critical $\hbox{\it Re}_{a}$ predicted before for the helical instability (where $\hbox{\it Re}_{a}= rv_{a}/\nu, r$ is the distance from the jet source, $v_a$ is the jet maximum velocity at a given $r$, and $\nu$ is the viscosity). Swirl, intensifying the divergence of streamlines, induces an additional, divergent instability (which occurs even in shear-free flows). For the swirl Reynolds number $\hbox{\it Re}_s$ (circulation to viscosity ratio) exceeding 3, the critical $\hbox{\it Re}_a$ for the single-helix counter-rotating mode becomes smaller than those for axisymmetric and multi-helix modes. Since the critical $\hbox{\it Re}_s$ is less than 10 for the near-axis jets, the boundary-layer approximation (used before) is invalid, as is Long's Type II boundary-layer solution (whose stability has been extensively studied). Thus, the non-parallel character of jets strongly affects their stability. Our results, obtained in a far-field approximation allowing reduction of the linear stability problem to ordinary differential equations, are more valid for short wavelengths.

The collapse dynamics of a diffusion-flame hole in the presence of a counterflow are studied. We construct unsteady solutions of the one-dimensional edge-flame model of Buckmaster (1996), in which heat and mass transverse losses are algebraic. The flame structure is determined in the classical limit of large activation energy. Solutions for both planar and axisymmetric strain geometry are considered for the particular case of unity Lewis number. It is shown that the final stage of the edge-flame collapse is determined by a dominant balance between the time rate of change of the mass fractions (and temperature) and diffusion, giving a self-similar structure in which the size of the edge-flame hole approaches zero, to leading (zeroth) order, as a $\frac{1}{2}$-power of time. This solution suggests an expansion of the full model equations in $\frac{1}{2}$-powers of time that allows detailed analysis of the effects of side losses and flow distribution in the edge-flame collapse process. It is found that side loss effects are apparent at the first order, whereas convection by the counterflow is first felt during collapse at the second order in the fractional-time expansion. Numerical integrations of the governing equations are found to verify the analytic results.

Journal of Fluid Mechanics, vol. 458 (2002), pp. 35–73

As pointed out by V. M. Shevtsova, the boundary condition (2.7$c$) for the interface $h(z)$ involving the hot-wall contact angle $\alpha$ contained an error.