ALBERT

Description: This note provides some explanation of the fact that, contrary to the requirements of local isotropy, the skewness S of the streamwise temperature derivative ∂θ/∂x1 has been observed to be a non-zero constant of magnitude of about unity in high-Reynolds-number and high-Péclet-number turbulent shear flows. Measurements in slightly heated homogeneous shear flows and in unsheared grid turbulence suggest that S is non-zero only when the mean shear dU1/dx2 and the mean temperature gradient dT/dx2 are both non-zero. The sign of S is given by –sgn (dU1/dx2).sgn (dT/dx2). For fixed dU1/dx2, S is of the form tanh (αdT/dx2), α being a constant, while for fixed dT/dx2, it is of the form S/S* = 1 − β1 exp (− β2τ), where S* is a characteristic value of S, β1 and β2 are positive constants, and τ can be interpreted as a ‘total strain’. The derivative skewness data in other (inhomogeneous) shear flows are also compatible with the latter relation. Predictions from a simplified transport equation for [formula omitted], derived in the light of the present experimental observations, are in reasonable agreement with the measured values of S. A possible physical mechanism maintaining S is discussed. © 1980, Cambridge University Press. All rights reserved.

Description: When pure solvent is separated from a solution of non-zero concentration Cb by a semi-permeable membrane, permeable to solvent (water) but not to solute, water flows osmotically across the membrane towards the solution. Its velocity J is given by J = PΔC, where P is a constant and ΔC is the concentration difference across the membrane. Because the osmotic flow advects solute away from the membrane, ΔC is usually less than Cb, by a factor γ which depends on the thickness of and flow in a concentration boundary layer. In this paper the layer is analysed on the assumption that the stirring motions in the bulk solution, which counter the osmotic advection, can be represented as two-dimensional stagnation-point flow. The steady-state results are compared with those of the standard physiological model in which the layer has a given thickness δ and the osmotic advection is countered only by diffusion. It turns out that the standard theory, although mechanistically inadequate, accurately predicts the value of γ over a wide range of values of the governing parameter β = PCbδ/D (where D is the solute diffusivity) if δ is given by where ν is the kinematic viscosity of the fluid and α is the stirring parameter. The final approach to the steady state is also analysed, and it is shown to be achieved in a time scale (D/ν)1/3/αk′ where k′ is a dimensionless number whose dependence on β is computed. Moreover, if β exceeds a certain critical value (≈ 10), the approach to the steady state is not monotonic but takes the form of a damped oscillation (in practice, however, β is unlikely to rise significantly above 1). The theory is extended to the case where the solute concentration is non-zero on both sides of the membrane and in that case it is shown that J is bounded as β → ∞. © 1980, Cambridge University Press. All rights reserved.

Description: A rational asymptotic theory describing the perturbed flow in a turbulent boundary layer encountering a small two-dimensional hump is presented. The theory is valid in the limit of very high Reynolds number in the case of an aerodynamically smooth surface, or in the limit of small drag coefficient in the case of a rough surface. The method of matched asymptotic expansions is used to obtain a multiple-structured flow, along the general lines of earlier laminar studies. The leading-order velocity perturbations are shown to be precisely the inviscid, irrotational, potential flow solutions over most of the domain. The Reynolds stresses are found to vary across a thin layer adjacent to the surface, and display a singular behaviour near the surface which needs to be resolved by an even thinner wall layer. The Reynolds stress perturbations are calculated by means of a second-order closure model, which is shown to be the minimum level of sophistication capable of describing these variations. The perturbation force on the hump is also calculated, and its order of magnitude is shown to depend on the level of turbulence closure; a cruder turbulence model gives rise to spuriously large forces. © 1980, Cambridge University Press. All rights reserved.

Description: Steady potential flow around a two-dimensional bubble with surface tension, either free or attached to a wall, is considered. The results also apply to a liquid drop. The flow and the bubble shape are determined as functions of the contact angle β and the dimensionless pressure ratio γ = (pb − ps)/½ρU2. Here pb is the pressure in the bubble, ps = p∞ + ½ρU2 is the stagnation pressure, p∞ is the pressure at infinity, ρ is the fluid density and U is the velocity at infinity. The surface tension σ determines the dimensions of the bubble, which are proportional to 2σ/ρU2. As γ tends to ∞, the bubble surface tends to a circle or circular arc, and as γ decreases the bubble elongates in the direction normal to the flow. When γ reaches a certain value γ0(β), opposite sides of the bubble touch each other. The problem is formulated as an integrodifferential equation for the bubble surface. This equation is discretized and solved numerically by Newton's method. Bubble profiles, the bubble area, the surface energy and the kinetic energy are presented for various values of β and γ. In addition a perturbation solution is given for γ large when the bubble is nearly a circular arc, and a slender-body approximation is presented for β ∼ ½π and γ ∼ γ0(β), when the bubble is slender. © 1980, Cambridge University Press. All rights reserved.

Description: The evolution of the shape of a slender inviscid drop in an axisymmetric straining motion is studied at low Reynolds numbers. It is found that the shape equation can be solved by polynommals with time-dependent coefficients. A global stability result can be used to show simply that only one possible equilibrium is stable. It is further shown that if the slender drop starts with a long-wavelength waist then it cannot go to this stable equilibrium and must either extend indefinitely or burst. In the class of trinomial shapes, it is shown that the drop either bursts or goes to the stable equilibrium, depending on whether or not the initial shape has a waist. © 1980, Cambridge University Press. All rights reserved.

Description: Conditions are found for the appearance of non-uniform progressive waves of permanent form from a long-wave modulation of a finite-amplitude Stokes wave on deep water. The waveheight at which the modulated waves can occur is a very slowly decreasing function of the modulation wavelength for values up to 150 times the original wavelength. Some qualitative remarks are made about the problem of determining the stability of the new waves. © 1980, Cambridge University Press. All rights reserved.

Description: Based on the parabolic approximation, a refraction—diffraction model for linear water waves is developed. With the assumption that the water depth (refraction index) is slowly varying, the model equation describes the forward-scattered wavefield. Two examples are considered in particular: (i) wave diffraction by a long thin barrier on a uniform slope, and (ii) wave convergence over a semicircular step shoal. For the former problem, a similarity solution in terms of Fresnel integrals is obtained for the wavefield in the neighbourhood of the shadow boundary. For the latter problem, the resulting Schrödinger equation is solved numerically. The wavefield near the caustics as well as in the shadow region is obtained and compared with experimental data. © 1980, Cambridge University Press. All rights reserved.

Description: The coherent structure dynamics in the near field of a circular jet has been experimentally explored by inducing ‘stable’ vortex pairing through controlled excitation (see Zaman & Hussain 1980) and applying phase-averaging techniques. Hot-wire measurements were made in a 7·62 cm air jet with laminar exit boundary layer at the Reynolds number ReD = 3·2 × 104, excited at the Strouhal number StD = 0·85. At a particular phase during the pairing process, spatial distributions of the phase-average longitudinal and lateral velocity perturbations (〈u)〉, 〈v〉), vorticity, streamlines, the coherent and background Reynolds stresses and turbulence intensities have been educed. These data have been obtained for four different locations occupied by the vortices at the same phase (preceding, during, and following the pairing event), in the region 0 〈 x/D 〈 5. Spatial distributions of these measures at four successive phases during the pairing process are also educed in an attempt to further understand the vortex-pairing dynamics. The flow physics is discussed on the basis of measurements over the physical extent of the vortical structures, phase-locked to specific phases of the pairing event and thus do not involve use of the Taylor hypothesis. The computed pseudostream functions at particular phases are compared with the corresponding streamlines drawn by the method of isoclines. Transition of the vortices is examined on the basis of vorticity diffusion, the superimposed random fluctuation field intensities and Reynolds stress and phase-locked circumferential correlation measurements. The peak vorticity drops rapidly owing to transition and interaction of the vortices during pairing but, farther downstream, the decay can be attributed to destruction of the coherent vorticity by the background turbulence Reynolds stress, especially at the locations of the latter's ‘saddle points’. Controlled excitation enhances the initial circumferential coherence of the vortical structures, but is ineffective in delaying turbulent breakdown near the end of the potential core; the breakdown appears to occur through evolution of the circumferential lobe structures. The coherent structure Reynolds stress is found to be much larger than the background turbulence Reynolds stress for 0 〈 x/D ≲ 3, but these two are comparable near the end of the jet potential core. The zone average of the coherent structure Reynolds stress over the cross-section of the merging vortex pair is much larger than that over a single vortical structure either before or after the completion of pairing. During the pairing process, such average correlations are found to be the largest at an early phase of the process while entrainment, turbulent breakdown as well as rapid diffusion of vorticity occur at a later phase. The regions of alternate positive and negative coherent Reynolds stresses associated with the structures and their interactions help explain ‘negative production’. © 1980, Cambridge University Press. All rights reserved.

Description: Fully developed periodic flow (with non-zero mean) of a Newtonian fluid in a rigid curved tube has been investigated both numerically and experimentally. Results are reported for the mean friction factor, the amplitude ratio and phase angle between flow rate and pressure drop, the axial velocity profile, and the wall shear stress distribution. The numerical results (obtained by a finite difference method) are restricted to rather slow flows (mean Dean number [formula omitted]), while the experimental results (extracted from instantaneous flow rate-pressure drop measurements) extend up to [formula omitted]. A ‘resonant’ interaction between the axial and secondary flows at intermediate frequencies appears to be a characteristic feature of periodic flow in a curved tube. © 1980, Cambridge University Press. All rights reserved.