We use a calculation of periodic homogeneous isotropic turbulence to simulate the experimental decay of grid turbulence. The calculation is found to match the experiment in a number of important aspects and the computed flow field is then treated as a realization of a physical turbulent flow. From this flow, we compute the large eddy field and the various averages of the subgrid-scale turbulence that occur in the large eddy simulation equations. These quantities are compared with the predictions of the models that are usually applied in large eddy simulation. The results show that the terms which involve the large-scale field are accurately modelled but the subgridscale Reynolds stresses are only moderately well modelled. It is also possible to use the method to predict the constants of the models without reference to experiment. Attempts to find improved models have not met with success.

The early attempts at calculating effective transport properties of suspensions of interacting spherical particles resulted in non-absolutely convergent expressions. In this paper we provide a physical interpretation for these convergence difficulties and we present a new method of determining the effective transport properties which clarifies difficulties in existing methods.

This method, which is described for simplicity in the context of the thermal conduction problem, is based on an expression that gives the temperature gradient ∇T at a point x in the matrix in terms of integrals over the surrounding particles and an integral over a large surface Γ which encloses x and which we term the ‘macroscopic boundary’. Without the integral over Γ, this expression for ∇T would be non-absolutely convergent, for the contribution to ∇T(x) from a distant particle is proportional to 1/r3, where r is the distance of the particle from x. On comparing the expression for ∇T with the formula used by Rayleigh (1892) in his investigation of the effective conductivity of a cubic array of spheres, we find that Rayleigh's convergence difficulties arose simply from an incorrect assessment of the macroscopic boundary integral.

By combining the expression for ∇T(x) with a formula for the dipole strength of a sphere placed in an ambient temperature field, we obtain a convergent expression relating the dipole strength of a sphere to integrals over the surrounding particles. An expression for the effective conductivity of a random suspension of spheres correct to O(ϕ2) is obtained simply by averaging this expression for the thermal dipole strength. By a similar procedure we obtain expressions for the effective viscosity and effective elastic moduli correct to O(ϕ2). Most of these results have been obtained by earlier workers using a ‘renormalization’ procedure due to Batchelor; the method presented here has the advantage that the renormalization quantity arises naturally from the macroscopic boundary integral referred to earlier, so there is no uncertainty about its choice.

The fluid mechanics of heart-valve motion is investigated experimentally and theoretically. From the experiments, it is found that the principal mechanism ensuring optimal mitral-valve closure is the pressure field induced by a strong flow deceleration through the valve. A complete theory, evolved from the Bellhouse & Talbot analytical model, is developed for both the mitral and the aortic valve to provide a relationship between valve motion and valve flow. Results predicted by this theory agree consistently well with those obtained from experiments.

The problem of subgrid modelling, that is, of representing energy transfers from large to small eddies in terms of the large eddies only, must arise in any large eddy simulation, whether the equations of motion are open or direct (unaveraged) or closed (averaged). Models for closed calculations are derived from classical closures, and these are used to determine the effect of filter shape, grid-scale spectrum and grid-scale anisotropy on the effective eddy viscosity: the Leonard or resolvable-scale stress is calculated separately and is found to account for 14% of the total drain in a typical high Reynolds number case.

The validity of using these eddy viscosities in an open calculation is considered. It is concluded that this is not unreasonable, but that the simulation would be much improved if the gross drain could be separated into net drain and backscatter.

The flow of an incompressible fluid contained between two infinitely long circular cylinders is considered when the inner cylinder performs small harmonic oscillations about the centre of the larger (fixed) cylinder. Numerical solutions are presented for the Navier–Stokes equations governing the steady-streaming component of the motion. Special attention is then paid to this flow when the Reynolds number of the steady streaming is large, and when the radius of the outer cylinder is much larger than that of the inner. Results obtained show an improved correlation with experimental results and indicate strongly that the finiteness of the domain is the major cause of the discrepancies between experiment and previous theoretical studies.

Mean velocity and turbulent intensity measurements are reported for five different plane turbulent mixing layers, each with a different velocity ratio. These experiments confirm that increasing free-stream turbulence causes increases in the growth rate and in the Reynolds stresses. Cross-correlation measurements with time delay made in the mixing layer with the lowest free-stream turbulence level show that the large-eddy structure had length scales in the two cross-stream directions which were roughly equal, unlike the results reported by Brown & Roshko (1974) and others. Further measurements showed that a vortex-street wake existed immediately downstream of the splitter plate and that transition occurred in the wake flow rather than in a normal laminar mixing layer. This is thought to have prevented the Brown–Roshko structures from forming. Comparison of the growth rate observed in this case with other measured results suggests that the essential or effective turbulent structure in mixing layers is independent both of velocity ratio and of the degree of two-dimensionality which exists in the largest scales of turbulence.

This paper reports the results of an experimental investigation to determine transition mechanisms and limits in gases at high pressure levels. We sought also to refine further the parameters for transition, in particular the role of kinematic viscosity. In flow adjacent to a vertical uniform-flux surface in nitrogen, pressures to 16 atm were used. Both mean and disturbance quantities for the temperature and velocity fields were measured for various values of the heat flux, downstream location and ambient pressure level. Hot-wire and fine thermocouple probes were used. We found that the velocity and thermal fields remain closely coupled. Velocity, or fluid-dynamic, transition is immediately followed by thermal transition. Each was detected as a decrease in the rate of increase of both the maximum velocity and the overall temperature difference, respectively, from the laminar downstream trends. Also, the ends of transition for the velocity and the thermal fields, respectively, signalled by no further appreciable change in the intermittency distributions, were simultaneous. These results re-affirm the finding that the events of transition are not correlated by the Grashof number alone. An additional dependence on both downstream location and pressure level arises. A fixed value of the parameter $Q_{BT} = qB^{\frac{2}{15}}= 290$ characterizes the beginning of transition, where q is the fifth root of the local non-dimensional wall heat flux and B is the unit Grashof number. The end of transition, on the other hand, is best correlated by $Q_{ET} = QB^{\frac{1}{30}} = 11.4$, where Q is the fifth root of the local non-dimensional total heat convected in the boundary region. A re-examination of other transition studies, in both gases and liquids, supports these correlations, although many such data were not determined with fast response to local sensors. There remains a small level of uncertainty in establishing exact limits for transition, since the apparently proper standards for determining them are very difficult to apply precisely in experiments. However, such limits are very important in separating regimes of different transport mechanisms.

Calculations of finite amplitude convection in cubic boxes containing fluid-saturated porous material are reported for Rayleigh numbers R as large as 150. Steady two- or three-dimensional convection can always be forced by appropriate choice of initial conditions. Randomly chosen initial conditions will result in either two- or three-dimensional convection. Although there is a non-uniqueness associated with initial conditions which makes possible either two- or three-dimensional convection, the non-uniqueness is limited in the sense that only one two-dimensional or one three-dimensional solution appears to be realizable. Two-dimensional flows have larger Nusselt numbers than three-dimensional flows for R [lsim ] 97; the opposite is true for R [gsim ] 97.

Surface tension provides a restoring force which cannot reasonably be ignored for water waves of short crest-to-crest length. Even for large wavelengths its presence precludes any sharp corner developing on the free surface. In this paper we begin an investigation of the effects of surface tension on steep water waves. The work of Longuet-Higgins (1975) is generalized to show how the integral properties of the wave train are affected. In particular it is shown that for pure capillary waves in deep water the mean fluxes of energy, mass and momentum are given by 3Tc, 2T/c and 4T − V respectively, where c is the phase velocity, T the kinetic energy and V the potential energy.

Also the exact solution for the wave profile of deep-water pure capillary waves (Crapper 1957) is used to obtain wave profiles, all with the same mean level. This yields the unexpected result that the height of the wave crest above the mean level is not a monotonic function of wave steepness.

With subsequent papers this work will form one limiting case of the general problem of deep-water gravity—capillary waves.

Two integral invariants of Shuto's (1974) generalization of the Korteweg—de Vries equation for a unidirectional wave in a channel of gradually varying breadth b and depth d are derived. The second-order (in amplitude) invariant measures energy, as expected, but the first-order invariant measures mass divided by b½d¼; accordingly, mass is conserved only if either the mean free-surface displacement vanishes or bd½ is constant. This difficulty is associated with the reflected wave that is excited by the channel variation but neglected in the KdV approximation. The total mass flux is resolved into a primary (KdV) flux and a residual flux that is proportional to the mean displacement of the primary wave. The reflected wave associated with the residual flux is constructed by neglecting both nonlinearity and dispersion (even though both are significant for the primary wave). The results are applied to a slowly varying cnoidal wave, which is fully determined by conservation of mass and energy and the known results for a uniform channel, and to a slowly varying solitary wave, for which mass is not conserved and both trailing and reflected residuals are excited. The development of the Boussinesq equations for a gradually varying channel and their reduction to Shuto's equation are sketched in an appendix.

A new model of surface drift currents is constructed using the full nonlinear equations of motion. This model includes the balance between Coriolis forces due to the mean and wave-induced motions and the surface wind stresses. The approach used in the analysis is similar to the work by Craik & Leibovich (1976) and Leibovich (1977), but the emphasis is on the mean motion rather than the small-scale time-dependent part of the Langmuir circulation. The final result indicates that surface currents can be generated by both the direct wind stresses, as in the classical Ekman model, and the Stokes drift, derived from the surface wave motion, in an interrelated fashion depending on a wave Ekman number E defined as \[ E = \Omega/\nu_ek^2_0, \] where Ω is the angular velocity of the earth's rotation, νe, the eddy viscosity and k0, the wavenumber of the surface wave at the spectral peak. When E [Lt ] 1, the Langmuir mode dominates. When E [Gt ] 1, inertial motion results. The classical Ekman drift current is a special case even under the restriction E ≃ 1. On the basis of these results, a new model of the surface-layer movements for future large-scale ocean circulation studies is presented. For this new model both the wind stresses and the sea-state information are crucial inputs.