Streamwise streaks are ubiquitous in transitional boundary layers, particularly when subjected to high levels of free-stream turbulence. Using the steady boundary-layer approximation, the upstream disturbances experiencing maximum spatial energy growth are numerically calculated. The calculations use techniques commonly employed when solving optimal-control problems for distributed parameter systems. The calculated optimal disturbances consist of streamwise vortices developing into streamwise streaks. The maximum spatial energy growth was found to scale linearly with the distance from the leading edge. Based on these results, a simple model for prediction of transition location is proposed. Available experiments have been used to correlate the single constant appearing in the model.

Optimal disturbances and bypass transition in boundary layers

Slow flow through a periodic array of spheres

Stokes' infinitesimal-wave expansion for steady progressive free-surface waves has been extended to high order using a computer to perform the coefficient arithmetic. Stokes' expansion has been found to be incapable of yielding the highest wave for any value of the water depth since convergence is limited by a square-root branch-point some distance short of the maximum. By reformulating the problem using a different independent parameter, the highest waves are obtained correctly. Series summation and analytic continuation are facilitated by the use of Pade approximants. The method is valid in principle for any finite value of the wavelength and solutions of high accuracy can be obtained for most values of the wave height and water depth.

Computer extension and analytic continuation of stokes expansion for gravity waves

Convection in a porous cavity driven by heating in the horizontal is analysed by a number of different techniques which yield a fairly complete description of the two-dimensional solutions. The solutions are governed by two dimensionless parameters: the Darcy-Rayleigh number R and the cavity aspect ratio A. We first find solutions valid for shallow cavities, A → 0, by using matched asymptotic expansions. These solutions are given up to O(A6R4). For A fixed, we find regular expansions in R by semi-numerical techniques, up to O(R30) in some cases. Series-improvement techniques then enable us to cover the range 0 ≤ R ≤ ∞. A limited result regarding bifurcations is noted. Finally, for R → ∞ with A fixed, we propose a self-consistent boundary-layer theory which extends previous approximate work. The results obtained by these different methods of solution are in good agreement with each other and with experiments.

Convection in a porous cavity

https://escholarship.org/content/qt1cn9f93h/qt1cn9f93h.pdf?t=p0hguc

Lubrication theory analysis of the permeability of rough-walled fractures

Analysis and improvement of perturbation series

Prandtl's boundary-layer theory is embedded as the first step in a systematic scheme of successive approximations for finding an asymptotic solution for viscous flow at large Reynolds number. The technique of inner and outer expansions is used to treat this singular-perturbation problem. Only analytic semi-infinite bodies free of separation are considered. The second approximation is analysed in detail for steady laminar flow past plane or axisymmetric solid bodies. Attention is restricted to low speeds and small temperature changes, so that the velocity field is that for an incompressible fluid, the temperature field being calculated subsequently. The additive effects are distinguished of longitudinal curvature, transverse curvature, external vorticity, external stagnation enthalpy gradient, and displacement speed. The effect of changing co-ordinates is examined, and the behaviour of the boundary-layer solution far downstream discussed. Application to specific problems will be made in subsequent papers.

Higher approximations in boundary-layer theory Part 1. General analysis

Dean's series for steady fully developed laminar flow through a toroidal pipe of small curvature ratio has been extended by computer to 24 terms Analysis suggests that convergence is limited by a square-root singularity on the negative axis of the square of the Dean number An Euler transformation and extraction of the leading and secondary singularities at infinity render the series accurate for all Dean numbers For curvature ratios no greater than , experimental measurements of the laminar friction factor agree with the theory over a wide range of Dean numbers In particular, they confirm our conclusion that the friction in a loosely coiled pipe grows asymptotically as the one-quarter power of the Dean number based on mean flow speed This contradicts a number of incomplete boundary-layer analyses in the literature, which predict a square-root variation

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112078001032

Extended Stokes series: laminar flow through a loosely coiled pipe

A uniformly valid asymptotic solution for large Reynolds number is constructed for plane steady laminar flow of a liquid into the channel between two semi-infinite parallel plates. The entry condition is taken as either that for a cascade of plates in a uniform oncoming stream, or uniform flow directly at the inlet. A paradox in the standard solution of Schlichting—that near the inlet the flow due to displacement would not be the accelerated uniform core on which his expansion is based—is resolved by showing that his series for small as well as large distance actually applies only to conditions far downstream, and matches with another expansion valid near the inlet. Good agreement is found with three independent numerical solutions of the full Navier-Stokes equations, except for a discrepancy in one solution for uniform entry that is traced to erroneous neglect of inlet vorticity.

Entry flow in a channel

The classical laminar boundary layer on a parabolic cylinder is calculated using the Blasius series, with modifications to improve convergence, and supplemented by an asymptotic expansion valid far downstream from the nose. The flow due to displacement thickness is thereby found with sufficient accuracy to permit evaluation of its second-order effect upon the boundary layer near the stagnation point. The skin friction and heat transfer are found to be reduced there by both displacement and curvature.

Milton Van Dyke

Papers

Analysis and improvement of perturbation series

Higher approximations in boundary-layer theory Part 1. General analysis

Extended Stokes series: laminar flow through a loosely coiled pipe

Entry flow in a channel

Higher approximations in boundary-layer theory Part 3. Parabola in uniform stream