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SUMMARYFractional-step methods solve the unsteady Navier-Stokes equations in a segregated manner, and can be implemented with only a single solution of the momentum=pressure equations being obtained at each time step, or with the momentum=pressure system being iterated until a convergence criterion is attained. The time accuracy of such methods can be determined by the accuracy of the momentum=pressure coupling, irrespective of the accuracy to which the momentum equations are solved. It is shown that the time accuracy of the basic projection method is ÿrst-order as a result of the momentum=pressure coupling, but that by modifying the coupling directly, or by modifying the intermediate velocity boundary conditions, it is possible to recover second-order behaviour. It is also shown that pressure correction methods, implemented in non-iterative or iterative form and without special boundary conditions, are second-order in time, and that a form of the non-iterative pressure correction method is the most e cient for the problems considered.

Direct numerical simulations (DNS) of turbulent stratified flow in an open channel with an internal heat source following the Beer-Lambert law from the surface are used to investigate the transition from neutral to strongly stable flow. Our buoyancy bulk parameter is defined through the ratio of the domain height δ to L , a bulk Obukhov length scale for the flow. We cover the range λ = δ/L = 0-2.0, from neutral conditions to the onset of the stable regime, with the Reynolds number range Re τ = 200-800, at a Prandtl number of 0.71. The result is a boundary layer flow where the effects of stratification are weak in the wall region but progressively stronger in the outer layer up to the free surface. At λ 1 the turbulent kinetic energy (TKE) budget is in local equilibrium over a region extending from the near-wall region to a free-surface affected region a distance l ν from the surface, with l ν /δ ∼ Re −1/2 . In this equilibrium region the flow can be characterised by the flux Richardson number R f and the local Obukhov length scale Λ. At higher λ local mixing limit conditions are observed over an extended region. At λ = 2 the flux Richardson number approaches critical limit values of R f ,c 0.18 and gradient Richardson number Ri c 0.2. At high λ, we obtain a flow field where buoyancy interacts with the smallest scales of motion and the turbulent shear stress and buoyancy flux are suppressed to molecular levels. We find that this regime can be identified in terms of the parameter Re L ,c = L u τ /ν 200-400 (where u τ is the friction velocity and ν the kinematic viscosity), which is related to the L * parameter of Flores and Riley (Boundary-Layer Meteorol., vol. 139 (2), 2011, pp. 241-259) and buoyancy Reynolds number R. With energetic equilibrium attained, the local buoyancy Reynolds number, Re Λ = Λ u w 1/2 /ν, is directly related to the separation of the Ozmidov (l O ) and Kolmogorov (η) length scales in the outer boundary layer by Re Λ R ≡ (l O /η) 4/3 . The inner wall region has the behaviour R ∼ Re L Re τ , in contrast to stratified boundary layer flows where the buoyancy flux is non-zero at the wall and R ∼ Re L .

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