The steady boundary-layer flow near the stagnation point on an impermeable vertical surface with slip that is embedded in a fluid-saturated porous medium is investigated. Using appropriate similarity variables, the governing system of partial differential equations is transformed into a system of ordinary differential equations. This system is then solved numerically. The features of the flow and the heat transfer characteristics for different values of the governing parameters, namely, the Darcy-Brinkman, , mixed convection, λ, and slip, γ , parameters, are analysed and discussed in detail for the cases of assisting and opposing flows. It is found that dual solutions exist for assisting flows, as well as those usually reported in the literature for opposing flows. A stability analysis of the steady flow solutions encountered for different values of the mixed convection parameter λ is performed using a linear temporal stability analysis. This analysis reveals that for γ = 0 (slip absent) and = 1 the lower solution branch is unstable while the upper solution branch is stable.

The planform patterns of meandering submarine channels and subaerial fluvial bends show many similarities that have given rise to strong analogies concerning the fluid dynamics of these channel types. Existing models of helical motion in open-channel bends depict flow that is characterized by surface flow towards the outer bank, and basal flow towards the inner bank. This paper investigates and compares, through an analytical model and physical experiment, flows within fluvial meanders, and submarine channel bends that contain density-driven gravity currents. The results indicate that the sense of helical motion can be reversed in submarine bends that contain density currents when compared with fluvial bends, and that the orientation of the helical flow is dependent on the vertical distribution of downstream velocity. Specifically, the sense of helical motion is reversed in bends when the maximum downstream velocity is near the bed, resulting in near-bed flow towards the outer bank. These findings suggest that the dynamics of sediment transport and deposition in curved channels with such velocity profiles will be fundamentally different to those currently assumed from sinuous openchannels.

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[1] This article describes the development and validation of a method for representing the complex surface topography of gravel bed rivers in high-resolution three-dimensional computational fluid dynamic models. This is based on a regular structured grid and the application of a porosity modification to the mass conservation equation in which fully blocked cells are assigned a porosity of zero, fully unblocked cells are assigned a porosity of one, and partly blocked cells are assigned a porosity of between 0 and 1, according to the percentage of the cell volume that is blocked. The model retains an equilibrium wall function and an RNG-type two-equation turbulence model. The model is combined with a 0.002 m resolution digital elevation model of a flume-based, waterworked, gravel bed surface, acquired using two-media digital photogrammetry and with surface elevations that are precise to ±0.001 m. The model is validated by comparison with velocity data measured using a three-component acoustic Doppler velocimeter (ADV). Model validation demonstrates a significantly improved level of agreement than in previous studies, notably in relation to shear at the bed, although the resolution of model predictions was significantly higher than the ADV measurements, making model assessment in the presence of strong shear especially difficult. A series of simulations to assess model sensitivity to bed topographic and roughness representation were undertaken. These demonstrated inherent limitations in the prediction of 3-D flow fields in gravel bed rivers without high-resolution topographic representation. They also showed that model predictions of downstream flux were more sensitive to topographic smoothing that to changes in the roughness parameterization, reflecting the importance of both mass conservation (i.e., blockage) and momentum conservation effects at the grain and bed form scale. Model predictions allowed visualization of the structure of form-flow interactions at high resolution. In particular, the most protruding bed particles exerted a critical control on the turbulent kinetic energy maxima typically observed at about 20% of the flow depth above the bed.

The problem of determining the steady axially symmetrical motion induced by a sphere rotating with constant angular velocity about a diameter in an incompressible viscous fluid which is at rest at large distances from it is considered. The basic independent variables are the polar co-ordinates (r, θ) in a plane through the axis of rotation and with origin at the centre of the sphere. The equations of motion are reduced to three sets of nonlinear second-order ordinary differential equations in the radial variable by expanding the flow variables as series of orthogonal Gegenbauer functions with argument μ = cosθ. Numerical solutions of the finite set of equations obtained by truncating the series after a given number of terms are obtained. The calculations are carried out for Reynolds numbers in the range R = 1 to R = 100, and the results are compared with various other theoretical results and with experimental observations.The torque exerted by the fluid on the sphere is found to be in good agreement with theory at low Reynolds numbers and appears to tend towards the results of steady boundary-layer theory for increasing Reynolds number. There is excellent agreement with experimental results over the range considered. A region of inflow to the sphere near the poles is balanced by a region of outflow near the equator and as the Reynolds number increases the inflow region increases and the region of outflow becomes narrower. The radial velocity increases with Reynolds number at the equator, indicating the formation of a radial jet over the narrowing region of outflow. There is no evidence of any separation of the flow from the surface of the sphere near the equator over the range of Reynolds numbers considered.

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