Ian P. Castro

The structure of strongly stratified flow over hills: dividing-streamline concept

In stably stratified flow over a three-dimensional hill, we can define a dividing streamline that separates those streamlines that pass around the hill from those that pass over the hill. The height Hs of this dividing streamline can be estimated by Sheppards simple energy argument; fluid parcels originating far upstream of a hill at an elevation above Hs have sufficient kinetic energy to rise over the top, whereas those below Hs must pass around the sides. This prediction provides the basis for analysing an extensive range of laboratory observations and measurements of stably stratified flow over a variety of shapes and orientations of hills and with different upwind density and velocity profiles. For symmetric hills and small upwind shear, Sheppards expression provides a good estimate for Hs. For highly asymmetric flow and/or in the presence of strong upwind shear, the expression provides a lower limit for Hs. As the hills become more nearly two-dimensional, these experiments become less well defined because steady-state conditions take progressively longer to be established. The results of new studies are presented here of the development of the unsteady flow upwind of two-dimensional hills in a finite-length towing tank. These measurements suggest that a very long tank would be required for steady-state conditions to be established upstream of long ridges with or without small gaps and cast doubt upon the validity of previous laboratory studies.

Rough-wall boundary layers: mean flow universality

Mean flow profiles, skin friction, and integral parameters for boundary layers developing naturally over a wide variety of fully aerodynamically rough surfaces are presented and discussed. The momentum thickness Reynolds number Reθ extends to values in excess of 47 000 and, unlike previous work, a very wide range of the ratio of roughness element height to boundary-layer depth is covered (0.03 0.2.

Channel flow over large cube roughness: a direct numerical simulation study

Computations of channel flow with rough walls comprising staggered arrays of cubes having various plan area densities are presented and discussed. The cube height h is 12.5% of the channel half-depth and Reynolds numbers (u? h/?) are typically around 700 – well into the fully rough regime. A direct numerical simulation technique, using an immersed boundary method for the obstacles, was employed with typically 35 million cells. It is shown that the surface drag is predominantly form drag, which is greatest at an area coverage around 15%. The height variation of the axial pressure force across the obstacles weakens significantly as the area coverage decreases, but is always largest near the top of the obstacles. Mean flow velocity and pressure data allow precise determination of the zero-plane displacement (defined as the height at which the axial surface drag force acts) and this leads to noticeably better fits to the log-law region than can be obtained by using the zero-plane displacement merely as a fitting parameter. There are consequent implications for the value of von K´arm´ an’s constant. As the effective roughness of the surface increases, it is also shown that there are significant changes to the structure of the turbulence field around the bottom boundary of the inertial sublayer. In distinct contrast to twodimensional roughness (longitudinal or transverse bars), increasing the area density of this three-dimensional roughness leads to a monotonic decrease in normalized vertical stress around the top of the roughness elements. Normalized turbulence stresses in the outer part of the flows are nonetheless very similar to those in smooth-wall flows.

The critical Reynolds number for rough-wall boundary layers

Rough-wall boundary layers become aerodynamically smooth if the ‘roughness Reynolds number’ Re*=u*z0/? becomes sufficiently small. Conventional wisdom is that Re* should exceed at least two and perhaps be as much as five before viscous effects are insignificant. This criterion is assessed for the types of rough wall commonly used in laboratory simulations of atmospheric boundary layers—arrays of sharp-edged obstacles with significant separation between each obstacle. It is demonstrated that for such surfaces the roughness length z0 remains constant for falling roughness Reynolds numbers down to at least Re*=1. Viscous effects are shown to change the near-wall Reynolds stresses only for Re* below similar values, so that z0-scaling (rather than scaling based on ?/u*) remains appropriate down to at least Re*=1. One of the practical implications of this result is that wind-tunnel simulations of atmospheric boundary layers can be successful at lower wind speeds (or with smaller roughness elements) than previously supposed.

Air flow and sand transport over sand-dunes

Developments in the modelling of turbulent wind over hills and sand dunes of different shapes by Hunt et al. [1], Carruthers et al. [2] are briefly described, and compared with earlier studies of Jackson and Hunt [3] and Walmsley et al. [4]. A new model (FLOWSTAR) is described; it has a more accurate description of airflow close to the surface, which is not in general logarithmic at typical measurement heights. Comparisons are made between the new model and the results of non-linear models using higher-order turbulence schemes, especially for surface shear stress.

Near-wall flow development after a step change in surface roughness

An experimental study of the initial flow field downstream of a step change in surface roughness is presented. The roughness length of the downstream surface was approximately tenfold that of the upstream roughness and, unlike all previous studies, attention was concentrated on the roughness sublayer region beneath the inertial (log-law) region. The experiments were conducted at a boundary layer Reynolds number of about 6 × 104 (based on layer thickness andfree-stream velocity) and around a longitudinal location where the (downstream) roughness length, zo2, was about 1% of the boundary-layer thickness atthe roughness change point.The thickness of the roughness sublayer was found for the two roughness. It was observed that the vertical profiles of mean velocity and turbulence characteristics started to show similarity after about 160z02 downstream of the roughness change. The presence of a shear stress overshoot is shown to depend strongly on the precise location (with respect to the roughness elements) at which the measurements are made and the thickness of the equilibrium layer is shown to be very sensitive to the way it is defined. It is demonstrated that the growing equilibrium layer has first to encompass the roughness sublayer before any thickness of inertial sublayer can be developed. It follows that, in somepractical cases, like flows across some urban environments, the latter(log-law) region may never exist at all.

Boundary layer development after a separated region

Measurements obtained in boundary layers developing downstream of the highly turbulent, separated flow generated at the leading edge of a blunt flat plate are presented. Two cases are considered: first, when there is only very low (wind tunnel) turbulence present in the free-stream flow and, second, when roughly isotropic, homogeneous turbulence is introduced. With conditions adjusted to ensure that the separated region was of the same length in both cases, the flow around reattachment was significantly different and subsequent differences in the development rate of the two boundary layers are identified. The paper complements, but is much more extensive than, the earlier presentation of some of the basic data (Castro & Epik, confirming not only that the development process is very slow, but also that it is non-monotonic. Turbulence stress levels fall below those typical of zero-pressure-gradient boundary layers and, in many ways, the boundary layer has features similar to those found in standard boundary layers perturbed by free-stream turbulence. Some numerical computations are used to assess the extent to which current turbulence models are adequate for such flows

A LIMITED-LENGTH-SCALE k-ε MODEL FOR THE NEUTRAL AND STABLY-STRATIFIED ATMOSPHERIC BOUNDARY LAYER

Analytical and numerical models of the neutral and stably-stratifiedatmospheric boundary layer are reviewed. Theoretical arguments andcomputational models suggest that a quasi-steady state is attainable in aboundary layer cooled from below and it is shown how this may be incorporatedwithin a time-steady, one-dimensional model. A new length-scale-limitedk-ε model is proposed for flows where a global maximum mixing length isimposed by the finite boundary-layer depth or, in stably-stratifiedconditions, by the Obukhov length, whilst still reducing to a form consistentwith the logarithmic law in the surface layer. Simulations compare favourablywith data from the Leipzig experiment and from Cardington airfield inEngland.

Stratified flow over three-dimensional ridges

An experimental study of the stratified flow over triangular-shaped ridges of various aspect ratios is described. The flows were produced by towing inverted bodies through saline-water solutions with stable (linear) density gradients. Flow-visualization techniques were used extensively to obtain measurements of the lee-wave structure and its interaction with the near-wake recirculating region and to determine the height of the upstream dividing streamline (below which all fluid moved around, rather than over the body). The Froude number F (= U/Nh ) and Reynolds number ( Uh /ν), where U is the towing speed, N is the Brunt–Vaisala frequency, h is the body height, and ν is the kinematic viscosity, were in the nominal ranges 0.2–1.6 (and ∞) and 2000–16000 respectively. The study demonstrates that the wave amplitude can be maximized by ‘tuning’ the body shape to the lee-wave field, that in certain circumstances steady wave breaking can occur or multiple recirculation regions (rotors) can exist downstream of the body, that vortex shedding in horizontal planes is possible even at F = 0.3, and that the ratio of the cross-stream width of the body to its height has a negligible effect on the dividing streamline height. The results of the study are compared with those of previous theoretical and experimental studies where appropriate.

Numerical considerations for simulations of flow and dispersion around buildings

Abstract This paper presents some aspects of computational work undertaken as part of a multipartner European Union project on flow and dispersion around buildings. Attention is concentrated on those features of the numerical methods which need particular care if adequate predictions are to be obtained. Some results from a few of the 15 or so test cases being computed by all the partners using a commercially available CFD code are used to illustrate the dangers that attend such calculations. It is shown that typical numerical solutions obtainable in an industrial context are likely to be strongly dependent not only on the turbulence model but also, and often more importantly, on mesh design and the numerical method. For example, a solution obtained with, say, the standard k − e turbulence model on a course grid can give results closer to experimental laboratory data than would be obtained with improved gridding and/or numerical schemes. Statements concerning apparent accuracy can therefore be misleading.