The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves

The Structure of Turbulent Shear Flow By Dr. A. A. Townsend. Pp. xii + 315. 8¾ in. × 5½ in. (Cambridge: At the University Press.) 40s.

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The Structure of Turbulent Shear Flow

In this article, Jive k-e, two-equation models are studied: the standard k-e model, a low-Reynolds-number k-e model, a two-layer k-e model, a two-scale k-e model, and a renormalization group (RNG) k-e model. They are evaluated for their performance in predicting natural convection, forced convection, and mixed convection in rooms, as well as an impinging jet flow. Corresponding experimental data from the literature are used for validation. It is found that the prediction of the mean velocity is more accurate than that of the turbulent velocity. These models are neither able to predict anisotropic turbulence correctly nor to pick up the secondary recirculation of indoor air flow; otherwise the performance of the standard k-e model is good. The RNG k-e model is slightly better than the standard k-e model and is therefore recommended for simulations of indoor airflow. The performance of the other models is not stable.

COMPARISON OF DIFFERENT k-ε MODELS FOR INDOOR AIR FLOW COMPUTATIONS

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

Natural-convection flow in a square cavity calculated with low-Reynolds-number turbulence models

Numerical study of laminar and turbulent natural convection in an inclined square cavity

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Natural-convection boundary layers

Natural-convection flow in an enclosure with adiabatic horizontal walls and isothermal vertical walls maintained at a fixed temperature difference has been investigated. At high values of the natural-convection parameter, the Rayleigh number, a recirculating pocket appears near the corners downstream of the vertical walls, and the flow separates and reattaches at the horizontal walls in the vicinity of this recirculation. There is also a considerable thickening of the horizontal layer. In some previous studies by different authors, this corner flow was considered to be caused by an internal hydraulic jump, and the jump theory was used to predict bifurcation of the steady flow into periodic flow. The present work examines the corner phenomenon closely to decide if it is indeed caused by a hydraulic jump. The results of the analysis reveal the oversimplification of the problem made in the previous studies: there is no connection of the corner phenomenon with a hydraulic jump. The separation of flow at the ceiling is not a feature of hydraulic jumps, and the essential energy loss associated with hydraulic jumps is not observed in the corner flow. It is shown that the corner structure is caused by thermal effects. Owing to the temperature undershoots in vertical boundary layer, which are known to be caused by the stable thermal stratification of the core, relatively cold fluid reaches the upper corner. This cold fluid detaches from the ceiling like a plume at high Rayleigh numbers, and causes the separation and recirculation.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112094000522

On the high-Rayleigh-number structure of steady laminar natural-convection flow in a square enclosure

Direct numerical simulations of the Navier–Stokes equations have been carried out with the objective of studying turbulent boundary layers in adverse pressure gradients. The boundary layer flows concerned are of the equilibrium type which makes the analysis simpler and the results can be compared with earlier experiments and simulations. This type of turbulent boundary layers also permits an analysis of the equation of motion to predict separation. The linear analysis based on the assumption of asymptotically high Reynolds number gives results that are not applicable to finite Reynolds number flows. A different non-linear approach is presented to obtain a useful relation between the freestream variation and other mean flow parameters. Comparison of turbulent statistics from the zero pressure gradient case and two adverse pressure gradient cases shows the development of an outer peak in the turbulent energy in agreement with experiment. The turbulent flows have also been investigated using a differential Reynolds stress model. Profiles for velocity and turbulence quantities obtained from the direct numerical simulations were used as initial data. The initial transients in the model predictions vanished rapidly. The model predictions are compared with the direct simulations and low Reynolds number effects are investigated.

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R.A.W.M. Henkes

Papers

Natural-convection flow in a square cavity calculated with low-Reynolds-number turbulence models

Numerical study of laminar and turbulent natural convection in an inclined square cavity

Natural-convection boundary layers

On the high-Rayleigh-number structure of steady laminar natural-convection flow in a square enclosure

Direct Numerical Simulation of Self-Similar Turbulent Boundary Layers in Adverse Pressure Gradients