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

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

/pdf/the-structure-of-turbulent-shear-flow-2yf3szhaqb.pdf

The Structure of Turbulent Shear Flow

A direct numerical simulation is applied to investigate three-dimensional unsteady flow characteristics around a finite wall-mounted square cylinder with an aspect ratio of 7 at a Reynolds number (Re) of 40-250. Determination of Re for the onset of vortex shedding and Re influence on the wake structure and integral parameters are the major objectives of the current research. The results show that the vortex shedding inception occurs within the range of 75 200.

Low-Reynolds-number flow around a wall-mounted square cylinder: Flow structures and onset of vortex shedding

A ghost-cell immersed boundary method for simulations of heat transfer in compressible flows under different boundary conditions

This article presents the numerical studies on predicting onset of flow separation and vortex shedding in flow past unconfined two-dimensional elliptical cylinders for various Axis Ratios (AR) and a wide range of Angles of Attack (AOA). An efficient Cartesian grid technique based immersed boundary method is used for numerical simulations. The laminar separation Reynolds number (Res) that marks separation of flow from surface and the critical Reynolds number (Recr) which represents transition from steady to unsteady flow are determined using diverse methods. A stability analysis which uses Stuart-Landau equation is also performed for calculating Recr. The shedding frequency (Stcr) that corresponds to Recr is calculated using Landau constants. The simulated results for circular cylinder are found to be in good agreement with the literature. The effects of AR and AOA on Res, Recr, and Stcr are studied. It is observed that the Res, Recr, and Stcr exhibit a direct/inverse relationship with AR depending upon th...

Onset of laminar separation and vortex shedding in flow past unconfined elliptic cylinders

This paper investigates the dynamics of velocity gradients for a spatially developing flow generated by a single square element of a fractal square grid at low inlet Reynolds number through direct numerical simulation. This square grid-element is also the fundamental block of a classical grid. The flow along the grid-element centreline is initially irrotational and becomes turbulent further downstream due to the lateral excursions of vortical turbulent wakes from the grid-element bars. We study the generation and evolution of the symmetric and anti-symmetric parts of the velocity gradient tensor for this spatially developing flow using the transport equations of mean strain product and mean enstrophy respectively. The choice of low inlet Reynolds number allows for fine spatial resolution and long simulations, both of which are conducive in balancing the budget equations of the above quantities. The budget analysis is carried out along the grid-element centreline and the bar centreline. The former is observed to consist of two subregions: one in the immediate lee of the grid-element which is dominated by irrotational strain, and one further downstream where both strain and vorticity coexist. In the demarcation area between these two subregions, where the turbulence is inhomogeneous and developing, the energy spectrum exhibits the best power-law slope. This is the same location where the experiments at much higher inlet Reynolds number show a well-defined spectrum over more than a decade of frequencies. Yet, the diagram, where and are the second and third invariants of the velocity gradient tensor, remains undeveloped in the near-grid-element region, and both the intermediate and extensive strain-rate eigenvectors align with the vorticity vector. Along the grid-element centreline, the strain is the first velocity gradient quantity generated by the action of pressure Hessian. This strain is then transported downstream by fluctuations and strain self-amplification is activated a little later. Further downstream, vorticity from the bar wakes is brought towards the grid-element centreline, and, through the interaction with strain, leads to the production of enstrophy. The strain-rate tensor has a statistically axial stretching form in the production region, but a statistically biaxial stretching form in the decay region. The usual signatures of velocity gradients such as the shape of diagrams and the alignment of vorticity vector with the intermediate eigenvector are detected only in the decay region even though the local Reynolds number (based on the Taylor length scale) is only between 30 and 40.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112017000544

Genesis and evolution of velocity gradients in near-field spatially developing turbulence

This article presents the numerical study of laminar forced convective heat transfer from elliptic cylinders of various axis ratios (AR = 0.1, 0.4, 0.6, 0.8, and 1.0), angles of attack (AOA = 30°, 45°, 60°, and 90°), and Reynolds numbers (Re = 50, 100, 150, and 200). Simulations are carried out for both isothermal and isoflux wall boundary conditions. A detailed study of flow field reveals distinct instantaneous and time-averaged flow patterns behind the elliptic cylinder. The effect of flow patterns on isotherms and, thus, on heat transfer, is analyzed in detail. Local and surface averaged Nusselt number (Nu and Nu avg ) is computed and their variation due to change in AR, AOA, and Re is studied. It is observed that increasing AR and Re increases Nu avg monotonically, while increasing AOA decreases Nu avg . Finally, correlations are proposed for Nu avg with respect to AR, AOA, and Re with minimum rms error.

Forced Convective Heat Transfer from Unconfined Isothermal and Isoflux Elliptic Cylinders

– The purpose of this paper is to study the effects of Angle of Attack (AOA), Axis Ratio (AR) and Reynolds number (Re) on unsteady laminar flow over a stationary elliptic cylinder. , – The governing equations of fluid flow over the elliptic cylinder are solved numerically on a Cartesian grid using Projection method based Immersed Boundary technique. This numerical method is validated with the results available in open literature. This scheme eliminates the requirement of generating a new computational mesh upon varying any geometrical parameter such as AR or AOA, and thus reduces the computational time and cost. , – Different vortex shedding patterns behind the elliptic cylinder are identified and classified using time averaged centerline streamwise velocity profile, instantaneous vorticity contours and instantaneous streamline patterns. A parameter space graph is constructed in order to reveal the dependence of AR, AOA and Re on vortex shedding. Integral parameters of flow such as mean drag, mean lift coefficients and Strouhal number are calculated and the effect of AR, AOA and Re on them is studied using various pressure and streamline contours. Functional relationships of each of integral parameters with respect to AR, AOA and Re are proposed with minimum percentage error. , – The results obtained can be used to explain the characteristics of flow patterns behind slender to bluff elliptical cylinders which found applications in insect flight modeling, heat exchangers and energy conservation systems. The proposed functional relationships may be very useful for the practicing engineers in those fields. , – The results presented in this paper are important for the researchers in the area of bluff body flow. The dependence of AOA on vortex shedding and flow parameters was never reported in the literature. These results are original, new and important.

Numerical analysis of laminar fluid flow characteristics past an elliptic cylinder: A parametric study

Non-canonical wakes of two-dimensional elliptic cylinders are analysed numerically for their near- and far-wake characteristics. The governing equations are solved using an immersed boundary method based projection scheme. The wakes are then classified into three distinct types according to diverse flow and thermal properties. An unexpected mean temperature evolution along the centreline of the wake is observed for certain wake states. In order to explain this unusual variation, novel heat transport models are constructed based on the vortex dynamics. These models are derived by considering vorticity is acted by flow, which has shear and swirl. Mechanisms of the primary vortex street breakdown and formation of the secondary vortex street are also proposed based on these models. A new phenomenon namely ‘dual near-wall instantaneous recirculation’ is observed, and its appearance is found to be a function of length of the primary von Karman vortex street. The same phenomenon is also found to be responsible for the secondary peak in the Nusselt number variation along the circumference of the cylinder. Despite varied differences between the wake types, it is observed that the transitions occur through a supercritical Hopf bifurcation in all of them, at least in the von Karman region of the wake. Low-frequency unsteadiness observed in the far wakes is examined through a signal decomposition method. Our results show that the secondary low frequency is resulting from the transition region which has a negative instability slope. Finally, onset of the primary vortex street breakdown and its scale in terms of Reynolds number is computed.

I. Paul

Papers

Onset of laminar separation and vortex shedding in flow past unconfined elliptic cylinders

Genesis and evolution of velocity gradients in near-field spatially developing turbulence

Forced Convective Heat Transfer from Unconfined Isothermal and Isoflux Elliptic Cylinders

Numerical analysis of laminar fluid flow characteristics past an elliptic cylinder: A parametric study

Analysis and characterisation of momentum and thermal wakes of elliptic cylinders