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

Code and data for "Aerosol and Surface Stability of SARS-CoV-2 as Compared with SARS-CoV-1"

Advanced Mathematical Methods for Scientists and Engineers

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

We analyse the currently popular vortex identification criteria that are based on point-wise analysis of the velocity gradient tensor. A new measure of spiralling compactness of material orbits in vortices is introduced and using this measure a new local vortex identification criterion and requirements for a vortex core are proposed. The inter-relationships between the different criteria are explored analytically and in a few flow examples, using both zero and non-zero thresholds for the identification parameter. These inter-relationships provide a new interpretation of the various criteria in terms of the local flow kinematics. A canonical turbulent flow example is studied, and it is observed that all the criteria, given the proposed usage of threshold, result in remarkably similar looking vortical structures. A unified interpretation based on local flow kinematics is offered for when similarity or differences can be expected in the vortical structures educed using the different criteria.

On the relationships between local vortex identification schemes

Direct numerical simulations of the incompressible Navier–Stokes equations are employed to study the turbulent wall-shear stress in a turbulent channel flow forced by lateral sinusoidal oscillations of the walls. The objective is to produce a documented database of numerically computed friction reductions. To this aim, the particular numerical requirements for such simulations, owing for example to the time-varying direction of the skin-friction vector, are considered and appropriately accounted for. A detailed analysis of the dependence of drag reduction on the oscillatory parameters allows us to address conflicting results hitherto reported in the literature. At the Reynolds number of the present simulations, we compute a maximum drag reduction of 44.7%, and we assess the possibility for the power saved to be higher than the power spent for the movement of the walls (when mechanical losses are neglected). A maximum net energy saving of 7.3% is computed. Furthermore, the scaling of the amount of drag reduction is addressed. A parameter, which depends on both the maximum wall velocity and the period of the oscillation, is found to be linearly related to drag reduction, as long as the half-period of the oscillation is shorter than a typical lifetime of the turbulent near-wall structures. For longer periods of oscillation, the scaling parameter predicts that drag reduction will decrease to zero more slowly than the numerical data. The same parameter also describes well the optimum period of oscillation for fixed maximum wall displacement, which is smaller than the optimum period for fixed maximum wall velocity, and depends on the maximum displacement itself.

/pdf/critical-assessment-of-turbulent-drag-reduction-through-48ayhit4xe.pdf

Critical assessment of turbulent drag reduction through spanwise wall oscillations

Waves of spanwise velocity imposed at the walls of a plane turbulent channel flow are studied by direct numerical simulations. We consider sinusoidal waves of spanwise velocity which vary in time and are modulated in space along the streamwise direction. The phase speed may be null, positive or negative, so that the waves may be either stationary or travelling forward or backward in the direction of the mean flow. Such a forcing includes as particular cases two known techniques for reducing friction drag: the oscillating wall technique (a travelling wave with infinite phase speed) and the recently proposed steady distribution of spanwise velocity (a wave with zero phase speed). The travelling waves alter the friction drag significantly. Waves which slowly travel forward produce a large reduction of drag that can relaminarize the flow at low values of the Reynolds number. Faster waves yield a totally different outcome, i.e. drag increase (DI). Even faster waves produce a drag reduction (DR) effect again. Backward-travelling waves instead lead to DR at any speed. The travelling waves, when they reduce drag, operate in similar fashion to the oscillating wall, with an improved energetic efficiency. DI is observed when the waves travel at a speed comparable with that of the convecting near-wall turbulence structures. A diagram illustrating the different flow behaviours is presented.

/pdf/streamwise-travelling-waves-of-spanwise-wall-velocity-for-11tdq3huwa.pdf

Streamwise-travelling waves of spanwise wall velocity for turbulent drag reduction

The transient behaviour of a turbulent channel flow suddenly subjected to spanwise harmonic oscillations of the walls is numerically studied by means of direct numerical simulations of the incompressible Navier-Stokes equations. It is well known that this movement of the walls produces a sustained and significant reduction in turbulent friction; in this paper we focus on the early stages of the motion after the start of the oscillations when the fully developed state has not yet established. It is found that at the very beginning of the oscillatory motion the streamwise wall shear-stress remains constant for a short time interval, the length of which depends on the parameters defining the oscillation. A spanwise velocity profile starts to develop, almost coincident with the analytical laminar solution for the sudden start-up of harmonic oscillations of the wall. The spanwise flow fully adapts to the new forcing after about one oscillation period, whilst the longitudinal flow is still evolving towards its ...

/pdf/initial-response-of-a-turbulent-channel-flow-to-spanwise-58g4o68hea.pdf

Initial response of a turbulent channel flow to spanwise oscillation of the walls

Three proposed local criteria for the identification of vortices are analyzed and discussed; they are based on the analysis of invariants of the velocity gradient tensor ▿u or invariants of the tensor S 2 +Ω 2 , where S and Ω are the symmetric and antisymmetric parts of ▿u. Moreover, a tentative non-local procedure is proposed, which takes advantage of the observation that vortices tend to be made up of the same fluid particles; this leads to the definition of a Galilean invariant quantity, which can be computed and used to identify vortical structures. Three analytical flow fields are used for a comparative evaluation of both local and non-local criteria, which allows a deeper understanding of the physical meaning of the considered techniques.

On the effectiveness and limitations of local criteria for the identification of a vortex

In the present work a technique is numerically investigated, which is aimed at reducing the friction drag in turbulent boundary layers and channel flows. A cyclic spanwise oscillation of the wall with a proper frequency and amplitude is imposed, allowing a reduction of the turbulent drag of up to 40%. The present work is based on the numerical simulation of the Navier-Stokes equations in the simple geometry of a plane channel flow. The frequency of the oscillations is kept fixed at the most efficient value determined in previous studies, while the choice of the best value for the amplitude of the oscillations is evaluated not only in terms of friction reduction, but also by taking into consideration the overall energy balance and the power spent for the motion of the wall. The analysis of turbulence statistics allows to shed some light on the way oscillations interact with wall turbulence, as illustrated by visual inspection of some instantaneous flow fields. Finally, a simple explanation is proposed for this interaction, which leads to a rough estimate of the most efficient value for the frequency of the oscillations.

Maurizio Quadrio

Papers

Critical assessment of turbulent drag reduction through spanwise wall oscillations

Streamwise-travelling waves of spanwise wall velocity for turbulent drag reduction

Initial response of a turbulent channel flow to spanwise oscillation of the walls

On the effectiveness and limitations of local criteria for the identification of a vortex

Turbulent drag reduction by spanwise wall oscillations