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

Superhydrophobic surfaces have drawn a lot of interest both in academia and in industry because of the self-cleaning properties. This critical review focuses on the recent progress (within the last three years) in the preparation, theoretical modeling, and applications of superhydrophobic surfaces. The preparation approaches are reviewed according to categorized approaches such as bottom-up, top-down, and combination approaches. The advantages and limitations of each strategy are summarized and compared. Progress in theoretical modeling of surface design and wettability behavior focuses on the transition state of superhydrophobic surfaces and the role of the roughness factor. Finally, the problems/obstacles related to applicability of superhydrophobic surfaces in real life are addressed. This review should be of interest to students and scientists interested specifically in superhydrophobic surfaces but also to scientists and industries focused in material chemistry in general.

What do we need for a superhydrophobic surface? A review on the recent progress in the preparation of superhydrophobic surfaces

A positive temperature coefficient is the term which has been used to indicate that an increase in solubility occurs as the temperature is raised, whereas a negative coefficient indicates a decrease in solubility with rise in temperature.

Effect of Temperature

Graphene-based materials are gaining heightened attention as novel materials for environmental applications The unique physicochemical properties of graphene, notably its exceptionally high surface area, electron mobility, thermal conductivity, and mechanical strength, can lead to novel or improved technologies to address the pressing global environmental challenges This critical review assesses the recent developments in the use of graphene-based materials as sorbent or photocatalytic materials for environmental decontamination, as building blocks for next generation water treatment and desalination membranes, and as electrode materials for contaminant monitoring or removal The most promising areas of research are highlighted, with a discussion of the main challenges that we need to overcome in order to fully realize the exceptional properties of graphene in environmental applications

Environmental applications of graphene-based nanomaterials.

In the present study, the overall economic impact of hull fouling on a mid-sized naval surface ship (Arleigh Burke-class destroyer DDG-51) has been analyzed. A range of costs associated with hull fouling was examined, including expenditures for fuel, hull coatings, hull coating application and removal, and hull cleaning. The results indicate that the primary cost associated with fouling is due to increased fuel consumption attributable to increased frictional drag. The costs related to hull cleaning and painting are much lower than the fuel costs. The overall cost associated with hull fouling for the Navy's present coating, cleaning, and fouling level is estimated to be $56M per year for the entire DDG-51 class or $1B over 15 years. The results of this study provide guidance as to the amount of money that can be reasonably spent for research, development, acquisition, and implementation of new technologies or management strategies to combat hull fouling.

Economic impact of biofouling on a naval surface ship

Predictions of full-scale ship resistance and powering are made for antifouling coating systems with a range of roughness and fouling conditions. The estimates are based on results from laboratory-scale drag measurements and boundary layer similarity law analysis. In the present work, predictions are made for a mid-sized naval surface combatant at cruising speed and near maximum speed. The results indicate that slime films can lead to significant increases in resistance and powering, and heavy calcareous fouling results in powering penalties up to 86% at cruising speed. The present estimates show good agreement with results from full-scale ship power trials.

Effects of coating roughness and biofouling on ship resistance and powering.

A review of predictive methods used to determine the frictional drag on a rough surface is presented. These methods utilize a wide range of roughness scales, including roughness height, pitch, density, and shape parameters. Most of these scales were developed for regular roughness, limiting their applicability to predict the drag for many engineering flows. A new correlation is proposed to estimate the frictional drag for a surface covered with three-dimensional, irregular roughness in the fully rough regime. The correlation relies solely on a measurement of the surface roughness profile and builds on previous work utilizing moments of the surface statistics. A relationship is given for the equivalent sandgrain roughness height as a function of the root-mean-square roughness height and the skewness of the roughness probability density function. Boundary layer similarity scaling then allows the overall frictional drag coefficient to be determined as a function of the ratio of the equivalent sandgrain roughness height to length of the surface. DOI: 10.1115/1.4001492

https://www.usna.edu/NAOE/_files/documents/Faculty/schultz/Flack,%20Schultz%20-%20Hydraulic%20Roughness%20Scales,%202010.pdf

Review of Hydraulic Roughness Scales in the Fully Rough Regime

Turbulence measurements for rough-wall boundary layers are presented and compared to those for a smooth wall. The rough-wall experiments were made on a three-dimensional rough surface geometrically similar to the honed pipe roughness used by Shockling, Allen & Smits (J. Fluid Mech. vol. 564, 2006, p. 267). The present work covers a wide Reynolds-number range (Re θ = 2180–27 100), spanning the hydraulically smooth to the fully rough flow regimes for a single surface, while maintaining a roughness height that is a small fraction of the boundary-layer thickness. In this investigation, the root-mean-square roughness height was at least three orders of magnitude smaller than the boundary-layer thickness, and the Karman number (δ+), typifying the ratio of the largest to the smallest turbulent scales in the flow, was as high as 10100. The mean velocity profiles for the rough and smooth walls show remarkable similarity in the outer layer using velocity-defect scaling. The Reynolds stresses and higher-order turbulence statistics also show excellent agreement in the outer layer. The results lend strong support to the concept of outer layer similarity for rough walls in which there is a large separation between the roughness length scale and the largest turbulence scales in the flow.

The rough-wall turbulent boundary layer from the hydraulically smooth to the fully rough regime

The Reynolds number similarity hypothesis of Townsend [The Structure of Turbulent Shear Flow (Cambridge University Press, Cambridge, UK, 1976)] states that the turbulence beyond a few roughness heights from the wall is independent of the surface condition. The underlying assumption is that the boundary layer thickness δ is large compared to the roughness height k. This hypothesis was tested experimentally on two types of three-dimensional rough surfaces. Boundary layer measurements were made on flat plates covered with sand grain and woven mesh roughness in a closed return water tunnel at a momentum thickness Reynolds number Reθ of ∼14000. The boundary layers on the rough walls were in the fully rough flow regime (ks+⩾100) with the ratio of the boundary layer thickness to the equivalent sand roughness height δ∕ks greater than 40. The results show that the mean velocity profiles for rough and smooth walls collapse well in velocity defect form in the overlap and outer regions of the boundary layer. The Reynolds stresses for the two rough surfaces agree well throughout most of the boundary layer and collapse with smooth wall results outside of 3ks. Higher moment turbulence statistics and quadrant analysis also indicate the differences in the rough wall boundary layers are confined to y<5ks. The present results provide support for Townsend’s Reynolds number similarity hypothesis for uniform three-dimensional roughness in flows where δ∕ks⩾40.

Michael P. Schultz

Papers

Economic impact of biofouling on a naval surface ship

Effects of coating roughness and biofouling on ship resistance and powering.

Review of Hydraulic Roughness Scales in the Fully Rough Regime

The rough-wall turbulent boundary layer from the hydraulically smooth to the fully rough regime

Experimental support for Townsend’s Reynolds number similarity hypothesis on rough walls