Friction Factor Comparison Between Mixing Length Theory and Empirical Correlation

Equivalent sand grain roughness is required for estimating friction factor for engineering applications from empirical relation via Haalands equation. The real surfaces are different from the sand grain profile. The correlations for friction factor were derived from use of discrete roughness elements with regular shapes such as cones, bars etc. The purpose of the paper is to derive analytical expression of friction factor for a 2 dimensional semi-cylindrical roughness (not exactly a 3 dimensional sand grain but for the circular profile of cross- section) using Navier Stoke equation and mixing length theory. This is compared with the modified series mathematical representation of Haalands equation for friction factor in terms of equivalent sand grain roughness. The comparison is valid for high Reynolds number where the velocity profile is almost flat beyond boundary layer and approximately linear all throughout the boundary layer. The high Reynolds number approximation for Haalands equation is derived and the series form of the friction factor compares approximately with the series form derived from first principles, where in the exponents of the series expansion are close.

Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface

In this paper, a century-old problem is solved; namely, to find a unified analytic description of the non-uniform distribution of mean velocity across the entire domain of turbulent flow for all Reynolds numbers within the framework of the Prandtl mixing length theory. This study obtains a closed form solution of the mean velocity profile of plane turbulent flow for the Prandtl theory, and as well an approximate analytical solution for the van Driest mixing length theory. The profiles of several useful quantities are given based the closed form solution, such as turbulent viscosity, Reynolds turbulent stress, Kolmogorov's scaling law, and energy dissipation density. The investigation shows that the energy dissipation density at the surface is finite, whereas Landau's energy dissipation density is infinite. Strictly speaking, the closed form solution reveals that the universality of the turbulent velocity logarithmic profile no longer holds, but the von K\'arm\'an constant is still universal. Furthermore, a new formulation of the resistance coefficient of turbulent flow in pipes is formulated in implicit form.

Generation of Gravity-Capillary Wind Waves by Instability of a Coupled Shear-Flow

The theory of surface wave generation, in viscous flows, is modified by replacing the linear-logarithmic shear velocity profile, in the air, with a model which links smoothly the linear and logarithmic layers through the buffer layer. This profile includes the effects of air flow turbulence using a damped mixing-length model. In the water, an exponential shear velocity profile is used. It is shown that this modified and coupled shear-velocity profile gives a better agreement with experimental data than the coupled linear-logarithmic, non smooth profile, (in the air)–exponential profile (in the water), widely used in the literature. We also give new insights on retrograde modes that are Doppler shifted by the surface velocity at the air-sea interface, namely on the threshold value of the surface current for the occurrence of a second unstable mode.

Rapid AC Electrokinetic Micromixer with Electrically Conductive Sidewalls

We report a quasi T-channel electrokinetics-based micromixer with electrically conductive sidewalls, where the electric field is in the transverse direction of the flow and parallel to the conductivity gradient at the interface between two fluids to be mixed. Mixing results are first compared with another widely studied micromixer configuration, where electrodes are located at the inlet and outlet of the channel with electric field parallel to bulk flow direction but orthogonal to the conductivity gradient at the interface between the two fluids to be mixed. Faster mixing is achieved in the micromixer with conductive sidewalls. Effects of Re numbers, applied AC voltage and frequency, and conductivity ratio of the two fluids to be mixed on mixing results were investigated. The results reveal that the mixing length becomes shorter with low Re number and mixing with increased voltage and decreased frequency. Higher conductivity ratio leads to stronger mixing result. It was also found that, under low conductivity ratio, compared with the case where electrodes are located at the end of the channel, the conductive sidewalls can generate fast mixing at much lower voltage, higher frequency, and lower conductivity ratio. The study of this micromixer could broaden our understanding of electrokinetic phenomena and provide new tools for sample preparation in applications such as organ-on-a-chip where fast mixing is required.

A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers

A recently developed mixing length model of the turbulent shear stress in pipe flow is used to solve the streamwise momentum equation for fully developed channel flow. The solution for the velocity profile takes the form of an integral that is uniformly valid from the wall to the channel centreline at all Reynolds numbers from zero to infinity. The universal velocity profile accurately approximates channel flow direct numerical simulation (DNS) data taken from several sources. The universal velocity profile also provides a remarkably accurate fit to simulated and experimental flat plate turbulent boundary layer data including zero and adverse pressure gradient data. The mixing length model has five free parameters that are selected through an optimization process to provide an accurate fit to data in the range $R_\tau = 550$ to $R_\tau = 17\,207$ . Because the velocity profile is directly related to the Reynolds shear stress, certain statistical properties of the flow can be studied such as turbulent kinetic energy production. The examples presented here include numerically simulated channel flow data from $R_\tau = 550$ to $R_\tau =8016$ , zero pressure gradient (ZPG) boundary layer simulations from $R_\tau =1343$ to $R_\tau = 2571$ , zero pressure gradient turbulent boundary layer experimental data between $R_\tau = 2109$ and $R_\tau = 17\,207$ , and adverse pressure gradient boundary layer data in the range $R_\tau = 912$ to $R_\tau = 3587$ . An important finding is that the model parameters that characterize the near-wall flow do not depend on the pressure gradient. It is suggested that the new velocity profile provides a useful replacement for the classical wall-wake formulation.

Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface

In this letter, a century-old problem is studied; namely, to find a unified analytic description of the non-uniform distribution of mean velocity across the entire domain of turbulent flow for all Reynolds numbers within the framework of the Prandtl mixing length theory. Considering the Prandtl mixing length model, a closed form solution of the mean velocity profile of plane turbulent flow is obtained, and approximate analytical solution of the van Driest mixing length theory is proposed. The profiles of several useful quantities are given, such as turbulent viscosity, Reynolds turbulent stress, Kolmogorov's scaling law, and energy dissipation density. It is shown that the energy dissipation density at the surface is finite, whereas Landau's energy dissipation density is infinite. The closed form solution reveals that the universality of the turbulent velocity logarithmic profile no longer holds, but the von K\'arm\'an constant is still universal. Furthermore, a new formulation of the resistance coefficient of turbulent flow in pipes is given in implicit form.

Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface

In this letter, a century-old problem is studied; namely, to find a unified analytic description of the non-uniform distribution of mean velocity across the entire domain of turbulent flow for all Reynolds numbers within the framework of the Prandtl mixing length theory. Considering the Prandtl mixing length model, a closed form solution of the mean velocity profile of plane turbulent flow is obtained. The profiles of several useful quantities are given, such as turbulent viscosity, Reynolds turbulent stress, Kolmogorov's scaling law, and energy dissipation density. It is shown that the energy dissipation density at the surface is finite, whereas Landau's energy dissipation density is infinite. The closed form solution reveals that the universality of the turbulent velocity logarithmic profile no longer holds, but the von K\'arm\'an constant is still universal. The closed form solution is validated by both direct numerical simulation and experiments. The studies confirm that the van Driest mixing length theory is suitable for smooth walls, and the Prandtl mixing length theory is suitable for rough walls. Furthermore, a new formulation of the resistance coefficient of turbulent flow in pipes is given in implicit form.