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.

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.

Supplying angular momentum to the jittering jets explosion mechanism using inner convection layers

We conduct one-dimensional stellar evolution simulations in the mass range 13 − 20M⊙ to late core collapse times and find that an inner vigorous convective zone with large specific angular momentum fluctuations appears at the edge of the iron core during the collapse. The compression of this zone during the collapse increases the luminosity there and the convective velocities, such that the specific angular momentum fluctuations are of the order of $j_{\rm conv} \simeq 5 \times 10^{15} {~\rm cm}^2 {~\rm s}^{-1}$. If we consider that three-dimensional simulations show convective velocities that are three to four times larger than what the mixing length theory gives, and that the spiral standing accretion shock instability in the post-shock region of the stalled shock at a radius of $\simeq 100 {~\rm km}$ amplify perturbations, we conclude that the fluctuations that develop during core collapse are likely to lead to stochastic (intermittent) accretion disks around the newly born neutron star. In reaching this conclusion we also make two basic assumptions with uncertainties that we discuss. Such intermittent disks can launch jets that explode the star in the frame of the jittering jets explosion mechanism.

Impact of modified turbulent diffusion of PM_2.5 aerosol in WRF-Chem simulations in Eastern China

Correct description of the boundary layer mixing process of particle is an important prerequisite to understanding the mechanism of heavy pollution episodes. Turbulent mixing process of particles is usually denoted by the turbulent diffusion relationship of heat, meaning that the turbulent transport of particles and heat are similar. This similarity has, however, never been verified. Here we investigate the dissimilarity between particles and heat, indicating that the unified treatment of all scalars in the model is questionable. Using mixing-length theory, the turbulent diffusion relationship of particle is established, embedded in the model and verified on a long-term scale. Simulated results of PM2.5 concentration were improved by 8.3 % (2013), 17 % (2014), 11 % (2015) and 11.7 % (2017) in Eastern China, respectively. However, under the influence of complex topography, the turbulent diffusion process is insensitive to the simulation of the pollutant concentration. In addition to the PM2.5 concentration, the simulation of the CO concentration has also been improved, which shows that the turbulent diffusion process is extremely critical to the change in the concentration of pollutants.

On the impact of the structural surface effect on global stellar properties and asteroseismic analyses

ABSTRACT In a series of papers, we have recently demonstrated that it is possible to construct stellar structure models that robustly mimic the stratification of multidimensional radiative magnetohydrodynamic simulations at every time-step of the computed evolution. The resulting models offer a more realistic depiction of the near-surface layers of stars with convective envelopes than parametrizations, such as mixing length theory, do. In this paper, we explore how this model improvement impacts on seismic and non-seismic properties of stellar models across the Hertzsprung–Russell diagram. We show that the improved description of the outer boundary layers alters the predicted global stellar properties at different evolutionary stages. In a hare and hound exercise, we show that this plays a key role for asteroseismic analyses, as it, for instance, often shifts the inferred stellar age estimates by more than 10 per cent. Improper boundary conditions may thus introduce systematic errors that exceed the required accuracy of the PLATO space mission. Moreover, we discuss different approaches for computing stellar oscillation frequencies. We demonstrate that the so-called gas Γ1 approximation performs reasonably well for all main-sequence stars. Using a Monte Carlo approach, we show that the model frequencies of our hybrid solar models are consistent with observations within the uncertainties of the global solar parameters when using the so-called reduced Γ1 approximation.

Sensitivity of the lower edge of the pair-instability black hole mass gap to the treatment of time-dependent convection

ABSTRACT Gravitational-wave detections are now probing the black hole (BH) mass distribution, including the predicted pair-instability mass gap. These data require robust quantitative predictions, which are challenging to obtain. The most massive BH progenitors experience episodic mass ejections on time-scales shorter than the convective turnover time-scale. This invalidates the steady-state assumption on which the classic mixing length theory relies. We compare the final BH masses computed with two different versions of the stellar evolutionary code $\tt{MESA}$: (i) using the default implementation of Paxton et al. (2018) and (ii) solving an additional equation accounting for the time-scale for convective deceleration. In the second grid, where stronger convection develops during the pulses and carries part of the energy, we find weaker pulses. This leads to lower amounts of mass being ejected and thus higher final BH masses of up to ∼$5\, \mathrm{M}_\odot$. The differences are much smaller for the progenitors that determine the maximum mass of BHs below the gap. This prediction is robust at $M_{\rm BH, max}\simeq 48\, \mathrm{M}_\odot$, at least within the idealized context of this study. This is an encouraging indication that current models are robust enough for comparison with the present-day gravitational-wave detections. However, the large differences between individual models emphasize the importance of improving the treatment of convection in stellar models, especially in the light of the data anticipated from the third generation of gravitational-wave detectors.

Impact of the convective mixing-length parameter α on stellar metallicity

Context. Mixing-length theory is used to treat stellar convection. As a simulation in one-dimensional stellar atmospheres models, the mixing-length parameter α is calibrated from the Sun and then applied to other stars. However, there is no strong evidence to suggest that α should be the same for stars of different evolutionary stages. Aims. We evaluate the impact of the α value on the metallicity of different types of stars and investigate the correlation between the metallicity discrepancy (Δ[Fe∕H]) and stellar parameters (Teff, log g). Methods. We selected ten well-studied field stars and one open cluster of nine members for which high-resolution and high signal-to-noise spectra are available. The model atmospheres were calculated with the code MAFAGS-OS. We derived iron abundances from Fe I and Fe II lines both under local thermodynamic equilibrium and non-LTE conditions using a spectrum synthesis method. After deriving [Fe/H] for each line, we calculated Δ[Fe∕H] with two different α values, fixed solar-calibrated α, and α obtained for each star individually. Finally, we investigated the correlation between Δ[Fe∕H] caused by revised α with stellar parameters. Results. For FGK dwarf stars, the Δ[Fe∕H] caused by the α correction is less than 0.02 dex, while for turn-off and giant stars, the Δ[Fe∕H] values are no more than 0.03 dex, which are lower than typical uncertainties in metallicity. For main-sequence stars, Δ[Fe∕H] versus Teff and Δ[Fe∕H] versus log g are well fit by linear relations.