Reynolds Number Formulation Research Articles

Viscosity-model-independent generalized Reynolds number for laminar pipe flow of shear-thinning and viscoplastic fluids

Understanding the flow dynamics of non-Newtonian fluids is crucial in various engineering, industrial, and biomedical applications. However, the existing generalized Reynolds number formulations for non-Newtonian fluids have limited applicability due to their dependencies on their specific viscosity models. In this study, we propose a new viscosity-model-independent generalized Reynolds number formulation for laminar pipe flow. The proposed method is based on the direct adaptation of the measurement principles of rotational viscometers for wall shear rate estimation. We assess the accuracy of this proposed formulation for power-law and Carreau-Yasuda viscosity models through robust friction factor experiments. The experimental results demonstrate the applicability and effectiveness of the proposed viscosity-model-independent Reynolds number, as the measured friction factor data align closely with our Reynolds number predictions. Furthermore, we compare the accuracy of our Reynolds number formulation against established generalized Reynolds formulations for pure shear-thinning (Carreau-Yasuda) and viscoplastic (Herschel-Bulkley-extended) models. The results of the comparative analysis confirm the reliability and robustness of this generalized Reynolds number in characterizing and interpreting flow behavior in systems with visco-inelastic non-Newtonian fluids. This unified generalized Reynolds number formulation presents new and significant opportunities for precise pipe flow characterization and interpretation as it is applicable to any visco-inelastic (time-independent) viscosity model without requiring additional derivations.

Numerical Investigation of Effect of Temperature Profile Imposed on the Crucible Surface on Oxygen Incorporated at the Crystal Melt Interface for 450 mm Diameter Silicon Single Crystal Growth in Presence of CUSP Magnetic Field Using Czochralski Technique

Numerical investigation has been carried out to study the effect of thermal boundary condition on turbulent melt flow and oxygen transfer inside a crucible used to grow 450 mm diameter silicon single crystal using Czochralski method. Two types of thermal boundary conditions, namely, isothermal crucible surface and experimentally measured temperatures on the crucible surface have been imposed on the crucible wall and crucible bottom. Melt motion owing to buoyancy, surface tension variation at the free melt surface as well as crystal and crucible rotation has been considered. Effect of CUSP magnetic field has also been investigated. Melt having aspect ratio of 1 and 0.5, representing high and moderate level of melt inside the crucible have been considered. K-epsilon turbulent model with low Reynolds number formulation has been used for turbulence modelling. Melt motion governed by natural convection, Marangoni convection and crystal as well as crucible rotation shows higher oxygen concentration at melt crystal interface for aspect ratio of 1, when experimental temperature profile is imposed at the crucible wall. For melt aspect ratio of 0.5, natural convection and Marangoni convection dependent melt show higher oxygen concentration for isothermal crucible wall boundary condition. Marangoni convection in absence of rotation of crystal and crucible leads to low oxygen concentration zone near the crystal outer surface, the size of which is larger for melt aspect ratio of 0.5. Maximum turbulent viscosity values for aspect ratio of 1, in case of experimental temperature at the crucible are higher compared to isothermal case, which is just the opposite of the trend for aspect ratio of 0.5. Imposing a CUSP magnetic field leads to similar distribution of turbulent viscosity for both types of thermal boundary conditions, irrespective of the type of thermal boundary condition. Oxygen concentration at the melt crystal interface is found to be higher in presence of CUSP magnetic field. Value of maximum turbulent viscosity for the two boundary condition are similar for melt aspect ratio of 1.0. Crucible surface temperature profile for aspect ratio of 1, on actual melt aspect ratio of 0.5 predicts lower oxygen concentration at melt crystal interface. Values and distribution of turbulent viscosity is however the same for both temperature profiles.

Nonlinear Theories for Shear Flow Instabilities: Physical Insights and Practical Implications

This article reviews the nonlinear stability theories that have been developed to explain laminar–turbulent transition processes in boundary and free shear layers. For such spatially developing shear flows, a high–Reynolds number approach is necessary to account for, in a systematic and self-consistent manner, multiple competing physical factors, such as nonlinearity, nonparallelism, nonequilibrium, and viscosity. While the basic ideas and fundamental mechanisms are rooted in the classical weakly nonlinear theory, which was formulated primarily for exactly parallel flows and on the basis of finite Reynolds number, the high–Reynolds number formulations lead to low-dimensional evolution systems, which differ significantly from the finite–Reynolds number counterparts and better describe the observations. Owing to efforts in the past 30 years or so, nonlinear evolution systems have been derived for inviscid Rayleigh modes, viscous Tollmien–Schlichting waves, (first and second) Mack modes, and cross-flow vortices. Theories have also been developed for nonlinear intermodal interactions, including oblique mode interaction, subharmonic resonance, phase-locked interactions, and fundamental resonance; these underpin many intriguing behaviors in the three-dimensional stages of transition. These theories and results explain several key nonlinear features observed and should play a useful role in guiding future experimental and numerical investigations.

Prediction of Turbulent Heat Transfer in Rotating and Nonrotating Channels With Wall Suction and Blowing

This paper reports on the prediction of heat transfer in a fully developed turbulent flow in a straight rotating channel with blowing and suction through opposite walls. The channel is rotated about its spanwise axis; a mode of rotation that amplifies the turbulent activity on one wall and suppresses it on the opposite wall leading to reverse transition at high rotation rates. The present predictions are based on the solution of the Reynolds-averaged forms of the governing equations using a second-order accurate finite-volume formulation. The effects of turbulence on momentum transport were accounted for by using a differential Reynolds-stress transport closure. A number of alternative formulations for the difficult fluctuating pressure–strain correlations term were assessed. These included a high turbulence Reynolds-number formulation that required a “wall-function” to bridge the near-wall region as well as three alternative low Reynolds-number formulations that permitted integration through the viscous sublayer, directly to the walls. The models were assessed by comparisons with experimental data for flows in channels at Reynolds-numbers spanning the range of laminar, transitional, and turbulent regimes. The turbulent heat fluxes were modeled via two very different approaches: one involved the solution of a modeled differential transport equation for each of the three heat-flux components, while in the other, the heat fluxes were obtained from an explicit algebraic model derived from tensor representation theory. The results for rotating channels with wall suction and blowing show that the algebraic model, when properly extended to incorporate the effects of rotation, yields results that are essentially identically to those obtained with the far more complex and computationally intensive heat-flux transport closure. This outcome argues in favor of incorporation of the algebraic model in industry-standard turbomachinery codes.

A novel implicit numerical treatment for non‐linear turbulence models using high and low Reynolds number formulations

AbstractNon‐linear turbulence models can be seen as an improvement of the classical eddy‐viscosity concept due to their better capacity to simulate characteristics of important flows. However, application of non‐linear models demand robustness of the numerical method applied, requiring a stable discretization scheme for convergence of all variables involved. Usually, non‐linear terms are handled in an explicit manner leading to possible numerical instabilities. Thus, the present work shows the steps taken to adapt a general non‐linear constitutive equation using a new semi‐implicit numerical treatment for the non‐linear diffusion terms. The objective is to increase the degree of implicitness of the solution algorithm to enhance convergence characteristics. Flow over a backward‐facing step was computed using the control volume method applied to a boundary‐fitted coordinate system. The SIMPLE algorithm was used to relax the algebraic equations. Classical wall function and a low Reynolds number model were employed to describe the flow near the wall. The results showed that for certain combination of relaxation parameters, the semi‐implicit treatment proposed here was the sole successful treatment in order to achieve solution convergence. Also, application of the implicit method described here shows that the stability of the solution either increases (high Reynolds with non‐orthogonal mesh) or preserves the same (low Reynolds number applications). Additional advantages of the procedure proposed here lie in the possibility of testing different non‐linear expressions if one considers the enhanced robustness and stability obtained for the entire numerical algorithm. Copyright © 2010 John Wiley & Sons, Ltd.

Heat‐Transfer Coefficient for Cellular Materials Modelled as an Array of Elliptic Rods

Convective heat-transfer coefficients in foam-like materials, modelled as an array of elliptic rods, are numerically determined. An incompressible fluid is considered, flowing through an infinite foam-like material with an arbitrary solid temperature. A repetitive cell is identified and periodic boundary conditions are applied. Turbulence is handled with both low and high Reynolds number formulations. The interfacial heat-transfer coefficient is obtained by volume integrating the distributed variables obtained within the cell. The results indicate that, for the same mass-flow rate, materials formed by elliptic rods have a lower interfacial heat-transfer coefficient compared to other media modelled as staggered arrays of square rods.

Numerical Simulations of Magnetic Flow Control in Hypersonic Chemically Reacting Flows

Hypersonic flows over blunt bodies subject to magnetic fields are numerically investigated. The magnetogasdynamic equations in the high magnetic Reynolds number formulation form an eight-equation system, with density, momentum, magnetic field, and total energy as unknowns. In the low magnetic Reynolds number approximation, the magnetic field induction is ignored, which leads to a five-equation system, where the magnetic interaction is represented by source terms in the momentum and energy equations. A four-stage modified Runge-Kutta scheme with the Davis-Yee symmetric total variation diminishing model as a postprocessing stage is used to solve the magnetogasdynamic equations. High-temperature effects are simulated by equilibrium and nonequilibrium chemistry models. The equilibrium model computes thermodynamic properties by interpolation from experimental data. The nonequilibrium model is a 1-temperature, 5-species, 17-reaction model solved by an implicit flux-vector splitting scheme. A loosely coupled approach is implemented to communicate between the magnetogasdynamic equations and the chemistry models