scholarly journals Reduction of Order: Analytical Solution of Film Formation in the Electrostatic Rotary Bell Sprayer

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 937 ◽  
Author(s):  
Mark Doerre ◽  
Nelson K. Akafuah
Keyword(s):  
Stokes Equations ◽  
Film Formation ◽  
Complex Function ◽  
Tangential Velocity ◽  
Velocity Profiles ◽  
Momentum Conservation ◽  
Navier Stokes ◽  
System Of Equations ◽  
Simple Boundary ◽  
Solution Spinning

This brief paper explains the slight differences in governing equations for a fluid film in a spinning cone, and the mechanism that reduces the order of a solution. Spinning cones with a centrally supplied fluid that spreads over its inner surface as a thin film have been the subject of interest for many years. Though often cast as a mathematical analysis, understanding this process is important, especially in the application of automotive painting. The analysis consists of a system of equations obtained from the Navier–Stokes equations along with simple boundary conditions that describe radial and tangential momentum conservation. Solutions to this system of equations are shown using several techniques. The connection between these techniques is slightly subtle. However, the conditions that enable reduction of order are clear once they are exposed. Directional velocity profiles in the film can be a combination of four roots in the complex plane. This system of roots also contains two diagonal axes of symmetry that are offset by 90 degrees. Alternatively, if the radial and tangential velocity profiles are expressed as a single complex function, a reduced order solution that is a combination of one set of diagonal set of roots can be found.

scholarly journals Determining the Tangential Velocity profile of a Rotating Fluid in a Static Container using Navier–Stokes equations

2020 ◽  
Author(s):  
RAJDEEP TAH ◽  
SARBAJIT MAZUMDAR ◽  
Krishna Kant Parida
Keyword(s):  
Angular Velocity ◽  
Velocity Profile ◽  
Velocity Gradient ◽  
Stokes Equations ◽  
Tangential Velocity ◽  
Navier Stokes ◽  
Rotating Fluid ◽  
Navier Stokes Equations ◽  
Similar Principle ◽  
Tangential Velocity Profile

The shape of the liquid surface for a fluid present in a uniformly rotating cylinder is generally determined by making a Tangential velocity gradient along the radius of the rotating cylindrical container. A very similar principle can be applied if the direction of the produced velocity gradient is reversed, for which the source of rotation will be present at the central axis of the cylindrical vessel in which the liquid is present. Now if the described system is completely closed, the angular velocity will decrease as a function of time. But when the surface of the rotating fluid is kept free, then the Tangential velocity profile would be similar to that of the Taylor-Couette Flow, with a modification that; due to formation of a curvature at the surface, the Navier-Stokes law is to be modified. Now the final equation may not seem to have a proper general solution, but can be approximated to certain solvable expressions for specific cases of angular velocity.

scholarly journals Experimental validation of wind energy estimation

2020 ◽  
Vol 24 (6 Part A) ◽  
pp. 3795-3806
Author(s):  
Predrag Zivkovic ◽  
Mladen Tomic ◽  
Vukman Bakic
Keyword(s):  
Wind Speed ◽  
Linear Model ◽  
Linear Models ◽  
Stokes Equations ◽  
Momentum Conservation ◽  
Electricity Production ◽  
Navier Stokes ◽  
Energy Estimation ◽  
Navier Stokes Equations ◽  
Non Linear

Wind power assessment in complex terrain is a very demanding task. Modeling wind conditions with standard linear models does not sufficiently reproduce wind conditions in complex terrains, especially on leeward sides of terrain slopes, primarily due to the vorticity. A more complex non-linear model, based on Reynolds averaged Navier-Stokes equations has been used. Turbulence was modeled by modified two-equations k-? model for neutral atmospheric boundary-layer conditions, written in general curvelinear non-orthogonal co-ordinate system. The full set of mass and momentum conservation equations as well as turbulence model equations are numerically solved, using the as CFD technique. A comparison of the application of linear model and non-linear model is presented. Considerable discrepancies of estimated wind speed have been obtained using linear and non-linear models. Statistics of annual electricity production vary up to 30% of the model site. Even anemometer measurements directly at a wind turbine?s site do not necessarily deliver the results needed for prediction calculations, as extrapolations of wind speed to hub height is tricky. The results of the simulation are compared by means of the turbine type, quality and quantity of the wind data and capacity factor. Finally, the comparison of the estimated results with the measured data at 10, 30, and 50 m is shown.

Mathematical Modelling of Thermo-Hydro-Mechanical Behaviour for Concrete under Elevated Temperature

2014 ◽  
Vol 670-671 ◽  
pp. 355-364
Author(s):  
Shao Bo Zhang ◽  
Xiao Chun Wang ◽  
Xin Pu Shen
Keyword(s):  
Elevated Temperature ◽  
Stokes Equations ◽  
Constitutive Relations ◽  
Gaseous Mixture ◽  
Mass Conservation ◽  
Momentum Conservation ◽  
Conservation Equation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Energy Conservation Equation

A hydro-thermo-mechanical model was presented for concrete at elevated temperature. Three phases of continuum were adopted in this model: gaseous mixture of water vapor and dry air, liquid water, and solid skeleton of concrete. Mass conservation equations, linear momentum conservation equation, and energy conservation equation were derived on the basis of the macroscopic Navier-Stokes equations for a general continuum, along with assumptions made for the purpose of simplification. Mathematical relationships between selected primary variables and secondary variables were given with existing data from references. Specifications of the constitutive relations were made for the kinetic variables and their conjugate forces.

Closed Form Solution for Outflow Between Corotating Disks

2016 ◽  
Vol 138 (5) ◽  
Author(s):  
Achhaibar Singh
Keyword(s):  
Closed Form ◽  
Stokes Equations ◽  
Closed Form Solution ◽  
Tangential Velocity ◽  
Form Solution ◽  
Navier Stokes ◽  
Published Data ◽  
Rotational Inertia ◽  
Viscous Losses ◽  
Present Solution

Mathematical expressions are derived for flow velocities and pressure distributions for a laminar flow in the gap between two rotating concentric disks. Fluid enters the gap between disks at the center and diverges to the outer periphery. The Navier–Stokes equations are linearized in order to get closed-form solution. The present solution is applicable to the flow between corotating as well as contrarotating disks. The present results are in agreement with the published data of other investigators. The tangential velocity is less for contrarotating disks than for corotating disks in core region of the radial channel. The flow is influenced by rotational inertia and convective inertia both. Dominance of rotational inertia over convective inertia causes backflow. Pressure depends on viscous losses, convective inertia, and rotational inertias. Effect of viscous losses on pressure is high at small throughflow Reynolds number. The convective and rotational inertia influence pressure significantly at high throughflow and rotational Reynolds numbers. Both favorable and unfavorable pressure gradients can be found simultaneously depending on a combination of parameters.

Numerical Simulation of Gas Flow Pattern in Cyclone Spray Scrubber

2011 ◽  
Vol 233-235 ◽  
pp. 701-706
Author(s):  
Bing Tao Zhao ◽  
Yi Xin Zhang ◽  
Kai Bin Xiong
Keyword(s):  
Numerical Simulation ◽  
Fluid Flow ◽  
Flow Pattern ◽  
Gas Flow ◽  
Stokes Equations ◽  
Tangential Velocity ◽  
Navier Stokes ◽  
Computational Domain ◽  
Spray Scrubber ◽  
Cfd Method

The numerical simulation of the fluid flow is presented by CFD technique to characterize the flow pattern of cyclone spray scrubber. In this process, the Reynolds-averaged Navier-Stokes equations with the Reynolds stress turbulence model (RSM) for fluid flow are solved by use of the finite volume method based on the SIMPLE pressure correction algorithm in the fluid computational domain. According to the computational results, the tangential velocity, axial velocity and turbulence intensity of the gas flow are addressed in the different flowrate. The results indicate that the CFD method can effectively reveal the mechanism of gas flow in the cyclone spray scrubber.

A PRIORI PRESSURE FIELD CONSTRUCTION ITERATIVE METHOD FOR THE SOLUTION OF THE STEADY-STATE NAVIER–STOKES EQUATIONS USING GENERAL FINITE-VOLUME CO-LOCATED GRIDS

2007 ◽  
Vol 04 (04) ◽  
pp. 567-601
Author(s):  
JOSE A. LAMAS
Keyword(s):  
Iterative Method ◽  
Pressure Field ◽  
Stokes Equations ◽  
A Priori ◽  
Mass Conservation ◽  
Momentum Conservation ◽  
Navier Stokes ◽  
Pressure Equation ◽  
Navier Stokes Equations ◽  
Correction Equations

An iterative method has been developed for the solution of the Navier–Stokes equations and implemented using finite volumes with co-located variable arrangement. A pressure equation is obtained combining algebraic momentum and mass conservation equations resulting in a self-consistent set of equations. An iterative procedure solves the pressure equation consistently with mass conservation and then updates velocities based on momentum equations without introducing velocity or pressure correction equations. The process is repeated until velocities satisfy both mass and momentum conservation. Tests demonstrate a priori pressure field solution consistent with mass conservation, and solution of hydrostatic problems in one iteration.

Computations of the Compressible Multiphase Flow Over the Cavitating High-Speed Torpedo

2003 ◽  
Vol 125 (3) ◽  
pp. 459-468 ◽  
Author(s):  
F. M. Owis ◽  
Ali H. Nayfeh
Keyword(s):  
High Speed ◽  
Multiphase Flows ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Multiphase System ◽  
System Of Equations ◽  
Cavitating Flows ◽  
Navier Stokes Equations ◽  
Compressible Multiphase Flow ◽  
Speed Flow

For high-speed cavitating flows, compressibility becomes significant in the liquid phase as well as in the vapor phase. In addition, the compressible energy equation is required for studying the effects of the propulsive jet on the cavity. Therefore, a numerical method is developed to compute cavitating flows over high-speed torpedoes using the full unsteady compressible Navier-Stokes equations. The multiphase system of equations is preconditioned for low-speed flow computations. Using the mass fraction form, we derive an eigensystem for both the conditioned and the nonconditioned system of equations. This eigensystem provides stability for the numerical discretization of the convective flux and increases the convergence rate. This method can be used to compute single as well as multiphase flows. The governing equations are discretized on a structured grid using an upwind flux difference scheme with flux limits. Single as well as multiphase flows are computed over a cavitating torpedo. The results indicate that the preconditioned system of equations converges rapidly to the required solution at very low speeds. The theoretical results are in good agreement with the measurements.

Instability and transitions of flow in a curved square duct: the development of two pairs of Dean vortices

1996 ◽  
Vol 314 ◽  
pp. 227-246 ◽  
Author(s):  
Philip A. J. Mees ◽  
K. Nandakumar ◽  
J. H. Masliyah
Keyword(s):  
Secondary Flow ◽  
Flow Pattern ◽  
Stokes Equations ◽  
Velocity Profiles ◽  
Navier Stokes ◽  
Flow State ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Measured Velocity ◽  
Dean Vortices

Steady developing flow of an incompressible Newtonian fluid in a curved duct of square cross-section (the Dean problem) is investigated both experimentally and numerically. This study is a continuation of the work by Bara, Nandakumar & Masliyah (1992) and is focused on flow rates between Dn = 200 and Dn = 600 (Dn = Re/(R/a)1/2, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension; the curvature ratio, R/a, is 15.1).Numerical simulations based on the steady three-dimensional Navier – Stokes equations predict the development of a 6-cell secondary flow pattern above a Dean number of 350. The 6-cell state consists of two large Ekman vortices and two pairs of small Dean vortices near the outer wall that result from the primary instability that is of centrifugal nature. The 6-cell flow state develops near θ = 80° and breaks down symmetrically into a 2-cell flow pattern.The apparatus used to verify the simulations had a duct dimension of 1.27 cm and a streamwise length of 270°. At a Dean number of 453, different velocity profiles of the 6-cell flow state at θ = 90° and spanwise profiles of the streamwise velocity at every 20° were measured using a laser-Doppler anemometer. All measured velocity profiles, as well as flow visualization of secondary flow patterns, are in very good agreement with the simulations, indicating that the parabolized Navier – Stokes equations give an accurate description of the flow.Based on the similarity with boundary layer flow over a concave wall (the Görtler problem), it is suggested that the transition to the 6-cell flow state is the result of a decreasing spanwise wavelength of the Dean vortices with increasing flow rate. A numerical stability analysis shows that the 6-cell flow state is unconditionally unstable. This is the first time that detailed experiments and simulations of the development of a 6-cell flow state are reported.

Inward Flow Between Stationary and Rotating Disks

2014 ◽  
Vol 136 (10) ◽  
Author(s):  
Achhaibar Singh
Keyword(s):  
Laminar Flow ◽  
Flow Field ◽  
Velocity Distribution ◽  
Stokes Equations ◽  
Tangential Velocity ◽  
Rotating Disk ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Rotating Disks ◽  
Navier Stokes Equations

The present study predicts the flow field and the pressure distribution for a laminar flow in the gap between a stationary and a rotating disk. The fluid enters through the peripheral gap between two concentric disks and converges to the center where it discharges axially through a hole in one of the disks. Closed form expressions have been derived by simplifying the Navier– Stokes equations. The expressions predict the backflow near the rotating disk due to the effect of centrifugal force. A convection effect has been observed in the tangential velocity distribution at high throughflow Reynolds numbers.

The Influence of Fluid Inertia in Unsteady Lubrication Films

1982 ◽  
Vol 104 (2) ◽  
pp. 180-186 ◽  
Author(s):  
A. Sestieri ◽  
R. Piva
Keyword(s):  
Computational Model ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Geometrical Shape ◽  
Fluid Inertia ◽  
Navier Stokes Equations ◽  
Film Boundary ◽  
Simple Boundary ◽  
Complete Set

The influence of the inertial forces in steady and unsteady lubrication films has been analyzed by means of an accurate computational model of the complete set of Navier Stokes equations for incompressible flows. A coordinate transformation accounts for the geometrical shape of the film boundary and its variation in time, reducing the numerical integration to a rectangular constant domain in the transformed plane. A dimensional analysis of the equations, both in steady and unsteady conditions, is performed to deduce the most significant nondimensional variables and parameters to be considered for a proper evaluation of the inertial effects. The applications are restricted to two-dimensional fields and simple boundary conditions to test the simulation capabilities of the computational model, by comparison with analytical solutions available in literature. Several geometrical and kinematic conditions are discussed and the effect of the convective and time dependent inertial terms is quantitatively evaluated.

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