scholarly journals A lubrication equation for a simplified model of shear-thinning fluid

2021 ◽  
Vol 70 ◽  
pp. 147-165
Author(s):  
Francois James ◽  
Meissa M’Baye ◽  
Khawla Msheik ◽  
Duc Nguyen
Keyword(s):  
Stokes Equations ◽  
Piecewise Linear ◽  
Vertical Direction ◽  
Slow Motion ◽  
Shear Thinning ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Lubrication Equation ◽  
Motion Scaling ◽  
Shear Thinning Fluid

A lubrication equation is obtained for a simplified shear-thinning fluid. The simplified rheology consists of a piecewise linear stress tensor, resulting in a two-viscosity model. This can be interpreted as a modified Bingham fluid, which can be recovered in a specific limit. The lubrication equation is obtained in two steps. First two scalings are performed on the incompressible Navier-Stokes equations, namely the long-wave scaling and the slow motion scaling. Second, the resulting equations are averaged along the vertical direction. Numerical illustrations are provided, bringing to light the different possible behaviours.

Roll and Heave Response of Hull Sections of Variable Shapes in Waves

2010 ◽  
Author(s):  
Yi-Hsiang Yu ◽  
Spyros A. Kinnas
Keyword(s):  
Numerical Scheme ◽  
Stokes Equations ◽  
Vertical Direction ◽  
Navier Stokes ◽  
Hull Form ◽  
Navier Stokes Equations ◽  
Beam Seas ◽  
Near Resonance ◽  
A Cell ◽  
Volume Method

This paper addresses the hull responses near resonance in beam seas. A 2-D analysis is performed, and the hull form is free to roll and to move in the vertical direction (2-DOF). A cell center based finite volume method is applied for solving the Navier-Stokes equations. The numerical scheme is utilized for analyzing the flow field around the hull section as well as for predicting the wave and floating hull interaction. The effect of the hull corner geometry and the effectiveness of using bilge keels on roll damping are examined. The results show that the maximum roll response is reduced when the hull is free to heave and to roll as compared to the roll-only case (1-DOF). In general, the maximum hull response decreases when the shed vortices are induced by the sharp edge, and the reduction increases as the keel width increases.

Numerical Simulations of Nonlinear Post-Critical Vibrations of an Airfoil After Flutter Instability

2011 ◽  
Author(s):  
Jaromi´r Hora´cˇek ◽  
Miloslav Feistauer ◽  
Petr Sva´cˇek
Keyword(s):  
Numerical Simulations ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Mass Matrix ◽  
Vertical Direction ◽  
Navier Stokes ◽  
Computational Domain ◽  
Arbitrary Lagrangian Eulerian ◽  
Navier Stokes Equations ◽  
Classical Flutter

The contribution deals with the numerical simulation of the flutter of an airfoil with three degrees of freedom (3-DOF) for rotation around an elastic axis, oscillation in the vertical direction and rotation of a flap. The finite element (FE) solution of two-dimensional (2-D) incompressible Navier-Stokes equations is coupled with a system of nonlinear ordinary differential equations describing the airfoil vibrations with large amplitudes taking into account the nonlinear mass matrix. The time-dependent computational domain and a moving grid are treated by the Arbitrary Lagrangian-Eulerian (ALE) method and a suitable stabilization of the FE discretization is applied. The developed method was successfully tested by the classical flutter computation of the critical flutter velocity using NASTRAN program considering the linear model of vibrations and the double-lattice aerodynamic theory. The method was applied to the numerical simulations of the post flutter regime in time domain showing Limit Cycle Oscillations (LCO) due to nonlinearities of the flow model and vibrations with large amplitudes. Numerical experiments were performed for the airfoil NACA 0012 respecting the effect of the air space between the flap and the main airfoil.

Mechanistic Study of Upper Ocean Turbulence Interacting With Surface Waves

2010 ◽  
Author(s):  
Xin Guo ◽  
Di Yang ◽  
Yi Liu ◽  
Lian Shen
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Stokes Equations ◽  
Vertical Direction ◽  
Navier Stokes ◽  
Mechanistic Study ◽  
Navier Stokes Equations ◽  
Random Forcing ◽  
Set Up ◽  
Pseudo Spectral Method

We perform direct numerical simulations to simulate the interaction between surface waves and the turbulence underneath. The Navier–Stokes equations are simulated using a pseudo-spectral method in horizontal directions and a finite-difference method in vertical direction, with fully nonlinear viscous free-surface kinematic and dynamic boundary conditions at the free surface. We set up the turbulence and the waves by a random forcing method in the bulk flow and a pressure forcing method at the surface, which were recently developed by [1]. It is found that there are surface waves generated on the free surface due to the excitation by the turbulence. The surface elevation is sensitive to the effect of gravity and surface tension. In the presence of progressive waves at the free surface, the turbulent vortical structure is turned, stretched, and compressed periodically by the strain field of waves.

scholarly journals An Assessment of Suitability of a SIMPLE VOF/PLIC-CSF Multiphase Flow Model for Rising Bubble Dynamics

2012 ◽  
Vol 4 (1) ◽  
pp. 65-83 ◽  
Author(s):  
S. Senthil Kumar ◽  
Y. M. C. Delauré
Keyword(s):  
Bubble Dynamics ◽  
Stokes Equations ◽  
Piecewise Linear ◽  
Navier Stokes ◽  
Simple Type ◽  
Two Phase ◽  
Simple Method ◽  
Navier Stokes Equations ◽  
Two Fluids ◽  
Fluid Properties

A Volume of Fluid (VOF) – Youngs' model for the solution of an incompressible immiscible two-phase flows is presented. The solver computes the flow field by solving the family of Navier Stokes equations on a fixed (Eulerian) Staggered Cartesian grid using the Finite Volume formulation of Semi-Implicit Pressure Linked Equation (SIMPLE) method and tracks the position of interface between two fluids with different fluid properties by Piecewise Linear Interface Construction (PLIC) Method. The suitability of the SIMPLE type implementation is assessed by investigating the dynamics of free rising bubbles for different fluid properties and flow parameters. The results obtained with the present numerical method for rising bubbles in viscous liquids are compared with reported numerical and experimental results.

scholarly journals Application of a Projection Method for Simulating Flow of a Shear-Thinning Fluid

Fluids ◽  
2019 ◽  
Vol 4 (3) ◽  
pp. 124 ◽  
Author(s):  
Masoud Jabbari ◽  
James McDonough ◽  
Evan Mitsoulis ◽  
Jesper Henri Hattel
Keyword(s):  
Projection Method ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Shear Thinning ◽  
Navier Stokes ◽  
Bottom Surface ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Main Vortex

In this paper, a first-order projection method is used to solve the Navier–Stokes equations numerically for a time-dependent incompressible fluid inside a three-dimensional (3-D) lid-driven cavity. The flow structure in a cavity of aspect ratio δ = 1 and Reynolds numbers ( 100 , 400 , 1000 ) is compared with existing results to validate the code. We then apply the developed code to flow of a generalised Newtonian fluid with the well-known Ostwald–de Waele power-law model. Results show that, by decreasing n (further deviation from Newtonian behaviour) from 1 to 0.9, the peak values of the velocity decrease while the centre of the main vortex moves towards the upper right corner of the cavity. However, for n = 0.5 , the behaviour is reversed and the main vortex shifts back towards the centre of the cavity. We moreover demonstrate that, for the deeper cavities, δ = 2 , 4 , as the shear-thinning parameter n decreased the top-main vortex expands towards the bottom surface, and correspondingly the secondary flow becomes less pronounced in the plane perpendicular to the cavity lid.

Interaction of a deformable free surface with statistically steady homogeneous turbulence

2010 ◽  
Vol 658 ◽  
pp. 33-62 ◽  
Author(s):  
XIN GUO ◽  
LIAN SHEN
Keyword(s):  
Free Surface ◽  
Surface Deformation ◽  
Stokes Equations ◽  
Pseudospectral Method ◽  
Vertical Direction ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Surface Boundary ◽  
Navier Stokes Equations ◽  
Surface Boundary Conditions

Direct numerical simulation is performed for the interaction between a deformable free surface and the homogeneous isotropic turbulent flow underneath. The Navier–Stokes equations subject to fully nonlinear free-surface boundary conditions are simulated by using a pseudospectral method in the horizontal directions and a finite-difference method in the vertical direction. Statistically, steady turbulence is generated by using a linear forcing method in the bulk flow below. Through investigation of cases of different Froude and Weber numbers, the present study focuses on the effect of surface deformation of finite amplitude. It is found that the motion of the free surface is characterized by propagating waves and turbulence-generated surface roughness. Statistics of the turbulence field near the free surface are analysed in detail in terms of fluctuations of velocity, fluctuations of velocity gradients and strain rates and the energy budget for horizontal and vertical turbulent motions. Our results illustrate the effects of surface blockage and vanishing shear stress on the anisotropy of the flow field. Using conditional averaging analysis, it is shown that splats and antisplats play an essential role in energy inter-component exchange and vertical transport.

Hydrodynamics and Heat Transfer of an Oblique Plane Jet Impinging Onto a Substrate

2002 ◽  
Author(s):  
Albert Y. Tong
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Stokes Equations ◽  
Piecewise Linear ◽  
Analytical Data ◽  
Jet Impingement ◽  
Navier Stokes ◽  
Maximum Pressure ◽  
Navier Stokes Equations ◽  
Plane Jet

The objective of the present study is to understand the hydrodynamics and heat transfer of the impingement process, particularly the complexities attributable to the asymmetric geometry of an oblique free liquid jet. The Navier-Stokes equations are solved using a finite-volume formulation with a two-step projection method on a fixed non-uniform rectangular grid. The free surface of the jet is tracked by the volume-of-fluid (VOF) method with a second order accurate piecewise-linear scheme. The energy equation is modeled by using an enthalpy-based formulation. The method provides a state-of-the-art comprehensive model of the dynamic and thermal aspects of the impinging process. Nusselt number plots and pressure distributions on the substrate are obtained. The locations of the maximum Nusselt number as well as maximum pressure on the surface are identified and compared with the geometric jet impingement point. Results for normal impingement are also obtained and are used as reference. The effects of several parameters are examined. These include jet Reynolds number, jet impingement angle and jet inlet velocity profile. Experimental and analytical data from the literature are also included for comparison.

I. Eigenmotion analysis

1965 ◽  
Vol 285 (1402) ◽  
pp. 382-399 ◽  
Keyword(s):  
Dihedral Angle ◽  
Stokes Equations ◽  
Slow Motion ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Flow Properties ◽  
Navier Stokes Equations ◽  
Weakly Singular ◽  
Complex Eigenvalues ◽  
Slow Motions

General wedge and comer problems lead to the introduction of complex Navier-Stokes equations of complex laminar motions the real parts of which describe real laminar flows. Under the non-slip condition at the surface of a dihedral angle the general solution of the complex Navier-Stokes equations is established on the basis of the corresponding integral of the Stokes equations of slow motions. The latter integration is accomplished in terms of slow-motion eigenfunctions with real and complex eigenvalues. The results render valuable information about the flow properties at the leading or trailing edge of a dihedral angle. Weakly singular solutions, which are characterized by an infinite pressure and a finite velocity at the edge of the dihedral angle, are shown to exist for all wedges under asymmetric attack and for sharp wedges under symmetric attack. The existence of critical and branching eigenvalues exhibits the non-analyticity of laminar flows around dihedral angles in their dependence upon the corresponding wedge or corner angles.

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