Inertial instability of flows on the inside or outside of a rotating horizontal cylinder

2013 ◽  
Vol 736 ◽  
pp. 107-129 ◽  
Author(s):  
E. S. Benilov ◽  
V. N. Lapin
Keyword(s):  
Stokes Equations ◽  
Liquid Films ◽  
Horizontal Cylinder ◽  
Thin Liquid Films ◽  
Navier Stokes ◽  
Lubrication Approximation ◽  
Navier Stokes Equations ◽  
Coating Flows ◽  
Inertial Instability ◽  
Rimming Flows

AbstractWe consider thin liquid films on the inside (rimming flows) or outside (coating flows) of a cylinder with horizontal axis, rotating about this axis. If the liquid’s net volume is small, such films are known to evolve towards a steady state with a smooth surface, whereas, for larger amounts, the flow develops a ‘shock’ similar to a tidal bore. In this work, smooth films are shown to be unstable. Since the strongest instability occurs at wavelengths comparable to the film’s thickness, our analysis is based on the full Navier–Stokes equations, not on the lubrication approximation (which has been traditionally used in this problem). It is also shown that, for cylinders of sufficiently small radii, the instability can be suppressed by surface tension.

scholarly journals NUMERICAL SIMULATION OF THE INTERACTION OF A REGULAR WAVE AND A SUBMERGED CYLINDER

2009 ◽  
Vol 8 (1) ◽  
pp. 78
Author(s):  
P. R. F. Teixeira
Keyword(s):  
Numerical Simulation ◽  
Free Surface ◽  
Stokes Equations ◽  
Horizontal Cylinder ◽  
Navier Stokes ◽  
Regular Wave ◽  
Viscous Effects ◽  
Navier Stokes Equations ◽  
Eulerian Formulation ◽  
Spectrum Distribution

A numerical simulation of the interaction between a regular wave and an immersed horizontal cylinder, whose axis is 3-radius deep, perpendicular to the direction of the wave propagation, is presented in this paper. The numerical model uses the semi-implicit two-step Taylor- Galerkin method to integrate Navier-Stokes equations in time and space. Arbitrary lagrangean-eulerian formulation is employed to describe the free surface movement. The free surface elevations near the cylinder and in some gauges along the channel, as well the spectrum distribution, are compared with experimental ones, and good agreement is obtained. The analysis shows that the viscous effects only affect the area that is very close to the cylinder.

Numerical and asymptotic methods for certain viscous free-surface flows

1992 ◽  
Vol 340 (1656) ◽  
pp. 1-45 ◽  
Keyword(s):  
Reynolds Number ◽  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Free Surface Flows ◽  
Navier Stokes ◽  
Lubrication Approximation ◽  
Navier Stokes Equations ◽  
Flow Rates

This paper concerns the two-dimensional motion of a viscous liquid down a perturbed inclined plane under the influence of gravity, and the main goal is the prediction of the surface height as the fluid flows over the perturbations. The specific perturbations chosen for the present study were two humps stretching laterally across an otherwise uniform plane, with the flow being confined in the lateral direction by the walls of a channel. Theoretical predictions of the flow have been obtained by finite-element approximations to the Navier-Stokes equations and also by a variety of lubrication approximations. The predictions from the various models are compared with experimental measurements of the free-surface profiles. The principal aim of this study is the establishment and assessment of certain numerical and asymptotic models for the description of a class of free-surface flows, exemplified by the particular case of flow over a perturbed inclined plane. The laboratory experiments were made over a range of flow rates such that the Reynolds number, based on the volume flux per unit width and the kinematical viscosity of the fluid, ranged between 0.369 and 36.6. It was found that, at the smaller Reynolds numbers, a standard lubrication approximation provided a very good representation of the experimental measurements but, as the flow rate was increased, the standard model did not capture several important features of the flow. On the other hand, a lubrication approximation allowing for surface tension and inertial effects expanded the range of applicability of the basic theory by almost an order of magnitude, up to Reynolds numbers approaching 10. At larger flow rates, numerical solutions to the full equations of motion provided a description of the experimental results to within about 4% , up to a Reynolds number of 25, beyond which we were unable to obtain numerical solutions. It is not known why numerical solutions were not possible at larger flow rates, but it is possible that there is a bifurcation of the Navier-Stokes equations to a branch of unsteady motions near a Reynolds number of 25.

scholarly journals Transport coefficients for low and high-rate mass transfer along a biological horizontal cylinder

2006 ◽  
Vol 10 (2) ◽  
pp. 441-447
Author(s):  
Alberto A. Barreto ◽  
Mauri Fortes ◽  
Wanyr R. Ferreira ◽  
Luiz C. A. Crespo
Keyword(s):  
Mass Transfer ◽  
Stokes Equations ◽  
Transport Coefficients ◽  
Horizontal Cylinder ◽  
Transport Processes ◽  
High Rate ◽  
Navier Stokes ◽  
Transfer Coefficients ◽  
Navier Stokes Equations ◽  
Drying Conditions

Knowledge of heat and mass transfer coefficients is essential for drying simulation studies or design of food and grain thermal processes, including drying. This work presents the full development of a segregated finite element method to solve convection-diffusion problems. The developed scheme allows solving the incompressible, steady-state Navier-Stokes equations and convective-diffusive problems with temperature and moisture dependent properties. The problem of simultaneous energy, momentum and species transfer along an infinite, horizontal cylinder under drying conditions in forced convection is presented, considering conditions normally found in biological material thermal treatment or drying. Numerical results for Nusselt and Sherwood numbers were compared against available empirical expressions; the results agreed within the associated experimental errors. For high rate mass transport processes, the proposed methodology allows to simulate drying conditions involving wall convective mass flux by a simple inclusion of the appropriated boundary conditions.

Numerical Model of Liquid Film Formation and Breakup in Last Stage of a Low-Pressure Steam Turbine

2017 ◽  
Vol 140 (3) ◽  
Author(s):  
Pietro Rossi ◽  
Asad Raheem ◽  
Reza S. Abhari
Keyword(s):  
Steam Turbine ◽  
Stokes Equations ◽  
Low Pressure ◽  
Liquid Films ◽  
Operating Conditions ◽  
Rotor Blades ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Last Stage ◽  
The Impact

Formation of thin liquid films on steam turbine airfoils, particularly in last stages of low-pressure (LP) steam turbines, and their breakup into coarse droplets is of paramount importance to assess erosion of last stage rotor blades given by the impact of those droplets. An approach for this problem is presented in this paper: this includes deposition of liquid water mass and momentum, film mass and momentum conservation, trailing edge breakup and droplets Lagrangian tracking accounting for inertia and drag. The use of thickness-averaged two-dimensional (2D) equations in local body-fitted coordinates, derived from Navier–Stokes equations, makes the approach suitable for arbitrary curved blades and integration with three-dimensional (3D) computational fluid dynamics (CFD) simulations. The model is implemented in the in-house solver MULTI3, which uses Reynolds-averaged Navier–Stokes equations κ – ω model and steam tables for the steam phase and was previously modified to run on multi-GPU architecture. The method is applied to the last stage of a steam turbine in full and part load operating conditions to validate the model by comparison with time-averaged data from experiments conducted in the same conditions. Droplets impact pattern on rotor blades is also predicted and shown.

scholarly journals Thin Liquid Film Dynamics on a Spinning Spheroid

Fluids ◽  
2021 ◽  
Vol 6 (9) ◽  
pp. 318
Author(s):  
Selin Duruk ◽  
Edouard Boujo ◽  
Mathieu Sellier
Keyword(s):  
Stokes Equations ◽  
Thin Liquid Film ◽  
Vertical Axis ◽  
Approximate Model ◽  
Navier Stokes ◽  
Lubrication Approximation ◽  
Navier Stokes Equations ◽  
Wide Range ◽  
Order Of Magnitude ◽  
The Impact

The present work explores the impact of rotation on the dynamics of a thin liquid layer deposited on a spheroid (bi-axial ellipsoid) rotating around its vertical axis. An evolution equation based on the lubrication approximation was derived, which takes into account the combined effects of the non-uniform curvature, capillarity, gravity, and rotation. This approximate model was solved numerically, and the results were compared favorably with solutions of the full Navier–Stokes equations. A key advantage of the lubrication approximation is the solution time, which was shown to be at least one order of magnitude shorter than for the full Navier–Stokes equations, revealing the prospect of controlling film dynamics for coating applications. The thin film dynamics were investigated for a wide range of geometric, kinematic, and material parameters. The model showed that, in contrast to the purely gravity-driven case, in which the fluid drains downwards and accumulates at the south pole, rotation leads to a migration of the maximum film thickness towards the equator, where the centrifugal force is the strongest.

Thermocapillary Convection in Thin Liquid Films on Walls With Microgrooves

2005 ◽  
Author(s):  
Alexander Alexeev ◽  
Tatiana Gambaryan-Roisman ◽  
Peter Stephan
Keyword(s):  
Numerical Model ◽  
Liquid Film ◽  
Thermocapillary Convection ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Liquid Films ◽  
Thin Liquid Films ◽  
Navier Stokes ◽  
Wall Structure ◽  
Film Rupture

Thermocapillary induced convection in thin liquid films on a horizontal wall with microgrooves is studied experimentally and numerically. To this end, we carry out experiments with silicon oil on a heated copper wall with parallel groves. The flow is visualized by tracking glass spheres seeded within the liquid film. The results of velocity measurements are reported. A numerical model for a liquid film on a structured wall is proposed. The full incompressible Navier-Stokes equations and the energy equation are integrated by a finite difference algorithm, whereas the mobile gas-liquid interface is tracked by the volume-of-fluid method. The numerical model is verified by comparison with the experimental data showing good agreement. The model is used to study flow patterns and film rupture caused by thermocapillary forces. We demonstrate that at any Marangoni number, either positive or negative, thermocapillary convection characterized by rolls develops within the film. In the experiments, two rolls in each groove are observed. The numerical solutions predict that at certain conditions the rolls are doubled under the influence of the wall structure guiding the flow. It is also found that an abrupt increase in wall temperature may rupture the liquid film near the structure crest. The results of this study may be applied to the design of microfluidic mixers and heat exchangers.

scholarly journals Modelling of droplet impacts on dry and wet surfaces using depth-averaged form

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Krish S. L. Hook ◽  
Sergii Veremieiev
Keyword(s):  
Stokes Equations ◽  
Liquid Films ◽  
Navier Stokes ◽  
Precursor Film ◽  
Crown Height ◽  
Navier Stokes Equations ◽  
Multigrid Algorithm ◽  
Three Phase ◽  
Impact Angles ◽  
Droplet Impacts

AbstractAn efficient time-adaptive multigrid algorithm is used to solve a range of normal and oblique droplet impacts on dry surfaces and liquid films using the Depth-Averaged Form (DAF) method of the governing unsteady Navier–Stokes equations. The dynamics of a moving three-phase contact line on dry surfaces is predicted by a precursor film model. The method is validated against a variety of experimental results for droplet impacts, looking at factors such as crown height and diameter, spreading diameter and splashing for a range of Weber, Reynolds and Froude numbers along with liquid film thicknesses and impact angles. It is found that, while being a computationally inexpensive methodology, the DAF method produces accurate predictions of the crown and spreading diameters as well as conditions for splash, however, underpredicts the crown height as the vertical inertia is not included in the model.

scholarly journals NEWTONIAN FLOW IN CONVERGING-DIVERGING CAPILLARIES

2013 ◽  
Vol 04 (03) ◽  
pp. 1350011 ◽  
Author(s):  
TAHA SOCHI
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Volumetric Flow Rate ◽  
Navier Stokes ◽  
Lubrication Approximation ◽  
Newtonian Flow ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
The One ◽  
Analytical Expressions

The one-dimensional Navier–Stokes equations are used to derive analytical expressions for the relation between pressure and volumetric flow rate in capillaries of five different converging-diverging axisymmetric geometries for Newtonian fluids. The results are compared to previously derived expressions for the same geometries using the lubrication approximation. The results of the one-dimensional Navier–Stokes are identical to those obtained from the lubrication approximation within a nondimensional numerical factor. The derived flow expressions have also been validated by comparison to numerical solutions obtained from discretization with numerical integration. Moreover, they have been certified by testing the convergence of solutions as the converging-diverging geometries approach the limiting straight geometry.

A three-equation model for thin films down an inclined plane

2016 ◽  
Vol 804 ◽  
pp. 162-200 ◽  
Author(s):  
G. L. Richard ◽  
C. Ruyer-Quil ◽  
J. P. Vila
Keyword(s):  
Stokes Equations ◽  
Random Noise ◽  
Mathematical Structure ◽  
Liquid Films ◽  
Equation Model ◽  
Navier Stokes ◽  
Neutral Stability ◽  
Inclined Plane ◽  
Navier Stokes Equations ◽  
Wave Limit

We derive a new model for thin viscous liquid films down an inclined plane. With an asymptotic expansion in the long-wave limit, the Navier–Stokes equations and the work–energy theorem are averaged over the fluid depth. This gives three equations for the mass, momentum and energy balance which have the mathematical structure of the Euler equations of compressible fluids with relaxation source terms, diffusive and capillary terms. The three variables of the model are the fluid depth, the average velocity and a third variable called enstrophy, related to the variance of the velocity. The equations are numerically solved by classical schemes which are known to be reliable and robust. The model gives satisfactory results both for the neutral stability curves and for the depth profiles of wavy films produced by a periodical forcing or by a random noise perturbation. The numerical calculations agree fairly well with experimental measurements of Liu & Gollub (Phys. Fluids, vol. 6, 1994, pp. 1702–1712). The calculation of the wall shear stress below the waves indicates a flow reversal at the first depth minimum downstream of the main hump, in agreement with experiments of Tihon et al. (Exp. Fluids, vol. 41, 2006, pp. 79–89).

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