The boundary layer on a spherical gas bubble

1963 ◽  
Vol 16 (2) ◽  
pp. 161-176 ◽  
Author(s):  
D. W. Moore
Keyword(s):  
Boundary Layer ◽  
Potential Flow ◽  
Bubble Radius ◽  
Terminal Velocity ◽  
Gas Bubble ◽  
Fractional Powers ◽  
Small Viscosity ◽  
Flow Round ◽  
Bubble Rising ◽  
Rear Stagnation Point

The equations governing the boundary layer on a spherical gas bubble rising steadily through liquid of small viscosity are derived. These equations are linear are linear and are solved in closed form. The boundary layer separates at the rear stagnation point of the bubble to form a thin wake, whose structure is determined. Thus the drag force can be calculated from the momentum defect. The value obtained is 12πaaUμ, where a is the bubble radius and U the terminal velocity, and this agrees with the result of Levich (1949) who argued from the viscous dissipation in the potential flow round the bubble. The next term in an expansion of the drag in descending fractional powers of R is found and the results compared with experiment.

The velocity of rise of distorted gas bubbles in a liquid of small viscosity

1965 ◽  
Vol 23 (4) ◽  
pp. 749-766 ◽  
Author(s):  
D. W. Moore
Keyword(s):  
Experimental Data ◽  
Boundary Layer ◽  
Gas Bubbles ◽  
Quantitative Agreement ◽  
Terminal Velocity ◽  
Dimensionless Group ◽  
Gas Bubble ◽  
Numerical Accuracy ◽  
Small Viscosity ◽  
Acceleration Of Gravity

The terminal velocity of rise of small, distorted gas bubbles in a liquid of small viscosity is calculated. Small viscosity means that the dimensionless group gμ4/ρT3 where g is the acceleration of gravity, μ the viscosity, ρ the density and T the surface tension, is less than 10−8. It is assumed—and the numerical accuracy of the assumption is discussed—that the distorted bubbles are oblate ellipsoids of revolution. The drag coefficient is found by extending the theory given recently (Moore 1963) for the boundary layer on a spherical gas bubble. The results are in reasonable quantitative agreement with the experimental data.

The quasi-static growth of CO2 bubbles

2014 ◽  
Vol 741 ◽  
Author(s):  
Oscar R. Enríquez ◽  
Chao Sun ◽  
Detlef Lohse ◽  
Andrea Prosperetti ◽  
Devaraj van der Meer
Keyword(s):  
Boundary Layer ◽  
Growth Rate ◽  
Early Phase ◽  
Bubble Growth ◽  
Bubble Radius ◽  
Combined Effects ◽  
Gas Bubble ◽  
Final Phase ◽  
Concentration Boundary ◽  
Bubble Detachment

AbstractWe study experimentally the growth of an isolated gas bubble in a slightly supersaturated water–CO2 solution at 6 atm pressure. In contrast to what was found in previous experiments at higher supersaturation, the time evolution of the bubble radius differs noticeably from existing theoretical solutions. We trace the differences back to several combined effects of the concentration boundary layer around the bubble, which we disentangle in this work. In the early phase, the interaction with the surface on which the bubble grows slows down the process. In contrast, in the final phase, before bubble detachment, the growth rate is enhanced by the onset of density-driven convection. We also show that the bubble growth is affected by prior growth and detachment events, though they are up to 15 min apart.

Deformation and Oscillations of a Single Gas Bubble Rising in a Narrow Vertical Tube

2006 ◽  
Author(s):  
Zhaoyuan Wang ◽  
Albert Y. Tong
Keyword(s):  
Terminal Velocity ◽  
Vertical Tube ◽  
Jump Discontinuity ◽  
Finite Volume Scheme ◽  
Computational Domain ◽  
Gas Bubble ◽  
Initial Stage ◽  
Bubble Rising ◽  
Bubble Oscillations ◽  
Stable Shape

A single gas bubble rising in a narrow vertical tube is investigated via a numerical model on a 3-D axisymmetric computational domain. The transient governing equations are solved by a finite volume scheme with a two-step projection method. The interface between the liquid and gas phase is tracked by a coupled level set and volume-of-fluid (CLSVOF) method. A surface tension modeling method, which preserves the jump discontinuity of pressure at the interface, is employed. The flow structure and terminal velocity obtained in the numerical simulation are in excellent agreement with experimental measurements. Special attention is paid to the bubble oscillations during the initial stage of ascent. It has been found that the bubble bottom undergoes severe oscillations while the nose maintains a stable shape. A parametric study is performed to identify the factors controlling the oscillations at the bubble bottom.

Survey on the Indirect Methods of Potential Flow Used for the Optimization of Airships Profile

2014 ◽  
Vol 554 ◽  
pp. 717-723
Author(s):  
Reza Abbasabadi Hassanzadeh ◽  
Shahab Shariatmadari ◽  
Ali Chegeni ◽  
Seyed Alireza Ghazanfari ◽  
Mahdi Nakisa
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Flow Field ◽  
Drag Coefficient ◽  
Body Surface ◽  
Potential Flow ◽  
The Body ◽  
Indirect Methods ◽  
Singularity Method ◽  
Singularity Distribution

The present study aims to investigate the optimized profile of the body through minimizing the Drag coefficient in certain Reynolds regime. For this purpose, effective aerodynamic computations are required to find the Drag coefficient. Then, the computations should be coupled thorough an optimization process to obtain the optimized profile. The aerodynamic computations include calculating the surrounding potential flow field of an object, calculating the laminar and turbulent boundary layer close to the object, and calculating the Drag coefficient of the object’s body surface. To optimize the profile, indirect methods are used to calculate the potential flow since the object profile is initially amorphous. In addition to the indirect methods, the present study has also used axial singularity method which is more precise and efficient compared to other methods. In this method, the body profile is not optimized directly. Instead, a sink-and-source singularity distribution is used on the axis to model the body profile and calculate the relevant viscose flow field.

Potential Flow Solution for Crossflow Impingement of a Slot Jet on a Circular Cylinder

1976 ◽  
Vol 98 (2) ◽  
pp. 249-255 ◽  
Author(s):  
H. Miyazaki ◽  
E. M. Sparrow
Keyword(s):  
Boundary Layer ◽  
Nusselt Number ◽  
Circular Cylinder ◽  
Potential Flow ◽  
Closed Form Solution ◽  
Form Solution ◽  
Stream Velocity ◽  
Cylinder Surface ◽  
Stagnation Region ◽  
Flow Solution

A closed-form solution has been obtained for the potential flow about a circular cylinder situated in an impinging slot jet. Among other results, the potential flow solution yields the free stream velocity for the boundary layer adjacent to the cylinder surface. A basic feature of the solution is the division of the flow field into subdomains, thereby making it possible to employ harmonic functions that are appropriate to each such subdomain. The boundary conditions on the free streamline and the conditions of continuity between the subdomains are satisfied by a combination of least squares and point matching constraints. Numerical evaluation of the solution was carried out for cylinder diameters greater or equal to the nozzle width and for a range of dimensionless separation distances between the nozzle and the impingement surface. Results are presented for the velocity and pressure distributions on the cylinder surface, for the position of the free streamline, and for the velocity gradients at the stagnation point. The latter serve as input information to the Nusselt number and skin friction expressions that are given by boundary layer theory. Comparisons were made with available experimental results for the pressure distribution, velocity gradient, and Nusselt number, and good agreement was found to prevail in the stagnation region.

Calculation of Diffuser Efficiency for Two-Dimensional Flow

1947 ◽  
Vol 14 (3) ◽  
pp. A213-A216
Author(s):  
R. C. Binder
Keyword(s):  
Boundary Layer ◽  
Incompressible Flow ◽  
Potential Flow ◽  
Layer Growth ◽  
Two Dimensional ◽  
Dimensional Flow ◽  
Check Calculation ◽  
Two Dimensional Flow ◽  
Boundary Layer Growth ◽  
Potential Velocity

Abstract A method is presented for calculating the efficiency of a diffuser for two-dimensional, steady, incompressible flow without separation. The method involves a combination of organized boundary-layer data and frictionless potential-flow relations. The potential velocity and pressure are found after the boundary-layer growth is determined by a trial-and-check calculation.

Analysis of the Motion and Deformation of a Bubble Rising in a Viscoelastic Fluid

2006 ◽  
Author(s):  
Shriram Pillapakkam ◽  
Pushpendra Singh ◽  
Denis L. Blackmore ◽  
Nadine Aubry
Keyword(s):  
Velocity Field ◽  
Newtonian Fluid ◽  
Viscoelastic Fluid ◽  
Bubble Radius ◽  
Polymer Concentration ◽  
Deborah Number ◽  
Two Phase ◽  
Bubble Volume ◽  
Recirculation Zones ◽  
Bubble Rising

A finite element code based on the level set method is developed for performing two and three dimensional direct numerical simulations (DNS) of viscoelastic two-phase flow problems. The Oldroyd-B constitutive equation is used to model the viscoelastic liquid and both transient and steady state shapes of bubbles in viscoelastic buoyancy driven flows are studied. The influence of the governing dimensionless parameters, namely the Capillary number (Ca), the Deborah Number (De) and the polymer concentration parameter c, on the deformation of the bubble is also analyzed. Our simulations demonstrate that the rise velocity oscillates before reaching a steady value. The shape of the bubble, the magnitude of velocity overshoot and the amount of damping depend mainly on the parameter c and the bubble radius. Simulations also show that there is a critical bubble volume at which there is a sharp increase in the bubble terminal velocity as the increasing bubble volume increases, similar to the behavior observed in experiments. The structure of the wake of a bubble rising in a Newtonian fluid is strikingly different from that of a bubble rising in a viscoelastic fluid. In addition to the two recirculation zones at the equator of the bubble rising in a Newtonian fluid, two more recirculation zones exist in the wake of a bubble rising in viscoelastic fluids which influence the shape of a rising bubble. Interestingly, the direction of motion of the fluid a short distance below the trailing edge of a bubble rising in a viscoelastic fluid is in the opposite direction to the direction of the motion of the bubble, thus creating a “negative wake”. In this paper, the velocity field in the wake of the bubble, the effect of the parameters on the velocity field and their influence on the shape of the bubble are also investigated.

Flow Separation

2009 ◽  
Author(s):  
Ahmad Fakheri
Keyword(s):  
Boundary Layer ◽  
Nusselt Number ◽  
Drag Coefficient ◽  
Numerical Solution ◽  
Flow Separation ◽  
Potential Flow ◽  
Experimental Results ◽  
Science Courses ◽  
Boundary Layer Equations ◽  
Solution Of Equations

In thermal science courses, flow over curved objects, like cylinders or spheres are generally discussed qualitatively, followed by the presentation of numerical or experimental results for the drag coefficient, Nusselt number, and flow separation. Rarely, there is much discussion of how solutions are obtained. In this paper the flow separation is first introduced by solving the Falkner-Skan flow. The process for numerical solution of equations is presented to show that the flow separates at a plate angle of about −18°. Comparisons are drawn between this and flow over a cylinder. The non-similar boundary layer equations are then solved flow over a cylinder, using potential flow results for the velocity outside of the boundary layer. This solution shows that the flow separates at 103.5°, which is significantly more than the experimental value of 80°. Using a more realistic velocity for flow outside of the boundary layer, the numerical solution obtained predicts flow separation at an angle of 79°, which is close to the experimental results. All the solutions are obtained using spreadsheets that greatly simplify the analysis.

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