The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

This study describes the development and validation of Computational Fluid Dynamics (CFD) methodology for the simulation of dispersed two-phase flows. A two-fluid (Euler-Euler) methodology previously developed at Imperial College is adapted to high phase fractions. It employs averaged mass and momentum conservation equations to describe the time-dependent motion of both phases and, due to the averaging process, requires additional models for the inter-phase momentum transfer and turbulence for closure. The continuous phase turbulence is represented using a two-equation k − ε−turbulence model which contains additional terms to account for the effects of the dispersed on the continuous phase turbulence. The Reynolds stresses of the dispersed phase are calculated by relating them to those of the continuous phase through a turbulence response function. The inter-phase momentum transfer is determined from the instantaneous forces acting on the dispersed phase, comprising drag, lift and virtual mass. These forces are phase fraction dependent and in this work revised modelling is put forward in order to capture the phase fraction dependency of drag and lift. Furthermore, a correlation for the effect of the phase fraction on the turbulence response function is proposed. The revised modelling is based on an extensive survey of the existing literature. The conservation equations are discretised using the finite-volume method and solved in a solution procedure, which is loosely based on the PISO algorithm, adapted to the solution of the two-fluid model. Special techniques are employed to ensure the stability of the procedure when the phase fraction is high or changing rapidely. Finally, assessment of the methodology is made with reference to experimental data for gas-liquid bubbly flow in a sudden enlargement of a circular pipe and in a plane mixing layer. Additionally, Direct Numerical Simulations (DNS) are performed using an interface-capturing methodology in order to gain insight into the dynamics of free rising bubbles, with a view towards use in the longer term as an aid in the development of inter-phase momentum transfer models for the two-fluid methodology. The direct numerical simulation employs the mass and momentum conservation equations in their unaveraged form and the topology of the interface between the two phases is determined as part of the solution. A novel solution procedure, similar to that used for the two-fluid model, is used for the interface-capturing methodology, which allows calculation of air bubbles in water. Two situations are investigated: bubbles rising in a stagnant liquid and in a shear flow. Again, experimental data are used to verify the computational results.

Computational fluid dynamics of dispersed two-phase flows at high phase fractions

A literature review on mechanisms and models for the coalescence process of fluid particles

Part I. The Early Years and the Influence of William G. Cochran: 1. William G. Cochran's contributions to the design, analysis, and evaluation of observational studies 2. Controlling bias in observational studies: a review William G. Cochran Part II. Univariate Matching Methods and the Dangers of Regression Adjustment: 3. Matching to remove bias in observational studies 4. The use of matched sampling and regression adjustment to remove bias in observational studies 5. Assignment to treatment group on the basis of a covariate Part III. Basic Theory of Multivariate Matching: 6. Multivariate matching methods that are equal percent bias reducing, I: Some examples 7. Multivariate matching methods that are equal percent bias reducing, II: Maximums on bias reduction for fixed sample sizes 8. Using multivariate matched sampling and regression adjustment to control bias in observational studies 9. Bias reduction using Mahalanobis-metric matching Part IV. Fundamentals of Propensity Score Matching: 10. The central role of the propensity score in observational studies for causal effects Paul R. Rosenbaum 11. Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome Paul R. Rosenbaum 12. Reducing bias in observational studies using subclassification on the propensity score Paul R. Rosenbaum 13. Constructing a control group using multivariate matched sampling methods that incorporate the propensity score Paul Rosenbaum 14. The bias due to incomplete matching Paul R. Rosenbaum Part V: Affinely Invariant Matching Methods with Ellipsoidally Symmetric Distributions, Theory and Methodology: 15. Affinely invariant matching methods with ellipsoidal distributions Neal Thomas 16. Characterizing the effect of matching using linear propensity score methods with normal distributions Neal Thomas 17. Matching using estimated propensity scores: relating theory to practice Neal Thomas 18. Combining propensity score matching with additional adjustments for prognostic covariates Part VI. Some Applied Contributions: 19. Causal inference in retrospective studies Paul Holland 20. The design of the New York school choice scholarships program evaluation Jennifer Hill and Neal Thomas 21. Estimating and using propensity scores with partially missing data Ralph D'Agostino Jr. 22. Using propensity scores to help design observational studies: application to the tobacco litigation Part VII. Some Focused Applications: 23. Criminality, aggression and intelligence in XYY and XXY men H. A. Witkin 24. Practical implications of modes of statistical inference for causal effects and the critical role of the assignment mechanism 25. In utero exposure to phenobarbital and intelligence deficits in adult men June Reinisch, Stephanie Sanders, and Erik Mortensen 26. Estimating causal effects from large data sets using propensity scores 27. On estimating the causal effects of DNR orders Martin McIntosh.

Matched Sampling for Causal Effects

Objective. —To test whether exposure to phenobarbital in utero is associated with deficits in intelligence scores in adult men and whether the magnitude of the postnatal effect is mediated by exposure parameters and/or postnatal environmental factors. Design. —Two double-blind studies were conducted on independent samples of adult men prenatally exposed to phenobarbital and matched control samples using different measures of general intelligence. Based on data from control subjects, regression models were built relating intelligence scores to relevant pre-exposure matching variables and age at testing. Models generated predicted scores for each exposed subject. Group mean differences between the individually predicted and observed scores estimated exposure effects. Setting. —Copenhagen, Denmark. Participants. —Exposed subjects were adult men born at the largest hospital in Copenhagen between 1959 and 1961 who were exposed to phenobarbital during gestation via maternal medical treatment and whose mothers had no history of a central nervous system disorder and no treatment during pregnancy with any other psychopharmacological drug. Study 1 included 33 men and study 2,81 men. Controls were unexposed members of the same birth cohort matched on a wide spectrum of maternal variables recorded prenatally and perinatally. Controls for studies 1 and 2 included 52 and 101 men, respectively. Main Outcome Measures. —In study 1: Wechsler Adult Intelligence Scale (Danish version); in study 2: Danish Military Draft Board Intelligence Test (Borge Priens Prove). Result. —Men exposed prenatally to phenobarbital had significantly lower verbal intelligence scores (approximately 0.5 SD) than predicted. Lower socioeconomic status and being the offspring of an "unwanted" pregnancy increased the magnitude of the negative effects. Exposure that included the last trimester was the most detrimental. Conclusion. —Phenobarbital exposure during early development can have long-term deleterious effects on cognitive performance. Detrimental environmental conditions can interact with prenatal biological insult to magnify negative outcomes. Physicians are urged to use increased caution in prescribing such medications during pregnancy. (JAMA. 1995;274:1518-1525)

https://jamanetwork.com/journals/JAMA/articlepdf/391346/jama_274_19_031.pdf

In Utero Exposure to Phenobarbital and Intelligence Deficits in Adult Men

Bubble coalescence in turbulent flows: A mechanistic model for turbulence-induced coalescence applied to microgravity bubbly pipe flow

Gas-liquid flow at microgravity conditions—I. Dispersed bubble and slug flow

The quasistatic injection of air bubbles at the wall in a viscous shear flow is experimentally investigated by high-speed video pictures and image processing. The bubble radius and center of gravity position are determined throughout the growth. Experimental results are then used to validate a force balance model during bubble growth and after detachment for small bubble Reynolds numbers. The contact angles at the bubble foot calculated from the force balance agree well with those experimentally obtained from image processing. Finally, the force balance model is used to predict the departure bubble radius versus the gas flow rate and liquid flow shear rate.

Dynamics of bubble growth and detachment in a viscous shear flow

https://hal.archives-ouvertes.fr/hal-00973210/document

Benchmarks and numerical methods for the simulation of boiling flows

Based on the experiments carried out over the past decade at microgravity conditions, an overview of our current knowledge of bubbly and slug flows is presented. The transition from bubble to slug flow, the void fraction and the pressure drop are discussed from the data collected in the literature. The transition from bubble to slug flow may be predicted by introducing a critical void fraction that depends on the fluid properties and the pipe diameter: however, the role of coalescence which controls this transition is not clearly understood. The void fraction may be accurately calculated using a drift-flux model: it is shown from local measurements that the drift of the gas with respect to the mixture is due to the non uniform radial distribution of void fraction. The pressuredrop happens to be controlled by the liquid flow for bubbly flow whereas for slug flow the experimental results show that pressure drops is larger than expected. From this study, the guidelines for future research in microgr...

/pdf/bubble-and-slug-flow-at-microgravity-conditions-state-of-2y2zudzjyv.pdf

Catherine Colin

Papers

Bubble coalescence in turbulent flows: A mechanistic model for turbulence-induced coalescence applied to microgravity bubbly pipe flow

Gas-liquid flow at microgravity conditions—I. Dispersed bubble and slug flow

Dynamics of bubble growth and detachment in a viscous shear flow

Benchmarks and numerical methods for the simulation of boiling flows

Bubble and slug flow at microgravity conditions: state of knowledge and open questions