Nearly equilibrium dissociating boundary-layer flows by the method of matched asymptotic expansions

1966 ◽  
Vol 26 (4) ◽  
pp. 793-806 ◽  
Author(s):  
George R. Inger
Keyword(s):  
Boundary Layer ◽  
Asymptotic Expansions ◽  
High Speed ◽  
Numerical Solutions ◽  
The Body ◽  
Matched Asymptotic Expansions ◽  
Form Analysis ◽  
Perturbation Problem ◽  
Valid Solution ◽  
Non Equilibrium

The approach to equilibrium in a non-equilibrium-dissociating boundary-layer flow along a catalytic or non-catalytic surface is treated from the standpoint of a singular perturbation problem, using the method of matched asymptotic expansions. Based on a linearized reaction rate model for a diatomic gas which facilitates closed-form analysis, a uniformly valid solution for the near equilibrium behaviour is obtained as the composite of appropriate outer and inner solutions. It is shown that, under near equilibrium conditions, the primary non-equilibrium effects are buried in a thin sublayer near the body surface that is described by the inner solution. Applications of the theory are made to the calculation of heat transfer and atom concentrations for blunt body stagnation point and high-speed flat-plate flows; the results are in qualitative agreement with the near equilibrium behaviour predicted by numerical solutions.

Asymptotic Methods for an Infinite Slider Bearing With a Discontinuity in Film Slope

1976 ◽  
Vol 98 (3) ◽  
pp. 446-452 ◽  
Author(s):  
J. A. Schmitt ◽  
R. C. DiPrima
Keyword(s):  
Boundary Layer ◽  
Film Thickness ◽  
Asymptotic Expansions ◽  
Asymptotic Expression ◽  
Asymptotic Methods ◽  
Matched Asymptotic Expansions ◽  
Slider Bearing ◽  
Trailing Edge ◽  
Slider Bearings ◽  
Special Cases

The method of matched asymptotic expansions is used to develop an asymptotic expression for the pressure for large bearing numbers for the case of an infinite slider bearing with a general film thickness that has a discontinuous slope at a point. It is shown that, in addition to the boundary layer of the pressure at the trailing edge, there is also a boundary layer in the derivative of the pressure at the point of discontinuity. The corresponding load formula is also derived. The special cases of the taper-flat and taper-taper slider bearings are discussed.

On the shape of small sessile and pendant drops by singular perturbation techniques

1991 ◽  
Vol 233 ◽  
pp. 519-537 ◽  
Author(s):  
S. B. G. O'Brien
Keyword(s):  
Boundary Layer ◽  
Singular Perturbation ◽  
Asymptotic Expansions ◽  
Numerical Results ◽  
Matched Asymptotic Expansions ◽  
Perturbation Techniques ◽  
Pendant Drop ◽  
Asymptotic Expressions ◽  
Good Agreement ◽  
Pendant Drops

The problem of obtaining asymptotic expressions describing the shape of small sessile and pendant drops is revisited. Both cases display boundary-layer behaviour and the method of matched asymptotic expansions is used to obtain solutions. These give good agreement when compared with numerical results. The sessile solutions are relatively straightforward, while the pendant drop displays a behaviour which is both rich and interesting.

Boundary-Layer Similarity Near the Edge of a Rotating Disk

1975 ◽  
Vol 42 (3) ◽  
pp. 584-590 ◽  
Author(s):  
R. J. Bodonyi ◽  
K. Stewartson
Keyword(s):  
Boundary Layer ◽  
Approximate Solution ◽  
Flow Field ◽  
Boundary Value Problem ◽  
Asymptotic Expansions ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Rotating Disk ◽  
Angular Speed ◽  
Rotating Fluid

Numerical solutions of the similarity equations governing the flow near the edge of a finite rotating disk are found to be possible only for −2.06626 ≤ α ≤ 1, where α is the ratio of the disk’s angular speed to that of the rigidly rotating fluid far from the disk. Furthermore, for α ≤ −1 the solutions of the boundary-value problem are not unique, and along one of the solution branches a singular structure of the flow field is approached as α → −1. Using the method of matched asymptotic expansions an approximate solution is found along the singular branch which explains some of the problems encountered in finding numerical solutions.

Triple-deck solutions for supersonic flows past flared cylinders

1987 ◽  
Vol 179 ◽  
pp. 469-487 ◽  
Author(s):  
Ph. Gittler ◽  
A. Kluwick
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Asymptotic Expansions ◽  
Numerical Solutions ◽  
Reynolds Numbers ◽  
Hysteresis Phenomenon ◽  
Two Dimensional ◽  
Planar Flow ◽  
Laminar Boundary Layers ◽  
External Flows

Using the method of matched asymptotic expansions, the interaction between axisymmetric laminar boundary layers and supersonic external flows is investigated in the limit of large Reynolds numbers. Numerical solutions to the interaction equations are presented for flare angles α that are moderately large. If α > 0 the boundary layer separates upstream of the corner and the formation of a plateau structure similar to the two-dimensional case is observed. In contrast to the case of planar flow, however, separation can occur also if α < 0, owing to the axisymmetric effect of overexpansion and recompression. The separation point then is located downstream of the corner and, most remarkable, a hysteresis phenomenon is observed.

Flow along a horizontal plate near a free surface

1993 ◽  
Vol 252 ◽  
pp. 399-418
Author(s):  
Milan Hofman
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Flat Plate ◽  
Viscous Liquid ◽  
Gravity Waves ◽  
Asymptotic Expansions ◽  
Small Amplitude ◽  
Surface Effect ◽  
Matched Asymptotic Expansions ◽  
Horizontal Plate

The problem of flow along a horizontal semi-infinite flat plate moving in its own plane through a viscous liquid just below the free surface is considered. The method of matched asymptotic expansions is used to analyse the interaction between the free surface and the boundary layer formed on the plate. It is found that, due to viscosity, small-amplitude gravity waves on the free surface can be formed. The formulae for the resistance of the plate containing the free-surface effect and for the lift, appearing as a new phenomenon, are derived.

On the nature of unsteady three-dimensional laminar boundary-layer separation

1978 ◽  
Vol 88 (2) ◽  
pp. 241-258 ◽  
Author(s):  
James C. Williams
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Boundary Layer Separation ◽  
The Body ◽  
Two Dimensional ◽  
Boundary Layer Equations ◽  
Unsteady Separation ◽  
Similar Solutions

Solutions have been obtained for a family of unsteady three-dimensional boundary-layer flows which approach separation as a result of the imposed pressure gradient. These solutions have been obtained in a co-ordinate system which is moving with a constant velocity relative to the body-fixed co-ordinate system. The flows studied are those which are steady in the moving co-ordinate system. The boundary-layer solutions have been obtained in the moving co-ordinate system using the technique of semi-similar solutions. The behaviour of the solutions as separation is approached has been used to infer the physical characteristics of unsteady three-dimensional separation.In the numerical solutions of the three-dimensional unsteady laminar boundary-layer equations, subject to an imposed pressure distribution, the approach to separation is characterized by a rapid increase in the number of iterations required to obtain converged solutions at each station and a corresponding rapid increase in the component of velocity normal to the body surface. The solutions obtained indicate that separation is best observed in a co-ordinate system moving with separation where streamlines turn to form an envelope which is the separation line, as in steady three-dimensional flow, and that this process occurs within the boundary layer (away from the wall) as in the unsteady two-dimensional case. This description of three-dimensional unsteady separation is a generalization of the two-dimensional (Moore-Rott-Sears) model for unsteady separation.

scholarly journals Ionized gas boundary layer on a porous wall of the body within the electroconductive fluid

2004 ◽  
Vol 31 (1) ◽  
pp. 47-71 ◽  
Author(s):  
Branko Obrovic ◽  
Slobodan Savic
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Gas Flow ◽  
Numerical Solutions ◽  
Porous Wall ◽  
The Body ◽  
Generalized Equations ◽  
Ionized Gas ◽  
Boundary Layer Equations ◽  
Longitudinal Coordinate

This paper investigates the ionized gas flow in the boundary layer, when the contour of the body within the fluid is porous. Ionized gas is exposed to the influence of the outer magnetic field induction Bm = Bm(x), which is perpendicular to the contour of the body within the fluid. It is presumed that the electroconductivity of the ionized gas is a function only of the longitudinal coordinate, i.e. ? = ?(x). By means of adequate transformations, the governing boundary layer equations are brought to a generalized form. The obtained generalized equations are solved in a four-parameter localized approximation. Based on the obtained numerical solutions, diagrams of important physical values and characteristics of the boundary layer have been made. Conclusions have also been drawn.

scholarly journals Monin–Obukhov Similarity and Local-Free-Convection Scaling in the Atmospheric Boundary Layer Using Matched Asymptotic Expansions

2018 ◽  
Vol 75 (10) ◽  
pp. 3691-3701 ◽  
Author(s):  
Chenning Tong ◽  
Mengjie Ding
Keyword(s):  
Surface Layer ◽  
Free Convection ◽  
Asymptotic Expansions ◽  
Atmospheric Surface Layer ◽  
Similarity Theory ◽  
Potential Temperature ◽  
Matched Asymptotic Expansions ◽  
Obukhov Length ◽  
Local Free Convection ◽  
Valid Solution

The Monin–Obukhov similarity theory (MOST) is the foundation for understanding the atmospheric surface layer. It hypothesizes that nondimensional surface-layer statistics are functions of [Formula: see text] only, where z and L are the distance from the ground and the Obukhov length, respectively. In particular, it predicts that in the convective surface layer, local free convection (LFC) occurs at heights [Formula: see text] and [Formula: see text], where [Formula: see text] is the inversion height. However, as a hypothesis, MOST is based on phenomenology. In this work we derive MOST and the LFC scaling from the equations for the velocity and potential temperature variances using the method of matched asymptotic expansions. Our analysis shows that the dominance of the buoyancy and shear production in the outer ([Formula: see text]) and inner ([Formula: see text]) layers, respectively, results in a nonuniformly valid solution and a singular perturbation problem and that [Formula: see text] is the thickness of the inner layer. The inner solutions are found to be functions of [Formula: see text] only, providing a proof of MOST for the vertical velocity and potential temperature variances. Matching between the inner and outer solutions results in the LFC scaling. We then obtain the corrections to the LFC scaling near the edges of the LFC region ([Formula: see text] and [Formula: see text]). The nondimensional coefficients in the expansions are determined using measurements. The resulting composite expansions provide unified expressions for the variance profiles in the convective atmospheric surface layer and show very good agreement with the data. This work provides strong analytical support for MOST.

Periodic boundary layer near a two-dimensional stagnation point

1970 ◽  
Vol 43 (3) ◽  
pp. 477-486 ◽  
Author(s):  
Hiroshi Ishigaki
Keyword(s):  
Boundary Layer ◽  
Skin Friction ◽  
Stagnation Point ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
The Body ◽  
Two Dimensional ◽  
Oncoming Flow ◽  
Mean Velocity ◽  
Velocity Amplitude

The time-mean characteristics of the laminar boundary layer near a two-dimensional stagnation point, when the velocity of the oncoming flow relative to the body oscillates are investigated analytically. First, when the amplitude of the oscillating velocity is small compared with the oncoming flow velocity, a series expansion is made and the obtained equations are solved numerically. The equations are also solved approximately in the extreme cases when the frequency is low and high. The obtained approximate solutions are compared with the numerical solutions in terms of skin friction. Next, when the frequency is high, the finite-velocity-amplitude case is treated. Time-mean velocity profiles and skin friction are obtained and compared with the small-amplitude case.

Laboratory-scale swash flows generated by a non-breaking solitary wave on a steep slope

2018 ◽  
Vol 847 ◽  
pp. 186-227 ◽  
Author(s):  
P. Higuera ◽  
P. L.-F. Liu ◽  
C. Lin ◽  
W.-Y. Wong ◽  
M.-J. Kao
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Solitary Wave ◽  
Numerical Simulations ◽  
Hydraulic Jump ◽  
High Speed ◽  
Laboratory Experiments ◽  
Numerical Solutions ◽  
Steep Slope ◽  
Bottom Shear Stress

The main goal of this paper is to provide insights into swash flow dynamics, generated by a non-breaking solitary wave on a steep slope. Both laboratory experiments and numerical simulations are conducted to investigate the details of runup and rundown processes. Special attention is given to the evolution of the bottom boundary layer over the slope in terms of flow separation, vortex formation and the development of a hydraulic jump during the rundown phase. Laboratory experiments were performed to measure the flow velocity fields by means of high-speed particle image velocimetry (HSPIV). Detailed pathline patterns of the swash flows and free-surface profiles were also visualized. Highly resolved computational fluid dynamics (CFD) simulations were carried out. Numerical results are compared with laboratory measurements with a focus on the velocities inside the boundary layer. The overall agreement is excellent during the initial stage of the runup process. However, discrepancies in the model/data comparison grow as time advances because the numerical model does not simulate the shoreline dynamics accurately. Introducing small temporal and spatial shifts in the comparison yields adequate agreement during the entire rundown process. Highly resolved numerical solutions are used to study physical variables that are not measured in laboratory experiments (e.g. pressure field and bottom shear stress). It is shown that the main mechanism for vortex shedding is correlated with the large pressure gradient along the slope as the rundown flow transitions from supercritical to subcritical, under the developing hydraulic jump. Furthermore, the bottom shear stress analysis indicates that the largest values occur at the shoreline and that the relatively large bottom shear stress also takes place within the supercritical flow region, being associated with the backwash vortex system rather than the plunging wave. It is clearly demonstrated that the combination of laboratory observations and numerical simulations have indeed provided significant insights into the swash flow processes.

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