Self-similar turbulent boundary layer in pressure gradient

Self-similar flows in a turbulent boundary layer when the free-stream velocity is specified as a power function of longitudinal coordinate are investigated. The self-similar formulation not only simplifies solving of the problem by reducing the equations of motion to ordinary differential equations but also provides a mean for formulating closure conditions. It is shown that for the class of flows under consideration that depend on three governing parameters the dimensionless mixing length is a function of the normalised distance from the wall and the exponent in the law specifying the free-stream velocity distribution in the outer region and a universal function of local Reynolds number in the wall region, the latter corollary being true even when the skin friction vanishes. In calculations this function is set to be independent of pressure gradient, which gives the results very close to experimental data. There exist four different self-similar flow regimes. Each regime is related to its similarity parameter, one of which is the well-known Clauser equilibrium parameter and the other three are established for the first time. In case of adverse pressure gradient when the exponent lies within certain limits, which depend on Reynolds number, the problem has two solutions with different values of the boundary layer thickness and skin friction, which points out the possibility of hysteresis in near-separating flow. Separation occurs not at the minimum value of the exponent that corresponds to the strongest adverse pressure gradient but at a higher one whose dependence on Reynolds number is calculated in the paper. The results of the theory are in good agreement with experimental data. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solutions of thermal boundary layer equations when temperature gradient at the moving flat plate in parabolic flow is prescribed

Solutions of steady as well as unsteady three-dimensional incompressible thermal boundary layer equations are studied when the temperature gradient at the moving flat plate in parabolic flow is prescribed. A general analysis is made and different cases are studied by giving values to β and Cx which determine the gradient and curvature of the outer flow steam lines. The components of velocity in boundary layer are discussed by Sarma and Gupta and those results are used to analyse the thermal boundary layer equations. The response of temperature of the plate are studied for large and small times and curves are drawn representing the variation of temperature with time for various cases. The limiting time is also calculated.     

A note on the ill-posedness of shear flow for the MHD boundary layer equations

For the two-dimensional Magnetohydrodynamics(MHD)boundary layer system,it has been shown that the non-degenerate tangential magnetic field leads to the well-posedness in Sobolev spaces and high Reynolds number limits without any monotonicity condition on the velocity field in our previous works.This paper aims to show that sufficient degeneracy in the tangential magnetic field at a non-degenerate critical point of the tangential velocity field of shear flow indeed yields instability as for the classical Prandtl equations without magnetic field studied by G′erard-Varet and Dormy(2010).This partially shows the necessity of the non-degeneracy in the tangential magnetic field for the stability of the boundary layer of MHD in 2D at least in Sobolev spaces.     

Solutions to fuzzy integral equations with arbitrary kernels

Using a parametric Riemann integral representation, a numerical algorithm for solving fuzzy Fredholm and Voltera integral equations of the second kind with arbitrary kernel is proposed and illustrated with examples.     

Effect of pressure gradient on coherent structures in a turbulent boundary layer

By using the idea of resonant triad of the theory of hydrodynamic stability, the effect of pressure gradient on coherent structures in a turbulent boundary layer is investigated. The favorable pressure gradient suppresses the generation of the coherent structure, while the adverse pressure gradient has the opposite effect. The scale, form, as well as the propagation speed of the coherent structures are different from those with zero pressure gradient. The theoretical results are, in general, in agreement with those found from experiments. From the calculated probability density curve of the circulation differences of the nearly streamwise vortex pairs, it is found that the adverse pressure gradient makes the vortex pair more symmetric. Project supported by the National Natural Science Foundation of China.     

Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions

Existence theory is developed for the equation ?(u)=F(u), where ? is a formally self-adjoint singular second-order differential expression and F is nonlinear. The problem is treated in a Hilbert space and we do not require the operators induced by ? to have completely continuous resolvents. Nonlinear boundary conditions are allowed. Also, F is assumed to be weakly continuous and monotone at one point. Boundary behavior of functions associated with the domains of definitions of the operators associated with ? in the singular case is investigated. A special class of self-adjoint operators associated with ? is obtained.     

Some Fredholm equations with boundary layer resonance

Certain singularly perturbed differential equations which exhibit boundary layer resonance are difficult to solve by the application of standard asymptotic methods. After reformulation as a singularly perturbed integral equation and treatment by a recently developed asymptotic methodology, the desired solution is obtained in a straightforward manner.     

Scaling laws for turbulent boundary layer flows in favorable and adverse pressure gradient

An analytical theory is proposed to describe incompressible plane and axisymmetric turbulent boundary layer flows in favorable and adverse pressure gradients for near-equilibrium conditions. Scaling laws for the mean velocity, the Reynolds stress components, and the skin friction have been established. A universal friction law makes it possible to represent the skin friction distributions corresponding to different Reynolds numbers and pressure gradients in terms of a function of one variable. The theory is based on general physical assumptions and does not involve any special hypotheses on the nature of the turbulent motion. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic theory of perturbations inducing a pressure gradient in a transonic flat-plate boundary layer

The role of asymptotic approaches to the study of viscous-inviscid interaction mechanisms in transonic outer flows is discussed. It is noted that there are several versions of multideck asymptotic constructions describing the self-induced pressure effect in transonic boundary layers. The asymptotic theory is used to uncover the internal structure of fluctuation fields, to treat instability-generating processes, and to analyze the behavioral features of linear and nonlinear wave fluctuations. Additionally, the properties of the eigenspectrum are described.     

The effect of suction on boundary layer for rotating flows with or without magnetic field

The effect of suction on the steady laminar incompressible boundarylayer flow for a stationary infinite disc with or without magnetic field, when the fluid at a large distance from the surface of the disc undergoes a solid body rotation, has been studied. The governing coupled nonlinear equations have been solved numerically using the shooting method with least square convergence criterion. It has been found that suction tends to reduce the velocity overshoot and damp the oscillation.     

Analysis of turbulent boundary layers with zero pressure gradient

Zusammenfassung Das hier entwickelte, formale, mathematische System beschreibt turbulente Grenzschichten an glatten und rauhen Platten und vermittelt eine zusammenhängende und folgerichtige Behandlung von solch groben Schichteigenschaften wie Durchschnittsgeschwindigkeitsprofil, Schubspannung, Wirbelviskosität und Grenzschichtdicke. Die erhaltenen Lösungen zeigen gute Übereinstimmung mit den Versuchsergebnissen. Die gegebene Formulierung kann in gewisser Hinsicht als eine neuartige Mischungsweghypothese betrachtet werden.Die Arbeit beschränkt sich auf Grenzschichten mit Nulldruckgradienten. Eine Verallgemeinerung auf von Null abweichende Druckgradienten ist einer späteren Mitteilung vorbehalten, in welcher gezeigt werden soll, dass die benutzte Methode zu mit dem Experiment übereinstimmenden Ergebnisse führt und keiner weiteren Hilfshypothesen bedarf.Die physikalische Bedeutung der Arbeit kann darin gesehen werden, dass ein allerdings nicht ganz genaues Mischungswegmodell dennoch auftretende Mischungsvorgänge näherungsweise beschreiben kann, was somit die Aufstellung einer Mischungsweghypothese ermöglicht, die mit genügender Genauigkeit ein grobes Bild verschiedener Aspekte der beobachteten Vorgänge gibt. Es ist somit möglich, zu einer handlichen und folgerichtigen mathematischen Formulierung zu gelangen, mit deren Hilfe allgemeine Eigenschaften von turbulenten Grenzschichten an festen Oberflächen untersucht werden können.

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Address for 1958: California Institute of Technology.

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Supported in part by a grant from the Rockefeller Foundation RF 56160.     

Flow of a second grade fluid over a sheet stretching with arbitrary velocities subject to a transverse magnetic field

The boundary layer flow of a second grade fluid over a permeable stretching surface with arbitrary velocity and appropriate wall transpiration is investigated. The fluid is electrically conducting in the presence of a constant applied magnetic field. An exact solution to the nonlinear flow problem is presented.     

The boundary layer on a wall nearly parallel to a magnetic field

This paper calculates the structure of the boundary layer which occurs at the wall of a nonconducting duct in fully-developed laminar flow of a conducting fluid when the wall is inclined at a small angle to the uniform transverse magnetic field and when the Hartmann numberM is high. When is large compared withM ^–1/2 the layer splits into a Hartmann layer on the inclined wall and a weak shear layer emanating from the duct corner along field lines. These two layers cause equal reductions in the total flow rate.

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Zusammenfassung Diese Arbeit behandelt die Struktur der Grenzschicht, die an der Wand eines nicht leitfähigen Kanals entsteht die eine vollentwickelte Laminarströmung einer leitfähigen Flüssigkeit führt, wobei die Wand einen kleinen Winkel mit dem gleichförmigen transversalen Magnetfeld bildet und die Hartmannsche ZahlM groß ist. Für große Werte von im Vergleich zuM ^–1/2 teilt sich die Grenzschicht in einen Hartmannsche Schicht an der geneigten Wand und eine schwache Scherschicht, die von der Ecke des Kanals entlang der Feldlinien ausgeht. Die beiden Schichten verursachen die gleichen Verminderungen in der Durchflußmenge.

     

Transition of MHD boundary layer flow past a stretching sheet

In this paper a study is carried out to understand the transition effect of boundary layer flow: (1) due to a suddenly imposed magnetic field over a viscous flow past a stretching sheet and (2) due to sudden withdrawal of magnetic field over a viscous flow past a stretching sheet under a magnetic field. In both the cases the sheet stretches linearly along the direction of the fluid flow. Governing equations have been non-dimensionalised and the non-dimensionalised equations have been solved using the implicit finite difference method of Crank–Nicholson type. Comparison between the steady state exact solutions and the steady state computed solutions has been carried out. Graphical representation of the dimensionless horizontal velocity, vertical velocity and local skin friction profiles of the steady state and unsteady state has been presented. Computation has been carried out for various values of the magnetic parameter M. The obtained results has been interpreted and discussed.     

Minimum error solutions of boundary layer equations

The minimum error solutions of boundary layer equations in the least square sense have been studied by employing the Euler-Lagrange equations. To test the method a class of problems,i.e., boundary layer on a flat plate, Hiemenz flow, boundary layer on a moving sheet and boundary layer in non-Newtonian fluids have been studied. The comparison of the results with approximate methods, like Karman-Pohlhuasen, local potential and other variational methods, shows that the present predictions are invariably better.