Semi-infinite Flat Plate Research Articles

Thermal-Boundary-Layer Response to Convected Far-Field Fluid Temperature Changes

Fluid flows of varying temperature occur in heat exchangers, nuclear reactors, nonsteady-flow devices, and combustion engines, among other applications with heat transfer processes that influence energy conversion efficiency. A general numerical method was developed with the capability to predict the transient laminar thermal-boundary-layer response for similar or nonsimilar flow and thermal behaviors. The method was tested for the step change in the far-field flow temperature of a two-dimensional semi-infinite flat plate with steady hydrodynamic boundary layer and constant wall temperature assumptions. Changes in the magnitude and sign of the fluid-wall temperature difference were considered, including flow with no initial temperature difference and built-up thermal boundary layer. The equations for momentum and energy were solved based on the Keller-box finite-difference method. The accuracy of the method was verified by comparing with related transient solutions, the steady-state solution, and by grid independence tests. The existence of a similarity solution is shown for a step change in the far-field temperature and is verified by the computed general solution. Transient heat transfer correlations are presented, which indicate that both magnitude and direction of heat transfer can be significantly different from predictions by quasisteady models commonly used. The deviation is greater and lasts longer for large Prandtl number fluids.

A note on the unsteady boundary layers over a flat plate

The unsteady boundary layer over a semi-infinite flat plate was investigated in this paper. The flow involves the unsteady flow over a flat plate with leading edge accretion or ablation. The momentum boundary layer was further analyzed and it was shown that the leading edge ablation had a similar effect to the wall mass injection or upstream wall movement making the fluid blown away from the wall. The thermal boundary layer of the same flow was also studied. Results show that the leading edge accretion or ablation can greatly change the fluid motion and the heat transfer characteristics.

Nonlinear free surface flows past a semi-infinite flat plate in water of finite depth

We consider the steady free surface two-dimensional flow past a semi-infinite flat plate in water of a constant finite depth. The fluid is assumed to be inviscid, incompressible and the flow is irrotational; surface tension at the free surface is neglected. Our concern is with the periodic waves generated downstream of the plate edge. These can be characterized by a depth-based Froude number F and the depth d (draft) of the depressed plate. Previous analytical studies have been restricted to small values of d and subcritical flows F<1, when linearized theory can be used. Here, our main concern here is with the critical regime, F≈1, where we use a weakly nonlinear long-wave analysis, For finite values of d, we solve the fully nonlinear problem numerically using a boundary integral equation method. Our results confirm previous studies and extend these to the critical regime. As d increases, our numerical results demonstrate that the downstream waves approach the highest possible wave. We also find some wave-free solutions, which when compared to the weakly nonlinear results are essentially just one-half of a solitary wave.

Authors’ reply to the comments on “Similarity analysis in magnetohydrodynamics: Hall effects on forced convective heat and mass transfer of non–Newtonian power law fluids past a semi-infinite vertical flat plate” by A. Pantokratoras (Acta Mech. 2007)

Authors’ reply to the comments on “Similarity analysis in magnetohydrodynamics: Hall effects on forced convective heat and mass transfer of non–Newtonian power law fluids past a semi-infinite vertical flat plate” by A. Pantokratoras (Acta Mech. 2007)

Comment on “Similarity analysis in magnetohydrodynamics: Hall effects on forced convective heat and mass transfer of non–Newtonian power law fluids past a semi-infinite vertical flat plate” by A.A. Afify, E.M. Aboeldahab and E.S. Mohamed (Acta Mech. 177, 71–87, 2005)

Comment on “Similarity analysis in magnetohydrodynamics: Hall effects on forced convective heat and mass transfer of non–Newtonian power law fluids past a semi-infinite vertical flat plate” by A.A. Afify, E.M. Aboeldahab and E.S. Mohamed (Acta Mech. 177, 71–87, 2005)

The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

The Blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick, semi-infinite flat plate. The Blasius function is the solution to $2 f_{xxx} + f f_{xx} = 0$ on $x \in [0, \infty]$ subject to $f(0)=f_{x}(0)=0, f_{x}(\infty) = 1$. We use this famous problem to illustrate several themes. First, although the flow solves a nonlinear partial differential equation (PDE), Toepfer successfully computed highly accurate numerical solutions in 1912. His secret was to combine a Runge-Kutta method for integrating an ordinary differential equation (ODE) initial value problem with some symmetry principles and similarity reductions, which collapse the PDE system to the ODE shown above. This shows that PDE numerical studies were possible even in the precomputer age. The truth, both a hundred years ago and now, is that mathematical theorems and insights are an arithmurgist's best friend, and they can vastly reduce the computational burden. Second, we show that special tricks, applicable only to a given problem, can be as useful as the broad, general methods that are the fabric of most applied mathematics courses: the importance of “particularity.” In spite of these triumphs, many properties of the Blasius function $f(x)$ are unknown. We give a list of interesting projects for undergraduates and another list of challenging issues for the research mathematician.

Crypto-steady supersonic pressure exchange: A simple analytical model

This paper is motivated by a long-range goal of developing a new class of turbomachine where fluid impellers, created through supersonic wave structures, replace mechanical impellers. This paper is an exploratory study to show how effectively supersonic wave structures can fulfill this goal, and to provide some fundamental insight as to the behavior. A comparison, at a fundamental level, of the spectrum of flow induction devices from mechanical turbomachinery to direct flow induction including steady flow ejectors, wave rotors, and pressure exchange devices is explored. While the authors have been studying various flow induction devices intended to lead to practical solutions, the details of the flow interactions are obscured by geometries and the complex interactions. This paper attempts to take a step backward and look at the simplest conceivable model that demonstrates the phenomena thought to be the key to a new technology. This model is a crypto-steady supersonic pressure exchange process occurring behind a supersonic frictionless semi-infinite flat plate having a pressure differential between both sides of the plate. It is found that very high compressor efficiencies and energy transfer rates are possible even with the presence of supersonic flow structure. Because of the high efficiencies, the analysis shows that there is a potential for the development of a new generation of direct flow induction devices far simpler and more compact than conventional turbomachinery, yet far more efficient than conventional ejectors and wave rotors. Fundamental information is provided which will facilitate exploiting supersonic pressure exchange in practical crypto-steady flow induction devices.

Homotopy analysis of a self-similar boundary-flow driven by a power-law shear

The boundary-layer flow driven over a semi-infinite permeable flat plate by a power-law shear with asymptotic velocity profile \(\hat{U}_\infty(\hat{y})=\hat{\beta} \hat{y}^\alpha\) is studied, where α and β are two constants. By means of the homotopy analysis method, a family of series solutions are obtained explicitly, which agree well with the analytic solutions or the numerical results. Especially, solutions in the range −1 < α < −2/3 seem to be overlooked in all previous publications.

A note on boundary-layer similarity flows driven by a power-law shear over a plane surface

The boundary-layer similarity flow driven over a semi-infinite flat plate by a power-law shear with asymptotic velocity profile u e ( y ) = β y α ( y → ∞ , β > 0 ) is considered. Theoretical analysis is reported to derive a range of exponents α and amplitudes β for which similarity solutions exist. The shear stress f ″ ( 0 ) parameter is determined as a function of α and β .

Far downstream analysis for the Blasius boundary-layer stability problem

In this paper, we examine the large Reynolds number (Re) asymptotic structure of the wave number in the Orr–Sommerfeld region for the Blasius boundary layer on a semi-infinite flat plate given by Goldstein (1983, J. Fluid Mech., 127, 59–81). We show that the inclusion of the term which contains the leading-order non-parallel effects, at O(Re− 1/2), leads to a non-uniform expansion. By considering the far downstream form of each term in the asymptotic expansion, we derive a length scale at which the non-uniformity appears, and compare this position with the position seen in plots of the wave number.

Stationary Flow Past a Semi-Infinite Flat Plate: Analytical and Numerical Evidence for a Symmetry-Breaking Solution

We consider the question of the existence of stationary solutions for the Navier Stokes equations describing the flow of a incompressible fluid past a semi-infinite flat plate at zero incidence angle. By using ideas from the theory of dynamical systems we analyze the vorticity equation for this problem and show that a symmetry-breaking term fits naturally into the downstream asymptotic expansion of a solution. Finally, in order to check that our asymptotic expressions can be completed to a symmetry-breaking solution of the Navier–Stokes equations we solve the problem numerically by using our asymptotic results to prescribe artificial boundary conditions for a sequence of truncated domains. The results of these numerical computations a clearly compatible with the existence of such a solution.

Crocco variable formulation for uniform shear flow over a stretching surface with transpiration: Multiple solutions and stability

The simultaneous effects of transpiration through and tangential movement of a semi-infinite flat plate on the self-similar boundary layer flow driven by uniform shear in the far field is considered. Difficulties with standard shooting techniques are overcome using Crocco variables which also serve to better elucidate the solution structure. The stabilities of dual, triple and even quadruple steady flow solutions encountered in different ranges of plate stretching and wall stress are determined using a linear temporal stability analysis for the self-similar flow.

Asymptotic receptivity analysis and the parabolized stability equation: a combined approach to boundary layer transition

We consider the interaction of free-stream disturbances with the leading edge of a body and its effect on the transition point. We present a method which combines an asymptotic receptivity approach, and a numerical method which marches through the Orr–Sommerfeld region. The asymptotic receptivity analysis produces a three-deck eigensolution which in its far downstream limiting form produces an upstream boundary condition for our numerical parabolized stability equation (PSE). We discuss the advantages of this method compared to existing numerical and asymptotic analysis and present results which justify this method for the case of a semi-infinite flat plate, where asymptotic results exist in the Orr–Sommerfeld region. We also discuss the limitations of the PSE and comment on the validity of the upstream boundary conditions. Good agreement is found between the present results and the numerical results of Haddad &amp; Corke (1998).

Method for analysis of showerhead film cooling experiments on highly curved surfaces

The transient liquid crystal technique has been extensively used for measuring the heat transfer characteristics in gas turbine applications. Thereby, the time evolution of the surface temperature is usually evaluated using the model of a semi-infinite flat plate. For experiments on cylinders, Wagner et al. [G. Wagner, M. Kotulla, P. Ott, B. Weigand, J. von Wolfersdorf, The transient liquid crystal technique: influence of surface curvature and finite wall thickness, ASME Paper GT2004-53553, 2004] showed, that curvature and finite thickness effects can have an influence on the obtained heat transfer coefficients. The aim of this study is to develop a time effective data reduction method that accounts for curvature and that is applicable to film cooling experiments with time varying adiabatic wall temperatures. To verify this method, transient liquid crystal experiments have been carried out on a blunt body model with showerhead film cooling. The experimental data was evaluated with the traditional semi-infinite flat plate approach and with the curvature correction using regression analysis.

Modified rational Legendre approach to laminar viscous flow over a semi-infinite flat plate

A numerical method for solving the classical Blasius’ equation is proposed. The Blasius’ equation is a third order nonlinear ordinary differential equation , which arises in the problem of the two-dimensional laminar viscous flow over a semi-infinite flat plane. The approach is based on a modified rational Legendre tau method. The operational matrices for the derivative and product of the modified rational Legendre functions are presented. These matrices together with the tau method are utilized to reduce the solution of Blasius’ equation to the solution of a system of algebraic equations. A numerical evaluation is included to demonstrate the validity and applicability of the method and a comparison is made with existing results.

Power series solutions of momentum and energy boundary layer equations for a power-law shear driven flow over a semi-infinite flat plate

The boundary layer similarity flow past an impermeable flat plate, driven by a power law velocity profile U = γy α , y → ∞ is considered and power series solutions of the momentum equation, valid for all the allowed range of the parameter α, are presented. The convergence radius of the proposed solutions is estimated and a comparison with numerical solutions is reported. The boundary layer energy equation is then considered for all the wall temperature profiles that admits similarity solutions and power series solutions are given for the full range of the wall temperature profile parameter n.

Effects of radiation and Hall current on MHD free convective heat-generating flow past a semi-infinite vertical flat plate with an aligned magnetic field

Effects of radiation and Hall current on MHD free convective heat-generating flow past a semi-infinite vertical flat plate with an aligned magnetic field

Series solutions of unsteady MHD flow of a non-newtonian fluid near the forward stagnation point

Unsteady flow of a viscous incompressible electrically conducting non-Newtonian power- law fluid near the forward stagnation point of a two-dimensional body in the presence of a magnetic field is studied by means of an analytic technique, namely, the homotopy analysis method. Accurate analytic approximations are obtained, which are uniformly valid for all dimensionless time in the whole region 0 ≤ η< ∞. Effects of integral power- law index of the non-Newtonian fluids for κ =1 , 2, 3 and magnetic number M ≤ 10 on the flow are considered. To the best of authors' knowledge, such kinds of analytic solutions have been never reported. Introduction. Non-Newtonian fluids are very important fluids, which are widely used in industry. Most particulate slurries (china clay and coal in water, sewage sludge, etc.), multiphase mixtures (oil-water emulsions, gas-liquid disper- sions, such as froths and foams, butter) are non-Newtonian fluids. Further exam- ples, including a variety of non-Newtonian characteristics, include pharmaceutical formulations, cosmetics and toiletries, paints, synthetic lubricants, biological fluids (blood, synovial fluid, saliva), and foodstuffs (jams, jellies, soups, marmalades). Indeed, behaviours of the non-Newtonian fluids are so widespread that it would be no exaggeration to say that the Newtonian fluid behaviours are an exception rather than the rule. Since the non-Newtonian fluids have more complicated equa- tions that relate the shear stresses to the velocity field than the Newtonian fluids have, additional factors must be considered in examining various fluid mechanics (see, Irvine and Karni (1)). Several models have been proposed to describe the non-Newtonian behaviour of fluids. Among these models, which are known to follow the empirical Ostwaald- de Waele model, the so-called power-law model, in which the shear stress varies according to a power function of the strain rate, has received wide acceptance. Boundary layer assumptions were successfully applied to this model and much work has been done on it. Schowalter (2) and Acrivos et al. (3) initially theoretically analyzed the steady boundary layer flow of incompressible non-Newtonian power- law fluids and found the existence of the similarity solutions. Kim et al. (4) made a detailed analysis to obtain possible similarity solutions of the steady boundary layer equations of a non-Newtonian fluid. Yukselen and Erim (5) considered the curvature effects on the boundary layer of a non-Newtonian fluid. Thompson and Snyder (6), and Kim and Eraslan (7) investigated the effect of wall mass injection on the flow of a non-Newtonian power-law fluid over a flat plate, near a stagnation point and past a wedge. Akcay and Yukselen (8) analyzed the boundary layer of a non-Newtonian fluid flow with fluid injection on a semi-infinite flat plate, which moves with a constant velocity in the direction opposite to that of the

On the three-dimensional alternative to the Blasius boundary-layer solution

Theoretical investigations have shown that there is a three-dimensional alternative to the Blasius solution for laminar boundary layer flow past a semi-infinite flat plate at zero incident. In this Note we show numerically that there exists a class of three-dimensional solutions confined to almost zero incident conditions. To cite this article: A. Ridha, C. R. Mecanique 333 (2005).

Periodic heat transfer by forced laminar boundary layer flow over a semi-infinite flat plate

The paper reports a study of periodic convection in a steady forced laminar boundary layer flow over a semi-infinite impermeable flat plate due to periodical variation of the wall heat flux. The Fourier transform based approach allows to obtain a transfer function for the boundary layer that can be used to solve also transient (non-periodic) heating problems, and examples are reported comparing with available studies in the open literature. The effect of periodic heating on the value of the the average heat transfer coefficient is analysed and it is found to be important for relatively high frequency fluctuations of the imposed heat flux, whereas fluctuation amplitude of the instantaneous heat transfer coefficient is non-negligible also for lower exciting frequency.