Falkner Skan Research Articles

Stochastic numerical treatment for solving Falkner–Skan equations using feedforward neural networks

In this article, the artificial intelligence techniques have been used for the solution of Falkner–Skan (FS) equations based on neural networks optimized with three methods including active set technique, sequential quadratic programming and genetic algorithms (GA) hybridization. Log-sigmoid activation function is used in artificial neural network architecture. The proposed techniques are applied to a number of cases for Falkner–Skan problems, and results were compared with GA hybrid results in all cases and were found accurate. The level of accuracy is examined through statistical analyses based on a sufficiently large number of independent runs.

Parabolized Stability Equations Code with Automatic Inflow for Swept Wing Transition Analysis

This paper aims at developing a parabolized stability equation transition analysis code coupled to the industrial in-house accurate Reynolds-averaged Navier–Stokes code developed at Israel Aerospace Industries for stability analysis of three-dimensional boundary layers over swept wings. In the first step, the parabolized stability equation derivation in a body-fitted coordinate system is exposed, as well as its classical initialization, using the solution procedure of the global Chebyshev eigenvalue problem for the linear stability equations. This initialization procedure demands large user interactions, and it prevents the parabolized stability equation method to be of practical use in an industrial environment. Therefore, one exposes in the second step a new way to automatically generate the inflow solution of the linear stability equations needed by the parabolized stability equations. This new robust approach paves the way for innovative automatic industrial use of the parabolized stability equation technology without the need to involve the user. This approach supplies the inflow conditions required for the parabolized stability equation marching procedure without the need to solve the standalone linear stability global eigenvalue problem. The solution of the linear stability equations at initialization is now obtained by iteratively converging the local linear stability solver starting from the new inflow empirical approximation obtained in two special coordinate systems for crossflow and Tollmien–Schlichting, respectively. These formulas represent some invariant functions of the stability characteristics of the exact Navier–Stokes laminar boundary-layer profiles instead of self-similar boundary-layer profiles like the Falkner–Skan–Cooke. Standard academic validation test cases and industrial results of the parabolized stability equation/Reynolds-average Navier-Stokes codes on a generic unmanned aerial vehicle swept wing are exposed.

Solution of the Falkner-Skan wedge flow by a revised optimal homotopy asymptotic method.

In this paper, a revised optimal homotopy asymptotic method (OHAM) is applied to derive an explicit analytical solution of the Falkner–Skan wedge flow problem. The comparisons between the present study with the numerical solutions using (fourth order Runge–Kutta) scheme and with analytical solution using HPM-Padé of order [4/4] and order [13/13] show that the revised form of OHAM is an extremely effective analytical technique.

Mixed convection in a Falkner–Skan system

The effects of mixed convection on the classical Falkner–Skan similarity solutions are considered, now involving a mixed convection parameter $$\lambda $$ as well as the exponent m associated with the outer flow. The forced convection solutions indicate a singularity in the temperature field as $$m \rightarrow 0.070722$$ . Numerical solutions for $$m>0$$ show the existence of a critical value $$\lambda _\mathrm{c}$$ with $$\lambda _\mathrm{c}<0$$ and solutions only for $$\lambda \ge \lambda _\mathrm{c}$$ . The nature of the solution for $$\lambda \gg 1$$ is investigated. For $$m=0$$ , there are solutions for all $$\lambda <0$$ , opposing flow, and only for a finite range of $$\lambda $$ in aiding flow with the asymptotic solution as $$\lambda \rightarrow -\infty $$ also being considered. Solutions for $$m<0$$ are obtained in the cases when there is a solution to the Falkner–Skan system and for a value of m when no solution to this system exists. In the former case, two completely separate parts to the solution are seen, whereas in the latter case, a solution exists only in aiding flow for a limited range of $$\lambda $$ . The variation of solution with the exponent m is also treated for both aiding and opposing flows. In both cases, a solution is seen to exist for all $$m>0$$ , which, however, is limited to a relatively small range of m when $$m<0$$ .

Combined effects of prescribed pressure gradient and buoyancy in boundary layer of turbulent Rayleigh–Bénard convection

Boundary layers of the velocity and temperature fields are essential for the transport of heat and momentum in turbulent Rayleigh–Bénard convection. Here we study the combined effects of a prescribed pressure gradient and buoyancy for a boundary layer flow by an extension of the two-dimensional, steady Falkner–Skan model. The resulting model, which couples the evolution of the velocity field to that of the temperature field, is used for a comparison with the dynamics in the vicinity of the isothermal bottom plate in a fully turbulent Rayleigh–Bénard convection flow in a closed cylindrical cell. The obtained set of boundary layer equations is solved numerically and the results are compared with direct numerical simulation data at a Rayleigh number Ra=3×109 and for Prandtl numbers Pr=0.7 and Pr=7. The additional buoyancy effects improve the agreement with the data for the lower of the two Prandtl numbers. The improvements remain however very small for the larger Prandtl number. In the latter case, the simulation data show a much weaker coherence and strength of the large-scale circulation. Pressure gradients are then generated by local impacts of colder fluid at the bottom rather than by a coherent circulation which fills the whole convection cell.

Impact of melting phenomenon in the Falkner–Skan wedge flow of second grade nanofluid: A revised model

This article investigates the magnetohydrodynamic (MHD) Falkner–Skan flow of a second grade nanofluid. The flow is caused by a stretching wedge with melting heat transfer and heat generation/absorption. A system of ordinary differential equations is obtained by using suitable transformations. Convergent series solutions are derived. The influence of various pertinent parameters on the velocity, temperature and concentration is evaluated. Analysis of the obtained results shows that fluid flow enhances with the increase of wedge and second grade fluid parameters. Also thermophoresis and Brownian motion parameters have a reverse behavior on the temperature and concentration fields.

Exact Multiple Solutions for the Slip Flow and Heat Transfer in a Converging Channel

A special case of Falkner–Skan flows past stretching boundaries is considered when the momentum and thermal slip boundary conditions are allowed at the boundary. Exact analytical solutions are found for the converging channel (wedge nozzle). The solutions are shown to be unique, double, or triple depending on the slip parameter and wall moving parameter. The provided closed-form analytical solutions are rare class of exact solutions for the Falkner–Skan flow equations. Thresholds of existence of multiple solutions are determined. For each flow solutions, the corresponding energy equation is also exactly solved when the internal heat generated by viscous dissipation can be neglected or numerically integrated when the viscous dissipation is significant. Analytic and numeric values of the rate of heat transfer affected by the presence of a surface temperature jump are also worked out. The possibility of realistic physical solution out of multiple solutions is finally discussed.

Enhancement of a Correlation-Based Transition Turbulence Model for Simulating Crossflow Instability

A new laminar–turbulence transition model for capturing three-dimensional boundary layers involving crossflow-induced transition has been developed by extending the existing transition model. In the present transition model, to resolve transition due to crossflow instability, the C1-criterion was evaluated locally using the Falkner–Skan–Cooke velocity profiles. Thus, only local flow quantities are used in determining the onset of three-dimensional transition. As a result, the extended transition model is compatible and works well with modern computational fluid dyanmics flow solvers, particularly for those based on unstructured meshes, under massive parallel environment. The new transition model was implemented in a three-dimensional incompressible unstructured mesh flow solver. Validations of the new transition model were made for an infinite swept-wing configuration and an inclined prolate spheroid by comparing the results with those of the baseline model and experiment. It was found that the new transition model works well by capturing the crossflow-induced transition inside three-dimensional boundary layers more accurately than the baseline model.

On existence and multiplicity of similarity solutions to a nonlinear differential equation arising in magnetohydrodynamic Falkner–Skan flow for decelerated flows

Previously, existence and uniqueness of a class of monotone similarity solutions for a nonlinear differential equation arising in magnetohydrodynamic Falkner–Skan flow were considered in the case of accelerating flows. It was shown that a solution satisfying certain monotonicity properties exists and is unique for the case of accelerated flows and some decelerated flows. In this paper, we show that solutions to the problem can exist for decelerated flows even when the monotonicity conditions do not hold. In particular, these types of solutions have nonmonotone second derivatives and are, hence, a distinct type of solution from those studied previously. By virtue of this result, the present paper demonstrates the existence of an important type of solution for decelerated flows. Importantly, we show that multiple solutions can exist for the case of strongly decelerated flows, and this occurs because of the fact that the solutions do not satisfy the aforementioned monotonicity requirements. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Application of the HAM-based Mathematica package BVPh 2.0 on MHD Falkner–Skan flow of nano-fluid

Many boundary-layer flows are governed by one or coupled nonlinear ordinary differential equations (ODEs). Currently, a Mathematica package BVPh 2.0 is issued for nonlinear boundary-value/eigenvalue problems with boundary conditions at multiple points. The BVPh 2.0 is based on an analytic approximation method for highly nonlinear problems, namely the homotopy analysis method (HAM), and is free available online. In this paper, the BVPh 2.0 is successfully applied to solve magnetohydrodynamic (MHD) Falkner–Skan flow of nano-fluid past a fixed wedge in a semi-infinite domain, and the influence of physical parameters on the considered flows is investigated in details. Physically, this work deepens and enriches our understandings about the magnetohydrodynamic Falkner–Skan flows of nano-fluid past a wedge. Mathematically, it illustrates the potential and validity of the BVPh 2.0 for complicated boundary-layer flows.

Global Stability Analysis of a Roughness Wake in a Falkner–Skan–Cooke Boundary Layer

A global stability analysis of a Falkner–Skan–Cooke boundary layer with distributed three-dimensional surface roughness is per- formed using high-order direct numerical simulations. Computations have been performed for different sizes of the roughness elements, and a time-stepping method has been used to find the instability modes. The study shows that a critical roughness height beyond which a global instability is excited does exist. Furthermore, the origins of this instability is examined by means of an energy analysis, which reveals the production and dissipation terms responsible for the instability, as well as the region in space where the instability originates.

Slip flow and heat transfer over a specific wedge: an exactly solvable Falkner–Skan equation

The traditional Falkner–Skan boundary layer equation is revisited in this paper with the aim of obtaining a further analytic solution. When a free stream has a certain form, an exact possible solution is already known for a moving permeable wall. The purpose here is to extend this case to the condition where a velocity slip at the surface of a wedge is in contact with the flow. Such a mechanism is found to enrich the physical properties of both the momentum and thermal boundary layers resulting from the Falkner–Skan equation. It is shown that above certain critical values of prescribed physical parameters, coexisting flow solutions exist. These solutions are later fed into the energy equation to derive coexisting exact analytic temperature fields largely influenced by the presence of a wall temperature jump parameter. The wall shear stress and heat transfer rate are also obtained in closed-form formulas ready to serve engineers working in the field.

Analysis of skin friction in MHD Falker–Skan flow problem

A new, accurate and general formula for evaluating the skin friction parameter in a Magneto-Hydrodynamics (MHD) Falkner–Skan flow over a permeable wall was obtained. The formula gives the value of the skin friction of the problem in an infinite interval (0,∞) for all various values of the parameters involved. Shooting method via Runge–Kutta method for solving two-point boundary value problem in a truncated interval (0,L) is used to compare the results obtained. It was observed that the percentage difference between the two sets of results is very small.

The iterative transformation method for the Sakiadis problem

In a transformation method the numerical solution of a given boundary value problem is obtained by solving one or more related initial value problems. This paper is concerned with the application of the iterative transformation method to the Sakiadis problem. This method is an extension of the Töpfer’s non-iterative algorithm developed as a simple way to solve the celebrated Blasius problem. As shown by this author (Fazio, 1997) the method provides a simple numerical test for the existence and uniqueness of solutions. Here we show how the method can be applied to problems with a homogeneous boundary conditions at infinity and in particular we solve the Sakiadis problem of boundary layer theory. Moreover, we show how to couple our method with Newton’s root-finder. The obtained numerical results compare well with those available in literature. The main aim here is that any method developed for the Blasius, or the Sakiadis, problem might be extended to more challenging or interesting problems. In this context, the iterative transformation method has been recently applied to compute the normal and reverse flow solutions of Stewartson for the Falkner–Skan model (Fazio, 2013).

Numerical and analytical solutions for Falkner–Skan flow of MHD Maxwell fluid

The present research examines the Falkne–Skan flow of magnetohydrodynamic (MHD) Maxwell fluid. Both analytic and numerical solutions are established for the governing problem. The variations of viscoelastic and magnetic parameters are emphasized.

Comment on the paper “Falkner–Skan wedge flow of a power-law fluid with mixed convection and porous medium, by T. Hayat, Majid Hussain, S. Nadeem and S. Mesloub, Computers & Fluids, 49 (2011) 22–28”

Comment on the paper “Falkner–Skan wedge flow of a power-law fluid with mixed convection and porous medium, by T. Hayat, Majid Hussain, S. Nadeem and S. Mesloub, Computers & Fluids, 49 (2011) 22–28”

Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals

The classical numerical treatment of boundary value problems defined on infinite intervals is to replace the boundary conditions at infinity by suitable boundary conditions at a finite point, the so-called truncated boundary. A truncated boundary allowing for a satisfactory accuracy of the numerical solution has to be determined by trial and errors and this seems to be the weakest point of the classical approach. On the other hand, the free boundary approach overcomes the need for a priori definition of the truncated boundary. In fact, in a free boundary formulation the unknown free boundary can be identified with a truncated boundary and being unknown it has to be found as part of the solution.In this paper we consider a different way to overcome the introduction of a truncated boundary, namely non-standard finite difference schemes defined on quasi-uniform grids. A quasi-uniform grid allows us to describe the infinite domain by a finite number of intervals. The last node of such grid is placed on infinity so that right boundary conditions are taken into account exactly. We apply the proposed approach to the Falkner–Skan model and to a problem of interest in foundation engineering. The obtained numerical results are found in good agreement with those available in literature. Moreover, we provide a simple way to improve the accuracy of the numerical results using Richardson’s extrapolation. Finally, we indicate a possible way to extend the proposed approach to boundary value problems defined on the whole real line.

Boundary layer flow and heat transfer of a Casson fluid past a symmetric porous wedge with surface heat flux

The aim of this paper is to investigate numerically the boundary layer forced convection flow of a Casson fluid past a symmetric porous wedge. Similarity transformations are used to convert the governing partial differential equations into ordinary ones. With the help of the shooting method, the reduced equations are then solved numerically. Comparisons are made with the previously published results in some special cases and they are found to be in excellent agreement with each other. The results obtained in this study are illustrated graphically and discussed in detail. The velocity is found to increase with an increasing Falkner–Skan exponent whereas the temperature decreases. With the rise of the Casson fluid parameter, the fluid velocity increases but the temperature is found to decrease in this case. Fluid velocity is suppressed with the increase of suction. The skin friction decreases with the increasing value of Casson fluid parameter. It is found that the temperature decreases as the Prandtl number increases and thermal boundary layer thickness decreases with the increasing value of Prandtl number. A significant finding of this investigation is that flow separation can be controlled by increasing the value of the Casson fluid parameter as well as by increasing the amount of suction.

Numerical solution of the Falkner Skan Equation by using shooting techniques

The aim of the paper is to examine the boundary value problems characterized by the well-known Falkner- Skan Equation.The Falkner -Skan Equation is governed by the third order non liner ordinary differential equation and then to solve numerically using the RungeKutta 4 th order with shooting techniques .and solve the Falkner Skan Equation for different parameters �� and the numerical results are obtain by using Mat lab Software and compare the results of the literature (1),(2),(3).

Second-law analysis of fluid flow over an isothermal moving wedge

In this study, entropy generation minimization (EGM) was employed to optimize fluid flow and heat transfer over a moving wedge. Governing partial differential equations including continuity, momentum and energy are reduced to ordinary ones using similarity variables and solved numerically. The novelty of this study is to consider the effects of the moving wedge parameter λ, to find the stable system via entropy generation minimization (EGM) method. The results indicated that as the slope of the wedge increases, the absolute values of the optimum moving wedge parameter λo grow as well. Moreover, it was found that the minimum value of entropy generation happens for the negative values of λo which gets smaller as Falkner–Skan power law parameter m increases.