Research Article  Open Access
Algebraic Type Approximation to the Blasius Velocity Profile
Abstract
For the Blasius velocity profile we propose a simple algebraic type approximate function which is uniformly accurate over the whole region. Moreover, for further improvement a correction method based on a weight function is introduced. The availability of the proposed method is shown by the result of numerical experiments.
1. Introduction
We consider the wellknown Blasius problem subject to the boundary conditions The socalled Blasius function describes the stream on the boundary layer over a flat plate. There are lots of analytical approximation methods to the Blasius function such as the variational iteration method [1–6], the Adomian decomposition method [7–9], and the homotopy analysis method [10–12]. Recently, spectral methods based on orthogonal functions have been applied in approximation of solutions for the nonlinear boundary value problems like the Blasius problem [13–17]. In addition, numerical solutions of the nonlinear differential equations for the boundary layer problems such as FalknerSkan equations, including the Blasius equation as a special case, have been studied by many researchers [18–25].
Concerning the streamwise velocity profile , we note an approximate analytical solution proposed in the literature [26] of the form where and are determined by the known properties of the Blasius function at the wall and far from the wall, respectively. The parameter is chosen by minimizing the residual function Recently, Savaş [27] introduced another approximate analytical solution for the streamwise velocity profile as for the constants or .
In the next section, motivated by the analytical solutions (3) and (5), we propose another algebraic type approximate analytical solution for the velocity profile as given by (6) and explore its properties with a method to determine the parameters therein. In Section 3, by using an appropriate weight function, we introduce a correction method to improve the accuracy of the presented approximation. Moreover, for further improvement we employ an auxiliary term which appropriately reflects the error of the presented approximation. Some numerical experiments are performed to demonstrate the efficiency of the presented method.
2. Approximation to the Velocity Profile
To approximate the velocity profile directly we suggest an algebraic type analytical function as for a constant and an exponent . We note that satisfies the boundary conditions and given in (2), and its derivative is Since , we may set which is a wellknown Blasius constant [28]. In addition, the velocity profile has an inversion of a simple form as
The related approximation to the Blasius stream function can be obtained by the formula In fact, using the symbolic computational software Mathematica (version 9), one can find the analytical form of as where is the hypergeometric function [29] whose series expansion is and is the shifted factorial defined by with .
For an appropriate parameter in we may choose a value at which the norm of the residual function , is minimized. To find one can use a package, Mathematica, for example, and we will obtain the local minimum in at the value .
Figure 1 shows the errors of the presented approximate velocity profile, , with integers and near the value . The error means difference between and the numerical solution for the velocity profile which is regarded as an exact solution. By numerical experiments for various values of , we can see that the accuracy of becomes better far from the wall as goes large while it becomes better near as goes small.
3. Improvement by a Weighted Average
In order to improve the accuracy of the proposed approximate velocity profile over the whole region, we introduce a weighted average for , where is a weight function defined as for and . It follows that for with . Moreover, it should be noticed that for a sufficiently large and thus This implies that the point plays the role of a threshold between two approximate velocity profiles and . On the other hand, the related approximate stream function can be obtained by numerical integration in the equation
Referring to Figure 1 for the cases of and , we may take in (16) which is a center of the points and at which and , respectively, have the maximum absolute errors. Thick lines in Figure 2 indicate errors (i.e., differences from the numerical solution) of the corrected approximate stream function and the velocity profile with and in the weight function . We can see that the maximum error is about 0.01 in the velocity profile and about 0.02 in the stream function. Comparing these with the errors of the approximate stream functions and in Figure 2(a) and the velocity profiles and in Figure 2(b), one can find distinct improvement of the corrected velocity profile defined in (15).
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(b)
In practice, by numerical experiments, we can find better case of parameters like , for example, which results in more accurate approximation with the maximum errors about 0.003 and 0.005 in the velocity profile and the stream function, respectively. However, this choice of the parameters looks rather ambiguous. Thus, for development of plausible further improvement, we refer to the correction method proposed in the literature [30] which uses an auxiliary term reflecting the error of the presented approximation. First, observing the behavior of the error given in Figure 2, for example, we can have the numerical values of the critical points and of . Then, to approximate appropriately, we suggest a function of the form where The value of in (20) can be determined by the condition which implies that We consider a corrected approximation One may expect that the accuracy of goes higher as becomes closer to the error .
For example, for the case of , , we can evaluate numerical values of the constants , , , and as given in Table 1. Figure 3 shows errors of and the velocity profile indicated by thick lines, compared with those of and indicated by thin lines, for the parameters . Additionally, dashed lines indicate Savas's approximations and . We can find that and with have the maximum errors about 0.002 and 0.005, respectively. This implies that the correction method (23) can highly improve the proposed method (15) as a result.

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Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF2013R1A1A4A03005079).
References
 J. H. He, “Approximate analytical solution of Blasius' equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 3, no. 4, pp. 260–263, 1998. View at: Publisher Site  Google Scholar
 J. H. He, “A review on some new recently developed nonlinear analytical techniques,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 1, no. 1, pp. 51–70, 2000. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. H. He, “A simple perturbation approach to Blasius equation,” Applied Mathematics and Computation, vol. 140, no. 23, pp. 217–222, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. Lin, “A new approximate iteration solution of Blasius equation,” Communications in Nonlinear Science & Numerical Simulation, vol. 4, no. 2, pp. 91–99, 1999. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. Y. Parlange, R. D. Braddock, and G. Sander, “Analytical approximations to the solution of the Blasius equation,” Acta Mechanica, vol. 38, no. 12, pp. 119–125, 1981. View at: Publisher Site  Google Scholar
 A.M. Wazwaz, “The variational iteration method for solving two forms of Blasius equation on a halfinfinite domain,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 485–491, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 G. Adomian, “Solution of the ThomasFermi equation,” Applied Mathematics Letters, vol. 11, no. 3, pp. 131–133, 1998. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. Biazar, M. Gholami Porshokuhi, and B. Ghanbari, “Extracting a general iterative method from an Adomian decomposition method and comparing it to the variational iteration method,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 622–628, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 F. M. Allan and M. I. Syam, “On the analytic solutions of the nonhomogeneous Blasius problem,” Journal of Computational and Applied Mathematics, vol. 182, no. 2, pp. 362–371, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S.J. Liao, “A uniformly valid analytic solution of twodimensional viscous flow over a semiinfinite flat plate,” Journal of Fluid Mechanics, vol. 385, pp. 101–128, 1999. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S.J. Liao, “An explicit, totally analytic approximate solution for Blasius' viscous flow problems,” International Journal of NonLinear Mechanics, vol. 34, no. 4, pp. 759–778, 1999. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 W. Bao and J. Shen, “A generalizedLaguerreHermite pseudospectral method for computing symmetric and central vortex states in BoseEinstein condensates,” Journal of Computational Physics, vol. 227, no. 23, pp. 9778–9793, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 K. Parand, M. Shahini, and M. Dehghan, “Rational Legendre pseudospectral approach for solving nonlinear differential equations of LaneEmden type,” Journal of Computational Physics, vol. 228, no. 23, pp. 8830–8840, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 K. Parand, M. Dehghan, A. R. Rezaei, and S. M. Ghaderi, “An approximation algorithm for the solution of the nonlinear LaneEmden type equations arising in astrophysics using Hermite functions collocation method,” Computer Physics Communications, vol. 181, no. 6, pp. 1096–1108, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 K. Parand, M. Dehghan, and A. Taghavi, “Modified generalized Laguerre function tau method for solving laminar viscous flow: the Blasius equation,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20, no. 67, pp. 728–743, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 J. Shen and L.L. Wang, “Some recent advances on spectral methods for unbounded domains,” Communications in Computational Physics, vol. 5, no. 24, pp. 195–241, 2009. View at: Google Scholar  MathSciNet
 S. Abbasbandy, “A numerical solution of Blasius equation by Adomian's decomposition method and comparison with homotopy perturbation method,” Chaos, Solitons and Fractals, vol. 31, no. 1, pp. 257–260, 2007. View at: Publisher Site  Google Scholar
 A. Asaithambi, “Numerical solution of the FalknerSkan equation using piecewise linear functions,” Applied Mathematics and Computation, vol. 159, no. 1, pp. 267–273, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 I. Hashim, “Comments on a new algorithm for solving classical Blassius equation,” Applied Mathematics and Computation, vol. 176, no. 2, pp. 700–703, 2006. View at: Publisher Site  Google Scholar  MathSciNet
 L. Howarth, “On the solution of the laminar boundary layer equations,” Proceedings of the Royal Society of London A, vol. 164, pp. 547–579, 1938. View at: Publisher Site  Google Scholar
 S. J. Liao, “A noniterative numerical approach for the twodimensional viscous flow problems governed by the FalknerSkan equation,” International Journal for Numerical Methods in Fluids, vol. 35, pp. 495–518, 2001. View at: Google Scholar
 C.S. Liu and J.R. Chang, “The Liegroup shooting method for multiplesolutions of FalknerSkan equation under suctioninjection conditions,” International Journal of NonLinear Mechanics, vol. 43, no. 9, pp. 844–851, 2008. View at: Publisher Site  Google Scholar
 L.T. Yu and C.K. Chen, “The solution of the Blasius equation by the differential transformation method,” Mathematical and Computer Modelling, vol. 28, no. 1, pp. 101–111, 1998. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 L. Wang, “A new algorithm for solving classical Blasius equation,” Applied Mathematics and Computation, vol. 157, no. 1, pp. 1–9, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 B. I. Yun, “Approximate analytical solutions using hyperbolic functions for the generalized Blasius problem,” Abstract and Applied Analysis, vol. 2012, Article ID 581453, 10 pages, 2012. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 Ö. Savaş, “An approximate compact analytical expression for the Blasius velocity profile,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 10, pp. 3772–3775, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. P. Boyd, “The Blasius function: computations before computers, the value of tricks, undergraduate projects, and open research problems,” SIAM Review, vol. 50, no. 4, pp. 791–804, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 R. Beals and R. Wong, Special Functions, Cambridge University Press, New York, NY, USA, 1st edition, 2010. View at: MathSciNet
 B. I. Yun, “Constructing uniform approximate analytical solutions for the Blasius problem,” Abstract and Applied Analysis, vol. 2014, Article ID 495734, 6 pages, 2014. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2014 Beong In Yun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.