**1. Introduction**

The advantage of the parietal suction is to delay the transition of the boundary layer to turbulence, reduce drag, and increase lift (avoid stalling) [1]. A method used in aviation is the multi-perforation of the walls. This type of boundary layer control is at the project stage on the wings of airplanes, such as the F-16XL and the A320, and starts to emerge in other industries. Equally, this technic is of interest in different engineering branches as the extraction of geothermal energy, nuclear reactor or electronics cooling system, filtration process, lubrication of ceramic machine parts, etc.

The theoretical and numerical study of laminar boundary layer over a flat plate without and/or with uniform suction is well introduced in the literature [2–8]. The modeling of the effect of this control technics can require analytical expressions of characteristic parameters of flow. Equally, theoretical investigations help understand the underlying physics of boundary layer suction and help to predict, with a minimum waste of time, its effect.

For the homogeneous suction, Schlichting and Bussmann [9] assumed that the longitudinal velocity gradient is null. Based on this hypothesis, he found an asymptotic solution which is expressed by the exponential function, but this profile is not valid in the region near the leading edge. To improve this solution, Preston [10] considered a family of parameters of velocity profiles having the Blasius profile and the asymptotic profile as a limit form. The solution obtained is more accurate by comparing it with the exact numerical solutions of Iglisch [11]. Palekar and Sarma [2] who applied the Bianchini approximate integral method [12] determined an analytical solution of the boundary layer profile expressed by the Error function in the case of nonuniform suction. This law was not compared with the real profile of Blasius. In the case of the uniform suction, the solution was obtained in an asymptotic form.

**2. Theoretical study**

*DOI: http://dx.doi.org/10.5772/intechopen.88637*

**2.1 Mathematical formulation**

The boundary conditions are:

(4) gives the following integral equation:

*U*2 ∞ *d dx* ð∞ 0

profile speed the following form:

**Figure 1.**

**123**

*The problem schematic.*

(**Figure 1**).

In this section, we have presented a new analytical approximation of boundary

We consider a horizontal flat plate placed in incompressible, two-dimensional, steady, and laminar flow with free-stream velocity *U* . The *x*-coordinate and *y*coordinate are measured from the leading edge and normal to the flat plate, respectively. In the case of permeable flat, the suction velocity *vp* is oriented to *y*-negative

> *∂*2 *u*

The integration of Eqs. (1) and (2) from 0 to ∞ with the conditions Eqs. (3) and

� �*dy* � *<sup>U</sup>*∞*vp* <sup>¼</sup> *<sup>υ</sup> <sup>∂</sup><sup>u</sup>*

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> <sup>0</sup>*,* (1)

*u*ð Þ¼ 0 0; *v*ð Þ¼ 0 *vp* (3) *u*ð Þ¼ ∞ *U*∞; *v*ð Þ¼ ∞ 0 (4)

> *∂y* � �

*y*¼0

*h x*ð Þ � � (6)

(5)

*<sup>∂</sup>y*<sup>2</sup> (2)

layer profiles of flow on the upper side of flat plate without and with uniform suction. Based on the approximate integral method of Bianchini and using Lambert

*Boundary Layer Theory: New Analytical Approximations with Error and Lambert Functions…*

*∂u ∂x* þ *∂v*

*u ∂u ∂x* þ *v ∂u <sup>∂</sup><sup>y</sup>* <sup>¼</sup> *<sup>υ</sup>*

*u U*<sup>∞</sup> <sup>1</sup> � *<sup>u</sup> U*<sup>∞</sup>

*u U*<sup>∞</sup>

The basic assumption of the integral method of Bianchini is to pose for the

<sup>¼</sup> *erf <sup>y</sup>*

and/or Error functions, we have achieved this solution.

Prandtl equations in the boundary layer are:

Kay [13] took velocity measurements out of a blower up a flat plate with uniform suction. The vertical velocity distribution was described by the exponential function. Aydin and Kaya [4] have considered finite difference approximations to resolve the boundary layer equations. Fang et al. [5] studied a similarity equation of the momentum boundary layer for a moving flat plate in a stationary fluid with mass suction at the wall surface. They provided a new solution branch for the Blasius equation. Recently, researchers have studied convergent and closed analytical solution of the Blasius Equation [14–16]. Wedin et al. [17, 18] have studied the effect of plate permeability on nonlinear stability of the asymptotic suction boundary layer. Zheng et al. [8] have proposed a solution of the Blasius equation expressed by two power series. They showed that the method for finding the closed analytical solution of Blasius equation was used in the regulation of the boundary layer injection and slip velocity.

The study of the flow over a flat plate requires a suitable geometry to avoid the transition to turbulence for low Reynolds numbers. Roach and Brierley [19] studied the flow over a flat plate with cylindrical leading edge of 2 mm diameter, tapered to 5° from the upper surface, to avoid instabilities and separation. Palikaras et al. [20] experimentally and numerically studied the effect of the semicircular leading edge on the transition laminar-turbulent from a flow on a flat plate without pressure gradient. They showed that this transition occurs in the presence of a pressure gradient in the region of the leading edge, resulting in the formation of the laminar bulb of separation. Several configurations were designed to avoid this phenomenon [21–26]. To avoid the influence of this disturbance, Walsh et al. [23] have designed and realized preliminary measurements from a new flat plate facility for aerodynamic research. This flat plate was consisted of a leading edge radius of 1 mm with a 5° chamfer in the intrados and adjustable positive or negative trailing edge flap deflections. The plate was made wof aluminum with 10 mm thick, 1 m long, and 0.29 m wide. Patten et al. [26] studied the design effectiveness of a new flat plate with trailing edge flap. They showed that the stagnation point anchored on the upper surface and the measurements along the flat plate were compared favorably to the Blasius profile.

This paper presents the steps to develop new laws of boundary layer profiles on a horizontal flat plate with uniform suction even in the impermeable case. In the first section, we analytically resolved the governing equations by using the integral method of Bianchini and by inserting particular solutions as Error and Lambert functions. Next, we numerically studied, by using CFD Fluent, the effects of the geometrical parameters of the flat plate (leading edge, trailing edge flap angle) on the boundary layer flow. Finally, the analytical solutions of boundary layer equations were validated with the present numerical results and the literature results in all cases with and without suctions.

*Boundary Layer Theory: New Analytical Approximations with Error and Lambert Functions… DOI: http://dx.doi.org/10.5772/intechopen.88637*
