**2. Theoretical study**

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 asymp-

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

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

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

totic form.

*Aerodynamics*

injection and slip velocity.

to the Blasius profile.

**122**

all cases with and without suctions.

In this section, we have presented a new analytical approximation of boundary 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 and/or Error functions, we have achieved this solution.

## **2.1 Mathematical formulation**

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 (**Figure 1**).

Prandtl equations in the boundary layer are:

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial v}{\partial \mathbf{y}} = \mathbf{0}, \tag{1}$$

$$
u \frac{\partial u}{\partial \mathbf{x}} + v \frac{\partial u}{\partial \mathbf{y}} = \nu \frac{\partial^2 u}{\partial \mathbf{y}^2} \tag{2}$$

The boundary conditions are:

$$
\mu(\mathbf{0}) = \mathbf{0}; \nu(\mathbf{0}) = \nu\_p \tag{3}
$$

$$u(\infty) = U\_{\infty}; v(\infty) = 0 \tag{4}$$

The integration of Eqs. (1) and (2) from 0 to ∞ with the conditions Eqs. (3) and (4) gives the following integral equation:

$$\int U\_{\infty}^{2} \frac{d}{d\mathbf{x}} \int\_{0}^{\infty} \frac{u}{U\_{\infty}} \left(1 - \frac{u}{U\_{\infty}}\right) d\mathbf{y} - U\_{\infty} v\_{p} = \nu \left(\frac{\partial u}{\partial \mathbf{y}}\right)\_{p=0} \tag{5}$$

The basic assumption of the integral method of Bianchini is to pose for the profile speed the following form:

$$\frac{u}{U\_{\infty}} = \text{erf}\left(\frac{y}{h(\infty)}\right) \tag{6}$$

**Figure 1.** *The problem schematic.*

where *erf <sup>y</sup> h x*ð Þ � � is the Error function and *h x*ð Þ is the unknown scale function to be determined. The choice of the Error function has the advantage of a good approximation of the exact solution of Blasius which will be proven below.

The insertion of Eq. (6) in Eq. (5), with the conditions Eqs. (3) and (4), gives a differential equation of *h*(*x*), with a boundary condition as below:

$$\begin{cases} U\_{\infty} a\_2 \frac{dh(\mathbf{x})}{d\mathbf{x}} - v\_p = \frac{\nu}{h(\mathbf{x})} a\_3 \\ \qquad h(\mathbf{0}) = \mathbf{0} \end{cases} \tag{7}$$

*2.2.2 Case with uniform suction*

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

a function of *x* to determine.

*<sup>A</sup>*<sup>1</sup> <sup>¼</sup> *<sup>A</sup>*<sup>2</sup> <sup>¼</sup> *<sup>b</sup>*<sup>1</sup>

¼ � <sup>2</sup>*<sup>υ</sup>* ffiffiffi *<sup>π</sup>* <sup>p</sup> *<sup>U</sup>*<sup>∞</sup>

> *vp U*<sup>∞</sup>

thickness, and the parietal friction coefficient:

2 6 4

*vp U*<sup>∞</sup> ; *g x*ð Þ¼� <sup>1</sup>

*, b*<sup>2</sup> <sup>¼</sup> <sup>ð</sup><sup>1</sup> <sup>þ</sup> ffiffi

<sup>1</sup> <sup>þ</sup> *<sup>W</sup>* � <sup>1</sup>

0

B@

particular solution:

the scaling function:

where

*<sup>b</sup>*<sup>1</sup> ¼ � *<sup>α</sup>*3*<sup>υ</sup> U*<sup>∞</sup>

> *u U*<sup>∞</sup>

Thus

**125**

plate with uniform suction is

<sup>¼</sup> *erf y<sup>=</sup> <sup>b</sup>*<sup>1</sup>

B@

recommended by Iglisch (1949) [11].

*Cf t*ð Þ 2 *vp U*<sup>∞</sup>

0

We considered the case of a flow on a permeable flat plate with uniform suction *vp* = �*v*<sup>0</sup> 6¼ 0. To simplify the resolution of the differential Eq. (7), we imposed the

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

where *A*<sup>1</sup> and *A*<sup>2</sup> are constant parameters, *W* is the Lambert function, and *g*(*x*) is

By inserting Eq. (14) in the differential Eq. (7), we supplied the parameters of

2 <sup>p</sup> <sup>Þ</sup>*πU*<sup>∞</sup>

Thus, the profile of the boundary layer velocity of the flow on the permeable flat

<sup>2</sup>*<sup>υ</sup>* exp � *<sup>b</sup>*<sup>2</sup>

These results enable us to determine the various characteristics of the boundary layer as the boundary layer thickness, the displacement thickness, the momentum

*<sup>δ</sup>*ð Þ¼ *<sup>x</sup>* <sup>1</sup>*,* <sup>82</sup> *h x*ð Þ; *<sup>δ</sup>*1ð Þ¼ *<sup>x</sup>* <sup>0</sup>*,* <sup>564</sup>*h x*ð Þ; *<sup>δ</sup>*2ð Þ¼ *<sup>x</sup>* <sup>0</sup>*,* <sup>231</sup>*h x*ð Þ;*Cf x*ð ÞRe*<sup>x</sup>* <sup>¼</sup> <sup>4</sup>

We can rewrite this friction coefficient in the universal form of law

¼ � <sup>1</sup> <sup>1</sup> <sup>þ</sup> *<sup>W</sup>* � <sup>1</sup>

<sup>¼</sup> *f t* ¼ � *vp*

*U*<sup>∞</sup>

<sup>2</sup>*<sup>υ</sup>* exp � <sup>1</sup><sup>þ</sup> ffiffi

ffiffiffiffiffiffiffiffi Re*<sup>x</sup>*

2 <sup>p</sup> ð Þ*<sup>π</sup>*

<sup>2</sup> *t*<sup>2</sup> þ *b*<sup>3</sup> � � � � (20)

<sup>p</sup> � � (19)

*Cf t*ð Þ 2 *vp U*<sup>∞</sup>

B@

0

*U*<sup>∞</sup> *vp*

� �<sup>2</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>b</sup>*<sup>3</sup>

<sup>2</sup>*<sup>υ</sup>* exp � *<sup>b</sup>*<sup>2</sup>

B@

0

*h x*ð Þ¼ *A*<sup>1</sup> þ *A*2*Wgx* ð Þ ð Þ (14)

*U*<sup>∞</sup> *vp*

� �<sup>2</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>b</sup>*<sup>3</sup>

<sup>2</sup>*<sup>υ</sup> ,* and *<sup>b</sup>*<sup>3</sup> <sup>¼</sup> ln 2ð Þ� *<sup>υ</sup>* 1 (16)

1

1

3 7 5

1

CA (17)

ffiffiffi *π* p

*x h x*ð Þ (18)

CA

CA

1

CA (15)

where

$$a\_2 = \int\_0^\infty \text{erf}(z)(1 - \text{erf}(z))dz = \frac{\sqrt{2} - 1}{\sqrt{\pi}},\tag{8}$$

$$a\_3 = \left(\frac{d\sigma f}{dz}\right)\_{z=0} = \frac{2}{\sqrt{\pi}}.\tag{9}$$

#### **2.2 Analytical solutions**

The analytical resolution of the differential Eq. (7) depends inevitably on the boundary conditions, in particular, the value of suction velocity *vp*, since we consider two cases: an impermeable flat plate when *vp* = 0 and a porous flat with uniform suction when *vp* = �*v*<sup>0</sup> 6¼ 0.
