**3.1 CFD study of the boundary layer**

Theoretically, the boundary layer equations were studied with hypothesis of zero pressure gradient flow. In order to validate and compare the numerical results with the new boundary layer laws, it is essential to avoid separation and instability of the flow. A critical part of the experiment that must be addressed is the leading edge of the flat plate. The boundary layer development is considerably influenced by the stagnation point location. A flat plate equipped with adaptable trailing edge flap ensures that the boundary layer is developed smoothly and a negligible stream-wise pressure gradient is achievable. So, the laminar separation bulb, which is one of the phenomena at the origin of the transition to turbulence, is avoided [23, 24, 26].

For this reason, we used a flat plate with a semicircular leading edge and provided with trailing edge flap deflected to an angle *β* = 40°. Its length and thickness are *L* = 0.9 m and *e* = 0.01 m, respectively, the flap chord l = 0.1 m. The leading edge is chamfered at an angle of 5° relative to the lower surface, and its diameter is *d* = 0.002 m (see **Figure 2**). This is similar to the flat plate designed by Patten et al. [26] for boundary layer research.

A judicious choice of the suction system is required to control the laminar flow [1, 27–29]. The suction area is placed in 0.1 m from the leading edge and extends to 0.7 m. It consists of holes of the same diameter *dsuc* = 0.2 mm and equidistant from 1 mm (**Figure 2**). This corresponds to a longitudinal dimensionless spacing *dp*/*dsuc* = 5 (where *dp* is the spacing between two successive holes), respecting the mechanical strength of the plate [30].

The quality of mesh has a great importance on the results of a numerical resolution. For this, we choose a structured and very tight mesh in the near-wall and in the area of the leading edge (Y+ = 1). This computational domain is made up of 53.970 cells in the case of the impermeable plate and 100.000 cells in the case of the

**Figure 2.** *Design characteristics of the flat plate.*

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

permeable plate, because of the mesh refinement above the suction zone to account for the velocity gradient. The Navier-Stokes equations for 2D, laminar, and incompressible flow were resolved by using the finite volume method (FVM). We used the algorithm "SIMPLE" for the pressure-velocity coupling.

In the case without suction, we studied the flow with free-steam velocity *U* = 5 m/s around the flat plate to compare their boundary layers with Blasius profiles. **Figure 3** shows the effect of the trailing edge flap angle on the position of the stagnation point. For *β* = 40°, the stagnation point is displaced to the upper surface of plate resulting in the reduction of the separation flow at the leading edge compared to the case of *β* = 0°. This result is compared to those obtained in the literature [26].

In **Figure 4**, we compared the Blasius profiles with the results from the CFD of the flow boundary layer on the upper side of the impermeable flat plate for *β* = 40°. It is shown that the boundary layer of the flat plate at different positions favorably follows the Blasius profile. Thus, the shapes of the leading edge and the deflected trailing edge have an effect to neglect the pressure gradient in the flow of the upper side of the plate which greatly influences the formation of the boundary layer. In the continuation of this work, we select the case of the trailing edge deflected to 40°.

#### **3.2 Validation and discussion**

Many solutions were found based on the Prandtl equations such as Blasius and Schlichting [9] profiles. The differential equations of Blasius have no solutions for

#### **Figure 3.**

**3. Validations of the new boundary layer theories**

CFD using the software package Fluent.

**3.1 CFD study of the boundary layer**

is avoided [23, 24, 26].

*Aerodynamics*

**Figure 2.**

**126**

*Design characteristics of the flat plate.*

[26] for boundary layer research.

mechanical strength of the plate [30].

In order to validate the new laws of boundary layer without and with suction, we studied the 2D, laminar, and incompressible flow around the flat plate by means of

Theoretically, the boundary layer equations were studied with hypothesis of zero pressure gradient flow. In order to validate and compare the numerical results with the new boundary layer laws, it is essential to avoid separation and instability of the flow. A critical part of the experiment that must be addressed is the leading edge of the flat plate. The boundary layer development is considerably influenced by the stagnation point location. A flat plate equipped with adaptable trailing edge flap ensures that the boundary layer is developed smoothly and a negligible stream-wise pressure gradient is achievable. So, the laminar separation bulb, which is one of the phenomena at the origin of the transition to turbulence,

For this reason, we used a flat plate with a semicircular leading edge and provided with trailing edge flap deflected to an angle *β* = 40°. Its length and thickness are *L* = 0.9 m and *e* = 0.01 m, respectively, the flap chord l = 0.1 m. The leading edge is chamfered at an angle of 5° relative to the lower surface, and its diameter is *d* = 0.002 m (see **Figure 2**). This is similar to the flat plate designed by Patten et al.

A judicious choice of the suction system is required to control the laminar flow [1, 27–29]. The suction area is placed in 0.1 m from the leading edge and extends to 0.7 m. It consists of holes of the same diameter *dsuc* = 0.2 mm and equidistant from

The quality of mesh has a great importance on the results of a numerical resolution. For this, we choose a structured and very tight mesh in the near-wall and in the area of the leading edge (Y+ = 1). This computational domain is made up of 53.970 cells in the case of the impermeable plate and 100.000 cells in the case of the

1 mm (**Figure 2**). This corresponds to a longitudinal dimensionless spacing *dp*/*dsuc* = 5 (where *dp* is the spacing between two successive holes), respecting the

*Streamline velocity colored by dimensionless velocity magnitude ((u<sup>2</sup> + v<sup>2</sup> )0,5/U ): (a) trailing edge flap angle β = 0°; (b) trailing edge flap angle β = 40°.*

**Figure 4.** *Comparison between Blasius and CFD profiles for impermeable flat plate for β = 40°.*

the case of uniform suction. Schlichting [9] neglected the dependency between the boundary layer velocity and the *x*-coordinate; this hypothesis is not acceptable in the region near the leading edge. In the case with uniform suction, Palekar and Sarma [2] found two solutions for boundary layer profiles: the first one nears to the leading edge and the second one far the leading edge. The advantage of the new solutions is that the Error function defines well the boundary layer near and far the leading edge in the cases with and without uniform suction, as well as, in the case of nonuniform suction.

**Figure 5** compares the boundary layer profile defined in Eq. (12), with the CFD, and the Blasius profiles for the impermeable flat plate. This result confirms the choice of the profile shape (*erf*(*y*/*h*(*x*)). This shows that the Error function has the advantage of a good approximation of the boundary layer theory. We compared in **Table 1** the new characteristic parameters for the boundary layer with those from the literature. The values found by this approximation are quite comparable to Blasius and generally better than the other approximations.

Concerning the case of uniform suction (*vp* = 0.0118 m/s), we compare the profile defined by Eq. (17) with the Palekar profiles for small and gross *x*. Our analytical solution verifies well both cases at once (**Figure 6**). As shown in **Figure 7**, the relation Eq. (20) well verified the universal law of friction found by Iglisch (1949) [11]. The comparison between the analytical profiles Eq. (17) with the numerical results (CFD) shows a little difference. This is caused by the nature of the parietal suction for each case, where in the theoretical study, the boundary condition at the wall is defined as a continuous suction along the plate; however, in the CFD study, the suction is discrete.

After validation of the new analytical laws, we presented the effect of the suction rate on the characteristics of the boundary layer. **Figure 8a, b**, and **c**, shows

#### **Figure 5.**

*Comparison of the analytic velocity profile Eq. (12) with the profile of Blasius and the profile obtained by using CFD Fluent for the impermeable flat plate.*

the velocity profiles, the parietal friction coefficients, and the boundary layer thicknesses for different suction rates. It is clear that when the suction rate increases, the thickness of the boundary layer decreases. As a result, the boundary

*<sup>2</sup> <sup>ν</sup><sup>p</sup> U*∞

*Validation of the new law in the case of uniform suction with the solutions found by Palekar (1984) [2], for*

*).*

*depending on t; (b) boundary layer profile (x = 0.5).*

*) and gross x (x = 0.5 m* � ¼ <sup>10</sup>�<sup>5</sup>

*u*

3 <sup>2</sup> *<sup>η</sup>* � <sup>1</sup>

**Table 1.**

**Figure 6.**

**Figure 7.**

**129**

*Representing curves of (a) the universal law of friction Cf*

*small x (x = 0.001 m* � ¼ <sup>10</sup>�<sup>5</sup>

*<sup>U</sup>*<sup>∞</sup> *δ*

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

ffiffiffiffiffiffi *Rex x*

*Comparative table of characteristic parameters of boundary layer.*

q *<sup>δ</sup><sup>1</sup>*

*δ*

<sup>2</sup>*<sup>η</sup>* � *<sup>η</sup>*<sup>2</sup> <sup>5</sup>*,* 4 0*,* <sup>33</sup> <sup>0</sup>*,* 4 0*,* <sup>36</sup>

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

<sup>2</sup> *η*<sup>3</sup> 4*,* 6 0*,* 375 0*,* 139 0*,* 33 <sup>2</sup>*<sup>η</sup>* � <sup>2</sup>*η*<sup>3</sup> <sup>þ</sup> *<sup>η</sup>*<sup>4</sup> <sup>5</sup>*,* 8 0*,* 3 0*,* <sup>121</sup> <sup>0</sup>*,* <sup>34</sup> *Blasius solution* 5 0*,* 344 0*,* 132 0*,* 332 *erf*ð Þ 0*;* 32*η* 5*,* 66 0*,* 31 0*,* 128 0*,* 36

*δ2 δ*

*Cf 2* ffiffiffiffiffiffiffiffi *Rex* p


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

#### **Table 1.**

the case of uniform suction. Schlichting [9] neglected the dependency between the boundary layer velocity and the *x*-coordinate; this hypothesis is not acceptable in the region near the leading edge. In the case with uniform suction, Palekar and Sarma [2] found two solutions for boundary layer profiles: the first one nears to the leading edge and the second one far the leading edge. The advantage of the new solutions is that the Error function defines well the boundary layer near and far the leading edge in the cases with and without uniform suction, as well as, in the case of

**Figure 5** compares the boundary layer profile defined in Eq. (12), with the CFD,

Concerning the case of uniform suction (*vp* = 0.0118 m/s), we compare the profile defined by Eq. (17) with the Palekar profiles for small and gross *x*. Our analytical solution verifies well both cases at once (**Figure 6**). As shown in **Figure 7**, the relation Eq. (20) well verified the universal law of friction found by Iglisch (1949) [11]. The comparison between the analytical profiles Eq. (17) with the numerical results (CFD) shows a little difference. This is caused by the nature of the parietal suction for each case, where in the theoretical study, the boundary condition at the wall is defined as a continuous suction along the plate; however, in the

After validation of the new analytical laws, we presented the effect of the suction rate on the characteristics of the boundary layer. **Figure 8a, b**, and **c**, shows

*Comparison of the analytic velocity profile Eq. (12) with the profile of Blasius and the profile obtained by using*

and the Blasius profiles for the impermeable flat plate. This result confirms the choice of the profile shape (*erf*(*y*/*h*(*x*)). This shows that the Error function has the advantage of a good approximation of the boundary layer theory. We compared in **Table 1** the new characteristic parameters for the boundary layer with those from the literature. The values found by this approximation are quite comparable to

Blasius and generally better than the other approximations.

nonuniform suction.

*Aerodynamics*

CFD study, the suction is discrete.

**Figure 5.**

**128**

*CFD Fluent for the impermeable flat plate.*

*Comparative table of characteristic parameters of boundary layer.*

#### **Figure 6.**

*Validation of the new law in the case of uniform suction with the solutions found by Palekar (1984) [2], for small x (x = 0.001 m* � ¼ <sup>10</sup>�<sup>5</sup> *) and gross x (x = 0.5 m* � ¼ <sup>10</sup>�<sup>5</sup> *).*

**Figure 7.** *Representing curves of (a) the universal law of friction Cf <sup>2</sup> <sup>ν</sup><sup>p</sup> U*∞ *depending on t; (b) boundary layer profile (x = 0.5).*

the velocity profiles, the parietal friction coefficients, and the boundary layer thicknesses for different suction rates. It is clear that when the suction rate increases, the thickness of the boundary layer decreases. As a result, the boundary

**Nomenclatures**

*η* ¼ *y*

ffiffiffiffiffi *U*<sup>∞</sup> *υx*

**Author details**

Chedhli Hafien<sup>1</sup>

Tunisia

**131**

Cf skin friction coefficient

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

e plate thickness (m) L length of the plate (m)

*U*<sup>∞</sup> velocity inlet (m/s)

vp suction velocity (m/s)

*d*d diameter of the leading edge of the plate (m)

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

*dpdp* distance between two successive holes (m)

l length of the trailing edge flap of the plate (m)

v0 absolute value of the suction velocity (m/s)

ν kinematic viscosity of the fluid (air) (m<sup>2</sup>

/s)

)

\*, Adnen Bourehla<sup>2</sup> and Mounir Bouzaiane<sup>1</sup>

1 Laboratory of Mechanics of Fluids, Faculty of Science of Tunis, Tunis Cedex,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*dasp*dasp thickness of the suction holes (m)

Re*<sup>x</sup>* Reynolds number based on *x u*u velocity component along *x* (m/s)

x longitudinal coordinate (m) y transversal coordinates (m) β trailing edge flap angle (°)

*δ* boundary layer thickness (m) δ<sup>1</sup> displacement thickness (m) δ<sup>2</sup> momentum thickness (m) *τp* parietal friction force (N/m<sup>2</sup>

q Blasius parameter

2 Aviation School of Borj El-Amri, Tunisia

provided the original work is properly cited.

\*Address all correspondence to: chedhli.hafien@gmail.com

*v*v velocity component along *y* (m/s)

#### **Figure 8.**

*Parameters of boundary layer for different values of suction rate (vp/U = 0; 5.10<sup>4</sup> ; 10<sup>3</sup> ; 1,5.10<sup>3</sup> ; 2.10<sup>3</sup> ) for U = 5 m/s; (a) velocity profiles; (b) boundary layer thicknesses; (c) parietal friction coefficients.*

layer profile flattens and the skin friction coefficient increases. This increase has no great effect on the total drag because it depends essentially on the form of drag. The contribution of the friction drag is negligible. This result is in accord with the literature.
