Heat Transfer and Fluid Dynamics on Inclined Smooth and Rough Surfaces by the Application of the Similarity and Integral Methods

Home , Stanton number

Advancement in Scientific and Engineering Research Vol. 4(1), pp. 1-16, May 2019 doi: 10.33495/aser_v4i1.19.101 ISSN: 2384-7336 Research Paper

Heat transfer and fluid dynamics on inclined smooth and rough surfaces by the application of the similarity and integral methods

Élcio Nogueira1* • Marcus V. F. Soares2 • Luiz Cláudio Pimentel3

1Faculty of Technology of the State University of Rio de Janeiro - FAT/UERJ, Brazil. 2Engineering Company - Engevale, Brazil. 3Institute of Geosciences of the Federal University of Rio de Janeiro - IGEO/UFRJ, Brazil.

*Corresponding author. E-mail: [email protected]

Accepted 29th March, 2019.

Abstract. The main objective of the analysis is to review and discuss the principles of the similarity method applied to the boundary layer on inclined surfaces, in laminar regime, and that can be extended to a turbulent regime. The emphasis applies to theoretical aspects related to the concept of similarity, but theoretical results were obtained in order to compare with empirical expressions and experimental results. Results are obtained for the hydrodynamic and thermal fields, such as coefficient of friction and Stanton number, as a function of the pressure gradient parameter and the Prandtl number. The fourth order Runge-Kutta method is applied, starting from the expansion in power series as the first approximation for the mathematical solution of hydrodynamic and thermal problems, in laminar regime. The Integral Method is applied to obtain an approximate solution for the flow in turbulent regime, by similarity variables method. Numerical and graphical results are presented in sufficient numbers to emphasize the consistency of the model developed in the determination of parameters related to thermal and hydrodynamic boundary layers on smooth and rough surfaces.

Keywords: Similarity method, fourth order Runge Kutta Method, hydrodynamic boundary layer; thermal boundary layer.

INTRODUCTION

At the beginning of the twentieth century, important preponderant, and an outer region where viscous forces developments occurred in the hydrodynamic theory of the can be neglected. Through this "artifice", simpler flows in the bodies, which provided explanations for the equations, derived from the Navier-Stokes equations observed discrepancies between the theory of hydraulic could be deduced and resolved with relative ease for flat and equations of Euler for an ideal fluid. Navier-Stokes plate flow Blasius (1908), Hager (2003) and sloping equations were already available, but the degree of surfaces Falkner and Skan (1930, 1931) in laminar difficulty imposed for complete solution of the equations, regime. even for simple problems, and low viscosity fluids such The boundary-layer equations made it possible to as air and water, has not provided a satisfactory solution rapidly advance the theory of fluid mechanics and its to the needs of the moment. In 1904 Prandtl developed application to airfoils; which is an important application for the concept of boundary layer, demonstrated that the flow aircraft design. The boundary layer theory has also around a body is divided into two regions: a thin layer brought advances in the solution of problems associated close to the body, where the viscous tensions are with the heat transfer around bodies. The classical 2 Adv. Sci. Eng. Res. / Nogueira et al.

mathematical method chosen to solve this class of nanoparticles is determined by Constant, Reynolds' and problems is called the "Similarity Method". Vogel's models. It is observed that velocity reduces at The similarity method makes it possible to transform a higher values of magnetic field intensity. nonlinear third order partial differential equation into an The problem of the steady, incompressible, three- ordinary differential equation. It is now considered that dimensional stagnation point flow of a micropolar fluid the main aspects of the similarity theory have already over an off centered infinite rotating disk in a porous been clarified. Extensions of the theory were applied to medium is studied Khan et al. (2017). Injection/suction is the turbulent regime, and the boundary layer concept applied uniformly throughout the surface of porous disk. associated with the similarity method continues to be The Darcy's resistance for the micropolar fluid is also used as a basis for more complex problems than those formulated. The partial differential equations are analyzed by Blasius and Falkner and Scan. These are converted into the set of ordinary differential equation by extensively published in traditional textbooks. However, utilizing the suitable transformation. The system of due to the complexity of the turbulent flows, a complete equations is analytically solved by the means of a non- theory, based on the principle of similarity, still does not perturbative technique, homotopy analysis method exist and research continues to be carried out in this area (HAM). The influence of rotational parameter, material (Abbasi et al., 2014; Bhattacharyya et al., 2016; Bognar parameter, spin gradient viscosity parameter, micro- and Hriczó, 2011; Castilho, 1997; Myers, 2010; Rahman, inertia density parameter, porosity parameter and 2011; Stemmer, 2010). suction/injection parameter on velocity functions is Analytical methods of solution of partial nonlinear presented in graphical form and discussed in detail. differential equations, based on stream function and Verification of the solutions is made by a numerical similarity method are being applied to solve numerous comparison with the previous study. engineering problems. Aziz et al. (2018) analyzes the heat transfer of a thin Buoyancy-driven, incompressible, two-dimensional flow film flow on an unsteady stretching sheet in nanofluids. of a micropolar fluid inside an inclined porous cavity in Three different types of nanoparticles are considered; the presence of magnetic field is investigated by Nazeer copper Cu, alumina Al2O3 and titania TiO2 with water as et al. (2017). The nonlinear partial differential equations the base fluid. The governing equations are simplified are solved by employing a robust Galerkin finite element using similarity transformations. The resulting coupled scheme. The code is validated and benchmarked with the nonlinear differential equations are solved by the previous numerical data available in the literature. Homotopy Analysis Method (HAM). The analytical series Sheikhzadeh and Abbaszadeh (2018) determined the solutions are presented and the numerical results solutions for the momentum and energy equations of obtained are tabulated. In particular, it shows that the laminar flow of a non-Newtonian fluid in an axisymmetric heat transfer rate decreases when nanoparticles volume porous channel using the least squares and Galerkin fraction increases. methods. The numerical solution is conducted using Combined influences of thermal radiation, inclined fourth order Runge-Kutta method. With comparing the magnetic field and temperature-dependent internal heat results obtained from the analytical and numerical generation on unsteady two-dimensional flow and heat methods, a good adaptation can be seen between them. transfer analysis of dissipative Casson-Carreau nanofluid It can also be observed that the results of the Galerkin over a stretching sheet embedded in a porous medium is method have further conformity with the numerical results investigated by Sobamowo et al. (2018). Similarity and the Galerkin method is simpler than the least square transformations are used to reduce the developed method and requires fewer computations. systems of governing partial differential equations to Ali et al. (2019) performed a numerical study for the nonlinear third and second orders ordinary differential mixed convection flow inside a triangular cavity. The equations which are solved using finite element method. rheological behavior of the fluid inside the cavity is In the study, kerosene is used as the base fluid which is modeled through the constitutive bi-viscosity equation. embedded with the silver (Ag) and copper (Cu) The governing nonlinear partial differential equations are nanoparticles. Also, effects of other pertinent parameters discretized using Galerkin finite element method and on the flow and heat transfer characteristics of the pressure is eliminated through the penalty method. The Casson-Carreau nanofluids are investigated and computations are presented graphically for a wide range discussed. of the bi-viscosity parameter, thermal radiation In this work, the aspects related to laminar regime are parameter, Hartman number, Grashof number, Reynolds associated, mainly, with the works of Evans (1968) and number, heat generation/absorption parameter and the aspects associated to the turbulent regime, smooth Prandtl number. and rough surface, are related to the works of Kays et al. Rabari et al. (2017) present an analytical study on the (1969). blood flow containing nanoparticles through porous blood The results obtained enable simplifications to be used vessels, in the presence of magnetic field, using the in numerous practical situations, such as modeling for Homotopy Perturbation Method (HPM). The viscosity of determination of micro meteorological parameters, as Adv. Sci. Eng. Res. / Nogueira et al. 3

recommended in Monin-Obukov's theory of similarity Stull where C is the value of U (x) where x is unitary, and the (1988). value of m depends on the pressure gradient in the main direction of the flow. However, according to Spalding and Pun (1962), it is convenient to impose that U (x) satisfies Objectives the following equation:

The main objective is to review and discuss the principles 푑푈 2(훽−1) = 퐶푈 훽 (2) of the method of similarity applied to boundary layer on 푑푥 sloped, smooth and rough surfaces, laminar and turbulent regimes. Emphasis applies to theoretical where β is a parameter that is associated with the aspects related to the concept of similarity and integral pressure gradient in the direction of the main flow. method, but results were obtained with the aim of The potential theory, applied around an angle wedge comparing with empirical expressions and experimental βπ/2 Evans (1988), satisfies Equation 1, above, where: results. 훽 A secondary objective is to present the main aspects of 푚 = (3) the problem analyzed by Falkner and Skan, focusing on (2 − 훽) the similarity method, which can be extended, as already stated, to more complex problems of flow and heat and, transfer in turbulent regime. 1 푑푈 푈 1 For comparison purposes, the problem of flux flow and = (4) heat transfer in turbulent regime in boundary layer on 훽 푑푥 푥 (2 − 훽) inclined surfaces, using velocity profile with an exponent one-seven (1/7), is applied as an extension of the However: solution with zero gradient of pressure, presented by 푑푝 푑푈 Kays and Crawford (1983) and Nogueira and Soares = −휌푈 (5) (2018). 푑푥 푑푥

The specification of U (x) is equivalent to specifying the METHODOLOGY pressure gradient, which is a function of β:

푑푝 훽 푈 The aspects related to laminar regime are based on the = −휌푈 (6) deep study carried out by Evans (1968), and in turbulent 푑푥 (2 − 훽) 푥 regime the text of Kays and Crawford (1966). It is assumed that there is no mass transfer through the The parameter β, as can be seen, depends only on the surface (without surface perspiration effect) and that the velocity distribution in the external region to the boundary perpendicular component of velocity is zero. In addition, layer, the variable x along the surface and the pressure the velocity component parallel to the surface is also gradient. zero, a condition called "no slip on the wall" in the We are interested in the flow conditions where -0.2 ≤ β specialized literature. All the presented solutions and ≤ 1.0, representing the limits of the boundary conditions results assume constant properties, unaffected by the for the pressure gradient parameter between the variation of temperature, and the velocities are sufficiently boundary layer detachment β=-0.2, and the two- low so that the viscous dissipation term can be neglected. dimensional stagnation flow β=1.0, in the laminar regime The basic equations for boundary layer similarity (Figure 1). conditions are widely discussed Schlichting (1968); The differential equation governing the velocity Evans (1968), Kays and Crawford (1983), Silva Freire distribution at a similar boundary layer for laminar regime (1990), and only essential details for the understanding of within the range of the already established pressure the arguments are presented in this work. Details on flat gradient parameter β is given by Schlichting (1968); plate flow are discussed by Nogueira and Soares (2018). Evans (1968), Kays and Crawford (1983), Silva Freire There are in the literature many ways of specifying the (1990): existence of similar solutions for the laminar boundary ′′′ ′′ ′2 layer equations. The main characteristic associated with 푓 + 푓. 푓 + 훽. (1 − 푓 ) = 0 (7) the concept of boundary layer similarity is that the undisturbed velocity distribution of the potential flow must with the following boundary conditions: satisfy the following expression, which follows the original suggestion of Falkner and Scan (1931): 휂 = 0, 푓 = 푓′ = 0 7.1 휂 → ∞, 푓′ → 1.0 7.2 푈(푥) = 퐶푥푚 (1) where f and η are defined by: 4 Adv. Sci. Eng. Res. / Nogueira et al.

Figure 1. Flow on an inclined surface of angle πβ.

푈푥 ( )1/2 푦 휐 휓/휐 휂 = 푒 푓 = 푈푥 (8) With the following recurrence rule: 푥 √2 − 훽 ( )√2 − 훽 휐 퐶푛+3 where x and y are, respectively, the primitive coordinates 1 = −푛! [ (훽. 퐶 . 퐶 ) along the surface and perpendicular to it. 1! (푛 − 1!) 2 푛 Choosing the coordinate η as a function of y / x, which 1 1 + (퐶 . 퐶 + 훽. 퐶 . 퐶 ) + (퐶 . 퐶 ) is very small, except at x = 0, by the square root of the 2! (푛 − 2!) 2 푛 3 푛−1 3! (푛 − 3!) 3 푛−1 푈푥 1/2 Reynolds number, 푅푒푥 = ( ) , which is very large, we 1 휈 + (퐶5. 퐶푛−3) ] (11) impose 푦/푥 is small, but η is not. 4! (푛 − 4!) From the definitions of η and f, we have expressions for the components of dimensionless velocities: for n≥4.

푑푓 1 2 푢 = 푈 푒 푣 퐶3 = −훽 ; 퐶5 = −2 (훽 + ) 퐶2 푒 퐶6 푑휂 2 1 1 휐 푑푈 2 푑푓 = 6훽 (훽 + ) 퐶2 (12) = − ( ) [푓 + (훽 − 1)휂 (9) 2 훽 푑푥 푑휂 The term C2 corresponds to f "(0), that is: The last boundary condition, Equation 7.2, means that ′ as η grows 푓 = 푢/푈 should approach the unit without 퐶2 ′′ exceeding it. The value of η, in this case, is called 휂∞, for = 푓 (0) (13) a given value of β. Due to the difficulty in solving the above boundary Through Shooting Method (Tannehill et al., 1997; condition problem with reasonable precision, we apply Oderinu, 2014), or other approach method, as the the 4th order Runge Kutta Method (Tannehill et al., "Bisection Method", we can obtain the value of C2, with 1997), with initial value of f''(0) given after application of the desired approximation. However, the approximation the Power Series Method. method through the series solution is slow, in order to The approximate solution by the power series method obtain the necessary solution for our purposes. In this with the Shooting Method (Tannehill et al., 1997; Oderin, sense, we apply the fourth Runge-Kutta method 2014), as an approximation procedure for the velocity (Tannehill et al., 1997), with initial value for f '' (0) from profile is obtained by assuming that the function f (η) the expansion in power series. As the Runge-Kutta satisfies the following expansion in series: method is a high-precision numerical method, coupled with the Newton-Raphson method, the final solution for 휂2 휂5 휂6 휂푛 the velocity field in the hydrodynamic boundary layer is 푓(휂) = 퐶 + 퐶 + 퐶 … + 퐶 (10) 2 2! 5 5! 6 6! 푛 푛! obtained in less time than necessary for the series Adv. Sci. Eng. Res. / Nogueira et al. 5

solution, with the same precision. Where The energy equation for determining the dimensionless ∞ temperature field is given by: 1 푎푞 퐸 = Γ ( ) + ∑ 푞 (20) 3 푑 푞=0 푃푟3 (휃′) + 푃푟. 푓. 휃′ = 0 (14) 푑휂 Γ is the gamma function: The temperature profile shall satisfy the following contour 1 conditions for the specified surface temperature: Γ ( ) = 2.6789385 (21) 3 휂 = 0, 휃 = 0 14.1 휂 → ∞, 휃 → 1.0 14.2 Expressions for 푎푞 contains the pressure gradient parameter β, the dimensionless viscous stress on the where Pr is the number of Prandtl, and wall 푓′′(0) , and numerical factors derived from the combination of gamma functions and are not presented. 푇 − 푇푊 The complete procedure for the exact determination is 휃 = (15) 푇∞ − 푇푊 found in Evans (1968). In addition, Equation 19, for determining the temperature gradient at the surface, It is assumed that 푇∞, temperature outside the boundary presents unsatisfactory results as β tends to the flow layer, is not affected by the heat rate removed outside of separation value (β = -0.2). Evans (1968) describes an the boundary layer. The value of 푇푊 corresponds to the alternative procedure for this case, but it will not be the surface temperature (reference!). subject of discussion in this analysis. The energy equation, Equation 14, is linear and less In turbulent regime there are no analytical solutions for complex than the velocity field equation. However, it the boundary layer equations. An alternative for the strongly depends on the solution of the velocity profile, determination of turbulent boundary layer parameters is since f appears explicitly in the second term. Therefore, the approximate solution of Von Kármán's equation: the greater the precision in the solution of f, the better the solution in θ. 푑훿2 퐶푓 휏푤 = = (22) The Runge-Kutta method is used for solution of the 푑푥 2 휌푈2 temperature field, but it is observed that the limit value for η, 휂 → ∞, is not necessarily the same as that obtained for Even in the zero-pressure gradient, flat plate flow, Von the velocity field, for a given β. As an alternative, in terms Kármán's equation has more unknowns than equations. of comparison, a second solution is obtained by directly Thus, it is necessary to relate the unknowns by specifying integrating the energy equation Kays and Crawford a dimensionless velocity profile. (1983); Evans (1968): For comparison purposes, in relation to the laminar regime, in this work, values for turbulent flow are 휂 휂 ′ determined on smooth and rough inclined surfaces, by 휃 = 휃0. ∫ exp [−푃푟. ∫ 푓. 푑휂] 푑휂 (16) means of an approximate theoretical model. Turbulent 0 0 flow with 1/7 power is used, and experimental results of 휂 Schultz-Grunow (1941), Pimenta et al. (1975), Schlicthing ′ ′ 휃 = 휃0. exp [−푃푟. ∫ 푓. 푑휂] (17) and Prandtl (1968), and Kays and Crawford (1983). 0 The theoretical procedure, in this case, corresponds to

the one recommended by Kays and Crawford (1983), for where 휃′ 'is the value of the surface temperature 0 flat plate flow, where the conditions of similarity are derivative: satisfied. In fact, the valid procedure is used for flat plate,

for determination of the profiles of speed and 휃 → 1, 휂 → ∞ 휃′ 0 temperature, and generalizes to situations where 훽 ≠ 0, 1 = ∞ 휂 (18) through the concept of the shape factor, 퐻12, and ∫0 exp [−푃푟. ∫0 푓. 푑휂] 푑휂 correction formulas obtained by Kays and Crawford (1983). However, the application of Equation 18, above, does not For the approximate determination of the turbulent provide adequate accuracy to obtain the surface velocity profile, associated with the integral equation of temperature gradient. In this sense, we chose to use momentum, a power law of type 1/7 is very convenient: ′ Evans's procedure, in 휃0. Therefore, 푢+ = 8.75푦+1/7 (23)

′′ 푑휃 3 푃푟. 푓0 The above expression represents the speed profile up to ( ) = ( )1/3 (19) 푑휂 0 퐸 3! 푦+ = 1500 a little better than the equation, much used in 6 Adv. Sci. Eng. Res. / Nogueira et al.

algebraic simulations, called "Logarithmic Law in the H increases from 1.29 to null pressure gradient β = 0, to Wall". approximately 2.7 in the separation condition 훽 ≅ 0.2 Simpson (1989). For accelerated flow the value of H 휏 increases again, as a function of the "tendency to laminar √ 푤⁄ 푢 휌 flow" effect, and tends to 1.47 for two-dimensional 푢+ = 푒 푦+ = 푦 (24) 휏 휐 stagnation flow, β = 1 Smith (1966). The velocity √ 푤⁄휌 distribution, U (x), must be known prior to the application of the integral momentum equation, Equation 28. If Equations 22 and 23 are valid throughout the boundary The displacement thickness, 훿1, has the effect of layer, and that the thickness δ corresponds to the displacing the undisturbed main flow current function with position where the velocity is equal to U, we have: respect to the value it should have for ideal, non-viscous fluid. The momentum thickness, 훿2, is the extent to which 푈 the amount of fluid movement in the boundary layer is below what should be for an ideal fluid. The viscous √휏푤 ⁄휌 thickness, 훿4, inverse of f "(0), is the measure of the resistance offered for transferring the amount of 휏 √ 푤⁄휌 movement of the main stream to the surface. = 8.75훿 (25) There are two predominant regions to be analyzed in a 휐 turbulent boundary layer:

The displacement, 훿1, and momentum thickness, 훿2, can be evaluated by the following expressions: 1. A predominantly viscous region close to the surface, where viscous stresses and molecular conduction prevail; ∞ 푢휌 2. A completely turbulent region where the amount of 훿1 = ∫ (1 − ) 푑푦 (26) movement and heat are transported in rates generally 0 푈휌∞ much higher than that of the viscous sublayer. and It is in the viscous sublayer, however, where events ∞ 휌푢 푢 associated with turbulence occur and are of greater 훿 = ∫ (1 − ) 푑푦 (27) importance than the fully turbulent region. The viscous 2 휌 푈 푈 0 ∞ forces, largely responsible for the characteristics of the

laminar flow, have the effect of restoring the laminar flow The integral equation of the momentum, in similar in turbulent flow and, otherwise, the inertial forces coordinates, is given by: associated with the local variations of the velocity field 1 1 푑 have the opposite effect. In fact, inertial forces tend to 푓′′(0) = = (푈2훿 ) amplify local disturbances. It is to be expected, therefore, 훿 휐푈 푥 2 4 that the stability of the laminar flow is associated with low 훿 푑푈 + 1 (28) numbers of Reynolds, ratio between the forces of inertia 휐 푑푥 by the viscous forces. Although instability is an essential or feature in the viscous sublayer, the turbulent boundary layer structure adjusts itself, constructing a relatively 훿 1 푈(푥) 푑훿 2 stable structure with stability characteristics (there is 2 = 2 regularity!). 훿4 2 휐 푑푥 At turbulent flow, along the surface, the laminar + (2 sublayer becomes narrow and becomes an increasingly 2 훿2 푑푈(푥) smaller fraction of the entire boundary layer. In essence, + 퐻 ) (29) 12 휐 푑푥 the turbulent boundary layer has the property of diffusing the amount of movement, and other properties of the where flow, much more rapidly than the simple molecular process. 퐻12 Equations 26 and 27, together with the integral 훿 = 1 푑푒푛표푚𝑖푛푎푡푒푑 푠ℎ푎푝푒 푓푎푐푡표푟 (30) equation of momentum, Equation 28, can be used to 훿2 obtain the coefficient of friction in the turbulent boundary layer. Note, however, that the velocity profile is valid for For similar boundary layer, each 훿푛 is a constant and null pressure gradients, that is, β = 0. For situations in therefore the shape factor is a constant. It is important to which the pressure gradient is different from zero, note that Equation 28 is valid for laminar and turbulent correction must be made. The expression for the regime. The shape factor increases in an adverse coefficient of friction, β = 0, is given by Kays and pressure field, β <0. For flow in turbulent boundary layer, Crawford (1983): Adv. Sci. Eng. Res. / Nogueira et al. 7

퐶푓 0.0594 RESULTS AND DISCUSSION = (31) 1/5 2 2푅푒푥 Results were obtained for velocity and temperature which can be compared with the experimental equation profiles, and associated values, such as friction obtained by Schultz-Grunow (1941): coefficient and Stanton number, as a function of the pressure gradient parameter and Prandtl number. 퐶푓 −2.584 Numerical results were computed using Fortran (1995) = 0.185(퐿표푔 (푅푒 )) (32) 2 10 푥 language and graphical results were obtained through software Grapher (2004). In turbulent boundary layer analysis, it is convenient to Figure 2 presents results for velocity profile, f', and define some type of similarity. However, the task is not as dimensionless viscous stress, f'', for laminar regime in simple as in laminar boundary layer. In turbulent flow, in a extreme situations, in β=-0.2 and β=1.0, in relation to the region very close to the surface, it is observed that u+= y+ results obtained for flat plate, β=0. These conditions, as e, logically, the principle of similarity applies. Outside this already pointed out, represent, respectively, the surface region and in the explicit coordinate system, the principle boundary layer detachment condition and the two- generally does not apply. However, there are some dimensional stagnation flow condition. It is observed that, classes of turbulent flow that have similarity, even outside for β=-0.2, the viscous stress is equal to zero on the the laminar sublayer. surface, as expected. Turbulent boundary layer that has similarity outside the In Equation 7, for 훽 = 0, since 푓′ = 0 on the wall, 푓′′′ is laminar sublayer is called the boundary layer in also zero and, as a consequence, 푓′′ has a maximum equilibrium. The equilibrium boundary layer is the one value on the wall. For 훽 < 0 values, 푓′′ also has a that satisfies the following velocity profile: maximum value, but the maximum point distances itself from the wall, and the values of 푓′′ on the wall are lower 푢 − 푈 푦 = 퐹 ( ) (33) than that of 훽 = 0.0. In fact, it can be observed that the 휏 훿 viscous tension in the wall decreases to negative 훽 √ 푤 3 휌 values and becomes zero in 훽 near -0.2. For accelerated flows, 훽 > 0, the maximum value also Where occurs on the wall, and these values increase with increasing acceleration of the flow. Since the viscous ∞ 푢 − 푈 stress is zero on the wall, where the separation of the 훿3 = − ∫ 푑푦 (34) 휏 flow occurs, 푓′ has a minimum at this point, as can be 0 √ 푤 휌 seen from Figure 2. For accelerated flow, 훽 = 1.0, a decrease in the displacement thickness, relative to flat For laminar boundary layer it was demonstrated that plate flow, 훽 = 0.0, can be observed. Equation 01 must be satisfied for similarity solutions to Table 1 present comparisons of results for exist. In turbulent boundary layer this same type of free- displacement thickness, 훿1, momentum thickness, 훿2, flow velocity profile must be satisfied, so that equilibrium and the inverse of shape factor, 퐻21, in laminar regime. boundary layer occurs, satisfying the similarity principle The consistency of the results obtained can be verified. (Kays and Crawford, 1983). For highly accelerated flows, better consistency is The turbulent coefficient of friction for the equilibrium achieved between models. boundary layer can be correlated with β through an Figure 3 presents results for temperature profile and empirical relation (Kays and Crawford, 1983): dimensionless temperature gradient in laminar regime and Pr = 1.0. The most important results, which should be 퐶푓 ⁄ 1 emphasized, are the values of the surface temperature 2 = (35) gradient, since the integral parameters associated to the 퐶푓 훽 ( ⁄ ) (1 + ) 2 5 temperature field are strongly associated with it. It can be 훽=0 observed that the temperature gradient decreases with

lower values of 훽, ie, the more accelerated the flow, the For turbulent boundary layer, assuming equilibrium greater the temperature gradients in the wall. Equivalent boundary layer, the Stanton number is determined for null to what occurs with the thickness of the hydrodynamic pressure gradient, β = 0, through the expression (Kays boundary layer, the thermal boundary layer thickness and Crawford, 1983): also decreases to higher 훽 values. In addition, another factor to be emphasized, the profile and the temperature 푆푡푥 퐶푓 gradient depend strongly on the solution of the velocity ⁄ field, according to Equations 16 and 17. = 2 (36) Figure 4 presents the results of a dimensionless √퐶푓 ⁄2 (13.2푃푟 − 10.16) + 0.9 coefficient of friction in the laminar regime determined by: 8 Adv. Sci. Eng. Res. / Nogueira et al.

1.2

Theoretical Solution - Laminar Regime f''(

f'(

0.8 f'(

f f'(

0.4 f''(

f''(

0 2 4 6



Figure 2. Solutions for dimensionless viscous velocity (f’’) and velocity profile (f’).

Table 1. Numerical comparisons in laminar regime for the inverse of Shape Factor (퐻21).

Results Evans (1968) β δ1 δ2 H21 δ1 δ2 H21 -0.2** 2.3587 0.5852 0.2400 2.3588 0.5854 0.2482 -0.019 2.0064 0.5762 0.2834 2.0068 0.5765 0.2873 -0.18 1.8714 0.5673 0.3006 1.8716 0.5677 0.3033 -0.15 1.6468 0.5449 0.3295 1.6470 0.5452 0.3310 -0.12 1.5111 0.5258 0.3472 1.5113 0.5263 0.3482 -0.10 1.4423 0.5147 0.3562 1.4427 0.5150 0.3570 0.0 1.2164 0.4695 0.3856 1.2168 0.4696 0.3859 0.2 0.9839 0.4081 0.4147 0.9842 0.4082 0.4148 0.3 0.9108 0.3856 0.4234 0.9110 0.3857 0.4234 0.4 0.8525 0.3666 0.4300 0.8527 0.3667 0.4301 0.6 0.7639 0.3359 0.4397 0.7640 0.3359 0.4397 0.8 0.6986 0.3119 0.4464 0.6987 0.3118 0.4463 1.0 0.6480 0.2924 0.4514 0.6480 0.2923 0.4513

**β = -0.19883768

퐶푓 푓′′(0) = (37) 2 1 퐶푓(훽 = 0) 0.332 푅푒 2(2 − 훽)1/2 = (39) 푥 2 1 푅푒푥2 And where 퐶푓(훽=0) 퐶푓 [퐹(훽) ] 2 퐹(훽) = 0.992436221 + 3.670315583훽 − 2.382778474훽2 ( )푒푚푝푖푟푖푐푎푙 = 훽 (38) 2 (1 + ) + 2.203613278훽3 (40) 5 Adv. Sci. Eng. Res. / Nogueira et al. 9

() -  = - 0.2 0.8 () -  = 1.0

() -  = 0.0

)

 0.6

(

 '() -  = 1.0

)

 0.4 Theoretical Solution

(

 Laminar Regime

0.2 '() -  = 0.0

'() -  = - 0.2

0 2 4 6 

Figure 3. Temperature profile (휃) and dimensionless temperature gradient (휃′) for laminar thermal layer (Pr = 1.0).

퐶푓 Figure 4. Dimensionless coefficient of friction ( ) as a function of the Reynolds number (푅푒 2 푥) for laminar regime.

퐹(훽) is obtained empirically from the data available in It is observed that the new empirical solution, Equation Evans (1968). 38, is very convenient, since it eliminates the need to 10 Adv. Sci. Eng. Res. / Nogueira et al.

퐶푓 Figure 5. Non-dimensional coefficient of friction ( ) as a function of Reynolds 2 number (푅푒푥) for turbulent regime on smooth surface.

solve the hydrodynamic boundary layer equation to boundary layer occurs primarily close to the surface, and obtain the coefficient of friction for the entire range of β. this leads to the definition of a rough Reynolds number: This is one of more significant and important result presented in this work, since the results differ significantly u k Re = τ s (43) only for values of β <-0.15, and low Reynolds number k ϑ values along the surface. Figure 5 shows the results obtained for the where ks is the absolute roughness. dimensionless friction coefficient for turbulent regime on Figure 6 shows the comparison between valid results smooth surface. The highlight corresponds to the for smooth and rough surfaces, for any range of values of Schultz-Grunow (1941) modified solution. The analytical the pressure coefficient previously defined, 0.2<β<1.0. solution, Equation 35, and Schultz-Grunow's empirical The characteristics of the hydrodynamic and thermal equation, Equation 42, associated with the modification boundary layer are controlled by important parameters proposed by Kays and Crawford (1993), for pressure such as speed and temperature, shape and surface gradient flows within the range analyzed in this work, -0.2 conditions. Surface conditions require special attention < 훽 < 1.0, are in good agreement. It is, therefore, a result where roughness is an inherent characteristic. compatible with that presented in Figure 4, since the Roughness usually increases the friction resistance and solution eliminates the need to solve the system of the heat transfer coefficient for a same Reynolds number, equations for velocity field where 훽 is different from zero. relative to the smooth surfaces. In fact, the roughness produces higher values for the friction factor and Stanton 퐶푓 0.0594 number, which result in speed and temperature deficits at = (41) 훽 the relative long distance of the surface when compared 2 2푅푒 1/5 ∗ (1 + ) 푥 5 to the smooth surface.

For Rek > 65 we have what is called the regime for a 퐶푓 −2.584 completely rough surface or even a completely rough ( )퐸푥푝 = [0.185(퐿표푔10(푅푒푥)) ] / (1 2 flow Pimenta et al. (1975). A completely rough regime is 훽 + ) (42) what is considered in this analysis and, for all intents and 5 purposes, we have Rek=70. It can be shown that the friction coefficient, for a completely rough regime, can be All previous discussion assumes boundary layer for obtained by the following expression, where the smooth surface. The effect of roughness on the turbulent correction factor for the pressure gradient effect is Adv. Sci. Eng. Res. / Nogueira et al. 11

퐶푓 Figure 6. Comparison for dimensionless friction coefficient ( ) between smooth and 2 rough surfaces in turbulent regime.

introduced, according to the empirical proposal of Kays n=0.20. and Crawford (1983): Introducing the proposed correction for inclined surfaces: 퐶푓 0.168 ( )푟푢푔 = (44) −0.4 −0.20 2 2 0.0287푃푟 푅푒푥 32.1푥 훽 푆푡 = (48) [ln ( )] (1 + ) 푇푢푟푏퐸푥푝 훽 1 5 1 + 푅푒푥5푘푠 5

The Stanton number, by definition, is obtained from the Figure 7 shows the theoretical-experimental comparison expression below: for Stanton number in laminar regime, for β=1.0. The results are quite satisfactory for Prandtl numbers close to 푁푢푥 the unit, and deviate to high values of the Prandtl 푆푡푥 = (45) number, as expected. Pr. 푅푒푥 Figure 8 presents theoretical and experimental data for For laminar regime turbulent regime, on smooth and rough surfaces, for the number of Pr=1.0. As expected, heat transfer on rough surfaces outweighs heat transfer to smooth surfaces, for 푑휃 푅푒 ( ) √ 푥 the same Reynolds number. 푑휂 0 2 The theoretical expression for determination of Stanton's 푁푢푥 = 1/2 (46) (2 − 훽) number is given by, following application of empirical modification suggested by Kays and Crawford (1983): In any situation, laminar or turbulent regime, for constant surface temperature and constant free-flow velocity, the 푆푡푟푢푔 Stanton number can be expressed in the form: 퐶푓⁄ ( 2)푟푢푔 −푛 = (49) 푆푡 = 퐶푅푒푥 (47) 퐶푓 훽 (√( ⁄ ) (13.2푃푟 − 10.16) + 푃푟푡) (1.0 + ) 2 푟푢푔 5 In turbulent regime, zero pressure gradient, smooth surface, Kays and Crawford present the following In a completely rough flow, the molecular thermal equation, which fits excellent with experimental results for conductivity remains as a significant variable, influence 5 6 -0.4 0.5

0.025

Laminar Regime Theoretical 0.02  Experimental Pr = 0.7

Pr = 1.0 0.015  Pr = 2.5

S Pr = 5.0 0.01 Pr = 10.0

0.005

1000 10000 100000 Re x

Figure 7. Stanton number (푆푡) for laminar regime as a function of Reynolds number.

Prandtl, 푃푟푡. The turbulent Prandtl number can be Experimental results, with correction factor, obtained by considered, for gases, to be 0.9, which in fact represents Pimenta et al. (1975) and Kays and Crawford (1983) an average value. were used for comparison, Equation (50). The data were The equation for determination of experimental Stanton taken from the table of Pimenta (1975), where U=130.63 number, with the correction for the pressure gradient, is ft/s and without surface perspiration effect. The value of given by: Ks is equal to 1.0 mm for all purposes in this analysis. It is important to emphasize that there is experimental 퐶푓 ( ) evidence that the shape factor tends to 1.47, for smooth 2 푟푢푔 푆푡퐸푥푝 = (50) surface and highly accelerated flows in completely 퐶푓 √( ) turbulent flow. This shows that despite the tendency to 2 푟푢푔⁄ 훽 the laminar regime, the flow remains turbulent because (푃푟푡 + )(1 + ) 푆푡푘 5 the value of the form factor for highly accelerated laminar flow is approximately 2.2, as presented through the −0.2 −0.44 푆푡푘 = 푅푒푘 푃푟 (51) results obtained in this work, Figures 9 and 10. There is theoretical and experimental evidence that the Where 푆푡푘 is a function of the roughness. detachment of the turbulent boundary layer is delayed in Figure 8 show that, for accelerated flow, the values relation to the laminar boundary layer detachment. The approximate the result obtained for laminar flow, for numerical results presented through Figure 9 smooth and rough surfaces. This effect is called demonstrate that in fact this occurs. Note, Figure 10, that laminarization of the boundary layer (Kays et al., 1969) the smallest value for the parameter that establishes the and demonstrates that the acceleration effect tends to pressure gradient is equal to 훽 = −0.19883768 ≅ −0.200, cause a "retransmission" of the turbulent boundary layer for laminar regime. to a pure laminar boundary layer. This is an effect In fact, as the results of Figure 11 show, the associated with decompression of the boundary layer, dimensionless viscous stress passes through a local where the roughness is immersed in the laminar maximum point, close to 훽 = −0.2, decreases sublayer. Table 2 shows the effect of the tendency for asymptotically to approximately 훽 = −0.24, where it laminar flow in the numerical determination of the Stanton becomes equal to zero, for turbulent regime on smooth number. surface. Adv. Sci. Eng. Res. / Nogueira et al. 13

Figure 8. Theoretical-experimental comparisons for Stanton number (푆푡) in turbulent regime, as a function of Reynolds number for smooth and rough surfaces.

Table 2. Stanton number (St) for turbulent regime in smooth and rough surfaces.

휷 = ퟎ. ퟎ 휷 = ퟏ. ퟎ

푹 (풙) Equation (48) Equation (50) Equation (48) Equation (50) 풆 Smooth Rough Smooth Rough 1.01E5 2.864E-3 4.721E-3 2.387E-3 2.977E-3 2.01E5 2.496E-3 3.988E-3 2.080E-3 2.804E-3 4.01E5 2.174E-3 3.412E-3 1.812E-3 2.398E-3 6.01E5 2.005E-3 3.132E-3 1.671E-3 2.199E-3 8.01E5 1.9893E-3 2.953E-3 1.577E-3 2.073E-3 1.00E6 1.810E-3 2.824E-3 1.509E-3 1.983E-3 1.20E6 1.746E-3 2.726E-3 1.455E-3 1.913E-3 1.40E6 1.693E-3 2.646E-3 1.411E-3 1.857E-3 1.60E6 1.648E-3 2.580E-3 1.373E-3 1.810E-3 1.80E6 1.610E-3 2.524E-3 1.341E-3 1.770E-3 2.00E6 1.576E-3 2.475E-3 1.314E-3 1.736E-3

CONCLUSIONS pressure gradient and extended through a correction factor, for a wide range of the pressure gradient The analysis, for flow and heat transfer in laminar regime parameter. and turbulent regime, on smooth and rough inclined The main result of the present work is associated with surface, includes theoretical aspects, experimental the fact that it is possible to obtain empirical solutions results and empirical correlations. An extensive review of compatible with analytical solutions for laminar and procedures associated to the boundary layer similarity turbulent flow in the whole range of values for the method, used for solution of nonlinear equations pressure parameter, 훽, considered in the analysis. This systems, was presented. result allows us to use reliable solutions for numerous In a turbulent regime, through the integral analysis of practical problems without having to solve the system of the momentum, the equations were first obtained for null nonlinear equations, which is the main source of 14 Adv. Sci. Eng. Res. / Nogueira et al.

3.5 H12Turb -0.21 2.70 -0.209 2.68 Smooth Surface -0.208 2.64 -0.207 2.60 3 -0.206 2.56 -0.205 2.52 -0.204 2.48  H12=B . ln()+A -0.203 2.44 -0.202 2.41 -0.201 2.37 B=0.069487 - A=1.47 2.5 -0.200 2.33

r -0.195 2.20 u -0.19 2.06 

2 -0.18 1.87

1 -0.17 1.75 -0.16 1.67 H 2 -0.15 1.61 -0.14 1.56 -0.12 1.51 -0.1 1.47 -0.05 1.35 0.00 1.29 1.5

0 0.4 0.8 

Figure 9. Shape Factor (퐻12) for turbulent regimen on smooth surface.

6 *

H H

 12Laminar 12Turb -0.200 4.02915 2.3333

 * -0.195 3.6414 2.1998

 -0.190 3.4806 2.0629  5 -0.180 3.2966 1.8719

 -0.170 3.1782 1.7493 Smooth Surface -0.160 3.0905 1.6673 = -0.150 3.0208 1.6104

2 -0.140 2.9632 1.5692 -0.120 2.8717 1.5120

1 -0.100 2.8010 1.4667

H 4 -0.050 2.6756 1.3532

0.000 2.5909 1.2943

- 0.050 2.5288 1.2978 0.100 2.4807 1.3133 r 0.200 2.4098 1.3566

o 0.300 2.3616 1.383

t 0.400 2.3251 1.4025 0.500 2.2963 1.4348 c 3 0.600 2.2735 1.4380

a 0.800 2.2398 1.4495 1.000 2.2157 1.4627

Laminar

p 2

h Turbulent

S * = -0.19883768 1

0 0.4 0.8 

Figure 10. Shape factors (퐻12) for laminar and turbulent regimes on smooth surfaces.

difficulties in the analysis performed. can be stated that problems associated to the determi- As a motivation for the development of future works, it nation of micrometeorological parameters, through the Adv. Sci. Eng. Res. / Nogueira et al. 15

5  f''(o)Laminar f''(0)Evans f''(0)Turb -0.200 9.345E-007 0.0000000 0.8163 -0.195 0.05518473 0.0551720 0.9125 -0.190 0.08571199 0.0856997 1.0083 -0.180 0.12864660 0.1286361 1.1283 -0.170 0.16212420 0.1621146 1.1885 4 -0.160 0.19078640 0.1907798 1.2165 -0.150 0.21636380 0.2163613 1.2278 -0.140 0.23973130 0.2397359 1.2307 -0.120 0.28175060 0.2817605 1.2224 -0.100 0.31926510 0.3192687 1.1947 -0.050 0.40030320 0.4003225 0.9851 0.000 0.46957790 0.4695999 0.7097 0.050 0.53110650 0.5311296 0.8054 3 0.100 0.58702400 0.5870351 0.9720

) 0.200 0.68686770 0.6867081 1.3274 0.300 0.77467040 0.7747545 1.5962 0 0.400 0.85429670 0.8544212 1.8303 Turbulent ( 0.500 0.92811050 0.9276800 2.1035

' 0.600 0.99597750 0.9958364 2.2536

0.800 1.12038900 1.1202676 2.5240

f 1.000 1.23282200 1.2325876 2.7322 2

Laminar 1

0 -0.24 -0.4 0 0.4 0.8



Figure 11. Dimensionless viscous surface tension (푓′′) for laminar and turbulent regimes.

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