Abstract
Even though Blasius’s flat plate boundary layer equation is considered an outstanding application of the boundary layer theory, it presents a series of inconsistencies both in its deduction and solution. This work reexamines, in detail, the fundamentals of the classical equation and the method to solve it to build correlations associated with the proposed new flat plate boundary equation and its solution. It deals with being well aware of avoiding or excluding the existing mathematical and erroneous physical considerations involved to improve the design, analysis, and solution of many practical problems in the fields of Fluid Mechanics and other scientific and technical areas. The new proposed flat plate boundary layer equation has a unique solution and satisfies Prandtl’s boundary layer concept; its inherent flow is sensitive to the transition phenomenon. In this sense, depending on the Reynolds number, it can generate perturbations that will justify the origin of the Tollmien–Schlitching waves.
1. Introduction
For many authors, for example, Leighton [1], Blasius’s solution of the flat plate boundary layer equation is “an outstanding early successful application of the boundary layer theory.” For others, e.g., Houghton et al. [2], it appears as “a model to gain insight into the laminar boundary layer concept.” However, despite its universal credibility, the flat plate boundary layer solution presented by Blasius [3] is permeated with two kinds of inconsistencies: mathematical and physical Jaguaribe [4, 5], namely: (1) a third-order differential equation is solved with four boundary conditions; (2) the resultant velocity profiles do not satisfy Prandtl’s boundary layer concept despite the use of the boundary conditions that, in principle, are well adequate. The reason for these inconsistencies stems from the following:(a)The misinterpretation that β = zηη(0) is a constant and an unknown parameter, z being the solution of the third-order nonlinear ordinary differential equation, originated by a transformation from the partial differential equation. As a consequence, a fourth boundary condition is required to “solve” the ODE, resulting in a “solution” that is not unique plus a series of other incompatibilities: there is just one value of η∞, i.e., the dimensionless variable at ∞, for all entire flat plate; it is necessary to use two artificial idealizations: the displacement thickness, δ∗, and the momentum thickness, δI, to justify unbalanced results.(b)The use of the continuity equation, in the differential form, to deduce the y-direction velocity component, engenders an inappropriate correlation that does not correctly define the well-outlined boundary layer profile.(c)The stream function is utilized to reduce partial to ordinary differential equations. This procedure naturally considers the continuity equation in the differential form and thus should be avoided.(d)In determining the standard form of the similarity solution using a transformation of the variables x and y and the stream function ψ, the original Blasius’s partial differential equation becomes independent of x. This implies that the dimensionless similarity variable, when y ⟶ ∞, i.e., η∞, remains invariant (see pages 661 and 662 in Katopodes [6]).
Given all mathematical and physical disparities, discussed in detail in Jaguaribe [4, 5], it becomes evident that the proposed series solution together with the classical set of three boundary conditions could never properly solve the centennial equation. The reason is that the mathematical model at x = 0 and y ≠ 0 does not make u equal to U∞, as shown in Figure 1. Therefore, the discontinuity between the liminal zone (x = 0) and the boundary layer region will affect the continuity and the momentum equations.
Another critical issue is related to one of the classical boundary conditions, i.e., , as explained in Jaguaribe [4, 5].
To solve the flat plate equation without the mathematical or physical discrepancies embodied in its design and solution, this work(1)proposes a series for the flat plate boundary layer solution, which, associated with the same three classical boundary conditions provides in return consistent results on the physical and dynamic aspects related to this particular flow.(2)It uses a new, practical, and compatible technique to transform the original partial differential equation into an ordinary differential equation, allowing an exact solution. The resultant equation is sensitive to the flow regime indicating a possible limit in the occurrence of the flow transition.(3)The solution presented does not make the similarity parameter at y ⟶ ∞, i.e., η∞, independent of x.(4)As it happens with Blasius’s dimensionless stream function, the new solution, z of the flat plate boundary layer, and its derivatives are expanded in a Maclaurin series, compatible with the existence of the uniform flow at x = 0.(5)The continuity equation is used in the integral form.(6)Prandtl’s boundary layer concept is fully satisfied.
2. The Basic Equations
2.1. The Fundamental Ordinary Differential Equation
The following partial differential equation is known as the classical flat plate boundary layer equation:solved by Blasius [3] in conjunction with the continuity equation and subjected to the following boundary conditions:where x and y are Cartesian coordinates; u is the x-direction component velocity; is the y-direction component velocity; and ν is the fluid kinematic viscosity.
The first procedure to solve the system of equations (1) to (3) is to transform the partial differential equation (1) into an Ordinary Differential Equation, ODE. This can be done by defining η = η(x, y), such aswhere p and A = A(U∞, ν) are constants, U∞ being the free stream velocity. If A is chosen asq, being another constant, (4) turns intowhere the Reynolds local number is defined as
Then, by dimensional analysis,
Thus,where ⟶ η∞ when y ⟶ ∞, i.e., y ⟶ δ.
On the other hand, from equation (9), the following equation is obtained.
Figure 2 shows a sketch of a portion of the boundary layer highlighting x-direction velocities and the boundary layer width. A similar sketch appears in Figure 3, showing nondimensional velocities and the surface area z, resulting from the integration of zη(η), i.e., the first derivative of z, concerning η.
Figure 3 shows the basis of the definition of the function z(η), where the first derivative, concerning η, is given by
Consequently, from equations (12), (9), and (11),where are, respectively, the shorthand notation for the first three derivatives of z in terms of , where p > 0. From equation (11),
2.2. The y-Direction Velocity Component and the Flat Plate Boundary Layer Equation
In most situations, the continuity equation is used in the classical form as follows:
Equation (18), however, does not comply with the particular aspect of the uniform velocity profile (uniform flow), intrinsic to the design of the flat plate flow, before it reaches the leading edge. Therefore, the mass balance in the integral form is determined over a finite portion of the flat plate, Δx, limited by the plate and the boundary layer outline. The resultant equation (19) guarantees, then, the dynamic accommodation of the mass of fluid inside the boundary layer, counterbalancing the retarded mass flow due to viscosity effects and ensuring the nullity of the y-direction component velocity at η∞.
Thus, using equations (11), (12), (16), (17), and (19) integrated from 0 to δ, after some mathematical manipulations, the new y-direction dimensionless velocity component iswhere p > 0. Now, replacing equation (1) with equations (12) to (14) and (20),where
2.3. The True Flat Plate Boundary Layer Equation
Considering Figure 2, it is possible to writewhere 0 < y´ ≤ δ.
Equation (23) plus (16) yieldwhich means that
Then,where 0 < η′ ≤ η∞.
There are two distinct situations that the x-direction velocity u should satisfy, and therefore that an adequate series solution of equation (21) will guarantee the following:(1)For all y ≠ 0,(2)For all x
Consequently, the unique series capable of meeting these requirements iswhere
In accordance with equation (28),with three natural boundary conditions, written aswhere are, respectively, the shorthand notation for the first three derivatives of z in terms of and N ⟶ ∞.
Condition equation (31), associated with equation (21), renders the following equation:
3. Solving Equation (36)
3.1. Calculating and
Using, equations (30), (31), and (36), one has a series expansion in such as
Hence,and then the odd terms are given byj = 0, 1, 2, …
Equations (28) and (30), at , lead to a0 = 0 and a1 = 0, respectively. And given equations (29) and (33), can be written asi.e.,
Given the results presented in equations (38), (39), and (41) yields a power series in , such as
Equation (42) leads towhere
Equation (43) is solved using the Mathcad root function with four arguments and 83 terms, determining λ = 5.61943 (It is clear that, in this case, only positive and real values may be valid as a solution. By the Fundamental theorem of algebra every nonconstant polynomial with complex coefficients has a root in the complex numbers - Fine and Rosenberger [7] as well as in Descartes's rule of signs-Meserve [8]. A polynomial such as the one in equation (43) has just one real root if an even number of terms is considered to determine the valid root λ.). Thus, the parameter ξ is evaluated in terms of the parameter λ for different values of x (see equation (44)) avoiding the solution through equation (42) and preventing difficulties produced by numbers greater than 10307.
3.2. The Coefficient a2
As shown in Jaguaribe [4, 5], a supplementary and inconceivable boundary condition (see (45)) is used to solve the Blasius equation, i.e.,where “c” is a guess when a shooting method is used, Nellis and Klein [9]. In the present model and given that a2 corresponds to the tangent of an angle α(x), which intersects a common point both in the curve u and on the plate: α(x) is formed by the tangent line itself and a line perpendicular to the plate, meeting the common point. Thus, 2a2(x) = β(x) = tan(α(x)). Then, from equation (29),where 0 . The angle α(x) is a result of the viscosity effect.
4. The Critical Reynolds Number, ReC
In hydrodynamics, “transition” means the change from the laminar to the turbulent regime, Schubauer and Klebanoff [10], where the Reynolds number serves to determine the limit at which the laminar regime ends. The first portion of the boundary layer, in its gradual development, is laminar, Emmons [11]. Factors such as anisotropy, length scale, and intensity turn the regime turbulent, Saraf [12]. However, following Kerswell [13], Jovanovic [14] and quoting Taylor [15] mentioned that in the absence of any external disturbances, the laminar regime can persist up to an infinite Reynolds number.
Burgers [16] and his student van der Hegge Zijnen [17], among others, were pioneers who studied experimentally the flat plate boundary layer. Hansen [18] and Dryden [19], after a series of technical procedures in the wind tunnel involving the plate itself and its mounting, used a mechanism to reduce the turbulence. Making several improvements after each run, Dryden [19] determined that the transition initiates at Rex of approximately 1.8 × 106. Seven years later, Schubauer and Skramstad [20] using the same wind tunnel as Dryden’s [19] improved the plate structure of fixation and with a six-wire damping screen reduced the turbulence stream to nearly its lowest level, determining the transition Reynolds number as 2.8 × 106. Quite ten years later, Spangler and Wells [21] almost doubled Schubauer and Skramstad’s Reynolds number [20]. Yousefi et al. [22] mentioned that the laminar flow over a smooth flat plate initiates the transit to turbulence at a Reynolds number of about 105, becoming fully turbulent after reaching values around 3 × 106. They considered and this is generally accepted that the Reynolds number of 5 × 105 is well established as the critical Reynolds number for flow over smooth flat plates.
The present paper develops its theory based on an ideal uniform and parallel flow advancing over a perfectly flat plate, slim, and smooth, where the velocity profiles do not present inflection points. Therefore, the critical Reynolds number, ReC, 5 × 105 is adopted.
5. The Reynolds Local Number, Rex, in terms of
The length, xC, corresponding to the critical distance on the plate, is considered the domain of an injective function that maps its elements to a domain of angle α. In this study, α assumes discrete and monotonous values in the laminar region, with magnitudes ranging from 0.965π/2 to –0.0038π/2 with a decrease of 0.2 π (see Figure 4). In Figure 4, xC is determined considering ReC = 5 × 105, and the flow velocity ratio . Particularly in the case of Figure 4, ν = 8.5 × 10−7 m2/s and = 1.7 × 10−5 m2/s. As can be noticed, a rational equation correlates α(x) and Rex as follows, which presents a standard error of estimate of 2.597 × 10−4. Figure 4 also shows the first point in the turbulent zone, the internal point P.
Figure 5 shows the distributions of the x and y dimensionless direction velocity components of a flat plate boundary layer, determined for Rex ≤ 500,000. The dimensionless direction curves, no matter the value of U∞, follow the principle of similarity, and as expected, they show a typical parabolic profile. Betchov and Criminale [23] determined the y-direction velocity component in an approximated way. Their curve is concordant with Prandtl's theory. It presents null values on the plate, as well as on the limit of the boundary layer. However, it seems too symmetric compared with the one in Figure 5. Figure 5 shows that the dimensionless x-direction velocity magnitudes are much larger than those in the y-direction.
5.1. The Present Model and the Sensitivity to Regime Transition
In page 132 in Schlichting [24], there is a peculiar note on the dimensionless boundary layer thickness : “this dimensionless thickness remains constant as long as the boundary layer is laminar, and its numerical value is nearly that given in Equation (7.35). At large Reynolds number U∞x/ν, the boundary layer ceases to be laminar and a transition to turbulent motion occurs.” Oddly, the classical Blasius’s flat plate equation does not indicate the transition limit, whatever the magnitude of the Reynolds number. Differently, the present solution (see Figure 6, which shows the x- and y-direction dimensionless velocities in terms of angle α) indicates that some points occur in the turbulent zone whereas in Figure 5, all points remain in the laminar zone; in Figure 6, there are points corresponding to the negative values of α. In Figure 6, we cannot see evidence of any perturbation inside the bulk of the fluid flow, inherent to the turbulent regime. On the other hand, in Figure 7, which presents the boundary layer thickness in terms of α, it is clear the configuration of an abrupt rupture in the laminar curve, at the point in which Rex overpasses ReC. The internal point P indicates the place of the occurrence. The fact that η∞ becomes negative may be considered unexpected, but it registers the transition regime, originating an inverted boundary layer, where the dimensionless velocities go on in perfect development to satisfy equations (20) and (36), simultaneously.
5.2. The Contradiction between the Existence of a Smooth Transition Considered by Schlichting and Kestin [25] and the Results Shown in Figure 7
We have earlier referred to Schlichting and Kestin’s [26] comment on the flat plate boundary layer transition. Although they did not mean a comparison with Blasius´s theoretical results, but with those by Hansen [18], they implicitly did so since Hansen’s theory of the laminar flow region is based on Blasius’s. In reality, Hansen [18] seems to believe that the transition to turbulence could be shown in a curve as described by Blasius’s theory (laminar region) and followed by another which incorporates the (1/7)th power velocity profile law. And, to promote experimentally, the turbulence “… a rounded strip of wood was applied to the front edge of the glass plate.” Thus, Hansen’s [18] graphic, presented in Figure 8, shows a smooth and continuous curve that begins in the laminar region and goes on through the turbulent region, exhibiting a geometric aspect that contrasts with the behavior of the curve in Figure 7.
See this curve in Hansen [18] and in Figure 2.19 in Schlitching and Kesting [26] and in Figure 2.4 in Schlitching and Gersten [24].
6. Practical Implications
6.1. Concerning the Persistence of a Laminar Flow up to Infinite
The spontaneous existence of a transition point between the laminar and the turbulent zone contradicts Taylor’s [15] statement that the laminar regime can persist up to an infinite Reynolds number in the absence of any external perturbation. It is somewhat difficult to understand how Pfenninger [25], keeping the regime laminar, artificially achieved a result that is considered the maximum Reynolds number Rex)T = 50.3 × 106, ever determined. He used the suction effect through individual slots, spaced relatively far apart, attending to the slots' design and their locations, to avoid disturbances around them. It has been common the use of external devices for keeping the flow in laminar condition. As explained by Jovanovic [14] and quoting Hinze [27], the elements introduced to stabilize the flow allowed the fluid to accelerate in the entry region of the pipe in such a way that the mentioned value of Rex)T should be reduced by at least a factor of ten, compared with the data obtained in the flat plate boundary layers.
6.2. The Natural Transition from Laminar to Turbulent Regime
Considering the enormous influence of the laminar boundary layer over extended areas of an airplane on the decrease of the associated friction drag, uncountable numerical and experimental works have been devoted to the theme (Alam and Sandham [28], Gopalarathnam and Selig [29], Amoignon et al. [30], and Xu et al. [31] among many others). Generally speaking, what we understand from the technical literature is that the laminar-turbulent transition often originates from an instability phenomenon induced by a mechanical intervention in the flow movement, such as a rough surface, or when a laminar flow over a solid surface encounters a strong enough adverse pressure gradient causing its separation from the surface, Alam and Sandham [28]. In a flat plate boundary layer, the growth of small amplitude can trigger perturbations on the laminar-turbulent transition zone, Xu et al. [31], Liu and Liu [32]. According to our results, however, even considering a smooth flat plate without any external disturbance agent, it is possible to originate a flow disorder that can justify the initialization of the Tollmien–Schlitching waves.
6.3. The Classical Relation between Fluid Friction and Heat Transfer
The shear stress on the wall of a flat plate, for example, can be expressed in terms of a friction coefficient, Cf, i.e.,or by the following expression:
Equation (48) associated with the velocity distribution of equation (50) (see page 41 in Rohsenow and Choi [33])becomes
The association of equation (51) with the relation derived from the boundary layer thickness (see page 124 in Holman [34]) i.e.,gives
For the plate heated over the entire length, it is possible to write(see Bennett and Myers [35], p. 332) where is the local Nusselt number.
Equation (54) may also be rewritten as
Comparing equations (53) and (55) and considering that both expressions are alike except for a difference of about 3% in the constant Holman [34], it is possible to write
Equation (55) expresses the relation between fluid friction and heat transfer, i.e., the Reynolds analogy. Based on what was discussed, however, the “magic” number 0.332, valid for the whole plate, does not make any sense. It is also strange to accept that equation (52), coming from an approximate method (von Kármán integral relation), can replace the results originated from the exact solution (Blasius equation). Therefore, this analogy should be reviewed.
6.4. The Classical way to Solve the Falkner–Skan Equation
Considering that the Blasius boundary layer solution can be generalized by taking a wedge at an angle of attack λπ/2 and then stating that the outflow has the formwhere L is a characteristic dimension and n is a dimensionless constant, we determine
If n = 0, it becomes analogous to the Blasius problem, where the angle of attack corresponds to zero radians. Therefore, not by coincidence does the solution of the Falkner–Skan equation follow the same steps used to solve the classical Blasius equation (see Asaithambi [36] or Fazio [37]), first a similarity variable η is defined; then, considering a stream function, the Falkner–Skan equation is written as a nonlinear ODE. From this point, there is an identity between the solution of the Blasius equation and that of Falkner–Skan. In this solution, a common step is to use a fourth boundary condition to solve an ODE of the third order. expresses the “missing” fourth boundary condition, where c is a constant. As discussed in Jaguaribe [4, 5], this mathematical choice turns the solution incorrect. The Falkner–Skan equation is related to a large number of industrial processes, Asaithambi [38], and is also involved in many other problems related to mixed convection and porous media, Hayat et al. [39], or Heat Transfer Analysis, Kuo [40], as well as in so many other technical and scientific fields. Thus, the Falkner–Skan solution certainly deserves reconsideration.
7. Conclusions
It is very easy to see that in Blasius’s classical solution,(1)β is considered an unknown parameter. This fact is enough to demonstrate that the flat plate Blasius’s solution could never be mathematically and physically admitted as correct.(2)The dimensionless thickness, η∞, has a unique value for the entire plate. This has paved the way for defining the limits of the boundary layer in terms of a percentual of the free stream velocity, i.e., 0.99 , most probably because of the lack of options.
On the other hand, nobody has admitted that in the design of a flow past a flat plate (similar to that of Blasius); it is necessary to associate a well-defined boundary condition with the uniform flow velocity at the leading edge and avoid the use of the continuity equation in the differential form.
To build up a consistent model and a mathematically coherent solution, the original system of PDE—equations (1) to (3)—was converted into another ODE system, using a new method that resolves many inconveniences of the classical solution. And, even though the resultant expressions were completely different, the new system of ODE allowed some of the classical equations, e.g., (9)–(11) to be reproduced, making p = q = ½. The new system of equations permits an exact, easy, and mathematically correct solution, presenting results in complete agreement with Prandtl's boundary layer concept.
In addition, it becomes evident that(a)It is inappropriate and nonsensical to define the boundary layer thickness δ, as the distance related to 0.99U∞.(b)The concepts of displacement thickness and momentum thickness lose their meaning when examined on the grounds of the present flat plate boundary layer theory.(c)η∞ is a function of α or Rex.(d)The plot of η∞ vs. Rex turns clear the existence of a transition point between the laminar and the turbulent zone.(e)Using equation (10), it is possible to outline the interface between the viscous fluid and the external potential flow region.(f)Prandtl’s boundary layer concept is fully adjusted to the y-direction dimensionless velocity.(g)It is remarkable that for a given Rex > ReC, there occurs a discontinuity in the laminar curve (see Figure 7) signaling the existence of perturbations that can justify the origin of the Tollmien–Schlitching waves.(h)The solution of the new flat plate boundary layer equation demystifies the number 0.332, implying the need to revisit the Reynolds analogy, as well as the von Kármán integral relation associated with the classical flat boundary layer equation.(i)Since the solution of the Falkner–Skan equation follows the same steps used to solve the classical Blasius equation, it should also be reevaluated.
Data Availability
All the data published in the article are available in the cited references.
Conflicts of Interest
The author declares that there are no conflicts of interest.