In this chapter, the theoretical description of forced laminar flow over a flat plate is revisited. In many texts (see, for example, Schlichting and Gersten [27], p. 148 or White [28], p. 466), an order of magnitude argument is used to claim that the momentum equation normal to the wall reduces to just the normal pressure gradient term equal to zero. This is not a correct statement of conservation of momentum. If the normal pressure gradient were zero, the normal velocity would be zero. This would mean that momentum conservation would not occur for boundary layer flow since the momentum lost due to the slower boundary layer flow on a wall is missing. Conservation of momentum, therefore, requires some of the momenta near the wall must be redirected as a normal velocity away from the wall and that the driving force for this must be a pressure gradient in the y-direction.

So why make this claim? To understand the reasoning behind this misrepresentation of the y-momentum equation, we need to look at the theoretical work of Falkner and Skan [26]. Falkner and Skan developed a similarity solution for the flow on a flat plate with an imposed pressure gradient using just the x‑momentum equation and the mass conservation equation. With the velocity solution in hand, the thinking has been that there is little need for the y-momentum equation. However, what is needed is a justification for Falkner and Skan’s use of the differential form of the Bernoulli equation in the x‑momentum equation. The apparent intent of indicating that the y‑momentum equation reduces to the y‑pressure gradient equal to zero is to justify changing the partial into a normal derivative for the x‑pressure gradient in the x‑momentum equation (White [28], p. 466). While this provides the justification for using the differential form of the Bernoulli equation, it also implies the normal velocity is assumed to be zero. Yet, when the Falkner-Skan x‑momentum equation is solved, the calculated normal velocity v(x,y) is certainly not zero. This y-pressure conundrum can be avoided by simply “assuming” the differential Bernoulli equation substitution is justified. Justification comes in the form of experimental verification (see Chapter 4.3 above, for example).

What has been lost in this widespread dismissal of the y-momentum equation is an understanding of the nature of the y-pressure gradient and the generation of the normal velocity. When fast-moving inlet flow encounters slow-moving boundary layer flow, a pressure imbalance is created in the boundary layer region. The slow-moving flow close to the wall induces a normal flow (y‑direction) away from the wall due to the pressure gradient formed by this slow flow - fast flow imbalance. Weyburne [25] pointed out that the y-momentum equation provides the means to determine the pressure gradient in the y‑direction once the x‑momentum equation has been solved.

To discern the true nature of the normal to the wall pressure gradient for laminar flow, we present a Falkner-Skan [26] analysis developed by Weyburne [25]. The Falkner-Skan solution obtained from the Prandtl x‑momentum equation (parallel to the wall, flow direction) is used to calculate the normal to the wall y‑pressure gradient using the y‑momentum equation. The y‑pressure gradient for the Blasius [9] as well as a more general Falkner-Skan [26] solution, can be obtained in this way.

The Falkner-Skan formulation is often identified with flow along a wedge since the inertial flow just above the boundary layer edge looks similar to the pure inertial flow past a displacement thickness broadened wedge. What is not widely appreciated is that the same Prandtl x‑momentum equation used by Falkner-Skan is also used to describe laminar flow along a flat plate that has an imposed pressure gradient in the flow direction, a flow commonly encountered in wind tunnels. The fast-moving free stream flow running into the slow-moving flow close to the wall induces a pressure imbalance along the wall which takes the form of x and y pressure gradients. Laminar flow with an imposed pressure gradient along the wall was first studied theoretically by V. M. Falkner and Sylvia Skan in 1930. Falkner-Skan [26] derived similarity solutions to the Prandtl momentum equations for this boundary layer flow situation. The Falkner-Skan analysis is outlined in most textbooks and on numerous online sites. However, we have found those discussions to be incomplete. As such, in what follows, the relevant equations are outlined in detail.

The Falkner-Skan analysis uses a stream function approach. A critical assumption to the whole theoretical development is that we assume the velocity in the flow direction, u(x,y), and the velocity normal to the floe direction, v(x,y), can be decomposed into a product of a length and velocity x-dependent functional times and a scaled y‑functional. That is, assume that a stream function ψ(x,y) exists (see Panton [29], p.543) such that

where f(η) is a dimensionless function that only depends on the scaled y-position, η= y/δ_s(x), where δ_s(x) is the length scaling parameter, and where u_s(x) is the velocity scaling parameter. The stream function must satisfy the velocity conditions

Falkner-Skan's [26] first step was to start with the Prandtl x-momentum equation given by

(where ν is the kinematic viscosity, ρ is the density, and p(x,y) is the pressure) and then replace the pressure gradient term with the differential form of the Bernoulli equation given by

where u_e(x) is the boundary layer edge velocity.

Falkner and Skan’s next key insight was to show that similarity solutions to the x-momentum equation are obtained by non-dimensionalizing the x-momentum equation using scaling parameters that are simple power functions of x of the form

where x₀, a, b, m, and n are constants. Taking the length scaling parameter as δ_s(x) and the velocity scaling parameter as u_s(x) lets Eqs. 5.3 (with 5.4) be completely non-dimensionalized. While Falkner and Skan assumed us(x) is equal to the ue(x), Weyburne [30] recently proved that if similarity is present in a set of velocity profiles, then the length scaling parameter δ_s(x) must the displacement thickness δ₁(x) and the velocity scaling parameter u_s(x) must be u_e(x), the boundary layer edge velocity. This holds for all 2-D bounded boundary layer flow (see Similarity Chapter).

Combining Eqs. 5.1, 5.4 and 5.5 means that the Prandtl x-momentum equation can be put into nondimensional form as

where α and β are simple functions of δ_s(x) and u_s(x), or in this case δ₁(x) and u_e(x). This is the well-known Falkner-Skan x-momentum equation. It is easily verified that α and β terms become constants when m and n in Eq. 5.5 satisfies the condition m+2n-1=0. Under this constraint, α and β reduce to

The normal approach to solve Eq. 5.6 is to set

which means the x-component of the momentum balance (Eq. 5.6) becomes

Programs to solve Eq. 5.9 are widely available (note that sometimes ab² /ν is set to 2/(m+1) so that α =1, β=2m/(m+1). This form is considered in Section 5.1 below). Once this equation has been solved for f₁ as a function of η₁, then one can recover the true η, f, f', and f'' by noting

and where f''' is recovered using Eq. 5.6. With f(η) and its derivatives as a function of η for a given α and β, it is a simple matter to back out u(x,y) and v(x,y).

Weyburne [25] pointed out that with the velocities in hand, one can use these calculated velocities in the Prandtl y-momentum equation to obtain the y-pressure gradient. The Prandtl y‑momentum equation for laminar flow is

Substituting in the Falkner-Skan stream function result from Eqs. 5.1-5.5 into Eq. 5.11, then the full Falkner-Skan version of the reduced y-momentum equation is given by

This is the Falkner-Skan y-pressure gradient expression. Putting this into the f₁ and its derivative version, together with the fact that for this case ν /(2ab² )=1/2, Eq. 5.12 becomes

The Blasius y-pressure gradient version of Eq. 5.13, corresponding to m=0 and u_e(x) equal to the inlet velocity u₀, is given by

The Blasius y‑pressure gradient asymptotes to a value of 0.4302 at large η-values (one-half of the normal velocity asymptote value). As η goes to zero, the scaled y-pressure gradient goes to 0.16603 (one-half of the wall shear stress numerical value). The Blasius y‑pressure gradient was shown to be in good agreement with computer simulation results for exterior-like laminar flow along a flat plate (see Fig. 2.3b and Weyburne [5]).

With f₁ and its derivatives in hand, the reduced velocities and reduced pressure gradient are calculated and plotted in Fig. 5.1 for the Blasius flow situation. The velocities are the well-known Blasius values. For the first time, the behavior of the Blasius y-pressure gradient in the boundary layer is also revealed. For the Falkner-Skan flow case where m is non-zero, Weyburne [25] presented example plots for various m-valued solutions.

5.1 Alternate y-pressure Gradient Expression

The y-pressure gradient equation, Eq. 5.13, was obtained because we set setting ab² /ν to 1. A slightly different expression is obtained when ab² /ν to is set to 2/(m+1) so that α =1, β=2m/(m+1). The α =1, β=2m/(m+1) case shows up in some texts. The ultimate u(x,y), v(x,y), and y-pressure gradient backed-out expressions will be the same for the two cases. So, to be complete, setting ab² /ν = 2/(m+1) in Eq. 5.12, the adjusted expression for the α =1, β=2m/(m+1) case is given by

where g is the solution to the differential equation given by

with boundary conditions g(0)=g'(0)=0 and g'(z )=1 as z goes to infinity.