Falkner-Skan Experimental Verification Failure for Airflow on a Wedge

Abstract

Velocity profiles along a wedge obtained by computational fluid dynamic simulation of laminar airflow are compared to the Falkner-Skan solution to boundary layer flow. The experimental flow direction velocity profiles appear to show a good match to the theoretical predictions but, on closer examination, the comparison fails. The results also indicate that the Falkner-Skan scaled experimental normal velocity and the scaled normal pressure gradient profiles do not show profile similarity for the laminar airflow on a wedge. The one exception that does show good profile similarity for both the velocities and the pressure gradient is when the flow direction pressure gradient is zero, i.e. the Blasius flat plate flow model. Examining various profile plots indicates that the failure is due to a non-constant pressure gradient along the wedge wall that fundamentally changes the shape of the profiles at the different stations along the wall. This makes airflow velocity profile similarity impossible.

1. Introduction

Since the introduction of the Falkner and Skan [1] paper in the 1930s, there have been thousands of journal articles written on different aspects of the Falkner-Skan solution to boundary layer flow (a Google Scholar search in May 2023 yielded 9800 article results). Along with the Blasius solution, it is covered in virtually every fluid flow textbook in existence. However, a search of the literature reveals the Falkner-Skan theoretical solution has never been verified experimentally. The fact that the Falkner-Skan solution has not been experimentally verified is surprising given that Falkner-Skan-designed wind tunnel experiments and measurements of the velocity profiles are widespread and even happen as part of undergraduate fluids courses.

It is perhaps the case that the flow community has felt that experimental verification was unnecessary given the routine nature of the measurements and the apparent agreement with Falkner-Skan theory. However, a closer look reveals that one reason for the lack of verification is due to the limitations of wind tunnel measurement capabilities. Velocity measurement noise limits high accuracy and pressure measurement probe size prevents measurements inside of the boundary layer from even being made. Furthermore, the normal velocity in a boundary layer is typically orders of magnitude smaller than the main flow which makes measurements of the normal velocity profile all but impossible. This means even with careful measurements of the velocity profile in the flow direction, the comparison to the Falkner-Skan theory would not be complete or fully definitive.

As a consequence of the wind tunnel difficulties, for the work herein, we turn to computational fluid dynamic (CFD) simulations of laminar flow on a wedge. The wedge simulation is chosen since the Falkner-Skan formulation is often identified as flow along a wedge in the literature, and in any case, represents a simple geometry that produces a pressure gradient in the flow direction along the wedge plate. Due to the simplicity of the flow model, we assume that the CFD simulations for laminar flow on a wedge are representative of the real-world situation without validation.

2. A Falkner-Skan review

Before proceeding to the simulation section, we first review the Falkner-Skan model for boundary layer flow. The Falkner and Skan [1] solution describes the steady boundary layer that forms for 2D laminar flow along a flat wall in which a pressure gradient is imposed in the flow direction. The Falkner-Skan theoretical approach to solving the boundary layer equations employs a series of key assumptions. The first key assumption, one that underlies the whole Falkner-Skan approach, is that there is a set of normal-to-the-wall velocity profiles at different stations along the wall that appear similar when scaled with a properly chosen length and velocity scaling parameters. That is, there must be a region along the wall in which plots of the scaled profiles collapse to a single curve. The velocity profile is defined as a series of velocity values taken from the wall out to the free stream in a direction normal to the wall. This non-dimensionalization process reduces the set of partial differential flow governing equations to an ordinary differential equation greatly simplifying the solution process.

Falkner-Skan employed a stream function approach to describe the velocity in the flow direction, u(x,y), and the normal velocity, v(x,y). The Falkner-Skan second key assumption is that a stream function Ψ(x,y) exists (see [2], p. 543) such that

$$\psi \left(x,y\right)={\delta }_s\left(x\right)u_s\left(x\right)f\left(\eta \right),$$E1

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 flow direction is along the wall in the x-direction with the y-direction as the normal to the wall direction. The stream function must satisfy the conditions

$$u\left(x,y\right)=\frac{\partial \psi \left(x,y\right)}{\partial y},\text{and}v\left(x,y\right)=-\frac{\partial \psi \left(x,y\right)}{\partial x}.$$E2

The third key assumption Falkner-Skan made was that the scaling parameters δ_s(x) and u_s(x) are simple power-law functions of the distance along the wall given by.

$${\mathit{u}}_{\mathit{s}}\left(\mathit{x}\right)=\mathit{a}{\left(\frac{\mathit{x}}{\mathit{L}}\right)}^{\mathit{m}}\text{and}{\delta }_{\mathit{s}}\left(\mathit{x}\right)=\mathit{b}{\left(\frac{\mathit{x}}{\mathit{L}}\right)}^{\mathit{n}},$$E3

where a, b, L, m, and n are constants. The constant L is the model reference length, e.g. the wedge length for the case where a wedge is used to induce the non-zero pressure gradient.

The fourth key assumption Falkner-Skan made was that Prandtl momentum equations apply rather than the full Navier-Stokes versions. Prandtl [3] used an order of magnitude argument to reduce the full Navier-Stokes momentum equations to a much simpler set of momentum equations known as the Prandtl momentum equations. Using these dimensional arguments, Falkner-Skan was then able to obtain velocity solutions using just the stream function based Prandtl x momentum conservation equation since the assumed y momentum contribution is negligible.

The Falkner-Skan fifth key assumption is related to the pressure gradient term in the Prandtl x-momentum conservation equation. Falkner-Skan assumed that the pressure gradient term can be replaced by the differential form of the Bernoulli equation in the high Reynolds number limit. This takes the form

$$-\frac{1}{\rho }\frac{\partial P\left(x,y\right)}{\partial x}=u_e\left(x\right)\frac{{\mathit{du}}_e\left(x\right)}{\mathit{dx}}.$$E4

Implicit in this substitution is that Falkner-Skan assumed the velocity scaling parameter u_s(x) is given by the velocity at the boundary layer edge u_e(x).

With the above assumptions, it is possible to reduce the boundary layer x-momentum equation to a simple ordinary differential equation that can be solved for f(η) and its derivatives by standard differential equation techniques. The resulting expressions for the u(x,y) profile, the scaled v(x,y) profiles, and the scaled y-pressure gradient profile are given by x-momentum by

$$\begin{array}{l}u\left(x,y\right)=u_s{f}^{\prime }\left(\eta \right),\\ v\left(x,y\right)=\frac{u_s{\delta }_s}{x}\left[-\frac{\mathrm{1}+m}{\mathrm{2}}f\left(\eta \right)+\frac{\mathrm{1}-m}{\mathrm{2}}\eta f^{\prime }\left(\eta \right)\right],\text{and}\\ \frac{\mathrm{1}}{\rho }\frac{\mathit{dP}}{\mathit{dy}}=\frac{u_s^{\mathrm{2}}{\delta }_s}{x^{\mathrm{2}}}\left[\begin{array}{l}-\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{3}m-\mathrm{1}\right)f^{″}\left(\eta \right)+\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}-m\right)\eta f^{‴}\left(\eta \right)-\frac{\mathrm{1}}{\mathrm{4}}{\left(m+\mathrm{1}\right)}^{\mathrm{2}}f\left(\eta \right)f^{\prime }\left(\eta \right)+\\ \frac{\mathrm{1}}{\mathrm{4}}{\left(m-\mathrm{1}\right)}^{\mathrm{2}}\eta {f^{\prime }}^{\mathrm{2}}\left(\eta \right)-\frac{\mathrm{1}}{\mathrm{4}}\left(m^{\mathrm{2}}-\mathrm{1}\right)\mathit{\eta }\mathit{f}\left(\eta \right)f^{″}\left(\eta \right)\end{array}\right],\end{array}$$E5

where m is the power coefficient from Eq. (3) and the primes represent derivatives of f(η) with respect to η. The y-pressure gradient expression was developed by substituting the Eq. (5) velocity expressions into the y-momentum conservation equation [4]. Eq. (5) represents the velocity and pressure gradient expressions that will be compared to the CFD simulation results.

3. Experimental design

The Falkner-Skan experimental verification process is based on OpenFOAM CFD simulations of 2D exterior-like laminar airflow along a wedge. The depiction of the simulation geometry is shown in Figure 1. The bottom solid line is the wedge wall surface of length L and has a wall boundary condition. The top and bottom black dashed lines are assigned a zero gradient boundary condition. The bottom dashed black line is the inlet region of length L_in. The flow inlet and outlet regions, height H, are slanted θ degrees to the wall normal. The velocity in the x-direction is desigmated u(x,y) and the velocity in the y-direction is designated v(x,y). The flat wedge geometry utilizes a 3300 × 1000 mesh that emphasizes the near wall and leading-edge regions. A total of five simulations are performed at five wedge angles, θ = −5°, 0°, 5°, 10°, and 20°. The inlet RT airflow is set to u₀ = 0.625 m/s resulting in a Re_x = 5.0 × 10⁵ at the end of the 12 m long wedge wall. The low velocity enabled the easier-to-converge incompressible option to be used. The simulation extents are based on a series of preliminary test cases. The height extent value is set so that exterior-like flow is generated as judged by the asymptotic approach of the normal velocity profile to zero in the free stream region above the wall. The inlet location was increased until there were minimal changes in the boundary layer along the wall. This resulted in the inlet (bottom black dashed line) being set to L_in = 80 m and the height being set to H = 500 m. The simulations are iterated until the velocity and pressure residuals all drop below a value of 1 × 10⁻⁶. To be consistent with the Falkner-Skan approach, the velocity and pressure profiles are measured normal to the wall.

Figure 1.

Depiction of the 2D laminar flow simulation geometry on a wedge.

4. Parameter extraction

An example of the simulation result is shown in Figure 2 for the of θ = 10° wedge angle case. Each velocity profile consists of a series of velocity values taken from a point on the wedge surface to a point deep into the free stream in a direction normal to the wedge plate wall (y-direction). u₀ is the inlet velocity and L is the wedge length. The experimental u(x,y) velocity profiles shown in Figure 2a show a much more nuanced boundary layer picture than is typically discussed in the literature. The question becomes what part of this complicated boundary layer picture corresponds to the Falkner-Skan boundary layer model. Falkner and Skan made it clear in their paper that the formulation is meant to describe the near-wall portion of the profiles where viscosity is important. Examination of Figure 2a reveals that after the initial jump from u(x,0) = 0, the profiles all asymptote to a velocity plateau near y/L = ∼0.01. This velocity corresponds to the velocity at the viscous boundary layer edge, u_e(x). For reproducibility, the actual u_e(x) values are extracted using a method based on calculating the viscous thickness. In this interpretation, the scaling parameter u_e(x) is given the value of the velocity, u(x,y), evaluated at y = δ_v(x), where δ_v(x) is the four-sigma viscous thickness. The four-sigma δ_v(x) location is calculated by the integral moment method [5]. It corresponds to the place where the majority of the viscous momentum contributions have vanished. The four-sigma δ_v(x) calculated location values ranged from y/L = 0.06 to y/L = 0.1 corresponding to the plateau regions in Figure 2a.

Figure 2.

(a) simulated 2D laminar airflow velocity profiles for u(x,y) at nine locations along the wedge plate for the θ =10° wedge. (b) nine normal velocity profiles, v(x,y), at the same locations.

The extracted u_e(x) values are plotted in Figure 3 as red circles for the θ = 10° case. The seven-profile subset x/L = 0.25 to x/L = 0.75 is chosen based on this set yielding the power law fit with the smallest adjusted R-square value. The blue line shows the power law fit, Eq. (3), with an extracted power-law fitted exponent, m = 0.07253, for this case. The extraction process for the θ = 0° Blasius case determined that, as expected, the fit to m ≅ 0 is very good for this case.

Figure 3.

The red circles are the extracted boundary layer edge velocity values, u_e(x), at seven locations along the wedge plate and the blue line is the power law fitted result.

With the u_e(x) power law fitting parameter in hand, the next step is to extract the relevant fitting parameters for the length scaling parameter δ_s(x) in Eq. (3). This parameter was never identified in the original Falkner-Skan article. However, to reduce the x-momentum equation to a simple differential equation, Falkner-Skan showed that power-law exponents in Eq. (3) are not independent but must satisfy the equation given by that

$$m+2n+1=1,$$E6

using the fitted m value from Figure 3, then for the θ = 10° case, n = 0.463735. The b parameter value is not fixed by the theory and can therefore be adjusted for best fit between the Falkner-Skan solution and the scaled experimental profiles.

5. Profile comparison

With the scaling values determined, the next step of the verification process is to verify profile similarity. This is accomplished by plotting the experimental scaled u(x,y) profiles, the scaled v(x,y) profiles, and the scaled y-pressure gradient profiles and comparing these to the Falkner-Skan theoretical profiles given by Eq. (5). Falkner-Skan verification confirmation requires that the set of experimental profiles plotted as y/δ_s(x) versus u(x,y)/u_s(x), for example, should all collapse to a single curve corresponding to the profile given in Eq. (5). In Figures 4–6, the results for the θ = 10° case and the θ = 0° cases are shown. In these figures, the Eq. (3) fitted values to the scaling parameter u_e(x) are used. The results for the θ = −5°, θ = 5°, and the θ = 20° simulations are not shown but show similar behavior to those in Figures 4a–6a. In each case, the Falkner-Skan f(η) and its derivatives are calculated with a standard Fortran program. The b parameter is adjusted to make the Falkner-Skan u(x,y) velocity and the experimental scaled profiles overlap in Figure 4a.

Figure 4.

(a) scaled u(x,y) velocity profiles at seven locations along the wedge plate for the θ=10° case, (b) scaled velocity profiles at seven locations along the wedge plate for the θ=0° Blasius case. The dashed red line is the Falkner-Skan profile.

Figure 5.

(a) scaled v(x,y) velocity profiles at seven locations along the wedge for the θ=10° case, (b) scaled velocity profiles along the wedge plate for the θ=0° case. The dashed red line is the Falkner-Skan profile and the velocity scale is δ_su_s/x.

Figure 6.

(a) scaled dP/dy profiles at seven locations along the wedge plate for the θ=10° case, (b), the scaled dP/dy profiles along the wedge plate for the θ=0° case. The dashed red line is the Falkner-Skan profile and the y-scale is δ_su_s²/x².

The Falkner-Skan scaled u(x,y) profiles in Figure 4a show reasonably good overlap with the theoretical curve whereas the Figure 4b Blasius θ = 0° case shows excellent overlap. However, a closer look at the Figure 4a result tells a different story. In Figure 7 we zoom in on a small region of the velocity profile curves from Figure 4. The θ = 10° curves no longer overlap at the expanded scale indicating profile similarity is absent.

Figure 7.

(a) a zoomed-in plot of the scaled u(x,y) velocity profiles from Figure 4a for the θ = 10° case, (b) a zoomed-in plot of the scaled u(x,y) velocity profiles from Figure 4b for the θ = 0° case.

The normal velocity and the y-pressure gradient experimental profiles, Figures 5a and 6a, also do not exhibit profile similarity and do not collapse to the Falkner-Skan profiles. All of the non-zero angle cases show the same type of non-similar behavior demonstrated in Figures 5a, 6a and 7a. The conclusion is that: 1) although the Falkner-Skan solution for laminar airflow does a reasonable job of approximating the u(x,y) profiles, it fails upon closer examination and 2) the normal velocity v(x,y), and the y-pressure gradient profiles do not show profile similarity using the Falkner-Skan scaling parameters.

6. The explanation

The key to understanding the nature of the failure is the one confounding case that does show good profile similarity for both the velocity and pressure gradients. When the flow direction pressure gradient is zero, i.e. the m = 0 Blasius flow model, profile similarity is obtained as shown in Figures 4b, 5b, and 6b. The fact that the similarity failure occurs for the non-zero-pressure gradient cases but similarity does occur for the zero-pressure gradient case implicates the flow direction pressure gradient as the problem area.

One possibility is that the power law-based Bernoulli equation approximation consisting of Eqs. (3) and (4) is a problem area. An examination of the scaled profiles reveals that the only way to force similar behavior is by allowing the Falkner-Skan δ_s(x) and u_s(x) scaling parameters a and b constants to change with x. This compromises the whole attempt to reduce the partial differential flow governing equations to the Falkner-Skan ODE. Even if one ignores the Falkner-Skan δ_s(x) and u_s(x) scaling parameters and tries different scalings to try to force similar behavior (for example V_max and y at V_max for the velocity and length scaling in Figure 5a), one discovers that the problem is that is no simple x and y shifts that can result in the velocity profiles or the y-pressure gradient profiles at different stations to overlap completely. The profiles have a different shape at each station along the wall!

Since the Blasius case does show similarity and the non-zero angle cases do not means that the x-pressure gradient is affecting the shape of the profile. What sets the x-pressure gradients apart for these cases? The x-pressure gradients for all five test cases are plotted along the wedge wall just above the viscous boundary layer in Figure 8. One of the clear observable differences is that only the θ = 0° case is showing a constant pressure gradient along the wall. This becomes important when one considers that the relationship between the second derivative of the velocity at the wall and the x-pressure gradient is given by the Prandtl x-momentum equation evaluated at the wall (y = 0) as

Figure 8.

The experimental x-pressure gradient curves for the five simulation cases measured at y/L = 0.012 along the wedge surface which is just above the viscous boundary layer thickness.

$$\mathit{v}\frac{{\partial }^{2}u\left(x,0\right)}{\partial y^2}=\frac{1}{\rho }\frac{\partial P\left(x\right)}{\partial x}.$$E7

This means that the wall value of the second derivative of the velocity can vary from a negative value to a positive value depending on whether we have an adverse pressure gradient or a favorable pressure gradient. The x-pressure gradient in the y-direction of the boundary layer region is known to be mostly constant. Hence, the non-constant pressure gradients observed at the boundary layer edge is changing the shape of the second derivative of the velocity as we move along the wall. This is illustrated in Figure 9 in which the second derivatives of the u(x,y) velocity profiles from Figure 4 are plotted for the θ = 10° and θ = 0° cases. No simple x and y scaling will make the θ = 10° profiles look similar whereas the θ = 0° profiles look Gaussian-like and visually appear to have a similar shape. If the second derivative profile shapes are different then velocity profiles obtained by twice integrating will be different. This is what leads to the non-overlapping behavior evident in Figure 7. The non-constant x-pressure gradient inducing profile changes explains all the behavior observed in Figures 4–7. The first key assumption of the Falkner-Skan model is that there must be a region along the wall where the shape of the profiles from different stations are the same. This is not the case for CFD laminar airflow cases on a wedge observed in this effort.

Figure 9.

(a) experimental scaled second derivative profiles for the θ=10° velocity profiles from Figure 2, (b) the scaled second derivative profiles for the θ=0° Blasius case. The y-scale in both plots is u₀/L²

7. Falkner-Skan is not wedge flow

The Falkner-Skan formulation is often identified as fluid flow along a wedge in the literature. The reason is that the potential flow solution to flow on a wedge appears similar to the outer flow predicted by the Falkner-Skan formulation (see [6], p. 156). It is certainly true that the wedge induces a pressure gradient in the flow direction along the wedge surface. However, the assumption in the literature has been that the viscous boundary layer shape remains the same for any induced pressure gradient. What is new here is the realization that laminar flow on a finite-sized wedge does not induce a constant pressure gradient along the wall and that the profile shape along the wall changes with a changing pressure gradient.

The trend lines in Figure 8 indicate that the pressure gradient changes become larger along the wedge as the wedge angle becomes larger but the changes are observable at even small angles. The leading edge induced pressure changes are expected to occur even if one moves the exit past the wedge endpoint. We note that the geometry used in this report is a geometry that is at least possible to mimic in a wind tunnel. Therefore, the failure cannot be attributed to the choice of the wedge angle or that the simulations are not representative of a realizable flow situation.

It appears that boundary layer similarity for airflow is limited to cases where the pressure gradient is constant. At present, the only case that can be experimentally generated that satisfies that condition is the unbounded exterior-like ZPG flow on a flat plate case. Other constant pressure gradient cases should also work but the problem is that it is not obvious how one would create a constant non-zero pressure gradient in a wind tunnel or in a simulation due to the leading-edge pressure bubble. We have verified that a thick flat plate also has a leading-edge pressure bubble but further along the wall, the pressure gradient becomes relatively constant. One geometry that is presently being explored is the case where the power law exponent in Eqs. (3) and (4) is m = 0.5. The Bernoulli equation approximation for the pressure gradient in Eq. (4), with m = 0.5 (half angle = 60°), would indicate the pressure gradient should be constant past the leading-edge pressure bubble.

This realization that the velocity profile shape along the wall changes with a changing pressure gradient laminar has implications for more than just the wedge flow case. There are other flow cases in the literature that assume profile similarity is possible where non-constant pressure gradients are present. Flow in a converging or diverging channel (sink flow) is an example. Thus, it is important that flow situations in the literature that assume the profile shape remains the same for cases in a non-zero pressure gradient need to be revisited.

8. Conclusions

Contrary to expectations, the simulated laminar airflow experiments along a wedge using computational fluid dynamic (CFD) do not compare well to the Falkner-Skan solution to boundary layer flow. While the experimental flow direction velocity profiles appear to show a good match to the theoretical predictions, on closer examination the comparison fails. The success of the one exception that does show good profile similarity for both velocities and the pressure gradient, i.e. the Blasius flat plate flow model, is attributed to having a constant pressure gradient in the flow direction. All other non-zero angle cases show non-constant pressure gradients. Theory, and the above experimental results, indicate that a non-constant pressure gradient along the wedge wall fundamentally changes the shape of the profiles at the different stations along the wall. This makes profile similarity impossible for airflow on a wedge.

Acknowledgments

The author acknowledges the support of the Air Force Research Laboratory and Gernot Pomrenke at AFOSR.