A New Boundary Layer Conceptual Model

The traditional boundary layer concept does not describe the boundary layer in a way that permits a complete accounting for the excess mass and momentum diverted around an aerodynamically thick wing. There is no way to theoretically describe aerodynamic lift, for example, if you cannot correctly account for the excess mass and momentum diverted by the wing. A new "unbounded" boundary layer concept is offered as a remedy. The new conceptual model, combined with the moment-based boundary layer thickness approach, provides a framework for accounting for the diverted mass and momentum flow around an aerodynamically thick object. The preliminary basis for a theoretical aerodynamic lift explanation is outlined at the end of the Chapter.

1.1 The Traditional Boundary Layer Concept

The equations that govern fluid flow require that mass, momentum, and energy must be conserved. A flow situation, like airflow over a wing, requires that this set of partial differential equations be solved as a grid of points. The iterative solution process is not straightforward in general. Ludwig Prandtl [1] simplified the equations for fluid flowing along a wall by dividing the flow into two regions: one close to the wall dominated by viscosity, and one outside this near-wall region where viscosity can be neglected. This near-wall region is called the boundary layer. The resulting simplified Prandtl equations for boundary layer flow are more amenable to numerical and theoretical solutions.

Ludwig Prandtl's [1] boundary layer concept for steady 2-D laminar flow along a wall is often depicted as shown in Fig. 1.1 (see, for example, Fig. 7.6 in Hermann Schlichting's [2] seminal book on boundary layer theory). This figure shows the flow velocity in the x-direction u(x,y) as the red line for fluid flow over a thin flat plate. A u(x,y) velocity profile is defined as the u(x,y) velocity values measured at a whole series of points from the wall (plate) surface to a point deep in the free stream perpendicular to the wall surface. The fluid's interaction with the wall induces a no-slip boundary condition (zero velocity at the wall). The u(x,y) velocity profile at a point x on the plate monotonically increases above the plate until it asymptotes to the velocity at the boundary layer edge which, in this case, is the free stream velocity u₀. The boundary layer thickness at a point x on the plate is δ(x), the dotted blue line, and is the point where the velocity just reaches u₀. The boundary layer thickness along the plate (dashed black line) slowly increases as viscosity affects slows down more and more fluid moving downstream (viscosity is important where the fluid's velocity is changing). Velocity profiles similar to those depicted in Fig. 1.1 are routinely measured on thin flat plates in wind tunnels.

While the velocity profiles similar to that depicted in Fig. 1.1 are often observed in wind tunnels, it is also a reasonable approximate depiction for exterior boundary layer airflow along a thin flat plate. Fig. 1.1 is only an approximate depiction for interior and exterior flow since, although not acknowledged in the literature, this depiction does not permit a complete accounting of the mass and momentum that is diverted due to the presence of the boundary layer (mass and momentum are not conserved). The diverted mass and momentum implicit in Fig. 1.1 apparently just disappear. Weyburne [3-5] pointed out that the diverted flow must result in a peak near the boundary layer edge before asymptoting to u₀ due to the mass deflected by the boundary layer and the plate. Support for this peaking behavior is offered in the next section. The reason the flow community has not acknowledged this peaking behavior is, in part, related to the fact that velocity peaks are not usually observed in wind tunnel experiments (the peaks are there, just not observable due to the way experiments are conducted). Whatever the reason, the result is that the majority of the flow literature depicts both interior and exterior boundary layers using an illustration similar to Fig. 1.1.

Unfortunately, this apparent similarity has led to the fluid flow community to treat interior and exterior boundary layers as equivalent (think wind tunnel experiments and flow on a wing in flight). This is reflected in the fact that it is impossible to find a discussion in the literature about the differences between interior and exterior boundary layers. For example, a Google search (on March 2020) of "boundary layer flow" yields 723,000 hits while a Google search of "interior boundary layer flow" yields zero hits and "exterior boundary layer flow" yields six hits. The problem with this equivalence thinking is that Fig. 1.1 does not come close to describing exterior boundary layer flows like flow along a wing in flight. As is shown below, the velocity peak on a wing near the boundary layer edge can easily be 15% higher than u₀ (Weyburne [3]).

1.2 The New Bounded and Unbounded Boundary Layer Concepts

It is easily verified that conservation of mass and momentum means that the velocity can never directly asymptote to u₀ as depicted in Fig. 1.1 (the displaced mass and momentum must go somewhere). Even if we replace u₀ with the velocity at the boundary layer edge, u_e(x), as is sometimes done, this updated depiction still does not enable a full accounting of the excess mass and momentum that accumulates as the fluid flows over an aerodynamically thick object like a wing. To correct these conceptual and depiction modeling problems, Weyburne [3-5] introduced the “bounded” and “unbounded” boundary layer concepts in a series of Air Force Tech Reports as a way to distinguish between interior and exterior boundary layers. The bounded and unbounded designations were adopted because certain wide-gap interior flows can behave as exterior flows and certain exterior flows can behave as an interior flow (e.g., aerodynamic ground effect). The differentiating property between bounded and unbounded is whether the boundary layer is being substantially influenced by more than one wall.

The main objective of the new conceptual model is to enable a full accounting of the mass and momentum around an aerodynamically thick object. The new conceptual model by itself is a good first step. Just as important as the new conceptual models is the adaption of the moment method for describing the thickness and shape of the fluid boundary layer. The moment method, developed by Weyburne [6-8] is reviewed in Chapter 3. It is the combination of the new model and new thickness description that provides the framework and characterization tools for a viable theoretical path to aerodynamic lift. In what follows, the new conceptual model is reviewed and the differences are discussed in terms of moment-based thickness and shape parameters.

1.2.1 The Unbounded Boundary Layer Concept

The biggest difference between the new conceptual model and the traditional conceptual model is that we acknowledge the difference between interior and exterior type boundary layer flows. The new "unbounded" boundary layer concept, as the name implies, applies to most exterior flows. The unbounded boundary layer concept is depicted for steady laminar flow along a flat plate in Fig. 1.2. For the unbounded boundary layer case, one of the important differentiating properties is that the velocity profile goes through a peak near the viscous boundary layer edge and then slowly asymptotes to the free stream velocity u₀. In Fig. 1.2, the lower dashed curve represents the location of the velocity peak u_max(x) and the upper dashed curve represents the location where u(x,y) essentially becomes u₀, i.e. the boundary layer thickness location, δ_m(x). This is the type of boundary layer encountered for exterior flow like that for airflow over a wing in flight. The slow rate at which the peak declines and asymptotes to the free stream velocity means that the calculated boundary layer thickness values are much larger than the traditional boundary layer case.

From a conceptualization standpoint, the peaking behavior means that the original Prandtl [1] boundary layer concept needs to be modified slightly. The new unbounded boundary layer model for low to moderate viscosity fluids is divided into three regions: 1) The near-wall viscous region that is located between the wall and the viscous thickness δ_v(x) given by the moment method. This region closely mimics the traditional viscous boundary layer region. 2) The mostly-inertial region is located from δ_v(x) out to the mostly-inertial region thickness described using the moment method and designated as δ_i(x). 3) Finally, the original Prandtl inertial region which now starts at δ_i(x) and has the properties of the free stream flow. The first two regions represent the boundary layer region. The total boundary layer thickness, δ_m(x), is the sum of the viscous thickness δ_v(x) and the inertial thickness δ_i(x).

The viscous inner region for unbounded external laminar flow along a flat plate can be shown to be well represented by the Blasius [9] theoretical model (New Blasius Chapter). Based on the limited laminar flow data (see Chapter 2.4) to date, the viscous thickness δ_v(x) is located close to the location δ_max(x), the location of the velocity maximum, u_max. The mostly-inertial region of the unbounded boundary layer, on the other hand, is a concept that is newly associated with boundary layers. It is not truly an inertial region since some viscous effects are present. The velocity change in the mostly-inertial region is u_max-u₀, which is usually relatively small compared to the velocity change in the near-wall region, u_max. This means the peak excess velocity dissipates over a longer distance. The mostly-inertial boundary layer region for laminar flows is hundreds to thousands of times larger than the viscous boundary layer region thickness (see Weyburne [3]). This slow return of the velocity to the free stream velocity from the peak value is the main defining characteristic of this region.

1.2.2. The Unbounded Boundary Peak Confirmation

At the beginning of the chapter, we noted that it is not well appreciated that the velocity profile goes through a peak before asymptoting to u₀. In fact, we could find no mention of this peaking behavior in any textbook or journal article. The peaking behavior only came to attention to the author after examining computer flow simulation results for airflow over a wing. {The flow community has allowed this false perception of non-peaking boundary layers to persist by simply not discussing it in the open literature. As a counter, we encourage the reader to do their own simulations to confirm the existence of boundary layer peaking. For example, in the Appendix, we show how to run a modified version of OpenFOAM’s validation-tutorial case Turbulent flow over NACA0012 airfoil (2D) and then plot the resulting peaked velocity profiles. }

The fact that the "unbounded" boundary layer must go through a peak before asymptoting to u₀ can be justified theoretically through the following simplified momentum balance argument: When fast-moving free stream flow encounters slow-moving boundary layer flow, a pressure imbalance is created in the boundary layer region. The Prandtl momentum equations tell us that this pressure imbalance manifests, in part, as a pressure gradient normal to the flow x-direction. This, in turn, generates a normal velocity thereby converting some of the incoming momentum into normal direction momentum. This momentum diversion accounts for some of the momentum loss due to the no-slip condition but not all of it. That is the case because the pressure imbalance is not unidirectional. This means that the same pressure imbalance that creates the normal velocity must also create an "excess" x-direction velocity in the form of a positive non-zero value for u(x,y)-u₀., i.e. a peak.

Experimental proof for the consequences of momentum balance is provided in Fig. 1.3. In this figure, the velocity profiles for exterior-like laminar channel flow are shown (Weyburne [3-5]). The simulation was part of a larger effort to simulate everything from wind-tunnel-like behavior to exterior flow-like behavior by simply changing the channel gap. The 200-meter channel gap spacing used to generate Fig. 1.3 displays exterior-like behavior as judged by the asymptotic behavior of the velocity profile tails. In Fig. 1.3a, the scaled normal velocity, v(x,y), at three different locations along the channel plate are shown. This confirms that a normal velocity is generated due to the pressure imbalance caused by the fast inlet flow encountering the slow boundary layer flow. Now consider the excess x‑direction velocity. In Fig. 1.3b, the excess velocity, u(x,y)‑u₀, is plotted as a function of the normal direction at the same three locations. The velocity peaks just above what is usually considered the traditional boundary layer thickness. This confirms the x-direction excess velocity generation. This excess x-direction velocity comes in the form of a positive velocity peak just above the viscous boundary layer edge as depicted in Fig. 1.2.

There are three takeaways from the simulation results depicted in Fig. 1.3. As already mentioned, an excess x-direction velocity exists which means that the u(x,y) velocity profile must peak. The second takeaway is that for exterior flows, the boundary layer is much thicker than traditionally believed. If we use the definition of the boundary layer as the region where the presence of the wall is affecting the fluid flow behavior, then the boundary thickness is hundreds to thousands of times thicker than the traditional interpretation. The third takeaway from this result is that the maximum velocities in both cases are about 0.1 % of u₀. This tells us the pressure imbalance affecting the y and excess x velocities are about the same just as one would expect from the above momentum balance argument.

This 0.1% excess velocity occurs for exterior airflow along a thin flat plate. Although the thin flat plate peak is small, the excess velocity value for a tilted plate, an aerodynamically thick plate, or a wing in flight, will be much larger. For example, a velocity profile for laminar airflow over a NACA_0012 wing section is shown in Fig. 1.4. This figure was extracted from Swanson and Langer’s[10] 4096 by 2048 mesh full compressible Navier-Stokes simulation of airflow around a NACA0012 airfoil at Re_c= 5000 and α =0°.

The u_max value at this position on the wing surface is 16% higher than the u₀ value (u_max values averaged 13% from the leading edge to the trailing edge). The large velocity peak demonstrates the inadequacy of Fig. 1.1 as a boundary layer depiction for aerodynamically thick objects.

Figure 1.4a includes the δ₉₉ and inertial δ_i thickness locations. The δ_max/δ₉₉ ratio value is ~2 and the δ_i/δ₉₉ ratio value is 311. The large difference between the δ₉₉ and the δ_i and δ_max values demonstrates the problem with using the traditional boundary layer thickness δ₉₉ for exterior flows. The 99% thickness, δ₉₉(x), for exterior flows does not correspond to a boundary layer location of consequence. The large δ_i/δ₉₉ ratio demonstrates that the laminar boundary layer is much thicker than previously thought. Fig. 2.4b is included to emphasize how close the peak is to the wing surface.

1.2.3 The Bounded Boundary Layer Concept

The bounded boundary layer concept is an attempt to reframe and correct some of the deficiencies encountered in the traditional boundary layer concept used to describe wind tunnel or pipe flow experiments. The first step is to replace the traditional figure shown in Fig. 1.1 with Fig. 1.5. The H/2 dashed line is added to denote this is an interior flow. All bounded boundary layers are interior flows.

The other major difference between Figs. 1.1 and 1.5 is that the asymptotic velocity value is changed to u_e(x) to acknowledge the fact that the boundary layer edge velocity can take on different values depending on the induced pressure gradients in the pipe/channel/wind tunnel. Most traditional wind tunnels all have a common feature in their implementation that allows the pressure gradient in the flow direction to be manipulated, for example, by adjusting the upper surface height, H, along the flow direction. In most wind tunnels, flows can be manipulated to have a zero-pressure gradient (ZPG), a favorable pressure gradient (FPG), or an adverse pressure gradient (APG) in the flow direction. For flow in a parallel walled channel wind tunnel, the walls induce a boundary layer pressure that becomes smaller as the flow moves along the wall toward the exit (atmospheric pressure) resulting in a FPG condition. To induce a ZPG Blasius [9]-like condition (depicted in Fig. 1.5 with u_e(x) a constant), a pressure gradient inducing mechanism, for example, changing the channel gap in the flow direction, needs to be adjusted to make the induced pressure gradient produced by the walls a constant along the flow direction.

The original “bounded” boundary layer concept detailed in the AF Tech Reports [3-5] has evolved somewhat due to recent revelations. When the original model was developed, it was thought that wind tunnel experiments only resulted in asymptotic velocities, u_e(x), that plateaued as is implied in the flow community’s literature. However, flow simulations indicate that boundary layers do not always asymptote to a plateau. The thin flat plate has a 0.1% peak and would not be observable in a wing tunnel. However, if a thick plate is used instead of a thin plate, then simulations indicate that a broad peak will result just above the traditional boundary layer edge (Weyburne, unpublished results). An example simulation result is shown in Fig. 1.6. The simulation is based on the thick plate OpenFOAM T3A turbulence transition simulation which is part of the OpenFOAM's tutorial-verification suite. To ensure the boundary layer is properly rendered, the number of mesh points was increased by a factor of four and the top boundary was turned into a wall boundary to mimic a wind tunnel. The T3A-like plate is placed on the centerline of the 1‑meter gap channel. The simulation with these changes is able to reproduce Roach and Brierlay’s [11] T3A experimental wind tunnel wall shear stress results.

The simulated velocity profiles shown in Fig. 1.6a are very similar to the Roach and Brierlay’s [11] wind tunnel experimental profiles (results not shown). The velocity profiles in Fig. 1.6a are intentionally cut off to match the actual maximum T3A experimental y-values. For comparison, in Figure 1.6b we show the complete velocity profile. The velocity profile scaling was switched to u₀ in order to show the full effects of the plate and walls on the profiles. The finite thickness T3A-like rounded nose plate results in velocity peaks that are about 2% higher than u₀.

It is the limited y-values of the wind tunnel experimental data that has led to the widespread impression that the profiles plateau to a fixed value. For the T3A wind tunnel data, for example, the maximum y-value is about three times δ₉₉ (Roach and Brierlay’s [11]). However, the results in Fig. 1.6b indicate the peak width is about ten times thicker than δ₉₉. This means the plateau-type behavior often observed in the wind tunnel experimental data (and in Fig. 1.6a) is due to the limited y-extent of the data. Hence, wind tunnel experiments involving aerodynamically thick objects should show peaking behavior of the velocity profile in a wind tunnel (depending on conditions) if the velocity is measured much further above the plate.

The traditional interpretation of the boundary layer edge velocity is that u_e(x) extends to the H/2 plane. This is approximately the case for a thin flat plate inserted on the centerline of a channel-type wind tunnel. However, as is evident from Fig. 1.6, aerodynamically thick objects divert mass and momentum around the nose of the object which, in turn, generates a velocity peak just above what is considered the traditional boundary layer. To acknowledge the fact this could be a boundary layer past an aerodynamically thick object-plate that might peak instead of plateauing, a space has been inserted between the velocity profile depiction and the H/2 dividing line. This is an important update to the original depiction in the earlier AF Tech Reports [3-5].

1.2.4 The Bounded-Unbounded versus Interior-Exterior Designations

There is no sharp division between the bounded and unbounded boundary layers. Laminar flow channel simulations (AF Tech Reports [3-5]) indicate that everything from a bounded to an unbounded-like boundary layer condition can be generated by changing the channel gap. The fact that an exterior-like flow can be generated in an interior flow situation and certain exterior flows can behave as an interior flow (e.g., aerodynamic ground effect) is the reason the bounded and unbounded designations were adopted. The choice of when a flow situation should be designated bounded or unbounded therefore comes down to a choice of an appropriate criterion. One possibility is to pick an outer region velocity gradient dv(x,y)/dy or d{u(x,y)‑u_H/2}/dy lower limit value to indicate when the correct asymptotic unbounded boundary layer behavior has been achieved. In Fig. 1.7 the results for the normal velocity, v(x,y), and u(x,y)‑u_H/2 at different gaps are shown for the laminar channel flow case (Weyburne [4]). The pictured y-scale is chosen to emphasize the tail regions return to zero behavior at the channel midgap. Notice that it is not until the gap reaches 200-meters that the velocity profiles start to show the asymptotic behavior of the tail region that you would expect for an exterior flow. The y-pressure gradient also shows the same type of behavior, only showing asymptotic behavior to zero for the 200-meter case (Weyburne [4]).

Given the complicated nature of the different boundary layer scenarios, the designations bounded and unbounded will take on the following descriptions: 1) the unbounded boundary layer and its associated depiction (Fig. 1.2) will be used to describe boundary layers on exterior walls (and very large gap interior flows), and 2) the bounded boundary layer and its associated depiction (Fig. 1.5) will be used to refer to most interior flows. The bounded designation is intended to be used for wind-tunnel type boundary layers taken at the typical wind tunnel experimental y‑extent, i.e. that show plateau type behavior. These boundary layer descriptions are not intended to encompass all possible boundary layer situations, just the most important and widely encountered versions.

The boundary layer thickness for a bounded boundary layer needs clarification. If the boundary layer thickness is defined as the point above the wall where the flow no longer feels the effect of the wall, then all interior bounded boundary layer thicknesses would be the channel or pipe width H or radius R. However, traditionally, the flow community has adopted the viscous boundary layer thickness, or the turbulent broadened viscous boundary layer thickness, as the definition instead. For interior flows, we will also adopt this designation to prevent confusion.

1.3 The Bounded and Unbounded Pressure Fields

The presence of the velocity peak speaks to the pressure field of the boundary layer. Unfortunately, the flow community’s reliance on the flawed boundary layer concept depicted in Fig. 1.1 has resulted in a general distorted and incorrect picture of the pressure fields involved in boundary layer flow. For example, there are many examples in the literature and textbooks that incorrectly assert that the normal to the flow y-pressure gradient is zero in the boundary layer region (see y‑momentum Chapter). The y‑pressure gradient may be small, but it is definitely nonzero since the velocity normal to the wall must be nonzero. Furthermore, given the flow community’s association of Fig. 1.1 with the Blasius theoretical flow situation (see Schlichting [2] Chapter VII, for example), one might also conclude that the x-pressure gradient in the flow direction is also zero. Hence, in the traditional interpretation of Fig. 1.1, there are no pressure effects at all. This type of thinking has led to the generally accepted belief that the boundary layer flow situation in a wind tunnel is equivalent to external boundary layer flow. Whereas the scaled u(x,y) velocity field in the flow direction can be made to look similar for the two cases, the pressure fields are NOT equivalent.

Ideally, one would like to do a direct comparison between the bounded and unbounded pressure fields. It is not possible to finely map the pressure spatially through the boundary layer region in a wind tunnel due to experimental difficulties. Fortunately, flow simulation via computational fluid dynamics for laminar flow can provide the missing insights. Weyburne [3-5] did a series of 2-D laminar flow simulations in a channel using a 8‑meter long plate in a 2-D flat wall channel. The no-wall inlet region varied from 0.6-meter to 20-meter to fully resolve the inlet pressure and velocity fields. Initially, the channel gap was set at 1-meter to mimic a standard wind tunnel with a thin flat plate along the centerline. In subsequent simulations, the gap was increased until the flow behaved as an exterior flow using the shape of the normal v(x,y) velocity tail region as a test. As is indicated in Fig. 1.7, asymptotic behavior of v(x,y) did not occur (Weyburne [3-5]) until the gap was increased to 200-meters for laminar air flows with an exit Reynolds number of Re_x=5x10⁵ (the critical laminar-turbulent transition Reynolds number)!

In Figs. 1.8 and 1.9, the pressure fields for the 1-meter gap interior-like and the 200-meter gap exterior-like 2-D channel airflow are shown. The inlet airflow in both cases is 0.9375 m/s resulting in an exit Reynolds number Re_x = 5x10⁵ at atmospheric pressure. It should be emphasized that both of the simulated channel flows are similar to what is obtained for a “thin flat plate” type of boundary layer in that both the velocity peaks at the walls are small (~0.1%).

The first difference to notice in the two figures is the pressure difference scales. The bounded flows pressure differences are more than an order of magnitude larger than the unbounded flow. To emphasize this, the pressure difference and the x‑pressure gradients near the viscous boundary layer edge (y=0.05-meter) along the plates for the two cases are shown in Fig. 1.10. The pressure difference at the mid-plate (x/L=0.5) boundary layer edge, the bounded (1‑meter gap) pressure is 49 times larger compared to the exterior-like unbounded (200-meter gap) mid-plate flow situation (0.01 atm versus -0.0002 atm). This reflects the fact that it requires more work to induce flow in a thin channel than a thick channel with a fixed exit pressure requirement.

The next point to notice in Fig. 1.10a is the pressure change just above the wall surface that is observable in the interior flow case whereas the exterior-like flow case shows almost no change along the channel wall. This is also observable in Fig. 1.8 and Fig. 1.9. In contrast to the wind tunnel like-result (1-meter gap), the unbounded exterior-like laminar flow (200-meter gap) along a zero-incidence angle flat plate appears to be naturally in a ZPG condition as shown in Fig. 1.10b (except at the front and end of the plate). The conclusion is that the pressure field affecting the external boundary layer (away from the front edge) in the flow direction is very different than the interior flow case. This pressure difference results in a noticeable velocity change (not shown) in the flow direction, du(x,y)/dy, for the internal wind tunnel-like flow but is small for the external-like flow.

The wind tunnel-like result (1-meter channel gap) pressure gradient shown in Fig. 1.10b does not behave like Blasius flow which is not unexpected given the FPG gradient condition shown in gradient inducing mechanism, for example, adjusting the gap along the channel, needs to be used to cancel out the naturally induced pressure gradient produced by the presence of the walls. The wind tunnel type channel simulation results confirm that a simple tilted upper wall can reduce the pressure gradient magnitude (Weyburne [5]). The simulation results indicate that a simple top wall tilt shifts the interior pressure gradient curve in Fig. 1.10b vertically towards a zero-pressure gradient condition. By finding the optimal tilt, the simulation scaled u(x,y) velocity profiles along the plate can be made to look almost identical along the plate using the Blasius scaling parameters. However, the normal velocity profiles and the normal pressure gradients generated under these conditions do not show similar behavior using the Blasius scaling parameters. Hence, a Blasius flow condition in a wind tunnel based on observation of only the u(x,y) velocity profiles does not guarantee a "known" flow condition for stability or transition experiments. More details of this ZPG condition are provided in the New Blasius Chapter.

Whereas the pressure behavior in the flow direction shows noticeable differences, the pressure differences in the normal direction are relatively small. The 1-meter and the 200-meter gap pressure differences between the near-wall values and the pressure at the channels mid-gap are only about 0.1% for all points along the plate. This means that the y-pressure gradients are small. The scaled pressure and the scaled y-pressure gradients at the mid-point on the plate are shown in Fig. 1.11. The scaling parameters used are the similarity scaling parameters discussed in the y-Momentum Chapter. Not unexpectedly, the 200-meter gap flow shows Blasius-type behavior consistent with the ZPG behavior in the flow direction (Fig. 1.10).

What is not so obvious in the above figures is the differences in the x-pressure gradient and y‑pressure gradient scales. In the 200-meter exterior-like flow, the x-pressure gradient and y‑pressure gradient values at the top of the viscous boundary layer (~y=0.05 meter) are about equal. For the wind tunnel-like result, the ratio of the x-pressure gradient to the y-pressure gradient is about a factor of 64 times bigger. This again points to the large differences between interior and exterior pressure fields.

1.4 Peaks and Valleys

The new boundary layer conceptual model needs to be modified to include the possibility of a boundary layer “valley” in the model. Recent revelations from the Aerodynamic Lift Chapter reveal that certain velocity profiles along a wing section at a 3° angle of attack do not peak but instead show near-wall region velocity profiles which are lower than the free stream velocity u₀. This “valley” persists over one-half a cord length below the wing surface. This behavior has been confirmed at several locations along the wing. The boundary layer moment-based thickness and shape descriptors are still usable, with appropriate caution, to apply to this new case. The added “valley” modification will ensure that the flow around an aerodynamically or hydrodynamically thick object in the path of the flow can be fully described and characterized.

1.5 Turbulent Boundary Layer Concept

The turbulent boundary layer adds another level of complexity to the boundary layer concept. Turbulent flows do not have a closed-form solution so the exact Navier-Stokes simulation approach used above cannot be easily applied to study the boundary layer behavior for this case. However, it has been confirmed that simulation results for turbulent flows on a wing using a Spalart-Allmaras shear stress transport model approximation do show the same type of peaking behavior discussed above (see Chapter 2.3).

It is known that the turbulent boundary layer can be divided into a near-wall region where viscosity is important and an outer region where it is not. The thickness and shape of the time-averaged viscous region and the time-averaged inertial region can be independently characterized and studied using the integral moment method described in the Boundary Layer Thickness Chapter. This time-averaged inertial region is not purely inertial since velocity peaking behavior also occurs in turbulent boundary layers (see Chapter 2.3). In one of the following chapters (The Origin of the Logarithmic Law of the Wall), we discuss the importance of the time-averaged nature of the velocity profile and how it impacts the viscous region of the turbulent boundary layer. This division into viscous and inertial regions means that the turbulent boundary layer may be conceptualized using a modification of that shown in Fig. 1.2. This would be accomplished by adding a third dashed line to depict the turbulent boundary layer thickness to the picture. However, at this time it is still not clear what is the relationship between the velocity peak, the viscous thickness, and the turbulent thickness. We will therefore postpone trying to present a complete turbulent boundary layer conceptual depiction until a later date.