Aerodynamic Lift

A graphical explanation for aerodynamic lift is presented based on the conservation of mass, momentum, and energy equations. To try to understand the forces resulting in lift in terms of mass and momentum conservation, a series of velocity profile plots along a wing are examined. The velocity profiles show peaking behavior in the near wing region. The momentum equation tells us that the resulting velocity changes must be accompanied by pressure changes. The important insights of this graphical exploration can be summarized as:

· The mass of the free stream air approaching the wing surface must be conserved by being redirected around the wing. The change in the air velocity and direction leads to a momentum change near the wing surface. The momentum conservation equations tell us that the incoming stream-wise momentum is partially converted into normal-to-the-flow momentum AND partially into a pressure change in the near wing region. The pressure changes induced by momentum conservation provide the critical connection between the velocity and pressure fields that has been missing in previous approaches.

· The difference between the above wing pressure and the below wing pressure is what produces aerodynamic lift. The normal momentum conservation equation can provide the needed framework for investigating this pressure change in the near wing areas.

· The boundary layer description, as presently practiced in the fluid flow literature, does NOT correspond to fluid flow around an aerodynamically thick object. The traditional boundary layer conceptual view in which the velocity asymptotes to a plateau simply does not fit with reality. The traditional boundary layer concept does NOT allow for the conservation of mass, momentum, and energy whereas the “peaking” concept discussed in Chapter 1 does.

· Conservation of mass means that the boundary layer around an aerodynamically thick object must include peaks and valleys in the near-wall velocity boundary layer. These peaks and valleys mean that the velocities are changing spatially which leads to changes in the momentum and, in turn, to pressure changes, the pressure changes that lead to lift.

· The same momentum-based arguments used herein to explain aerodynamic lift also explain the Coandă effect and the fluid pumping action of rotating machinery.

2.1 The Aerodynamic Lift Problem

For as long as man has explored flight, there have been attempts to develop a simple explanation as to why planes fly without much success. The problem is not with the science of how airplanes fly. Scientists have had no trouble accurately simulating aircraft flight. By treating the flow as a continuum, a set of conservation of mass, momentum, and energy equations have been developed that appear to describe aerodynamic fluid flow situations very well. While the simulations are straightforward, trying to interpret the solution to gain a simplified understanding of what causes aerodynamic lift, for example, has proven difficult. Current explanations based on the Bernoulli equation, Newton’s 3rd law of motion, or the Coandă effect are incomplete or unsatisfactory in one way or another [12].

We contend that one of the reasons for this is that until recently, there has not been a way to accurately describe the boundary layer situation for flow along a wing. All previous practitioners have incorrectly adopted the thin flat plate wind tunnel boundary layer concept as to how they believed flow over a wing behaved. This equivalence of thin flat plate and external boundary layer flow is deeply embedded in the fluid flow literature. However, for the last forty years, computer flow simulations have been available which, when examined at even a coarse level, easily demonstrate that the traditional boundary layer concept does not correctly describe boundary layer flow on a wing. The thin flat plate wind tunnel model with a non-peaking velocity boundary layer simply cannot describe what happens to the diverted mass and momentum that occurs when the flow encounters an aerodynamically thick obstruction in the flow environment. As pointed out by Weyburne [3-5], this diverted mass results in velocity peaks in the near-wall region. The peaking behavior is described using a new boundary layer conceptual model, which when combined with the new moment-based boundary layer thickness and shape method, allows one to describe and characterize the peaks as well as the slow return of the fluid to the free stream velocity. Peaks and valleys of the velocity profile on an airfoil surface are something the flow community has simply not discussed in any form. Yet, there is no possibility of describing aerodynamic lift without incorporating velocity profile peaks and valleys into the discussion.

In the following chapter, we use this new conceptual model and boundary layer description to graphically explain aerodynamic lift. We start by reviewing the flow governing equations. Mass conservation on a wing is demonstrated by plotting a series of velocity profiles along a NACA0012 wing section. The velocity profiles show spatial changes in the near-wall region. The momentum conservation equation tells us that these velocity changes must be accompanied by pressure changes. The resulting pressure differences along the wing are shown which makes it possible to calculate the lifting force per unit area. To demonstrate this approach, a 2‑D NACA0012 airfoil laminar flow simulation by Swanson and Langer [10] is used. This low Reynolds number (Re_c = 5000 based on the chord length), alpha =3° laminar flow simulation's main attributes, for our purposes, are that: 1) it is an exact Navier-Stokes solution as opposed to a RANS-based approximate turbulent flow simulation, and 2) the laminar momentum balance equation is much simpler to deal with then the turbulent version. The approach is based on the graphical demonstration of the conservation of mass and momentum since the energy conservation contributions are largely hidden from view. We will discuss this latter contribution in Section 2.5 at the end of the chapter.

2.2. Laminar Airflow along a NACA0012 Airfoil

The airflow encountering an aerodynamically thick object, such as a NACA0012 airfoil, must be redirected around the object. This redirection must proceed in a manner that results in the conservation of mass, momentum, and energy. Conservation of mass for a two-dimensional (2‑D) isothermal steady-state fluid flow requires that

where ρ is the fluid density, u(x,y) is the x-direction velocity, v(x,y) is the y-direction velocity, the free stream is in the x-direction, and y is normal to the x-direction. The velocity at a point x,y is the vector sum of the u(x,y) and v(x,y) velocities. This equation specifies that the spatial redirection of the mass must be compensated by velocity changes (momentum is given by mass times velocity). The momentum changes accompanying these velocity and mass changes must also be conserved. For laminar flow along an airfoil, the momentum equations that dictate how this happens are given by the Prandtl boundary layer x and y components of the momentum balance. The x‑component Prandtl momentum equation for 2D laminar flow requires that

where is the kinematic viscosity, and P(x,y) is the pressure. The equivalent Prandtl y‑component of the boundary layer momentum balance (normal to the free stream flow direction) is given by

The unusual arrangement of the terms in this equation is done to emphasize that changes in the fluid's velocity and direction (left side of Eq. 2.3) must result in a pressure change to conserve momentum (right side of Eq. 2.3). This momentum conservation is the key to understanding how a wing develops low-pressure regions above and below the wing.

The mass and momentum conservation equations together with the energy conservation equation are routinely used to simulate airflow around a wing. This is done by dividing the flow region into a grid of points and then solving the conservation equations at each point by an iteration process. An example from Swanson and Langer [10] showing the pressure distribution around a NACA0012 wing section is shown in Fig. 2.1. As the free stream airflow approaches the wing surface, the fluid’s velocity must necessarily go to zero at the wing surface (no-slip boundary condition). The incoming mass must be redirected and that is accomplished, in part, by the generation of a normal to the flow velocity v(x,y). However, as the momentum conservation equations make clear (Eqs. 2.2 and 2.3), some of that momentum change must also result in a pressure change. The pressure buildup in front of the wing’s leading edge, the red region, illustrated in Fig. 2.1 is one consequence of this process. This pressure buildup is intuitively obvious; trying to force an aerodynamic thick object through air will result in a pressure resistance we call form drag. What is not as clear is what causes the dark blue areas in Fig. 2.1 located above and below the wing surface. Ultimately, aerodynamic lift is dependent on the pressure difference above the wing and below the wing. Hence, the challenge is to try to generate a graphical explanation that explains the origin of the observed pressure distribution that ultimately leads to the generation of aerodynamic lift.

This process of illustrating aerodynamic lift is usually done using streamline and/or contour plots of the velocity and pressure around a wing (e.g., Fig. 2.1). However, these types of plots are not useful when trying to illustrate the conservation of mass and momentum. Instead, we turn to velocity profile plots taken at several locations along the wing surface. The reason velocity profiles are preferred will become obvious shortly. The boundary layer velocity profile for u(x,y) is defined as a whole series of u(x,y) velocity values taken above (or below) a point x on the wing surface and is usually illustrated by a plot of these velocity values as a function of y. The velocity profile at the wing surface is zero due to friction (no-slip boundary condition) and monotonically increases to the free stream value. The portion of the velocity profile for which the velocity has not yet at the free stream value is called the boundary layer.

So, what does the boundary layer velocity profile on a wing look like? Traditionally, the boundary layer velocity profile on a wing is assumed to be a simple transpose of the boundary layer on the thin flat plate. The thin flat plate boundary layer as depicted in most fluid flow textbooks is shown in Fig. 1.1. As we discussed in the Boundary Layer Concept Chapter, this traditional boundary layer depiction has a fatal flaw that has been ignored in the literature: there is no possibility of momentum balance using the traditional boundary layer model. The depicted momentum change due to the presence of the boundary layer is NOT conserved. To make matters worse, this traditional boundary layer description does not correspond to reality. It is not what a boundary layer on any aerodynamically thick object looks like. Based on actual experimental data, Weyburne [3-5] developed a new boundary layer "peaking" conceptual model to account for the diverted mass and momentum encountered in flow around an aerodynamically thick object. This peaking behavior of the boundary layer is simply not discussed anywhere in the literature. However, it is easily demonstrated using CFD simulations. For example, in Fig. 2.2, we show the boundary layer velocity profiles from the Swanson and Langer [10] flow that produced the pressure contour plot depicted in Fig. 2.1. In this figure, c is the chord length and u₀ is the

free stream velocity. The u(x,y) profiles all start at zero at the wing surface but, to emphasize the peaking behavior, only the tops of the profiles are shown. The velocity at a point x,y is the vector sum of the u(x,y) and v(x,y) velocities which means that for this simulation case, the average peak velocity over the majority of the wing surface is 10% to 20% above the free stream velocity u₀. This CFD simulation has a laminar separation bubble near the trailing edge starting at ~x/c = 0.8. Since the intent is to illuminate the mechanism of lift, we do not show profiles near the trailing edge to avoid confusing the explanation. Their contribution to aerodynamic lift, in any case, is small.

The excess mass deflected by the wing leads to the peaking behavior (e.g., Fig. 2.2). Peaking behavior means that the velocity values are continuously changing in the y‑direction in the near-surface region. Furthermore, the y-profiles for u(x,y) and v(x,y) velocities are not identical at different x-positions which means that they are also changing in the x-direction. The conservation of momentum equations, Eqs. 2.2 and 2.3, tell us that changing velocity values and direction must lead to pressure changes. For the velocity data depicted in Fig. 2.2, the associated y-pressure gradient profiles are shown in Fig. 2.3.

This is what we are looking for: a connection between the velocity field and the pressure field. This is what has been missing from the previous explanations of aerodynamic lift. It is the momentum conservation equations that provide the critical link.

The pressure gradient profile plots are another way to show the pressure field behavior along the wing surface. The pressure gradient plots indicate that most of the pressure effects are happening near the leading edge of the wing. It is also evident the pressure gradients are larger above the wing compared to below the wing. By integrating the pressure gradient from the wing surface to deep into the free stream, we obtain the pressure difference between the wing surface and the free stream. For the profiles in Fig. 2.3, the integration will result in low-pressure areas in the near-wall region. This explains the appearance of the dark blue areas in Fig. 2.1. Although both regions have a low-pressure cloud-like region, a closer examination of Fig. 2.3 reveals that the pressure above the wing will be lower than the pressure below the wing. The aerodynamic lift force acting on the wing is the pressure difference above and below the wing times the wing’s surface area. The normalized above and below pressure difference for the a=3° case is shown in Fig. 2.4. Integrating the pressure difference over the surface contour would yield the aerodynamic lift per unit wing length. The curve in Fig. 2.4 indicates the overall aerodynamic lift force per unit wing length for this flow case will be positive.

2.3. Turbulent Airflow along a NACA0012 Airfoil

The above laminar flow results are certainly convincing but for most airflow over wings, the flow rapidly turns turbulent at even moderate Reynolds numbers. The turbulent flow closure problem means that approximations must be used to obtain airfoil airflow simulation solutions. With these limitations in mind, it is therefore gratifying that the turbulent flow simulations for airflow along a wing also show velocity peaking behavior. Visual confirmation for the peaking behavior in turbulent airflows over wings is provided in Fig. 2.5. This figure was generated using a modified version of OpenFOAM’s validation-tutorial Turbulent flow over NACA0012 airfoil (2D) case (see Appendix). Fig. 2.5a shows the compressible turbulent simulation of airflow on a NACA0012 airfoil and Fig. 2.5b shows the incompressible turbulent simulation result under the same conditions. The compressible version shows much sharper and larger peaks than the incompressible version. To encourage the reader to do simulations to confirm the existence of boundary layer peaking for themselves, the Appendix provides all the steps necessary to run this modified version of OpenFOAM’s validation-tutorial case. For the turbulent case, it is possible to do the same presentation of the velocity and pressure gradient profiles to confirm near wing low-pressure regions as we did above for the laminar case. We will leave this to the reader to verify for themselves after completing the simulations in the Appendix.

2.4. Aerodynamic Lift Explanation

Any theory of aerodynamic lift must ultimately be based on the physics dictating how the fluid flows around a wing. The currently accepted theory for fluid flow is based on treating the fluid as a continuum and using the fluid flow governing equations consisting of the mass, momentum, and energy conservation equations as the theoretical basis for fluid flow. The new graphical approach to aerodynamic lift incorporates these equations as the fundamental building block of the explanation. The momentum conservation equations provide the critical connection between the velocity and pressure fields that has been missing in previous approaches.

While the conservation equations supply the theoretical basis, the key to the approach is the understanding that near-wall peaking of the velocity is critical to the process. The diverted mass flowing around the wing results in increased velocity values in the near-surface region, i.e. the peaking behavior. The momentum conservation equations, Eqs. 2.2 and 2.3, tell us that the changing velocity values associated with a peak must be accompanied by pressure changes. None of this would occur if the flow behaved as the traditional asymptoting-to-a-plateau boundary layer model was used instead of the new peaking model. With the diverted mass and induced momentum changes in mind, the examination of the velocity and pressure gradient profiles along the wing surface provides that intuitive link to a simplified explanation aerodynamic lift that the flow community has been looking for.

What is remarkable about the peaking behavior is that it is that both the laminar and turbulent peaks are closely following the wing surface even though the wing surface is not flat. This Coandă-like behavior can be quantified using the viscous second derivative boundary layer moments described in Chapter 3. For fluid flow along a solid surface, viscosity forces the velocity to go to zero at the surface (no-slip boundary condition) and then rapidly increases toward the free stream value. The upper bound to the boundary layer region influenced by viscosity is given by the viscous boundary layer thickness, δ_v. It is normally given by the second derivative moment-based viscous thickness defined as the mean location, μ₁, plus 2-sigma (δ_v is ~δ₉₉ for laminar flow on a flat plate according to Weyburne [8]). In Fig. 2.6, the viscous boundary layer thickness, δ_v^2.6, and the y-value at the peak for the u(x,y) profile, given by δ_max(x), are shown for the Swanson and Langer's [10] a=0° airflow case along a NACA0012 wing.

The values above ~x/c = 0.8 are not shown since this simulation case also has a laminar separation bubble. The 2.6 sigma value means that the velocity peak location occurs just above where the viscous forces essentially vanish. Fig. 2.6b shows the same data but normalized by the displacement thickness and plotted along the wing surface. Fig. 2.6b also shows the Blasius laminar flow on a thin flat plate two-sigma viscous thickness, δ_v(x), divided by the Blasius displacement thickness, δ₁(x). What this tells us is that the linearly increasing velocity peak seen in Fig. 2.6a is partly fueled by the accumulation of the excess mass resulting from the displaced fluid due to the viscous forces and partly by the mass and momentum being redirected by the wing. Overall, this viscous thickness and peak location correlation offers a plausible explanation as to why the potential flow solution to a displacement thickness broadened airfoil appears to work fairly well.

This hugging of the wing surface by the velocity peaks brings into focus the role of the energy conservation equation. The physics dictates that the fluid flow must conserve mass, momentum, and energy. What is also evident, but rarely discussed, is that this happens in a way that not only conserves but minimizes the overall energy in the system through the action of an imposed equation of state. The equation of state is a thermodynamic equation relating the fluid variables such as the pressure, density, and temperature. It appears that the CFD solution in which the velocity profile peak is right at the viscous boundary layer thickness location corresponds to the minimum energy state. This, in turn, puts the minimum pressure at the wing surface. At present, all we can say is this situation is what leads to a minimum system energy. Perhaps in a future endeavor, it will be possible to flesh out why the velocity peaks hug the wall.

Stepping back and examining the results in Fig. 2.2c reveals that the new peaking boundary layer conceptual model has to be modified to include the possibility of a boundary layer “valley” in the model. Examination of Fig. 2.2c shows that the x/c = 0.06 profile does not peak but instead shows a u(x,y) near-wall region which is lower than the free stream velocity u₀. This “valley” persists over one-half a cord length below the wing. This behavior has been confirmed at other nearby locations along the wing. 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.

The thickness and shape of the boundary layer “peaks” and “valleys” can be characterized using the moment-based description technique developed by Weyburne. For example, the very slow return of the airflow to the free stream value from the velocity peaks is characterized by the 2‑sigma inertial thicknesses, δ_i . For the Swanson and Langer's [10] simulations considered above, δ_i is about five cord lengths at points all along the wing. This very slow return of the airflow to the free stream value from the peak value is a consequence of the mostly-inertial nature of the fluid in these regions. This slow return is masked by the fact that 90% of the peak area is within ~0.5 chords of the airfoil surface.

2.5 Application of the Momentum Balance Approach to Other Flow Situations

It should be mentioned that attempts have been made to explain aerodynamic lift using the Coandă effect (see reference [12]). The Coandă effect itself does not have an accepted simplified explanation as to how it works nor is there a detailed explanation of how it applies to aerodynamic lift. Thus, saying the Coandă effect explains aerodynamic lift doesn't actually explain anything. It does appear that the flow-hugging-the-wall behavior for the two cases is somewhat similar. In the former case, you have an air jet directed at a curved object and, in the latter, a uniform air blanket directed at the curved object. It may be possible to adapt the same momentum-based graphical arguments based on velocity profile peaking used above to develop a simplified explanation for the Coandă effect. It may also be possible to explain the fluid pumping action of rotating machinery. In all these cases, it is the fluid’s change in direction and velocity that leads to momentum changes, changes that necessarily require pressure changes.