Engineering Turbulence – Part II: Lifting The Wright Way!

“It’s easy to explain how a rocket works, but explaining how a wing works takes a rocket scientist…” – Philippe Spalart (senior technical fellow – the Boeing company)

The intention of this set of posts is to embark on a journey of connecting the dots between CFD and turbulence modeling with the phenomenological and practical concepts of engineering aerodynamics.
Even though it will by no means be all encompassing nor highly accurate I hope for the essence to be captured, and even more so communicated

In Part I of “Engineering Turbulence”, I have tried to somewhat connect the dots between turbulence and its phenomenological aspects. This will prove very valuable for the description of drag, and especially, to understand how flow control mechanics actually work.

But before turning to control the drag, It is important to actually define and explain notions like lift and drag. Without carrying out an extensive and scientific survey, I would bet that if there were ever an aerodynamic concept that every laymen would like to grasp, it would be how on earth these supremely heavier than air, humongous objects like a 747 are able to carry us from continent to continent without dropping to the ground. Understanding LIFT.


LIFT

Before trying to control the forces applied on flying vehicles, it is mandatory to define which forces we are actually trying to control. To do that effectively, we resolve the forces in ways which we find most appropriate to the application at hand. In the case of flying vehicles it would be most convenient first to switch into a frame of reference bound to the the flying object, such that the flying object is now considered stationary with the flow driven from the far-field in the opposite direction to what used to be the flight direction in the “real” frame of reference. Then to resolve the component perpendicular to the far-field flow, and call it lift. Again, it is a matter of convenience to resolve the force in that manner such that the lift would generally be the force countering gravity and/or providing maneuvering forces for our flying objects (it may be responsible to a whole different kind of concept when encountered in sails for example, but we are focused on flying now… 🤓).

Ingredients of Lift

In essence we could claim that lift is described by a solution to Navier-Stokes equations. This sounds somewhat a little too obvious and somewhat unsatisfactory though. It’s not even much of an explanation but more as an agreed upon fact, and most importantly, it doesn’t improve our understanding of how lift comes about.
We may also claim that something like the “Kutta–Joukowski theorem” which relates the lift to the circulation around the airfoil, provides a good explanation for lift. But again, this also amounts mostly to jargon and math, and in any case shifts us from explaining how lift comes about to explaining how did the circulation got there… We would like a simple, yet more complete way to explain lift once there.

The ingredients for our lift explanation then, shall be as follows:

  • A lifting-surface (or airfoil) in a certain range of an angle of attack such that lift is produced.
  • A fluid which flows as a continuous material, deforms to follow the airfoils shape, affects the direction and/or speed and at the same time the pressure field in the flow, and it does so in a wide, yet “confined” region around the airflow.
  • A reciprocal 2-way cause-and-effect relationship between pressure difference and velocity. 

Airfoil and the Angle of Attack

A lifting surface shaped as in the figure below is called an Airfoil. Although some airfoils could be much more effective than others, almost any shape of an (not too thick) airfoil would produce lift when the angle of attack (AOA), being positive when the front of the lifting surface (leading edge) is above its back (trailing edge), is in a certain range.

lift
Here are some examples of airfoils oriented to the far-field incoming flow such that (as long as the flow is mostly attached) lift is produced:

airfoils

As you may evidently see, the shape of a lift producing airfoil may even be a flat plate, and as we will soon see, such a shape somewhat removes the rug under lift explanations such as “Longer Path Equal Transient Time”, as described in the figure below (taken from Wikipedia which actually presents this explanation as a fallacy, then follows by providing a somewhat incomplete simplified lift explanation of its own), actually flawed in more way than one (there’s also a nice explanation from NASA as to this incorrect lift theory).

equal transit time

In essence, there is no reason why fluid parcels that split at the leading edge must rejoin at the trailing edge. When we choose to track fluid parcels (outside the immediate boundary layer), we find that under lifting conditions, parcels that traverse the upper surface make the trip in less time and get to the trailing edge before the corresponding parcels that traverse the lower surface. Parcels that started close together near the attachment streamline ahead of the airfoil end up permanently displaced from each other after they pass the trailing edge.
Even if it weren’t wrong, simply saying that a fluid parcel must go faster than another originating from some location to get to the back of the airfoil at the same time does not explain why it goes faster (and changes direction all along), for example, by identifying a physical force that accelerates it to a higher velocity (and forces changes to its direction).

Newton’s Third Law: for every (force) action there’s an equal and opposite (force) reaction

When the airfoil shape is in a lift producing AOA range, the airfoil pushes downward on the fluid which pushes back with an equal and opposite force in the downward direction. This is lift.
The reason for the fact that a push back occurs is of course Newton’s third law, stating that for every (force) action there is an equal and opposite (force) reaction. Meaning that lift is an interaction in which the airfoil and the fluid exchange equal and opposite forces. 

What is the Push?

Fluids always pushes against themselves, and against every surface they interact with. This push is Pressure. So when an airflow is surrounded by a stagnant fluid, the pressure is, for all practical purposes, the same everywhere, and the push downward on the airfoil’s top surface, and upwards on the airfoil’s bottom surface is the same, the integrated forces due to this exerted pressure is zero, and there is no lift. When the fluid is moving, and the airfoil is in a certain range of AOA, the integrated forces are such that lift occurs, it is then directly due to a pressure difference which comes about as (on average) a higher than ambient on the lower surface (usually), and lower than ambient on the top surface (always). 

How Does The Push Comes About?

Fluid flows as a continuous material. I have somewhat explored this shift from the molecular view to the continuum in Engineering Turbulence – PART I: Phenomenology. Do to this continuum view, the fluid appears to be gradually deforming around an airfoil such that it follows its contour, but also the flow keeps being affected on a wide area, producing pressure changes predominantly in a somewhat “confined” region, such that when lift is produced there is always a confined (but not sharply) region of lower than ambient pressure above the airfoil, and there is usually a confined (but not sharply) region of higher than ambient pressure below the airflow.

NSE liftPressure and velocity fields for an AOA of 7°  NACA  0015 airfoil

This (non sharp) “confinement” in both horizontal and vertical directions of the pressure difference is a critical for the “push” to come about. When a (small) fluid parcel with a (small) mass is passing along somewhere in this “confined” region over the top or below the bottom surfaces of the airfoil it interacts directly with other fluid parcel’s which exert a net force on it through the differences in pressure. The forces pushing on the fluid parcel are not balanced since the pressure on one side of the fluid parcel is higher than on its other side. This is Newton’s Second Law in action, according to which if a net force is imposed on a fluid parcel it must cause a change in its speed and/or direction.
So there is also a confined effect on the velocity field, such that besides the deflection downwards of the fluid, the flow on the upper and lower surfaces speeds up, and slows down. This speeding up and slowing down is also confined, and fades further away from the airfoil.

The figure below illustrates for the (lift producing) NACA  0015 airfoil in a 7° AOA how these fluid parcels would be acted upon by the pressure field in the “confined” region. The arrows are of the net force exerted on such parcels, and it is easy to see that whether above or below the airfoil, fluid parcels see a higher pressure above than below them, hence a net force which is mostly downward. 

Pressure difference and net force

We can also see the flow is entering above the airfoil the “confined” region of a lower than ambient pressure with a net force from left to right (as the arrow shows) which will speed it up, then exiting on the other side of the “confined” region, where the pressure difference is in the opposite direction (and hence so is the net force), which will slow it down.
The reverse sequence is seen for the flow entering the “confined” region below the airfoil. A fluid parcel passing through a higher than ambient pressure, is slowed down and then speed back up. This description is actually the Beroulli’s theorem, and since it applies to regions of steady-flows which are not affected significantly by viscous effects (e.g. friction), we should add that this description is valid in the “confined” region I’ve described, but predominantly outside the boundary layer).

reciprocal cause-and-effect relationship between pressure difference and velocity

Already by looking at Navier-Stokes equation we see that both pressure gradients and the velocity fields are coupled, and there is no 1-way causation arrow between them in either direction. And indeed, as explained in the above paragraph, the pressure difference in the “confined” region causes a change in the fluid parcel’s direction and speed, and, at the same time, the resistance of the fluid parcel to this change in speed and/or direction sustains the pressure difference. Or in other words:

  • The pressure differences cause the flow to change speed and/or direction.
  • The change of speed (acceleration) and/or direction of the flow + its inertia causes the pressure differences to be sustained.

In sum, because the fluid is a considered a continuous material it appears to be gradually deforming around an airfoil, and it keeps being affected on a wide , yet “confined” region around the airfoil, producing pressure changes this “confined” region. When the airfoil is a certain range of AOA such that there is always a confined (but not sharply) region of lower than ambient pressure above the airfoil, and there is usually a confined (but not sharply) region of higher than ambient pressure below the airflow, then the airfoil pushes downward on the fluid which pushes back with an equal and opposite force in the downward direction (in accordance with Newton’s third law).
At the same time, these pressure differences in the “confined” region exert a net force which is mostly downward on the flow (in accordance with newton’s second law) such that it changes the speed and/or direction of the flow mostly downwards.
These pressure differences also act in the horizontal direction, speeding up or slowing down the fluid in a mechanism formally described by the Bernoulli’s theorem.
The change of speed (acceleration) and/or direction of the flow along with its its inertia causes the pressure differences to be sustained completing a 2-way cause-and-effect reciprocal relationship between pressure difference and velocity.


This concludes Part II of the “Engineering Turbulence” series. In Part III: “What A Drag!” , the concept of Drag shall be somewhat explored and explained, along with its reduction by flow control methodologies which are based on certain fundamental aspects of transition and turbulence coherent structures.

MSI Engineering Software and Mechanical Analysis Consultancy Services

One thought on “Engineering Turbulence – Part II: Lifting The Wright Way!

  1. Pingback: This Week in CFD | Another Fine Mesh

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