Theory of Flight Helicopter v1.4 33

[image]. PHYSICAL BACKGROUND. [image]. Bernoulli’s Principle Lift Generation Venturi Theory Equal Transit Theory fallacy Coanda effect Theory Newton’s Laws of Motion Momentum Theory Skipping Stone Theory Lift in Mathematics.

[Audio] Physical Background Lift Generation Though we don’t always think of it this way, Air is a fluid, just less dense than liquids like water. Objects that are lighter than air are buoyant within it, while heavier objects require an upward force, called Lift, to stay aloft. For planes, this force is mostly generated by the Wings. As for helicopters, it’s generated by rotor blades. But how? Well, wings and Rotor blades are made in the shape of an Airfoil, and airfoils produce lift when a fluid flows over them. Then again, Why and How? While some state that the upward force (Lift) is a result of a differential in pressure between the lower and the upper surface of an airfoil in accordance with Bernoulli’s principle, others see to it a matter of momentum between the airfoil and the moving air compliantly to Newton’s laws of motion. Let’s check out each standpoint! Lift Lift.

[Audio] Physical Background Bernoulli’s principle Conservation of Energy : The concept of conservation of energy states that : energy cannot be created or destroyed, it can only be transformed. So the amount of energy entering a system is the same that must exit. In fluid dynamics, the “energy” referred to in a is Total Pressure. In a streamline, This energy can be broken down into: Dynamic pressure (the fluid’s kinetic energy more velocity, more kinetic energy); and Static pressure (or simply Pressure energy). These energies will change among themselves, but the Total Pressure energy remains constant inside the system..

[Audio] Bernouilli’s Principle : As a statement of the law of conservation of energy, Bernoulli’s principle describes the relationship between internal fluid pressure P (static) and its velocity 5 (dynamic). As it applies to Aerodynamics (air being a fluid), Bernoulli's principle states that if there is an increase in air velocity, then there will be a proportional decrease in its pressure. Put otherwise : “the faster the air moves the lower pressure it creates due to the conservation of energy”. This statement, as it is, holds true, however, a misapplication of Bernoulli’s principle and equations would lead to wobbly explanations such as the Venturi theory, or to utterly false hypotheses such as the Equal Transit Time theory. P plus ½ (ρV² )= cst Dynamic Pressure (Kinetic energy) Static Pressure Pressure Velocity.

[Audio] Venturi Theory : Venturi effect is a direct application of Bernoulli’s principle to a fluid that flows through a tube with a constriction in it. As the fluid flows through the constriction, the fluid speeds up, but the static pressure the fluid exerts on the tube’s internal walls decreases. Venturi Theory assumes* that airflow around an airfoil performs similar to airflow through a constricted tube. As velocity of the airflow increases, static pressure decreases above and below the airfoil. With an upper curved airfoil, there is a greater velocity increase and greater static pressure decrease over the upper surface than the lower surface.. This differential in pressure develops an upward aerodynamic force. * As air tends to want to move in a straight line, the tube here is assumedly formed with the unaffected free straight streamlines. However, this assumption is not that accurate which is the main weakness of Venturi theory besides its inability to explain why an aircraft can fly upside down and why a flat foil can generate lift. Free Streamline Lift Free Streamline.

[Audio] Equal transit time theory fallacy : D2 The “equal transit time” theory applies Bernoulli in a false way: it supposes that air molecules that separate at the front edge of an upper curved airfoil must rejoin simultaneously at the rear edge. Because the upper air molecules travel a greater distance than the lower ones in a given amount of time, they would have to go faster to catch up with the lower molecules at the rear edge. This theory is wrong and the fallacy here is that there is no physical reason that the two parcels must reach the rear edge simultaneously. And indeed, empirically, they do not. Actually, the air above moves much faster than this theory would suggest. Bernoulli equations can only be applied in a single streamline. Meanwhile, the upper airflow and lower airflow are two different streamlines. D1 D2 > D1 V2 > V1.

[Audio] Coanda effect Theory : Another Bernoulli based explanation uses Coanda Effect. Coanda effect is the tendancy of a fluid jet to stay attached to a convex surface. How is this related to lift? Let’s see: When moving forward, the airfoil affects the air movement around it. As the air meets the airfoil solid surface, a thin layer sticks to the airfoil curvature under Coanda effect pulling surrounding air with it. Another standpoint supports that a vaccum is created on the upper surface of the airfoil pulling air into that void. (this at least tries to explain why the upper flow sticks to the upper surface). Either way, The air molecules experience centripetal accelerations as they make their way around any curvature. More acceleration means more speed. And since the upper surface has more curvature than the lower, the aiflow above then gathers more speed, and thus, less pressure than airflow travelling below. The difference in pressure therefore develops an upward aerodynamic force . Coanda Effect Vacuum (void).

[Audio] While air does flow smoothly around curved airfoils, curvature is not a key factor in this theory, because, it’s not the cause of lift. In fact, a flat airfoil, tilted upwards, can also create Lift as long as the air bends around it, reinforcing the pressure difference. This explains why flat wings planes fly and why some planes can fly upside down. Notwithstanding, an airfoil too curved or steeply angled won’t work, since the airflow above may detach from the upper surface of the airfoil and become turbulent. This is probably what happened to Einstein’s “cat’s back” airfoil design. While this theory can also be combined with Newton’s laws to explain lift, its direct extension to airfoil is debatable in first place..

[Audio] The air flowing over the top of an airfoil is definitely moving faster than below. This is demonstrated by Computational Fluid Dynamics C-F-D simulations. Regardless of what causes the air to move faster over the top of the airfoil, and how low pressure above the wing is ultimately formed; low pressure above the wing has been determined to be a rather small component of total lift generated by an airfoil. So, the pressure difference is only a small contributor to the total generated lift. There are, actually, other contributors that helps explaining lift generation..

[Audio] Wrap up : Compliantly to Bernoulli’s principle, Lift is a result of a pressure difference between the upper and lower surfaces since the upper airflow goes faster. As for why air on the upper surface moves faster, Bernoulli’s equation itself cannot provide an answer. Many theories tried to unveil that conundrum but failed somehow to give a correct, complete and satisfactory explanation. Regardless, as proved by the C-F-D simulation, the air above an airfoil does go faster incurring a pressure difference and creating an upward aerodynamic force. However, this force seems only a small part of the total generated Lift as there are other factors that contribute to create Lift..

[Audio] Newton’s laws of motion Newton's laws are arguably the primary contributors that explain airfoil lift. They are also the ones that we have direct control over while we fly our helicopters. Let’s see what these laws say: 1st law of Newton: an object will not change its motion unless a force acts on it. This is the law of Inertia. 3rd Law of Newton: for every action (force) there is an equal and opposite reaction. This is the law of Action & Reaction. 2nd Law of Newton: if there is a motion there is a force. that force is equal to the change in momentum* per time. * Physically speaking, Momentum is defined to be the mass of an object multiplied by its velocity. In simple terms, it’s a “Force” that a moving object gain by virtue of its mass and motion. Like Energy, Momentum is conserved through time within an isolated system: It is neither created nor destroyed, but only changed through the action of forces. P = m . 5.

[Audio] Momentum Theory: P-I-N Now, let’s apply Newton’s motion Laws on an airfoil moving through air: According to 1st Newton Law: the movement of the parcel of air has changed as it passed the airfoil, so there must be some force applied by the airfoil that changed the movement of air. According to 2nd Newton’s Law: if there’s a change of motion there is a force which is equal to the change of momentum per time. So, there is actually a force applied by the airfoil on air, and that force is equal to the change of air momentum per time. According to 3rd Newton’s Law: as the airfoil applies a downward force on air, air applies in its turn an equal and opposite, thus, upward force on the airfoil. This aerodynamic force is what we call Lift. With this theory, you don’t even need to worry about pressure difference. F pout Lift P-I-N F pout F = dP/dt = m.dV/dt = m.a.

[Audio] A natural follow up question is : How the airflow is turned down? Well, while it’s obvious and easy to accept for the airflow below the airfoil, the airflow above needs some hypotheses such as Coanda effect or viscosity effects to explain why the upper flow follows the curvature of the airfoil. As comprehensive as it may seems, Momentum Theory loses applicability in autorotative flight. Note that in the same way some explanations misapplied Bernoulli’s principle to explain lift, some theories applied Newton’s Law in a flawed way. The most famous one is the Skipping Stone theory..

[Audio] Total Lift Skipping Stone Theory : L1 Skipping Stone theory (a.k.a Particles Kinetic theory) is based on Newton’s 3rd law stating that for every action, there is an equal and opposite reaction. It says that Lift is the reaction force to air molecules striking the bottom of the airfoil as it moves through the air. Because this is similar to the way in which a flat rock thrown at a shallow angle skips across a body of water, it is called the "Skipping Stone" theory of lift. While 3rd Newton’s law holds true, Skipping Stone theory is incomplete as it overlooks the air molecules above airfoil and the interaction between molecules. L2 L3.

[Audio] Wrap up : Lift According to Newton’s Laws of motion and conservation of momentum, Lift is a reaction force experienced by the airfoil due to its turning the flow (change of momentum). The airflow receives downward momentum from the airfoil at a rate that depends on how much air is affected (mass) and how much turning is done (dV/dt). Symmetrically, the airfoil receives upward momentum at the same rate creating Lift. Due to Coanda Effect, The airflow above the airfoil sticks to the upper surface incurring a laminar flow and turning the airflow downward. Laminar Flow m.VIN P M.vout.

[Audio] Lift in Mathematics With all the aerodynamic engineers on here arguing with one another, it is amazing they can even design aircraft that work in the first place! Unlike the popular explanations of lift, the mathematical theory of lift must be precise and predictive. Otherwise, no one would trust a plane, let alone a helicopter! Classical aerodynamics used to calculate Lift based on K-J theorem which gave insightful results and helped designing various airfoil way before C-F-D methods. Now engineers use a set of formulas called Navier stokes Equations to precisely model airflow around an airfoil and detail how lift is generated. These equations are difficult to solve analytically and are only treated with C-F-D-..

[Audio] In a nutshell… While experts agree that airfoils generate lift when air flows through them, their explanation for how can vary : Some may point up the air behavior at the airfoil surface as a result of a pressure differential (Bernoulli based theories), while others emphasize the momentum matter referring the upward force creation to air downward deflection (Newton’s Laws theories). However there is no controversy when it comes to math: engineers use Navier stokes Equations to precisely model airflow around an airfoil and detail how Lift is generated thanks to C-F-D-..