276225-86042500right-123190American University of Science and TechnologyMath DepartmentMAT 201 ” Calculus IFall 2017 ” 201800American University of Science and TechnologyMath DepartmentMAT 201 ” Calculus IFall 2017 ” 2018righttop012100center818008745855 941009200 Calculus And Aerodynamicscenter16065500Elie-Jose Nemer12170166Bilal Zahalan12170203Bilal Sraj12170293Amir Dlykan12170128Table of ContentsThe Definition of Aerodynamic.2Introduction to the Air Induced drag….3What is Lift?…4What is Down Force?…………………………….6The Physics Behind Aerodynamics…7Application of Calculus in Aerodynamics.9Reference……11The Definition of AerodynamicAerodynamics is a branch of dynamics that deals with the motion of air and other gaseous fluids and with the forces acting on bodies in motion relative to such fluids.
This branch of dynamics is used in everyday life. It is used in many vehicles, such as cars and planes. And its main role is to keep balance in certain vehicles.All physical objects on Earth are subject to gravity, but gravity is not the only force that tends to keep them pressed to the ground. The air itself, though it is invisible, operates in such a way as to prevent lift and to keep cars pinned to the road or ground.
As engines became more powerful and cars became faster and lighter, engineers realized that air resistance is affecting the speed of these powerful cars. So, for this reason, racing cars now have different designs than normal cars. These cars have been designed to move fast. The design of these cars is based of having a sharp body form which reduces this air resistance, thus increasing a car’s speed. They are also made in a way to balance between performance, handling and fuel consumption.Many of the same aerodynamic principles used in racing also apply to regular cars and trucks. Automotive engineers use computer simulations and wind tunnel experiments with scale models and actual vehicles to fine-tune the aerodynamics of automobiles so they generate the optimum amount of downward force to the front and back wheels with the least possible amount of unnecessary drag. right346628Introduction to the Air Induced DragThe power on a protest that opposes a solid’s movement through a liquid or fluid in general, is called drag. At the point when the liquid is a gas like air, it is called streamlined drag. At the point when it’s a liquid like water it is called hydrodynamic drag. Liquids and fluids are portrayed by their capacity to stream. This makes them difficult to hold yet simple to pour, mix, and spread. Liquids have no unequivocal shape however go up against the state of their compartment. Liquids are neighborly one might say. They yield their space generally simple to other material things; at any rate when contrasted with solids. A liquid will escape your direction on the off chance that you ask it. All fluids must be told with damaging power, if collided with at high speed.Liquids may not be strong, but rather they are definitely material. The basic property of being material is to have both mass and volume. Material things oppose changes in their speed and no two material things may involve a similar space due to their volume. In order for a solid to move through a liquid it’s supposed to push the liquid aside of its path. The liquid flow bending generates and opposing force to the movement of the liquid, thus decreasing its kinetic energy or initial speed. Shooting a bullet (solid) underwater (liquid/ fluid) is a perfect example to illustrate drag.water is capable of stopping a high speed bullet, due to its density, as the bullet has to push the relatively heavy water out of the way in order to continue moving, and does not, in fact, travel very far.330200329244What is Lift?Lift is a mechanical aerodynamic force which is opposite to weight. For an aircraft to fly, lift is required. Lift is generated by all the parts of a plane, especially the wings. Lift acts through the center of pressure of the object and directed perpendicular to the flow direction. Lift is generated when a moving flow of gas is turned by a solid object. Lift will be generated in almost the opposite direction of the flow (Newton’s Third Law of action and reaction).877415984600 According to Bernoulli’s Equation (Conservation of Energy), when the velocity of a fluid increases, its pressure decreases by the same amount to have the same overall energy. When the wing moves through stationary air, a Lift force will be formed which is perpendicular to the motion of the wing, and it is greater than the gravitational force on the wing. Lift is accompanied by drag which represents the air resistance against the wing and it forces its way to the air. Bernoulli’s principle, which says that if air speeds increases the pressure decreases, then the wing generates lift because the air goes forward over the top and creates a region of lower pressure and thus lift. 225565102367There are many types of wings: conventional, symmetric… In flight, the early biplane wings that looked like warped boards, and even the proverbial “barn door”. In all these types the wing forces the air to go down or to pull the air down from above. All the wings have an angle of attack with respect to the oncoming air in common. Angle of attack is the parameter to determine the lift. The wing can be explained by its angle of attack. Lift requires power and this power is given by the airplane’s engine or by gravity.What is Down Force?Downforce, also known as negative lift, is an Aerodynamic force which is produced by the airflow around an object (body of a vehicle). This force creates a stronger pressure between the tires of a vehicle and the road surface. One common method to increase the downforce of a vehicle is to reduce the air pressure under the vehicle. center1016000Forced high speed is required to create downforce. The main purpose of downforce is to allow a vehicle to travel faster through the corners. This is achieved by creating a huge vertical force on a vehicle’s tires. This force will give more grip and traction to the tires via the chassis and the suspensions. Downforce will help in improving your vehicle’s stability and it’s handling when you’re going fast. Air dams, Splitters, Spoilers and Wings are all used for increasing the downforce. When the air hits a car’s surface, downforce is created. Air passes over a car and pushes it towards the ground. Furthermore, higher downforce means higher air resistance. This means that as downforce increases, speed decreases. This being said, a too much down force would mean more air resistance, more fuel consumption and more emissions for less speed. Balancing down force is in conclusion crucial, in order to find the sweet spot between handling, and performance.The Physics Behind AerodynamicsAirplane wings, flat plates, spinning balls, For example, generate Lift as stated earlier.That Force that keeps these objects in the air, is no exception to the second law of newton.Car chassis and wings, are made to generate, down force using the force generated by airflow bending, to push the car towards the ground to maintain its stability. These 2 forces are generated due to the drag the air causes, when it hits the surface at a certain velocity.458752016217300Isaac newton is a physicist born in 1642, that established classic mechanical rules, and imposed 3 fundamental laws of movement.The second law of newton states that: m for massa for accelerationAcceleration is a change in velocity with time.411480698500so in other words, the acceleration is the derivative of the Velocity and can be calculated 533903149860by: To sum everything up, by applying the second law of newton we can calculate the force of lift.39543341110000Another physician more specific to the aerodynamic field is Daniel Bernoulli, born in 1700, he established the equation of the pressure in a fluid. To simplify, we can take into account less variables, and write the simplest, yet valid form of this equation:left339090Ps for static pressure for dynamic pressurePt for total pressureThis equation grants us a way to calculate the total pressure, exercised by a fluid.These principles, are cores to the final and most useful equation in the aerodynamic field, which is no other than the drag equation:D for DragCd for Drag coefficient for density of the fluidV for VelocityA for the reference areaThe coefficient Cd contains all the complex dependencies and is usually determined experimentally.Choice of the reference area A affects the value of Cd.Application of Calculus in Aerodynamics4762581216500Around the wing of an airplane for example, the pressure in a fluid decreases as the fluid’s velocity increases, according to Bernoulli’s equation.The pressure on top of the wing, is than the pressure on bottom.The Bernoulli equation, for the pressure in a Fluid:P + gy + Ѕv2 = constant along a given stream line. P for Pressure outside the fluidgy for the gravitational contribution to the pressureЅv2 for the dynamic or kinetic contribution to pressureThis equation is relatable to the initial Bernoulli principle and was built depending on it.it enables us to find the pressure.The lift Coefficient is a dimensionless number, used to find the lift force generated, according to the density of the fluid, the surface of the area, and the velocity. It’s determined by various complex dependencies and is often determined experimentally.The relation goes by the equation: L for the lift ForceCl for the lift coefficient for the densityV for the velocityA for the Surface The lift force found, is applied against the gravitational pull, as its generated on a moving solid through a fluid in this case the air.The lift force depends also, on the design of the wings, but all wings have a curve on top for the Lift force. When the curve of the wing is minimal to negligible, the pressure difference is 0, and there is no lift force.Bernoulli’s equation is used to calculate the pressure at any point, to find the Lift force.336931037211000According to other multiple forces applied, like the weight, the Lift force, the engine thrust induced force, and the total drag (found using the drag equationof Bernoulli) we can use the Newton’s second law:Ft = m.aFt is the sum of all the forcesm is the massa is the accelerationWe can also type Ft ” m = aas a’ is the derivative of the Velocity.we can integrate a to find the velocity of the wings.This whole path, puts in use various physics notions, and calculus methodology like, derivatives and integrals, and basic math to study the interaction between the fluid air, and the airplane.the same methodology is also used to study the down force produced by a car shape or wings. The down force is similar to the lift, its uses the force induced by the airflow bending, but towards the ground, as it adds up the gravitational pull to push the car on the ground. Reference