03 Top Notch Notes Supersonic Aerodynamics R01-1.pdf

Top-Notch Aeronautics Top-Notch Notes Supersonic Aerodynamics Copyright Notice The notes and discussion, except the materials from other sources, are copyright of Top-Notch Aeronau...

Top-Notch Aeronautics Top-Notch Notes Supersonic Aerodynamics Copyright Notice The notes and discussion, except the materials from other sources, are copyright of Top-Notch Aeronautics. Unpublished Work © 2024 Top-Notch Aeronautics. All rights reserved. Any redistribution or reproduction of part or all of the contents in any form is prohibited other than the following: you may print or download to a local hard disk extracts for your personal and non-commercial use only you may copy the content to individual third parties for their personal use, but only if you acknowledge the author as the source of the material You may not, except with the express written permission, distribute or commercially exploit the content. Nor may you transmit it or store it in any other website or other form of electronic retrieval system. Engr. Jewel Christian P. Puse LECTURER R01 MAY/2024  Top-Notch Notes in Supersonic Aerodynamics Tips 1. Try to read the books as much as possible. They are the most raw and detailed information available. 2. When you cannot grasp a certain topic, try to search it in the internet, especially Youtube. I highly suggest: The Efficient Engineer (Lift, Drag, Bernoulli, Fluid Mechanics Playlist, and even StreMa) FlightInsight (Very good explanation especially in Flight Performance) Flight-club You must learn how to effectively search on the Internet, use keywords. “The information there is unlimited, but you must know how to access (search) it.” 3. In certain topics where the formula slightly changes depending on the condition (magkakamukha), make a table so that you can easily see the differences in the formula. With this, you can provide more “context” for memorization. 4. After each learning session, try to explain what you have learned. Teach it to a friend, your mama / papa, your cat, or talk to yourself. Unpublished Work © 2024 Top-Notch Aeronautics 1|Page  Top-Notch Notes in Supersonic Aerodynamics Introductory Information Roadmap Aerodynamics Syllabus C. References: 1. Elements of Practical Aerodynamics – Bradley Jones 2. Technical Aerodynamics - K D Wood 3. Engineering Supersonic Aerodynamics - Bonney 4. Aerodynamic Theory (Volumes I to VI) - Durand 5. FAA Code of Federal Regulations, Parts 1 to 59 – US Printing Office 6. Basic Helicopter Handbook - US Printing Office 7. Wind Tunnel Testing - Pope 8. Aerodynamics - L.S. Clancy Unpublished Work © 2024 Top-Notch Aeronautics 2|Page  Top-Notch Notes in Supersonic Aerodynamics Review of Thermodynamics Isentropic Relations Perfect Gas (Equation of State) Again, for isentropic flows, Gas in which the intermolecular forces are An adiabatic process is one in which no heat is neglected. added or taken away: δq = 0. 𝑷 = 𝝆𝑹𝑻 A reversible process is one in which no frictional or other dissipative effects occur. Since 1 An isentropic process is one that is both adiabatic 𝜈= and reversible. 𝜌 Therefore No heat exchange nor any effect due to friction. 𝑷𝝂 = 𝑹𝑻 𝒌 𝑷𝟐 𝑻𝟐 𝒌−𝟏 𝝆𝟐 𝒌 𝝂𝟏 𝒌 =[ ] =[ ] =[ ] 𝑷𝟏 𝑻𝟏 𝝆𝟏 𝝂𝟐 Internal Energy and Enthalpy Where: Sum of all the energies (translational, rotational, k = Specific Heat ratio kinetic, and vibrational) of molecules of a gas. Why is it frequently used? Why are we so interested in an isentropic process when it seems so restrictive—requiring both adiabatic and reversible conditions? The answers rest on the fact When the internal energy is per unit mass, it is that a large number of practical compressible flow called the specific internal energy e, and is related problems can be assumed to be isentropic. to enthalpy by the formula, ℎ = 𝑒 + 𝑝𝑣 Relating these two quantities to the specific heats, Total Conditions for a calorically perfect gas, What is “Total”?? 𝑒 = 𝑐𝑣 𝑇 Fluid element adiabatically (no heat transfer) ℎ = 𝑐𝑝 𝑇 slowed to zero velocity. Relating to the specific heat ratio and specific gas Remember, constant, Total Enthalpy 𝑘𝑅 𝑐𝑝 = 𝑽𝟐 𝑘−1 𝒉𝟎 = 𝒉 + 𝟐 𝑅 𝑐𝑣 = Where: 𝑘−1 h0 = Total Enthalpy h = Enthalpy (Static) V = Velocity Unpublished Work © 2024 Top-Notch Aeronautics 3|Page  Top-Notch Notes in Supersonic Aerodynamics However, since the fluid is brought to rest *** DON’T BE CONFUSED!!! adiabatically, An isentropic flow is also an adiabatic flow, 𝒉𝟎 = 𝒉 therefore, T0 is still constant! 𝒉𝟎 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 What then??? Total Temperature 𝑻𝑻 (𝒌 − 𝟏) 𝟐 =𝟏+ 𝑴 𝑻 𝟐 Total Pressure 𝒌 𝑷𝑻 (𝒌 − 𝟏) 𝟐 𝒌−𝟏 = [𝟏 + 𝑴 ] 𝑷 𝟐 Total Density 𝟏 𝝆𝑻 (𝒌 − 𝟏) 𝟐 𝒌−𝟏 = [𝟏 + 𝑴 ] 𝝆 𝟐 Features: Shockwaves At adiabatic - Total Enthalpy is constant Propagation of Pressure Waves flow, - Total Temperature is constant The greater the Mach number of the object, the more acute the upwash angle and the fewer the number of air particles that can move out of the path of the object. Air will begin to build up in front of the object and the density of the air will increase. Features: At isentropic - Total Pressure is constant flow, - Total Density is constant Unpublished Work © 2024 Top-Notch Aeronautics 4|Page  Top-Notch Notes in Supersonic Aerodynamics Flow Through a Nozzle Flow in a converging-diverging nozzle is used on many purposes like supersonic wind tunnels, rockets/missile engines, Athodyds (shortened term for aerothermodynamic duct), and gas turbine engine. “XXX “– Mach Number Relation Temperature – Mach Number Relation 𝑻𝟐 (𝒌 − 𝟏)𝑴𝟏 𝟐 + 𝟐 = 𝑻𝟏 (𝒌 − 𝟏)𝑴𝟐 𝟐 + 𝟐 Pressure – Mach Number Relation 𝒌 𝑷𝟐 (𝒌 − 𝟏)𝑴𝟏 𝟐 + 𝟐 𝒌−𝟏 Shockwave =[ ] 𝑷𝟏 (𝒌 − 𝟏)𝑴𝟐 𝟐 + 𝟐 An extremely thin region across which flow Density – Mach Number Relation properties change drastically. 𝟏 Oblique shock wave: 𝝆𝟐 (𝒌 − 𝟏)𝑴𝟏 𝟐 + 𝟐 𝒌−𝟏 =[ ] 𝝆𝟏 (𝒌 − 𝟏)𝑴𝟐 𝟐 + 𝟐 Area – Mach Number Relation 𝒌+𝟏 𝑨𝟐 𝑴𝟏 (𝒌 − 𝟏)𝑴𝟐 𝟐 + 𝟐 𝟐(𝒌−𝟏) = [ ] 𝑨𝟏 𝑴𝟐 (𝒌 − 𝟏)𝑴𝟏 𝟐 + 𝟐 Normal shock wave: Throat – Section Area – Mach Number Relation 𝒌+𝟏 𝑨 𝟏 (𝒌 − 𝟏)𝑴𝟐 + 𝟐 𝟐(𝒌−𝟏) = [ ] 𝑨∗ 𝑴 (𝒌 − 𝟏) + 𝟐 Unpublished Work © 2024 Top-Notch Aeronautics 5|Page  Top-Notch Notes in Supersonic Aerodynamics Personal Note!!! 6 Possible Conditions of flow in a C-D Nozzle Condition 1. Subsonic Flow along the nozzle As earlier suggested, the format used to memorize the isentropic equations can now be easily used to get Supersonic Isentropic and “xxxx” – Mach Condition 2. Subsonic Flow along the nozzle but Number Relationships. choked at the throat Flow Behavior on a C-D Nozzle Condition 3. Supersonic Flow after the throat, known as “Correctly Expanded” nozzle Unpublished Work © 2024 Top-Notch Aeronautics 6|Page  Top-Notch Notes in Supersonic Aerodynamics Condition 4. Supersonic Flow on the nozzle on a Over Expanded vs Correctly Expanded vs condition known as “Overexpanded Nozzle” Under Expanded Condition 5. Supersonic Flow in an “Underexpanded” condition. Condition 6. Formation of a Shock Wave inside the Nozzle. Speed Regimes Subsonic M is less than 1 at every point Transonic M slightly below unity Pockets of superso at top and bottom Normal shockwave M slightly above unity o Bow shock Unpublished Work © 2024 Top-Notch Aeronautics 7|Page  Top-Notch Notes in Supersonic Aerodynamics Normal Shock Waves Supersonic Mfs > 1 at every point If theta is large enough, o Oblique will detach and turn curved bow Hypersonic Given that theta is fixed, o As M increases, oblique shock Three Physical Facts: moves closer to body 1. The flow is steady. 2. The flow is adiabatic. 3. There are no viscous effects. 4. (+1) There are no body forces. Examples of NSW: On blunt bodies, bow shock are expected to form in front. Looking closer at the “leading edge” of the bow shock, we can consider it as normal to the flow or as we know, a normal shock. Unpublished Work © 2024 Top-Notch Aeronautics 8|Page  Top-Notch Notes in Supersonic Aerodynamics Temperature Ratio 𝑻𝟐 [𝟐𝒌𝑴𝟏 𝟐 − (𝒌 − 𝟏)][𝑴𝟏 𝟐 (𝒌 − 𝟏) + 𝟐] = 𝑻𝟏 𝑴𝟏 𝟐 (𝒌 + 𝟏)𝟐 Or, since k = 1.4, and following the equation of state, 𝑻𝟐 𝑷𝟐 𝑽𝟐 = 𝑻𝟏 𝑷𝟏 𝑽𝟏 NSW Formulas Cheat Sheet!!! Mach Number behind shock M-V-P-T 𝟐 𝑴𝟏 𝟐 (𝒌 − 𝟏) + 𝟐 𝑴𝟐 = 𝟐𝒌𝑴𝟏 𝟐 − (𝒌 − 𝟏) Or, since k = 1.4, 𝟐 𝟓 + 𝑴𝟏 𝟐 𝑴𝟐 = 𝟕𝑴𝟏 𝟐 − 𝟏 Velocity Ratio 𝑽𝟐 𝑴𝟏 𝟐 (𝒌 − 𝟏) + 𝟐 = 𝑽𝟏 𝑴𝟏 𝟐 (𝒌 + 𝟏) Or, since k = 1.4, 𝟓 + 𝑴𝟏 𝟐 𝑴𝟐 𝟐 = 𝟕𝑴𝟏 𝟐 − 𝟏 Pressure Ratio 𝑷𝟐 𝟐𝒌𝑴𝟏 𝟐 − (𝒌 − 𝟏) = 𝑷𝟏 𝒌+𝟏 Or, since k = 1.4, 𝟕𝑴𝟏 𝟐 − 𝟏 𝑴𝟐 𝟐 = 𝟔 Unpublished Work © 2024 Top-Notch Aeronautics 9|Page  Top-Notch Notes in Supersonic Aerodynamics Oblique Shock Waves Whenever a supersonic flow is “turned to itself”, an oblique shock will occur. This commonly happens when a flow encounters a concave corner. As discussed on the topic of speed of sound, we know that due to the movement of the body in a medium (air), and especially when it reaches Parts of an OSW supersonic speeds, a standing wave is formed in front of the body due to the air upstream not able to “react and work its way” out of the moving body. Mach Angle Wave Angle – Angle of shockwave / oblique shock A function of the local Mach number wave because of its deflection. Corresponds to the angle created by the Mach Wave Mach angle – Angle of the Mach Wave with respect to the horizontal. Remember that Mach Wave is 𝟏 only a function of local Mach number. 𝝁 = 𝐬𝐢𝐧−𝟏 𝑴 Deflection angle – Angle of the body / surface / airfoil that causes the supersonic flow to deflect. Unpublished Work © 2024 Top-Notch Aeronautics 10 | P a g e  Top-Notch Notes in Supersonic Aerodynamics An oblique shock wave has two components: Effect of increasing upstream Mach Number 1. Normal component of the velocity 2. Tangential component of the velocity The tangential component of the flow velocity is constant across an oblique shock. Therefore, changes across an oblique shock wave are governed only by the component of velocity normal to the wave. Attached & Detached Shock Features: As upstream wave angle decreases Mach number increases, Effects of increasing deflection angle Weak vs Strong Shock Features: As deflection wave angle increases angle increases, OSW Formulas Features: A more is a stronger shock. vertical shock, Upstream component of Velocity and Mach Number 𝑽𝑵𝟏 = 𝑽𝟏 𝐬𝐢𝐧 𝜽 𝑽𝑻𝟏 = 𝑽𝟏 𝐜𝐨𝐬 𝜽 𝑴𝑵𝟏 = 𝑴𝟏 𝐬𝐢𝐧 𝜽 Unpublished Work © 2024 Top-Notch Aeronautics 11 | P a g e  Top-Notch Notes in Supersonic Aerodynamics Downstream component of Velocity and Mach Number 𝑽𝑵𝟐 = 𝑽𝟐 𝐬𝐢𝐧(𝜽 − 𝜹) 𝑽𝑻𝟐 = 𝑽𝟐 𝐜𝐨𝐬(𝜽 − 𝜹) 𝑴𝑵𝟐 = 𝑴𝟐 𝐬𝐢𝐧(𝜽 − 𝜹) Downstream Mach Number Normal to Shockwave MN2 𝟐 𝑴𝑵𝟐 𝟐 (𝒌 − 𝟏) + 𝟐 𝑴𝑵𝟐 = 𝟐𝒌𝑴𝑵𝟏 𝟐 − (𝒌 − 𝟏) Expansion Waves Downstream Mach Number of the Whenever a supersonic flow is “turned away from Shockwave M2 itself”, an expansion wave will occur. This 𝑴𝑵𝟐 𝟐 (𝒌 − 𝟏) + 𝟐 commonly happens when a flow encounters a 𝟐 𝑴𝑵𝟐 = 𝒄𝒔𝒄𝟐 (𝜽 − 𝜹) convex corner. This is sometimes called the 𝟐𝒌𝑴𝑵𝟏 𝟐 − (𝒌 − 𝟏) Prandtl-Meyer Expansion Wave. Deflection Angle – Shockwave Angle Main feature: ISENTROPIC 𝟐 𝟐 𝟐 𝑴𝟏 𝒔𝒊𝒏 𝜽 − 𝟏 𝐭𝐚𝐧 𝜹 = [ ] 𝐭𝐚𝐧 𝜽 𝒌𝑴𝟏 𝟐 + 𝑴𝟏 𝟐 𝐜𝐨𝐬 𝟐𝜽 + 𝟐 Practical Application: Inlets Visualized as infinite number of Mach waves Unpublished Work © 2024 Top-Notch Aeronautics 12 | P a g e  Top-Notch Notes in Supersonic Aerodynamics Prandtl – Meyer Function 𝑽(𝑴𝟏 ) = 𝑽𝟏 𝒌+𝟏 𝒌−𝟏 =√ 𝒕𝒂𝒏−𝟏 √ (𝑴𝟏 𝟐 − 𝟏) 𝒌−𝟏 𝒌+𝟏 − 𝒕𝒂𝒏−𝟏 √𝑴𝟏 𝟐 − 𝟏 Fan Angle 𝑭𝑨 = 𝝁𝟏 + 𝜽 − 𝝁𝟐 𝑭𝑨 = 𝝁𝟏 − 𝝁𝟐 + 𝑽𝟐 − 𝑽𝟏 Approximate Methods Lift Coefficient 𝟒𝜶 𝑪𝑳 = √𝑴𝟐 − 𝟏 Drag Coefficient 𝟒𝜶𝟐 𝑪𝑫 = √𝑴𝟐 − 𝟏 Compressible Flow through Nozzles, Diffusers, and Wind Tunnels Flow Characteristics Remember that subsonic and supersonic flow characteristics in a wind tunnel is very different! Unpublished Work © 2024 Top-Notch Aeronautics 13 | P a g e  Top-Notch Notes in Supersonic Aerodynamics Additional Theories / Concepts Supercritical Airfoil Effect of Altitude to Speed of Sound An airfoil that, principally, has been designed to delay the onset of wave drag in the transonic speed Since temperature decreases as altitude increases, range. the speed of sound also decreases. Critical Mach Number The free stream Mach number at which the local velocity first reaches Mach 1.0 (sonic). Features: As AoA Critical Mach Number increases, decreases. As t/c Critical Mach Number increases, decreases. Both due to increase in acceleration in top surface. Past the Critical Mach Number, The airfoil will experience a small region of supersonic airflow on the upper surface, terminated by a shock wave. Drag Divergence Mach Number The Mach number at which the aerodynamic drag begins to increase rapidly. Caused by: The drag directly associated with the trailing edge shock waves (energy loss). Shock Stall Separation of the boundary layer. The formation of the bow shock wave Stall caused by the formation of shockwave on the above M 1.0. upper surface of the airfoil which causes the boundary layer to be separated. Unpublished Work © 2024 Top-Notch Aeronautics 14 | P a g e  Top-Notch Notes in Supersonic Aerodynamics Aerodynamic Buffeting vs Flutter Buffet Vibration of aircraft caused by turbulent airflow over the surface of the airplane. Although it may be repetitive, it is not cyclic or harmonic. Flutter An un-commanded, self-perpetuating and cyclic movement of any part of an aircraft. This is a dangerous phenomenon caused when the structure is exposed to airflow that creates the phenomenon. Unpublished Work © 2024 Top-Notch Aeronautics 15 | P a g e