AE3 Aerodynamics I UNIT A: Fundamental Concets ROAD MAP... A-: Engineering Fundamentals Review A-: Standard Atmoshere A-3: Governing Equations of Aerodynamics A-4: Airseed Measurements A-5: Aerodynamic Forces and Moments AE3 Aerodynamics I : List of Subjects Seed of Sound Mach number Measurement of Airseed Incomressible Flow Comressible Flow What s Incomressible?
Page of Seed of Sound APPLICATION OF CONTINUITY ON A SOUND WAE Let us consider a coordinate system attached to (and thus, moving with the same seed with) the sound wave. Continuity equation ( m m) yields: Aa ( d) A( a da) a ( d)( a da) a ad da dda da Therefore, a (eqn. ) d SPEED OF SOUND () d Recall, the Euler s equation (in terms of seed of sound): d ada => da (eqn. ) a d d Substituting eqn. into eqn. yields: a => a da d The flow through a sound wave involves no heat addition, and the effect of friction is negligible: means, it is isentroic flow). Therefore: d a d isentroic
Page of Mach Number a RT ISENTROPIC FLOW Seed of Sound (Sea-Level Standard alue) SI Units: 34.3 m/s or,5.8 km/h U.S. Customary Units:,6.5 ft/s or 76.5 mh or 66.58 knots For isentroic flow: => constant c, or, simly: c (eqn. ) SPEED OF SOUND () Starting from d a d isentroic From eqn., the ressure of isentroic flow can be exressed as: c d d Therefore, ( c ) c (eqn. ) d d isentroic d Substituting eqn. back into eqn., and simlifying: d isentroic d Therefore, the seed of sound is: a d isentroic For an ideal gas: RT => RT ; therefore, the seed of sound becomes: a RT
Page 3 of Measurement of Airseed Pitot Tube: senses total ressure (subtract) Static Pressure Orifice: senses static ressure Pitot-Static Probe AIRSPEED MEASUREMENT DEICE Pitot-static robe measures both stagnation (or total) ressure and static ressure: rovides ressure difference between them ( ) STATIC, DYNAMIC, AND TOTAL (OR STAGNATION) PRESSURES Static ressure () at a given oint is the ressure we would feel if we were moving along with the flow at that oint. Total ressure () at a given oint in a flow is the ressure that would exist if the flow was slowed down isentroically to zero velocity: therefore, < (for a stagnant air: = ). Dynamic ressure is a ressure due to the added energy into the moving fluid (air). The difference between total and static ressures ( ) is dynamic ressure. Dynamic ressure is zero for a stagnant air ( = ). Stagnation oint is where = : so at stagnation oint, the ressure becomes close to the total ressure: stagnation ressure total ressure.
Page 4 of Incomressible Flow () (Subsonic: M <.3) ( ) () BERNOULLI S EQUATION For incomressible flow, we can emloy Bernoulli s equation. Along a streamline: constant = Let us define a dynamic ressure: q Then, the Bernoulli s equation becomes: q constant = AIRSPEED MEASUREMENT FOR SUBSONIC INCOMPRESSIBLE FLOW (M <.3) Let us define: location being the flow far ustream (called, the freestream ) and location being the location of zero velocity, the ti of the Pitot-Static tube (called, the stagnation oint ). Alying Bernoulli s equation between freestream ( ) and the ti of the Pitot-Static tube ( ti ): ti ti At the freestream: =, because = (this is freestream). At the ti: ti =, because ti = (this is stagnation oint). Therefore, =>
Page 5 of Incomressible Flow () (Subsonic: M <.3) Pitot-static robe TRUE AND EQUIALENT AIRSPEEDS The air density is difficult to measure. For small (low subsonic and low altitude cruise) airlanes, often the equivalent airseed is indicated on its airseed indicator: Equivalent (or indicated) airseed is the airseed that uses the standard sea-level air density value for the airseed calculation: ( ) e (Equivalent Airseed) s As long as the altitude is low (close to the sea-level), the equivalent (or indicated) airseed is fairly accurate. The true airseed is the airseed that uses the actual air density value for a given flight altitude for the airseed calculation: true ( ) (True Airseed) Have you heard about KEAS = Knots in Equivalent AirSeed? ( knot =.5 mh)
Page 6 of Class Examle Problem A-4- Related Subjects... Airseed Measurement: M <.3 The altimeter on a low-seed rivate aircraft (M <.3) reads 3, ft. If a Pitot-static robe (as shown in the figure) measures a ressure of 53.3 lb/ft, what is the equivalent airseed of the airlane? Suose, if you know the outside air temerature (through an indeendent measurement) is 5 ºF, what is the true airseed? Calculate the error of equivalent airseed. The equivalent airseed can be calculated (by using the air density at standard sea-level). Using s =.3769 slugs/ft 3 and 53.3 lb/ft : ( ) (53.3) e.774 ft/s s.3769 If the temerature is known, it is ossible to calculate the true air density: Using,896.7 lb/ft (ressure altitude 3, ft):,896.7 =.676 slugs/ft 3 RT (, 76)(5 46) Using this true air density: ( ) (53.3).78 ft/s true.676.78.774 The error of equivalent airseed: 4.5 %.78
Page 7 of Comressible Flow () (Subsonic: > M >.3) ENERGY EQUATION For comressible flow, we can no longer use Bernoulli s equation. Let us look at the energy equation one more time. Recall, the energy equation: ct constant (along the streamline) AIRSPEED MEASUREMENT FOR SUBSONIC COMPRESSIBLE FLOW (M >.3) Alying the energy equation for a Pitot tube (freestream and stagnation oint ): T ct ct => (eqn. ) T c T Also, the definition of secific heat can be given by: R Substituting this into eqn. : T T [ R / ( )] T RT Note that from seed of sound, a RT: thus the equation becomes, T T M => M T a T T Using the isentroic relationshi: T M and M c
Page 8 of Comressible Flow () (Subsonic: > M >.3) s cal s a a TRUE AND CALIBRATED AIRSPEEDS () Starting from: M Solving this equation for M: M (Note: M ) a a a (True Airseed) TRUE AND CALIBRATED AIRSPEEDS () True airseed requires information of a (i.e., T) and. The static temerature and ressure in the air surrounding the airlane is often very difficult to measure (in high-seed flight). Therefore, all high-seed airseed indicators are calibrated. For examle, assuming that a and are both equal to the standard sea-level value (as = 34.3 m/s =,6.5 ft/s and s =.3 5 N/m =,6. lb/ft ), the calibrated airseed (based on the standard sea-level condition) becomes: a s cal s (Calibrated Airseed)
Page 9 of Class Examle Problem A-4- Related Subjects... Airseed Measurement: M >.3 A jet aircraft is cruising high seed (high subsonic: M >.3) at km cruising altitude. If a Pitot-static robe (as shown in the figure) measures a ressure of 5.5 3 N/m, what is the calibrated airseed (and associated Mach number) of the airlane? Suose, if you know that the outside air temerature (through an indeendent measurement) is 45 ºC, what is the true airseed (and associated Mach number)? Calculate the error of calibrated airseed. Pitot-static robe measurement: = 5.5 3 N/m At standard sea-level: s =.3 5 N/m and as = 34.3 m/s.4 3.4 a s (34.3) 5.5 cal 5 s.4.3 cal 93.878 = > Calibrated airseed: cal 93.878 m/s M.76 a s 34.3 Pressure altitude of km: =.65 4 N/m Measured temerature is: T = 45 C = 45+73 = 8 K Seed of sound is: a RT (.4)(87)(8) 3.67 m/s.4 3.4 a (3.67) 5.5 4.4.65 59.43 = > True airseed: 59.43 m/s M.56 a 3.67 59.43 93.878 Error of calibrated airseed 4.5 % 59.43
Page of What s Incomressible? DEFINITION OF INCOMPRESSIBLE FLOW So far, we emloyed the rule of thumb (M <.3) as an indicator of incomressible flow. But, why this is valid? Recall, for isentroic flow, with calorically erfect ideal gas, the ratio of density between location (freestream) and location (stagnation oint) can be given as: M Note that the freestream is the location where the density is lowest within the flow field, while stagnation oint is the location where the density is highest (most comressed). Hence, this equation is the density variation within the given flow field (from lowest to highest density). For isentroic flows with Mach numbers less than about.3, the density variation within the flow field is less than 5 ercent. The variation is small, and thus the flow can be treated as incomressible.