AE301 Aerodynamics I UNIT A: Fundamental Concets ROAD MAP... A-1: Engineering Fundamentals Reiew A-: Standard Atmoshere A-3: Goerning Equations of Aerodynamics A-4: Airseed Measurements A-5: Aerodynamic Forces and Moments AE301 Aerodynamics I : List of Subjects Continuity Princile Incomressible Flow Momentum Princile Bernoulli s Equation Energy Princile Secific Heats Isentroic Flow Energy Equation Summary of Equations
Page 1 of 16 Continuity Princile m 1 V 1 m V Time rate of change of mass within the control olume + Flux of mass across the control surface = 0 CONTINUITY PRINCIPLE Continuity rincile = conseration of mass Density () will change as a small element of air is comressed. Howeer, the mass will neer change. 1 Secific olume will also be changed: Let us define the mass flow rate: dm m VA dt A STREAM TUBE A stream tube is an aerodynamic analytical concet: a tube within a flow field, defined by streamlines (streamlines are the wall of the stream tube) For a gien stream tube, the continuity equation is defined by: 1AV 1 1 AV
Page of 16 Incomressible Flow M < 0.3 1 (Incomressible) COMPRESSIBLE & INCOMPRESSIBLE FLOW Comressible flow is the flow in which the density of the fluid elements can change from oint to oint. Incomressible flow is the flow in which the density of the fluid elements is always constant. Let us look at a duct (conerging or dierging, or combined). If the flow is incomressible, the continuity rincile can be alied as: A 1 1AV 1 1 AV => AV 1 1 AV => V V1 A 1 In aerodynamics, the low subsonic seed is considered incomressible. The rule of thumb for incomressible flow is: M < 0.3 (Why?) INCOMPRESSIBLE AERODYNAMICS You must understand that air (as a fluid substance), is not incomressible. This is fundamental and called, material (fluid) roerty. Howeer, if the flow field is sufficiently low seed (we usually emloy Mach number to measure that), we can ignore the effect of comressibility so that our analysis can dramatically be simlified. Hence, under the assumtion of M < 0.3 (we ll discoer the reasoning of this assumtion later), we urosefully ignore the change in density within the flow field. This is, in fact, flow field roerty. Equations we emloy in aerodynamic analysis are built under assumtions: understanding the assumtions means understanding the limitations of each equation (ery imortant).
Page 3 of 16 Class Examle Problem A-3-1 Related Subjects... Continuity Princile Consider a conergent circular duct with an inlet diameter d 1 = 10 m. Air enters this duct with a elocity V 1 = 0 m/s and leaes the duct exit with a elocity V = 5 m/s. What is the area of the duct exit? Velocity Pressure Temerature Velocity Pressure Temerature At sea-leel, the elocity 0 m/s and 5 m/s corresond to the Mach numbers of 0.059 and 0.073, resectiely (M << 0.3). Therefore, we can assume that the flow is incomressible (constant density: 1 ). Alying the continuity rincile: V 1 AV 1 1 AV => A A1 V V 1 0 Thus, A d1 (10) 4 V 4 5 6.83 m
Page 4 of 16 Momentum Princile Time rate of change of momentum within the control olume + Flux of momentum across the control surface = Sum of all the forces in a gien direction MOMENTUM PRINCIPLE Momentum rincile = conseration of momentum In aerodynamics, we consider three tyes of forces: Pressure, Body, and Viscous forces EULER S EQUATION ALONG A STREAMLINE Let us consider a small cubic element of air (with the olume: dx dy dz) moing steadily along the streamline. If we ignore body force and iscous force, the momentum equation will become Euler s equation. F ma along the streamline: d dv dydz dxdydz ( dxdydz) dx dt Note that: dv dv dx dv d dv V, thus: F ma => ( dxdydz) ( dxdydz) V dt dx dt dx dx dx Therefore, d VdV (Euler s equation along a streamline)
Page 5 of 16 Bernoulli s Equation BERNOULLI S EQUATION From Euler s equation: d 3 V1 V V3 1 3 (3 is not along the same streamline of 1 & ) VdV If we assume that the density is constant along a streamline (incomressible), we can integrate this equation along a streamline: V d VdV => 1 V1 V V1 1 ( ) V1 V => 1 or simly, (along a streamline) V constant LIMITATIONS OF BERNOULLI S EQUATION The Bernoulli s equation is the most conenient, but the most commonly abused (in other words, mistakenly used ) equation in aerodynamics. Assumtions required for deriing Bernoulli s equation includes: From Euler s equation: 1. The flow is steady (streamline).. The body force is ignored (no body force: common assumtion in aerodynamics, but NOT in hydrodynamics). 3. The iscous force is ignored (iniscid). Added onto the Euler s equation (secific for Bernoulli s equation): 4. The density is assumed to be constant (incomressible). Note: the assumtion 4 (incomressible) is usually not true in aerodynamics... only if M < 0.3 (called, incomressible subsonic flow).
Page 6 of 16 Class Examle Problem A-3- Related Subjects... Bernoulli s Equation Consider an aircraft wing in a flow of air, where far ahead ustream of the wing (called a freestream ), the ressure, elocity, and density are measured as: 1 atm (absolute), 100 mh, and 0.004 slugs/ft 3, resectiely. At a gien oint A on the wing, the measured ressure is,070 lb/ft. What is the elocity at oint A? A First, let us check to see if we can really use Bernoulli s equation to sole this roblem. 1. Is this steady flow? => Yes, since the roblem only roides a single alue of roerties in the flow field (time indeendent: steady flow).. Is this iniscid flow? => Not really, but the iscosity of air is ery low (if you comare against a liquid). We can assume that the flow of air is iniscid, as long as we do not consider details of boundary layer. 3. Is this incomressible flow? => The freestream airseed is: 88 ft/s 100 mh 146.667 ft/s 60 mh At the sea-leel, this is Mach 0.131 < 0.3, so we can assume incomressible flow. V VA So, let us aly Bernoulli s equation: A V ( A) (,116., 070) V (146.667) 44.97 ft/s 167.06 mh 0.004 A
Page 7 of 16 Class Examle Problem A-3-3 Related Subjects... Bernoulli s Equation Consider the same conergent duct and conditions as in Class Examle Problem A-3-1. If the air ressure at the inlet is 1 = 1 atm, what is the ressure at the exit? Velocity Pressure Temerature Velocity Pressure Temerature From the roblem A-3-1, recall: V1 = 0 m/s and V = 5 m/s. Let us assume the standard sea-leel air density: 1.5 kg/m 3. V1 V Using Bernoulli s equation: 1 5 5 1.5 1 ( V1 V ) 1.01310 [(0) (5) ] 1.0116 10 N/m
Page 8 of 16 Energy Princile (1) 1st Law of Thermodynamics: ENERGY PRINCIPLE Energy rincile = conseration of energy Let us consider a system (a unit mass of air). The first law of thermodynamics can be alied to this system (in thermodynamics, this is called the control mass) as: q w de Energy added or subtracted due to Heat Energy added or subtracted due to Work Change of Internal Energy This equation says that energy added or subtracted by heat and work is balanced against the change of internal energy: energy must be balanced. This is the first law of thermodynamics. d : exact differential (means infinitesimally small and rocess indeendent quantities = internal energy / enthaly) : infinitesimally small quantities, but not exact differential (means rocess or ath deendent = heat/work energy transfer. Often uer case is also used: this is called inexact differential )
Page 9 of 16 Energy Princile () 1st Law of Thermodynamics (in terms of internal energy) : Moing boundary work: 1st Law of Thermodynamics (in terms of enthaly): FIRST LAW OF THERMODYNAMICS In aerodynamics, work due to ressure (in thermodynamics, this is called dv or moing boundary work) is the main comonent of energy transfer consider work energy transfer er unit mass: 1 w sda d (Note that:, called the secific olume ) A The first law of thermodynamics (alied for aerodynamics) becomes: q w de => q de w => q de d (eqn. 1) ALTERNATIVE FORM OF FIRST LAW (IN TERMS OF ENTHALPY) Let us define a new roerty, called the ehthaly (er unit mass) as: h e e (eqn. ) Let us differentiate the eqn., such that: dh de d d or de dh d d (eqn. 3) Substituting the eqn. 3 into eqn. 1 yields: q de d ( dh d d) d This results in the alternatie form of the equation of the first law of thermodynamics: q dh d
Page 10 of 16 Secific Heats (1) Constant Volume Heat Addition Thermally erfect: de c ( ) T dt Calorically erfect: e c T SPECIFIC HEATS Definition of secific heats: heat energy added er unit change in temerature of the system: Secific heats are the measure of energy that can be stored (er unit mass) of a substance. q c dt There are two different kinds of secific heats: Secific heat at constant olume: c q dt constant => q cdt Secific heat at constant ressure: c q dt constant => q cdt CONSTANT VOLUME SPECIFIC HEAT Recall, the equation of the first law of thermodynamics: q de d Since the olume is held constant during the heat addition rocess: = constant => d 0. 0 Therefore, q de d de From the definition of secific heat ( q cdt ): de cdt
Page 11 of 16 Secific Heats () Constant Pressure Heat Addition Calorically erfect ideal gas relations: Thermally erfect: dh c ( ) T dt c c R Calorically erfect: h c T = 1.4 (Air at Standard Condition) c c CONSTANT PRESSURE SPECIFIC HEAT Recall, the first law of thermodynamics (alternatie form): q dh d Since the ressure is held constant during the heat addition rocess, = constant => d 0 0 Therefore, q dh d dh From the definition of secific heat ( q cdt ): dh cdt AIR AS AN IDEAL GAS: TWO MODELS Thermally erfect ideal gas: secific heats are not constant, but can be modeled as a simle functions of temerature. c c T c c T and Further, under a certain (limited) temerature range, the secific heats can be assumed constant. This is called calorically erfect ideal gas. c constant(1) and c constant() These constants can be found by the gas constant (R). For a calorically erfect ideal gas, the ratio between c and c is constant. c This is called the secific heat ratio: (for standard air, = 1.4) c
Page 1 of 16 Class Examle Problem A-3-4 Related Subjects... Secific Heats (a) Exlain the difference between: (i) thermally erfect ideal gas and (ii) calorically erfect ideal gas. (b) For calorically erfect ideal gas, derie the relationshi between two secific heats (c and c ) as: c = c + R (a) As we learned, there are two models for air as an ideal gas. (i) An ideal gas means at least thermally erfect ideal gas in aerodynamics, means that secific heats are functions of temerature only. Thus, de c ( T) dt and dh c ( T) dt (ii) A calorically erfect ideal gas is thermally erfect ideal gas and if the secific heats are constant. Thus, e c T and h ct For examle, under the standard atmoshere (1 atm), the secific heats are constant in the range of from 50C to 50 C (i.e., the room temerature): c = 1,005 J/kgK = 6,006 ftlb/slugr. (b) For calorically erfect ideal gas, e c T and h ct. (eqn. 1) Recall, the definition of enthaly: h e. If we assume air as an ideal gas, RT (equation of state) can be used, so: h e RT. Let us lug-in eqn. 1 into this equation: c T c T RT => c c R
Page 13 of 16 Isentroic Flow Isentroic Relations: ISENTROPIC FLOW Isentroic flow is the flow, in which the rocess is both adiabatic and reersible. As you know well at this oint, the Bernoulli s equation cannot be used for comressible flow. Instead of Bernoulli s equation, you may be able to emloy isentroic relationshi, if the flow is still isentroic. Isentroic flow assumtion (adiabatic and reersible) is alid for wide ariety of flow regimes, een including suersonic flows. Howeer, tyical non-isentroic flows include: High-seed flows across the shock waes (shock waes are non-isentroic comression) Flows with combustion and/or chemical reactions (examle: rocket combustions) Flows under the iscous effects (i.e., boundary layer) Isentroic flows will be discussed in-deth in AE30 (Aerodynamics II), so let us ski the deriation of the equation at this oint the equation, called isentroic relations:
Page 14 of 16 Energy Equation FIRST LAW OF THERMODYNAMICS (IN TERMS OF ENTHALPY) Staring from the alternatie form of the first law of thermodynamics: q dh d If the flow is isentroic, there is no heat transfer (adiabatic or q 0 ): dh d 0 => dh d (eqn. 1) ENERGY EQUATION Recall, the Euler s equation: d VdV (eqn. ) Combining eqns. 1 & yields: dh VdV => dh VdV (Note: 1 ) Integrating this equation between any two oints along the streamline: h V V V 1 dh VdV => ( h h1) h V 1 1 V1 V h1 h or For calorically erfect ideal gas: V h constant (along the streamline) h c T V1 V V ct1 ct or ct constant (along the streamline)
Page 15 of 16 Class Examle Problem A-3-5 Related Subjects... Energy Equation A high-seed business jet aircraft is flying at 10 km altitude with 750 km/h. The temerature and ressure at a oint on the wing is 45 C and.8 10 4 N/m, resectiely. What is the elocity at this oint? Is Bernoulli s equation still alicable to sole this roblem? Proerty of Air ( = 1 atm) First, let s check the flight Mach number. At 10 km altitude, the seed of sound is 99.5 m/s. The flight Mach number is: km 1,000 m 1 h 750 V h 1 km 3,600 s M = 0.696 > 0.3 a 99.5 m/s Bernoulli s equation is not alid. V1 V Let us aly energy equation, instead: ct 1 ct (along the streamline, behind the shockwae) => V V1 c( T1 T ) At 10 km altitude, the temerature is: 3.3 K. Also, V1 750 km/h = 08.333 m/s Therefore, V (08.333) [1,005 (J/kg K)][3.3 ( 45 73)] 184.7 m/s (M = 0.615)
Page 16 of 16 Summary of Equations EQUATIONS AND RESTRICTIONS OF USE 1. Bernoulli s equation (M < 0.3: Rule of Thumb) = Euler s equation + incomressible assumtion Steady flow Incomressible ( = constant) Iniscid (no iscosity, no friction). Energy equation (M > 0.3) = First law of thermodynamics + Euler s equation + Isentroic flow + Ideal Gas Steady flow Comressible ( constant), but isentroic (adiabatic and reersible) Iniscid (no iscosity, no friction: this is required for reersibility)