High Speed Aerodynamics
High Speed Aerodynamics AE 3021
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Welcome to AE3021 High Speed Aerodynamics Fall 2006 Prerequisites AE 2020 Aerodynamics AE 3450 Compressible Flow 1 Review of Low Speed Aerodynamics and Gas Dynamics Copyright 2006 Narayanan Komerath AE 3021 High Speed Aerodynamics Fall 2006 Dr Narayanan M Komerath Professor School of Aerospace Engineering Narayanankomerathaegatechedu I check email a lot more frequently than I check my of ce voicemail 4048943017 Text Either Fundamental of Aerodynamics by Anderson or Aerodynamics for Engineers Bertin and Smith Use texts that you used for Aerodynamics and Gas Dynamics Learn the class notes rst then read the texts Copyright 2006 Narayanan Komerath Basic Concepts and Results in Aerodynamics Freestream Vector Velocity of the fluid far ahead of the object in the ow undisturbed by the presence of the object Aerodynamic Lift Force perpendicular to the freestream exerted by the flow on an object L39ft d 39c ress re Ianform area I39ft coeff39c39ent l ynamlp up I ll L Drag Force along the freestream acting on the aircraft Drag dynamic pressurepanform areadrag coef cient D qOOS 2 Dynamic Pressure in lowspeedflow 1 U 2 goo 00 Copyright 2006 Narayanan Komerath LiftCurve Slope of an airfoil thwame mew w c w Cupyngqcznna Nzrzyznan Kumerath Effect of Mach Number on Lift Curve Slope In subsonic flow 1C1 dcl do M 1 1Mquot In supersonic flows dc1 4 10 M J39ME 1 Copyright 2006 Narayanan Komerath KuttaJowkowskv Theorem Derived from Newton s 2nd Law Derived by observing that there are 2 ways that lift can be explained 1 Due to circulation flow turning gt Lift per unit span Z3 pU Cx X f Here P is the Bound Circulation ME 2 Due to compression I expansion of the flow density changes due to velocity changes this cannot happen in incompressible flow but we will see this type in this course gt Copyright 06 6 Narayanan Komerath Finite Mng Vortex system 2 Aspect Ratio AR 7 where b is the span and S is the planfonn area ofthe wing For an 2D airfoil AR 2 CO Spanwise Load Distribution Tipv0 rtices Copyn39ghtZOO 7 Narayanan Komerath P 39 anu Iuwer 39 39r the wing There are Mo effects ofihis a Loss in Ii compared to 2D airfoils b Induced drag ie drag induced by the force veciortilting backwards as a result ofihe induced downwash Drag D Lift L Normal Force F Vinf Vind Veff Copyngm 2006 8 arayanan Komerath Points to Note on Finite Wings 1 Since induced drag is directly related to the lift it can be calculated by the same mathematical formulation used to calculate lift 2 Does not require consideration of viscosity 3 No induced drag on 2D airfoils under steady conditions 4 At finite Aspect ratios 2 C CDl 4 J39L AR6 5 Ideal elliptic lift distribution implies minimum induced drag ie spanwise ef ciency 9 1 6 At low Mach numbers drag on well designed aircraft is primarily due to induced drag 7 Note In the 2D limit airfoil there is no liftinduced drag in incompressible ow There is always pro le drag due to viscous effects friction drag and ow separation pressure drag 8 Potential Flow inviscid analysis cannot capture the pro le drag in subsonic owBut it can capture the liftinduced drag on nite wings Copyright 2006 9 Narayanan Komerath Effect of Aspect Ratio on Wing Lift Curve Slope Copyright 2006 10 Narayanan Komerath Review of Gas Dynamics Compressible Flow Speed ofsound in a medium Speed at which in nitesimal disturbances are propagated into an undisturbed medium Mach cones and quotzones ofsilencequot In a flow where the velocity is lt a ie subsonic ow disturbances are felt everywhere When u gt a ie supersonic ow disturbances cannot propagate upstream y is the quotMach Anglequot Copyright zoos 11 Narayanan Komerath Mr 393 3111 1i 339 3111 439 J Bigger disturbances propagate faster eg shocks This can be seen by considering the dependence of a on local properties a yRT If T is substantially higher downstream of the disturbance a downstream is gt a upstream Also constant entropy Copyright 2006 12 Narayanan Komerath Isentropic Flow Relations r 1 P 1 quot39F 393 1 M where fquot Cr 77 9 ADD lo 3 p1 I III m II 1quotquot I39 p R 8313 I I I Jar III R J II In S units for aIr Rquot where Ivlolecular wt 235 T v39 C 39 specnflc heat at constant pressure P Ir 1 3 1 I speci c heat at constant volume 1 quot39l3939 v39 C 39 enthalpy 2 Copyright 2006 Narayanan Komerath Quasi1dimensional ow Thus thgtt supersohtc 0 so that do gt 0 expahstoh dA lt 0 so that do lt 0 compresstoh and W ltt subsohtc dA gt 0 so that do lt 0 compresstoh dA lt 0 so that do gt 0 expahstoh Copyright 2m Narzyznzn Kumerath Mass flow rate through a choked throat P t PM I D A m 6qu r warm 5 31 jltij I Jl rifjl 39 tum 1 KT E T IJ39TE R warn choked throat M1 r l i 1139KAI39R L PD m M39t1 M j F t Jt J 1 I quotquot 2 mlquot 0 Po cc 7 40 Copyright 2006 15 Narayanan Komerath Normal Shocks Copyright 2006 Narayanan Komerath These changes in properties can be calculated using the normal shock relations Normal shock relations 141 102142 2 2 1 us 1 1 2 31 A 7C 2 26 H IP IP 2 39 2 p 1 Ml p1 tr 1 2 2 P1 71141 P2 92142 Mf 3113 2 3 1 A 9 9 Mf 1 r 1 MIME pl 9 DMIQ 2 Copyright 2006 Narayanan Komerath Obhgue shocks Ache above ho dtmefmthe norma componem owemcmy m addmon hetangenha componem ofve oc y remams unchanged across he obhque shoe tan 9u2 tani We Cupyngm 2m Namyunan Kumemm Limits on for an obli ue shock ShOCk H lt B lt 90 deg U gta Vgt1 Copyright 2006 19 Narayanan Komerath Prandtl Meyer Exgan n Isentropic T0 P0 constant Nara Copyngm 2006 yanan Komer ath Isentropic T0 P0 constant did u 212 M2 2quot At any point where the Mach is M 15 152 1 a5 JM39R 1 2 1 3 1 1 a 1 2quot a 2 1 2 142 gt A 41 ljtan I ljtzh 1 tan iM 431 1 quot e2 61 00 M1 39 00 M2 00 is tabulated as the PrandtIMeyer function M l Copyn39ght 2006 21 Narayanan Komerath 2 REVIEW OF CONSERVATION EQUATIONS Copyright 2006 Narayanan Komerath 22 CONSERVATION EQUATIONS These equatrorrs are expressrorrs otthetaws otphysros Wrrtterr m forms approprratetor ows r Mass rs rrerther created rror destroyed oorrtrrrurty equatrorr or oorrservatrorr of mass tr Rate otohange of momertum Net force oorrservatrorr of momentum Hr Energy rs conserved though rt may ohangetorm oorrservatrorr oterrergy Integral forms for a control volume Cnpynm 2006 23 Narayanan Komerath 3dVjj d3 frwkazeverrcrmges 0dV 0 d3 03quot at 0 3 CVOP C39V Mass firma w frmr o w 39 n at C3 Momentum a if rpujdm f aujlfu ems fpnd3 fpde F5 2 C3 CE ZV The terms on the rhs are I pressure forces acting normal to the surface per unit area ll body forces per unit mass lll shear forces acting parallel to the surface per unit area hear Copyright 2006 24 Narayanan Komerath Energy 2 2 45 2 p 6 dV gs p 6 fidS glipq39dV g chW CV at CS 2 CV CV J 4 2 Note u 397 Energy per unit volume volume 9 9 13 plnternal energy per unit mass kinetic energy per unit mass In addition to these specific conservation laws speci c equations relating the state variables are needed to solve problems for given kinds of fluids To solve very complicated flow problems the boundary conditions are specified and all of these equations are solved simultaneously all over the flowfield for each step in time This is a task which usually requires fast computers with large memory because we have to keep track of a large number of variables and perform a large number of calculations Long before this became quotpossiblequot people gured out more restricted ways of solving specific problems needed to build airplanes These quotsmart analytical methodsquot form the subject of this course Copyright 2006 25 Narayanan Komerath Relations Used to Reduce the Conservation Equations to Differential Form 1 Stokes39 Theorem f o d Vgtlt o d3 C A where 1 is the vector quantity of interest if is the vector along the closed contour of integration C d is the unit vector normal to the area enclosed by C A x rt39 s p A 19 JV My 1 V f quot quot quot x gt quot 1 p u x a l r gums a quot quot1 gt H xrc 39 quot h P p p c Copyright 2006 26 Narayanan Komerath 2 Divergence Theorem od gsvo dv A V 3 Gradient Theorem If p is a scalar eld then gSpdA g VpdV A V 4 Vorp apvo ovp where p isa J scalar and u is a vector Copyright 2006 27 Narayanan Komerath Substantial Derivative The Eulerian Frame of Reference is the one xed to the control volume The Lagrangian frame of reference is the one fixed to a packet of uid a fluid element The rate of change of any property as seen by the uid element is M 2ui 9 w ar Dr 8 8 8y 82 Copyright 2006 28 Narayanan Komerath The substantial derivative is I 9 it ammuth Dr 8 The rst term on the rhs is the quotlocalquot or quotunsteadyquot term The second is the quotconvectivequot term The rate of change DDt is for two reasons 1 Things are changing at the point through which the element is moving unsteady local 2 The element is moving into regions with different properties Copyright 2006 29 Narayanan Komerath Using the vector identity 4 above the conservation equations can be rewritten Continuity fvorp io gpVOQJOVp n 29 DIpVou J In terms of veclocity components this can be written as a scalar equation D an 82 8w 9 Dtpx8yamp0 Copyright 2006 30 Narayanan Komerath Momentum Conservation Differential Form Du 8p 3935 33 ltI I39T m39scous X DV Q F351 5 pf Fv1 scou5y Dw 8p 3935 p sz scousz Copyright 2006 31 Narayanan Komerath Knowing the properties of the particular fluid and problem being considered the body force term and the viscous force term can be expanded One very useful form is where the viscous stresses are related to the rate of strain of the uid through a linear expressionThis is valid for quotNewtonian Fluidsquot This is further simplified using the Stokes hypothesis which permits us to delete the normalstrain terms from the strain terms leaving only shearstrain terms The resulting form of the momentum equation is called the Navier Stokes equation This is often used as the general starting point to solve problems in uid mechanics Energy Conservation ul D 9 DT pgquot 39r39 pu p pg 11 Q viscous W viscous Copyright 2006 32 Narayanan Komerath The Euler equation If the Reynolds number R E w Inertial Force divided by Viscous Force gtgt1 in our ow problem we can neglect the viscous stress terms Thus the differential form of the momentum equation reduces to 391 4 3539 J I I W 3 I quotf E p14 o pl 314 9 Ill 3 r3 39 a 7 r 393 as n l J II 1 e quot 399 t w W t a iu r 4 an 39il l39 39o1r v I quot39 I I I Jquot 5 Copyright 2006 3 3 Narayanan Komerath