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High Speed Flow

by: Demond Hoppe
Demond Hoppe

GPA 3.79

Narayanan Komerath

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Narayanan Komerath
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This 0 page Class Notes was uploaded by Demond Hoppe on Monday November 2, 2015. The Class Notes belongs to AE 6020 at Georgia Institute of Technology - Main Campus taught by Narayanan Komerath in Fall. Since its upload, it has received 30 views. For similar materials see /class/234305/ae-6020-georgia-institute-of-technology-main-campus in Aerospace Engineering at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15
AE 6020 Compressible Flow Spring 2008 Instructor Dr Narayanan Komerath Professor Daniel Guggenheim School of Aerospace Engineering Georgia Institute of Technology Atlanta GA 303320150 Komeratl igatechedu Lecture 1 Introduction Welcome to AE 6020 The course website is at httpwwwadlqatecheduclassesae6020 The required textbook is Anderson JD Modern Compressible Flow With Historical Perspective McGrawHill 2nd edition or later Panicked over prerequisites Steady Aerodynamics for all speed regimes at the undergraduate level Engineering Calculus at the undergraduate level Some computer programming skills See the course website for notes on prerequisite material SYLLABUS Transonic Aerodynamics 07019953 Introduction Compressible Potential Flow Review Transonic flow physical features and approaches Transonic Small Disturbance TSD Equations Full Potential Equations FPE and solution Supercritical Airfoil Design Discussion Laminar Flow Wing and Fuseage Design Supersonic flow problems 1 2 Discussion Aerodynamic design for reduced sonic boom footprint Hypersonic Aerodynamics P Fquot39quot N Conical Flow Blunt Body Solution Introduction Hypersonic Shock and Expansion Relations Local Surface Inclination Methods Small Disturbance Theory and Applications CFD Methods for Inviscid Flow Boundary Layer Theory CFD Methods for Viscous Flow High Temperature Effects Advanced Supersonic Aerodynamics Topics of interest to supersonic flight GRADING Assignments 30 including signi cant weightage for class participation 2 Quizzes 5 each 2 Midterm Tests 15 each Final 30 General Policy on Assignments Ask anyone about methods but do all work yourself or within your assigned team Acknowledge sources of help Quiz 1 Covers review of compressible flow and introduction to transonics Closed book and notes Loads on an Airfoil v L1 i1 1 ppoopoo KPoo CP 1 2 2 Y 2 EPooVoo Moo Moo xc X cN cpjower cpupper1kzj x xc CX I x0 C1 CN cosoc 7 OX sinoc Cd CXcosoc Cstnoc 1 4 xc x cmc4 I cpupper cglowerI x 0 i m K C pupper upper cp10werk dY dx Normai Force N Drag ChordWiSe Force X i x jlower M where Yx is the airfoil shape and a is the angle of attack Continuity Equation 2E 3 31quot DIPEEJ 3 5 Here aswerecall D Eg hiu39mi DE 5 a a a E39 E39 E39 9 Steady 20 5 a H amj v3 0 1 u component v component Momentum Conservation in Differential Form Energy equation Energy equation for steady inviscid flows may be written as PUhox PVho y PWho z 0 When continuity is used the above equation becomes u h0xvh0y wh0z 0 Dho Dt Where DDt t u X v y W z In terms of substantial derivative DDt ie specific total enthalpy hO is constant along a streamline assuming adiabatic flows no heat addition and no body forces doing work on the fluid 2 ho CpT u2 V W22 Constant Compressible Potential Flow The Full Potential Equation W Recall that for incompressible ow conditions velocity is not large enough to cause density changes so density is known Thus the unknowns are velocity and pressure We need two equations The Continuity equation is enough to solve for velocity as a function of time and space Pressure can be obtained from the Bernoulli equation which comes from Momentum Conservation If the ow is irrotational as well we can de ne a potential and reduce the continuity equation to the form of the Laplace equation qu O HereIis defined such that 7 8CD u v W 5 9x 9y Highspeed flows can also be irrotational Obiectives 1Derive the potential equation for compressible flow 2Reduce to linearized form for smallperturbation analysis 3Apply results to thin airfoils and slender axisymmetric bodies m Generally ight vehicles are designed to create as little quotperturbationquot as possible because large disturbances cause large drag We will derive the potential equation for 2 D flows At the end it will be obvious how to extend it to 3D flow so we will just write down the 3D equation Assume 1lsentropic ow 2Steady flow Speed of sound 2 510 I 5 Le 632 agaiaggl za23a Q ap ay wax 893 ix Substitute 3a in 2 3 an at 12 u v za t4 u v A a ax 6y 0 6y ex 5139quot ax For homework 1Multiply 4a by u and 4b by v and add Call this eqn 5 2Expand eqn 1 and substitute in eqn 5 Solution This equation contains derivatives of u and v If we could de ne a potential we could reduce the number of variables 0 Assume Irrotational Flow V X U 84 If this is true then a potential fcan be de ned such that gt UV ampCD g 7 v 9y 2 2 2 2 239 iii i E IFFH 3 ti Hit l a Six 1 Qua a a ti Substituting in eqn 6 or using the notation 2 33quot xax y 1 M 2m 12 i 1 1 i 4 Si 12 1 a a yy I 39 What is 39a39 the local speed of sound To see this go to the energy equation for steady adiabatic flow hu2 132 zea zh a 9 hchrz E lra n quot l 2 2 yRTTu 1 j mn a2 7T1U2 constz 61002 771Uoo2 6102 Thus 1 a2 2 6102 7 U2 OI 2 a at VT losa3w Extension to 3D ow By inspection equations 8 and 11 can be written for 3D potential ow as IN 2 1 xx 1 yp 1 323 2Mv gl vw yz w ax 0 39 I I I 391 Eq 12 is the 3D Steady FuII Potential Equation where 2 1 cai 3 T1 E gt13 Note 9501quot This is still an exactequation We have not made any approximations In other words if the ow satisfies our assumptions steady irrotational isentropic the solution of this will still give exact results Note the very important distinctions between this and the approximate equations that we will presently derive based on quotengineering estimatesquot We have not made the assumption that disturbances are small although we have implicitly assumed that the shocks are absent or quite weak Thus this equation can be and is used to calculate transonic flows over very complex configuration such as fighter aircraft and rotor blades This equation is highly nonlinear in I Superposition of solutions will not work Equally valid for subsonic and supersonic ows We have used h EFT and 3x from These can be easily modi ed and the resulting equations can be used to analyze high temperature flows rocket exhaust nozzles Classic Linear Supersonic Aerodynamic Design Process us SST and Pre 1990 Linear Desrgn Loop Prior Knowledge Design Mission Objectives Con guration Features Initial Component Sizes Linear Theory Design integration Design Optimization Linear Theory Cp Calculation Apply Real Flow Satisfy Design Constraints V nd Tunnel Test Programs Validate Supersonic Performance Off Design Flap Optimization Model to Airplane Geometry Differences Excrescence Miscellaneous Drag Power Effects Scale Viscous Drag to Full Scale Conditions Using Flat Plate Skin Friction Theory Full Scale Performance 39 Courtesy Dr B Kufan BOEING COMPANY 2004 Re ned Linear Supersonic Aerodynamic Design Process Initial HSCT Technology 1990 Prior Knowledge Design Mission Objectives Con guration Features Initial Component Sizes V ng Leading Edge Design Linear Theory Design integration Design Optimization CFD Anal es Inviscid Ascous Apply Real Flow Design Constraints Off design Flap Parametric Optimization Satisfy Design Constraints Validate V nd Tunnel Test Programs Performance Validate Supersonic Performance o Off Design Flap Optimization Model to Airplane Geometry Differences Excrescence Miscellaneous Drag Power Effects Scale scous Drag to Full Scale Conditions Using Flat Plate Skin Friction Theory or Navier Stokes Predictions I Courtesy Dr B Kufan BOEING COMPANY 2004 Current NonLinear Supersonic Aerodynamic Design Process HSR Technology 1998 Prior Knowledge Design Mission Objectives Linear Theory Design integration Design Optimization Initial Component Sizes V ng Leading Edge Design CFD Anal es Inviscid Analyses scous Analyses Ascous Determination of the ow characteristics Satisfy Design Constraints i i Con guration Features I i i i i i i i i i i i i NonLinear Design Optimization Point Design Off Design Flap Optimization MultiPoint Optimization V nd Tunnel Test Programs Validate Supersonic Performa ce n Off Design Flap Optimization Model to Airplane Geometry Differences Excrescence Miscellaneous Drag Full Scale Performance Pow r Effects e Scale Ascous Drag Scale Pressure Drag Courtesy Dr B Kufan BOEING COMPANY 2004 Performance 9 Viscous Flows NavierStokes Equation gaoaRg t xayaxay Cl E CDJUQ39n X 339 o Flow properties vector given by p pU pv E Total Internal Kinetic energy per unit volume pCVTpU2V22 pupU2ppuvpuho pv puvpV2ppvho o 01 TXX Txy U CXX V CXy k TX Xy Tyy urxy v ryy k Ty Specific total enthalpy CIo T u2 V 2 Conductivity molecular plus turbulent Euler Equation Application inviscid region of compressible flow over airfoils and wings Problems with weak shock waves amp small angle of attack allow modeling inviscid region separately from boundary layer then doing a viscousinviscid interaction aq 8F ac at ax ay Linearized Potential Equation Preview The full potential equation is nonlinear and requires numerical solution We can however obtain some nice simple engineering results by restricting our consideration to small changes in ow variables In this case we can neglect terms involving products of small fractions and simplify the potential equation Applications Analyze performance of highspeed airfoils and slender pointed body shapes at small angles of attack in the subsonic and supersonic regimes Perturbation Potentials Let us de ne the potential 1 such that ItI1 Um where UOO3 Uoou39 8x 14 81 8CD v v w w 3y 3y tat qua iai aiwaigs agcpai gm ax 255152 25 3Z2 y y 39 Equation 8 for 2D ows becomes Li2 ELJ39LJ2 Li392 Umv39w39w39 at 3 1 II 2a 2 Iy a 2 i wn ii5 Note This is still an exact equation no different from equation 8 Using 14 11 becomes 612 L1 Li 1 61002 2 61002 61002 a2 y l U002 U002 2u39 U002 u392 v392 I 2 21 2 2 6100 2 dog aOO 1U2 2w 39339 v39i 1 F Tx M 16 am Um U Uni Linearization I I 1 I i 1 1 qtj ml g 1M i a U Hi a I 2 U00 U03 III III Suppose that M 2 2 ME lt1 3 f n addton 0 I am H Um This is true if M m is not too large ie excluding hypersonic flow Eq 15 with 39a39 replaced by I am I 2 I 1 iv vIE 2 2 2 2 t Mm 1Mthm Ugh mmiUm U 1 MtUm 11 0 gt 1 7 Let us look at each of the terms The second derivatives of f are of the same order of magnitude no reason to expect otherwise 1The coefficient of 43 XX can be written as M2 214 393 mfg 11 m Mi t If t m Mi1Um U3 m J M u39 2 lt1 Mm IUW This is true if we exclude the transonic regime where Mm m 1 The coefficient of 13 W can be written as 392 1 m 1 if M is not too large EDI exclude hypersonic flow But then we already did that The coefficient of 43 W is 1 u39 v39 1339 u39 2M2 2JiaIrg 1 Um UmUm Umir Um Mgi i we have to throw this out or we won39t get a linear equation So let us assume that W U D the other coefficients This means that the airfoil must be very thin if m is high or M else must be very low if the airfoil is not so thin The linearized equation then is rm 41 aw 0 418 for his gt1 t l Mijda w 314419 with lt1 Similarly for 3D u Miiwrww Ur2m Mm for These equations can be used to describe small perturbations in subsonic but not transonic and supersonic but not transonic or hypersonicO flow Note the difference between equations 18 and 19 The sign of the rst term of each depends on whether MI E lt or gt 1 If we choose the equations so that the first term is positive then there is a vast difference between equations 18 and 19 The character of the solutions to these equations is vastly changed when the sign of the second term changes from to Equation 19 describes subsonic flow where the solution is quotellipticquot ie changes at any point affect the solution at every other point Equation 18 describes supersonic ow where the solution is quothyperbolicquot It is also called the quotwave equationquot since it can be used to describe the propagation of waves eg sound One feature of the solution is that changes cannot be felt upstream Pressure Coefficient for small perturbations Recall the de nition of pressure coefficient for incompressible flows P Pm top 1 2 U When the flow is compressible it makes more sense to use M m as the normalizing parameter Thus cp can be written as P 3P R 3M 2 show this for homework and also de ne limiting values and the corresponding flow conditions Let us derive a linearized form of cp to use along with out smallperturbation equations From the isentropic flow relations L P D 3 H T IIIZI DJ From the energy equation 2 2 U U where U 2 U00 uv2 vv2 or 321 I Zu39Umu392v392 where C Ry Tm QCPTW P 3f1 T 1 2w he 1 LMi T 2 Um U2 on DJ The second term on the RHS is ltlt 1 Thus 3quot P T3 r1 PM I30 L is ofthe form 539 F31 where E 39C 3913 1 This can be expanded using the Binomial theorem H n a I a 1a 1x Jan 232 as H H E 1 239EE smallerterms where H H E F TIME F Qua H 1H or for U mkU m 1 Thus 395 quot39U m This is valid for a 2D or 3D flow over flat wings Boundarv Conditions for use with smallperturbation theorv under mamemancaii mucn impiei lU Ui e Asin h a we piupei ifasoiution of to a iinearized potentiai the i L L physics of p the equation is to make sense it must satisfy the boundary conditions that We specify o angie ofatt ck unit vector normai to the upper surface at a given point 0 siope ofthe surface at a given point is tan 0 The linearized potential eqn is a 2nd order partial differential equation 2 boundary conditions are required The perturbation must die away as we move far from the source from the airfoil E im a W x im u39a 3 There can be no flow through the surface of the airfoil The normal component of the velocity is zero at the surface U 0 O atthe surface Another way of saying this is that Thus the tocat ow vetoctty at the surface must be tahgehttat to the surface 1 Vts the equattoh ofthe atrfott surface v39 7 d Thus Um u39 itan 6 7 a v d u39 m Approxtmatety Um dx sthce Um tsvery smatt Planar Wing Approximation For thin wings and airfoils with small camber at small angles of attack it does not make much difference whether the boundary surface is taken at the airfoil surface or at the y0 plane When you calculate 3923 1i sg39 you will nd that the values quotI39 are small and you will wonder why you went through the tro ble of calculating it Thus the boundary condition becomes U quot U 9 oo 00 3y 2 for small 6in radians or vvyzo 8 U00 Q 8y yzo dx Note that we have only applied the be at y0 We have not assumed that the slope of the surface is zero Similarity Relations for Subsonic Compressible Flow Airfoilsl2 D ow ObiectiveFind the pressure distribution around an airfoil in subsonic compressible ow Method Transform the linearized potential equation into an equation describing a quotrelatedquotincompressible ow This will be the Laplace equation Its solution will give the pressure distribution around the quotrelatedquot airfoil in incompressible flow Alternately the pressure distribution in incompressible flow can be obtained from the extensive wind tunnel data available in the literatureincompressible flow wind tunnels are far cheaper to run than highMach number tunnels Transform this solution back to the compressibleflow case Linearized potential equation for 2D subsonic flow 1 M 1 in Ellyn l x andysubscripts denote derivatives 2 or 32 ij 51 l where viiM 395 3 U The Laplace equation describing incompressible flow is w U cng Him Ut2i Equation 1 can then be transformed to2 by setting ix and nBiri3i and titan 51 m jl where 39m39 is some constant 0 Note The text by Anderson refers to i3 as LP and m B Thus equation 1 becomes 2 III E III ci i U iTI i39i39l SENS warmers Note that 39m39 is still undetermined We can decide later what we want it to be Boundary Conditions Atin nity i x 5 Aquot U 2 tingqigt39i391i Surface tangency Eff Subsonic compressible ow 2 H le Um Um d 3 where y f x is the surface n dill I I n ncompressibe ow 5 inn ITFD Uml 39 LEI d where T describesthesurface Note that the same lJED is used in both flows If we used different ones everything cancels out eventuallyit makes no difference Now t an 39jf l39ilrl lJEiiiu at m an 43933 391quot 1 III or hf umk um an i3 i3 die However the boundary condition in the incompressible ow is a D d i liq DUuEl39 Um g an 315 r19 m u m a tit 8 d5 5 dirt or 5 E 9 Relation between surface slopes of the original airfoil and the related incompressible ow airfoil lTI Note that the slope is a function of chordwise location on the airfoil Different choices of will give different relations between the slopes l5 Pressure Coef cients Eu39 2 as LP I Lima I I Incompressibleflow i I 2af39b r it D Cu PUMBE Ilealgrn39j C L F39 rn Depending on the problem to be solvedwe can make various choices of m Consider the choice m1 This implies that CI Cp 3 ie the chordwise pressure distributions in the incompressible and compressible ows are the same However we see also that it implies that a can ri1 M leg Note 0ltb lt1 The compressible flow airfoil must have a lower surface slope than the incompressible ow airfoil in order for the pressure distributions to be the same gt 4 C C compressible flow incompressible flow The other usuai question is How does the pressure distribution over a given airfoii change as the iiach is increased Toanswerthismakeachoiceof rn thatwtiigive 6 63 This ismb 3 C 1 C p E p 0 C PD cp increases With iiach PrandtirGiauert Transformation We have so far seen the general idea of transforming the linearized potential equation and the boundary conditions for our problem to an equivalent Laplace equation with appropriate boundary conditions in another coordinate system We saw that by leaving an quotundetermined constantquot in the transformationwe have the flexibility to pick useful transformations Below we extend this basic concept to the problems of a thin wing and then to the case of an axisymmetric body a body of quotrevolutionquot This is in fact a 2D problem because things look the same everywhere around a given section Critical Mach Number Conswder the foHong swtuahon M lt1 gt u Uri u V M T M We As subsonic air flow over an airfoil or wing it accelerates reaches a maximum speed and then decelerates toward the trailing edge Since T0 constant 3 g 1 u H 3quot r r T 2 2R3 P Thus the Mach number of the flow increases and then decreases The magnitude of this change depends on the airfoil shape and the angle of attack Thus it is evident that as you increase Mm the highest local M on the surface may exceed 1 long before M 3 reaches 1 The value of M at which the highest M on the airfoil rst reaches 1 is called the critical ED Mach number Mcr m 1Mor is a value of the freestream Mach number 2Mor is less than 1 for anything with thickness at all Calculation of Mg We know that 1 g 1 M 21 r Isentropicflow relation III L aE lm T PD 2 P Pm 2 I P l39l ivyQPW wm Pm or 2 EightIE L i 31 331 P 3M 11M2 I232 At M 2 Ma M reaches 1 somewhere The value of cp at this point can be found by setting M1 gr1 a L A 2 1 EP M2 gt H m 1EM 2 r Note t For any gtveh atrfott the vatue of MN can be found om the vatue ofthe rhthtrhurhcp oh the atrott 2 W e t t thee to 39 1 ttawhumuet u mgwer rGtauert t Airfoils in Transonic Flow What happens when M gt M subsonic normal shook supersonic region SUbSDi IIC flow The ow accelerates over the front part of the airfoil Before the point of minimum pressure is reached it goes supersonic Once the point of minimum pressure is passed the flow experiences an adverse pressure gradient pressure increases downstream Several things happen The boundary layer begins to get thicker Note that information can move upstream through the boundary layer because the velocity is subsonic has to reach zero at the wall The flow is forced to turn because of the thickening boundary layer Compression waves are formed These merge into one or more oblique shocks or a normal shock depending on the Mach number and surface curvature Reynolds number of the boundary layer etc nurmal ShDEH Dblique Situchr 7 quotLambda Shockquot The pressure rises suddenly across the shocks The boundary layer thickens much more It may separate The supersonic region ends in a normal shock and the flow becomes subsonic and decelerates Drag increases greatly because of the shocks The wing may stall because of the boundary layer separation Supercritical Airfoils The preceding discussion shows that if M m gt M increased 1Use a very thin airfoil with a sharp leading edge This is impractical for airliners where would fuel be stored 2Reduce the curvature on the upper surface This reduces the acceleration and deceleration of the flow so that any shocks formed will be relatively weak drag rises greatly How can Mor be Cl Such airfoils were among the rst to be designed using detailed mathematical computation Comparison of transonic ow over a quotusualquot NACA 64A series airfoil with transonic ow over a supercritical airfoil strung shock supersonic WSW Ivveak shock separated flow 1 BF lower surface V From Bertin amp Smith 2nd Edition p 366 Transonic Drag Rise We have seen that f MD gt MU shocks form above the avfo Thws causes a arge ncrease m drag CDw Merit Mdd In addition the expressions for the supersonic wave drag coefficient have and 39l in the denominator Thus the supersonic wave drag coef cient also becomes very high in the transonic range Some people believe that linear theory could be close to the truth at M F3 1 so that t d a m Thus they hypothesized the existence of the quotsound barrierquot In practice Cd does become quite high at Mm l but fortunately stays finite so that powerful engines can still accelerate the aircraft through l y lm 1 without needing infinite thrust Conclusions 1Good aerodynamic design dictates that the surface slopes should be smooth and gentle to avoid sharp peaks in the pressure distribution 2The value of Mg must be kept high for transonic flight 3Flight at M as 1 U should be kept to a short duration Variation of QMMOO Wake 4gt M6075 a Shock Shocks 4 a Moo 0 89 MW 0 98 Shock 5 d shock shocks 4 M6140 S From Bertin amp Smith p 361 Swee Obviously it is desirable to reduce the Mach number of the flow over the airfoil section A long time ago it was discovered that the ow could be quotfooledquot by simply yawing the wing Moo ltwas discovered that the characteristics of the yawed wing at M Mm were similar to those of the straight wing at Mr 2 Mr 2 Mr CD5 Q Supersonic and Subsonic leadinq edqes H K a 1 MR K 15 Consider the yingwing shown If J1EE 3 1 the compression waves from the apex will be felt within the angle m mach angle However in the gure every point on the leading edge of the wing is in the undisturbed supersonic flow and cannot feel the compression and flow deceleration due to the other points This is a wing with a quotsupersonic leading edgequot Now consider the arrowwing shown below Here every point on the leading edge is within the region of disturbed decelerated flow caused by the apex As a result the Mach number at the leading edge is slightly smaller than the freestream Mach number This is a quotsubsonic leading edgequot Note that the ow is still supersonic Transonic Area Rule Within the limitations of small perturbation theory at a given transonic Mach number aircraft with the same longitudinal distribution of crosssectional area including fuselage wings and all appendages will at zero lift have the same wave drag Why Mach waves under transonic conditions are perpendicular to flow Things You Should Know By Now or Find Out Right Away An Essential But By No Means Exhaustive List AE Senior Yr Prof Narayanan Komerath What is a fluid How are fluids different from solids What is hydrostatic pressure What is dynamic pressure What is static pressure What is quotshear stressquot What is quotArchimedes39 Principlequot What is the Avodagro Number What does it indicate What is the quotcenter of pressurequot on an airfoil Where is is located on an airfoil in lowspeed flow does it depend on airfoil geometry and if so how A Pitot tube is used to measure the stagnation pressure in an air stream where the flow speed is 10ms Given that the static temperature is 280K and the static pressure is 101300 Nm2 find the stagnation pressure at the probe tip What is quotincompressible flowquot An airplane is flying at 500 mph at 30000 feet Would you consider this to be an incompressible flow situation Why Why does the density of the atmosphere decrease as you go up What is the quotdivergence theoremquot What has this got to do with fluid mechanics What is a vector ls hydrostatic pressure a vector or a scalar What aboutvelocity temperature density What is a quotstream functionquot What is quottwodimensional flowquot What are quotstreamlinesquot Two lines AB and CD intesect at point E Can these be streamlines Why Consider the flowfield within 01mm of the surface of the wing of a Boeing 727 under cruise conditions Is this a quotpotential flowquot Why Consider the flowfield 05 above the wing ofa Boeing 787 under cruise conditions Can this be treated as potential flow Why Can you predict the drag of an airfoil using potential theory Why What good is potential flow theory Can you write a potential function for the flow in the boundary layer of a flat plate State Helmholtz39s vortex theorems What is quotlocal accelerationquot What is quotconvective accelerationquot Write down Bernoulli39s equation for incompressible flow Does the relation between static and stagnation pressure in a compressible flow conform to the same equation Is the q the same as the difference between static and stagnation pressure in a flow at Mach 02 How about at Mach 23 Why What is quotLaplace39s equationquot What is a body of revolution What is the Kutta condition What is it used for What is a source panel Given a vortex sheet how would you find the velocity jump across it What are the quotboundary layer approximationsquot What is the vmomentum equation for an incompressible laminar boundary layer in the 2D Cartesian coordinate system where x is along the surface and y is perpendicular to the surface What is Reynolds number A circular cylinder of radius 10mm is moved at a speed of 10ms through air at standard sea level conditions What is the Reynolds number based on cylinder diameter Do you expect the boundary layer on the cylinder to be laminar or turbulent How would you estimate the force required to keep the cylinder moving at a constant velocity of 10 ms What do you think will happen if the cylinder is made to start spinning about its axis while it is moving You have to design a flat plate surface which will be exposed to high speed flight lfthe Mach number is expected to reach about 08 at sealevel what is the approximate value ofthe highest surface temperature to be expected Assume that there is a layer of insulating material right below the flat plate so that very little heat is conducted away What is the first law of thermodynamics What is the second law of thermodynamics What is entropy What is enthalpy What is quotelliptic loadingquot on a wing Why is this desirable undesirable A coop designs a jet engine using isentropic flow equations and claims that his engine will produce a net thrust of 30000 lbs at sea level Will the actual thrust be greater or less Why Why do people use the Laplace equation in aerodynamics when we all know that the full Navier Stokes equations are more exact How do you solve the Laplace equation for the flowfield around an airfoil What is a perfect gas Why are shocks formed in supersonic flows and not in fully subsonic flows Why are diverging ducts used to 39 39 flows and g39 g ducts to accelerate subsonic flows How is a Mach angle different from a shock angle Calculate Mach angle at Mach 2 Plot the static pressure velocity stagnation pressure static temperature and stagnation temperature changes across a normal shock Why is a shock called a quotwavequot What happens to the gas molecules at a particular point in still air is a shock passes across them Will they stay unaffected Will they move with the shock or away from the shock and if so how fast Will their speed of random motion be affected in any way Will the number of molecules per unit volume change Plot the section lift coefficient as a function of angle of attack for a 2D lowspeed symmetric airfoil Also plot the lift coefficient versus angle of attack for a 3D rectangularwing with a symmetric section incompressible flow What is the slope of this line Why physically are the two slopes similar different What happens when the angle of attack gets large What is a lifting line A trailing vortex sheet Define induced and effective angles of attack for a wing section What is induced drag What does its magnitude depend on Prove that the speed for minimum drag occurs when the induced drag coefficient is equal to the coefficient of liftindependent drag ls Mach number behind a normal shock greater or less than 1 How about an oblique shock Describe changes in static and stagnation pressure as an inviscid flow traverses a Prandtl Meyer expansion Does a shock quotreflectquot from a solid wall as a shock or as an expansion How about a shock encountering a free surface Calculate the speed of sound at the surface of the planet Xylon pressure 01 million Newtons per square meter where the atmosphere is 100 Xenon and the temperature is 400K Calculate the section lift coefficient of a thin symmetrical airfoil of chord 12 m at 5 degrees angle of attack at a velocity of 30 ms at 10 km altitude on a standard day above Earth What is the effect of increasing aspect ratio on the drag of a wing in incompressible flow Calculate the maximum turning angle through a PrandtlMeyer expansion from Mach 30 Calculate the induced velocity at a point on the lifting line located 20 feet away from a vortex shed from a point on the wing where the lift per unit span changes from 20Nm to 30 Nm suddenly The flight speed is 50 ms at standard sea level conditions Calculate the pressure coefficient at a point on the wing where the velocity is 25 times the freestream value Sketch the streamline pattern in the logitudinal section through the flow over a conical missile nose flying horizontally at Mach 2 What is meant by the term quotstrong oblique shock solutionquot How in words would you derive the Glauert Solution to the quotmonoplane equationquot in incompressible flow What is the stagnation pressure at a missile nose at Mach 08 at 10km pressure altitude if the Prandtl number is 078 How in words do you derive the normal shock relations Estimate the oblique shock angle for a turn of 2 degrees at Mach 245 Consider the flow at the exit of a slightly underexpanded 2D nozzle The exitplane pressure is 107 times the outside pressure and the Mach number is 20 a sketch the flow patterns through deceleration to subsonic velocity b calculate the distance downstream ofthe exit where the first disturbance waves converge at the plane of symmetry in terms of the values of the nozzle height H and discuss any other essential parameters Derive the full potential equation and the linearized equations for steady compressible flow A 5 thick airfoil placed at A degrees angle of attack produces a pressure coefficient of 03 at a given chordwise location X at a freestream Mach number of 05 Find the thickness of an airfoil of the same family required to given the same pressure coefficient at the same chordwise location at Mach 075 What method would you use to analyze the lift and induced draq of a tapered twisted wing of aspect ratio 8 at low angle of attack in incompressible flow How would you modify the results above for a Mach number of05 What method would you use to analyze the lift and induced draq of a highly swept cambered wing of aspect ratio 18 at low angle of attack in incompressible flow How would you modify the results above for a Mach number of 05 What method would you use to analyze the lift and induced draq of a highly swept cambered wing of aspect ratio 18 at low angle of attack at Mach 05 Do you need to modify your answer to the above two questions based on this Can lifting line theory give you the pitching moment variation of a finite wing with angle of attack How Can slender wing theory be used to find the pressure drag of a finite wing Why How How do you find the wave drag of a supersonic airfoil section How do you find the laminar skin friction drag of an airfoil in incompressible flow How do you find turbulent skin friction on a flat plate in incompressible flow How do you find the skin friction drag of a wing in supersonic flow What is a SearsHaack body What is a Karman Ogive A supersonic air flow has a freestream Mach number of20 What is the maximum expansion turning angle possible with this flow ifthe static pressure cannot go below 1 of freestream pressure What is the pressure coefficient at this point Find the drag experienced due to a strip 01 m wide of a flatplate supersonic wing at Mach 25 The upper surface is at zero angle of attack relative to the freestream and there is no shock or expansion at the leading edge The aircraft is flying at 40000 feet ISA The chord at this spanwise location is 30m How does the above problem change if the wall is held at 250K by heat transfer to the fuel contained in the wing How will you calculate the contour of an axisymmetric nozzle with an area ratio of 80 for a gas of specific heat ratio 13 and molecular weight of 18 How will you predict the aerodynamic load distribution on a tapered twisted wing of aspect ratio 8 at low angle of attack at Mach 02 How about at Mach 07 How about at Mach 17 How are the linearized potential equations for steady supersonic and subsonic flow different What is a quotcharacteristic directionquot in the above context The critical Mach number of an airfoil is 08 for a given angle of attack How much should the leading edge of a wing made of this airfoil be swept to bring the critical Mach number of the wing to 088 How will consideration of 3D aerodynamics affect the answer to the above Where does the idea that ideal lift curve slope of an airfoil is 2Pii come from What happens to lift curve slope in low Reynolds number flow Why Can lifting line theory be used to find the pressure drag of a finite wing Why How Given a 2 D velocity field how would you decide if it can represent a physically possible flow Calculate the speed of sound at 10000 feet International Standard Altitude A supersonic fighter passes 3000 meters above your head going at Mach 14 Assuming that the temperature of the atmosphere is constant at 270K up to 3500 meters estimate the time that you have to cover your ears before the sound from the aircraft first hits them How far will the fighter have gone in this time What is the difference between an equation and an identity What is the Oswatitsch Equivalence Rule How does this help predict drag of a configuration What are Newton39s Laws of Motion What is the continuity equation What is pressure Why does it occur in a gas Derive the 1D form of the steadyflow energy equation A stagnation probe is placed in an air flow where the velocity is 200 ms static temperature is 500K and static temperature is 1 atmosphere What is the static enthalpy ofthe flow What is the stagnation enthalpy What have these things to do with the first law of thermodynamics Two lines AB and CD appear to originate from the same point Can these be streamlines Why What is stagnation pressure What is total pressure Why do airplanes have a nonzero speed for minimum drag How do you predict that speed Why does the pressure of the atmosphere decrease as you go up How do you estimate the thrust of a rocket motor given stagnation pressure and temperature gas composition and a conical nozzle with a 20 degree angle How do you calculate the lift of a wing in supersonic flow What is a thermal boundary layer Is it the same thickness as the viscous boundary layer Why Is the friction coefficient for a turbulent boundary layer greater than or less than that for a laminar boundary layer What is the approximate value of the ratio of the two in an incompressible flow Sketch the chordwise variation of the pressure coefficient on the upper and lower surfaces of a cambered wing section if the wing is at moderate angle of attack at essentially incompressible flow conditions What are the Hemholtz vortex theorems What do they mean Why do we care How do you calculate the angle of attack required for a rectangular wing in order that it support a given aircraft in steady level flight Prove that the tangential velocity component is unchanged across an oblique shock wave Sketch the velocity profiles for a laminar and a turbulent boundary layer Explain physically why they have different shapes What is quotboundary layer transitionquot What factors influence the transition Reynolds number Define Prandtl Number and Recovery Factor What is a Blasius Profile What assumptions are made in arriving at the solution for this profile Explain the concepts of boundary layer displacement thickness and momentum thickness What is the usefulness ofthe Karman Momentum lntegral Define friction coefficient What are favorable and adverse pressure gradients Contrast the behavior ofa viscous boundary layer under the influence of each of these types of pressure gradient in turn When tested in a lowspeed tunnel the pressure coefficient at the suction peak was 05 Find the critical Mach number How is a quotvortex sheetquot similar to different from a quotshear layerquot What is the ideal lift curve slope of a symmetric thin airfoil in incompressible flow What is the ideal lift curve slope of a rectangular wing of aspect ratio 10 in incompressible flow Does the lift curve slope of an airfoil increase or decrease with Mach number in subsonic flow What is the lift curve slope of an airfoil at Mach 2 Does the lift curve slope of an airfoil increase or decrease with supersonic Mach number How can air flow be called quotincompressiblequot when everyone knows that air can be compressed How can you increase the lift curve slope of an airfoil How do you solve for the lift coefficient of a thick airfoil in incompressible flow What is a boundary layer What is the highest pressure on the blunt nose of the Space Shuttle at 10km pressure altitude if the Shuttle is flying at Mach 3 Transonic Small Disturbance Equation The quasilinear form of the potential equation takes on the following simpler form 2 2 1 M00 y 1Moo x xx pyy 0 Divergence form used in computations 2 l 1 2 2 1Moo x oo x y 0 2 x y Axisymmetric form of the transonic small disturbance equation 1 M020 y 1M020 x xx r rr o Transonic Similarit y T 3 I x The TSD can be non dimen5onalized as follows x y 3 Z C C C and L 2 CU 001 3 where L Is the thicknesstochord ratio or slenderness ratio The TSD then becomes 2 T K Moo V Fl x xx yy ZZ 0 2 where K is the transonic similarity parameter 13 Replacing Mach number approximately with 1 K V1XXx 3yy 22 0 If two transonic ows have the same K the solution fjj is the same C Pressure coeffts are related through T 2 C fK7yZ 3 T Boundary Conditions Solid Boundary Flow Tangency V y ll Y u Uoo x dx 0r dY x dY yUoo x dx Uoo1UOOdx dY y z U00 Boundary conditions may be applied at yYX Oronaslit yo Boundarv Conditions alonq a cut downstream of the airfoil trailinq edqe n r Around Airfoil We will assume that the disturbance potential jumps by an unknown amount F along a cut that starts at the airfoil trailing edge and ends at downstream in nity 7 d W d 1161 Jyj 39 dx dyj dx dy Cut across which ax 8y igniting d f A Contour enclosing airfoil Far Field Boundary Conditions Subsonic freestream the disturbance velocity vanishes at the farfield boundary Supersonic Freestream Upstream Boundary At this boundary p is set to zero Downstream Boundary On this boundary it is not appropriate to specify any boundary conditions because in supersonic flow information can only flow downstream In other words the behavior of p is determined by the flow over the airfoil itself Thus we should determine p from the nonlinear governing equation or its linearized form Lateral Boundaries Characteristic 1 d1 4 dX ngo 1 p constant Characteristic 2 1 dx M 1 p Constant Numerical Solution Of The 2D Transonic Small Disturbance Equa on 1971 Murman and Cole solution technique for solving the TSD equation recognized the fact that the TSD equation changes in character from an elliptic equation to hyperbolic in supersonic regions allowed the shock waves to be captured as part of the numerical procedure 1972 transonic ow over lifting airfoils could be studied 1973 Improved MurmanCole scheme 1975 Bailey Steger and Ballhaus NASA Ames techniques for lifting transonic flow over isolated wings and wingbody con gurations using the 3D TSD equation Overview of the TSD Solution Procedure 3P 3Q 7 7 0 dx y where M2 P 1 Mirpx y 17 px2 Qfpy steps Select quotnodesquot in the xy plane where the governing equation will be solved The region of interest is a large region surrounding the airfoil represented as a slit Because the disturbances associated with the airfoil are felt for large distances away from the airfoil this region must be 5 to 6 chord lengths wide in all directions surrounding the slit Use a numerical approximation which converts the PDE into a set of nonlinear algebraic equations at the nodes selected in step 1 This step is known as discretization Solve nonlinear algebraic equations for the numerical value of p at the nodes This is usually iteratively done starting with some arbitrary guess for the solution at the node The process by which p converges from an arbitrary initial guess to the solution which satisfies the discretized form developed in step b is known as the relaxation procedure Once the values of q are known at all the nodes we can find other quantities such as disturbance velocities surface pressure Cp distribution the lift drag and pitching moment experienced by the airfoil the total number of supersonic points in the flow field etc This process is known as the postprocessing step Grid Generation From experience it is known that there must be at least 50 points over the slit spaced apart by a distance c50 Geometrically increase grid spacing both in the x and y directions away from the slit Use Ax 1le next previous Aynext K2 Ayprevious K1 and K2 are stretching factors between 105 and 115 Fine Grid Coarse Grid near Slit Away from slit Algebraically stretched grid for the 2D TSD solution Grid numbering Identify each node with two indices ij i refers to the vertical line on which the node lies whilej refers to the horizontal grid line on which the node lies 0141 lt gt w gt hi V Hi i i1 j 7 Li1 Distance separating the nodes id and i1j is Axi Horizontal distance separating the node id and i1j is AXM vertical distances that separate ij from ij1 and ij1 are ij and A yj1 respectively SITt 39JBODY1 g The slit is placed halfway between lines j JBODY and jJBODY1 jJBODY Discretization of the TSD equation on a stretched Cartesian grid Three common approaches to discretization are a finite volume method b finite element method and c finite difference method Books by Anderson Tannehill and Pletcher P J Roache Hirsch Here finite difference method is used for its simplicity and also because Murman and Cole39s original work used this type of discretization MurmanCole Scheme Essentials 1 At a subsonic point 97 i139i 139 xij 01xlj JZAX J 92 ij ax 2 IEW l Information at point ij comes from both upstream and downstream MurmanCole Scheme Essentials 2 At a supersonic point 97 i39 i 239 WW 82 ij 2 i 1j i 2j 1 P ax 2 w My Upwind differences are used point ij is influenced only by points upstream MurmanCole Scheme Essentials 3 At both subsonic and supersonic points the y derivative uses central differences 5 7i11 2 7w 7w1 1 W2 W M The equation solved is the approximated nondimensional form K r 1lt3xq3xx 1 4322 0 Substituting the derivatives for the subsonic point the equation can be expressed as A i B Supersonic point 2 C ij D ij E where A BC D and E are known numbers Start with guessed values at all grid points Examine each and decide if supersonic or subsonic Solve the appropriate equation for the new value of 451 j Stop when the values stop changing Detailed Process Expand each term using a Taylor series in terms of the values of the dependent variable stored at the nodal points surrounding and including a typical node ij The order of accuracy depends on how many terms in the series expansion are kept Assumption is that the individual terms in the PDE are smooth differentiable functions of X and y in the vicinity of the node ij This assumption is not always valid in the vicinity of the shocks A i12j Change in F i1 39 Change in x xv Graphical Representation of the Derivative of Fx dFdx a AFAx K110Ax P P p x M Axi1 Mi 2 Thumb rule x Denvative of F FRight FLeft quotJ D1stance between Nodes to theR1 ght and Left GAbove GBelow Denvative of G y W Distance between Nodes Above and Below The term 6P6X Derivative is computed by expanding the function P in the vicinity of the node ij Expand P at the halfnodes i12j and i12j about the node ij ii1 ij12 i1i i12j u i12i i1i ij12 Li1 Half Nodes are used to discretize the TSD equation 2 PiHZJ PM Pxij AXI81PXXLJgt 0Ax3m Taylor series truncated at three terms Creates truncation error 6P6x becomes PH Pi Ax Axi Pxij 14 Pmij0Ax2 2 fKV Iquotquot d Wma ZIf a rCV gXV Z RV 0 f ZI WWW cl 1 1 XV W Z XV m szWIM szIF z f ZI x Z x 00 1 ltZZW dzw1fZId Z KW f K Z In summary second order accurate numerical approximation to the 2D transonic small disturbance equation is P P z12j 1 12j Qij12 Qij 12 0 Axi Axi1 ij A3941 Reason for using half points to compute P and Q leads to compact expressions for PX and Qy at node ij involving only the node and its four neighbors If we had computed P and Q at the full nodes l 1 j and i j 1 then our nal form would have involved nodes far away such as l 2j and i j 2 In general use of a compact form compact quotstencilquot involving just a handful of nodes closely spaced together is likely to yield better results than over a large stencil involving nodes far apart Shock Fitting vs Shock Capturing Shock Capturing Shocks are modeled as large but differentiable changes in the flow velocity quotcapturedquot across a small number of points rather than smeared over a large number of points Shock fitting involves treating the shock surface as a discontinuity and applying special jump conditions in lieu of the governing equations Shock capturingquot approach used by Murman amp Cole and many subsequent researchers because of its inherent simplicity Shock fitting is more accurate but is somewhat cumbersome to program Murman and Cole Scheme Subsonic Region Numerical Domain of Physical domain of Dependence Depndence In elliptic regions the node ij depends on the surrounding region In supersonic regions point P is not in uenced by points outside the quotMachquot cone a wedge formed by the two characteristic lines discussed in Chapter III Wedge formed by Supersonic Region Characteristic Lines il1 Li quot1 i2i i1j Ll1 Physical Domain of Dependence Numerical Domain of Depdendence In supersonic regions the node ij should depend primarily on the information within the characteristic cone In supersonic regions the i indices in the equation are shifted to the left resulting in the following approximation Pi 12j P Axi Axi1 ij A i 32j Qij12 Qij 12 0 j1 The rst term is only a rst order approximation to Eat the node ij even on uniformly spaced grids Show 3 6 Becomes equivalent to subsonic eqn in subsonic regions and equivalent to supersonic eqn in supersonic regions The sign ofthe quantity A given by A 1 M y 1WOZOCDJC determines if the TSD equation is elliptic or hyperbolic MurmanCole evaluates it as As 1 M y AXi IA XHI Drawbacks of the MurmanCole Scheme 1 Only rst order accurate in supersonic regions A large number of points are needed in the x direction to achieve an acceptable level of accuracy because the error is proportional to Ax and not to Ax 2 Predicted shock waves that were weaker than expected from an Euler analyses Mach number ahead of the shock was lower and the pressure jump across the shock was weaker than expected from an Euler analysis The shock location was predcited to be upstream of the expected location from an inviscid analysis The second issue mimics what would happen if a boundary layer were included So it matches actual results better though this happens fortuitously The problem is that mass is not conserved in the scheme We expect our numerical solution to satisfy the divergence theorem which states PX Q ixdy f dS Entire Domain Outer Boundaries Switching between equations within just a few cells does not allow mass Conservation Numerical schemes which satisfy the Divergence theorem in a numerical sense are called conservative discussing quotconservation of massquot because our TSD is a descendent of the continuity equation Murman39s Conservative Form It is the last term that abruptly appears only in supersonic regions which destroys the telescopic sum property Murman proposed that this last term be slightly modified leading to the following discrete approximation to the 2D TSD equation Pi12j Pi 12j Qij12 Qij 12 l Pi12j Pi 12j Pi 12j Pi 32j M Mi 1j 0 Axi Axi1 ij1 l Axi Axi l Axi Axi l Application of Solid Wall Boundary Conditions LCP U00 dY 07y dx iJBODY Slit the terms involving Q in equation 417 may be written as dY QiJBODY12 Uood x iUPPER AYj l39ijH Application of Murman Conservative Form across the Cut The velocity potential has a discontinuity equal to the circulation F along cut This cut is chosen to be the line strating from the trailing edge and ending at the downstream boundary Across this out p is discontinuous iJBODY1 GJBODY Slit GUI auBOqu Treatment of Nodes above the Slit Subtract the jump in p before computing the y derivative The points underneath the cut j JBODY are handled in a similar manner Application of Far Field Boundary Conditions Subsonic Freestream At all the points on the outer upstream downstream top and bottom boundaries the following equation is applied F CpFar Fie1d 7 arctanK 1 M 2 717 K x Usually applied following every iterative step in the relaxation procedure after the values of p at the interior points are evaluated Supersonic freestream upstream boundary i 1 Setj to be zero at the upstream boundary Supersonic freestream downstream boundarv i lMAXl Two Choices Apply the compatibility equations derived from the method of Characteristics Apply the governing equations treat the downstream boundary like ny other interior point IMAXj IMAX2j IMAX1J supersonic outflow modified MurmanCole TSD equation depends only on nodes shown Sugersonic freestream too boundarv i JMAX use the method of charcateristics to arrive at the following compatibility relation LLJMAX JMAX JMAXJ Characteristic of slope 1 At the top boundary the method of characteristics may be used Supersonic freestream bottom boundary go constant Characteristic of slope 1 L2 Relaxation Procedure for solving the Discretized form of the TSD Equa on Iterative approach similar to the NewtonHaphson iterative method used to find the roots of nonlinear algebraic equations n n1 Acpzj plyj plyj If our iterative process converges then will go to zero at all the interior nodes after a sufficiently large number of iterations Starting guess for g set p at all the interior points to zero to start our calculations That is CPL O Linearization of the Discretized form of the TSD Equation Arrive at a system of linear equations for Acp one equation per interior node Consider a term such as Did 2 n1 n1 Q 1 0 quot1 M ij12 yl 12 J ij1 goirfjH 903 Agog41 Agog ij1 mm m n Q zJ12 ij1 y n1 1 M px T M px 2 i12j 1 M Xpquot ACpx yT1MPquot A x2 i12j n E 1 M 21MPx2 1 M y 1MpAPX 39 12 i12j 1 J n n Pi12j Ai12j ACpL 1412 j Pquot Aquot W i12j i12j Axi1 Linear system of equations for Acp BijApi 2j CijApi 1j DijApij ELJ ACPHLJ FijApij 1 GijApij1 R is called the residual and is simply the negative of the left side of equation 417 computed at the previous iteration 39n39 1 1 1XV 1XV W 1 IDIjHIILV I f1nv 1Z 1 SH XV gny1 XV XV Fl1 I m 1 V V HIKV VIV k f IKV 1 f 39 J 11XV 1W 11x 1 kl ZIH V IHI393 V mm 1W I1XV1XVI Iw I ms AV ZILV xv 1xv zxv 1 1 WHV IHI ID z D smouo se puno M399 9 SJ 39019 Q 9 sluegoggeoo eqi Can be written as M HM R M is a sparse banded matrix with six diagonals one corresponding to each of the coefficients defined in the previous slide This matrix may be stored in the memory of a computer system simply by storing the six diagonals defined above Numerical Solution We can simplify our solution process by replacing the matrix M on the left hand side with another easily invertible matrix N which is closely related to M In other words we solve instead N Arp R What criteria should be used to select the matrix N From numerical analysis it is known that any matrix N will do provided the absolute values of all the eigenvalues of the matrix I N391M is less than unity Some commonly used N matrices are discussed next We will not attempt to prove that these matrices satisfy the criterion l N391M is less than unity Point Jacobi Scheme Matrix N contains only the diagonal elements of M In other words we solve DijApij Rij Or A R 90 Du The Point Jacobi scheme converges in all situations although very slowly For a typical lifting transonic flow several thousand iterations are usually needed for R and Acp to be driven to zero Gauss Seidel Scheme N is either the lower triangular part of M or the upper triangular part of M If the lower triangular part is chosen we solve BijApi 2j CijApi 1j DijApij FijApij 1 Rm 0r BijApi 2j iJ39Ami IJ FijApij 1 1 D M Easily implemented in a double Doloop over i and j in any high level programming language If the upper triangular part is Chosen we solve DijApij EijApi1j GijApij1 0r R j EijACpi1j GLjACPLjH 1 In supersonic flows this may diverge Point Successive OverRelaxation SOR lScheme Identical to the GaussSeidel scheme except the residual R on the right hand side is multiplied by an overrelaxation factor 00 between 1 and 2 This factor is chosen by trial and error for best convergence Successive Line OverRelaxation Scheme 5 of the six diagonals that constitute the M matrix are kept and only the term involving the coefficient Fi j is neglected The right hand side is multiplied by a relaxation factor 00 between 1 and 2 We therefore solve CUR ij1 ij BijApi 2j CijAp 0r DUAcpid FljApij1 GiJArpiJ1 wig thArp i1j DijApij FijApij 1 i 2j QjApi 1j Solve for the Acp values on all the nodes on a vertical grid line i constant simultaneously When we are solving for Acp at an i station the Acp values at the previous stations i1j and i2j are already known and may be brought to the right side as shown in equation 439 The resulting form is a tridiagonal system of equations linking Acp at the nodes ij ijt and ij1 Right Hand 2 Side 439 The Thomas algorithm is simply the Gaussian elimination scheme intelligently implemented taking advantage of the g RfHS fact that the matrix N has d 239 many zero elements In f this scheme the matrix N is factored into a product of a lowertriangular L matrix and an upper triangular matrix U as shown The elements f d and g of the LU form are recursively related to the original elements of matrix on the left side of 440 These may be stored as onedimensional vectors or arrays fj Fij dj 123139 fjgj l GM 6 gj Approximate Factorization Methods the N matrix we choose is one that is easily factored into smaller tridiagonal or LU matrices Approximate factorization methods are much faster than the SLOR scheme but are somewhat more difficult to program For an excellent approximate factorization scheme for solving the 2D TSD equation the reader is referred to an article by Ballhaus Jameson and Albert in the AIAA Journal 1978 Transonic Full Potential Equation Used in the aircraft industry Supercritical wing sections are thick FPE allows interference from adjoining components such as fuselage nacelle etc to be modeled without assuming that such interference effects are small The FPE in quasilinear form 2 2 2 2 a u qu 2uv xya v ng 0 characteristic equation 212 rapt20 dx dx Here dydx is the slope of a characteristic line at a given point and 2 2 2 2 Aa u B2uv Ca v d uvcmu2v2 a2 7 dx a2 u2 M gt1 the FPE is hyperbolic M lt1 no real characteristics exist and the FPE is called elliptic At sonic points where M1 only one characteristic exists and the FPE is parabolic ln supersonic regions equivalent expression is dy 7 t 6 dx an M where 6 arctanv M arcsin 1 u M Note that the characteristic lines are symmetric about the velocity vector U and not about the x axis You may recall that in the case of the TSD equation the characteristic lines were symmetric with respect to the x axis A simple shift of the x derivatives upstream is no longer adequate in general both the x and y derivative terms must be properly shifted when solving the full potential equation whether in the quasilinear form or the divergence form Divergence form of the full potential equation is p xx p yy 0 where 1y11421uzv2 11 1 pa 2 V Jameson39s Rotated Difference Scheme Jameson J Communications in Pure and Applied Mathematics Vol 27 1974 pp 283309 Quasilinear form in a new Cartesian coordinate system sn where s is aligned with the flow direction naxis is normal to it Angle between the x axis and s axis equal to the flow angle 6 is a X constant and is not a function of s Strea mnne S X COS 6 ySiIl 6 n ycosl9 xsin 6 gtx and xscosl9 nsin6 y ssin 6ncosl9 Derivatives i aasaan 90056 asin 9 9x 9s 926 M 926 9s 9n 9 9 9S 9 9n 9 9 7 s1n60056 9y 9s 9y 9n 9y 9s 9n And 92 9 9 9 9 9 9 72 GOSH SID 6009 7SIII 9 9x 9x 9x 99 9n 95 9 2 2 a sin2 6 a 9S9n 9n 2 coszt9jz 200565in9 2 s Also 2 2 2 2 2 coszt9 92 200565in6 00526 2 9y 9n 9S9n 9n 2 a acost9 asin6asin6acost9 9x65 9s 9n 9s 9n Quasilinear FPE becomes a2 of PSS a209quot 0 Jameson Only terms that contribute to IDSS need be shifted upstream in the direction of the flow vector in supersonic regions The term involving Dnn need not be shifted Similar to the MurmanCole scheme for the TSD equation where we shifted the PX derivative upstream but modeled Qy using a symmetric unshifted form To apply Jameson39s scheme we need to a Identify the terms in the quasilinear form that contribute to 188 and PM b Shift only the terms that contribute to 1338 in the direction of the flow c Collect the 1383 and PM terms Terms that contribute to ltIgtss and Dnn E EE a ay cos9isin93 8s8x 8s5g 8x u 8 v 8 7375 82 8 8 u 8 128 u 8 v 8 aszasltasgtltmmgtltmqaygt a 82 382 q28x2 q28x8y q28y2 Likewise 82 u 82 W 82 v2 82 7 7 01n2 qZ 072 qZ qZ 01x2 FPE can be written as Rotated Differences Consider subsonic fowat grid point ii on a uniform orid i1j 2 ij i 1j XX LJ AXZ ij1 2 ij pm 1 ij Ayz z i1j1 i1j 1 i 1j1 i 1j 1 m 39 4AXAy 13737 Xy These finite difference expressions may be derived by expanding Did about its neighbor points using a Taylor series Supersonic flow The DXX DXy and lbw terms appearing within 138 must be shifted in the direction of the flow while the terms appearing in PM must be symmetrically differenced 2 2 211V L2 1 a q ijlq2 XXi 1j q2 XYi 12j 12 q2 yyiJ1J 112 211V V2 1 31 yyij q2 pxhj 124 th O Extension of Jameson39s Rotated Difference Scheme to Divergence Form of the Full Potential equation We note that the various derivatives of I that constitute 138 may be expanded about the node ij xxi1j E yy2j1 a yy Ay yyyij AX A XY1 12j 1 2 E XY1j Xxyij 2y ny1j 212 u2iXX 2uvciKy a2 V2 pny 2 2 a2 q21jAX2 Xxx XxyL az q2ijAyl2 pm nyL 0 Divergence Form MK r3ltrgty y 0 Where i p MApr r3 p MAypy Retarded densities or biased densities Physical densities modified by a density gradient term The factor it is set to be zero in subsonic regions It reduces to our original divergence form The factor u is set to 1 1M2 in supersonic regions Numerical Solution of the Full Potential Equation for Lifting Transonic Flow over Airfoils Grid Generation A curvilinear grid is used composed of two families of lines that are nearly orthogonal to each other One of these lines belonging to one of these two families wraps around the airfoil Bodyfitted grids may be generated around 2D or 3D configurations For isolated airfoils grids may be generated using simple algebraic transformations The grids may be one of the following three kinds H grids where one family of grid lines resemble streamlines and flow around the airfoil O grids where one family of lines resemble deformed circles that wrap around the airfoil and C grids where one family of lines resemble the letter C Hgrid over isolated airfoil Family Family H Gird over an isolated airfoil Transformation from physical plane to computational plane lMAX1JMAX Coordinate Cut lMAXt 2 Transformation is purely numerical There is no analytical relationship linking these two planes Every grid point in the physical plane maps onto a unique point in the transformed plane 6X 6y Yag 14 ii X nX 6 6 an EX nx gy and my are known as the metrics of transformation d de aydy dn TIde nydy Or dE dn ax an m m dX dy dX ngE XndT l dy yng yndn Or EH 2 J is called the Jacobian of 11X nyJ yE ynJ transformation and is interpreted as the ratio of the Or area of an infinitesimal element surrounding the node ij in the E A Jy Em plane to the area of its X 39 39 he Xy plane Xy Xy n Imageint E n n E X Ey n JXn XEyn Xnyg X ngn XnyE 3 x E Xi1j Xi 1j g 2A y yi1j yi 1j g 2A X Xij1Xij 1 quot 2A1 yij1yij 1 yquot 2A1 let us consider a very general equation 6F 6G 0 9X 6y FpuandGpv PEEK anx Ggay Gnny O gt F yn 1wa GEXH GHXE O


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