Aerothermodynamics of high speed flows
|
|
- Christal Black
- 5 years ago
- Views:
Transcription
1 Aerothermodynamics of high speed flows AERO Lecture 6: D potential flow, method of characteristics Thierry Magin, Greg Dimitriadis, and Johan Boutet Thierry.Magin@vki.ac.be Aeronautics and Aerospace Department von Karman Institute for Fluid Dynamics Aerospace and Mechanical Engineering Department Faculty of Applied Sciences, University of Liège Room B5 +1/443 Wednesday 9am 1:00pm February May 017 Magin (AERO ) Aerothermodynamics / 6
2 Outline 1 Governing equations Method of characteristics 3 Exercises: exact theory of oblique shocks / expansion waves Magin (AERO ) Aerothermodynamics / 6
3 Governing equations Outline 1 Governing equations Method of characteristics 3 Exercises: exact theory of oblique shocks / expansion waves Magin (AERO ) Aerothermodynamics / 6
4 Governing equations Entropy eq. (Navier-Stokes) First law of thermodynamics for open system: T ds =de + pd( 1 ) Mass and thermal energy (Navier-Stokes) eqs. ) Entropy eq. D Dt (s) = T D (e) = pr u r q : ru Dt D ( ) Dt = r u D Dt (e) p T D Dt ( )= 1 T r q 1 T : ru Transport fluxes: = S and q = rt Conservative t ( s)+r ( us)+r ( q T )= The entropy production rate is nonnegative = T S:S + T rt 0 In smooth inviscid flows, the entropy is constant along a trajectory (pathline) D (s) =0 Dt Magin (AERO ) Aerothermodynamics / 6
5 Governing equations Total enthalpy eq. Total energy E = e + 1 u D (E)+r (pu)+r q + r ( u) =0 Dt Total enthalpy H = h + 1 u = E + p/ p D D Dt ( p )= D Dt (p) Dt ( ) =@ tp + u rp + pr u t p + r (pu) ) D Dt tp + r q + r ( u) =0 In steady inviscid flows, the total enthalpy is constant along a trajectory (pathline) D (H) =0 Dt Magin (AERO ) Aerothermodynamics / 6
6 Governing equations Crocco s eq. Momentum t u + u ru = 1 rp 1 r Lagrange identity: u ru =(r u) u + 1 r(u u) Vorticity vector:! = r u First law of thermodynamics: dh = T ds + 1 dp ) 1 rp = rh T t u +! u + 1 r(u u) =T rs rh 1 r Crocco s t u +! u = T rs rh 1 r rs In steady inviscid flows, with constant H (rh = 0), the entropy is constant in irrotational flows (! = 0)! u = T rs Conversely, if the flow is isentropic rs = 0, it must be irrotational unless! k u everywhere (Beltrami flow) In smooth inviscid and steady flows, the entropy vanishes on a pathline: u rs = 0. This property does not imply that rs =0 Magin (AERO ) Aerothermodynamics / 6
7 Governing equations Rotational flows Exemples of rotational flow: boundary layer detached shock! = r u 6= 0)rs 6= 0 Magin (AERO ) Aerothermodynamics / 6
8 Governing equations Irrotational flows Steady flow Uniform upstream conditions Isentropic flow Inviscid flow (no dissipation) No discontinuities ) Potential (irrotational) flow! = r u =0 u = r Exemples of irrotational flow: attached shock (except shock region), nozzle, expansion Magin (AERO ) Aerothermodynamics / 6
9 Governing equations Alternative form of continuity eq. D Continuity eq.: Dt ( )+ r u =0 Let us consider a smooth inviscid and steady flow The material derivative reads D Dt ( ) =u r The entropy is conserved on a path line: u rs =0 By definition of the speed of sound: u r = 1 a u rp Multiplying by u the momentum eq.: ) D Dt ( ) = a u u : ru u rp = u ( u ru) = u u : ru Alternative form of the continuity eq. valid for both rotational and irrotational flows a r u u u : ru =0 Magin (AERO ) Aerothermodynamics / 6
10 Method of characteristics Outline 1 Governing equations Method of characteristics 3 Exercises: exact theory of oblique shocks / expansion waves Magin (AERO ) Aerothermodynamics / 6
11 Method of characteristics D potential flow: method of characteristics Non linear potential eq. a r u u u : ru =0 Irrotational flow! = r u =0 x u x1 u =0 ) System of partial di erential eqs. a u1 u 1 u 0 1 x1 u with Q = u1 u + u1 u a u 1 x1 Q + x Q =0 1 A = a u1 x =0 u u 1 u a u (a u1 ) 0 Magin (AERO ) Aerothermodynamics / 6
12 Method of characteristics Eigen vectors and eigen values Left eigen vector l and eigen value : l A = l Characteristic equation: l (A I) = 0 ) A I = 0 (a u 1) Discriminant: = a (u1 + u a ) M > 1! > 0 : hyperbolic eq. M =1! = 0 : parabolic eq. M < 1! < 0:ellipticeq. For M > 1(" = ±) +u 1 u + a u =0 Real eigen values: " = u1u/a +" p M 1 u1 /a 1 Eigen vectors: (l 11 l 1 ) A = + (l 11 l 1 )and(l 1 l ) A = (l 1 l ) l 1 /l 11 =( + a 11 )/a 1 and l /l 1 =( a 11 )/a 1 After some algebra, one gets l 1 /l 11 = and l /l 1 = + Magin (AERO ) Aerothermodynamics / 6
13 Method of characteristics Physical interpretation of the characteristic lines Change of variables u1 /a = M cos u /a = M sin " = M sin cos + " p M 1 M cos 1 = M tan + " p M 1(1 + tan ) (" p M 1) tan = ("p M 1tan +1)(" p M 1+tan ) (" p M 1 tan )(" p M 1+tan ) = = tan + "/ p M 1 1 " tan / p M 1 : 1/ cos =1+tan tan +tan("µ) 1 tan tan("µ) : sinµ =1/M, tan µ =1/p M 1 ) " =tan( + "µ) with positive in the counterclockwise direction Limits: lim M!1 µ =0andlim M!1 + µ = / Magin (AERO ) Aerothermodynamics / 6
14 Method of characteristics tan " = " =tan( + "µ) The streamline at one point makes an angle with the x 1 axis Two characteristics are passing at this point: one at the angle µ above the streamline, and the other at the angle µ below the streamline The characteristic lines are Mach lines No discontinuity in the velocity or any other fluid property along the characteristics, Mach lines are patching lines for continuous flows Magin (AERO ) Aerothermodynamics / 6
15 Method of characteristics Characteristic eqs. for M > l11 l Using LA = L with = and L = 1 0 l 1 l The system of partial di erential eqs. reads L@ x1 Q + x Q =0) L@ x1 Q + x Q =0 After explicit calculation l11 (@ x1 + x )u 1 + l 1 (@ x1 + x )u = 0 l 1 (@ x1 x )u 1 + l (@ x1 x )u = 0 The terms between ( ) can be interpreted as directional derivatives in the x 1, x plane. Using the characteristic directions s " = (cos ", sin " ) ( d d l11 u ds l 1 u ds + = 0 d d l 1 u ds 1 + l u ds = 0 one obtains a system of ordinary di erential eqs. (hodograph plane) along the directions s + and s ) ( l11 + l 1 du du 1 = 0 on s + l 1 + l du du 1 = 0 on s in the u 1, u plane Magin (AERO ) Aerothermodynamics / 6
16 Method of characteristics Solution method Cauchy problem: u 1 and u are given on an arc compute the flow downstream of that arc and one wishes to Discretized solution The characteristic slope direction + and can be computed at each point of arc, in particular + A =tan + A and B =tan B The position of point P is defined by the intersection of the characteristic lines approximately numerically by their tangent at A and B The velocity components at P can be computed by integrating (numerically) the characteristic equations ( u1 P u 1 A u 1 P u s + + P u A A s + = 0 on s + u 1 B u s + + P u B B s = 0 on s Magin (AERO ) Aerothermodynamics / 6
17 Method of characteristics Zone of influence and zone of silence Consider left- and right-running characteristics through point P Zone of influence for P: areabetweenthetwodownstream characteristics. This region is influenced by any disturbances or information at point P Zone of silence for P: all points outside the zone of influence will not be a ected by disturbances or information at P Zone of dependence for P: areabetweenthetwoupstream characteristics. Properties at point P depend on any disturbances or information in the flow within this upstream region In steady supersonic flow, disturbances do not propagate upstream Magin (AERO ) Aerothermodynamics / 6
18 Method of characteristics Riemann invariants The characteristic eqs. in the hodograph plane ( 1+ du du 1 = 0 on s du du 1 = 0 on s have only one solution through the point of the plane These solutions are called Riemann invariants F1 (u 1, u ) = constant on s + F (u 1, u ) = constant on s After change of variables (u 1, u )! (M, ), one obtains (M) = constant on s+ (M)+ = constant on s with the the Prandtl-Meyer function (M) introduced in lecture Magin (AERO ) Aerothermodynamics / 6
19 Method of characteristics Proof u1 = u cos ) du Change of variables 1 =cos du u sin d u = u sin ) du =sin du + u cos d du 1 + " du = 0 (cos + " sin )du + u( sin + " cos )d = 0 (cos( "µ)cos +sin( "µ)sin )du +u( cos( "µ)sin +sin( "µ)cos )d = 0 cot µ 1 du "d = 0 u 8 >< H = c pt + 1 u 1 =( ( 1)M + 1 )u Change of variable: 1 dh = ( ( 1)M >: + 1 )udu dm u 1 M 3 = 0 p M 1 1 g(m)dm "d = 0 : with g(m) = M 1+ 1 M Z M (M) " = constant : with (M) = g(m 0 )dm 0 0 Magin (AERO ) Aerothermodynamics / 6
20 Method of characteristics Prandtl-Meyer expansion Let us consider an incoming flow following a curved (convex) surface It is possible to compute exactly the flow quantities at each point P of the wall (the angle is negative in the clockwise direction) On s emanating from the uniform incoming flow region (M P )+ P = (M A ) For any point Q on s + emanating from point P (MQ ) Q = (M P ) P (M Q )+ Q = (M A ) = (M P )+ P! (M Q )= (M P )and Q = P :theslopeofs + is identical at all point (straight line) The angle + µ decreases: the characteristics diverge and form a fan Magin (AERO ) Aerothermodynamics / 6
21 Method of characteristics Formation of a shock wave Let us consider an incoming flow following a curved (concave) surface, the characteristics converge The characteristics from the same family might therefore intersect Such an intersection indicates the breakdown of the method of characteristics, because there would be too much information to calculate the flow quantities at the point of intersection In reality, the intersection of characteristics in not possible, it rather indicates the appearance of a shock wave As opposed to characteristics, shocks are patching lines for discontinuous flows Magin (AERO ) Aerothermodynamics / 6
22 Method of characteristics In practice, finite-volume representation In the finite-volume representation of the method of characteristics, the solution, supposed piecewise constant over elementary cells, is obtained as follows: Divide the Cauchy arc into a number of segments on which the flow conditions are assumed to be constant Approximate by straight lines the characteristics emanating from the segment edges Based on the characteristics, define a set of elementary cells on which the flow quantities will be determined Compute the slopes of the characteristics On s : (M 3 )+ 3 = (M 1 )+ 1 On s + : (M 3 ) 3 = (M ) ) ( (M 3 ) = (M 1)+ (M ) = (M 1) (M ) + 1+ ) ( 13 = µ 1+µ 3 3 = + 3 µ +µ 3 Magin (AERO ) Aerothermodynamics / 6
23 Method of characteristics When characteristics hit physical boundaries Wall: flow angle imposed Free jet: pressure and thus Mach number imposed (provided that isentropic assumption is valid) Example: wall On s + : (M 3 ) 3 = (M ) 3 = wall ) ) (M 3 )= (M )+ wall 3 = + 3 µ +µ 3 Magin (AERO ) Aerothermodynamics / 6
24 Exercises: exact theory of oblique shocks / expansion waves Outline 1 Governing equations Method of characteristics 3 Exercises: exact theory of oblique shocks / expansion waves Magin (AERO ) Aerothermodynamics / 6
25 Exercises: exact theory of oblique shocks / expansion waves Exercise: forces on D airfoils Consider a symmetrical diamond-shaped airfoil flying at an angle of attack to the free stream of 8 when the upstream Mach number is. The atmospheric pressure of the free stream is equal to 101,35 Pa. The ratio of the thickness t to the chord c of the airfoil is equal to 0.1. Calculate the lift and drag forces exerted on the airfoil assuming that the chord is equal to 1m (R = 87 J/ (kg K), =1.4, T 1 =188K). Magin (AERO ) Aerothermodynamics / 6
26 Exercises: exact theory of oblique shocks / expansion waves Solution The airfoil half-angle is determined from =arctan(t/c) = The pressure coe cient for region i =1,, 3, 4isdefinedas C pi = p i p 1 = 1 1u 1 M1 ( p i 1) p 1 The forces exerted on the airfoil in the local reference frame are T = (p 1 + p 3 p p 4 )l sin = M 1 p1 N = (p 1 + p p 3 p 4 )l cos = M 1 p1 t (C p1 + C p3 C p C p4 ) c (C p1 + C p C p3 C p4 ) Magin (AERO ) Aerothermodynamics / 6
27 Exercises: exact theory of oblique shocks / expansion waves The lift and drag forces exerted on the airfoil in the flow reference frame are L = T sin + N cos D = T cos + N sin Region 1: oblique shock theory Deflection angle =8 + = Shock-wave angle: = Normal Mach: M n1 = Static pressure: p 1 /p 1 =.0575 Pressure coe cient: C p1 =0.377 Static temperature: T 1 T 1 = Normal Mach number: Mn1 = 1+ 1 h (M n1 1) i apple +( 1)M n1 ( +1)M n1 M n1 Mn1 1 = q Tangentiel Mach number: M t1 = M T1 t1 T1 =1.98 q Mach number: M 1 = Mn1 + M t1 =1.498 Region : Prandtl-Meyer expansion Mach number: (M )= (M 1 )+ =3.734 ) M = p Static pressure: 1+ M = 1 1 p 1 1 ) p = p p 1 = M p 1 p 1 p 1 Pressure coe cient: C p =0.051 =1.4 Magin (AERO ) Aerothermodynamics / 6
28 Exercises: exact theory of oblique shocks / expansion waves Region 3: Prandtl-Meyer expansion Mach number: (M 3 )= (M 1)+ = ) M 3 = p Static pressure: 3 1+ M = 1 1 p 1 1 = M 3 Pressure coe cient: C p3 = Region 4: Prandtl-Meyer expansion Forces Mach number: (M 4 )= (M 3 )+ = ) M 4 = p Static pressure: 4 1+ M = 1 1 p 1 1 = M 4 Pressure coe cient: C p4 = C T = 1 t c (C p1 + C p3 C p C p4 )=0.05( ) = C N = 1 (C p1 + C p C p3 C p4 )=0.5( ) = T = N = M1 p1 C T =6891N/m M1 p1 C N =95893N/m Magin (AERO ) Aerothermodynamics / 6
29 Exercises: exact theory of oblique shocks / expansion waves L = T sin + N cos = N/m D = T cos + N sin = N/m For a supersonic inviscid flow over an infinite wing, the drag per unit span is finite. This wave drag is inherently related to the loss of total pressure and increase of entropy across the oblique shock waves created by the airfoil. Magin (AERO ) Aerothermodynamics / 6
Aerothermodynamics of high speed flows
Aerothermodynamics of high speed flows AERO 0033 1 Lecture 4: Flow with discontinuities, oblique shocks Thierry Magin, Greg Dimitriadis, and Johan Boutet Thierry.Magin@vki.ac.be Aeronautics and Aerospace
More informationSteady waves in compressible flow
Chapter Steady waves in compressible flow. Oblique shock waves Figure. shows an oblique shock wave produced when a supersonic flow is deflected by an angle. Figure.: Flow geometry near a plane oblique
More informationThin airfoil theory. Chapter Compressible potential flow The full potential equation
hapter 4 Thin airfoil theory 4. ompressible potential flow 4.. The full potential equation In compressible flow, both the lift and drag of a thin airfoil can be determined to a reasonable level of accuracy
More information1. (20 pts total 2pts each) - Circle the most correct answer for the following questions.
ME 50 Gas Dynamics Spring 009 Final Exam NME:. (0 pts total pts each) - Circle the most correct answer for the following questions. i. normal shock propagated into still air travels with a speed (a) equal
More informationHigh Speed Aerodynamics. Copyright 2009 Narayanan Komerath
Welcome to High Speed Aerodynamics 1 Lift, drag and pitching moment? Linearized Potential Flow Transformations Compressible Boundary Layer WHAT IS HIGH SPEED AERODYNAMICS? Airfoil section? Thin airfoil
More informationShock and Expansion Waves
Chapter For the solution of the Euler equations to represent adequately a given large-reynolds-number flow, we need to consider in general the existence of discontinuity surfaces, across which the fluid
More information6.1 According to Handbook of Chemistry and Physics the composition of air is
6. Compressible flow 6.1 According to Handbook of Chemistry and Physics the composition of air is From this, compute the gas constant R for air. 6. The figure shows a, Pitot-static tube used for velocity
More informationModule3: Waves in Supersonic Flow Lecture14: Waves in Supersonic Flow (Contd.)
1 Module3: Waves in Supersonic Flow Lecture14: Waves in Supersonic Flow (Contd.) Mach Reflection: The appearance of subsonic regions in the flow complicates the problem. The complications are also encountered
More informationSeveral forms of the equations of motion
Chapter 6 Several forms of the equations of motion 6.1 The Navier-Stokes equations Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationAerothermodynamics of High Speed Flows
Aerothermodynamics of High Speed Flows Lecture 5: Nozzle design G. Dimitriadis 1 Introduction Before talking about nozzle design we need to address a very important issue: Shock reflection We have already
More informationIntroduction to Fluid Mechanics. Chapter 13 Compressible Flow. Fox, Pritchard, & McDonald
Introduction to Fluid Mechanics Chapter 13 Compressible Flow Main Topics Basic Equations for One-Dimensional Compressible Flow Isentropic Flow of an Ideal Gas Area Variation Flow in a Constant Area Duct
More informationIX. COMPRESSIBLE FLOW. ρ = P
IX. COMPRESSIBLE FLOW Compressible flow is the study of fluids flowing at speeds comparable to the local speed of sound. This occurs when fluid speeds are about 30% or more of the local acoustic velocity.
More informationAOE 3114 Compressible Aerodynamics
AOE 114 Compressible Aerodynamics Primary Learning Objectives The student will be able to: 1. Identify common situations in which compressibility becomes important in internal and external aerodynamics
More informationHIGH SPEED GAS DYNAMICS HINCHEY
HIGH SPEED GAS DYNAMICS HINCHEY MACH WAVES Mach Number is the speed of something divided by the local speed of sound. When an infinitesimal disturbance moves at a steady speed, at each instant in time
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationVarious lecture notes for
Various lecture notes for 18311. R. R. Rosales (MIT, Math. Dept., 2-337) April 12, 2013 Abstract Notes, both complete and/or incomplete, for MIT s 18.311 (Principles of Applied Mathematics). These notes
More informationGiven the water behaves as shown above, which direction will the cylinder rotate?
water stream fixed but free to rotate Given the water behaves as shown above, which direction will the cylinder rotate? ) Clockwise 2) Counter-clockwise 3) Not enough information F y U 0 U F x V=0 V=0
More informationFundamentals of Aerodynamics
Fundamentals of Aerodynamics Fourth Edition John D. Anderson, Jr. Curator of Aerodynamics National Air and Space Museum Smithsonian Institution and Professor Emeritus University of Maryland Me Graw Hill
More informationAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Hierarchy of Mathematical Models 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 2 / 29
More informationThe conservation equations
Chapter 5 The conservation equations 5.1 Leibniz rule for di erentiation of integrals 5.1.1 Di erentiation under the integral sign According to the fundamental theorem of calculus if f is a smooth function
More informationFLUID MECHANICS 3 - LECTURE 4 ONE-DIMENSIONAL UNSTEADY GAS
FLUID MECHANICS 3 - LECTURE 4 ONE-DIMENSIONAL UNSTEADY GAS Consider an unsteady 1-dimensional ideal gas flow. We assume that this flow is spatially continuous and thermally isolated, hence, it is isentropic.
More informationAerothermodynamics of high speed flows
Aerothermodynamics of high speed flows AERO 0033 1 Lecture 2: Flow with discontinuities, normal shocks Thierry Magin, Greg Dimitriadis, and Johan Boutet Thierry.Magin@vki.ac.be Aeronautics and Aerospace
More informationINSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad
INTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, yderabad - 500 043 AERONAUTICAL ENGINEERING COURE DECRIPTION FORM Course Title Course Code Regulation Course tructure Course Coordinator Team
More informationLifting Airfoils in Incompressible Irrotational Flow. AA210b Lecture 3 January 13, AA210b - Fundamentals of Compressible Flow II 1
Lifting Airfoils in Incompressible Irrotational Flow AA21b Lecture 3 January 13, 28 AA21b - Fundamentals of Compressible Flow II 1 Governing Equations For an incompressible fluid, the continuity equation
More informationFUNDAMENTALS OF AERODYNAMICS
*A \ FUNDAMENTALS OF AERODYNAMICS Second Edition John D. Anderson, Jr. Professor of Aerospace Engineering University of Maryland H ' McGraw-Hill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas
More informationContinuity Equation for Compressible Flow
Continuity Equation for Compressible Flow Velocity potential irrotational steady compressible Momentum (Euler) Equation for Compressible Flow Euler's equation isentropic velocity potential equation for
More informationFundamentals of Aerodynamits
Fundamentals of Aerodynamits Fifth Edition in SI Units John D. Anderson, Jr. Curator of Aerodynamics National Air and Space Museum Smithsonian Institution and Professor Emeritus University of Maryland
More informationTo study the motion of a perfect gas, the conservation equations of continuity
Chapter 1 Ideal Gas Flow The Navier-Stokes equations To study the motion of a perfect gas, the conservation equations of continuity ρ + (ρ v = 0, (1.1 momentum ρ D v Dt = p+ τ +ρ f m, (1.2 and energy ρ
More informationIV. Compressible flow of inviscid fluids
IV. Compressible flow of inviscid fluids Governing equations for n = 0, r const: + (u )=0 t u + ( u ) u= p t De e = + ( u ) e= p u+ ( k T ) Dt t p= p(, T ), e=e (,T ) Alternate forms of energy equation
More informationME 6139: High Speed Aerodynamics
Dr. A.B.M. Toufique Hasan Professor Department of Mechanical Engineering, BUET Lecture-01 04 November 2017 teacher.buet.ac.bd/toufiquehasan/ toufiquehasan@me.buet.ac.bd 1 Aerodynamics is the study of dynamics
More informationNotes #4a MAE 533, Fluid Mechanics
Notes #4a MAE 533, Fluid Mechanics S. H. Lam lam@princeton.edu http://www.princeton.edu/ lam October 23, 1998 1 The One-dimensional Continuity Equation The one-dimensional steady flow continuity equation
More informationCompressible Flow. Professor Ugur GUVEN Aerospace Engineer Spacecraft Propulsion Specialist
Compressible Flow Professor Ugur GUVEN Aerospace Engineer Spacecraft Propulsion Specialist What is Compressible Flow? Compressible Flow is a type of flow in which the density can not be treated as constant.
More informationChapter 3 Bernoulli Equation
1 Bernoulli Equation 3.1 Flow Patterns: Streamlines, Pathlines, Streaklines 1) A streamline, is a line that is everywhere tangent to the velocity vector at a given instant. Examples of streamlines around
More informationFlight Vehicle Terminology
Flight Vehicle Terminology 1.0 Axes Systems There are 3 axes systems which can be used in Aeronautics, Aerodynamics & Flight Mechanics: Ground Axes G(x 0, y 0, z 0 ) Body Axes G(x, y, z) Aerodynamic Axes
More informationFluid Mechanics - Course 123 COMPRESSIBLE FLOW
Fluid Mechanics - Course 123 COMPRESSIBLE FLOW Flow of compressible fluids in a p~pe involves not only change of pressure in the downstream direction but also a change of both density of the fluid and
More informationFundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics
Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/
More informationLecture 5.7 Compressible Euler Equations
Lecture 5.7 Compressible Euler Equations Nomenclature Density u, v, w Velocity components p E t H u, v, w e S=c v ln p - c M Pressure Total energy/unit volume Total enthalpy Conserved variables Internal
More informationMULTIGRID CALCULATIONS FOB. CASCADES. Antony Jameson and Feng Liu Princeton University, Princeton, NJ 08544
MULTIGRID CALCULATIONS FOB. CASCADES Antony Jameson and Feng Liu Princeton University, Princeton, NJ 0544 1. Introduction Development of numerical methods for internal flows such as the flow in gas turbines
More informationNAPC Numerical investigation of axisymmetric underexpanded supersonic jets. Pratikkumar Raje. Bijaylakshmi Saikia. Krishnendu Sinha 1
Proceedings of the 1 st National Aerospace Propulsion Conference NAPC-2017 March 15-17, 2017, IIT Kanpur, Kanpur NAPC-2017-139 Numerical investigation of axisymmetric underexpanded supersonic jets Pratikkumar
More information1. Fluid Dynamics Around Airfoils
1. Fluid Dynamics Around Airfoils Two-dimensional flow around a streamlined shape Foces on an airfoil Distribution of pressue coefficient over an airfoil The variation of the lift coefficient with the
More informationUNIT IV BOUNDARY LAYER AND FLOW THROUGH PIPES Definition of boundary layer Thickness and classification Displacement and momentum thickness Development of laminar and turbulent flows in circular pipes
More informationDetailed Outline, M E 320 Fluid Flow, Spring Semester 2015
Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015 I. Introduction (Chapters 1 and 2) A. What is Fluid Mechanics? 1. What is a fluid? 2. What is mechanics? B. Classification of Fluid Flows 1. Viscous
More information2013/5/22. ( + ) ( ) = = momentum outflow rate. ( x) FPressure. 9.3 Nozzles. δ q= heat added into the fluid per unit mass
9.3 Nozzles (b) omentum conservation : (i) Governing Equations Consider: nonadiabatic ternal (body) force ists variable flow area continuously varying flows δq f ternal force per unit volume +d δffdx dx
More informationIntroduction to Aerospace Engineering
4. Basic Fluid (Aero) Dynamics Introduction to Aerospace Engineering Here, we will try and look at a few basic ideas from the complicated field of fluid dynamics. The general area includes studies of incompressible,
More informationComplex Analysis MATH 6300 Fall 2013 Homework 4
Complex Analysis MATH 6300 Fall 2013 Homework 4 Due Wednesday, December 11 at 5 PM Note that to get full credit on any problem in this class, you must solve the problems in an efficient and elegant manner,
More information6.1 Momentum Equation for Frictionless Flow: Euler s Equation The equations of motion for frictionless flow, called Euler s
Chapter 6 INCOMPRESSIBLE INVISCID FLOW All real fluids possess viscosity. However in many flow cases it is reasonable to neglect the effects of viscosity. It is useful to investigate the dynamics of an
More informationViscous flow along a wall
Chapter 8 Viscous flow along a wall 8. The no-slip condition All liquids and gases are viscous and, as a consequence, a fluid near a solid boundary sticks to the boundary. The tendency for a liquid or
More informationIntroduction and Basic Concepts
Topic 1 Introduction and Basic Concepts 1 Flow Past a Circular Cylinder Re = 10,000 and Mach approximately zero Mach = 0.45 Mach = 0.64 Pictures are from An Album of Fluid Motion by Van Dyke Flow Past
More informationTHE ability to estimate quickly the properties of a supersonic
JOURNAL OF PROPULSION AND POWER Vol. 26, No. 3, May June 2010 Reduced-Order Modeling of Two-Dimensional Supersonic Flows with Applications to Scramjet Inlets Derek J. Dalle, Matt L. Fotia, and James F.
More informationDetailed Outline, M E 521: Foundations of Fluid Mechanics I
Detailed Outline, M E 521: Foundations of Fluid Mechanics I I. Introduction and Review A. Notation 1. Vectors 2. Second-order tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic
More informationFluid Mechanics Prof. T. I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay
Fluid Mechanics Prof. T. I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay Lecture No. # 35 Boundary Layer Theory and Applications Welcome back to the video course on fluid
More informationGiven a stream function for a cylinder in a uniform flow with circulation: a) Sketch the flow pattern in terms of streamlines.
Question Given a stream function for a cylinder in a uniform flow with circulation: R Γ r ψ = U r sinθ + ln r π R a) Sketch the flow pattern in terms of streamlines. b) Derive an expression for the angular
More informationAEROSPACE ENGINEERING
AEROSPACE ENGINEERING Subject Code: AE Course Structure Sections/Units Topics Section A Engineering Mathematics Topics (Core) 1 Linear Algebra 2 Calculus 3 Differential Equations 1 Fourier Series Topics
More informationSection 2: Lecture 1 Integral Form of the Conservation Equations for Compressible Flow
Section 2: Lecture 1 Integral Form of the Conservation Equations for Compressible Flow Anderson: Chapter 2 pp. 41-54 1 Equation of State: Section 1 Review p = R g T " > R g = R u M w - R u = 8314.4126
More information4.1 LAWS OF MECHANICS - Review
4.1 LAWS OF MECHANICS - Review Ch4 9 SYSTEM System: Moving Fluid Definitions: System is defined as an arbitrary quantity of mass of fixed identity. Surrounding is everything external to this system. Boundary
More information+ d = d o. "x ) + (d! v2 ) "v. "y = 0. (4.4.2)
4.4 Expansion Fans Compressions: Formal Theory. The ideas presented in the previous section can be formalized using the method of characteristics for steady, 2D, shallow flow. This methodology has been
More informationSupersonic Aerodynamics. Methods and Applications
Supersonic Aerodynamics Methods and Applications Outline Introduction to Supersonic Flow Governing Equations Numerical Methods Aerodynamic Design Applications Introduction to Supersonic Flow What does
More informationThe Riemann problem. The Riemann problem Rarefaction waves and shock waves
The Riemann problem Rarefaction waves and shock waves 1. An illuminating example A Heaviside function as initial datum Solving the Riemann problem for the Hopf equation consists in describing the solutions
More informationThe Hopf equation. The Hopf equation A toy model of fluid mechanics
The Hopf equation A toy model of fluid mechanics 1. Main physical features Mathematical description of a continuous medium At the microscopic level, a fluid is a collection of interacting particles (Van
More informationWaves in a Shock Tube
Waves in a Shock Tube Ivan Christov c February 5, 005 Abstract. This paper discusses linear-wave solutions and simple-wave solutions to the Navier Stokes equations for an inviscid and compressible fluid
More informationAnalytical reconsideration of the von Neumann paradox in the reflection of a shock wave over a wedge
PHYSICS OF FLUIDS 20, 046101 2008 Analytical reconsideration of the von Neumann paradox in the reflection of a shock wave over a wedge Eugene I. Vasilev, 1 Tov Elperin, 2 and Gabi Ben-Dor 2 1 Department
More informationFluid Dynamics Exercises and questions for the course
Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r
More informationIntroduction to Aerodynamics. Dr. Guven Aerospace Engineer (P.hD)
Introduction to Aerodynamics Dr. Guven Aerospace Engineer (P.hD) Aerodynamic Forces All aerodynamic forces are generated wither through pressure distribution or a shear stress distribution on a body. The
More informationCompressible Potential Flow: The Full Potential Equation. Copyright 2009 Narayanan Komerath
Compressible Potential Flow: The Full Potential Equation 1 Introduction Recall that for incompressible flow conditions, velocity is not large enough to cause density changes, so density is known. Thus
More informationn v molecules will pass per unit time through the area from left to
3 iscosity and Heat Conduction in Gas Dynamics Equations of One-Dimensional Gas Flow The dissipative processes - viscosity (internal friction) and heat conduction - are connected with existence of molecular
More informationIn which of the following scenarios is applying the following form of Bernoulli s equation: steady, inviscid, uniform stream of water. Ma = 0.
bernoulli_11 In which of the following scenarios is applying the following form of Bernoulli s equation: p V z constant! g + g + = from point 1 to point valid? a. 1 stagnant column of water steady, inviscid,
More information2 Equations of Motion
2 Equations of Motion system. In this section, we will derive the six full equations of motion in a non-rotating, Cartesian coordinate 2.1 Six equations of motion (non-rotating, Cartesian coordinates)
More informationPreviously, we examined supersonic flow over (sharp) concave corners/turns. What happens if: AE3450
Preiously, we examined supersonic flow oer (sharp) concae corners/turns oblique shock allows flow to make this (compression) turn What happens if: turn is conex (expansion) already shown expansion shock
More informationPropulsion Thermodynamics
Chapter 1 Propulsion Thermodynamics 1.1 Introduction The Figure below shows a cross-section of a Pratt and Whitney JT9D-7 high bypass ratio turbofan engine. The engine is depicted without any inlet, nacelle
More information1. For an ideal gas, internal energy is considered to be a function of only. YOUR ANSWER: Temperature
CHAPTER 11 1. For an ideal gas, internal energy is considered to be a function of only. YOUR ANSWER: Temperature 2.In Equation 11.7 the subscript p on the partial derivative refers to differentiation at
More informationFluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition
Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition C. Pozrikidis m Springer Contents Preface v 1 Introduction to Kinematics 1 1.1 Fluids and solids 1 1.2 Fluid parcels and flow
More informationIntroduction to Aerospace Engineering
Introduction to Aerospace Engineering Lecture slides Challenge the future 3-0-0 Introduction to Aerospace Engineering Aerodynamics 5 & 6 Prof. H. Bijl ir. N. Timmer Delft University of Technology 5. Compressibility
More informationAEROSPACE ENGINEERING DEPARTMENT. Second Year - Second Term ( ) Fluid Mechanics & Gas Dynamics
AEROSPACE ENGINEERING DEPARTMENT Second Year - Second Term (2008-2009) Fluid Mechanics & Gas Dynamics Similitude,Dimensional Analysis &Modeling (1) [7.2R*] Some common variables in fluid mechanics include:
More informationFundamentals of Gas Dynamics (NOC16 - ME05) Assignment - 8 : Solutions
Fundamentals of Gas Dynamics (NOC16 - ME05) Assignment - 8 : Solutions Manjul Sharma & Aswathy Nair K. Department of Aerospace Engineering IIT Madras April 5, 016 (Note : The solutions discussed below
More informationGasdynamics 1-D compressible, inviscid, stationary, adiabatic flows
Gasdynamics 1-D compressible, inviscid, stationary, adiabatic flows 1st law of thermodynamics ρ const Kontrollfläche 1 2 m u 2 u 1 z Q 12 +P 12 = ṁ } h 2 h {{} 1 Enthalpy Q 12 + 1 2 (u2 2 u2 1 }{{} ) +
More informationApplied Gas Dynamics Flow With Friction and Heat Transfer
Applied Gas Dynamics Flow With Friction and Heat Transfer Ethirajan Rathakrishnan Applied Gas Dynamics, John Wiley & Sons (Asia) Pte Ltd c 2010 Ethirajan Rathakrishnan 1 / 121 Introduction So far, we have
More informationAnswers to Problem Set Number 04 for MIT (Spring 2008)
Answers to Problem Set Number 04 for 18.311 MIT (Spring 008) Rodolfo R. Rosales (MIT, Math. Dept., room -337, Cambridge, MA 0139). March 17, 008. Course TA: Timothy Nguyen, MIT, Dept. of Mathematics, Cambridge,
More informationAA210A Fundamentals of Compressible Flow. Chapter 5 -The conservation equations
AA210A Fundamentals of Compressible Flow Chapter 5 -The conservation equations 1 5.1 Leibniz rule for differentiation of integrals Differentiation under the integral sign. According to the fundamental
More informationBoundary-Layer Theory
Hermann Schlichting Klaus Gersten Boundary-Layer Theory With contributions from Egon Krause and Herbert Oertel Jr. Translated by Katherine Mayes 8th Revised and Enlarged Edition With 287 Figures and 22
More informationTopics in Fluid Dynamics: Classical physics and recent mathematics
Topics in Fluid Dynamics: Classical physics and recent mathematics Toan T. Nguyen 1,2 Penn State University Graduate Student Seminar @ PSU Jan 18th, 2018 1 Homepage: http://math.psu.edu/nguyen 2 Math blog:
More informationFundamentals of compressible and viscous flow analysis - Part II
Fundamentals of compressible and viscous flow analysis - Part II Lectures 3, 4, 5 Instantaneous and averaged temperature contours in a shock-boundary layer interaction. Taken from (Pasquariello et al.,
More informationFundamentals of Gas Dynamics (NOC16 - ME05) Assignment - 10 : Solutions
Fundamentals of Gas Dynamics (NOC16 - ME05) Assignment - 10 : Solutions Manjul Sharma & Aswathy Nair K. Department of Aerospace Engineering IIT Madras April 18, 016 (Note : The solutions discussed below
More informationAA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow
AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 1 1.2 Conservation of mass Mass flux in the x-direction [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit
More informationThe Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III)
Calculus Vector Calculus (III) Outline 1 The Divergence Theorem 2 Stokes Theorem 3 Applications of Vector Calculus The Divergence Theorem (I) Recall that at the end of section 12.5, we had rewritten Green
More informationChapter 17. For the most part, we have limited our consideration so COMPRESSIBLE FLOW. Objectives
Chapter 17 COMPRESSIBLE FLOW For the most part, we have limited our consideration so far to flows for which density variations and thus compressibility effects are negligible. In this chapter we lift this
More information0.3.4 Burgers Equation and Nonlinear Wave
16 CONTENTS Solution to step (discontinuity) initial condition u(x, 0) = ul if X < 0 u r if X > 0, (80) u(x, t) = u L + (u L u R ) ( 1 1 π X 4νt e Y 2 dy ) (81) 0.3.4 Burgers Equation and Nonlinear Wave
More informationNumerical Investigation of Supersonic Nozzle Producing Maximum Thrust For Altitude Variation
Numerical Investigation of Supersonic Nozzle Producing Maximum Thrust For Altitude Variation Muhammad Misbah-Ul Islam 1, Mohammad Mashud 1, Md. Hasan Ali 2 and Abdullah Al Bari 1 1 Department of Mechanical
More informationShock Waves. 1 Steepening of sound waves. We have the result that the velocity of a sound wave in an arbitrary reference frame is given by: kˆ.
Shock Waves Steepening of sound waves We have the result that the velocity of a sound wave in an arbitrary reference frame is given by: v u kˆ c s kˆ where u is the velocity of the fluid and k is the wave
More informationIntroduction to Gas Dynamics All Lecture Slides
Introduction to Gas Dynamics All Lecture Slides Teknillinen Korkeakoulu / Helsinki University of Technology Autumn 009 1 Compressible flow Zeroth law of thermodynamics 3 First law of thermodynamics 4 Equation
More informationAerodynamics. Lecture 1: Introduction - Equations of Motion G. Dimitriadis
Aerodynamics Lecture 1: Introduction - Equations of Motion G. Dimitriadis Definition Aerodynamics is the science that analyses the flow of air around solid bodies The basis of aerodynamics is fluid dynamics
More informationWilliam В. Brower, Jr. A PRIMER IN FLUID MECHANICS. Dynamics of Flows in One Space Dimension. CRC Press Boca Raton London New York Washington, D.C.
William В. Brower, Jr. A PRIMER IN FLUID MECHANICS Dynamics of Flows in One Space Dimension CRC Press Boca Raton London New York Washington, D.C. Table of Contents Chapter 1 Fluid Properties Kinetic Theory
More informationDEVELOPMENT OF A COMPRESSED CARBON DIOXIDE PROPULSION UNIT FOR NEAR-TERM MARS SURFACE APPLICATIONS
DEVELOPMENT OF A COMPRESSED CARBON DIOXIDE PROPULSION UNIT FOR NEAR-TERM MARS SURFACE APPLICATIONS Erin Blass Old Dominion University Advisor: Dr. Robert Ash Abstract This work has focused on the development
More informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:
More informationRocket Propulsion Prof. K. Ramamurthi Department of Mechanical Engineering Indian Institute of Technology, Madras
Rocket Propulsion Prof. K. Ramamurthi Department of Mechanical Engineering Indian Institute of Technology, Madras Lecture 11 Area Ratio of Nozzles: Under Expansion and Over Expansion (Refer Slide Time:
More informationMAS. Modelling, Analysis and Simulation. Modelling, Analysis and Simulation. Building your own shock tube. J. Naber
C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and Simulation Building your own shock tube J. Naber REPORT MAS-E5 JANUARY 5 CWI
More informationContinuum Mechanics Lecture 5 Ideal fluids
Continuum Mechanics Lecture 5 Ideal fluids Prof. http://www.itp.uzh.ch/~teyssier Outline - Helmholtz decomposition - Divergence and curl theorem - Kelvin s circulation theorem - The vorticity equation
More informationEntropy generation and transport
Chapter 7 Entropy generation and transport 7.1 Convective form of the Gibbs equation In this chapter we will address two questions. 1) How is Gibbs equation related to the energy conservation equation?
More informationVORTEX LATTICE METHODS FOR HYDROFOILS
VORTEX LATTICE METHODS FOR HYDROFOILS Alistair Fitt and Neville Fowkes Study group participants A. Fitt, N. Fowkes, D.P. Mason, Eric Newby, E. Eneyew, P, Ngabonziza Industry representative Gerrie Thiart
More informationOutlines. simple relations of fluid dynamics Boundary layer analysis. Important for basic understanding of convection heat transfer
Forced Convection Outlines To examine the methods of calculating convection heat transfer (particularly, the ways of predicting the value of convection heat transfer coefficient, h) Convection heat transfer
More informationWings and Bodies in Compressible Flows
Wings and Bodies in Compressible Flows Prandtl-Glauert-Goethert Transformation Potential equation: 1 If we choose and Laplace eqn. The transformation has stretched the x co-ordinate by 2 Values of at corresponding
More information