AA210A Fundamentals of Compressible Flow. Chapter 13 - Unsteady Waves in Compressible Flow
|
|
- Basil Casey
- 5 years ago
- Views:
Transcription
1 AA210A Fundamentals of Compressible Flow Chapter 13 - Unsteady Waves in Compressible Flow
2 The Shock Tube - Wave Diagram
3
4 13.1 Equations for irrotational, homentropic, unsteady flow ρ t + x k ρ U i t ( ρu k ) = 0 + ρ x i P = ρ P 0 ρ 0 γ U k U k 2 + P = 0 x i
5 13.2 The acoustic equations Consider the case where fluctuations in flow variables are very small compared to the mean. There is no mean velocity and the velocity disturbance is small. Linearize the equations
6 Define the condensation The equations of motion in terms of the condensation are.
7 Replace the pressure with the density. where
8 Differentiate both equations. Subtract one from the other The condensation satisfies the linear wave equation.
9 Consider the velocity disturbance. Again, subtract one from the other Use the identity
10 Similarly, all other variables of the flow satisfy the linear wave equation.
11 13.3 Propagation of acoustic waves in one space dimension In one dimension the acoustic equations are The wave equation Can be solved in the general form where F and G are arbitrary functions
12 Right propagating F waves Left propagating G waves
13 The pressure disturbance generated by the wave is In differential form
14 The particle velocity induced by an acoustic disturbance can also be written in a very general form Substitute the expressions for the condensation and the velocity into the acoustic equations.
15 Let The equations for r and U become Add and subtract these relations. The result is From which we can conclude
16 This result gives us the relationship between density and particle velocity in left and right running waves. Now In a right running wave In a left running wave In differential form Recall
17 13.4 Isentropic, finite amplitude waves In a general 1-D isentropic flow Locally, the velocity disturbance can be assumed to be Integrate this result from an initial to a final state beginning at U 1 =0 Thus
18 The local acoustic speed is The wave speed at any point is
19 We can put this result on a somewhat more rigorous foundation for a finite amplitude wave as follows. The equations for 1-D isentropic unsteady flow are: Assume The derivatives of the density are
20 Substitute the derivatives of the density into the 1-D equations. Rearrange The continuity and momentum equations become identical except for the coefficient of the last term. To have a solution the coefficients must be equal.
21 Rearrange Take the square root Substitute Result
22 13.5 Centered expansion wave Consider a piston withdrawn from a compressible fluid at rest. The speed of the fluid in region three is equal to the piston speed. Note that the piston speed is negative. In effect, this gives us the sound speed in region 3.
23 The leading characteristic propagates to the right with the speed of sound in region 4, the undisturbed gas. The tail of the disturbance moves to the right with wave speed
24 The density ratio across the wave is given by the isentropic relation. or and the pressure ratio is
25 13.6 Compression wave Suppose the piston motion is into the gas. x=c 2 t x=c 1 t The wave speed at the surface of the piston is Once the compression waves catch up the isentropic assumption is no longer valid and there is the formation of a shock wave.
26 13.7 The shock tube
27 The conditions at the contact surface are In a frame of reference moving with the shock The shock jump conditions give
28 The shock Mach number can be written in terms of the piston speed. where U p is positive. This can be written in terms of the shock pressure ratio.
29 The velocity behind the expansion is Note that Use the conditions at the contact surface. and
30 The result is the basic shock tube equation
31 The shock Mach number is determined from or
32
33
34
35 Taylor Wave This is an expansion wave that occurs behind a detonation wave. Recall from Chapter 11 the properties of a Chapman-Jouget wave can be determined from the heat addition jump conditions. Recall the ZND model in a frame of reference moving with the detonation.
36 Detonation Wave Mach number Subsonic and supersonic roots. We assume gamma does not change through the wave Pressure, temperature and density directly behind the wave come from Rayleigh theory for thermally choked heat addition.
37 Suppose a detonation is initiated in a tube filled with a fuel/oxidizer mixture. t < 0 Δh / C p T 1 (1) P 1,T 1,ρ 1 Ignition t = 0 (1) (2) t > 0 (3) c detonation U 3 = 0 U 1 = 0 (1) P 2 = P Chapman Jouget U p c 3 = a 3 t P P 3 t (3) Taylor wave (2) P 1 x (1) C-J detonation x
38 Properties of the Taylor wave connecting regions 2 and 3 (2) U 3 = 0 U 1 = 0 (3) c detonation (1) U ' 2 = c detonation +U p = a 2 U p = c detonation a 2 = M 1 a 1 a 2 U ' 1 = c detonation Velocities in a frame of reference moving with the detonation U ' 1 = ( 1+ γ )M U ' 2 1+ γ M 1 ( ) = ρ 2 2 ( ) ρ 1 T 2 ( ) 2 = 1+ γ M 2 1 T 1 1+ γ ( ) 2 M ( ) a 2 = 1+ γ M 1 P 2 = 1+ γ M a 1 ( 1+ γ )M 1 P 1 1+ γ a 3 = a 2 γ 1 2 U p ρ 3 = a γ 1 3 γ 1 ρ 2 = 1 2 U p a 2 P 3 = 1 γ 1 P 2 2 U p 2 a 2 2γ γ 1 a 2 ( ) 2 γ 1 Pressure and temperature in region 2 just behind the detonation wave comes from steady flow Rayleigh line theory. Pressure and temperature in region 3 comes from unsteady expansion wave and isentropic flow theory.
39 Example - mixture of 5% hydrogen and 5%fluorine in 90% nitrogen. Behind the Taylor wave expansion. U p = 700 m / sec a 2 = 1050 m / sec γ = 1.4 a 3 = a 2 γ 1 2 U p = = 910 m / sec ρ 3 = a γ 1 3 γ 1 ρ 2 = 1 2 U p γ = a 2 2 P 3 = 1 γ 1 P 2 2 U p γ = a 2 2γ a = = T 3 T 2 = P 3 P 2 ρ 2 ρ 1 = 0.751
40 Properties of the reflected shock Incident shock c s > 0 P 4 = f γ 1,γ 4, a 1, P 2 P 1 a 4 P 1 +x U 2 = U p ( 2) c s ( 1) U = 0 M s = c s a 1 = M 1 M 1 = M s = γ γ 1 1/2 P 2 + γ 1 1 P 1 γ /2 M 2 U p = a γ M 1 a 2 a 1 = 1/2 γ 1 M 2 1 γ γ γ M 1 M 1 2 1/2 P 4 P 1 P 2 P 1 M 1 U p, a 2 a 1 c rs < 0 Reflected shock +x U 2 = U p ( 2) c rs ( 5) U = 0 M rs = c rs +U p a 2 = M 2 M 2 U p = a γ M 2 Take the positive root M 2 2 U p γ 1 +1 a 2 2 M 2 1 = 0 U p, a 2 a 1 M 2 a 5 a 2, P 5 P 2 a 5 a 2 = 1/2 γ 1 M 2 2 γ γ γ M 2 2γ 1 P 5 γ = 1 1 M P 2 γ 1 +1 γ 1 1 M 2 2 1/2
41 Relected shock with a high Mach number incident shock Take the limit Take the positive root M 2 2 U p γ 1 +1 a 2 2 M 2 1 = 0 M γ 1 +1 γ M 1 2 M 2 2 γ 1 M 2 1 γ 1/ γ γ M 1 lim M 1 M 2 2 γ 1 γ ( ) 1/2 γ M 1 2 1/2 M 1 = 0 1/2 2 M 2 1 = 0 U p, a 2 a 1 M 2 a 5 a 2, P 5 P 2 M 2 2 M 2 = γ 1 γ ( ) 1/2 γ γ 1 a 5 a 2 = 2 γ ( ) 1/2 γ 1 1 2γ M 2 = 1 γ 1 1 1/2 2 M 1 = 0 1/2 2 1/2 γ 1 M 2 2 γ γ γ M 2 2γ 1 P 5 γ = 1 1 M P 2 γ 1 +1 γ 1 1 ( ) 2 + 8( γ 1 )( γ 1 1) ( )( γ 1 1) + 1 γ /2 2 2 γ 1 M 2 2 1/2 1/2
42 Problem 13.5
43 From a previous final exam
44
45
46
47 13.8 Problems
48
49
50
51
52
53
54
55
Introduction 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 informationGAS DYNAMICS. M. Halük Aksel. O. Cahit Eralp. and. Middle East Technical University Ankara, Turkey
GAS DYNAMICS M. Halük Aksel and O. Cahit Eralp Middle East Technical University Ankara, Turkey PRENTICE HALL f r \ New York London Toronto Sydney Tokyo Singapore; \ Contents Preface xi Nomenclature xiii
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 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 informationUnsteady Wave Motion Shock Tube Problem - Shock Reflection
Unsteady Wave Motion Shock Tube Problem - Shock Reflection Niklas Andersson Division of Fluid Dynamics Department of Applied Mechanics Chalmers University of Tecnology 8 februari 09 Shock Reflection When
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 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 informationTHERMODYNAMIC ANALYSIS OF COMBUSTION PROCESSES FOR PROPULSION SYSTEMS
2nd AIAA Aerospace Sciences Paper 2-33 Meeting and Exhibit January -8, 2, Reno, NV THERMODYNAMIC ANALYSIS OF COMBUSTION PROCESSES FOR PROPULSION SYSTEMS E. Wintenberger and J. E. Shepherd Graduate Aeronautical
More informationLecture 7 Detonation Waves
Lecture 7 etonation Waves p strong detonation weak detonation weak deflagration strong deflagration / 0 v =/ University of Illinois at Urbana- Champaign eflagrations produce heat Thermal di usivity th
More informationPHYS 643 Week 4: Compressible fluids Sound waves and shocks
PHYS 643 Week 4: Compressible fluids Sound waves and shocks Sound waves Compressions in a gas propagate as sound waves. The simplest case to consider is a gas at uniform density and at rest. Small perturbations
More informationA MODEL FOR THE PERFORMANCE OF AIR-BREATHING PULSE DETONATION ENGINES
A MODEL FOR THE PERFORMANCE OF AIR-BREATHING PULSE DETONATION ENGINES E. Wintenberger and J.E. Shepherd Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125 An analytical
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 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 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 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 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 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 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 informationModel for the Performance of Airbreathing Pulse-Detonation Engines
JOURNAL OF PROPULSION AND POWER Vol. 22, No. 3, May June 2006 Model for the Performance of Airbreathing Pulse-Detonation Engines E. Wintenberger and J. E. Shepherd California Institute of Technology, Pasadena,
More informationApplication of a Laser Induced Fluorescence Model to the Numerical Simulation of Detonation Waves in Hydrogen-Oxygen-Diluent Mixtures
Supplemental material for paper published in the International J of Hydrogen Energy, Vol. 30, 6044-6060, 2014. http://dx.doi.org/10.1016/j.ijhydene.2014.01.182 Application of a Laser Induced Fluorescence
More informationDetonation of Gas Particle Flow
2 Detonation of Gas Particle Flow F. Zhang 2.1 Introduction Fine organic or metallic particles suspended in an oxidizing or combustible gas form a reactive particle gas mixture. Explosion pressures in
More informationIsentropic Duct Flows
An Internet Book on Fluid Dynamics Isentropic Duct Flows In this section we examine the behavior of isentropic flows, continuing the development of the relations in section (Bob). First it is important
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 informationGravity Waves Gravity Waves
Gravity Waves Gravity Waves 1 Gravity Waves Gravity Waves Kayak Surfing on ocean gravity waves Oregon Coast Waves: sea & ocean waves 3 Sound Waves Sound Waves: 4 Sound Waves Sound Waves Linear Waves compression
More informationTHE INTERACTION OF TURBULENCE WITH THE HELIOSPHERIC SHOCK
THE INTERACTION OF TURBULENCE WITH THE HELIOSPHERIC SHOCK G.P. Zank, I. Kryukov, N. Pogorelov, S. Borovikov, Dastgeer Shaikh, and X. Ao CSPAR, University of Alabama in Huntsville Heliospheric observations
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 information1. Introduction Some Basic Concepts
1. Introduction Some Basic Concepts 1.What is a fluid? A substance that will go on deforming in the presence of a deforming force, however small 2. What Properties Do Fluids Have? Density ( ) Pressure
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 informationFundamentals of Rotating Detonation. Toshi Fujiwara (Nagoya University)
Fundamentals of Rotating Detonation Toshi Fujiwara (Nagoya University) New experimental results Cylindical channel D=140/150mm Hydrogen air; p o =1.0bar Professor Piotr Wolanski P [bar] 10 9 8 7 6 5 4
More informationSPC 407 Sheet 5 - Solution Compressible Flow Rayleigh Flow
SPC 407 Sheet 5 - Solution Compressible Flow Rayleigh Flow 1. Consider subsonic Rayleigh flow of air with a Mach number of 0.92. Heat is now transferred to the fluid and the Mach number increases to 0.95.
More informationNotes #6 MAE 533, Fluid Mechanics
Notes #6 MAE 533, Fluid Mechanics S. H. Lam lam@princeton.edu http://www.princeton.edu/ lam October 1, 1998 1 Different Ways of Representing T The speed of sound, a, is formally defined as ( p/ ρ) s. It
More informationRichard Nakka's Experimental Rocketry Web Site
Página 1 de 7 Richard Nakka's Experimental Rocketry Web Site Solid Rocket Motor Theory -- Nozzle Theory Nozzle Theory The rocket nozzle can surely be described as the epitome of elegant simplicity. The
More informationPreparation of the concerned sectors for educational and R&D activities related to the Hungarian ELI project
Preparation of the concerned sectors for educational and R&D activities related to the Hungarian ELI project Ion acceleration in plasmas Lecture 10. Collisionless shock wave acceleration in plasmas pas
More informationDriver-gas Tailoring For Test-time Extension Using Unconventional Driver Mixtures
University of Central Florida Electronic Theses and Dissertations Masters Thesis (Open Access) Driver-gas Tailoring For Test-time Extension Using Unconventional Driver Mixtures 006 Anthony Amadio University
More informationDetonation Diffraction
Detonation Diffraction E. Schultz, J. Shepherd Detonation Physics Laboratory Pasadena, CA 91125 MURI Mid-Year Pulse Detonation Engine Review Meeting February 10-11, 2000 Super-critical Detonation Diffraction
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 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 informationAn analytical model for direct initiation of gaseous detonations
Issw21 An analytical model for direct initiation of gaseous detonations C.A. Eckett, J.J. Quirk, J.E. Shepherd Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena CA 91125,
More informationApplication of Steady and Unsteady Detonation Waves to Propulsion
Application of Steady and Unsteady Detonation Waves to Propulsion Thesis by Eric Wintenberger In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology
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 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 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 information4 Compressible Fluid Dynamics
4 Compressible Fluid Dynamics 4. Compressible flow definitions Compressible flow describes the behaviour of fluids that experience significant variations in density under the application of external pressures.
More informationSound Waves Sound Waves:
Sound Waves Sound Waves: 1 Sound Waves Sound Waves Linear Waves compression rarefaction 2 H H L L L Gravity Waves 3 Gravity Waves Gravity Waves 4 Gravity Waves Kayak Surfing on ocean gravity waves Oregon
More informationUNIVERSITY OF CALGARY. Stability of Detonation Wave under Chain Branching Kinetics: Role of the Initiation Step. Michael Anthony Levy A THESIS
UNIVERSITY OF CALGARY Stability of Detonation Wave under Chain Branching Kinetics: Role of the Initiation Step by Michael Anthony Levy A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT
More informationAME 513. " Lecture 8 Premixed flames I: Propagation rates
AME 53 Principles of Combustion " Lecture 8 Premixed flames I: Propagation rates Outline" Rankine-Hugoniot relations Hugoniot curves Rayleigh lines Families of solutions Detonations Chapman-Jouget Others
More informationthe pitot static measurement equal to a constant C which is to take into account the effect of viscosity and so on.
Mechanical Measurements and Metrology Prof. S. P. Venkateshan Department of Mechanical Engineering Indian Institute of Technology, Madras Module -2 Lecture - 27 Measurement of Fluid Velocity We have been
More informationLecture-2. One-dimensional Compressible Fluid Flow in Variable Area
Lecture-2 One-dimensional Compressible Fluid Flow in Variable Area Summary of Results(Cont..) In isoenergetic-isentropic flow, an increase in velocity always corresponds to a Mach number increase and vice
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 informationDevelpment of NSCBC for compressible Navier-Stokes equations in OpenFOAM : Subsonic Non-Reflecting Outflow
Develpment of NSCBC for compressible Navier-Stokes equations in OpenFOAM : Subsonic Non-Reflecting Outflow F. Piscaglia, A. Montorfano Dipartimento di Energia, POLITECNICO DI MILANO Content Introduction
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 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 informationGas Dynamics: Basic Equations, Waves and Shocks
Astrophysical Dynamics, VT 010 Gas Dynamics: Basic Equations, Waves and Shocks Susanne Höfner Susanne.Hoefner@fysast.uu.se Astrophysical Dynamics, VT 010 Gas Dynamics: Basic Equations, Waves and Shocks
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 informationAn analytical model for the impulse of a single-cycle pulse detonation tube
Preprint, see final version in Journal of Propulsion and Power, 19(1):22-38, 23. http://dx.doi.org/1.2514/2.699. See also the erratum (JPP 2(4) 765-767, 24) and responses to comments by Heiser and Pratt
More informationSUPERSONIC WIND TUNNEL Project One. Charles R. O Neill School of Mechanical and Aerospace Engineering Oklahoma State University Stillwater, OK 74078
41 SUPERSONIC WIND UNNEL Project One Charles R. O Neill School of Mechanical and Aerospace Engineering Oklahoma State University Stillwater, OK 74078 Project One in MAE 3293 Compressible Flow September
More informationAerothermodynamics 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 informationWebster s horn model on Bernoulli flow
Webster s horn model on Bernoulli flow Aalto University, Dept. Mathematics and Systems Analysis January 5th, 2018 Incompressible, steady Bernoulli principle Consider a straight tube Ω R 3 havin circular
More informationRocket Thermodynamics
Rocket Thermodynamics PROFESSOR CHRIS CHATWIN LECTURE FOR SATELLITE AND SPACE SYSTEMS MSC UNIVERSITY OF SUSSEX SCHOOL OF ENGINEERING & INFORMATICS 25 TH APRIL 2017 Thermodynamics of Chemical Rockets ΣForce
More informationOPAC102. The Acoustic Wave Equation
OPAC102 The Acoustic Wave Equation Acoustic waves in fluid Acoustic waves constitute one kind of pressure fluctuation that can exist in a compressible fluid. The restoring forces responsible for propagating
More informationPlease welcome for any correction or misprint in the entire manuscript and your valuable suggestions kindly mail us
Problems of Practices Of Fluid Mechanics Compressible Fluid Flow Prepared By Brij Bhooshan Asst. Professor B. S. A. College of Engg. And Technology Mathura, Uttar Pradesh, (India) Supported By: Purvi Bhooshan
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 informationChapter 1. Introduction to Nonlinear Space Plasma Physics
Chapter 1. Introduction to Nonlinear Space Plasma Physics The goal of this course, Nonlinear Space Plasma Physics, is to explore the formation, evolution, propagation, and characteristics of the large
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 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 informationEnergy Budget of Cosmological First Order Phase Transitions
Energy Budget Cosmological First Order Phase Transitions Jose Miguel No King's College London Making the EW Phase Transition (Theoretically) Strong Motivation: Why Bubbles? Cosmological First Order Phase
More informationAeroelasticity. Lecture 9: Supersonic Aeroelasticity. G. Dimitriadis. AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 9
Aeroelasticity Lecture 9: Supersonic Aeroelasticity G. Dimitriadis AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 9 1 Introduction All the material presented up to now concerned incompressible
More informationImpulse of a Single-Pulse Detonation Tube
Impulse of a Single-Pulse Detonation Tube E. Wintenberger, J.M. Austin, M. Cooper, S. Jackson, and J.E. Shepherd Graduate Aeronautical Laboratories California Institute of Technology Pasadena, CA 91125
More informationTutorial Materials for ME 131B Fluid Mechanics (Compressible Flow & Turbomachinery) Calvin Lui Department of Mechanical Engineering Stanford University Stanford, CA 94305 March 1998 Acknowledgments This
More informationAcoustic Energy Estimates in Inhomogeneous Moving Media
Subject: FORUM ACUSTICUM 1999 Abstract Acoustic Energy Estimates in Inhomogeneous Moving Media, NASA Langley Research Center, Hampton, Virginia Mark Farris, Midwestern State University, Wichita Falls,
More informationRadiative & Magnetohydrodynamic Shocks
Chapter 4 Radiative & Magnetohydrodynamic Shocks I have been dealing, so far, with non-radiative shocks. Since, as we have seen, a shock raises the density and temperature of the gas, it is quite likely,
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 informationContents. I Introduction 1. Preface. xiii
Contents Preface xiii I Introduction 1 1 Continuous matter 3 1.1 Molecules................................ 4 1.2 The continuum approximation.................... 6 1.3 Newtonian mechanics.........................
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 informationThe Euler Equations! Advection! λ 1. λ 2. λ 3. ρ ρu. c 2 = γp/ρ. E = e + u 2 /2 H = h + u 2 /2; h = e + p/ρ. 0 u 1/ρ. u p. t + A f.
http://www.nd.edu/~gtryggva/cfd-course/! Advection! Grétar Tryggvason! Spring! The Euler equations for D flow:! where! Define! Ideal Gas:! ρ ρu ρu + ρu + p = x ( / ) ρe ρu E + p ρ E = e + u / H = h + u
More informationPEMP ACD2505. M.S. Ramaiah School of Advanced Studies, Bengaluru
Governing Equations of Fluid Flow Session delivered by: M. Sivapragasam 1 Session Objectives -- At the end of this session the delegate would have understood The principle of conservation laws Different
More informationFUNDAMENTALS OF GAS DYNAMICS
FUNDAMENTALS OF GAS DYNAMICS Second Edition ROBERT D. ZUCKER OSCAR BIBLARZ Department of Aeronautics and Astronautics Naval Postgraduate School Monterey, California JOHN WILEY & SONS, INC. Contents PREFACE
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 informationChemical inhibiting of hydrogen-air detonations Sergey M. Frolov
Chemical inhibiting of hydrogen-air detonations Sergey M. Frolov Semenov Institute of Chemical Physics Moscow, Russia Outline Introduction Theoretical studies Classical 1D approach (ZND-model) Detailed
More informationDrag of a thin wing and optimal shape to minimize it
Drag of a thin wing and optimal shape to minimize it Alejandro Pozo December 21 st, 211 Outline 1 Statement of the problem 2 Inviscid compressible flows 3 Drag for supersonic case 4 Example of optimal
More informationDETONATION DIFFRACTION THROUGH AN ABRUPT AREA EXPANSION. Thesis by. Eric Schultz. In Partial Fulfillment of the Requirements.
DETONATION DIFFRACTION THROUGH AN ABRUPT AREA EXPANSION Thesis by Eric Schultz In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena,
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 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 informationSteady Deflagration Structure in Two-Phase Granular Propellants
12th ICDERS, Ann Arbor, Michigan July 23-28, 1989 Steady Deflagration Structure in Two-Phase Granular Propellants Joseph M. Powers 1, Mark E. Miller2, D. Scott Stewart3, and Herman Krier4 July 23-28, 1989
More informationPARAMETRIC AND PERFORMANCE ANALYSIS OF A HYBRID PULSE DETONATION/TURBOFAN ENGINE SIVARAI AMITH KUMAR
PARAMETRIC AND PERFORMANCE ANALYSIS OF A HYBRID PULSE DETONATION/TURBOFAN ENGINE By SIVARAI AMITH KUMAR Presented to the Faculty of the Graduate School of The University of Texas at Arlington in Partial
More informationIntroduction to Physical Acoustics
Introduction to Physical Acoustics Class webpage CMSC 828D: Algorithms and systems for capture and playback of spatial audio. www.umiacs.umd.edu/~ramani/cmsc828d_audio Send me a test email message with
More informationSPC 407 Sheet 2 - Solution Compressible Flow - Governing Equations
SPC 407 Sheet 2 - Solution Compressible Flow - Governing Equations 1. Is it possible to accelerate a gas to a supersonic velocity in a converging nozzle? Explain. No, it is not possible. The only way to
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 informationReview of Fundamentals - Fluid Mechanics
Review of Fundamentals - Fluid Mechanics Introduction Properties of Compressible Fluid Flow Basics of One-Dimensional Gas Dynamics Nozzle Operating Characteristics Characteristics of Shock Wave A gas turbine
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 informationFundamentals of Fluid Dynamics: Waves in Fluids
Fundamentals of Fluid Dynamics: Waves in Fluids Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/ tzielins/ Institute
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 informationThe Stochastic Piston Problem
The Stochastic Piston Problem G. Lin, C.-H. Su and G.E. Karniadakis Division of Applied Mathematics 18 George Street Brown University Providence, RI 91 Classification: Physical Sciences: Applied Mathematics
More informationIntroduction to Chemical Engineering Thermodynamics. Chapter 7. KFUPM Housam Binous CHE 303
Introduction to Chemical Engineering Thermodynamics Chapter 7 1 Thermodynamics of flow is based on mass, energy and entropy balances Fluid mechanics encompasses the above balances and conservation of momentum
More informationLongitudinal Waves. waves in which the particle or oscillator motion is in the same direction as the wave propagation
Longitudinal Waves waves in which the particle or oscillator motion is in the same direction as the wave propagation Longitudinal waves propagate as sound waves in all phases of matter, plasmas, gases,
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 informationInfluence of Molecular Complexity on Nozzle Design for an Organic Vapor Wind Tunnel
ORC 2011 First International Seminar on ORC Power Systems, Delft, NL, 22-23 September 2011 Influence of Molecular Complexity on Nozzle Design for an Organic Vapor Wind Tunnel A. Guardone, Aerospace Eng.
More informationSimulation of Condensing Compressible Flows
Simulation of Condensing Compressible Flows Maximilian Wendenburg Outline Physical Aspects Transonic Flows and Experiments Condensation Fundamentals Practical Effects Modeling and Simulation Equations,
More informationShock Wave Boundary Layer Interaction from Reflecting Detonations
Shock Wave Boundary Layer Interaction from Reflecting Detonations J. Damazo 1, J. Ziegler 1, J. Karnesky 2, and J. E. Shepherd 1 1 Introduction The present work is concerned with the differences in how
More information