Steady waves in compressible flow
|
|
- Benjamin Maxwell
- 6 years ago
- Views:
Transcription
1 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 shock wave. We can think of the deflection as caused by a planar ramp at this angle although it could be generated by the blockage produced by a solid body placed some distance away in the flow. In general, a 3-D shock wave will be curved, and will separate two regions of non-uniform flow. However, at each point along the shock, the change in flow properties takes place in a very thin region much thinner than the radius of curvature of the shock. If we consider a small neighborhood of the point in question then within the small neighborhood, the shock may be regarded as locally planar to any required level of accuracy and the flows on either -
2 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW - side can be regarded as uniform. With the proper orientation of axes the flow is locally two-dimensional. Therefore it is quite general to consider a straight oblique shock wave in a uniform parallel stream in two-dimensions as shown below. Balancing mass, two components of momentum and energy across the indicated control volume leads to the following oblique shock jump conditions. u = u P + u = P + u u v = u v (.) h + u + v = h + u + v Since u is constant, v = v and the jump conditions become u = u P + u = P + u v = v (.) h + u = h + u. When the ideal gas law P = RT is included, the system of equations (.) closes allowing all the properties of the shock to be determined. Note that, with the exception of the additional equation, v = v, the system is identical to the normal shock jump conditions. The oblique shock acts like a normal shock to the flow perpendicular to it. Therefore almost all of the normal shock relations can be converted to oblique shock relations with the substitution M! M Sin( ) M! M Sin( ). (.3) Recall the Rankine-Hugoniot relation
3 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -3 = P P + + P P (.4) P P + P P plotted in figure.. Figure.: Plot of the Hugoniot relation (.4) This shows the close relationship between the pressure rise across the wave (oblique or normal) and the associated density rise. The jump conditions for oblique shocks lead to a modified form of the very useful Prandtl relation u u =(a ) v (.5) + where (a ) = RT. From the conservation of total enthalpy, for a calorically perfect gas in steady adiabatic flow C p T t = C p T + U = a + U = + ( ) (a ). (.6) The Prandtl relation is extremely useful for easily deriving all the various normal and oblique shock relations. The oblique shock relations generated using (.3) are
4 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -4 P = M Sin ( ) ( ) P + = u = ( + ) M Sin ( ) u ( ) M Sin ( )+ T = M Sin ( ) ( ) ( ) M Sin ( )+ T ( + ) M Sin ( ) (.7) M Sin ( ) = ( ) M Sin ( )+ M Sin ( ) ( ). The stagnation pressure ratio across the shock is P t ( + ) M Sin ( ) = P t ( ) M Sin ( )+ +. (.8) M Sin ( ) ( ) Note that (.8) can also be generated by the substitution (.3)... Exceptional relations One all new relation that has no normal shock counterpart is the equation for the absolute velocity change across the shock. U U = 4 M Sin ( ) M Sin ( )+ ( + ) M 4 Sin ( ) (.9) Exceptions to the substitution rule (.3) are the relations involving the static and stagnation pressure, P t /P and P t /P across the wave. The reason for this is as follows. Consider P t P = P t P t P t P = P t + P t M. (.0) Similarly P t P = P t P t P t P = P t + P t M. (.)
5 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -5 The stagnation to static pressure ratio in each region depends on the full Mach number, not just the Mach number perpendicular to the shock wave... Flow deflection versus shock angle The most basic question connected with oblique shocks is: given the free stream Mach number, M, and flow deflection,, what is the shock angle,? The normal velocity ratio is u = ( ) M Sin + u ( + ) M Sin = u v. u v (.) From the velocity triangles in figure. Tan( )= u v Tan( ) = u v. (.3) Now ( ) M Sin ( )+ Tan( ) =Tan( ) ( + ) M. (.4) Sin ( ) An alternative form of this relation is 0 Tan( ) =Cot( + + M Sin ( ) A. (.5) M M Sin ( ) The shock-angle-deflection-angle relation (.5) is plotted in figure.3 for several values of the Mach number. Corresponding points in the supersonic flow past a circular cylinder sketched below are indicated on the M =.5 contour. At point a the flow is perpendicular to the shock wave and the properties of the flow are governed by the normal shock relations. In moving from point a to b the shock weakens and the deflection of the flow behind the shock increases until a point of maximum flow deflection is reached at b. The flow solution between a and b is referred to as the strong solution in figure.3. Notice that the Mach number behind the shock is subsonic up to point c where the Mach number just downstream of
6 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -6 Figure.3: Flow deflection versus shock angle for oblique shocks. the shock is one. Between c and d the flow corresponds to the weak solutions indicated in figure.3. If one continued along the shock to very large distances from the sphere the shock will have a more and more oblique angle eventually reaching the Mach angle! µ = Sin (/M ) corresponding to an infinitesimally small disturbance. Figure.4: Supersonic flow past a cylinder with shock structure shown. Note that as the free-stream Mach number becomes large, the shock angle becomes independent of the Mach number.
7 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -7 lim M! Cos( ) Sin( ) Tan( ) = (.6) Sin +. Weak oblique waves In this section we will develop the di erential equations that govern weak waves generated by a small disturbance. The theory will be based on infinitesimal changes in the flow and for this reason it is convenient to drop the subscript 0 0 on the flow variables upstream of the wave. The sketch below depicts the case where the flow deflection is very small d <<. Note that M is not close to one. Figure.5: Small deflection in supersonic flow. In terms of figure.3 we are looking at the behavior of weak solutions close to the horizontal axis of the figure. For a weak disturbance, the shock angle is very close to the Mach angle Sin(µ) =/M. Let and make the approximation Sin( )= M + " (.7) Using (.8) we can also develop the approximation M Sin ( ) = +M". (.8) Cot( ) = M / M 3 M ". (.9) Using (.8) and (.9) the (, ) relation (.5) can be expanded to yield
8 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -8 Tan(d ) = d = 4 + M / ". (.0) M The velocity change across the shock (??) is expanded as U U + = 4 U M M + " M M + " + ( + ) M 4 M + ". (.) Retaining only terms of order " the fractional velocity change due to the small deflection is du U + = 8 ". (.) ( + ) M Equation (.) is approximated as Write (.3) in terms of the deflection angle du U = 4 ". (.3) ( + ) M or du U = 4 ( + ) M " = 4 ( + ) M + 4 M d (.4) (M / ) du U = d (.5) (M / ) where d is measured in radians. Other small deflection relations are dp P = M d (M / ) d = M d (M / ) (.6) dt T = ( ) M (M ) / d
9 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -9 and dp t P t = 3( + ) M Sin 3 = 6 M 3 3( + ) "3 = ds R (.7) or using (.0) dp t P t = ( + ) M 6 (M ) 3/ (d )3 = ds R. (.8) Note that the entropy change across a weak oblique shock wave is extremely small; the wave is nearly isentropic. The Mach number is determined from dm M = du U dt T = (M ) / d ( ) M d. (.9) (M / ) Adding terms dm + M = M d. (.30) (M / ) Eliminate d between (.5) and (.30) to get an integrable equation relating velocity and Mach number changes. du U = + M The weak oblique shock relations (.6) are, in terms of the velocity. dm M (.3) dp P = M du U dt T = M du U (.3) d = M du U
10 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -0 These last relations are precisely the same ones we developed for one dimensional flow with area change in the absence of wall friction and heat transfer in chapter 9. From that development we had M du U = da A M dm + M M = da A. (.33) If we eliminate da/a between these two relations, the result is du U = + M which we just derived in the context of weak oblique shocks. dm M (.34).3 The Prandtl-Meyer expansion The upshot of all this is that du U = d (.35) (M / ) is actually a general relationship valid for steady, isentropic flow. In particular it can be applied to negative values of d. Consider flow over a corner. Figure.6: Supersonic flow over a corner.
11 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW - Express the angle in terms of the Mach number. d = M / dm + M M (.36) Now integrate the angle between the initial and final Mach numbers. Z 0 d 0 = Z M / M M + M dm M (.37) Let! be the angle change beginning at the reference mach number M =. The integral (.37) is! (M) = + / Tan! / M / + Tan M /. (.38) This expression provides a unique relationship between the local Mach number and the angle required to accelerate the flow to that Mach number beginning at Mach one. The straight lines in figure.5 are called characteristics and represent particular values of the flow deflection. According to (.38) the Mach number is the same at every point on a given characteristic. This flow is called a Prandtl-Meyer expansion and (.38) is called the Prandtl-Meyer function, plotted below for several values of. Figure.7: Prandtl-Meyer function for several values of.
12 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW - Note that for a given there is a limiting angle at M!.! max = + /! (.39) For =.4,! max =.45. The expansion angle can be greater than90. If the deflection is larger than this angle there will be a vacuum between! max and the wall..3. Example - supersonic flow over a bump Air flows past a symmetric -D bump at a Mach number of 3. The aspect ratio of the bump is a/b = p 3. Figure.8: Supersonic flow over a bump. Determine the drag coe cient of the bump assuming zero wall friction. C d = Drag force per unit span U b (.40) Solution The ramp angle is 30 producing a 5 oblique shock with pressure ratio P P = (.4) The expansion angle is 60 producing a Mach number M =.406! =9.6. (.4) The stagnation pressure is constant through the expansion wave and so the pressure ratio over the downstream face is
13 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -3 M 3 =4.68! = 69.6 (.43) and P 3 P = + + M M 3 A! 3.5 = +0.(.406) +0.(4.68) = =0.049 (.44) and P 3 P = P 3 P P P = = (.45) The drag coe cient becomes C d = P b Sin(30) P 3 bsin(30) M P b = (9) = (.46).4 Problems Problem - Use the oblique shock jump conditions (.) to derive the oblique shock Prandtl relation (.5). Problem - Consider the supersonic flow past a bump discussed in the example above. Carefully sketch the flow putting in the shock waves as well as the leading and trailing characteristics of the expansion. Problem 3 - Consider a streamline in compressible flow past a -D ramp with a very small ramp angle. Figure.9: Supersonic flow past a -D ramp.
14 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -4 Determine the ratio of the heights of the streamline above the wall before and after the oblique shock in terms of M and d find the unknown coe cient in (.47). Pay careful attention to signs. H H =+(???)d (.47) Problem 4 - Consider a body in subsonic flow. As the free-stream Mach number is increased there is a critical value, M c, such that there is a point somewhere along the body where the flow speed outside the boundary layer reaches the speed of sound. Figure.0 illustrates this phenomena for flow over a projectile. Figure.0: Projectile in high subsonic flow. In this figure.0 the critical Mach number is somewhere between and as evidenced by the weak shocks that appear toward the back of the projectile in the middle picture. The local pressure in the neighborhood of the body is expressed in terms of the pressure coe cient. C P = P P U (.48) Show that the value of the pressure coe cient at the point where sonic speed occurs is
15 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -5 C Pc = + Mc + M c. (.49) State any assumptions needed to solve the problem. Problem 5 - Consider frictionless (no wall friction) supersonic flow over a flat plate of chord C at a small angle of attack as shown in figure.. Figure.: Supersonic flow past a flat plate at a small angle of attack. The circulation about the plate is defined as I = Uds. (.50) where the integration is along any contour surrounding the plate. ) Show that, to a good approximation, the circulation is given by = U C M / (.5) where the integration is clockwise around the plate. ) Show that, to the same approximation, Liftperunitspan = U. Problem 6 - Consider frictionless (no wall friction) flow of air at M = over a flat plate of chord C at 5 angle of attack as shown in figure.. Figure.: Supersonic flow over a flat plate at 5 angle of attack.
16 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -6 Evaluate the drag coe wave approximation. cient of the plate. Compare with the value obtained using a weak Problem 7 - Consider frictionless (no wall friction) supersonic flow of Air over a flat plate of chord C at an angle of attack of 5 degrees as shown in figure.3. Figure.3: Supersonic flow over a flat plate at 5 angle of attack. Determine the lift coe cient L C L = U C (.5) where L is the lift force per unit span. Problem 8 - Figure.4 shows a symmetrical, diamond shaped airfoil at a 5 attack in a supersonic flow of air. angle of Figure.4: Supersonic flow past a diamond shaped airfoil. Determine the lift and drag coe cients of the airfoil. C L = C D = Lift per unit span U C Drag per unit span U C (.53)
17 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -7 What happens to the flow over the airfoil if the free-stream Mach number is decreased to.5? Compare your result with the lift and drag of a thin flat plate at 5 angle of attack and free-stream Mach number of 3. Problem 9 - The figure below shows supersonic flow of Air over a 30 a 0 wedge. The free stream Mach number is 3. wedge followed by Figure.5: Supersonic over a wedge with a shoulder. ) Determine M, M 3 and the included angle of the expansion fan,. ) Suppose the flow was turned through a single 0 wedge instead of the combination shown above. Would the stagnation pressure after the turn be higher or lower than in the case shown? Why? Problem 0 - Figure.6 shows a smooth compression of a supersonic flow of air by a concave surface. The free-stream Mach number is.96. The weak oblique shock at the nose produces a Mach number of.93 at station. From station to station the flow is turned 0 degrees. Figure.6: Supersonic flow compressed by a concave surface. ) Determine the Mach number at station. ) Determine the pressure ratio P /P. 3) State any assumptions used.
18 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -8 Problem - Figure.7 shows supersonic flow of air in a channel or duct at a Mach number of three. The flow produces an oblique shock o a ramp at an angle of 6 degrees. The shock reflects o the upper surface of the wind tunnel as shown below. Beyond the ramp the channel height is the same at the height ahead of the ramp. Figure.7: Mach 3 flow in a duct with a ramp. ) Determine the Mach number in region. ) Determine the Mach number in region 3. 3) Describe qualitatively how P t and T t vary between regions, and 3. 4) Suppose the channel height is 0 cm. Precisely locate the shock reflections on the upper and lower walls. 5) Suppose the walls were lengthened. At roughly what point would the Mach number tend to one? Problem - Figure.8 shows supersonic flow of air turned through an angle of 30. The free stream Mach number is 3. Figure.8: Supersonic flow turned 30. In case (a) the turning is accomplished by a single 30 wedge whereas in case (b) the turning is accomplished by two 5 degree wedges in tandem. Determine the stagnation pressure change in each case,(p t /P t ) (a) and (P t3 /P t ) (b) and comment on the relative merit of one design over the other. Problem 3 - Figure.9 shows the flow of helium from a supersonic over-expanded round jet. If we restrict our attention to a small region near the intersection of the first two oblique shocks and the so-called Mach disc as shown in the blow-up, then we can use
19 CHAPTER. STEADY WAVES IN COMPRESSIBLE FLOW -9 oblique shock theory to determine the flow properties near the shock intersection (despite the generally non-uniform 3-D nature of the rest of the flow). The shock angles with respect to the horizontal measured from the image are as shown. Figure.9: Supersonic flow from an over expanded round jet. ) Determine the jet exit Mach number. Hint, you will need to select a Mach number that balances the pressures in regions and 4 with a dividing streamline that is very nearly horizontal as shown in the picture. ) Determine the Mach number in region. 3) Determine the flow angles and Mach numbers in regions 3 and 4. 4) Determine P /P and P 4 /P. How well do the static pressures match across the dividing streamline (dashed line) between regions and 4? Problem 4 - Figure.0 shows the reflection of an expansion wave from the upper wall of a -D, adiabatic, inviscid channel flow. The gas is helium at an incoming Mach number, M =.5 and the deflection angle is 0. The flow is turned to horizontal by the lower wall which is designed to follow a streamline producing no reflected wave. Determine M, M 3 and H/h. Figure.0: Supersonic flow in an expansion.
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 informationAerothermodynamics of high speed flows
Aerothermodynamics of high speed flows AERO 0033 1 Lecture 6: D potential flow, method of characteristics Thierry Magin, Greg Dimitriadis, and Johan Boutet Thierry.Magin@vki.ac.be Aeronautics and Aerospace
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 Duct Flow with Friction
Compressible Duct Flow with Friction We treat only the effect of friction, neglecting area change and heat transfer. The basic assumptions are 1. Steady one-dimensional adiabatic flow 2. Perfect gas with
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 informationLab Reports Due on Monday, 11/24/2014
AE 3610 Aerodynamics I Wind Tunnel Laboratory: Lab 4 - Pressure distribution on the surface of a rotating circular cylinder Lab Reports Due on Monday, 11/24/2014 Objective In this lab, students will be
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 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 informationSPC Aerodynamics Course Assignment Due Date Monday 28 May 2018 at 11:30
SPC 307 - Aerodynamics Course Assignment Due Date Monday 28 May 2018 at 11:30 1. The maximum velocity at which an aircraft can cruise occurs when the thrust available with the engines operating with the
More information6.1 Propellor e ciency
Chapter 6 The Turboprop cycle 6. Propellor e ciency The turboprop cycle can be regarded as a very high bypass limit of a turbofan. Recall that the propulsive e ciency of a thruster with P e = P 0 and f
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 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 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 informationAerodynamics. Basic Aerodynamics. Continuity equation (mass conserved) Some thermodynamics. Energy equation (energy conserved)
Flow with no friction (inviscid) Aerodynamics Basic Aerodynamics Continuity equation (mass conserved) Flow with friction (viscous) Momentum equation (F = ma) 1. Euler s equation 2. Bernoulli s equation
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 informationThe ramjet cycle. Chapter Ramjet flow field
Chapter 3 The ramjet cycle 3. Ramjet flow field Before we begin to analyze the ramjet cycle we will consider an example that can help us understand how the flow through a ramjet comes about. The key to
More informationDepartment of Mechanical Engineering
Department of Mechanical Engineering AMEE401 / AUTO400 Aerodynamics Instructor: Marios M. Fyrillas Email: eng.fm@fit.ac.cy HOMEWORK ASSIGNMENT #2 QUESTION 1 Clearly there are two mechanisms responsible
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 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 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 informationP 1 P * 1 T P * 1 T 1 T * 1. s 1 P 1
ME 131B Fluid Mechanics Solutions to Week Three Problem Session: Isentropic Flow II (1/26/98) 1. From an energy view point, (a) a nozzle is a device that converts static enthalpy into kinetic energy. (b)
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 informationIran University of Science & Technology School of Mechanical Engineering Advance Fluid Mechanics
1. Consider a sphere of radius R immersed in a uniform stream U0, as shown in 3 R Fig.1. The fluid velocity along streamline AB is given by V ui U i x 1. 0 3 Find (a) the position of maximum fluid acceleration
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 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 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 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 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 information1 One-dimensional analysis
One-dimensional analysis. Introduction The simplest models for gas liquid flow systems are ones for which the velocity is uniform over a cross-section and unidirectional. This includes flows in a long
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 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 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 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 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 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 informationNumerical Investigation of Wind Tunnel Wall Effects on a Supersonic Finned Missile
16 th International Conference on AEROSPACE SCIENCES & AVIATION TECHNOLOGY, ASAT - 16 May 26-28, 2015, E-Mail: asat@mtc.edu.eg Military Technical College, Kobry Elkobbah, Cairo, Egypt Tel : +(202) 24025292
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 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 informationUncertainty in airflow field parameters in a study of shock waves on flat plate in transonic wind tunnel
Journal of Physics: Conference Series OPEN ACCESS Uncertainty in airflow field parameters in a study of shock waves on flat plate in transonic wind tunnel To cite this article: L C C Reis et al 03 J. Phys.:
More informationUOT Mechanical Department / Aeronautical Branch
Chapter One/Introduction to Compressible Flow Chapter One/Introduction to Compressible Flow 1.1. Introduction In general flow can be subdivided into: i. Ideal and real flow. For ideal (inviscid) flow viscous
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 informationApplied Fluid Mechanics
Applied Fluid Mechanics 1. The Nature of Fluid and the Study of Fluid Mechanics 2. Viscosity of Fluid 3. Pressure Measurement 4. Forces Due to Static Fluid 5. Buoyancy and Stability 6. Flow of Fluid and
More informationUNIT 1 COMPRESSIBLE FLOW FUNDAMENTALS
UNIT 1 COMPRESSIBLE FLOW FUNDAMENTALS 1) State the difference between compressible fluid and incompressible fluid? 2) Define stagnation pressure? 3) Express the stagnation enthalpy in terms of static enthalpy
More information10.52 Mechanics of Fluids Spring 2006 Problem Set 3
10.52 Mechanics of Fluids Spring 2006 Problem Set 3 Problem 1 Mass transfer studies involving the transport of a solute from a gas to a liquid often involve the use of a laminar jet of liquid. The situation
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 information58:160 Intermediate Fluid Mechanics Bluff Body Professor Fred Stern Fall 2014
Professor Fred Stern Fall 04 Chapter 7 Bluff Body Fluid flows are broadly categorized:. Internal flows such as ducts/pipes, turbomachinery, open channel/river, which are bounded by walls or fluid interfaces:
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 informationLecture-4. Flow Past Immersed Bodies
Lecture-4 Flow Past Immersed Bodies Learning objectives After completing this lecture, you should be able to: Identify and discuss the features of external flow Explain the fundamental characteristics
More informationOblique Shock Visualization and Analysis using a Supersonic Wind Tunnel
Oblique Shock Visualization and Analysis using a Supersonic Wind Tunnel Benjamin M. Sandoval 1 Arizona State University - Ira A. Fulton School of Engineering, Tempe, AZ, 85281 I. Abstract In this experiment,
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 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 informationThe E80 Wind Tunnel Experiment the experience will blow you away. by Professor Duron Spring 2012
The E80 Wind Tunnel Experiment the experience will blow you away by Professor Duron Spring 2012 Objectives To familiarize the student with the basic operation and instrumentation of the HMC wind tunnel
More informationFanno Flow. Gas Dynamics
Fanno Flow Simple frictional flow ( Fanno Flow Adiabatic frictional flow in a constant-area duct * he Flow of a compressible fluid in a duct is Always accompanied by :- ariation in the cross sectional
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 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 informationDepartment of Energy Sciences, LTH
Department of Energy Sciences, LTH MMV11 Fluid Mechanics LABORATION 1 Flow Around Bodies OBJECTIVES (1) To understand how body shape and surface finish influence the flow-related forces () To understand
More informationFluid Mechanics Qualifying Examination Sample Exam 2
Fluid Mechanics Qualifying Examination Sample Exam 2 Allotted Time: 3 Hours The exam is closed book and closed notes. Students are allowed one (double-sided) formula sheet. There are five questions on
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 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 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 (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 informationNumerical Investigation of Shock wave Turbulent Boundary Layer Interaction over a 2D Compression Ramp
Advances in Aerospace Science and Applications. ISSN 2277-3223 Volume 4, Number 1 (2014), pp. 25-32 Research India Publications http://www.ripublication.com/aasa.htm Numerical Investigation of Shock wave
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 informationFundamentals of Fluid Mechanics
Sixth Edition Fundamentals of Fluid Mechanics International Student Version BRUCE R. MUNSON DONALD F. YOUNG Department of Aerospace Engineering and Engineering Mechanics THEODORE H. OKIISHI Department
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 informationAirfoils and Wings. Eugene M. Cliff
Airfoils and Wings Eugene M. Cliff 1 Introduction The primary purpose of these notes is to supplement the text material related to aerodynamic forces. We are mainly interested in the forces on wings and
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 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 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 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 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 informationCompressible Fluid Flow
Compressible Fluid Flow For B.E/B.Tech Engineering Students As Per Revised Syllabus of Leading Universities in India Including Dr. APJ Abdul Kalam Technological University, Kerala Dr. S. Ramachandran,
More informationJet Aircraft Propulsion Prof. Bhaskar Roy Prof. A.M. Pradeep Department of Aerospace Engineering
Jet Aircraft Propulsion Prof. Bhaskar Roy Prof. A.M. Pradeep Department of Aerospace Engineering Indian Institute of Technology, IIT Bombay Module No. # 01 Lecture No. # 08 Cycle Components and Component
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 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 informationPART 1B EXPERIMENTAL ENGINEERING. SUBJECT: FLUID MECHANICS & HEAT TRANSFER LOCATION: HYDRAULICS LAB (Gnd Floor Inglis Bldg) BOUNDARY LAYERS AND DRAG
1 PART 1B EXPERIMENTAL ENGINEERING SUBJECT: FLUID MECHANICS & HEAT TRANSFER LOCATION: HYDRAULICS LAB (Gnd Floor Inglis Bldg) EXPERIMENT T3 (LONG) BOUNDARY LAYERS AND DRAG OBJECTIVES a) To measure the velocity
More informationSome Basic Plane Potential Flows
Some Basic Plane Potential Flows Uniform Stream in the x Direction A uniform stream V = iu, as in the Fig. (Solid lines are streamlines and dashed lines are potential lines), possesses both a stream function
More informationDetailed Derivation of Fanno Flow Relationships
Detailed Derivation of Fanno Flow Relationships Matthew MacLean, Ph.D. Version. October 03 v. fixed a sign error on Eq. 5 ! My motivation for writing this note results from preparing course notes for the
More informationRayleigh processes in single-phase fluids
Rayleigh processes in single-phase fluids M. S. Cramer Citation: Physics of Fluids (1994-present) 18, 016101 (2006); doi: 10.1063/1.2166627 View online: http://dx.doi.org/10.1063/1.2166627 View Table of
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 informationMasters in Mechanical Engineering Aerodynamics 1 st Semester 2015/16
Masters in Mechanical Engineering Aerodynamics st Semester 05/6 Exam st season, 8 January 06 Name : Time : 8:30 Number: Duration : 3 hours st Part : No textbooks/notes allowed nd Part : Textbooks allowed
More information