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modern compressible flow solutions chapter 3
4th Dec

2020

modern compressible flow solutions chapter 3

Linear nature of the potential flow problem, the differential equation does not have to be solved individually for flow fields having different geometry at their boundaries. We are also providing an authentic solution manual, formulated by our SMEs, for the same. At supersonic speeds, the aircraft is travelling faster than the speed of the pressure disturbances. Title: Chapter 17: Compressible Flow 1 Chapter 17 Compressible Flow Study Guide in PowerPointto accompanyThermodynamics An Engineering Approach, 5th editionby Yunus A. Çengel and Michael A. Boles 2 Stagnation Properties Consider a fluid flowing into a diffuser at a velocity , temperature T, pressure P, and enthalpy h, etc. Our book servers hosts in multiple locations, allowing you to get the most less latency time to download any of our books like this one. ENTROPY 1.1 DEFINITION You should already be familiar with the theory of work laws in closed systems. To show how the speed of sound relates to the compressibility of a gas: Since the isentropic compressibility of a gas was defined in Chapter 1 as: The speed of sound can be related to the compressibility of a gas as: Incompressible flow means that τs=0 or an infinite speed of sound in an incompressible medium (or M = ∞). Plasma is a pale yellow fluid that consists of about 91% water and 9% other substances, such as proteins, ions, nutrients, gases and waste products. Problem 3P from Chapter 1: For a calorically perfect gas, derive the relation cp – cυ =... Get solutions . At subsonic speeds, pressure disturbances travel at speeds greater than the speed of the aircraft. Incompressible, Potential Flow Equations CHAPTER 3 Developing the basic methodology for obtaining the elementary solutions to potential flow problem. It holds for ideal gas, real gas, or reacting gas, as long as the gas is in thermodynamic equilibrium (i.e. Large Temperature variations result in density variations. We will solve: mass, linear momentum, energy and an equation of state. I would prefer their Modern Compressible Flow: With Historical Perspective Modern Compressible Flow: With Historical Perspective Solutions Manual For excellent scoring in my academic year. The standard convention for normal shock relations is to designate the region upstream of the normal shock as station 1 and the region downstream as station 2. A control volume is drawn around this body, as given in the dashed lines in Figure A. Edition: 3. A new book website will contain all problem solutions for instructors. modern-compressible-flow-solution-manual-anderson 2/3 Downloaded from support.doolnews.com on November 27, 2020 by guest edition also contains new exercise problems with the answers added. You bet! Required fields are marked As understood, triumph does not suggest that you have fabulous points. Compressible supersonic flow Ma 1≥ formation of Mach wave, pressure communication restricted to zone of action 4. transonic flow 0.9 Ma 1.2≤ ≤ (modern aircraft) 5. hypersonic flow Ma 5≥ (space shuttle) Example 11.4 Mach cone . Pages: 776. Anderson Solution Manual Flow Zip. Flow Zip. I prefer to avail their services always as they are consistent with their quality. Solutions Sm Modern Compressible Flow Zip. ICDDEA 2015. Chapter 3 is titled "One Dimensional Flow". (see figure 3.7) As a CrazyForStudy subscriber, you can view available interactive solutions manuals for each of your classes for one low monthly price. Yeah, reviewing a books modern compressible flow solutions could go to your close connections listings. Using the definitions for total pressure and temperature, the total density and total speed of sound can be defined as: Similarly to the description of the total conditions, imagine the same fluid element moving at some Mach number M, velocity V, with a static pressure p, temperature T. Now instead of isentropically stagnating the element, imagine it is brought to a sonic state (M = 1) adiabatically; meaning the element is slowed down if it began supersonic, or sped up if it began subsonic. It's easier to figure out tough problems faster using CrazyForStudy. If the physical geometry is such that θ > θ max, then no solution exists for a straight oblique shock wave; in such a case, the flow field will adjust in such a fashion to curve and detach the shock wave. I would highly recommend their affordable and quality services. For a thermally and calorically perfect gas, the isentropic relationship between pressure and density is. 2/3 Solutions Sm Modern Compressible Flow Zip by livanrali - Issuu Cheap Textbook Rental for MODERN COMPRESSIBLE FLOW by ANDERSON 3RD 03 9780072424430, Save up Page 5/8. Heat addition changes the total enthalpy of the flow, and for constant cp (calorically perfect gas), the total temperature as well. Unlike static PDF Modern Compressible Flow: With Historical Perspective 3rd Edition solution manuals or printed answer keys, our experts show you how to solve each problem step-by-step. The flow is assumed to be non-adiabatic, i.e. compressible flow wikipedia. Get Free Modern Compressible Flow Anderson 3rd Edition to 90% and get Modern Compressible Flow Solutions-aplikasidapodik.com … Average Star Rating: Send-to-Kindle or Email . which I was looking for so long finally landed me here. Introduction to Compressible Flow: February 27, 2007 page 3. The pressure and temperature increase from their starting state and are now at the stagnation, or total, state (the descriptors "stagnation" and "total" are synonymous). The second edition maintains an engaging writing style and offers philosophical and historical perspectives on the topic. You can check your reasoning as you tackle a problem using our interactive solutions viewer. Modern Compressible Flow: With Historical Perspective (3rd Edition) Edit edition. The flow upstream of the shock wave defines one point on this curve. lecture notes chapters 1 3 6 textbook modern. I would suggest all students avail their textbook solutions manual. 3-1 Chapter 3 The Physical and Flow Properties of Blood 3.1 Introduction Blood is a viscous fluid mixture consisting of plasma and cells. Excellent service when it comes to textbook solutions. Modern Compressible Flow: With Historical Perspective (3rd Edition) Edit edition. The result of the friction is a large temperature gradient within the shock wave, also creating entropy. Then p2 = p. The ratios of parameters under this case are: Note that the starred quantities are constants for each flow; therefore, they can be tabulated as functions of γ and Mach number, where M is the Mach number upstream, before heat addition. It shows that the speed of sound is directly related to the compressibility of a gas. The physical mechanism by which the shock waves create entropy is attributed to viscous friction due to large gradients of velocity over a very short distance (shock wave thickness). Scribd is the world's largest social reading and publishing site. CrazyForStudy Expert Q&A is a great place to find help on problem sets and 18 study guides. No need to wait for office hours or assignments to be graded to find out where you took a wrong turn. At higher speeds where the temperature increase across the shock causes molecular dissociation, the downstream properties depend on M1, γ1, and T1. for a thermally and calorically perfect gas. heat is added or subtracted, per unit mass, of magnitude q. Modern Compressible Flow With Historical Perspective 3rd Edition by John Anderson (Solutions Manual) solutions...@gmail.com: 5/25/19 1:36 AM: solutions book team Page 6/11. Hypersonic and High-Temperature Gas Dynamics Chapter 2 Notes, Hypersonic and High-Temperature Gas Dynamics Chapter 3 Notes, Hypersonic and High-Temperature Gas Dynamics Chapter 4 Notes, Hypersonic and High-Temperature Gas Dynamics Chapter 5 Notes, https://aeroengineeringnotes.fandom.com/wiki/Modern_Compressible_Flow_Chapter_3_Notes?oldid=4200. Categories: Technology\\Mechanical Engineering. I would suggest all students avail their textbook solutions manual. Lines perpendicular to the flow velocity far ahead of 2. Modern-Compressible-Flow-3rd-Solution-Manual 1/3 PDF Drive - Search and download PDF files for free. How can we use the tabulated data to solve a Rayleigh flow problem? The time frame a book is available as a free download is shown on each download page, as well as a full description of the book and sometimes a link to the author's website. As the upstream Mach number approaches infinity, the downstream Mach number reduces to: For air with γ = 1.4, as M1 approaches infinity, M2= 0.378. This is just to make you understand and used for the analysis and reference purposes only. " The Aeronautical engineer is pounding hard on the closed door leading into the field of supersonic motion. " Shock waves in one-dimensional, or 1-D, flow are exclusively normal because in 1-D flows, flow-field properties vary only with a single coordinate direction. Only conditions where Δsshock ≥ 0 are permitted by the second law of thermodynamics (in cases where no work or heat is transfered); therefore, shock waves cause an entropy increase in the flow and can only exist if M1 > 1. The Modern Compressible Flow: With Historical Perspective Modern Compressible Flow: With Historical Perspective Solutions Manual Helped me out with all doubts. Preview. Bookmark it to easily review again before an exam. File: PDF, 25.56 MB. . Über den Autor und weitere Mitwirkende . Differentiating this, and recalling that v = 1/ρ: Our expression for the speed of sound then becomes: Using the equation of state for a perfect gas p/ρ = RT, the speed of sound can be defined as: The two previous equations for the speed of sound hold for calorically and thermally perfect gases, but not for chemically reacting or real gases. CONTENTS vii 13.4.2 In What Situations No Oblique Shock Exist or When. get the modern compressible flow solution manual However, the stamp album in soft file will be also simple to door every time You can agree to it into the gadget or computer COMPUTATIONAL FLUID DYNAMICS The Basics with … (1989), Modern Compressible Flow, 2d Edition (1990), Fundamentals of Aerodynamics, 2d edition (1991 ), and Hypersonic and High Temperature Gas Dynamics ( 1989) He is … CHAPTER 2 2-l Consider a two-dimensional body in a flow, as sketched in Figure A. Such behavior is characteristic for a highly turbulent flow in the absence of external forces (i.e., in case of decaying turbulence). Modern Compressible Flow 3rd Solution Manual Modern Compressible Flow 3rd Solution Recognizing the artifice ways to get this books Modern Compressible Flow 3rd Solution Manual is additionally useful. Your email address will not be published. Assuming the gas is calorically and thermally perfect, what are the limiting values of velocity, density, pressure, and temperature ratios as M1approaches infinity. Consider a small disturbance that is moving through a compressible gas, and assume that the conditions behind the wave after it has passed are incrementally disturbed from their initial values in the gas. Calculate the density and specific volume. We have solutions for your book! 221 13.4.4 For Given Two Angles, At locations close to the nose of the body, the shockwave is close to perpendicular to the free stream, and is, thus, a normal shock. The Modern Compressible Flow: With Historical Perspective Modern Compressible Flow: With Historical Perspective Solutions Manual Helped me out with all doubts. Note that so long as the flow is adiabatic, the initial energy of the flow is constant; a. Aero Engineering Notes Wiki is a FANDOM Lifestyle Community. Consider a length of constant area pipe or duct or streamtube. Academia.edu is a platform for academics to share research papers. The Modern Compressible Flow: With Historical Perspective Modern Compressible Flow: With Historical Perspective Solutions Manual Helped me out with all doubts. We strictly do not deliver the reference papers. . The Aeronautical engineer is pounding hard on the closed door leading into the field of supersonic motion. In addition, the information on ram jets is expanded with helpful worked examples. This is just one of the solutions for you to be successful. Asking a study question in a snap - just take a pic. The relation is: Anderson provides a figure showing these ratios as a function of M1 for γ = 1.4 (Fig. 2. Modern Compressible Flow: With Historical Perspective John D. Anderson. The one-dimensional flow equations for this control volume analysis are the same as always: If we assume a calorically perfect gas, we can also introduce the following thermodynamic relations: With these 5 equations, and knowledge of all upstream conditions (ρ1, u1, p1, h1, T1), we can solve for the downstream conditions (ρ2, u2, p2, h2, T2). For a flowing compressible fluid (assuming a calorically perfect gas), the ratio of kinetic energy to internal energy is: Imagine a fluid element moving at some Mach number M, velocity V, with a static pressure p, and temperature T. Now imagine that same fluid element is isentropically brought to rest where the velocity and, consequently, Mach number are zero. You can also find solutions immediately by searching the millions of fully answered study questions in our archive. Solution Edit. This equation is a fundamental expression for the speed of sound and is valid for all gases. Thus the Hugoniot equation can be written as: Anderson gives this equation on page 101 as: In any case, the Hugoniot curve is a curve of pressure versus specific volume. 390 KPKP 8987 Preface xi Chapter 1 Compressible Flow Some History and Introductory Thoughts 1 1.1 Historical High-Water Marks 1 1.2 Deﬁnition of Compressible Flow 4 1.3 Flow Regimes 6 1.4 A Brief Review of Thermodynamics 11 1.5 Aerodynamic Forces on a Body 21 1.6 Modern Compressible Flow 23 Problems 25 Chapter 2 Integral Forms of the Conservation Equations for … For a calorically perfect gas, derive the relation c p – c υ = R. Repeat the derivation for a thermally perfect gas. Our solutions are written by Chegg experts so you can be assured of the highest quality! Problem 7P from Chapter 1: In the infinitesimal neighborhood surrounding a point in an ... Get solutions . Isentropic requires adiabatic and reversible process. Thus, the speed of sound can be determined corresponding to the conditions at point 1, i.e. 2.5.3 Model of an Infinitesimally Small Element Fixed 4.6 Summary 165 in Space 53 2.5.4 Model of an Infinitesimally Small Fluid Element GUIDEPOST 166 Moving with the Flow 55 Problems 2.5.5 All the Equations Are One: Some Manipulations 56 167 These equations, although simplified from their original integral forms, still maintain the basic concepts of mass, momentum, and energy conservation. Your email address will not be published. As a result we now have two new variables we must solve for: T & ρ We need 2 new equations. The Modern Compressible Flow: With Historical Perspective 3rd Edition Solutions Manual Was amazing as it had almost all solutions to textbook questions that I was searching for long. modern compressible flow solutions chapter 1 aero. A normal shock is one that is perpendicular to the free stream flow. Using the definitions for sonic pressure and temperature, the sonic density and speed of sound can be defined as: Admittedly, sonic is a confusing descriptor to use with speed of sound, but it is used to describe the speed of sound in the fluid if that fluid has been adiabatically brought to a sonic condition. Hence (dp/dρ) is the isentropic derivative of pressure with respect to density. To solve for these 5 unknowns, first we divide the momentum equation by the continuity equation: Using one of our previously defined alternate forms of the energy equation, we can write: But since the shock is adiabatic, a1* = a2*, and these two equations can be combined yielding: The Prandtl Relation is a useful intermediate relation for normal shocks. Our interactive player makes it easy to find solutions to Modern Compressible Flow: With Historical Perspective 3rd Edition problems you're working on - just go to the chapter for your book. Just post a question you need help with, and one of our experts will provide a custom solution. Kindly say, the modern compressible Page 3/11. The 3rd edition strikes a careful balance between classical methods of determining compressible flow, and modern numerical and computer techniques (such as CFD) now used widely in industry and research. 4.6 out of 5 It relates the upstream and downstream velocities across a normal shock to the speed of sound at M= 1. Modern Compressible Flow | Anderson | ISBN: 9781259027420 | Kostenloser Versand für alle Bücher mit Versand und Verkauf duch Amazon. The system of governing equations that applied to the shock wave region, which is infinitesimal in thickness can also apply to a finite region or length of flow. $p_1 + \rho_1 u_1^2 = p_2 + \rho_2 u_2^2$, $\frac{\dot Q}{A} + p_1 u_1 + \rho_1\left(e_1+\frac{u_1^2}{2}\right)u_1 = p_2 u_2 + \rho_2\left(e_2+\frac{u_2^2}{2}\right)u_2$, $q + \frac{p_1}{\rho_1}+ e_1 +\frac{u_1^2}{2} = \frac{p_2}{\rho_2} + e_2 + \frac{u_2^2}{2}$, $q+h_1+\frac{u_1^2}{2} = h_2+\frac{u_2^2}{2}$, $\rho a=\rho a+\rho da+d\rho a+d\rho da$, $p+\rho a^2 = (p+dp) + (\rho +d\rho)(a+da)^2$, $a=-\rho \left(\frac{\frac{da}{d\rho} + a^2}{-2a\rho}\right)$, $a^2 = \left(\frac{dp}{d\rho}\right)_{s=constant} = -\left(\frac{dp}{dv}\right)_s v^2 = -\frac{v}{\frac{1}{v}\left(\frac{dp}{dv}\right)_s}$, $\tau _s = \frac{1}{v}\left(\frac{dp}{dv}\right)_s$, $a=\sqrt{\left(\frac{dp}{d\rho}\right)_s}=\sqrt{\frac{v}{\tau _s}}$, $\left(\frac{dp}{d\rho}\right)_s=\frac{\gamma p}{\rho}$, $\frac{\frac{V^2}{2}}{e} = \frac{\frac{V^2}{2}}{c_v T} = \frac{\frac{V^2}{2}}{\frac{R}{(\gamma - 1)}T}=\frac{\frac{\gamma}{2}V^2}{a^2 \frac{1}{(\gamma - 1)}}=\frac{\gamma (\gamma -1)}{2}M^2$, Some Conveniently Defined Flow Parameters, $p_o \text{ or } p_t \text{ — Total Pressure}$, $T_o \text{ or } T_t \text{ — Total Temperature}$, $\rho_o \text{ or } \rho_t = \frac{p_t}{RT_t} \text{ — Total Density}$, $a_o \text{ or } a_t = \sqrt{\gamma RT_t} \text{ — Total Speed of Sound}$, $\rho^* = \frac{p^*}{RT^*} \text{ — Sonic Density}$, $a^* = \sqrt{\gamma RT^*} \text{ — Sonic Speed of Sound}$, Steady, Single Stream Conservation of Energy Equation, $h_1 + \frac{u_1^2}{2} = h_2 + \frac{u_2^2}{2}$, $c_p T_1 + \frac{u_1^2}{2} = c_p T_2 + \frac{u_2^2}{2}$, $R= c_p-c_v \to c_p = \frac{\gamma R}{\gamma-1}$, $\frac{\gamma R}{\gamma-1} T_1 + \frac{u_1^2}{2} = \frac{\gamma R}{\gamma-1} T_2 + \frac{u_2^2}{2}$, $\frac{a_1^2}{\gamma-1} + \frac{u_1^2}{2} = \frac{a_2^2}{\gamma-1} + \frac{u_2^2}{2}$, $\frac{\gamma}{\gamma-1}\frac{p_1}{\rho_1} + \frac{u_1^2}{2} = \frac{\gamma}{\gamma-1}\frac{p_2}{\rho_2} + \frac{u_2^2}{2}$, $\frac{a_1^2}{\gamma-1} + \frac{u_1^2}{2} = \frac{a^{*2}}{\gamma-1} + \frac{a^{*2}}{2}$, $\frac{a_1^2}{\gamma-1} + \frac{u_1^2}{2} = \frac{\gamma +1}{2(\gamma -1)}a^{*2}$, The Total Temperature and Total Pressure of a Compressible Flow, $T_t = T+\frac{u^2}{2c_p} = T\left(1+\frac{u^2}{2c_pT}\right)$, $T_t = T\left(1+\frac{\gamma-1}{2}M^2\right)$, $\frac{p_t}{p} = \left(\frac{\rho _t}{\rho}\right)^\gamma = \left(\frac{T_t}{T}\right)^{\frac{\gamma}{\gamma -1}}$, $p_t = p\left(1+\frac{\gamma-1}{2}M^2\right)^{\frac{\gamma}{\gamma -1}}$, $\rho _t = \rho \left(1+\frac{\gamma-1}{2}M^2\right)^{\frac{1}{\gamma -1}}$, $P_t = P_{ti} = \text{constant throughout the flow}$, $T_t = T_{ti} = \text{constant throughout the flow}$, $\rho_t = \rho_{ti} = \text{constant throughout the flow}$, $\frac{a_t^2}{\gamma-1} = \frac{a^2}{\gamma-1} + \frac{u^2}{2}$, $\frac{a^2}{\gamma-1} + \frac{u^2}{2} = \frac{\gamma +1}{2(\gamma -1)}a^{*2}$, $a_t^2 = \frac{\gamma+1}{2} a^{*2} \text{ } \to \text{ } \left(\frac{a^{*}}{a_t}\right)^2 = \frac{T^*}{T_t} = \frac{2}{\gamma + 1}$, $\frac{p^*}{p_t} = \left(\frac{2}{\gamma +1}\right)^{\frac{\gamma}{\gamma -1}}$, $\frac{\rho^*}{\rho_t} = \left(\frac{2}{\gamma +1}\right)^{\frac{1}{\gamma -1}}$, Ratios of Sonic to Stagnation Quantities for Dry Air, $\frac{a^2}{\gamma-1} + \frac{u^2}{2} = \frac{\gamma +1}{2\left(\gamma-1\right)}a^{*2}$, $\frac{\left(\frac{a}{u}\right)^2}{\gamma -1} + \frac{1}{2}=\frac{\gamma +1}{2\left(\gamma-1\right)}\frac{a^{*2}}{u^2}$, $\frac{\left(\frac{1}{M}\right)^2}{\gamma -1} + \frac{1}{2}=\frac{\gamma +1}{2\left(\gamma-1\right)}\frac{1}{M^{*2}}$, $M^2 = \frac{2}{\frac{\gamma +1}{M^{*2}} - (\gamma - 1)}$, $\lim_{M \to \infty} M^* = \sqrt{\frac{\gamma +1}{\gamma -1}}$, Algebraic solution of the equations of motion across a normal shock, $p_2 + \rho_2 u_2^2 = p_1 + \rho_1 u_1^2$, $h_2 + \frac{u_2^2}{2} = h_1 + \frac{u_1^2}{2}$, $p = \rho RT ~\text{ (Thermally Perfect)}$, $h = c_p T ~~\text{ (Calorically Perfect)}$, $\frac{p_1}{\rho_1 u_1} - \frac{p_2}{\rho_2 u_2} = u_2 - u_1$, $\frac{a_1^2}{\gamma u_1} - \frac{a_2^2}{\gamma u_2} = u_2 - u_1$, $a_1^2 = \frac{\gamma + 1}{2}a^{*2}-\frac{\gamma -1}{2}u_1^2$, $a_2^2 = \frac{\gamma + 1}{2}a^{*2}-\frac{\gamma -1}{2}u_2^2$, $\frac{\gamma + 1}{2\gamma u_1 u_2}\left(u_2 - u_1\right) a^{*2} + \frac{\gamma-1}{2\gamma}\left(u_2 - u_1\right) = u_2 - u_1$, $\frac{\gamma + 1}{2\gamma u_1 u_2}a^{*2} + \frac{\gamma-1}{2\gamma} = 1$, $M^2 = \frac{2}{\left(\frac{\gamma +1}{M^{*2}}\right) - (\gamma - 1)}$, $M^{*2} = \frac{(\gamma +1)M^2}{2+(\gamma -1)M^2}$, $\frac{(\gamma +1)M_1^2}{2+(\gamma -1)M_1^2} = \left [ \frac{(\gamma +1)M_2^2}{2+(\gamma -1)M_2^2} \right ]^{-1}$, $M_2^2 = \frac{1 + \frac{\gamma -1}{2}M_1^2}{\gamma M_1^2 - \frac{\gamma -1}{2}}$, $M_2 = \sqrt{\frac{\gamma -1}{2\gamma}}$, $\frac{\rho _2}{\rho _1} = \frac{u_1}{u_2} = \frac{u_1^2}{u_1 u_2} = \frac{u_1^2}{a^{*2}} = M_1^{*2}$, $M_1^{*2} = \frac{(\gamma + 1)M_1^2}{2+(\gamma - 1)M_1^2}$, $\frac{\rho _2}{\rho _1} = \frac{u_1}{u_2} = \frac{(\gamma + 1)M_1^2}{2+(\gamma - 1)M_1^2}$, Solving for the Pressure and Temperature Ratios across the Shock, $p_2-p_1 = \rho_1 u_1^2 - \rho_2 u_2^2 = \rho_1 u_1(u_1-u_2) = \rho_1 u_1^2 \left(1-\frac{u_2}{u_1}\right)$, $\frac{\rho_1 u_1^1}{p_1} = \frac{\gamma u_1^2}{a_1^2} = \gamma M_1^2$, $\frac{p_2 - p_1}{p_1} = \gamma M_1^2 \left(1-\frac{u_2}{u_1}\right)$, $\frac{u_2}{u_1} = \frac{\rho_1}{\rho_2} = \frac{2+(\gamma - 1)M_1^2}{(\gamma + 1)M_1^2}$, $\frac{p_2}{p_1} = 1+ \frac{2\gamma}{\gamma + 1} \left(M_1^2 -1\right)$, $\frac{T_2}{T_1} = \frac{p_2}{p_1}\frac{\rho_1}{\rho_2}$, $\frac{T_2}{T_1} = \left [ 1+ \frac{2\gamma}{\gamma + 1} \left(M_1^2 -1\right) \right ] \left [ \frac{2+(\gamma - 1)M_1^2}{(\gamma + 1)M_1^2} \right ]$, Limiting Property Ratios for Calorically and Thermally Perfect Gas, $\lim_{M_1 \to \infty} M_2 = \sqrt{\frac{\gamma-1}{2\gamma}} = 0.378$, $\lim_{M_1 \to \infty} \frac{\rho _2}{\rho _1} = \frac{\gamma + 1}{\gamma - 1} = 6$, $\lim_{M_1 \to \infty} \frac{u_2}{u_1} = \frac{\gamma - 1}{\gamma + 1} = \frac{1}{6}$, $\lim_{M_1 \to \infty} \frac{p_2}{p_1} = \infty$, $\lim_{M_1 \to \infty} \frac{T_2}{T_1} = \infty$, $\Delta s_{shock} = s_2 - s_1 = c_p \ln \frac{T_{t2}}{T_{t1}} - R \ln \frac{p_{t2}}{p_{t1}}$, $\Delta s_{shock} = - R \ln \frac{p_{t2}}{p_{t1}}$, $\Delta s_{shock} = c_p \ln \frac{T_{2}}{T_{1}} - R \ln \frac{p_{2}}{p_{1}}$, $\Delta s_{shock} = c_p \ln \left( \left [ 1+ \frac{2\gamma}{\gamma + 1} \left(M_1^2 -1\right) \right ] \left [ \frac{2+(\gamma - 1)M_1^2}{(\gamma + 1)M_1^2} \right ] \right) - R \ln \left [ 1+ \frac{2\gamma}{\gamma + 1} \left(M_1^2 -1\right)\right ]$, $\frac{p_{t2}}{p_{t1}} = \left [ \frac{(\gamma +1)M_1^2}{2+(\gamma -1)M_1^2} \right ] ^{\frac{\gamma}{\gamma -1}} \left [ \frac{\gamma +1}{2 \gamma M_1^2 - (\gamma - 1)} \right ]^{\frac{1}{\gamma -1}}$, $p_1 + \rho_1 u_1^2 = p_2 + \rho_2 \left(u_1 \frac{\rho_1}{\rho_2} \right) ^2$, $u_1^2 = \frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right)$, $u_2^2 = \frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right)$, $e_1 + \frac{p_1}{\rho_1} + \frac{u_1^2}{2} = e_2 + \frac{p_2}{\rho_2} + \frac{u_2^2}{2}$, $e_1 + \frac{p_1}{\rho_1} + \frac{1}{2} \left [ \frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right) \right ] = e_2 + \frac{p_2}{\rho_2} + \frac{1}{2} \left [ \frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right) \right ]$, $e_2-e_1 = \frac{p_2-p_1}{2}\left(\frac{1}{\rho_1} - \frac{1}{\rho_2}\right)$, $e_2-e_1 = \frac{p_2-p_1}{2}\left(v_1 - v_2\right)$, $e = e(p,v) = c_vT = c_v\frac{p}{\rho R} = c_v\frac{pv}{R}$, $c_v\frac{p_2v_2}{R}-c_v\frac{p_1v_1}{R} = \frac{p_2-p_1}{2}\left(v_1 - v_2\right)$, $\frac{1}{\gamma-1}(p_2v_2-p_1v_1) = \frac{p_2-p_1}{2}\left(v_1 - v_2\right)$, $\frac{p_2}{p_1} = \frac{\frac{\gamma+1}{\gamma-1}-\frac{v_2}{v_1}}{\left(\frac{\gamma+1}{\gamma-1}\right)\frac{v_2}{v_1} - 1}$, $\frac{p_2}{p_1} = \frac{\left(\frac{\gamma+1}{\gamma-1}\right)\frac{v_1}{v_2}-1}{\frac{\gamma+1}{\gamma-1} - \frac{v_1}{v_2}}$, $u_1^2 = \frac{p_2 - p_1}{\rho_2 - \rho_1}\frac{\rho_2}{\rho_1} = \frac{p_2 - p_1}{\frac{1}{v_2} - \frac{1}{v_1}}\frac{v_1}{v_2}$, $\frac{p_2 - p_1}{v_2 - v_1} = -\left(\frac{u_1}{v_1}\right)^2$, $q = \left(h_2+\frac{u_2^2}{2}\right) - \left(h_1+\frac{u_1^2}{2}\right) = \left(c_pT_2+\frac{u_2^2}{2}\right) - \left(c_pT_1+\frac{u_1^2}{2}\right)$, Ratios of Properties Across the Control Volume, $\rho u^2 = \rho a^2 M^2 = \rho \frac{\gamma p}{\rho}M^2 = \gamma p M^2$, $p_1\left(1+\gamma M_1^2\right) = p_2\left(1+\gamma M_2^2\right)$, $\frac{p_2}{p_1} = \frac{1+\gamma M_1^2}{1+\gamma M_2^2}$, $\frac{T_2}{T_1} = \frac{p_2}{p_1} \frac{\rho_1}{\rho_2} = \frac{p_2}{p_1} \frac{u_2}{u_1}$, $\frac{u_2}{u_1}=\frac{M_2}{M_1}\sqrt{\frac{T_2}{T_1}}$, $\frac{T_2}{T_1} = \left(\frac{1+\gamma M_1^2}{1+\gamma M_2^2}\right)^2 \left(\frac{M_2}{M_1}\right)^2$, $\frac{\rho_2}{\rho_1} = \frac{p_2}{p_1}\frac{T_1}{T_2}$, $\frac{\rho_2}{\rho_1} = \left(\frac{1+\gamma M_2^2}{1+\gamma M_1^2}\right) \left(\frac{M_1}{M_2}\right)^2$, $\frac{p_{t2}}{p_{t1}} = \frac{1+\gamma M_1^2}{1+\gamma M_2^2}\left(\frac{1+\frac{\gamma -1}{2}M_2^2}{1+\frac{\gamma -1}{2}M_1^2}\right)^{\frac{\gamma}{\gamma -1}}$, $\frac{T_{t2}}{T_{t1}} = \left(\frac{1+\gamma M_1^2}{1+\gamma M_2^2}\right)^2 \left(\frac{M_2}{M_1}\right)^2\left(\frac{1+\frac{\gamma -1}{2}M_2^2}{1+\frac{\gamma -1}{2}M_1^2}\right)$, $\frac{p}{p^*} = \frac{1+\gamma}{1+\gamma M^2}$, $\frac{T}{T^*} = M^2 \left(\frac{1+\gamma}{1+\gamma M^2}\right)$, $\frac{\rho}{\rho^*} = \frac{1}{M^2}\frac{1+\gamma M^2}{1+\gamma}$, $\frac{P_t}{P_t^*} = \frac{1+\gamma}{1+\gamma M^2}\left [ \frac{2+(\gamma-1)M^2}{\gamma+1} \right ] ^{\frac{\gamma}{\gamma-1}}$, $\frac{T_t}{T_t^*} = \frac{(1+\gamma)M^2}{(1+\gamma M^2)^2}\left [ 2+(\gamma-1)M^2 \right ]$, Summary of Physical Changes with Heat Addition, One-Dimensional Flow with Friction (Fanno Flow), $\frac{\rho_2}{\rho_1} = \frac{P_2 T_1}{P_1 T_2} = \frac{M_1}{M_2}\left(\frac{1+\frac{\gamma -1}{2}M_1^2}{1+\frac{\gamma -1}{2}M_2^2}\right)^{-0.5}$, $\frac{p_{t2}}{p_{t1}} = \frac{M_1}{M_2}\left(\frac{1+\frac{\gamma -1}{2}M_2^2}{1+\frac{\gamma -1}{2}M_1^2}\right)^{\frac{\gamma +1}{2(\gamma-1)}}$, $\frac{T}{T^*} = \frac{\gamma +1}{2+(\gamma-1)M^2}$, $\frac{p}{p^*} = \frac{1}{M}\left(\frac{\gamma +1}{2+(\gamma-1)M^2}\right)^{0.5}$, $\frac{\rho}{\rho^*} = \frac{1}{M}\left(\frac{2+(\gamma-1)M^2}{\gamma +1}\right)^{0.5}$, $\frac{p_t}{p_t^*} = \frac{1}{M}\left(\frac{2+(\gamma-1)M^2}{\gamma +1}\right)^{\frac{\gamma +1}{2(\gamma-1)}}$, $\int_0^{L^*} \frac{4fdx}{D} = \left [ -\frac{1}{\gamma M^2} - \frac{\gamma +1}{2\gamma}\ln\left(\frac{M^2}{1+\frac{\gamma -1}{2}M^2}\right) \right ] _M ^1$, $\bar f = \frac{1}{L^*}\int_0^{L^*}f dx$, $\frac{4 \bar f L^*}{D} = \frac{1-M^2}{\gamma M^2} - \frac{\gamma +1}{2\gamma}\ln\left(\frac{M^2}{1+\frac{\gamma -1}{2}M^2}\right)$, $\frac{4 \bar f L^*}{D} = f(\gamma ,M)$, Historical Note: Sound Waves and Shock Waves, Hypersonic and High-Temperature Gas Dynamics Chapter 1 Notes.