lecture_1Velocity and Rates of Change

Differential equationNot to be confused with Difference equation.Stokes differential equations used to simulate airflow around an obstruction.ClassificationSolutionVisualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.A differential equation is amathematical equation that relatessome function of one or more variableswith its derivatives. Differentialequations arise whenever adeterministic relation involving somecontinuously varying quantities(modeled by functions) and their ratesof change in space and/or time(expressed as derivatives) is known orpostulated. Because such relations areextremely common, differentialequations play a prominent role inmany disciplines includingengineering, physics, economics, andbiology.Differential equations aremathematically studied from severaldifferent perspectives, mostlyconcerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have beendeveloped to determine solutions with a given degree of accuracy.ExampleFor example, in classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time.In some cases, this differential equation (called an equation of motion) may be solved explicitly.An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.Directions of studyThe study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.Mathematicians also study weak solutions (relying on weak derivatives), which are types of solutions that do not have to be differentiable everywhere. This extension is often necessary for solutions to exist.The study of the stability of solutions of differential equations is known as stability theory.NomenclatureThe theory of differential equations is well developed and the methods used to study them vary significantly with the type of the equation.Ordinary and partial•An ordinary differential equation (ODE) is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to considering a system of ordinary differential equations for a single function.Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are first-order and second-order differential equations. For example, Bessel's differential equation(in which y is the dependent variable) is a second-order differential equation. In the classical literature a distinction is also made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form. Also important is the degree, or (highest) power, of the highest derivative(s) in the equation (cf. : degree of a polynomial). A differential equation is called a nonlinear differential equation if its degree is not one (a sufficient but unnecessary condition).• A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second-order linear equations, is of utmost importance. Some partial differentialequations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.Linear and non-linearBoth ordinary and partial differential equations are broadly classified as linear and nonlinear.• A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products of the unknown function and its derivatives are not allowed) and nonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of an appropriate function space, which results in much more developed theory of linear differential equations. Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a constant coefficient linear differential equation.•There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit verycomplicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness).However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).ExamplesIn the first group of examples, let u be an unknown function of x, and c and ω are known constants.•Inhomogeneous first-order linear constant coefficient ordinary differential equation:•Homogeneous second-order linear ordinary differential equation:•Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:•Inhomogeneous first-order nonlinear ordinary differential equation:•Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a pendulum of length L:In the next group of examples, the unknown function u depends on two variables x and t or x and y.•Homogeneous first-order linear partial differential equation:•Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:•Third-order nonlinear partial differential equation, the Korteweg–de Vries equation:Related concepts• A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.• A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion equations.• A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.Connection to difference equationsSee also: Time scale calculusThe theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve approximation of the solution of a differential equation by the solution of a corresponding difference equation.Universality of mathematical descriptionMany fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.Notable differential equationsPhysics and engineering•Newton's Second Law in dynamics (mechanics)•Euler–Lagrange equation in classical mechanics•Hamilton's equations in classical mechanics•Radioactive decay in nuclear physics•Newton's law of cooling in thermodynamics•The wave equation•Maxwell's equations in electromagnetism•The heat equation in thermodynamics•Laplace's equation, which defines harmonic functions•Poisson's equation•Einstein's field equation in general relativity•The Schrödinger equation in quantum mechanics•The geodesic equation•The Navier–Stokes equations in fluid dynamics•The Diffusion equation in stochastic processes•The Convection–diffusion equation in fluid dynamics•The Cauchy–Riemann equations in complex analysis•The Poisson–Boltzmann equation in molecular dynamics•The shallow water equations•Universal differential equation•The Lorenz equations whose solutions exhibit chaotic flow.Biology•Verhulst equation – biological population growth•von Bertalanffy model – biological individual growth•Lotka–Volterra equations – biological population dynamics•Replicator dynamics – found in theoretical biology•Hodgkin–Huxley model – neural action potentialsEconomics•The Black–Scholes PDE•Exogenous growth model•Malthusian growth model•The Vidale–Wolfe advertising modelReferences•P. Abbott and H. Neill, Teach Yourself Calculus, 2003 pages 266-277•P. Blanchard, R. L. Devaney, G. R. Hall, Differential Equations, Thompson, 2006• E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955• E. L. Ince, Ordinary Differential Equations, Dover Publications, 1956•W. Johnson, A Treatise on Ordinary and Partial Differential Equations[2], John Wiley and Sons, 1913, in University of Michigan Historical Math Collection [3]• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.•R. I. Porter, Further Elementary Analysis, 1978, chapter XIX Differential Equations•Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems[4]. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.[1]/w/index.php?title=Template:Differential_equations&action=edit[2]/cgi/b/bib/bibperm?q1=abv5010.0001.001[3]/u/umhistmath/[4]http://www.mat.univie.ac.at/~gerald/ftp/book-ode/External links•Lectures on Differential Equations (/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/) MIT Open CourseWare Videos•Online Notes / Differential Equations (/classes/de/de.aspx) Paul Dawkins, Lamar University•Differential Equations (/diffeq/diffeq.html), S.O.S. Mathematics•Differential Equation Solver (/tools/differential_equation_solver/) Java applet tool used to solve differential equations.•Introduction to modeling via differential equations (/mat/u-u/en/ differential_equations_intro.htm) Introduction to modeling by means of differential equations, with critical remarks.•Mathematical Assistant on Web (http://user.mendelu.cz/marik/maw/index.php?lang=en&form=ode) Symbolic ODE tool, using Maxima•Exact Solutions of Ordinary Differential Equations (http://eqworld.ipmnet.ru/en/solutions/ode.htm)•Collection of ODE and DAE models of physical systems (/research/models.htm) MATLAB models•Notes on Diffy Qs: Differential Equations for Engineers (/diffyqs/) An introductory textbook on differential equations by Jiri Lebl of UIUC•Khan Academy Video playlist on differential equations (/math/ differential-equations) Topics covered in a first year course in differential equations.•MathDiscuss Video playlist on differential equations (/category/courses/ solutions-differential-equations/homogeneous-linear-systems/)Article Sources and Contributors8Article Sources and ContributorsDifferential equation Source: /w/index.php?oldid=610771276 Contributors: 17Drew, After Midnight, Ahoerstemeier, Alarius, Alfred Centauri, Amahoney, AndreiPolyanin, Andres, AndrewHowse, Andycjp, Andytalk, AngryPhillip, Anonymous Dissident, Antoni Barau, Antonius Block, Anupam, Apmonitor, Arcfrk, Asdf39, Asyndeton, Attilios,Babayagagypsies, Baccala@, Baccyak4H, Bejohns6, Bento00, Berland, Bidabadi, Bigusbry, BillyPreset, Bob.v.R, Bolatbek, Brandon, Bryanmcdonald, Btyner, Bygeorge2512,Callumds, Charles Matthews, Christian75, Chtito, Cispyre, Cmprince, Coginsys, ConMan, Cxz111, Cybercobra, DAJF, Danski14, Dbroadwell, Ddxc, Delaszk, DerHexer, Dewritech, Difu Wu, Djordjes, DominiqueNC, Donludwig, Dpv, Dr sarah madden, Drmies, DroEsperanto, Duoduoduo, Dysprosia, EconoPhysicist, Elwikipedista, Epicgenius, EricBright, Erin.Annette.Brown,Estudiarme, F=q(E+v^B), Fintor, Fioravante Patrone, Fioravante Patrone en, Flameturtle, Friend of the Facts, FutureTrillionaire, Gabrielleitao, Gandalf61, Gauss, Genedronek, Geni, Giftlite,GoingBatty, Gombang, Grenavitar, Haham hanuka, Hamiltondaniel, Harry, Haruth, Haseeb Jamal, Heikki m, Holmes1900, Ilya Voyager, Iquseruniv, Iulianu, Izodman2012, J arino, J.delanoy, Ja 62, Jak86, JamesBWatson, Jao, Jarble, Jauhienij, Jayden54, Jeancey, Jersey Devil, Jim Sukwutput, Jim.belk, Jim.henderson, JinJian, Jitse Niesen, JohnOwens, Johndoeisnotmyname, JorisvS,Julesd, K-UNIT, Kayvan45622, KeithJonsn, Kensaii, Khalid Mahmood, Klaas van Aarsen, Kr5t, Krushia, LOL, Lambiam, Lavateraguy, Lethe, LibLord, Linas, Lumos3, Madmath789, Mandarax, Mankarse, MarSch, Martastic, Martynas Patasius, Maschen, Math.geek3.1415926, Matqkks, Mattmnelson, Maurice Carbonaro, Maxis ftw, Mazi, McVities, Mduench, Mets501, Mh, MichaelHardy, Mindspillage, MisterSheik, Mohan1986, Mossaiby, Mpatel, MrOllie, Mtness, Mysidia, Nik-renshaw, Nkayesmith, Norm mit, Okopecz, Oleg Alexandrov, Opelio, Pahio, Parusaro, Paul August, Paul Matthews, Paul Richter, PavelSolin, Pgk, Phoebe, Pine, Pinethicket, Pratyya Ghosh, PseudoSudo, Qwerty Binary, Qzd800, R'n'B, Rama's Arrow, Randomguess, Reallybored999, RexNL, Reyk, RichMorin, Robin S, Romansanders, Rosasco, Ruakh, SDC, SFC9394, SakeUPenn, Salix alba, Sam Staton, Sampathsris, Sardanaphalus, Senoreuchrestud, Silly rabbit, Siroxo,Skakkle, Skypher, SmartPatrol, Snowjeep, Spirits in the Material, Starwiz, Suffusion of Yellow, Sverdrup, Symane, TVBZ28, TYelliot, Tannkrem, Tbhotch, Tbsmith, TexasAndroid, Tgeairn, The Hybrid, The Thing That Should Not Be, Timelesseyes, Tranum1234567890, Tsirel, Tuseroni, User A1, Vanished User 0001, Vishwanathnm, Vthiru, Waffleguy4, Waldir, Waltpohl, Wavelength, Wclxlus, Wihenao, Willtron, Winterheart, Wsears, XJaM, Yafujifide, Zepterfd, ﺪﺟﺎﺳ ﺪﺠﻣﺍ ﺪﺟﺎﺳ, 363 anonymous editsImage Sources, Licenses and ContributorsFile:Airflow-Obstructed-Duct.png Source: /w/index.php?title=File:Airflow-Obstructed-Duct.png License: Public Domain Contributors: Original uploader was User A1 at en.wikipediaFile:Elmer-pump-heatequation.png Source: /w/index.php?title=File:Elmer-pump-heatequation.png License: Creative Commons Attribution-Sharealike 3.0Contributors: Christian1985, Crimerob, Kri, User A1, 2 anonymous editsLicenseCreative Commons Attribution-Share Alike 3.0///licenses/by-sa/3.0/。

比例变化英语作文Title: The Dynamics of Proportional Change: Exploring its Impact and Implications。

Introduction。

Proportional change is a fundamental concept in various aspects of life, encompassing economics, science, mathematics, and even social dynamics. This essay delves into the multifaceted nature of proportional change, its significance, and the implications it carries across different domains.Understanding Proportional Change。

Proportional change refers to alterations in quantities or values relative to their initial states, maintaining a consistent ratio throughout the transformation. It is often expressed in terms of percentages, ratios, or rates of change. This concept underpins numerous phenomena, fromeconomic growth and scientific discoveries to demographic shifts and technological advancements.Economic Perspectives。

16.333:Lecture# 14Equations of Motion in a Nonuniform AtmosphereGusts and Winds1Fall2004 16.33312–2Equations of Motion• Analysis to date has assumed that the atmosphere is calm andfixed –Rarely true since we must contend with gusts and winds–Need to understand how these air motions impact our modelingof the aircraft.• Must modify aircraft equations of motion since the aerodynamic forces and moments are functions of the relative motion between the aircraft and the atmosphere,and not of the inertial velocities.F–Thus the LHS of the dynamics equations(�=m�a)must bewritten in terms of the velocities relative to the atmosphere.–If u is the aircraft perturbation velocity(X direction),and u g isthe gust velocity in that direction,then the aircraft velocity withrespect to the atmosphere isu a =u−u g• Now rewrite aerodynamic forces and moments in terms of aircraft velocity with respect to the atmosphere(see4–11)∂X∂X ∂X ΔX=(u−u g)+(w−w g)+∂˙(˙w˙g)∂U∂W W w−∂X∂X∂X g+(q−q g)+...+θ+...+θ+ΔX c∂Q ∂Θ∂Θ–The gravity terms∂X g and control terms∂ΘΔX c=Xδe δe+Xδpδpstay the same.• The rotation gusts p g,q g,and r g are caused by spatial variations in the gust components⇒rotary gusts are related to gradients of the vertical gustfield∂w g ∂w gp g=and q g=∂y ∂xFigure1:Gust Field creating an effective pitching gust.� � • The next step is to include these new forces and moments in the equations of motion ⎤⎡⎤⎡⎤⎡⎤⎡ 0 0 0 u ˙X u X w 0 −mg cos Θ0m u ⎢⎢⎢⎣ ⎥⎥⎥⎦ ⎥⎥⎥⎦ ⎢⎢⎢⎣ ⎢⎢⎢⎣ ⎥⎥⎥⎦ 0 m − Z ˙w 0 0 0 −M ˙w I yy 0 0 0 0 1 w ˙ q ˙ Z u Z w Z q + mU 0 −mg sin Θ0 0M u M w M q w q = θ˙001 0 θ ⎤⎡ X δe X δp Z δe Z δp M δe M δp ⎥⎥⎥⎦ δe δt⎢⎢⎢⎣ + 00 0−X u −X w ⎤⎡ ⎤ ⎡ ⎢⎢⎢⎣ ⎣ ⎥⎥⎥⎦ u g 0−Z u −Z w −M u −M w −M q ⎦ + w g q g 000ˆB u u +ˆEx˙= Ax +ˆB w w ⇒• Multiply through by E −1 to get new state space modelx ˙= Ax + B u u + B w wwhich has both control u and disturbance w inputs.– A similar operation can be performed for the lateral dynamics in terms of the disturbance inputs v g , p g , and r g .• Can now compute the response to specific types of gusts, such as a step or sinusoidal function, but usually are far more interested in the response to a stochastic gust field⎥⎥⎥⎦ ⎢⎢⎢⎣Atmospheric Turbulence• Can develop the best insight to how aircraft will behave with gust disturbances if we treat the disturbances as random processes.–What is a random process?Something(signal)that is random sothat a deterministic description is not practical!!–But we can often describe the basic features of the process(e.g.mean value,how much it varies about the mean).• Atmospheric turbulence is a random process,and the magnitude of the gust can only be described in terms of statistical properties.–For a random process f(t),talk about the mean square� T1f2(t)=lim f2(t)dtT 0T→∞as a measure of the disturbance intensity(how strong it is).• Signal f(t)can be decomposed into its Fourier components,so can use that to develop a frequency domain measure of disturbance strength –Φ(ω)≈that portion of f2(t)that occurs in the frequency band→ωω+dω–Φ(ω)is called the power spectral density• Bottom line:For a linear system y=G(s)w,thenΦy(ω)=Φw(ω)|G(jω)2|⇒Given an input disturbance spectral density(e.g.gusts),quite sim­ple to predict expected output(e.g.ride comfort,wing loading).Implementing a PSD in Matlab•Simulink has a Band­Limited White Noise block that can be used for continuous systems.–Primary difference from the Random Number block is that this block produces output at a specific sample rate,which is related to the correlation time of the noise.• Continuous white noise has a correlation time of0⇒flat power spectral density(PSD),and a covariance of infinity.–Non­physical,but a useful approximation when the noise distur­bance has a correlation time that is small relative to the natural bandwidth of the system.–Can simulate effect of white noise by using a random sequence with a correlation time much smaller than the shortest time con­stant of the system.• Band­Limited White Noise block produces such a sequence where the correlation time of the noise is the sample rate of the block.–For accurate simulations,use a correlation time t c much smaller than the fastest dynamics f max of the system.12πt c≈100f max�• Power spectral densities for Von Karman model (see MIL­F­8785C)−5/6 Φu g (Ω) = σ2 2L u � 1 + (1.339L u Ω)2u πwhere σu is the intensity measure of the disturbance, Ω is the spatial frequency variable, and L u is a length scale of the disturbance.– All scale parameters depend on day, altitude, and turbulence type. – Different models available for each direction.– Scale length parameter L (in feet) varies with altitude – MIL­F­8785C model valid up to 1000 feeth L u = (0.177 + 0.000823h )1.2– Turbulence intensity for low altitude flight MIL­F­8785C as0.1W 20σu = (0.177 + 0.000823h )0.43 where W 20 is the wind speed as measured at 20 ft3 W 20 < 15 knots is classified as “light” turbulenceW 20 ≈ 30 knots is “moderate”W 20 > 45 knots is “heavy”–Standard approach:assume turbulencefieldfixed(frozen)in space ⇒aircraft is justflying into it.3Then can relate turbulence spatial frequencyΩto the temporal frequencyωthat the aircraft would feelω⇒ω=ΩU0Ω≡U0–Used to model how the aircraft will behave in bumpy air.Figure2:Wind gust examples� � Fall 2004 16.333 12–9Wind Shear• Wind shear is of special interest because it involves significant local changes in the vertical and horizontal velocity (e.g. downdraft) – Particularly important near airports during landings.• Simple analysis – consider the case shown, where the change in (hor­izontal) wind velocity is represented by du u g =Δh dhwhere– Δh corresponds to changes in altitude (what we previously just called h )– Value du/dh gives magnitude of wind shear , which is 0.08–0.15s −1 for moderate and 0.15–0.2s −1 for strong.Figure 3: Aircraft descending into a horizontal windshearBut can write the changes in altitude (use h˙=Δh/Δt ) as • h˙=U 0(θ − α)� � � Fall 2004 16.333 12–10 • Now have a coupled situation that is quite interesting– Forces on the aircraft change with altitude (h ) because height changes the gust velocity– Changes in the forces on the aircraft impact the height of the airplane since both θ and α will change.• Analyze the coupling by looking at the longitudinal equations��T x = u w q θh h = 00001 x = C h xwith controls fixed (zero) and input u gx ˙= Ax +˜˜B w (:, 1)˜B w (:, 1)u g = Ax +˜du h dh du ˜˜= Ax + B w (:, 1)C h x � dh du ˜= A ˜+ B w (:, 1)C h x dh⇒ dynamics have been modified because of the coupling between the change in altitude and the change in forces (with u g ).• Closed this loop on the B747 dynamics to obtaindu Phugoid Polesdh 0 0.08 0.15 0.2 ­0.0033 ± 0.0672i­0.0014 ± 0.1150i0.0002 ± 0.1442i0.0014 ± 0.1619i– Clearly this coupling is not good, and an unstable Phugoid mode is to be avoided during landing operations.Fall200416.33312–11 Wind Code1%Gust modeling2%16.333,Fall20043%Jonathan P.How45Xu=­1.982e3;Xw=4.025e3;Zu=­2.595e4;Zw=­9.030e4;Zq=­4.524e5;Zwd=1.909e3;6Mu=1.593e4;Mw=­1.563e5;Mq=­1.521e7;Mwd=­1.702e4;7%8g=9.81;theta0=0;S=511;cbar=8.324;9U0=235.9;Iyy=.449e8;m=2.83176e6/g;cbar=8.324;rho=0.3045;10Xdp=.3*m*g;Zdp=0;Mdp=0;11Xde=­3.818e­6*(1/2*rho*U0^2*S);Zde=­0.3648*(1/2*rho*U0^2*S);12Mde=­1.444*(1/2*rho*U0^2*S*cbar);;13%14Ehat=[m000;0m­Zwd00;0­Mwd Iyy0;0001];15Ahat=[Xu Xw0­m*g*cos(theta0);[Zu Zw Zq+m*U0­m*g*sin(theta0)];16[Mu Mw Mq0];[0010]];17Bhat=[Xde Xdp;Zde Zdp;Mde Mdp;00];18%19%form the gust input matrix20Bwhat=[­Xu­Xw0;­Zu­Zw0;­Mu­Mw­Mq;000];21%22%add height state23Ehat(5,5)=1;%\dot h state not coupled24Ahat(5,5)=0;Ahat(5,[1:4])=[0­10U0];%add height state25Bhat(5,2)=0;%noinput26Bwhat(5,3)=0;%no input2728%form the full model\hat B_w w29%E\dot x=\hat A x+\hat B u_controls+30%==>\dot x=A x+B u_controls+B_w w31%32A=inv(Ehat)*Ahat;33B=inv(Ehat)*Bhat;34Bw=inv(Ehat)*Bwhat;35%36%set u_controls=037%assume that w_g=q_g=0and38%u_g=(du/dh)h39%40du_dh=[0.08.15.2];41A1=A+Bw(:,1)*[00001]*du_dh(1);42A2=A+Bw(:,1)*[00001]*du_dh(2);43A3=A+Bw(:,1)*[00001]*du_dh(3);44A4=A+Bw(:,1)*[00001]*du_dh(4);4546ev1=eig(A1);ev2=eig(A2);ev3=eig(A3);ev4=eig(A4);47plot([ev1ev2ev3ev4],’x’)48[ev1ev2ev3ev4]。

ResearchGreen Chemical Engineering—ArticleLaminar-to-Turbulence Transition Revealed Through a Reynolds NumberEquivalenceXiao Dong ChenSchool of Chemical and Environmental Engineering,Soochow University,Suzhou 215123,Chinaa r t i c l e i n f o Article history:Received 3August 2018Revised 20September 2018Accepted 27September 2018Available online 23March 2019Keywords:Local Reynolds number equivalenceFlow transition from laminar to turbulent modeUniversal Law of the Wall Pipe flow Plate flow Modelinga b s t r a c tFlow transition from laminar to turbulent mode (and vice versa)—that is,the initiation of turbulence—is one of the most important research subjects in the history of engineering.Even for pipe flow,predicting the onset of turbulence requires sophisticated instrumentation and/or direct numerical simulation,based on observing the instantaneous flow structure formation and evolution.In this work,a local Reynolds number equivalence c (ratio of local inertia effect to viscous effect)is seen to conform to the Universal Law of the Wall,where c =1represents a quantitative balance between the abovementioned two effects.This coincides with the wall layer thickness (y +=1,where y +is the dimensionless distance from the wall surface defined in the Universal Law of the Wall).It is found that the characteristic of how the local derivative of c against the local velocity changes with increasing velocity determines the onset of turbu-lence.For pipe flow,c %25,and for plate flow,c %151.5.These findings suggest that a certain combina-tion of c and velocity (nonlinearity)can qualify the source of turbulence (i.e.,generate turbulent energy).Similarly,a re-evaluation of the previous findings reveals that only the geometrically narrow domain can act locally as the source of turbulence,with the rest of the flow field largely being left for transporting and dissipating.This understanding will have an impact on the future large-scale modeling of turbulence.Ó2019THE AUTHOR.Published by Elsevier LTD on behalf of Chinese Academy of Engineering and Higher Education Press Limited Company.This is an open access article under the CC BY-NC-ND license(/licenses/by-nc-nd/4.0/).1.IntroductionAt present,predicting the onset of turbulence,even for pipe flow,requires sophisticated instrumentation and/or direct numer-ical simulation (DNS)[1–4],based on observations of the detailed instantaneous flow structure formation and evolution.However,all of this modern research is conducted around the classical criti-cal Reynolds number (Re c ).Osborne Reynolds (1842–1912)carried out thorough laboratory investigations on the behavior of Newto-nian fluids [5,6].His most remarkable discovery was the identifica-tion of the two modes of flow phenomena:laminar flow and turbulent flow [5–7].The experimental methodology and theory proposed by Reynolds to investigate the transition from one type of flow to another have inspired numerous researchers over gener-ations.The transition between these two types of flow is marked by a dimensionless parameter attributed to Reynolds—that is,the Reynolds number (Re ):Re ¼q Ud lð1Þwhere q is the fluid density (kg Ám À3),l is the fluid viscosity (Pa Ás),U is a characteristic velocity (m Ás À1),and d is a characteristic dimen-sion of the object with which the fluid is in contact (m).In a pipe,d is the inner pipe diameter;however,if the fluid flows around the pipe outside (cross-flow),d becomes the outer diameter.This number has often been said to represent the ratio of the inertia forces to the viscous forces.As the most important parameter,the Re ,together with other fluid-related dimensionless parameters,provides a powerful foundation for many friction,heat,and mass-transfer correlations in fluid flow-related problems.These are par-ticularly useful in designing process equipment and process opti-mizations [7].While appreciating the experiments carried out by Reynolds,it is notable that the diameter of the pipe was limited;hence,a large Re might be obtained mainly by changing the fluid viscosity and/or increasing the fluid velocity.Flow visualization took place in the central region of the pipe (the ink fluid was injected at the center location),so it would have been the result of the integrated or cumulative effect of turbulence generation along the pipe wall,transport,and dissipation.These three aspects would have been intertwined in the visualization in the experi-ments,and the Re should be viewed as a global parameter.In 1952,measurements in the proximity of the pipe wall showed aE-mail address:xdchen@very significant result:that using the friction velocity(u r)and the product of the kinematic viscosity and the friction velocity(vu r) to scale velocity and distance,respectively,away from the wall,a unique dimensionless velocity profile in the near-wall region was obtained.As calculated,based on the measurements,the rate of tur-bulence generation reaches a sharp maximum at the sub-layer thickness(y+%11.5,where y+is the dimensionless distance from the wall surface defined in the Universal Law of the Wall(ULW)) [8].From a rational perspective,the broad peak as shown may be better qualified as y+%11.5±5.The commonly acknowledged divide between the laminar sub-layer and the buffer layer is marked at y+=5for a fully developed turbulent wall layer[8,9].Micro-transient details of how afluid transitions from being disturbed by localized perturbation into full-blown turbulence in a(long)pipe have only been captured very recently[1,2].Sampling stations for local behaviors have been set up,facilitated by advanced computing power and modern experimental techniques. Experiments have been conducted below and above the well-known Re c for pipeflow—that is,Re c=2300,where the subscript c represents critical.In a small-diameter pipe in a laboratory setting,turbulence that is transient at low Re becomes sustained after a distinct Re c;how-ever,this phenomena was captured locally(unlike the general type of observation originally made by Reynolds)[1,2].The critical point for transiting to sustained turbulence is decided when the local proliferation of puffs outweighs their decay.Experimentally artifi-cial puffs were generated at precision to trigger turbulent behavior. Two timescales were captured(partly established through DNS)for the decay and spreading of the puffs.Plotting the Re dependence of the mean time until a second puff is nucleated and the turbulence fraction increases(declining with increasing Re),and the Re depen-dence of the mean time until the turbulence decays and theflow relaminarizes(increasing with increasing Re),creates a very sharp intersect at Re c=2040±10,marking the onset of laminar-to-sustained turbulence in pipeflow[1].To explain this transition from laminarflow to turbulence,a bi-stability analysis with nonlinear propagation(advection)of turbu-lent fronts has been executed[2].The interesting phenomena of destabilizing turbulence in pipeflow were subsequently studied using the same experimental strategies and DNS[3,4].It is worth noting that most practical problems in thisfield, including airplane design,are simulated with semi-empirical tur-bulent models for turbulent kinetic energy and the Reynolds stres-ses.These models make computation more efficient.Although DNS is seen to be the ultimate way to directly generate images of turbu-lence,our knowledge about turbulence still mostly comes from intuitive prospects,whether reported or taught in classes.In the present work,a dimensionless number is reported that is deduced intuitively from the concept of Re but applied to the local fluidflow.This dimensionless number is the ratio of the inertia effect to viscous effect,and its definition allows for an alternative analysis of the onset of turbulence,which has not previously been seen.Three classical cases influid mechanics are employed to show the effectiveness of the approach:the ULW,flow in a smooth circular pipe,and parallelflow on a smooth plate[9].The analytical velocity profiles of these cases are well known[9–13],allowing derivations to be made to demonstrate the intended arguments precisely.This philosophy is in line with what Churchill[11] reported in his famous American Institute of Chemical Engineers Institute Lecture—that is,elucidating the fundamentals of trans-port phenomena without computationalfluid dynamics.Given this new number,beyond capturing the onset of turbu-lence,the author points to a significant possibility that turbulence (i.e.,turbulent energy)originates from a very narrow domain(s) (defined by c(ratio of local inertia effect to viscous effect)and velocity),leaving the rest of theflowfield for transporting and dissipating turbulent energies.This perspective creates consider-able scope for controlling turbulentflow and provides an idea for future improvements in turbulence-modeling effectiveness on large scales.2.Main analyses2.1.Defining the local ratio of inertia effect to viscous effectTo introduce the new dimensionless number,for simplicity,a semi-infinite Cartesian(x,y)parallelflow scheme,with one side bounded by a smoothflat solid wall(the smooth plate)is consid-ered.Taking u as the local velocity in the x-direction(parallel to the plate),and recognizing that the predominant velocity gradient occurs in the y-direction,the prominent shear stress can be expressed as s¼Àl@u=@y.The no-slip condition is applied at the plate surface;hence,the scaling consideration leads tos%Àlðu chÀ0Þ=D ch.Here,D ch is a characteristic distance corre-sponding to the representative velocity change of interest and u ch is a characteristic velocity,where the subscript ch represents the characteristic value of the system.Taking the above to represent the viscous effects,a new dimensionless number is deduced: c¼q l2=ðl u ch=D chÞ.As D ch?0,the new local dimensionless num-ber can be expressed locally:c¼q u2l@u=@yj jð2ÞIn Eq.(2),the absolute value is employed to avoid any confu-sion.Based on the derivation,it can be seen that this number is conceptually similar to the Re.It is argued that this number holds important physical meaning when interpretingfluid behavior at a finite point(x,y).If c becomes very large,the viscous effect becomes negligible,and thefluid at that point should be able to maintain its pathway without changing direction.If thefluidflow becomes turbulent,the instantaneous velocity u in Eq.(2)may be replaced with the time-averaged local velocity u,according to con-ventional wisdom.An appropriate value of c must be attained in order to produce an eddy or eddies.On the other hand,theflow must be energetic enough to begin with,if turbulence can be sus-tained(i.e.,u must be large).When the two effects are comparable, it should be found that c%1.It is envisaged that for different direc-tions in a generalflow domain,c is directionally dependent.It is further noted that c is in fact different from Re,because when the Re increases over a critical value,turbulence must occur.On the other hand,c can vary from zero to infinity,even for laminar flows.2.2.Conforming c to the ULWFirst,it is found that c conforms to the ULW,thus demonstrat-ing significant physical meaning.For a large Re,the wall-bounded turbulentflows exhibit boundary layers that fall within the dimen-sionless velocity distribution of an approximately universal nature. Many measurements have demonstrated the ULW[7–9].Large-scale(industrial)turbulence modeling often takes advantage of the ULW to create a wall function in order to avoid detailed com-putations near the wall and thereby reduce the computing effort. In the ULW,thefluid boundary layer is divided into three regions: a(pure)viscous sub-layer(also called the wall layer),a buffer layer,and an overlap layer[9].For the viscous sub-layer, 0y+\5and u+=y+,where y+=y/d v.The wall layer thickness (y+=1)is d v=v/u r.The friction velocity u r is defined as u r¼ffiffiffiffiffiffiffiffiffiffiffiffis w=qp,where s w is the time-averaged shear stress at the wall (NÁmÀ2).u+is defined as u+= u=u r.X.D.Chen/Engineering5(2019)576–579577This leads to the following:c ¼yd v2¼y þðÞ2ð3ÞEq.(3)presents a significant finding that,in essence,when y is equal to the wall layer thickness,c =1,the inertia forces are com-parable to the shear forces.It is also interesting to note that y +=5is the conventional divide for the viscous sub-layer and the buffer layer,at which c =25.y +=11.5may also be important,as this is where turbulence production has been reported to peak in the ULW.At this point,it is intuitive to think that there may be a crit-ical c that corresponds to the range marked by the above number moving from laminar to turbulence,provided that the fluid in this location has sufficient energy (or sufficiently high velocity).As mentioned above,in both laminar and turbulent flows,c can vary from zero to infinity.Therefore,it is obvious that a single c value cannot be a sole marker for generating turbulence.2.3.Pipe flowThe strategy,then,is to seek the relationship between c and changing velocity.As shown in Appendix A in the Supplementary data ,c can be obtained as a function of r/R (r is the radial coordi-nate and R is the radius of pipe)first for different Re (s),where Re is defined using the mean velocity u m .Then,via the relationship between u and r in this classic case,c can be further obtained as a function of u/u m .Differentiating this u/u m dependence function against u/u m ,for both the fully developed laminar and the fully developed turbulent regimes,yields a useful characteristic of c varying with velocity.For laminar flow,it is the parabolic velocity profile;for turbulent flow,it is the 1/7th power velocity profile.It can be shown that at point c xx %25(corresponding to y +=5),where the subscript xx indicates that the inertia effects in the x -direction interact with the shear effects applied in the x -direction as well,the crossover of the two derivatives against u/u m yields Re %2083.At this point,u/u m %0.597(see Fig.S1in the Supplementary data ,which demonstrates the obtainment of this result).When the time-averaged velocity profile for turbulence is generalized to be of the 1/N th order,where N is the power of the classic approximation of the time-averaged velocity distribution in turbulence regime (dimensionless),especially with N =11,it can be shown that Re =2005.75is critical (see Appendix A in the Sup-plementary data ).Here,the velocity at c xx =25and u=u m =0.650(see Fig.S2in the Supplementary data ).This analysis indicates that when c xx =25,if the fluid flow at that location has sufficient power,turbulence occurs.2.4.Plate flowIn contrast to the flow in a pipe,the flow parallel to and above a flat plate is at least two-dimensional.It is well known that the solution to a laminar velocity distribution in the plate boundary layer can be obtained accurately through similarity solution proce-dures [9];that is, u=U 1=f (g ),where g is the dimensionless trans-formation variable g ¼y =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq U 1=2l x p and U 1is the velocity of the bulk fluid.The laminar velocity profile can also be expressed in a parabolic format,while the turbulent profile can be expressed approximately as a 1/7th power format when scaled using the boundary layer thickness (see Appendix B in the Supplementarydata ).Once again,d c xx /d( u=U 1)can be obtained for both flow modes,respectively.The crossover is set at Re x ,c =5.5Â105,which is the oft-mentioned Re c for the onset of turbulence in plate flow.It is found that c xx %154.6(y +%12.4)(see Appendix B in the Supple-mentary data ).At this point, u=U 1=0.441.The obtainment of the critical parameter is shown in Fig.S3in the Supplementary data .In fact,the literature tends to suggest a range for the onset Re for turbulence of 105–106.The low estimate,Re x ,c =105,yieldsc xx =89.24(y +=9.45),for which u=U 1=0.487.This result actually aligns well with the critical condition for pipe flow.2.5.Reversing back from turbulence to laminar flowIf the velocity profile is changed while the flow rate is kept the same in a smooth and straight pipe,it is interesting to see whether Re c for the onset of turbulence changes or not.It is possible to pre-fix c to be 25,and then see whether Re c is influenced by altering the velocity profile.It is shown that when the law of 1/7th for turbulence is changed to the 1/20th law for turbulence,where the velocity profile becomes flatter,Re c becomes 2485,in contrast to 2083for the 1/7th law.In other words,it is possible to reverse the already turbulent situation ‘‘back”to the laminar situation if the velocity profile is somehow forced to be flatter in the gap of Re =2485–2083.In general,when N becomes greater than 7,Re c also increases.This result aligns well with the original work reported recently [3].3.Further remarksIt has been successfully shown that the onset of turbulence can be interpreted through the introduction of c ,which is the ratio of local inertia effect to viscous effect.Based on the well-established velocity profiles,it is possible to evaluate the critical transition Re through the relationship of how the local derivative of c against velocity changes with velocity.The sensitive region for flow transi-tion is narrow,based on the analyses given in this work (see Figs.S1–S3in the Supplementary data ;beyond the crossover points,the change in the local derivative of c against velocity (as well as that of c )increases rapidly with increasing velocity,and no further crossover can be found).Increasing c would dam the tur-bulence,even it was already generated.It is probable that only a very thin or narrow geometrical region (i.e.,a line or a shell)is cap-able of sustaining turbulence generation.Upon a further analysis of previous results [1],albeit not elaborated in that study (see Fig.5in Ref.[1]),where the mean time of a puff from the wall before decay-ing or splitting as a function of Re in the pipe is shown,the occur-rence of a very sharp critical phenomena is suggested.A lesser or greater Re than the Re c has a much lower chance of producing sus-tained randomness (Fig.1).A previous work [2](see Fig.3in Ref.[2])also shows that the level of turbulence reflected by the cross-stream velocity fluctuations v 0/U does not actually increasemuchFig.1.Mean lifetime of a puff before decaying (solid line)or splitting (dashed line),plotted using the mean lifetime functions of the Re created previously [1].578X.D.Chen /Engineering 5(2019)576–579with increasing Re,once past the Re c.It is thus highly probable that the source of(strongest)turbulence is located in a narrow region (s);furthermore,if this were true,then the rest of theflowfield would largely be left for the transportation and dissipation of tur-bulence energy.This perspective would have a profound influence on the modeling of turbulence.In future studies,it will be helpful to visualize and compare the c distribution for both laminar and turbulent regimes with the sameflow arrangement and in the same device.Finally,it is emphasized again that despite the vast difference between the Re c found for pipeflow and for plateflow,respectively,the current dimensionless parameters under critical conditions for the two cases are not that different.AcknowledgementsThe author is grateful to his father,Prof.Naixing Chen(1933–2018),who was thefirst to introduce him to thefield offluid mechanics over35years ago;the author had discussed the initial ideas of this paper with him not long before he fell terminally ill. Some17months were spent working on and off as a research assis-tant in Prof.Lixing Zhou’s laboratory at Tsinghua University in 1985–1987,on a code for simulating a two-dimensional multi-phaseflow in a sudden-expansion combustion chamber.The per-sonal knowledge of Dr.Tuoc Trinh of Canterbury University and later of Fonterra New Zealand in the late1980s to early1990s, respectively,was a real inspiration in thinking about wall turbu-lence.Dr.Trinh wrote a remarkable PhD thesis in the early2000s on his original ideas on boundary layer turbulence. Nomenclatured characteristic dimension of the object(m)Re Reynolds number(dimensionless)U characteristic velocity(mÁsÀ1)u, u local velocity and time-averaged local velocity,respectively(mÁsÀ1)U1velocity of the bulkfluid in plateflow(mÁsÀ1)u m mean velocity(mÁsÀ1)u r friction velocity(mÁsÀ1)as defined in the ULW[8–12]v kinematic viscosity(m2ÁsÀ1)r the radial coordinate(m)R the radius of pipe(m)N the power of the classic approximation of thetime-averaged velocity distribution in turbulence regime(dimensionless)u ch a characteristic velocity(mÁsÀ1)x x-coordinate in the Cartesian systemy y-coordinate in the Cartesian systemy+dimensionless distance from the wall surface defined in the ULWd v wall layer thickness(m)[8,13]D c characteristic distance corresponding to the representativevelocity change(m)c ratio of local inertia effect to viscous effect(dimensionless) g dimensionless transformation variable in the classicsimilarity solution of plateflowlfluid viscosity(PaÁs)qfluid density(kgÁmÀ3)s, s shear stress and time-averaged shear stress,respectively(Pa)Appendices A and B.Supplementary dataSupplementary data to this article can be found online at https:///10.1016/j.eng.2018.09.013.References[1]Avila K,Moxey D,de Lozar A,Avila M,Barkley D,Hof B.The onset of turbulencein pipeflow.Science2011;333(6039):192–6.[2]Barkley D,Song B,Mukund V,Lemoult G,Avila M,Hof B.The rise of fullyturbulentflow.Nature2015;526(7574):550–3.[3]Hof B,de Lozar A,Avila M,Tu X,Schneider TM.Eliminating turbulence inspatially intermittentflows.Science2010;327(5972):1491–4.[4]Kühnen J,Song B,Scarselli D,Budanur NB,Ried M,Willis AP,et al.Destabilizingturbulence in pipeflow.Nat Phys2018;14(4):386–90.[5]Reynolds O.An experimental investigation of the circumstances whichdetermine whether the motion of water shall be direct or sinuous,and of the law of resistance in parallel channels.Philos Trans R Soc Lond1883;174: 935–82.[6]Tokaty GA.A history and philosophy offluid mechanics.New York:Dover;1971.[7]Bird RB,Stewart WE,Lightfoot EN.Transport phenomena.2nd ed.NewYork:John Wiley&Sons;2002.[8]Laufer J.The structure of turbulence in fully developed pipeflow.NACAtechnical report.United States:National Bureau of Standards;1953Jun.Report No.:NACA-TN-2954.[9]Schlichting H,Gersten K.Boundary-layer theory.8th ed.Berlin:Springer;2003.[10]Churchill SW.Progress in the thermal sciences:AIChE Institute Lecture.AIChE J2000;46(9):1704–22.[11]Nikuradse J.Gesetzmassigkeiten der turbulenten stromung in glattenrohren.Berlin:VDI Verlag;1932.German.[12]Pai SI.On turbulentflow in circular pipe.J Franklin Inst1953;256(4):337–52.[13]Çengel YA,Cimbala JM.Fluid mechanics—fundamentals and applications.2nded.New York:McGraw-Hill Higher Education;2006.X.D.Chen/Engineering5(2019)576–579579Engineering 2 (2016) xxx–xxxResearchGreen Chemical Engineering—Article通过一个等价雷诺数揭示层流到湍流的转捩区域陈晓东School of Chemical and Environmental Engineering, Soochow University, Suzhou 215123, Chinaa r t i c l e i n f o摘要Article history:Received 3 August 2018Revised 20 September 2018Accepted 27 September 2018Available online 23 March 2019流动从层流到湍流的转捩现象或其逆过程是工程科学中的最重要研究课题之一。