Advanced fluid mechanics (II)Course content:1.Compressible Fluid Mechanics Textbook: Modern Compressible Flow, 2nd ed. , by John D Anderson, Jr. Reference Book1. Gas Dynamics, 2nd ed., by James. E. A. John2. Compressible Fluid Dynamics by Philip A. Thompson3. Elements of Gasdynamics by H. W. Liepmann and A. Roshko4. Compressible Fluid Flow. , 2nd ed. , by Michel A. SaadGrading: 1. Homework 60% 2. Final Project 40%

Chapter I1. Introduction and Review of ThermodynamicsWhat is Compressible Flow?1. 2. Energy transformation and temperature change are important considerations Importance of Thermodynamics e.q Flow of standard sea level conditions, Specific internal energy

Specific kinetic energy

1.1 Definition of Compressible Flow Incompressible flow compressibility effect can be ignored.

is the specific volume &

Compressibility of the fluid

Physical meaning: the fractional change in volume of the fluid element per unit change in pressureChapter INote: dp(+) dv(-)

. Isothermal compressibility ..isentropic compressibility (speed of sound)Compressibility is a property of the fluidLiquids have very low values of

e.g for water = at 1atm Gases have high e.g for air =10-5 m2/N at 1 atm, Alternate form of

Chapter I

Chapter IGeneral speakingMa >0.3 Compressible effect can not be ignoredMa < 0.3 Incompressible flowFor most practical problem compressible

1.2. Regimes of compressible flow

Streamline deflected far upstream of the bodyFlow is forewarned of the presence of the body

Subsonic flowChapter I

Chapter ITransonic flow is less than 1 , but high enough to produce a pocket of locally supersonic slow

LooselyDefined as the Transonic regimeIf is increased to slightly above 1 , the shock will move to the trailing edge of the airfoil , and bow shock appears upstream of the leading edge.(Highly unstable)

Chapter I Supersonic FlowEverywhere Behind the shock+ Parallel the free stream flow is not forewarned of presence of the body until the shock is encountered+ Both flow of upstream of the shock and downstream of the shock are supersonic+ Dramatic physical and mathematical difference between subsonic and supersonic flows.(We will mostly focus on this regimes)

High enough to excite the internal modes of energy dissociate or even ionize the gas.

Chapter IHypersonic FlowReal gas effect !!! Chemistry comes in

Chapter IIncompressible flow is a special case of subsonic flow limiting case Trivial , no flowFor incompressibilityFlow can be also be classified as ViscousinviscidViscous flow: + Dissipative effects : Viscosity, thermal conduction, mass diffusion.+ Important in regions of large gradients of V, T and Ci e.g. Boundary layerFlows

Chapter IInviscid flows: - ignore dissipative effects outside of B.L

(We will treat this kind of flow )Also consider the gas to be Continuum Mean free path

R - specific gas constant 1.3 A Review of Thermodynamics 1.3.1 Ideal gas intermolecular force are negligible 8314 (J/kg.mole.k)Molecular weightsFor air at standard conditionsBoltzmann constant =

Isothermal compressibilityLdL > 10d , for most compressible flows

Chapter I-Translational-Rotational No of collisions > 5 equilibrium-Vibration : No of collisions > 0 (100 ) equilibrium Add one more time scale or length scale-Electronic excitation + nuclear1.3.2. Internal Energy and EnthalpyIf the particles of the gas (called the system) are rattling about their state of maximum disorder, the system of particle will be in equilibrium.Statistical Thermodynamics +Quantum mechanics

Let be specific internal energyLet be specific enthalpy

For both a real gas and a chemically reacting mixture of perfect gases.

Thermally perfect gas

Chapter IReturn to macroscopic view continuum

Chapter ICalorically perfect gasare const Will be assumed in the discussion of this classRatio of specific heat , =1.4 for a diatomic gas =5/3 for a monatoinic gas Air, T

Consider caloriacally perfect gas + thermally perfect gas Note: specific heat at constant pressure specific heat at constant volume

Chapter IIdeal gas Perfect gas

Conservation of EnergyConsider a system, which is a fixed mass of gas separated from the surroundings by a flexible boundary. For the time being, assume the system is stationary, i.e., it has no directed kinetic energy

e is state variable, de is an exact differential depends only on the initial and final states of the system1.3.3. First law of the thermodynamicsAn incremental amount of heat added to the system across the boundary The work done on the system by the surrondings

Chapter IFor a given , there are in general an infinite different ways (processes) of We will be primarily concerned with 3 types of processes:Adiabatic processReversible process no dissipative phenomena occur, i.e,. Where the effects of viscosity, thermal conductivity, and mass diffusion are absent (see any text on thermodynamic)3. Isentropic process - both adiabatic & reversible2nd law of thermodynamic

A contribution from the irreversible dissipative phenomena of viscosity thermal conductivity, and mass diffusion occurring within the systemChapter I Define a new state variable, the entropy,

The actual heat added/T, These dissipative phenomena always increase the entropyFor a reversible processIf the process is adiabatic, 2nd lawIn summary, the concept of entropy in combination with the 2nd law allow us to predict the direction that nature takes.1.3.4 Entropy and the Second Law of Thermodynamic or

Chapter IAssume the heat is reversible, 1st law becomes For a thermally perfect gas, If the gas also obey the ideal gas equation of stateIntegrate Note

1.3.5. Isentropic realtionsFor an adiabatic process and for a reversible process Hence, from eq ,i.e.,

the entropy is constant. Chapter IIf we further assume a calorically perfect gas,

Chapter I

Outside B.L-Isentropic relations prevaile.g. T=1350KP=?T=2500 KP=15atmM=12, Cp=4157 J/kg.K

Chapter I1.3.6. Aerodynamic forces on a BodyMain concerns : Lift & dragForces on a body of airfoil-Surface forces: pressure shear stress-Body forces : gravity ; electric-magneticSources of aerodynamic force, resultant force and its resolution into lift and drag

Drag D is the component of parallelIn our plot. L// , D//Lift L is the component of perpendicular to the relative wind Chapter ILet be unit vectors perpendicular and parallel, respectively to the element ds,inviscid

Chapter IPressure drag -> wave drag, e.g slender supersonic shapes with shock wavesSkin friction drag-We consider only inviscid flows and both pressure and skin-friction drags are important-In the most cases, we can not predict the drag accuratelyFor blunt bodies, Dp dominatesFor streamlined bodies, Dskin dominateswith shock wave, Dwave drag dominate and Dskin can be neglectedD can be predicted reasonably

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