3/8/2007 Compressible Fluid Flow Indo‐GermanWinterAcademy2006

Compressible Fluid Flow Indo‐German Winter Academy 2006

Presented By :

Puneet Kumar Department of Chemical Engineering Indian Institute of Technology Madras Tutor : Dr. Sanjay Mittal 3/8/2007

Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Contents • • • • • • • •

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Compressible flow – History Basics of Compressible flow Speed of Sound A Brief Review of Thermodynamics Propagation of sound source Shock waves and Normal shock Oblique Shock Summary

Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Historical Prospective • Convergent – divergent steam nozzles of ‘de-Laval’ • Advent of jet propulsion and high speed flights as ‘Bell XS-1’ • Dealing with high temperature, chemical reactive gases associated with rocket engines

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Compressible flow • Compressible flow – Variable density flow • Compressibility of fluid 1 dv 1 dρ τ =− = − v dp ρ dp • Gas velocities less than 0.3 of the speed of sound are considered as incompressible flow • Compressible flows are high energy flow • Shock waves in all disturbed supersonic flows • Examples : High speed airplanes and jet engines

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Flow Regimes • Subsonic Flow – Flow velocity everywhere less than the speed of sound

• Transonic Flow – Flow velocity is close to the speed of sound

• Supersonic Flow – Flow velocity is everywhere greater than the speed of sound

• Hypersonic Flow – Properties of flow increases explosively across the shock wave

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Basic Conservation Equations

Conservation Equations

Three Fundamental Principles

Models of Flow

Continuity equation Momentum equation

Some Applications

Energy equation

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Balance Equations • Continuity equation

− ∫∫ S

ur ur ∂ ρ V.d S =  ρ dV ∫∫∫ ∂t V

• Momentum equation ur ur uur ur ur ur ∂ ( ρ V) dV =  pd S + Fvis ρ f dV −  ∫∫S ρ (V.d S ) V + ∫∫∫ ∫∫∫ ∫∫ ∂t V V S • Energy equation ur uur ur ur ∫∫∫ q ρ dV − ∫∫ ρ (V.d S ) + ∫∫∫ ρ ( f .V)dV = V

S

∫∫∫ V

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V

⎛ ∂ ⎡ ⎛ V2 ⎞⎤ V 2 ⎞ ur uur ρ ⎜e+ ⎢ρ ⎜ e + ⎟ ⎥ dV +  ⎟ V .dS ∫∫ ∂t ⎣ ⎝ 2 ⎠⎦ 2 ⎠ ⎝ S

Puneet Kumar, Department of Chemical Engineering, IIT Madras

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One Dimensional Flow

One Dimensional Flow

Normal shock waves

Speed of sound

One dimensional flow with heat addition

One dimensional flow with friction

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Wave Propagation • • • •

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Waves carry information in flow Travel at local speed of sound For incompressible flow speed of sound is infinite Speed of sound is finite for compressible flow

Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Speed of Sound • Air molecules in random motion • Perfect gas properties are only T dependent • Make the sound wave stationary for analysis

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Speed of Sound… • By applying Mass and Momentum balance to the CV

• Neglect the higher order terms • Replacing the term momentum equation

in the

which simplifies to

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Speed of Sound… • By combining two equations • For an isentropic flow we get • For a perfect gas • Mach number

M

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=

v a

Puneet Kumar, Department of Chemical Engineering, IIT Madras

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A Brief Review of Thermodynamics • Perfect Gas – Intermolecular forces are neglected – Valid in the low pressures and high temperatures

PV = M RT P = ρ RT • For a thermally perfect gas all the properties like e, h, cp, cv are functions of Temperature only

• Using First and Second Law of thermodynamics

T2 P2 s2 − s1 = c p ln − R ln T1 P1 3/8/2007

Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Isentropic Relations • Relation among density, pressure and temperature in an isentropic flow

• All the properties will become a function of γ and mach number Μ • By using basic energy equation for an adiabatic process = constant

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Isentropic Relations Differentiate the energy equation For a thermally perfect gas For a calorific perfect gas In stagnant conditions, energy equation becomes which is equal to

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Isentropic Relations… • Eliminate T using

• Now multiply by

we get

and get

• By using P,T relations for an isentropic flow

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Propagation of Source of Sound a. b. c. d.

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Stationary Source Source moving at Subsonic Speeds Source moving at the Speed of Sound Source moving at Supersonic Speeds

Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Propagation of Source of Sound…

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Response of Subsonic and Supersonic Flows to an Obstacle

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Shock Waves • Spontaneous change in a flow • Shocks that are oriented perpendicular to the flow Normal Shock waves • Detached shock wave

⇒

• Attached shock wave

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Formation of a Shock Wave • Give jerk at t = 0 which emits a weak wave • The wave propagates and sets the gas into motion • The pressure jump across the stronger wave is not dp1 but is dp1+dp2.

• This phenomenon where the waves merge is called Coalescence 3/8/2007

Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Formation of a Shock Wave…

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Normal Shock Wave • Shocks which are stationary and normal to the flow • Shock thickness is very small • Balance equations across the shock

• Substitute for the term in momentum equation and get

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Normal Shock Wave Equations • Total enthalpy is constant across the shock h01 = h02

which is for a thermally perfect gas T01 = T 02

• By using isentropic relation

• By eliminating ρ and u from the continuity equation

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Normal Shock Wave Equations… • Substitute for the temperature ratio

• Solutions to this equation are

• By neglecting the imaginary and trivial solution

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Normal Shock Wave Equations… • Using this relation we can relate all the properties across the shock • Relations for the total properties are

• Entropy change across the shock

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Characteristics of Normal Shock Wave • If M1 > 1 , then M2< 1 • If M1 < 1 , then M2> 1

Possible Mathematically

• Shocks with M1 < 1 are physically impossible 3/8/2007

Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Traffic Rules for Compressible Flow

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Applications of 1-D flow • Flow through ducts and nozzles • Subsonic flow responds to area changes in the same manner as an incompressible flow • Supersonic flow behaves in an opposite manner

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Flow through a Converging Nozzle • Back Pressure, pb is equal to the reservoir pressure,p0 Î No flow • In sonic range flow increases with decreasing pb • After reaching sonic conditions nozzle get choked

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Flow through a Converging-Diverging Nozzle

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Two-Dimensional Compressible Flow

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Oblique Shock Wave • Normal Shock – a special case of oblique shock • Change in flow direction across an oblique shock

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Oblique Shock Wave Formation • For subsonic flow beeper always stays inside the circular sound wave fronts • For supersonic flow beeper will move outside the circular wave fronts

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Oblique Shock Relations • Additional tangential velocity component • Tangential component remains unchanged across the shock • Normal component changes according to the normal shock relations • Flow gets deflected towards the shock wave

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Oblique Shock Relations… • β is shock angle and θ is deflection angle

• Define mach no. for the normal velocity component

• For a given Mach Number, M1 , we have a minimum shock angle, and , the maximum inclination is

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Oblique Shock Relations…

• There exists two solutions for this equation • The smaller value gives what is called a Weak Solution. The other solution with a higher value of is called a Strong Solution.

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Oblique Shock Relations… • Relationship between θ and β

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Summary • Under subsonic conditions compressible and incompressible flow behaves similarly • Speed of sound is infinite for incompressible flow whereas finite for compressible flow • Shock forms in supersonic compressible flows • Incoming supersonic flow will become subsonic after the shock

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Further Reading • Modern Compressible Flow 3rd edition– » John D. Anderson

• Compressible Fluid Flow » Patrick H. Oosthuizen, William E. Carscallen

• www.aeromech.usyd.edu.au/aero/gasdyn

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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Puneet Kumar, Department of Chemical Engineering, IIT Madras

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