22-B 1 Lect-22-B In this lecture ... • Shock Waves and Expansion – Normal Shocks – Oblique Shocks – Prandtl–Meyer Expansion Waves • Duct Flow with Heat Transfer and Negligible Friction (Rayleigh Flow) – Property Relations for Rayleigh Flow • Duct flow with friction without heat transfer (Fanno flow) 2 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Shock waves • Sound waves are caused by infinitesimally small pressure disturbances, and travel through a medium at the speed of sound. • Under certain flow conditions, abrupt changes in fluid properties occur across a very thin section: shock wave. • Shock waves are characteristic of supersonic flows, that is when the fluid velocity is greater than the speed of sound. • Flow across a shock is highly irreversible and cannot be approximated as isentropic. 3 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Throat Pe P0 Pb P Pb A P P0 B A Subsonic flow at nozzle exit PB C No shock PC P* D Subsonic flow at nozzle exit PD Shock in nozzle P E Supersonic flow at nozzle exit Sonic flow P F No shock in nozzle at throat PG Inlet Throat Exit x M Shock in nozzle Supersonic flow at nozzle exit No shock in nozzle 1.0 Subsonic flow at nozzle exit D Shock in nozzle C Subsonic flow at nozzle exit B A No shock x 4 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Normal shocks • Shock waves that occur in a plane normal to the direction of flow: Normal shocks. • A supersonic flow across a normal shock becomes subsonic. • Conservation of energy principle requires that the enthalpy remains constant across the shock. h01 = h02 • For an ideal gas, h = h(T) and thus T01 = T02 5 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Normal shocks Control volume V1 V2 M > 1 P1 P2 M < 1 h1 h2 ρ1 ρ2 s1 s2 Shock wave Flow across a normal shock wave 6 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Normal shocks • Across the normal shock we apply the governing equations of fluid motion: • Mass: ρ1A1V1= ρ2A2V2 Energy: h = h • 01 • 02 • Momentum: A(P1 – P2)= m (V2 – V1) • Entropy: s2 – s1 ≥ 0 • If we combine mass and energy equations and plot them on h-s diagram: Fanno line • Similarly combining mass and momentum gives: Rayleigh line 7 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Normal shocks h 2 V2 /2 h01 h02 h01=h02 Subsonic flow 2 (M<1) h 2 b a 2 M=1 V1 /2 Fanno line Rayleigh line Shock wave h1 Supersonic flow 1 (M>1) s1 s2 s The h-s diagram for flow across a normal shock. 8 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Normal shocks • To derive expressions before and after the shock T01 γ −1 2 T02 γ −1 2 =1+ M1 and =1+ M 2 T1 2 T2 2 Since T01 = T02 , and simplifying, P M 1+ M 2 (γ −1) / 2 2 = 1 1 P 2 1 M 2 1+ M 2 (γ −1) / 2 • This is the Fanno line equation for an ideal gas with constant specific heats. 9 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Normal shocks • Similarly if we combine and simplify the mass and momentum equations, we can get an equation for Rayleigh line. M 2 + 2 /(γ −1) 2 = 1 M 2 2 2M1 γ /(γ −1) −1 • This represents the intersections of the Fanno and Rayleigh lines • This equation relates the Mach number upstream of the shock with that downstream of the shock. 10 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Normal shocks P increases P0 decreases M > 1 V decreases M decreases T increases T0 remains constant s increases Normal shock Variation of flow properties across a normal shock. 11 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Oblique shocks • Shock waves that are inclined to the flow at an angle: oblique shocks. • In a supersonic flow, information about obstacles cannot flow upstream and the flow takes an abrupt turn when it hits the obstacle. • This abrupt turning takes place through shock waves. • The angle though which the fluid turns: deflection angle or turning angle, θ. • The inclination of the shock: shock angle or wave angle, β. 12 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Oblique shocks Oblique shock M1 > 1 θ M2 β δ Flow across an oblique shock 13 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Oblique shocks • Oblique shocks are possible only in supersonic flows. • However, the flow downstream of the shock can be either supersonic, sonic or subsonic, depending upon the upstream Mach number and the turning angle. • To analyse an oblique shock, we decompose the velocity vectors upstream and downstream of the shock into normal and tangential components. 14 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Oblique shocks Oblique shock P1 V P2 1,t V1,n V2 Across the shock, the V θ 1 tangential component V2,t V2,n of velocity does not change, V = V β 1,t 2,t Velocity vectors through an oblique shock of shock angle β and deflection angle θ. 15 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Oblique shocks P1 P2 M2,n <1 M1,n >1 β-θ M1,n = M1 sin β and M 2,n = M 2 sin(β −θ ) θ Where, M1,n = V1,n / c1 and M 2,n = V2,n / c2 V2 V2,t If we use normal components V1,n of velocity, all the equations, V1,t β V2,n tables etc. For a normal shock V 1 can be used for an oblique Oblique shock shock as well. 16 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Oblique shocks θ–β–M relationship 17 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Oblique shocks • There are two possible values of β for θ<θmax. • θ=θmax line: Weak oblique shocks occur to the left of this line, while strong oblique shocks are to the right of this line. • M=1 line: Supersonic flow to the left and subsonic flow to the right of this line. • For a given value of upstream Mach number, there are two shock angles. • β=βmin represents the weakest possible oblique shock at that Mach number, which is called a Mach wave. 18 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Prandtl-Meyer expansion waves • An expanding supersonic flow, for eg, on a two-dimensional wedge, does not result in a shock wave. • There are infinite Mach waves forming an expansion fan. • These waves are called Prandtl-Meyer expansion waves. • The Mach number downstream of the expansion increases (M2>M1), while pressure, density, and temperature decrease. 19 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Prandtl-Meyer expansion waves Expansion waves M1 > 1 M2 μ1 μ2 θ δ Oblique shock Flow across an oblique shock 20 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Prandtl-Meyer expansion waves • Prandtl–Meyer expansion waves are inclined at the local Mach angle μ. • The Mach angle of the first expansion wave -1 μ1 = sin (1/M1) -1 • Similarly, μ2 = sin (1/M2) • Turning angle across an expansion fan is θ = ν(M2) - ν(M1) • ν(M) is called the Prandtl–Meyer function γ +1 − γ +1 − ν (M ) = tan 1 (M 2 −1) − tan 1( M 2 −1) γ −1 γ −1 21 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Duct flow with heat transfer and negligible friction • Many engineering problems involve compressible flow which involve chemical reactions like combustion, may involve heat transfer across the system boundaries. • Such problems are usually analysed by modelling combustion as a heat gain process. • The changes in chemical composition are neglected. 22 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Duct flow with heat transfer and negligible friction • 1-D flow of an ideal gas with constant specific heats through a duct of constant area with heat transfer and negligible friction: Rayleigh flows. • Q P1, T1, ρ1, V1 P2, T2, ρ2, V2 Control volume 23 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Duct flow with heat transfer and negligible friction • For a gas whose inlet properties P1, T1, ρ1, V1 and s1 are known, the exit properties can be calculated from the five governing equations: • Mass, momentum, energy, entropy and equation of state. • The Rayleigh flow on T-s diagram is called the Rayleigh line. • The Rayleigh line is the locus of all physically attainable downstream states corresponding to an initial state. 24 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-22-B Duct flow with heat transfer and negligible friction T M =1/√γ T b max b M =1 Cooling a a M 0 Heating M 1 smax Heating M 1 Cooling M ∞ s T-s diagram for Rayleigh flow 25 Prof.