1. The figure shows that the minimum area which can occur in a given isentropic duct flow is the sonic, or critical throat area. Choking For γ=1.4, this reduces to : For Subsonic Flow M≈ For Supersonic Flow M≈ Prof. Dr. MOHSEN OSMAN1

2. Chapter III Normal–Shock Wave A common irreversibility occurring in supersonic internal or extern- al flows is the normal shock wave sketched in figure. Except at near vacuum pressures such shock waves are very thin (a few micromet- ers thick) and approximate a discontinuous change in flow propert- ies. We select a control volume just before and after the wave; as shown in figure. Prof. Dr. MOHSEN OSMAN2

3. The analysis is identical to that of control-volume analysis of a finite-strength pressure-wave, that is, a shock wave is a fixed strong pressure wave. To compute all property changes rather than just the wave speed, we use all our basic one-dimensional steady-flow relations, letting section 1 be upstream and section 2 be downstream. 1 – Continuity equation: (3.1a) 2 – X-Momentum equation: (3.1b) 3 – Energy equation: (3.1c) 4 – Perfect gas: (3.1d) 5 – Constant specific heat C p : (3.1e) If we assume that the upstream conditions (P 1,V 1,ρ 1,h 1,T 1 ) are known, equation (3.1) are five algebraic relations in the five unknowns (P 2,V 2, ρ 2,h 2,T 2 ). Because of the velocity–squared term, two solutions are found and the correct one is determined from the second law of thermodynamics, which requires that S 2 ˃ S 1. Prof. Dr. MOHSEN OSMAN 3

4. The velocities V 1 and V 2 can be eliminated from eqns. (3.1a) to (3.1c) to obtain the Rankine–Hugoniot relation. From (3.1c) (I) From (3.1b) But from (3.1a) Multiply both sides of equation by (II) Compare equations I & II we have: (3.2) This contains only thermodynamics properties and is independent of the equation of state. Introducing the perfect gas law Prof. Dr. MOHSEN OSMAN 4

5. Substitute into equation (3.2) Prof. Dr. MOHSEN OSMAN5

6. Divide by P 1 (3.3) We can compare this relation with the isentropic-flow relation for a very weak pressure wave in a perfect gas (3.4) Also, the actual change in entropy across the shock can be computed from the perfect-gas relation: Prof. Dr. MOHSEN OSMAN6

7. ] (3.5) Prof. Dr. MOHSEN OSMAN7 0.50.61540.6095- 0.0134 0.90.9275 - 0.00005 1.0 0.0 1.11.007041.007050.00004 1.51.33331.33590.0027 2.01.62501.64070.0134

8. We see that the entropy change is negative if the pressure decreases across the shock, which violates the second law. Thus a rarefaction shock is impossible in a perfect gas. We see also that weak-shock waves are very nearly isentropic Mach-Number Relations For a perfect gas all the property ratios across the normal shock are unique functions of γ and the upstream Mach number M 1. For example, if we eliminate ρ 2 and V 2 from Eqns. (3.1a) to (3.1c) and introduce (3.1a) (3.1b) (3.1c) Prof. Dr. MOHSEN OSMAN8

9. Multiply both sides by (γ-1) (3.6) Since (3.7) P 2 ˃ P 1 if and only if: From eqn (3.7) we see that, for any γ, P 2 ˃ P 1 only if M 1 ˃ 1.0. Thus for flow through a normal- shock wave the upstream Mach number must be supersonic to satisfy the second law of thermodynamics. Prof. Dr. MOHSEN OSMAN9

10. What about the downstream Mach number? From the perfect-gas identity, we can write equation (3.1b) as: (3.8) Equating equations (3.7) & (3.8) Prof. Dr. MOHSEN OSMAN10

11. Accordingly, (3.9) For γ = 1.4 Since M 1 must be supersonic, this equation predicts for all γ ˃ 1 that M 2 must be subsonic. Thus a normal shock wave decelerates a flow almost discontinuously from supersonic to subsonic conditions. Prof. Dr. MOHSEN OSMAN11

12. Further manipulation of the basic relations (3.1) for a perfect gas gives additional equations relating the change in properties across a normal- shock wave in a perfect gas. (3.10) (3.11) Prof. Dr. MOHSEN OSMAN12

13. All these relations are given in Normal – Shock Relations for a perfect gas’ Table and plotted versus upstream Mach number M 1 in figure, for γ = 1.4. We see that pressure increases greatly while temperature and density increases moderately. The effective throat area A* increases slowly at first and then rapidly. The failure of students to account for this change in A* is a common source of error in shock calculations. The stagnation temperature remains the same, but the stagnation pressure and density decrease in the same ratio, i.e., the flow across the shock is adiabatic but non isentropic. Other basic principles governing the behavior of shock waves can be summarized as follows: 1. The upstream flow is supersonic and the downstream flow is subsonic. 2. For perfect gases (and also for real fluids except under bizarre thermodynamic conditions) rarefaction shocks are impossible, and only a compression shock can exist. 3. The entropy increases across a shock with consequent decreases in Prof. Dr. MOHSEN OSMAN13

14. stagnation pressure and stagnation density and an increase in the effective sonic-throat area. 4. Weak shock waves are very nearly isentropic. Moving Normal Shock The preceding analysis of the fixed shock applies equally well to the moving shock if we reverse the transformation in finite – strength pressure wave control – volume analysis. To make the upstream conditions simulate still fluid, we move the shock of fig. 3.1 to the left at speed V 1 ; that is, we fix our coordinates to control volume moving with the shock. The downstream flow then appears to move to the left at slower speed V 1 - V 2 following the shock. The thermodynamic properties are not changed by this transformation, so that all our Eqs. (3.2) to (3.11) are still valid. Prof. Dr. MOHSEN OSMAN14