1. Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras
Advanced Transport Phenomena
Module 4 Lecture 13
Momentum Transport: Shock Waves
Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras

2. Momentum Transport: Shock Waves

3. STEADY 1D COMPRESSIBLE FLUID FLOW
Steady, frictionless flow of a nonreacting gas mixture in a constant-area duct with heat addition:
Conservation equations may be written as:

4. STEADY 1D COMPRESSIBLE FLUID FLOW
Mass & momentum conservation equations yield: Since , This locus on the p-v (or corresponding T-s) plane  Rayleigh line (locus)

5. STEADY 1D COMPRESSIBLE FLUID FLOW
Local stagnation (or total) temperature For a constant cp-gas mixture between any two duct sections 1 & 2, change in T0 is governed by heat addition per unit mass: Since Entropy change & Ma at each point along Rayleigh line may be calculated.

6. STEADY 1D COMPRESSIBLE FLUID FLOW
Steady one-dimensional flow of a perfect gas (with g=1 .3) in a constant area duct,
frictionless flow with heat addition

7. STEADY 1D COMPRESSIBLE FLUID FLOW
Steady compressible flow of a nonreacting gas mixture in a constant-area duct with friction but without heat addition:
Conservation equations may be written as:
(Fanno Locus)

8. STEADY 1D COMPRESSIBLE FLUID FLOW
Steady one-dimensional flow of a perfect gas (with g=1 .3) in a constant area duct, adiabatic flow with friction

9. SHOCK WAVES Discontinuity separating two adjacent continua
e.g., mixture of perfect gases, same EOS valid on both sides of discontinuity
How do we apply conservation laws on field variables?
Assume locally planar discontinuity, fixed in space, fed by a gas stream with known velocity normal to it, known thermodynamic state properties

10. SHOCK WAVES
Control volume and station nomenclature for applying conservation principles across a gas dynamic discontinuity separating two regions of flow in which diffusion processes can be neglected

11. SHOCK WAVES
Consider a macroscopic control volume, shrunk down to a “pillbox” of unit area straddling the figure
Field variables “jump” across discontinuity
“jump operator”

12. SHOCK WAVES
Conservation equations across a discontinuity without chemical reaction:

13. SHOCK WAVES
Mass flux ,

14. SHOCK WAVES
In general, G, wi, h0 are continuous (no jump) across discontinuity; u, p, T, v (≡ 1/r), s jump.
Compatible with conservation principles, relevant EOS
Combining total mass & normal momentum relations:

15. SHOCK WAVES
When discontinuity becomes a sufficiently weak compression wave, positive entropy jump is negligible; hence
For a perfect gas, then:

16. SHOCK WAVES
For a discontinuity of arbitrary strength, final state must lie on intersection of Fanno and Rayleigh loci passing through initial state on T-s diagram, corresponding to common mass flux G
Rayleigh line links all states with same p + Gu (irrespective of heat addition)
Fanno locus links all states with same stagnation enthalpy irrespective of viscous dissipation

17. SHOCK WAVES
Fanno and Rayleigh loci for the same mass flux G, displayed on the T-s plane. The normal shock transition goes from the supersonic intersection to the subsonic intersection

18. SHOCK WAVES Rankine – Hugoniot interrelation:
Defines a locus (“shock adiabat”) on p-v plane along which final state must lie
For a perfect gas, this relation is given by

19. SHOCK WAVES
For strong compression waves, upstream flow is supersonic, downstream subsonic
Rules out possibility of rarefaction shocks

20. SHOCK WAVES
Rankine-Hugoniot “shock adiabat” on the p-v plane

21. SHOCK WAVES
In terms of upstream (normal) Mach number:
and
As Ma1  ∞, p2/p1 and T2/T1 also  ∞; however, r 2 /r 1 approaches the finite limit (g+1)/(g-1)

22. SHOCK WAVES
Normal shock property ratio as a function of upstream (normal) Mach number Ma ( for =1.3 ) g

23. DETONATION / DEFLAGRATION WAVES
Abrupt transitions accompanied by chemical reactions
h must include chemical contributions
Reaction may be seen as adding heat q per unit mass to a perfect gas mixture of constant specific heat g
e.g., many fuel-lean/ air mixtures

24. DETONATION / DEFLAGRATION WAVES
Generalized R-H conditions then become:

25. DETONATION / DEFLAGRATION WAVES
Detonation adiabat is above shock adiabat by an amount depending on heat release q
Detonations propagate with an end-state at or near Chapman-Jouguet point CJ (figure on next slide)
Singular point at which combustion products have minimum possible entropy, and normal velocity of products is exactly sonic (Ma 2 = 1)

26. DETONATION / DEFLAGRATION WAVES
Rankine -Hugoniot “detonation adiabats” on the p-v plane

27. DETONATION / DEFLAGRATION WAVES
Imposing Ma2 = 1 yields:
{+ sign  upstream Mach number for a CJ-detonation (compression wave)
- sign  upstream Mach number for a CJ-deflagration (subsonic combustion wave)}
where

28. MULTIDIMENSIONAL INVISCID STEADY FLOW
Even neglecting diffusion & non-equilibrium chemical reaction, equations governing local conservation of mass, momentum & energy for steady flow of a perfect gas remain PDE’s
In field variables
Must be solved subject to conditions “at infinity” & vn along body surface

29. DETONATION / DEFLAGRATION WAVES
Simpler procedure:
Reduce PDEs to one (higher order) PDE involving only one unknown– velocity potential, ,where:
All inviscid compressible flows admit such a potential, with constant gradient far from the body
everywhere along body surface

30. MULTIDIMENSIONAL INVISCID STEADY FLOW
Scalar function must satisfy non-linear 2nd order PDE: where when a2  ∞(e.g., incompressible liquid): (Laplace’s Equation)

31. MULTIDIMENSIONAL INVISCID STEADY FLOW
Hence, many simple inviscid flows can be constructed using “potential theory” & analytical methods (rather than numerical)
Nature of solution depends on whether local flows are supersonic or subsonic
In case of upstream supersonic conditions, shock waves can appear within flow field– piecewise continuous
Energy can then be dissipated even in inviscid fluids
Interplay of diffusion & convection determines structure of discontinuities