Advanced Fluid Mechanics

Winter Quarter 2020

Stefan LLEWELLYN SMITH
EBUII 574
x23475
http://mae.ucsd.edu/~sgls

Homework VII

Due

Friday March 6, 2020, in class (or before).

Problems (MYO is the textbook; 8th edition)

  1. MYO 11.28
  2. MYO 11.37
  3. MYO 11.39
  4. MYO 11.41
  5. MYO 11.48
  6. MYO 11.52
  7. Write 2-3 paragraphs about the Concorde supersonic passenger aircraft, including design, technology, political climate and fate.

Comments

I will lecture Section 11.5 after Section 11.9.

Section 11.1 reviews the ideal gas law. This should be familiar, but is a good time to review enthalpy, the ratio of specific heats and entropy.

Section 11.2 concerns stagnation properties, that is the properties a fluid would have if brought to rest without any addition or loss of energy. The resulting formulas link temperature to stagnation temperature, pressure to stagnation pressure and so on.

Section 11.3 discusses the speed of sound. Sound is the basic linear response of a compressible fluid: fluid particles expand and contract, leading to a propagating signal.  Since this is the result of  the interplay of density and pressure, the speed of sound comes out as the square root of the derivative of pressure with respect to density. Hence incompressible fluid corresponds to infinite sound speed: the fluid adjusts instantaneously everywhere. This is not possible in reality, but is a good approximation if the speed of sound is much larger than the characteristic velocity of fluid in the system, i.e. if the Mach number, V/c, is small.

Section 11.4 explores different flow regimes at different Mach numbers. One can distinguish incompressible flow, compressible subsonic flow, transonic flow, (compressible) supersonic flow, and hypersonic flow. In supersonic flow, the disturbance induced by a body is localized inside a Mach cone

Section 11.5 presents shock waves (normal shocks in fact). A shock wave is an infinitesimally thin discontinuity in fluid properties. The underlying conservation laws of mass, momentum and energy can be used to obtain a relation between the upstream and downstream Mach numbers. The stagnation temperature does not change across a shock. The second law leads to the result that the upstream flow must be supersonic.

Section 11.6 discusses isentropic flow along a duct. We consider the case with no shaft work; then the condition of constant entropy leads to the result that all stagnation properties are constant. The critical state corresponds to exactly sonic flow.

Section 11.7 uses these results to examine flow in a variable-area duct. The most important application is to rocket engines.

Section 11.8 and 11.9 discuss Fanno and Rayleigh flow, namely compressible flow in ducts with friction and heat addition respectively.

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