Mechanics of Fluids - short answer questions from AMIE exams (Summer 2019)

Explain the following in brief (5 marks each)

Normal shock

A shock wave can be considered as a discontinuity in the properties of the flow field. The fluid crossing a shock wave, normal to the flow path, will experience a sudden increase in pressure, temperature, and density, accompanied by a sudden decrease in speed, from a supersonic to a subsonic range.
Normal shocks occur, for example, in supersonic internal and jet flows.
Expressing the Prandtl relation in terms of 𝑀 = 𝑢/𝑎, we obtain:
M2 = 1/M1
This leads to an important conclusion that the velocity change across a normal shock must be from
supersonic to subsonic.

Boundary layer

A boundary layer is a thin layer of a flowing gas or liquid in contact with a surface, such as that of an aeroplane wing or of the inside of a pipe. The fluid in the boundary layer is subjected to shearing forces. A range of velocities exists across the boundary layer, from maximum to zero.
As an object moves through a fluid, or as a fluid moves past an object, the molecules of the fluid near the object are disturbed and move around the object. This creates a thin layer of fluid near the surface, in which the velocity changes from zero at the surface to the free stream value away from the surface. Engineers call this layer the boundary layer because it occurs on the boundary of the fluid. 

Boundary layers may be either laminar (layered), or turbulent (disordered) depending on the value of the Reynolds number. For lower Reynolds numbers, the boundary layer is laminar and the streamwise velocity changes uniformly as one moves away from the wall, as shown on the left side of the figure. For higher Reynolds numbers, the boundary layer is turbulent and the streamwise velocity is characterized by unsteady (changing with time) swirling flows inside the boundary layer.

Fanno line flow

  • Fanno flow is the adiabatic flow through a constant area duct, where the effect of friction is considered.
  • For this model, the duct area remains constant, and no mass is added within the duct. 
  • The Fanno flow model is considered an irreversible process because friction transfers energy from the flow by thermal conduction to the wall of the duct. 
  • For a flow with an upstream Mach number greater than 1 in a sufficiently long enough duct, deceleration occurs, and the flow can become choked. It can be shown that for calorically perfect flows, the maximum entropy occurs at M = 1. 
  • Fanno flow is named after Gino Girolamo Fanno. 

Von Karman momentum integral equation

Applying the basic integral conservation principles of mass and momentum to a length of boundary layer, ds, yields the Karman momentum integral equation that will prove very useful in quantifying the evolution of a steady, planar boundary layer, whether laminar or turbulent. 

We consider the control volume, ABCD, sketched in Figure 1 and bounded by two boundaries normal to the solid surface that are ds apart, by the edge of the boundary layer and by the solid surface. 

We obtain following Von Karman equation.

τw = shear stress
δM = momentum thickness

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