Lecture T11: Stagnation Quantities Part 2

The lecture reviewed the material presented in T12 regarding stagnation properties, and then continued with four different example problems (one, two , three, four). Each of these illustrates the basics of the frame dependence of the stagnation quantities. Please read over the explanations of these problems. Remember, the following points. 1) You can only apply the steady flow energy equation in a reference frame where the flow appears as steady. 2) Flows stagnate on surfaces. 3)If two points are not moving relative to one-another (i.e. they are in the same reference frame), and the flow moves from one point to another without heat addition or external work, then the total enthalpy is constant and the stagnation temperatures at the two points are the same. 4) If a flow moves without heat addition or external work, the enthalpy is reduced ( and the temperature along with it) as the kinetic energy is increased (because total enthalpy is constant). 5) If a reference frame is moving relative to a stationary frame, flows that stagnate in the moving frame have a higher stagnation temperature (the stagnation or total enthalpy in the moving frame is higher).

Responses to 'Muddiest Part of the Lecture Cards'

(33 respondents out of 64 students in class)

1) General fuzziness about frame dependency, unclear about 3 PRS at end, and related questions. (30 students more or less) You have a lot of company. We will work on this more in recitation. I will make a few comments below and answer a few of the specific questions which I think may be particularly enlightening.

Second, the wind-tunnel model example and the engine sitting on the ground example (they are the same case-- the flow starts out stagnant someplace and moves to a new location with no heat or external work where it stagnates again on a surface -- either the wall of the engine inlet or the model. The two reference frames are the same with no relative motion, therefore the stagnation temperature is the same.) Why didn't the model get hotter just as the skin of the airplane? Isn't it just backwards, air is moving instead of the body? It did get hotter--hotter than the static temperature of the freestream. This is the same thing that happened with the supersonic airplane, except the static temperatures were different. In the wind-tunnel, the freestream flow is very cold (when it is moving at M=2.5). When it is stagnated on the model it reaches its stagnation temperature. The whole process (from stagnated flow in the high pressure cylinders, to moving flow in the pipes and wind-tunnel, and back to stagnated flow on the surface of the wind-tunnel model) can be approximated as adiabatic and there is no external work, and the model and the high pressure cylinders are motionless with respect to one another. Therefore the stagnation temperature is constant. Enthaly (cpT) is just traded for kinetic energy (c^2/2) as the flow accelerates and decelerates. So as the flow moves faster, its static temperature drops. As the flow moves slower, its static temperature increases.

2) So just to clarify, T1 is the temperature I would measure if I placed a thermometer somewhere in space and TT1 is the temperature that the chunk of air attains under the conditions specified by stagnation? (1 student) Sort of. Your statement is correct except for the thermometer part. If you place a thermometer in the flow you always get the stagnation temperature (as long as q and ws are zero) since the flow always stagnates on the surface of the thermometer. However, if you measured the temperature with a non-intrusive laser-based method, you would get T1, the static temperature.

3) After an aircraft wing is heated by passing airflow as the result of stagnation, is there any cooling effect from the lower temperature of the passing airflow as well? (1 student) No the wing reaches the temperature of the fluid which stagnates on its surface (to first approximation, in general there is a small amount of heat transfer to the flow which does not fully stagnate and this changes the surface temperature). Relative to the air in the atmosphere after the airplane flies through, the airflow that doesn't have work put into it (that is outside the viscous boundary layer) retains its stagnation temperature (that of the atmosphere). But the flow that is heated by the passage of the airplane does mix with the flow around it and the temperature wake gradually dissapates through this process of mixing and the process of thermal conduction.