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.

First, the rotor blade and supersonic airplane examples (they are the same case--instead of blades spinning around, think of tiny supersonic airplanes spinning around in the engine). There are two ways to think about the problems. First, in the reference frame of the blade or the airplane. They see flow coming at them with kinetic energy associated with the moving flow, and with an internal energy (or more appropriately enthalpy) associated with its temperature far away in the atmosphere. When the flow is stagnated on the blade, both of these sum into the stagnation enthalpy. Thus the temperature is higher than the ambient temperature. The second way to look at it is in the fixed reference frame (not moving with respect to the atmosphere). In this case the flow has some stagnation temperature as it comes in the inlet which is equal to the ambient atmospheric temperature. Then the blade or airplane comes along and the fluid sticks to the blade or airplane (work is done on the fluid particle) gaining kinetic energy. Thus the total or stagnation temperature on the surface of the blade is greater than the stagnation temperature the flow had when coming in the inlet. But this is an unsteady problem (the blades and the airplane are moving), so you can only think about it this way conceptually -- you can't apply the steady flow energy equation in this frame. You need to apply the steady flow energy equation in a reference frame where the flow is steady.

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.

4) If air comes in contact with a moving wing, it picks up heat and kinetic energy. How can stagnation enthalpy be constant if T goes up and c goes up.? (1 student) If the stagnation temperature goes up it is supplied by shaft work right? (1 student) For air stagnating on the blade is there shaft work? (1 student) These are very good questions. The answer is a little complicated, so read it carefully. There must be external work done to change the stagnation enthalpy of the flow (for the case with no heat transfer that is--heat transfer will change the stagnation enthalpy also). But it is possible to stagnate a steady flow without external work (although there will be flow work) and this is a good approximation for many flow fields of practical importance when viewed from the reference frame where the flow field is steady. But these arguments depend on the frame in which the problem is being addressed (which is why stagnation quantities are called frame-dependent quantities). From the perspective of a reference frame moving with a high speed airplane, the steady flow stagnates on its surface with a very high temperature -- accounting for the sum of the kinetic energy of the relative flow speed and the static enthalpy. The same problem from the perspective of a ground-fixed reference frame (with the airplane flying through it) is not steady and the steady flow energy equation does not apply. You will see in the propulsion lectures in the spring that an unsteady process like this is a pre-requisite for changing the stagnation enthalpy of a flow through external work (with no heat transfer). When we apply the steady flow energy equation in the frame where the flow is steady (that attached to the airplane), the external work that is done to move the fluid (in the unsteady frame attached to the ground) is accounted for in the relative kinetic energy of the flow field (in the flow-steady frame attached to the airplane).

5) Does the air stuck to the blade or wing get hotter or does the blade get hotter? (1 student) They both get hotter. Why is the stagnation temperature the skin temperature? (1 student) After a sufficient period of time the body will reach the temperature of the flow that sticks to it (the stagnation temperature).

6) How does the relationship for stagnation and static pressure relate to Bernoulli's equation and what is Bernoulli's equation? (1 student) You will learn more about this in fluids soon. You don't need to know about it for the thermo part of the course.

7) No mud (3 students). Good.