Stagnation and Sonic Condition Relationship Question

[Red_CCF]

Hi

I was reading Anderson's Modern Compressible Flow and two of his equations were confusing. I attached the relevant pages on this post.

He defined two conditions or state the sonic and stagnation state used to define flows. The sonic state was defined as an adiabatic transition of the flow to M = 1 while the stagnation state was an isentropic transition of the flow to stagnation. However, what confused me was his formulation of Eq. 3.35 and 3.36 on 2.jpg. There he related the stagnation and sonic states of the flow with an isentropic relationship. However, since the sonic state is an adiabatic transition as defined, the entropy may not necessarily be equal to that of the original flow and thus entropy may not be equal to that of the stagnation state?

Any clarification is appreciated.

Thanks

 

[boneh3ad]

The two states are being related through total temperature, which is constant as long as the flow is adiabatic (it does not require the flow to be isentropic). Take for example the fact that the total temperature is constant across a shock. In other words, the quantities being related only require the flow to be adiabatic, and by definition, all isentropic flows are also adiabatic.

 

[Red_CCF]

The two states are being related through total temperature, which is constant as long as the flow is adiabatic (it does not require the flow to be isentropic). Take for example the fact that the total temperature is constant across a shock. In other words, the quantities being related only require the flow to be adiabatic, and by definition, all isentropic flows are also adiabatic.

I get that the total temperature remains constant, but I do not get how the pressure and density ratios (between that of the sonic and stagnation state) for the same point in the flow can be represented by an isentropic relationship since the imagined transition from the point in the flow to the sonic condition is adiabatic but not necessarily isentropic while the stagnation state is.

Thanks very much

 

[boneh3ad]

The equations used (28, 30, 31) are isentropically relations, meaning the processes they represent are both adiabatic and reversible. If you plug in M=1 to the equation, by definition it is adiabatic so moving the flow to M=1 in those equations satisfies the adiabaticity requirement of the starred values by default and relates stagnation conditions to sonic conditions.

 

[Red_CCF]

The equations used (28, 30, 31) are isentropically relations, meaning the processes they represent are both adiabatic and reversible. If you plug in M=1 to the equation, by definition it is adiabatic so moving the flow to M=1 in those equations satisfies the adiabaticity requirement of the starred values by default and relates stagnation conditions to sonic conditions.

Hi

I was just thinking about this again. If I have an actual flow with actual conditions p, T, M and use Eq. 3.28, 3.30 to find the stagnation state T0, p0, and then use those to find the sonic states p* (Eq. 3.35) and by extension T*, would the 3 states found using this method have the same entropy?

I get that subbing M=1 into the isentropic relationships guarantees adiabaticity, but it also constrains that the entropy of the stagnation, sonic, and by extension of the real state is the same. Since the sonic state definition only constrains adiabaticity, does this mean multiple sonic states are possible and the formulated equations are for a specific sonic state that is brought isentropically from the real state?

Thanks very much

 

[boneh3ad]

Given that they were all found for the same point in the flow, yes, they would all have the same entropy. Each single point in space in the flow has one sonic reference state and one stagnation reference state. The reason you need to distinguish between whether those states are for adiabatic or isentropic changes to the flow is because depending on which type of process is involved, the reference state can change throughout a given flow, for example, across a shock.

For example, stagnation conditions are identical throughout the entirety of a supersonic flow passing through a supersonic nozzle provided the nozzle is started (excluding the effects of the boundary layer, of course). Even as the flow accelerates, the stagnation conditions remain the same. On the other hand, if there is a shock present, the stagnation states change. Shocks are adiabatic, of course, so the stagnation temperature will remain the same across the shock, as will the sonic conditions, but they are entropy-generating processes as well, so other stagnation quantities will change.

 

[Red_CCF]

Given that they were all found for the same point in the flow, yes, they would all have the same entropy. Each single point in space in the flow has one sonic reference state and one stagnation reference state. The reason you need to distinguish between whether those states are for adiabatic or isentropic changes to the flow is because depending on which type of process is involved, the reference state can change throughout a given flow, for example, across a shock.

For example, stagnation conditions are identical throughout the entirety of a supersonic flow passing through a supersonic nozzle provided the nozzle is started (excluding the effects of the boundary layer, of course). Even as the flow accelerates, the stagnation conditions remain the same. On the other hand, if there is a shock present, the stagnation states change. Shocks are adiabatic, of course, so the stagnation temperature will remain the same across the shock, as will the sonic conditions, but they are entropy-generating processes as well, so other stagnation quantities will change.

Hello

I'm wondering why the sonic reference state is defined as an adiabatic but not necessarily isentropic change. Why not just define the sonic reference state as an isentropic change to M=1 like the stagnation state to begin with since the equations derived for the sonic-stagnation reference state relationship already assumes that both states are reached isentropically from the actual condition? Also, is it only the sonic temperature T* that is constant across a shock and not other properties?

Thanks very much for your help