Stagnation and Sonic Condition Relationship Question

In summary, the conversation discusses Anderson's Modern Compressible Flow and the confusion surrounding two equations related to the sonic and stagnation states. The expert explains that the quantities being related only require the flow to be adiabatic, and all isentropic flows are also adiabatic. They also discuss how the reference state can change throughout a given flow, and that the sonic and stagnation reference states are reached isentropically from the actual condition. The expert also clarifies that the sonic temperature remains constant across a shock, but other properties may change.
  • #1
Red_CCF
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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
 

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  • #2
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.
 
  • #3
boneh3ad said:
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
 
  • #4
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.
 
  • #5
boneh3ad said:
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
 
  • #6
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.
 
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  • #7
boneh3ad said:
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
 

Related to Stagnation and Sonic Condition Relationship Question

1. What is stagnation in relation to sonic conditions?

Stagnation is the condition of a fluid when it is brought to rest by converting its kinetic energy into pressure energy. Sonic conditions refer to the speed of sound in a medium, which is the maximum speed that a disturbance can travel through the medium.

2. How are stagnation and sonic conditions related?

Stagnation and sonic conditions are related through the conservation of energy. When a fluid reaches sonic conditions, its kinetic energy is converted into pressure energy, resulting in a stagnation state.

3. What are the implications of stagnation and sonic conditions in fluid dynamics?

Stagnation and sonic conditions play a crucial role in fluid dynamics, particularly in high-speed flows. The conversion of kinetic energy to pressure energy at sonic conditions can lead to supersonic flow, shock waves, and other complex phenomena.

4. How can stagnation and sonic conditions be calculated?

Stagnation and sonic conditions can be calculated using the stagnation and sonic equations, which take into account factors such as the fluid's temperature, density, and velocity. These equations are often used in aerodynamics and gas dynamics to predict flow behavior.

5. What are the practical applications of understanding the relationship between stagnation and sonic conditions?

Understanding the relationship between stagnation and sonic conditions is crucial in various fields, including aeronautics, meteorology, and chemical engineering. It allows for the design and analysis of high-speed vehicles, prediction of weather patterns, and optimization of industrial processes involving compressible fluids.

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