Are Total and Static Temperatures Equivalent for Incompressible Fluids?

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Discussion Overview

The discussion revolves around the relationship between total and static temperatures in incompressible fluids. Participants explore whether these two temperatures can be considered equivalent under various conditions, particularly focusing on the implications of incompressibility and Mach number.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant suggests that total and static temperatures of an incompressible fluid are the same, seeking confirmation of this understanding.
  • Another participant presents a formula relating static and total temperatures, indicating that under low Mach number conditions, the assumption of equivalence may hold, but accuracy depends on context.
  • A different viewpoint asserts that for incompressible flow, total and static temperatures are exactly the same, regardless of velocity, and seeks validation of this claim.
  • One participant challenges the equivalence by referencing the definition of total properties and questioning how they can be the same at high Mach numbers, even in incompressible fluids.
  • Another participant argues that the equations referenced are based on perfect gas laws and may not apply to all fluids, emphasizing that for incompressible flow, entropy is solely a function of temperature.
  • One participant highlights that all fluids are compressible to some extent, including water, and suggests that the assumption of incompressibility may not fully capture the behavior of fluids.
  • A participant reiterates that the relationship between total and static temperatures under the assumption of incompressible flow remains unclear, questioning their equivalence.
  • Another participant states that static and total temperatures are not equal under all conditions, emphasizing their dependence on Mach number and suggesting that closed-form solutions for real gases may not exist.

Areas of Agreement / Disagreement

Participants express differing views on the equivalence of total and static temperatures in incompressible fluids, with no consensus reached. Some argue for their equivalence under certain assumptions, while others challenge this notion based on compressibility and the nature of fluid dynamics.

Contextual Notes

Participants note that the discussion relies on assumptions about fluid behavior, including the ideal gas approximation and the definition of incompressibility. The implications of Mach number and the conditions under which the equivalence may hold are also highlighted as significant factors.

defunc
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Am I correct if I say the total and static temperatures of an incompressible fluid are the same?

Thanks.
 
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Generally, the ratio between static and total temperature is:
[tex] \frac{T_0}{T} = 1 + \frac{\gamma -1}{2}M^2[/tex]
So if one is assuming low Mach incompressible flow then it's a reasonable assumption. It depends on the accuracy you're looking for though.
 
Yep. But from my understanding, for incompressible flow the total and static temperatures are exactly the same. Regardless the velocity. Just want to confirm this.

Regards.
 
How can they be the same given what Minger pointed out to you? Imagine hydraulic fluid or water (both incompressible) at a high Ma. The definition of a total or stagnation property is in bringing a moving fluid to rest isentropically. Granted, most places you see total properties are in compressible flow areas, but there is nothing (at least that I can think of) that doesn't say you can't use them for incompressible.

What is it in your understanding makes them the same based on compressibility?
 
Mingers equations are based on perfect gas laws. For incompressible flow entropy is only a function of temperature. From there my conclusion...
 
Incompressible is an assumption.

Everything is compressible. Water is compressible. It may be essentially incompressible but its not. That's what you're not fully understanding.
 
The equations you gave is also based on an assumption: perfect gas behaviour. So its not applicable to any fluids. So my question still remains, under the assumption of incompressible flow, what's the relation between total and static temperatures? Is it the same?
 
Static and total temperature are no for equal at any condition than static and total pressure are. They are both simply functions of Mach. If you want a closed-form solution assuming a real gas, then I'm sorry, I don't think any exist. Run a CFD using real gas propreties...which you typically can't either, unless you hardwire it yourself.

Sure suuuuree, under your assumption that for incompressible flow, Mach = 0, they are the same. How same are they actually? I don't know, plug and chug. You'll have orders of magnitude less error assuming ideal gas than assuming imcompressible.
 

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