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Hi guys,
I'm trying to delve into compressible flow and in the various textbooks I'm reading I've found two school of thoughts when it comes to how to calculate the threshold (or error compared to incompressible flow) of 'M' when to consider a flow compressible.
In Fundamentals of Aerdnyamics (Anderson) this equation is used to calculate the 'error' between considering the fluid compressible and not
$$ \frac{\rho_0} {\rho}= (1+ \frac {\gamma -1}{2} M^2)^{\frac{1} {\gamma-1}} $$
For air (gamma = 1.4), this would give a deviation of (nearly) 5% between ##\rho## and ##\rho_0## at M = 0.3. Anderson uses this to show with an error of merely 5% you might as well consider the flow incompressible. Whereas in other textbooks (White; Kundu, Cohen) this equation is used to determine said deviation
$$ -M^2 \frac {dV}{V} = \frac {d\rho}{\rho}$$
This would give us a 9% error at M = 0.3 (for some reason literally every textbook rounds this up to 10%. I haven't found a single textbook that said otherwise). In these textbooks the authors derive at the conclusion that with an error of 10% (or less) the flow can be considered incompressible and ignore all the extra kerfuffle that compressible flow brings to the table.
My question is: What differentiates the two approaches to calculate the error? I mean obviously in the first equation the fluid is important (gamma value), but other than that? I think the two approaches describe two different "errors", but I can't seem to pinpoint what exactly the difference is.
If there is a link or a textbook that describes or answers my question, there is no need to spoonfeed me, just point me in the right direction.
Thanks!
I'm trying to delve into compressible flow and in the various textbooks I'm reading I've found two school of thoughts when it comes to how to calculate the threshold (or error compared to incompressible flow) of 'M' when to consider a flow compressible.
In Fundamentals of Aerdnyamics (Anderson) this equation is used to calculate the 'error' between considering the fluid compressible and not
$$ \frac{\rho_0} {\rho}= (1+ \frac {\gamma -1}{2} M^2)^{\frac{1} {\gamma-1}} $$
For air (gamma = 1.4), this would give a deviation of (nearly) 5% between ##\rho## and ##\rho_0## at M = 0.3. Anderson uses this to show with an error of merely 5% you might as well consider the flow incompressible. Whereas in other textbooks (White; Kundu, Cohen) this equation is used to determine said deviation
$$ -M^2 \frac {dV}{V} = \frac {d\rho}{\rho}$$
This would give us a 9% error at M = 0.3 (for some reason literally every textbook rounds this up to 10%. I haven't found a single textbook that said otherwise). In these textbooks the authors derive at the conclusion that with an error of 10% (or less) the flow can be considered incompressible and ignore all the extra kerfuffle that compressible flow brings to the table.
My question is: What differentiates the two approaches to calculate the error? I mean obviously in the first equation the fluid is important (gamma value), but other than that? I think the two approaches describe two different "errors", but I can't seem to pinpoint what exactly the difference is.
If there is a link or a textbook that describes or answers my question, there is no need to spoonfeed me, just point me in the right direction.
Thanks!