Relation Between Ma and Re for Isentropic Expansion

[Ankith]

Homework Statement

Consider the isentropic expansion of air from a fixed given reservoir (i.e. total pressure and temperature). Investigate the behaviour of the value of the Reynolds number of the flow, as a function of the Mach Number M of the expanded flow.
For small values of M, the thermodynamic properties of the flow will not deviate signifciantlz from the reservoir conditions, hence Re increases linearly with U and therefore with M.

How do I show that for increasing expansion (i.e. increasing M) the value fo Re will eventually decrease, by investigating the limit behaviour for M>>1. I must emplease the temperature dependence of viscosity using the Power Law.
Sutherlands Constant, S=111 K

Homework Equations

Power Law: mu/mu_0=[(T/T_0)][w] , where w=3/2 - T_0/T_0+S

The Attempt at a Solution

T_0=288K
w=0.778
Substituting in Power Law,
mu=1.81*10^(-5)*(T/288)^0.778

Now Re=rho*U*L/mu

Re=rho*sqrt(R*gamma*T_0*M)/mu

For different values of M, I can calculate T using the isentropic expansion equation, subsititute that value of T in the equation to calculate mu, and further substitute the Mach number and value of 'mu' in the Reynolds number equation. This can thus give me the distribution of the Reynolds number for different Mach numbers.

However, after plotting what I observe is that the Re continues to linearly increase, which is not what the question states, as I must observe a reduction in the value of Re after Ma=1.

 

[Chestermiller]

What exactly it the geometry? Can you please provide a sketch?

 

[Ankith]

There is no geometry provided. The question is very specific to using the power law for the temperature dependence of viscosity. Which is why an area ratio is not really meant to be considered.

 

[Ankith]

What exactly it the geometry? Can you please provide a sketch?

2 / 0

There is no geometry provided. The question is very specific to using the power law for the temperature dependence of viscosity. Which is why an area ratio is not really meant to be considered.

 

[Chestermiller]

There is no geometry provided. The question is very specific to using the power law for the temperature dependence of viscosity. Which is why an area ratio is not really meant to be considered.

So you have air flow through a pipe of increasing cross sectional diameter?

 

[Ankith]

So you have air flow through a pipe of increasing cross sectional diameter?

I believe that would be our best bet. That is what I have assumed as well.