# Relation Between Ma and Re for Isentropic Expansion

• Ankith
In summary, the conversation discusses the isentropic expansion of air from a fixed given reservoir and investigates the behavior of the Reynolds number as a function of the Mach number. It is mentioned that for small values of Mach number, the thermodynamic properties of the flow will not deviate significantly from the reservoir conditions, resulting in a linear increase in Reynolds number. The question then asks to investigate the limit behavior for larger Mach numbers and take into consideration the temperature dependence of viscosity using the Power Law. A value for Sutherland's constant is provided, and the equations for Power Law and Reynolds number are discussed. The proposed solution involves calculating the distribution of Reynolds number for different Mach numbers by substituting the calculated temperature and Mach number values into the equations. The
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.

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

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 said:
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.

Ankith said:
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?

Chestermiller said:
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.

## 1. What is the relation between Ma and Re for isentropic expansion?

The relation between Ma (Mach number) and Re (Reynolds number) for isentropic expansion is given by the equation Re = Ma * sqrt(gamma / (gamma - 1)), where gamma represents the specific heat ratio. This equation describes the ratio of inertial forces to viscous forces in a flow field and is used to characterize the properties of a compressible fluid undergoing isentropic expansion.

## 2. How is Ma related to the speed of sound in a fluid?

Ma is defined as the ratio of the flow velocity to the speed of sound in a fluid. It is a dimensionless quantity that indicates the compressibility of the fluid. A Ma value less than 1 indicates subsonic flow, while a Ma value greater than 1 indicates supersonic flow.

## 3. Can the relation between Ma and Re be used for all types of fluids?

The relation between Ma and Re is specifically used for compressible fluids undergoing isentropic expansion. It may not be applicable for all types of fluids, such as incompressible fluids or those undergoing non-isentropic processes.

## 4. How does the relation between Ma and Re affect the flow behavior of a fluid?

The Ma-Re relation plays a significant role in determining the flow behavior of a fluid. For example, as the Ma number increases, the flow becomes more compressible and shock waves can form. Additionally, as the Re number increases, the flow becomes more turbulent and the boundary layer thickness decreases, resulting in increased drag and heat transfer.

## 5. What is the practical application of the Ma-Re relation?

The Ma-Re relation is used in various engineering applications, such as in the design and analysis of compressible flow systems, such as gas turbines and jet engines. It is also used in aerodynamics to study the behavior of high-speed flows around objects, such as aircraft and rockets.

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