One Dimension Normal Shock

Fluid Flow Regimes

Various flow regimes are classified based on the definition of Mach number.

1. Subsonic Flow: When the fluid velocity is lower than the acoustic speed (M<1) then the fluid flow is called as subsonic. However Mach number of the flow changes while passing over an object or through a duct. Hence for simplicity, flow is considered as subsonic if Mach number is in the range 0-0.8. All small amplitude disturbances travel with acoustic speed and speed of the flow in the subsonic regime is less than acoustic speed hence presence of the disturbance is felt by the whole fluid domain.

Therefore subsonic flow is pre-warned or prepared to face the disturbance.

2. Transonic flow: When the flow Mach number is in the range 0.8-1.2 it is called transonic flow.
 
Highly unstable and mixed subsonic and supersonic flows are the main features of this regime.

3. Sonic flow: When flow Mach number is 1 it is called sonic flow.

4. Supersonic Flow: When the flow Mach number is more 1 everywhere in the domain then it is called as supersonic flow. This flow is not pre-warned since the fluid speed is more than the speed of sound.

5. Hypersonic Flow: As per the thumb rule, when the flow Mach number is more than 5 then it is called as hypersonic flows. This is not the fixed definition for hypersonic flow since hypersonic flow is defined by certain characteristics of flow.

Isentropic Relations for Reference Conditions

Flow is said to be stagnant when its velocity is zero. Here we are interested to predict the flow properties at the stagnation conditions. Let's imagine that a fluid flow is decelerated from its existing state isentropic ally to zero velocity which is termed as the stagnation condition as shown in Fig. 1. All the properties of the flow at stagnation condition are called as stagnation properties. Similarly if we decelerate the supersonic flow or accelerate the subsonic flow isentropic ally so that the fluid particles reach sonic velocity, then flow properties are called as star properties. Both the stagnation properties and star properties are the reference properties of the flow and are constant in the fluid domain if the flow is isentropic. Let's apply the 1D energy conservation principle to derive the relation initially between stagnation and static properties.
 


Consider that the fluid particle is isentropically brought to zero as shown in above figure. We know that 1D form of energy conversion equation is

 
Here subscript 1 stands for initial state of the fluid and subscript 2 stands for final decelerated state of fluid. Since, V₂=0, lets represent T₂=T₀ is above equation. Then,

 
Dividing the above equation by 

 
Here subscript 0 represents the stagnation condition. Its evident from this equation that the stagnation temperature to static temperature ratio is dependent on Mach number & specific heat ratio.

The Mach number in this expression is the Mach number of the flow before commencement of isentropic deceleration.

Since the process is isentropic and we already have derived isentropic relations, we can find out stagnation pressure to static pressure relation and the same for density also.
 
 

From the expression for stagnation pressure to static pressure, it can be seen that the stagnation pressure and static pressure are almost equal if Mach number is zero. However for the incompressible flows with Mach number less than 0.3, it can be evaluated that the difference between static pressure and stagnation pressure is equal to the dynamic pressure. But this isn't the case for compressible flows.

Adiabaticity of the flow is sufficient for the definition of total temperature since in the process of deriving the ratio of stagnation temperature to the static temperature we have assumed steady, adiabatic and inviscid flow.

There is no assumption of reversibility made while deriving the 1D energy equations. Therefore adiabaticity of the process is sufficient for calculation of the stagnation to static temperature ratio. Flow past the normal shock (to be learnt soon) is an irreversible process like friction where we will prove constancy of total temperature across the normal shock. Unlike stagnation to static temperature ratio, pressure and density ratios need adiabatic and reversible assumption since we have used explicit isentropic relations while obtaining these ratios. In this derivation we have seen that, an imaginary adiabatic or isentropic process of deceleration should be followed for evaluation of reference quantities at a point in the flow field, however actual flow field might not be adiabatic or isentropic. Therefore, in reality, stagnation properties of the fluid change from point to point.

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