P4 Basic Aerodynamic Equations

Question 1

What are the considerations for compressible 1D airflow?

What are the relevant equations?

Answer 1

Airflow is in STEADY STATE, there is NO HEAT EXCHANGE and is IDEAL GAS;
CONTINUITY equation;
MOMENTUM/EULER equation;
ENERGY equation;
IDEAL GAS law;
CRITICAL POINT

Question 2

What does the energy equation tell us about the relationship between v and a?

Answer 2

When SPEED is at a MAXIMUM, the TEMPERATURE and SPEED of SOUND is 0;
When SPEED is 0, the TEMPERATURE is STAGNATION temperature and the SPEED of SOUND is at MAXIMUM;
When SPEED is EQUAL to SPEED of SOUND, M=1 also known as the CRITICAL POINT

Question 3

What is M*?

What is the purpose of it?

Answer 3

The SPEED COEFFICIENT which is a RATIO of SPEED of airflow to CRITICAL SPEED of SOUND;
Gives an INDICATION of whether airflow is SUBSONIC or SUPERSONIC since ac does NOT CHANGE with v along flow stream;
If M=1 then airflow is SONIC and M=1;
If M<1 then airflow is SUBSONIC and M1 then airflow is SUPERSONIC and M>M*>1

Question 4

Using equations, describe the air properties of a subsonic airflow at the inlet of a chamber with increasing and decreasing areas respectively?

Answer 4

CONVERGING NOZZLE:
dA/A = (M^2 - 1)dv/v: When AREA is DECREASING dA<0 (negative) and M<1 (subsonic) dv>0 SPEED INCREASES;
dA/A = (1 - M^2)/(γM^2)dp/p: When AREA is DECREASING dA<0 (negative) and M<1 (subsonic) dp<0 PRESSURE DECREASES;
PRESSURISING TUBE:
dA/A = (M^2 - 1)dv/v: When AREA is INCREASING dA<0 (positive) and M<1 (subsonic) dv<0 SPEED DECREASES;
dA/A = (1 - M^2)/(γM^2)dp/p: When AREA is INCREASING dA<0 (positive) and M<1 (subsonic) dp>0 PRESSURE INCREASES;

Question 5

Using equations, describe the air properties of a supersonic airflow at the inlet of a chamber with increasing and decreasing areas respectively?

Answer 5

DIVERGING NOZZLE:
dA/A = (M^2 - 1)dv/v: When AREA is INCREASING dA>0 (positive) and M<1 (supersonic) dv>0 SPEED INCREASES;
dA/A = (1 - M^2)/(γM^2)dp/p: When AREA is INCREASING dA>0 (positive) and M<1 (supersonic) dp<0 PRESSURE DECREASES;
PRESSURISING TUBE DIFFUSER:
dA/A = (M^2 - 1)dv/v: When AREA is DECREASING dA<0 (negative) and M>1 (supersonic) dv<0 SPEED DECREASES;
dA/A = (1 - M^2)/(γM^2)dp/p: When AREA is DECREASING dA<0 (negative) and M>1 (supersonic) dp>0 PRESSURE INCREASES

Question 6

Describe the process of airflow passing through a converging-diverging nozzle where the air is subsonic at the inlet?
Ensure to add equations and expressions

Answer 6

INLET SUBSONIC flow: DECREASE AREA, INCREASE VELOCITY, DECREASE PRESSURE;
THROAT SONIC flow: AREA local MINIMUM therefore DA/A = 0 so M = 1 to keep equation true. The THROAT is also the CRITICAL POINT so using ENERGY EQUATION v = ac therefore M = 1. AIRSPEED is GREATER than INLET but LESS than OUTLET and PRESSURE is LESS than INLET but GREATER than OUTLET;
OUTLET SUPERSONIC flow: INCREASE AREA, INCREASE VELOCITY, DECREASE PRESSURE

Question 7

Describe the process of airflow passing through a converging-diverging nozzle where the air is supersonic at the inlet?
Ensure to add equations and expressions

Answer 7

INLET SUPERSONIC flow: DECREASE AREA, DECREASE VELOCITY, INCREASE PRESSURE;
THROAT SONIC flow: AREA local MINIMUM therefore DA/A = 0 so M = 1 to keep equation true. The THROAT is also the CRITICAL POINT so using ENERGY EQUATION v = ac therefore M = 1. AIRSPEED is LESS than INLET but GREATER than OUTLET and PRESSURE is GREATER than INLET but LESS than OUTLET;
OUTLET SUBSONIC flow: INCREASE AREA, DECREASE VELOCITY, INCREASE PRESSURE

Question 8

What is a shockwave?

How does it form?

Answer 8

A shockwave is an ACCUMULATION of DISTURBANCES that form a WAVE FRONT;
When airflow is SUPERSONIC, air PARTICLES travel at speed GREATER than SPEED of SOUND;
If a COMPRESSION DISTURBANCE is experienced in the SUPERSONIC airflow the DISTURBANCE will PROPAGATE at the SPEED of SOUND as the PRESSURE WAVE CANNOT match SPEED of AIR;
When airflow is SUBSONIC any CHANGE in air PROPERTIES can take place CONTINUOUSLY, however since the CHANGE CANNOT match the SPEED of air in SUPERSONIC flow the PROPERTIES are NOT UNIFORM or CONTINUOUS;
This DISCONTINUITY ACCUMULATES to form WAVE FRONT which is regarded as the shockwave;
The SHOCKWAVE is very THIN (1mm) and air PROPERTIES CHANGE SUDDENLY across it

Question 9

What is the condition for airflow properties across a normal shockwave?

Answer 9

It is an IRREVERSIBLE, ADIABATIC process

Question 10

Using equations, describe how Mach number and speed changes across a normal shockwave?

Answer 10

Assuming a 1-DIMENSIONAL, NON-VISCOUS flow:
CONTINUITY, IMPULSE MOMENTUM and ENERGY equations can be combined to find that M1.M2 = 1;
Since the FORMATION of SHOCKWAVE requires that M1* or M1 > 1 (SUPERSONIC) before shock, M*2 or M2 must be SUBSONIC to satisfy this equation;
The HIGHER M1 is, the LOWER M2 will be which means the HIGHER the AIRSPEED before a normal shock, the LOWER the AIRSPEED after the normal shock

Question 11

Describe how the air properties change across a normal shockwave (T, p, ρ, To, po)?

Answer 11

STAGNATION TEMPERATURE is CONSTANT;
STAGNATION PRESSURE DECREASES after normal shockwave;
TEMPERATURE, PRESSURE and DENSITY all INCREASE after normal shockwave