P4 Supersonic Flow

Question 1

Give an example of a reversible adiabatic process?

Explain how it is reversible and what this process is called?

Answer 1

CONVERGENT-DIVERGENT NOZZLE;
As SUBSONIC air flows in one way it becomes SUPERSONIC by the OUTLET, if SUPERSONIC air flows through the same initial OUTLET towards initial INLET it will RETURN to SUBSONIC airflow with the SAME PROPERTIES as originally provided the PRESSURE at inlet and outlet remain CONSTANT through both processes;
NO ENERGY is TRANSFORMED into different forms and the ENTROPY of the air (THERMODYNAMIC PROPERTY) has not CHANGED either meaning it is a ISENTROPIC process

Question 2

Give an example of an irreversible adiabatic process?

Explain why it is irreversible?

Answer 2

AIRFLOW travelling THROUGH a NORMAL SHOCKWAVE;
COMPRESSIBLE SUPERSONIC airflow passes through a NORMAL SHOCKWAVE and becomes SUBSONIC with a PRESSURE INCREASE;
The shockwave RECEIVES ENERGY from SUPERSONIC flow, but the SUBSONIC flow CANNOT pass back through shockwave and GAIN the ENERGY LOST in this process to become SUPERSONIC;
In this process NO HEAT was TRANSFERRED but the ENTROPY CHANGED so it is an IRREVERSIBLE process

Question 3

Describe the basic air property changes of a normal shockwave (Stagnation Temp and Pressure)?
How can them be calculated?

Answer 3

STAGNATION (TOTAL) PRESSURE DECREASES;
STATIC PRESSURE INCREASES;
STAGNATION TEMPERATURE is CONSTANT;
ADIABATIC equations or RAYLEIGH supersonic pitot tube equation

Question 4

Explain how pressure changes across a normal shockwave?

Answer 4

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Question 6

Describe the condition for an oblique shockwave to form?

Where can they form?

Answer 6

When SUPERSONIC air flows over a CONCAVE corner a SHOCKWAVE is formed at the corner with ANGLE to INCOMING AIRFLOW, the ANGLE between SHOCKWAVE and DIRECTION of INCOMING AIRFLOW is LESS than 90° and called OBLIQUE shockwave;
PRESSURE INCREASES and there is a POSITIVE PRESSURE DISTURBANCE, with accumulating WAVES;
LEADING EDGE of supersonic aircraft, inside BENT DUCK/PIPE or at EXIT of OVER-EXPANDED airflow

Question 7

Explain the air properties before and after an oblique shockwave (Pressure, velocity and Mach)?

Answer 7

Since the shockwave is a COMPRESSIVE SUPERSONIC wave so PRESSURE INCREASES across shockwave;
VELOCITY is broken into 2 COMPONENTS, PARALLEL to shockwave (τ) and PERPENDICULAR to shockwave (n);
In (τ) DIRECTION air does NOT CROSS the shockwave so there is NO CHANGE in VELOCITY assuming NO PARTICLE SLIP;
In (n) DIRECTION air PASSES shockwave as if passing through NORMAL SHOCKWAVE which causes a DECREASE in VELOCITY as V1n.V2n = ac^2 or M1n.M2n = 1;
Since TEMPERATURE/SPEED of SOUND are the SAME in both PARALLEL and PERPENDICULAR DIRECTIONS M1 and M2 can be used to calculate both components in their respective regions. M2n will be SUBSONIC (<1) however, M2 may be SUPERSONIC (>1) depending on SHOCK ANGLE (β)

Question 8

What are the equations for the parallel and perpendicular velocities of oblique shockwaves?

Answer 8

V1τ = V1 cos β
V1n = V1 sin β
V2τ = V2 cos (β - θ)
V2n = V2 sin (β - θ)

Question 9

What does the θ-β-M equation tell us?

What do the black lines represent?

Answer 9

For a given MACH number there is a RELATIONSHIP between DEFLECTION ANGLE θ and SHOCK ANGLE β, the SHOCK ANGLE of oblique shock depends on the DEFLECTION ANGLE;
Each INCOMING MACH number (M1) has a CURVED LINE that is CONCAVE to the β axis with a TURNING POINT which represents MAXIMUM DEFLECTION θmax

Question 10

What does the red line on the θ-β-M diagram show?

What about the blue line?

Answer 10

The RED line CONNECTS the MAXIMUM DEFLECTION ANGLES for each INCOMING MACH number. If DEFLECTION ANGLE for a given MACH number EXCEEDS the MAXIMUM, the oblique shock will be DETACHED from LEADING EDGE of deflected CORNER;
If the DEFLECTION ANGLE is LESS than MAXIMUM there are 2 possible value of SHOCK ANGLE β. The SMALLER SHOCK ANGLE indicates the oblique shock is a WEAK SHOCK M2>1 and airflow AFTER is SUPERSONIC. The GREATER SHOCK ANGLE indicates the oblique shock is a STRONG SHOCK M2<1 and airflow AFTER is SUBSONIC. The blue line indicates where M2=1 (CRITICAL MACH LINE) which is the DIVIDE between STRONG and WEAK SHOCKWAVE

Question 11

If the deflection angle is 0° what is the shock angle?

Answer 11

STRONG SHOCK ANGLE = 90° ALWAYS (NORMAL shockwave);

WEAK SHOCK ANGLE = sin^-1.(1/M1)

Question 12

Give examples of where strong and weak shocks occur?

Answer 12

STRONG: INTERNAL SUPERSONIC flow ie: inside TURBINE or at EXIT of JET ENGINE;
WEAK: EXTERNAL SUPERSONIC flow ie: airflow OVER AEROFOIL or FUSELAGE

Question 13

What is the assumption when incoming Mach number is large?

Answer 13

When Mach number is at MAXIMUM DEFLECTION ANGLE, M2 = 1

Question 14

Explain the condition for expansion wave formation?
What is an expansion region?
What determines the size of expansion region?

Answer 14

COMPRESSIBLE airflow EXPANDS when travelling to VOLUME INCREASING location or encounters PRESSURE DECREASING DISTURBANCE;
A MACH WAVE forms at DEFLECTED SURFACE which is the EXPANSION WAVE;
The EXPANSION WAVE CANNOT PROPAGATE wit the airflow so a SERIES of WAVES form at DIFFERENT STAGES called EXPANSION REGION usually at a CONVEX CORNER;
DEFLECTION ANGLE and INCOMING MACH number

Question 15

What happens to the air properties across expansion wave?
What type of process is this?
What equation can be used?

Answer 15

Air PRESSURE DECREASES;
Air SPEED INCREASES;
INCOMING MACH SMALLER than OUTGOING MACH;
NO HEAT TRANSFER so is ISENTROPIC (REVERSIBLE ADIABATIC) PROCESS;
ENERGY equation

Question 16

What is the Mach angle?

How can we use this to find the expansion region?

Answer 16

ANGLE between INCOMING AIRFLOW and MACH WAVE (u = sin^-1(1/M));
Find the MACH ANGLE of FIRST WAVE using INCOMING MACH number and MACH ANGLE of LAST WAVE using OUTGOING MACH number

Question 17

What equation gives a definite relationship between incoming and outgoing Mach?
What function is needed for this?

Answer 17

θ = v(M1) - v(M2)

PRANDTL-MEYER FUNCTION to find v(M)

Question 18

Explain why is there a limit to maximum deflection angle for expansion waves?

Answer 18

THEORETICALLY when air EXPANDS PRESSURE DECREASES to 0 and the AIRSPEED will reach HIGHEST possible value;
If P2 is 0 the M2 will be INFINITY therefore, MAXIMUM DEFLECTION ANGLE is associated with M1 = 1, P2 = 0 and M2 = INFINITY;
V(1) = 0° and θmax = 130°, beyond this ANGLE air CANNOT EXPAND FURTHER;
If the DEFLECTION ANGLE is GREATER than the MAXIMUM, the line of θmax is called SLIPSTREAM and the AREA beyond that is the STAGNATION REGION;
In THEORY, PRESSURE at SLIPSTREAM is 0 and AIRSPEED is MAXIMUM however AIRSPEED in STAGNATION REGION is 0. A sudden CHANGE like this CANNOT be SUSTAINED due to VISCOSITY and in practice VORTICES would form in REGION which form VORTEX STREET which cause LOCAL OSCILLATION of airflow