Review of Fundamentals -Fluid Mechanics - Amazon S3 · PDF fileReview of Fundamentals -Fluid Mechanics ... • In compressible flow, ... mass flow rates and pressure distribution for

Review of Fundamentals - Fluid

Mechanics

© 2015 SIM University. All rights reserved.

Introduction

• Properties of Compressible Fluid Flow

• Basics of One-Dimensional Gas Dynamics

• Nozzle Operating Characteristics

• Characteristics of Shock Wave

• A gas turbine cycle through the use of H-K Diagram


• Approximate Mach zones:

< M0.3 Subsonic, incompressible

M0.3 – M0.8 Subsonic, compressible

M0.8 – M1.2 Transonic, shock waves appear

M1.2 – M3 Supersonic

�M3 Hypersonic

• Normal to specify P, T, V to describe state

• In compressible flow, V is often replaced by Mach number, total pressure

and total temperature.


Compressible Flow Properties

Without gravity effects, the Steady Flow Energy Equation (SFEE) is

For calorically perfect (constant cp, cv) gas,

The stagnation or total enthalpy ht is defined as

The stagnation or total temperature Tt is defined as


Total Enthalpy / Total Temperature

• For an aircraft in flight at velocity Va, the airstream velocity at the

leading edge stagnation point is negligibly small

• Kinetic energy is brought to rest and produces a rise in temperature

(aerodynamic heating)

Watch video on aerodynamic heating:

http://www.youtube.com/watch?v=RChlt5wdqBs

Adapted: “Elements of Propulsion: Gas Turbines and

Rockets” by Jack D. Mattingly



Inserting the total enthalpy into the SFEE:

For calorically perfect gas,

If there is no heat transfer and no work interactions,

(i.e. q - wx = 0, or q = wx = 0), then

and, for a calorically perfect gas,



The total pressure Pt of a flowing gas is defined as the pressure obtained

when the gas is brought to rest isentropically (sy – s1 = 0)

© 2015 SIM University. All rights reserved.Source: Soon Kim Tat

(Note: recall that )

Total Pressure

• Stagnation temperature ratio

• Stagnation pressure ratio

• At M=1 (choked nozzle) for air (isentropic),


Stagnation Temperature and Pressure

© 2015 SIM University. All rights reserved. Source: Soon Kim Tat

Stagnation Temperature and Pressure

• Perfect gas is brought to stagnation (V2 = 0)

• Under adiabatic (q = 0), no-shaft-work (wx = 0)

• Same final stagnation temperature will be attained whether it is

irreversible or reversible process, i.e.

Tt2, irreversible = Tt2,reversible

• However, the final total pressure

will be lower, i.e.

Pt2, irreversible < Pt2, reversible

• Py depends on the entropy increase ,

(sy – s1) - a measure of the degree of

irreversibility

Source: p 98,“Elements of Propulsion: Gas

Turbines and Rockets” by Jack D. Mattingly


Total Pressure (Irreversible)

• Total pressure of air passing through an engine inlet and nozzle or a

shock wave cannot increase and must decrease because of the

irreversible effects of friction.

Schlieren Imaging of Supersonic Inlet

shock Waves

Source: http://en.wikipedia.org/wiki/Unstart


Total Pressure (Irreversible)

For a one-dimensional flow where q = wx = 0,

For a calorically perfect gas,

Rewrite in terms of dimensionless static enthalpy

and dimensionless kinetic energy as

We obtain

or H + K = 1


One-Dimensional Gas Dynamics

• Dimensionless static enthalpy (H) and dimensionless kinetic

energy (K)

• Useful for

– explaining the more complex internal flow behaviour of air

breathing engines

– visualising the operation of propulsion devices

• Not a state diagram


H-K Diagram

M<1 M>1

1

1

Adapted: “Elements of Propulsion: Gas Turbines and Rockets” by Jack D. Mattingly© 2015 SIM University. All rights reserved.

H-K Diagram

Key: 0 = freestream reference state.

Point c =choked condition at constant impulse.

Points u and d denote end states of normal shock.

Circled numbers denote isolines of constant property as follows:

1. Static enthalpy, static temperature

2. Kinetic energy, velocity, pressure (for frictionless heating or cooling only)

3. Isoline of constant Mach number

4. Total enthalpy, total temperature (adiabatic),

5. Post-heat release adiabatic, τ > 1

6. Impulse function / stream thrust, area (for frictionless flow with heating or

cooling only), case I = I0 ;

7. Impulse function, case φ > φ0

0

1t

t

T

Tτ ≡ =


H-K Diagram

Adapted: “Elements of Propulsion: Gas Turbines and Rockets” by Jack D. Mattingly


Scramjet H-K Diagram

Source: The Jet Engine by Rolls Royce


Nozzle Design

• From a large chamber, a gas flows through a nozzle with mass flow

rate ṁc

– chamber pressure Pc = Pt

– chamber temperature Tc = Tt

Graphics: Soon Kim Tat

Nozzle Gas Relationships:


Nozzle Design

Nozzle design – To pass a given mass flow with minimum frictional

losses between 2 regions of different pressure. (Independent

variable:– P)

Nozzle operating characteristics – Given a nozzle, determine the

mass flow rates and pressure distribution for various nozzle

pressure. (Independent variable: A)

• 4 variables: P, T, V, A

• Select one variable as independent and find the remaining


Nozzle Design Approaches

Design Objective: To expand exhaust gas to a target static pressure

Source: “Elements of Propulsion: Gas Turbines and Rockets” by Jack D. Mattingly


Nozzle Design - Example

• Consider a wind-tunnel nozzle (next slide)

• As air flows from storage chamber into evacuated receiver:

– raises the pressure in the nozzle exhaust region Pa

– decreases the nozzle pressure ratio Pn = Pc / Pa.

• 7 possible distinct nozzle pressure ratio operating conditions.


Nozzle Flow and Shock Waves

Nozzle Pressure Ratio

Pn=Pc/Pa

1. Underexpanded

Pn>Pṅ

2. Design expansion

Pn=Pṅ

3. Overexpanded

Pn<Pṅ

4. Normal shock at

exit

5. Normal shock

inside

6. Sonic at throat,

subsonic

elsewhere

7. Subsonic flow

everywhere (mass

flow below max)

Adapted: “Elements of Propulsion: Gas Turbines and Rockets” by Jack D. Mattingly© 2015 SIM University. All rights reserved.

Nozzle Flow and Shock Waves

Summary

• Total enthalpy, total temperature and total

pressure, and their relationships

• Cycle of an air-breathing jet engine using H-K

diagram

• Evaluating the issues related to nozzle design


Reflection Question

• Evaluate what happens to the gas pressure,

temperature and velocity as it passes through a

convergent nozzle and a divergent duct, if the

initial velocity is:

a. Subsonic

b. Supersonic


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