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Gas Dynamics



2 Author: Dr. Sushil Chaturvedi

Old Dominion University

Objective

This module introduces students to applications of laws of thermodynamics to steady one-

dimensional compressible flows. Applications such as subsonic and supersonic nozzles and

diffusers are considered in some details for both isentropic and non-isentropic (shock embedded)

flows.

Motivation and Relevance

The basic concepts, definitions and analysis tools introduced here will help students analyze

subsonic and choked converging nozzles in jet engines and supersonic converging-diverging

rocket nozzles. Consideration of under-expanded and over-expanded flows in converging and

diverging nozzles will help the reader relate nozzle operating parameters to observed

phenomenon of shock-diamonds in these flows.

Concepts, Definition and Analysis

Compressible Flows and Speed of Sound

In a compressible fluid the density (ρ) is significantly affected by pressure changes in the fluid.

Conversely, density in incompressible fluids is not affected by pressure changes. It should be

noted that a compressible fluid such as air can behave as incompressible fluid in some flow

situations if pressure changes are small. These flows are known as incompressible flows (of a

compressible fluid). More precise criterion for characterizing flows as compressible or

incompressible will be introduced later. The concept of speed of sound or acoustic speed is

particularly important in compressible flows. The speed of sound is the speed with which an

infinitesimal pressure disturbance dP travels through a compressible medium. As noted earlier,

this pressure disturbance will create a small density disturbance (dρ) in the medium. The concept

of speed of sound is illustrated in Fig. 10.1.





Figure 10.1 Illustration of velocity of sound

Consider a small pressure disturbance set up by a small but sustained movement (dV) of the

piston enclosed in a cylinder containing a compressible medium such as air. The sudden

movement of piston creates a sudden crowding of molecules near the piston surface, that raises

the pressure level from P to P+dP. This creates a compression wave that travels in the

undisturbed medium at a velocity C, known as the velocity of sound. Ahead of the moving wave

front, the medium is undisturbed, while behind the front, the medium is disturbed, with

properties at P+dP, h+dh, T+dT and ρ+dρ levels. The medium also moves towards the right at a

velocity dV, thus satisfying the no penetration condition at the piston face, which is also moving

to the right with velocity dV.

In order to analyze this wave front propagation phenomenon in the steady state formulation, the

phenomenon is viewed by an observer who is moving with wave front at a velocity C. To put itdifferently, the observer jumps on the wave and rides with it, thus reducing the relative velocity

between the two to a zero value (moving frame reference). The same phenomenon seen by the

moving observer, is shown in Fig. 10.2. Note that the wave front has been made stationary with

respect to the moving observer and consequently undisturbed air flows toward the stationary

front at velocity C.

Figure 10.2 The stationary compression wave front

T+dT

P+dP

h+dh

ρ+dρ

T

P

h

ρ

CdV

dV Air

(undisturbed)

Compressionwave front

T+dT

P+dP

h+dh

ρ+dρ

TP

h

ρ

CC - dV

undisturbed

medium moving at

velocity c

Stationary wave front

12

disturbed

medium

Control volume





We now apply the first and second laws to a control volume drawn around the stationary front.

First Law: Control Volume

Neglecting potential energy changes and since qcv = 0 (adiabatic process) and wcv = 0, Eq. (10.1)

becomes,

ℎ +2

2= ℎ + ℎ +

−2

2

or,

ℎ + 2

2 = ℎ + ℎ + 2

2 + 2

2 −

Neglecting dV 2 /2 compared to other terms, the first law becomes

Mass Conservation

For steady state case being considered here, the incoming mass flow rate must be equal to the

outgoing mass flow rate.

1 = 2

= + − = + −−

Neglecting small order term (dρdV) compared to other terms, one gets,

Second Law: Control Volume

The second law for adiabatic processes can be expressed as

≥

=

=

+ + + = + + + 10.1

10.2

10.3

10.4





The equality is for a reversible process while the inequality applies to an irreversible process.

The compression wave is of infinitesimal (small) strength and does not involve friction and other

factors that make a process irreversible. Consequently, the compression wave front is reversible

and the process is reversible adiabatic or isentropic.

T-ds Relation

Applying the second T-ds relation and invoking the isentropic condition (ds = 0), one gets

= ℎ−

Eliminating dh from Eqs. (10.2) and (10.6), one gets

From mass conservation equation (Eq. (10.3)) one gets,

Eliminating dV from Eqs. (10.8) and (10.9), one gets,

2

=

=

=

=

= =

= =

= 10.5

10.7

10.6

10.8

10.9

10.10





The partial derivative is introduced with a subscript since P = P(ρ, s), and s is constant in this

derivation. Equation (10.10) applies to any pure substance in solid, liquid or vapor phase. For

ideal gases with constant specific heats undergoing an isentropic process, the pressure-volume

relationship is given as,

= 1 ; ≡ 0 0

or,

= 1 , =1

From above equation, the derivative ∂P ∂ρ s can be evaluated as

= 1−1 =1 =

Since, P = ρRT for ideal gases,

Steady, One-Dimensional Isentropic Flow with Area Change

Consider flow of an ideal gas in a geometry with area change shown in Fig. 10.3. Note that all

properties shown in figure, except entropy which is constant, are variable and function of only x,

the axial co-ordinate (h = h(x), P = P(x) etc.). A very small control (elemental) volume is chosen

across which the first and second laws of thermodynamics and the mass conservation law are

applied. The analysis is very similar to the one presented for the speed of sound derivation

except that the present case is more complicated due to area change or A being variable, A =

A(x).

Figure 10.3 Flow of gas in a pipe of changing area

V

Ρ

h

PT

A

s

V + dV

ρ + dρ

h + dh

P + dPT + dT

A + dA

s + ds

1 2

x

= 10.11





Note that as flow passes from inlet to exit, properties change. The increase or decrease would be

determined by the sign of dv, dρ, etc. If dv is positive, it would mean that the flow velocity has

increased in the increasing x or the flow direction.

First Law (Control Volume)

Neglecting the changes in potential energy, the first law can be expressed as,

+ ℎ1 +1

2

2= ℎ2 +

22

2+

ℎ +2

2= ℎ + ℎ +

+ 2

2

or,

ℎ + 2

2 = ℎ + ℎ + 2

2 + 2

2 +

Neglecting higher order term dV 2 /2 compared to other terms one gets,

ℎ + = 0

For ideal gases with constant specific heat

Equation (10.13) indicates that if velocity increases across the control volume (dV > 0), enthalpy

and temperature decrease (dh < 0; dT < 0) in the flow direction. This would happen in nozzles.

However, the opposite will happen in a diffuser since velocity decreases in the flow direction

(dV< 0) and this will imply dh > 0 and dT > 0.

The Second Law and Property Relations

= ℎ−

For reversible adiabatic or isontropic flows.

= 0

0 = ℎ−

= = −

= −

0 0

10.12

10.13





Equation (10.14) indicates that if pressure increases in the flow direction (dP > 0), as it would

happen in a diffuser, enthalpy must also increase in the flow direction (dh > 0). Eliminating dh

from Eqs. (10.13) and (10.14), one gets

Equation (10.15) governs the following two cases.Nozzles : dV > 0; velocity increases in the flow direction

dP < 0; pressure (P) decreases in the flow direction.

dT < 0; temperature decreases in the flow direction.

Diffusers : dV < 0; velocity decreases in the flow direction

dP > 0; pressure (P) increases in the flow direction.

dT > 0; temperature increases in the flow direction.

Mass Conservation Equation

In steady state flows, the mass conservation requires incoming mass flow rate be equal to

outgoing mass flow rate across the control volume.

Taking log of the above equation results in,

log + log + log = log 1

Differentiation of the log equation yields,

+ +

=

=

= = −

=

10.14

10.15

10.16

10.17





Special Case

For incompressible flows dρ = 0 (no change in density due to pressure changes),

Above equation indicates that in incompressible flows or fluids, as area decreases (dA < 0), as in

case of a nozzle, the velocity increases (dV > 0) and vice-versa.

For compressible flow case, solving Eq. (10.17) for dA/A, one gets

Using the chain rule one can express dρ as

Since C2 =∂P

∂ρs

Substituting Eq. (10.20) in Eq. (10.19), one gets

Also from Eq. (10.15)

Eliminating dV from Eqs. (10.21) and (10.22), one gets

= −

= − −

= =

= − −

=

−

10.18

10.19

10.20

10.21

10.22





=2

− 2

We define Mach number (M) as,

Equation (10.23) becomes

Equation (10.25) is a result of combined application of the first, second and mass conservation

laws to this problem. It is a very important equation that describes characteristics of subsonic (M

< 1) and supersonic (M > 1) nozzles and diffusers.

Four Configurations for Nozzles and DiffusersEquation (10.25) allows for four combinations based on the signs of dP and (1 – M

2). First we

note from Eq. (10.25) that at M = 1 (sonic condition), dA = 0.

Equation (10.26) indicates that M = 1 occurs where A has a maximum or a minimum. In fact,

one can show that its the minimum area condition that applies to M = 1 condition. In other

words, if M = 1 ever occurs in a nozzle or a diffuser, it must occur at A = Amin. This area is

generally designated as A*.

Table 10.1 shows the four combinations, namely subsonic and supersonic nozzle and subsonic

and supersonic diffuser. In diffusers pressure increases in the flow direction (dP > 0) while in the

= =

= −

≡

= −

10.23

10.24

10.25

10.26





nozzles, pressure decreases in the flow direction (dP < 0). An interesting aspect noted from Table

10.1 is that, in supersonic nozzles the area in the flow directions must increase while in

supersonic diffusers, the area in the flow direction decreases. This can be explained from Eq.

(10.25). For instance, for supersonic nozzles,

< 0 ; > 1

Equation (10.25) indicates that

> 0

The property changes along flow direction are also indicated in the table. Variations of all

properties except M were discussed earlier. Mach number change along flow direction can be

expressed as,

=

=

Since V increases and T decreases in the flow direction, M increases in the flow direction in a

nozzle (dM > 0). The reverse (dM < 0) is true in diffusers.

Table 10.1

dP < 0 dP > 0

M < 1 at

inlet

Subsonic Nozzle

dA< 0

dP< 0

dT < 0dV > 0

dM > 0

Subsonic Diffuser

dA> 0

dP> 0

dT > 0

dV < 0dM < 0

M < 1 at

inlet

Supersonic Nozzle

dA> 0

dP< 0

dT < 0dV > 0

dM> 0

Supersonic Diffuser

dA< 0

dP> 0

dT > 0

dV < 0

dM < 0





Can supersonic flow exist in a converging nozzle? Equation (10.25) can be applied to rule out

this possibility. Since dP < 0 for nozzles, for dA < 0 (converging nozzle) and M > 1, one notes

that

= 2 × 1 −2

negative negative negative

Clearly, a negative number on the left side cannot equal a positive number on the right side.

Hence supersonic Mach numbers cannot exist in a converging nozzle. This indicates tha t M ≤ 1

in a converging nozzle. Equation (10.25) also indicates that in order to accelerate flow from

subsonic to supersonic, the flow must transition from a converging area to a diverging area

through the minimum cross-sectional area A = A* where M = 1.

Stagnation Properties (P0, T0 and ρ0)

It can be shown that isentropic flows in converging-diverging nozzles and diffusers possess two

set of properties, namely (T, P) and (T0, P0). The first set (T, P) describes the thermodynamic

state, while the second set (T0, P0) characterize the state of motion (T0) and condition of

reversibility or irreversibility (P0). This is illustrated through Fig. 10.4 in which flow moving

with a velocity V is brought to rest isentropically to rest at a point designated as a stagnation

point.

Fig 10.4 Stagnation point

We note that stagnation enthalpy (h0) and stagnation temperature (T0) can be defined by applying

the first law.

First Law (CV):

The flow is brought to rest isentropically (reversible adiabatic process).

CV

V, h,

P, TStagnation

point V0 = 0





+ ℎ +2

2+ = ℎ0 +

02

2+ 0 +

Above equation indicates that a fluid element, moving with velocity V, has a total energy content

(h0) consisting of enthalpy (h) and kinetic energy of the moving fluid element. For ideal gases

with constant specific heats, Eq. (10.27) can be used to define stagnation temperature T 0.

ℎ0 − ℎ = 00 − =2

2

or,

Equation (10.28) indicates that stagnation temperature is the sum of (static) temperature (T) and

the dynamic temperature2

20, a manifestation of kinetic energy content of the moving fluid

element.

The stagnation pressure (P0) is calculated from the isentropic relationship, applied between the

moving state (T, V, P) and the stagnation state (T0, P0, V0 = 0).

Note that Eq. (10.29) is valid for isentropic flows (s = s0). Using the definition of speed of sound

and Mach number, Eq. (10.28) can be expressed as,

0 = 1 +2

20 = 1 +2

20

Since C2

= KRT, K = cp0 /cv0 and R = cp0 – cv0, the expression for T0 becomes

= −

= +

= +

10.27

10.28

10.29





Substituting Eq. (10.30) in Eq. (10.29), one gets an expression for P0 /P.

We note that P0 is defined using Eq. (10.29) which is valid for an isentropic (reversible adiabatic)

process. It also represents the maximum pressure (P0) that is achieved if a moving fluid in a state

(P, V, T) is brought to rest isentropically. Since in real life processes irreversibilities will cause

entropy to increase in an adiabatic process, the pressure realized at stagnation point would be less

than P0.

Thermodynamic States on a T-s DiagramWe have demonstrated that a fluid moving at a velocity, V, possesses two set of properties, (P,

T) and (P0, T0), that one must deal with in compressible flows. At low speeds as V→0 or M→0,

Eqs. (10.30) and (10.31) show that P0/P→1 and T0/T→1. In other words in low Mach number

flows the difference between the two set of properties vanishes. Conversely, the higher the Mach

number of the flow, the wider is the gap between the two set of properties. This is illustrated in

Fig. 10.5.

It is noted that the stagnation state (P0, T0) sits directly above the thermodynamic state (P, T)

because of the isentropic constraint (s = s0) used in calculating P0,

= = +

− −

= +− −

= +− 10.30

10.31

10.32





Figure 10.5 Stagnation state on a T-s diagram

Representation of states on a T-s Diagram for Isentropic Flows in Nozzles and Diffusers

Consider a converging nozzle in which air flow accelerates from velocity V1 to velocity V2 as

shown in Fig. 10.6.

Figure 10.6 Air flow in a converging nozzle

Thermodynamic and stagnation states at both sections are shown in Fig. 10.7.

Figure 10.7 Stagnation and thermodynamic states on a T-s diagram

(P0, T0)

T

s

P0

P

(P, T)T

V2/2cP0

(V2/2cP0)+T

s = s0

P1

T1

V1

M1

P2

T2

V2

M2

V1 V2

12

2

(P01, T01)

T

s

P1

T2

V22 /2cP0

P2

P01 = P02

(P02, T02)

1

T1

V12 /2cP0

T02 = T01





One notes from Fig. 10.7 that states 01 and 02 are located directly over states 1 and 2

respectively. In an adiabatic process,

or,

For ideal gas with constant specific heats,

ℎ01 − ℎ02 = 001 − 02 = 0

or,

Stagnation temperature (T0) is constant in an adiabatic process. This constraint of T 01 = T02 and

s1 = s2 (isentropic process), leads to the fact that state points 01 and 02 are located at the

intersection of constant T0 and constant entropy line. This proves that P 01 = P02 and leads to a

very important conclusion that in an isentropic process stagnation properties P0, T0 and ρ0 remain

constant. Although we have used a converging (subsonic) nozzle as an example, the same

conclusion is also valid for isentropic flows in subsonic diffusers and supersonic nozzles and

diffusers.

Mass Flow Rate through an Area Change – Expression for A/A*

Consider isentropic flow of air through a nozzle shown in Fig. 10.8.

Figure 10.8 Calculation of mass flow rate

Aρ

V

=

=

+

=

+

10.33

10.34

10.35





Mass flow rate , constant through the nozzle due to the steady state steady flow assumption,

can be expressed as,

=

The next step is to substitute for P and T in terms of P0 and T0 from Eqs. (10.30) and (10.31).

= 0 1 + − 1

22 ; = 0 1 +

− 1

22 −1

Equation (10.36) becomes,

Applying above equation to a section A = A* where Mach number M = 1, one gets,

It is to be noted that in a converging nozzle A* occurs, if it occurs at all, at the exit section since

it is the minimum area section in the nozzle. Dividing Eq. (10.38) by Eq. (10.37), one gets a very

important equation relating area A with M.

∗ = + +

− +−

∗ = + +−

=

+− +−

= = = = 10.36

10.37

10.38

10.39





Isentropic Gas Table

From Eqs. (10.30), (10.31), (10.32) and (10.39) it is evident that ratios (T/T 0), (P/P0), (ρ/ρ0) and

(A/A*), for a given gas (or K), are functions of Mach number only. Since T 0, P0, ρ0 and A* are

constant in isentropic flows, these equations are in effect relationships between T, P, ρ and A and

Mach number (M). These relationships are tabulated in Table 10.2 for easy access in analysis

involving isentropic flow in nozzles and diffusers. The isentropic table is for K = 1.4 and applies

to all diatomic gases such as air, H2, O2, N2 etc.

Graphical Representation of A/A* and Mach Number Relationship

The Eq. (10.39) describes the relationship between the area ratio A/A* and M. The symbol A

represents area at any location, x, in the nozzle and A* is the sonic area at a location where

M = 1 as shown in Fig. 10.9. This figure also shows A/A* plotted against Mach number.

Figure 10.9 Variation of A/A* with M in isentropic flows

A

M

B

A*

M = 1

C

M

A/A*

M<1

(subsonic)

M>1

(supersonic)

M = 0.3 M = 2.21.0

1.0

2.0B C





It is noted that A/A* is a double valued function of M. In other words, every value of A/A* (say

2.0) yields two solutions, one subsonic (M = 0.3) and the other supersonic (M = 2.2). These

results come from the solution of Eq. (10.39) for A/A* = 2.0. Referring to the nozzle geometry, it

means that the subsonic solution occurs at section B in the converging section while the subsonic

or supersonic solution occurs at the section C in the diverging section where A/A* is also 2.0.

Which of the two possibilities (subsonic or supersonic) will occur at section C will depend on the

level of prevailing back pressure. This point will be discussed further in connection with the

choking phenomenon. Nature as reflected in Eq. (10.25) allows only subsonic solution to exist in

the converging section, while allowing the possibility that either subsonic or supersonic solution

may exist at section C in the diverging section, depending on the exhaust flow back pressure.

Choking Phenomenon in Converging Nozzles

This phenomenon refers to the fact that under certain conditions the Mach number at the exit

section becomes equal to one and the mass flow rate handled by the nozzle reaches a maximum

value. In order to establish ground work to analyze the choking phenomenon, we consider a

conceptual experiment in which a converging nozzle operates with fixed stagnation properties P0

and T0 but the back pressure (Pb) is reduced in steps (Fig. 10.10). For every back pressure

chosen, the pressure distribution P(x) inside the nozzle is measured, and the ratio P/P 0 is plotted

as shown in Fig. 10.10.

Figure 10.10 Choking phenomenon in converging nozzles.

PE

V ≈ 0

P0

T0

x directionBack Pressure

PB

a

bc

d

P/P0

x

PB/P0 = 1 (No Flow)

PB/P0 < 1 (Flow)PB/P0 = 0.528 (ME = 1)

PB/P0 < 0.528 (ME = 1)





For a given P0 and T0 and exit area AE as the back pressure is lowered from level a to level b,

flow is established in the nozzle, pressure drops through the nozzle and velocity increases,

reaching a maximum at the nozzle exit. The Mach number at the exit ME is less than one and the

exit pressure is matched to the back pressure exactly as shown in Fig. 10.10. As the back

pressure is lowered to a critical value, M E becomes equal to unity (sonic condition) as illustrated

by case c. This is the maximum Mach number that can occur in a converging nozzle. Note that

exit pressure PE is exactly matched to the back pressure PB. Also the mass flow rate is greater

than for the case b. In fact, mass flow rate has reached its maximum value ( = ) for

the case c. Lowering the back pressure further (case d) produces two surprising results. First, the

pressure distribution inside the nozzle remains the same as it was in the case c. Also for the case

d, the back pressure, PB, is not exactly matched to the exit pressure, PE. The exit pressure is

always greater than PB once the sonic condition (ME = 1) takes hold at the nozzle exit section. In

fact, fluid comes out of the exit section oblivious of the fact that the back pressure has been

lowered. The exit Mach number (ME) for the case d is still equal to one, consistent with earlier

results that showed that supersonic flow cannot be established in a converging nozzle if

incoming flow is subsonic. This is further illustrated graphically in Fig. 10.11.

Figure 10.11

From Fig. 10.11 it is noted that for cases a, b and c, PE = PB but for case d, PE > PB. Also, the

mass flow rate reaches a maximum ( ) once exit Mach number becomes one and remains at

that value. This is also known as the the choked flow condition. Reducing the back pressure

sends a signal that travels upstream at speed of sound C. But once exit condition becomes sonic

(ME = 1 or cE = VE) the pressure disturbance created by further lowering of back pressure is not

able to travel past the exit section because of the condition CE = VE (or ME = 1). Consequently,

d (ME = 1)

a

B (ME < 1)

c (ME = 1)a

bcd

PB/P0 PB/P0

PE/P0

1.0

1.0

1.00.528 0.528

0.528

45°





upstream section of nozzle remains unaware of the reduced back pressure and continues to pass

the same (maximum) mass flow rate through the nozzle.

Critical Pressure Ratio

The critical pressure ratio PB / P0 = 0.528 is obtained by applying the ME = 1 to Eq. (10.31).0

=1

1 + − 1

22 −1 = 0.528

Explanation of the ‘Choking’ Phenomenon

As the back pressure (PB) is lowered while keeping upstream pressure P0 constant, one observes

expected increase of mass flow rate through the nozzle since pressure difference is the potential

for mass flux. However, this trend ends once PB reaches the critical value 0.528P0. The question

is why? The answer is illustrated in Fig. 10.12.

Figure 10.12 Explanation of the ‘Choking’ phenomenon

As the back pressure is reduced slowly a pressure disturbance wave propagates upstream with

the speed of sound CE past the exit section, where the prevailing flow velocity is VE. For

operating conditions a and b, ME is less than one (VE < CE). This allows the disturbance to

propagate upstream since pressure disturbance velocity CE is always relative to the fluid flow

velocity. This transmission of pressure disturbance wave to upstream location allows the

upstream parts of the nozzle to become aware of changed downstream condition of reduced back

pressure and allows adjustments to changing back pressure conditions by increasing the mass

flow rate and decreasing pressure levels in the nozzle. This process works until the Mach number

ME becomes unity (VE = CE). Once this condition is reached at the exit section, any pressure

disturbance arising from further reduction of back pressure (case d) is not able to pass through

P0 PBVC





the bottleneck at the exit section due to the VE < CE condition. This means that upstream parts of

the nozzle are unable to adjust to the changed operating conditions downstream. As a result,

changes in mass flow rate and nozzle pressure distribution ceases and the nozzle operates at the

maximum mass flow rate (choked) condition.

Criterion for Choking

If a converging nozzle with P0 and T0 and a given geometry is operating in an environment with

a back pressure of PB, how can one determine if the flow is choked (M E = 1) or unchoked (ME <

1)? There are two methods that can be used. The first method assumes choking (M E = 1) as the

starting point. The second method assumes unchoked flow condition (M E < 1). Both methods are

consistent in that they give same results.

Method 1

a. Assume choked flow (ME = 1).

b. Calculate PE from Eq. (10.31) or the isentropic table. For K = 1.4, PE /P0 = 0.528.

c. Calculate PE = 0.528P0.

d. Check if PB ≤ PE. If yes, then the flow is choked. If PB > PE, the flow is not choked.

Method 2

a. Assume unchoked flow (ME < 1).

b. Then PB = PE (note: pressures are matched).

c. Calculate PE /P0.

d. Check if PE /P0 < 0.528 (0.528 is the critical value for choked flow condition, M E = 1).

e. If yes, then the flow is unchoked and ME < 1.

f. If PE /P0 > 0.528, the flow is choked.

Choking Phenomenon in Converging-Diverging Area Changes

An experiment similar to that for converging nozzle involving lowering of back pressure PE for

given P0 and T0 is used here. Results are shown in Fig. 10.13. As the back pressure is lowered

(case b), it leads to a subsonic flow regime throughout the nozzle. The pressure drop in the

nozzle (upstream part) reaches a minimum (also a velocity maximum) at the throat. However, the





Mach number at the throat, though a minimum, is not equal to one. As a result, the diverging

section acts as a diffuser and pressure rises (Mach number drops) and matches the back pressure

(PB = PE). When PB is lowered further (case c in Fig. 10.13), the Mach number at the throat

becomes one (Mt = 1) and the throat becomes the sonic area (A t = A*). At this point, the

isentropic flow theory predicts two solutions for the given area ratio AE /At. In the subsonic case

c, the pressure rises and matches the back pressure at the exit section. Alternatively, if a much

lower back pressure is maintained (case d), the flow becomes supersonic in the diverging section

and the entire geometry (converging as well as diverging sections) acts as a nozzle. The exit

pressure again matches the back pressure. These two cases (c and d) belong to the choked flow

domain since establishment of Mt = 1 at the throat section precludes any more increase in mass

flow rate. The nozzle is handling maximum flow rate for all back pressures P B ≤ Pc.

Under-Expanded Flow Regime

If the back pressure is lowered below pressure level Pd (case h) the pressure and Mach number

distributions inside the nozzle do not change because the nozzle is operating under choked flow

condition. Consequently, a discontinuity of pressure occurs at the exit section (P E > PB). The

flow inside the nozzle still remains isentropic but the flow outside goes into a convulsion due to

the pressure discontinuity. It is reflected by the shock-diamond phenomenon in which the jet

exiting the nozzle goes through an expansion fan followed by oblique shocks compressing the

flow. This pattern of expansion (jet pressure drop) and compression (jet pressure rise) continues

until the jet pressure eventually matches the back pressure (PB). This phenomenon is visible to

the naked eye as in case of space shuttle lift-off.

Normal Shock inside the Nozzle

In this flow regime, occurring for Pc < PB ≤ Pe, a normal shock occurs in the diverging part of the

nozzle. The nozzle is still choked (Mt = 1) and the location of the shock depends on the pressure

level PB. As PB decreases from Pc level towards Pe level, the normal shock moves from the

vicinity of the throat towards the exit section. At P B = Pe, the normal shock occurs at the exit

section. For all cases between c and e, the exit Mach number is subsonic. Consider the case g

(Pg < Pc). The normal shock inside the nozzle causes pressure spike at the normal shock location.

Since Mach number behind a normal shock is always less than one, the flow behind the shock is





subsonic. As a result, the flow in the remainder diverging section behind the shock is like

isentropic flow in a diffuser, raising pressure gradually upto the exit section where exit pressure

PE matches prevailing back pressure Pg.

Over-Expanded Flow Regime (Pe < PB < Pd)

At PB = Pe, the normal shock stands at the exit section. In the back pressure range between P e and

Pg, the flow inside the nozzle remains isentropic and the flow exits at the same supersonic flow

Mach number as in the case d. But there exists a discontinuity of pressure at exit with exit

pressure PE being lower than PB, the prevailing back pressure. This flow is referred to as the

over-expanded regime (PE < PB). The jet, upon leaving the exit section, goes through a series of

oblique shock waves followed by expansion fans. This process continues until the jet pressure

equilibrates with the prevailing back pressure.

Figure 10.13 Flow regimes in converging-diverging nozzles

The regimes described earlier are summarized here:

1. Subsonic flow throughout (M <1). (cases between a and c) – Isentropic process

2. Converging section subsonic, throat sonic (M = 1) and diverging section subsonic (case c)

– Isentropic process

PB

ef

PE

V ≈ 0

P0

T0

Throat

ab

c

d

P/P0

x

g

h

xg




ld i i i i

3. Converging section subsonic, throat sonic and diverging section supersonic (case d) –

Isentropic process

4. Converging section subsonic, throat sonic and normal shock in the diverging section (cases

between c and e) – Non-isentropic process (up to shock the flow is isentropic, everything

same as case d, but normal shock in the diverging section of the nozzle)

5. Same as case d but normal shock at exit – Isentropic process in the entire nozzle (case d) but

shock at exit section

6. Same as the case d but oblique shocks outside the nozzle (cases between e and d) –

Isentropic process in the nozzle but oblique shocks outside the nozzle

7. Same as case d but expansion waves and oblique shock waves outside the nozzle (cases

below d).

Determination of Flow Regimes in Converging-Diverging Nozzles

These regimes (pressure ranges) can be determined by knowing the exit area (AE) and the throat

area (At). The ratio AE /At from isentropic gas tables determines the subsonic case c and

supersonic case d. The pressure ratio Pc /P0 and Pd /P0 can be determined from gas tables or Eq.

(10.31). The pressure ratio Pe /P0 (for case e) represents the case for which normal shock exists at

the nozzle exit section. Using Pd /P0 and normal shock relations or normal shock table one can

find Pe. If the pressure is between P

cand P

e, say P

gthen a normal shock occurs in the nozzle at

location xg as shown in Fig. 10.13. The flow ahead of the shock is supersonic and behind is

subsonic. After that the nozzle section behaves as a diffuser and the flow is isentropic with the

Mach number decreasing further as the exit section is approached. As the back pressure is raised

to level g, the shock becomes weaker and shifts towards the throat section. At pressure level c,

the shock disappears at the throat where Mach number is one and the shock strength (pressure

spike) goes to zero. This is the isentropic case c. Lowering the back pressure below Pd gives the

same pressure and Mach number profile as in case d but outside one witnesses the shock-

diamond phenomenon.

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