Flow through Converging- Diverging passages

Convergent nozzles are used for subsonic and sonic flows. They can also be used as flow measuring and flow
regulating devices. Convergent-divergent nozzles are used for supersonic flows. It is comparatively difficult to
obtain convergent-divergent passages in compressors and turbine blade rows.

Convergent Nozzles
Figure 1 shows the flow from an infinite reservoir to an exhaust chamber through a convergent nozzle. The
stagnation conditions (po ,To etc.) in the reservoir are kept constant while the back pressure, i.e., the exhaust
chamber pressure can be varied.

Fig 1 Isentropic flow through a convergent nozzle

Isentropic flow through a convergent nozzle

The pressure distributions along the nozzle for various values of the pressure ratio (pe/po,) across the nozzle are
shown in curves a, b, c, d and e. Curves a and b correspond to values of the pressure ratio more than the
critical. Curve c corresponds to the critical pressure ratio: pb/po = 0.528 (for 𝛾 = 1.4). For these curves the
pressure pe, at the exit of the nozzle is same as the pressure in the exhaust chamber. The nozzle exit pressure
does not decrease when the exhaust chamber pressure is further reduced below the critical value. This
condition is depicted by the curves d and e; here the nozzle exit pressure is still pe, but pb< pe. The change of
pressure from pe, to pb, takes place outside the nozzle exit through an expansion wave. Figure 2 shows the
variation of the nozzle pressure ratio (pe/po) and the mass flow parameter against the pressure ratio pb/po, for
𝛾= 1.4. Points corresponding to the a, b, c, d and e of Fig. 1 have also been shown on these curves. The
maximum mass flow occurs at point c after which both the mass flow parameter and the nozzle pressure ratio
cease to vary. The numerical valués in the figure are for 𝛾 = 1.4.

Fig 2. Variation of pressure ratio and the mass flow parameter for a convergent nozzle (𝜸 = 1.4)

Convergent-divergent Nozzles
Figure 3 shows the flow from an infinite reservoir through a convergent-divergent nozzle to an exhaust
chamber. Though the main subject here is isentropic flow, it will be observed that flow through a convergent-
divergent nozzle is not isentropic under all condition. The static pressure distributions for various values of the
pressure ratio pb/po are shown in curves a, b, c, .., h, i and j.
In curves a and b the pressure ratio pe/po , across the nozzle is such that the flow is accelerating only up to the
throat; the diverging part acts as a diffuser through which the pressure rises to the back pressure p. In curve c
critical conditions (pt =p*; Mt = 1) are reached at the throat but the back pressure is such that the diverging part
still acts as a diffuser.
Curve h corresponds to the design value of the back pressure. All other curves are for off design conditions.
Here the flow is supersonic in the entire divergent part of the nozzle. The flow in the curves a, b, c, .., h, i and j
is isentropic but non-isentropic in other curves. After the conditions of curve c, when the back pressure is
further lowered expansion takes place to supersonic velocity beyond the throat to a point where a discontinuity
in the flow occurs: this is on account of the "off design" value of the back pressure and the supersonic flow as
in Curves d and e. In these cases at some point the minimum pressure reached in supersonic expansion has to
rise to the back pressure. But equation for Mach number variation for an isentropic flow is
𝑑𝐴 𝑑𝜌
= (1 − 𝑀 )
𝐴 𝜌𝑐
As per the above equation it is known that for the deceleration of supersonic flow the passage should be
convergent. Therefore, the existing shape the nozzle downstream of this point is incompatible with the
required process, as a result of this the flow readjusts itself to the shape of the passage by suddenly becoming
subsonic. This satisfies the above equation. The pressure, temperature and density suddenly rise to values
which are compatible with the subsonic flow immediately downstream of the supersonic flow region. Such a
sudden change of supersonic flow to subsonic occurs through a plane of discontinuity between the supersonic
and subsonic flow regions. The flow through the shock wave is no longer isentropic.

Fig 3. Flow through a convergent-divergent nozzle

When the back pressure is lowered further the shock wave moves downstream till it reaches the exit as in
curves f and g; here the back pressure is still higher than the design value. The nozzle exit pressure p e rises to
the back pressure pb, through a shock wave outside the nozzle exit section as shown in curves f and g. Curves i
and j correspond to back pressure lower than the nozzle exit pressure. The conditions
in the nozzle remain unchanged and the adjustment of the nozzle exit pressure to the exhaust chamber pressure
occurs through an expansion wave outside the nozzle.
Figure 4 shows the variation of the throat pressure ratio with the pressure ratio pb,/po. It may be seen that after
choking conditions (Mt = 1, pt =p*) are reached (point c) at the throat, further variation of the back pressure
does not affect the throat pressure.
 Fig.4 Variation of throat pressure ratio in a convergent-divergent nozzle (𝜸 = 1.4)

Figure 5 shows the variation of the nozzle pressure ratio (p/Po) with py/Po, In curves a, b, c, d, e and h (Figure
4) pe, is identical with pb, This is shown by the points a, b, c, d, e and on the line inclined at 45o. At point f and
beyond ie., at g, i and j there is no variation in pe. with further reduction in pb.

Fig.5 Variation of exit pressure ratio in a convergent-divergent nozzle (𝛾 = 1.4)

Variation of the mass flow parameter 𝑚̇ ∗ with the pressure ratio pb/po, is shown in Figure 6. The maximum
mass flow conditions are reached (at point c) when the throat pressure ratio achieves critical value (p t =p*);
there is no further increase in the mass flow with decrease in the back pressure after this point. This condition
is called "choking".
 Fig 6. Variation of the mass flow parameter in a convergent-divergent nozzle (𝜸 = 1.4)

FLOW THROUGH DIFFUSERS

Diffusion of flow, i.e., deceleration of the flow with rise in pressure is a common feature of the flow machines
and systems. Figure 7 shows both reversible and irreversible diffusion of supersonic flow. In the isentropic
diffusion there is a continuous rise in static pressure.

Fig 7 Reversible and irreversible flows through a diffuser

The supersonic part of the diffuser is convergent, and the subsonic part is divergent. In many applications’
diffusion occurs through a shock wave (irreversible diffusion). The pressure rise across the shock wave is
sudden and is governed by the upstream supersonic Mach number. The flow before and after the shock may
still be isentropic. Since the Mach number is minimum at the throat it is profitable to have the shock at this
section.

MACH NUMBER VARIATION

The isentropic energy Equation (2.60) gives
𝑑𝑝 = −𝜌𝑐 𝑑𝑐
From continuity equation mass flow rate is given by
𝑚̇ = 𝜌 𝐴𝑐 = constant
Taking logs and differentiating
In ρ + In 𝐴 + In 𝑐 = constant
𝑑𝜌 𝑑𝐴 𝑑𝑐
+ + =0
𝜌 𝐴 𝑐
𝑑𝜌 𝑑𝐴
𝑑𝑐 = −𝑐 +
𝜌 𝐴
Equations (4.1) and (4.2) yield
𝑑𝜌 𝑑𝐴
𝑑𝑝 = 𝜌 𝑐 +
𝜌 𝐴
𝑑𝐴 𝑑𝑝 𝑑𝜌
= 1− 𝑐
𝐴 𝜌𝑐 𝑑𝑝
For an isentropic process from Equations
𝑑𝑝 𝜕𝑝
= =𝑎
𝑑𝜌 𝜌𝑐

Equations (4.3) and (4.4) give

𝑑𝐴 𝑑𝑝 𝑐 𝑑𝑝
= 1− = (1 − 𝑀 )
𝐴 𝜌𝑐 𝑎 𝜌𝑐
Equation (4.5) can now be considered for both accelerating and decelerating passages
for various values of Mach number.

4.2.1 Expansion in Nozzles

Gases and vapours are expanded in nozzles by providing a pressure ratio across
them. It is shown here with the aid of Equation (4.5), that the shape of the passage
depends on the local Mach number. Since the purpose of a nozzle is to accelerate
the flow by providing a pressure drop, dp is Equation (4.5) is always negative.
Following three possible conditions are considered for Equation
(4.5):
(a) For 𝑀 < 1, 𝑑𝐴 = −𝑣𝑒.
This shows that for nozzles the area decreases in the range 𝑀 = 0 𝑡𝑜 𝑀 = 1 giving a
convergent passage.
(b) For 𝑀 = 1 (sonic velocity)
𝑑𝐴 = 0
Which implies that there is no change of passage area at the point where the Mach
numbers is unity. This section is referred to as the throat of the passage.
(c) For 𝑀 > 1, 𝑑𝐴 = +𝑣𝑒
This shows that M>1 the area of the nozzle increases continuously giving a
divergent passage.
-:i
 4.2.2 Compression in Diffusers
Diffusers are employed to obtain pressure rise in flowing gases for a given value of the
initial kinetic energy. The static pressure rise is at the cost of the deceleration of flow in
the diffuser passage. Here dp in Equation (4.5) is always positive. Equation (4.5) is
now considered for such a case for three possible values of the Mach number.
(a) For𝑀 < 1, 𝑑𝐴 = +𝑣𝑒.
This shows that for subsonic diffusers 𝑀 = 1 𝑡𝑜 𝑀 = 0 the area increases with decreasing Mach
number giving a divergent passage.
(b) For 𝑀 = 1, 𝑑𝐴 = 0
There is no change in the passage area.
(c) For 𝑀 > 1, 𝑑𝐴 = −𝑣𝑒
The area of the diffuser passage decreases with Mach number giving a convergent
passage.
The passages for expansion and compression of gases in nozzles and diffusers are
shown in Figures 4.3 and 4.4 respectively It may be noted that while a continuous
acceleration of

Figure 4.3 Isentropic flow of a gas in a nozzle (decreasing pressure)

Real flow from subsonic to supersonic Mach numbers is possible the converse of this ,i.e.,
continuous deceleration of a real flow from supersonic to subsonic Mach numbers is
impossible. It will be seen in Sections 4.7.2and6.15
thata1·ealsupersonicflowdeceleratestoa subsonic flow only through a shock wave. The
results obtained in this section on the basis of Equation (4.5) are summarized inTable4.1

Figure 4.4. Isentropic flow of a gas in a diffuser (increasing pressure)

4.3 AREA RATIO AS FUNCTION OF MACH NUMBER
Like temperature, pressure and density ratios, area ratio at a given section of a passage is
also a useful quantity. Here the ratio of area (A) at the given Mach number (M) and the
reference area (A*) at sonic conditions (M=Mt= 1) is expressed as a function of the local
Mach number (M).
From continuity Equation
𝜌𝐴𝑐 = 𝜌∗ 𝐴∗ 𝑐 ∗
∗ ∗
∗ = (4.13)

From Equation (2.52)

𝑐 ∗ 𝑎∗ 1 1+ 𝑀
= = =
𝑐 𝑐 𝑀 𝑀

∗
= + 𝑀 (4.14)

Substituting Equation (4.12) and (4.14) in (4.13) gives

( )/ ( )
∗ = + 𝑀 (4.15)

The variation of area ratio for subsonic and supersonic isentropic acceleration and deceleration
is shown in figure 4.7

FIGURE 4.7 Variation of area ratio with Mach number

In many compressible flow calculations the function ∗ occurs frequently. This can also be
obtained as a function of Mach number from Equations (2.27) and (4.15)
( )/ ( ) /( )
𝐴 𝑝 1 2 𝛾−1 𝛾−1
∗
= + 𝑀 × 1+ 𝑀
𝐴 𝑝 𝑀 𝛾+1 𝛾+1 2
This on simplification and rearrangement yields
( )/ ( )

∗ = (4.16)

4.4 IMPULSE FUNCTION

The quantities 𝑝𝐴 and 𝑝𝐴𝑐 occur frequently in some compressible flow problems.
Since the unit of both these quantities are the units of force they are conveniently
expressed together as a important gas dynamic parameter.
This is referred to as the impulse function (F) or the wall force function..
One-dimensional flow through a control surface (a symmetrical straight duct) is
shown in Figure 4.8.The thrust or wall force experienced by the duct in the
direction shown is a resul change in pressure and Mach number between the cross-
sections1and 2. By momentum equation the thrust (𝜏) is given. by
𝜏 = 𝑝2 𝐴2 + 𝜌2 𝐴2 𝑐22 − 𝑝1 𝐴1 + 𝜌1 𝐴1 + 𝜌1 𝐴1 𝑐21 (4.17)

Figure.4.8. Directions of flow and thrust in a duct

For a perfect gas

𝜌𝑐 = 𝑐 = 𝛾𝑝 = 𝛾𝑝𝑀 (4.18)

The impulse function is 𝐹 = 𝑝𝐴 + 𝜌𝐴𝑐 (4.19)

Equations (4.18) and (4.19) give
𝐹 = 𝑝𝐴 + 𝛾𝑝𝐴𝑀
𝐹 = 𝑝𝐴 + (1 + 𝛾𝑀 ) (4.20)
Substituting Equation (4.20) in (4.17) yields
𝜏 =𝐹 −𝐹
 𝜏 = 𝑝 𝐴 (1 + 𝛾𝑀 ) − 𝑝 𝐴 (1 + 𝛾𝑀 ) (4.21)

This Equation demonstrates that the use of impulse function is very convenient in
obtaining the thrust exerted by the flowing fluids; The thrust exerted by the fluid due
to its flow between two sections of a duct can be obtained by the change of the
impulse function between these sections. Equations (4.17) to(4.21) are independent of
the type of process. However, for obtaining a relation between the non-dimensional
impulse function and the Mach number the flow is assumed isentropic.
𝐴𝑡 𝑀 = 1, 𝐹 = 𝐹 ∗ and Equation (4.20) reduces to
𝐹 ∗ = 𝑝∗ 𝐴∗ (1 + 𝛾) (4.22)

Equations (4.20) and (4.22) yield the expression for the non-dimensional impulse function

∗ = ∗ ∗ (4.23)
Substituting from Equations (4.11) and (4.15) for ∗ and ∗ in Equation (4.23)

−𝛾/(𝛾−1) (𝛾+1)/2(𝛾−1)
𝐹 2 𝛾−1 1 2 𝛾−1 1 + 𝛾𝑀
∗ = + 𝑀 × + 𝑀 ×
𝐹 𝛾+1 𝛾+1 𝑀 𝛾+1 𝛾+1 1+𝛾
/
𝐹 1 2 𝛾−1 1 + 𝛾𝑀
∗ = + 𝑀 ×
𝐹 𝑀 𝛾+1 𝛾+1 1+𝛾
On simplifying and rearranging the above relation reduces to
𝐹
= (4.24)
𝐹∗ 𝛾−1
(1+𝛾)

Another non-dimensional expression for the impulse function may be obtained as follows;
𝐹
= (4.25)
𝑝0 𝐴∗ 𝑝0 𝐴∗

𝑝𝐴
Substituting for from Equation (4.16)
𝑝0 𝐴∗

𝐹 (𝛾+1)/2(𝛾−1)
∗ = (4.26)
𝑝0 𝐴 𝛾+1 𝛾−1