PHYSICAL REVIEW E 105, 045107 (2022)
Critical pressure in liquids due to dynamic choking
Jiˇ rí Vacula
*
and Pavel Novotný
Institute of Automotive Engineering, Faculty of Mechanical Engineering, Brno University of Technology,
Technická 2896/2, 616 69 Brno, Czech Republic
(Received 20 December 2021; accepted 31 March 2022; published 21 April 2022)
The existence of a critical pressure ratio due to gas-dynamic choking is well known for an ideal gas. It is
reasonable to assume that liquids whose compressibility is defined by the bulk modulus also have a critical
pressure ratio. The problem discussed here is a fundamental one because it deals with the basic principles of
the compressible flow of liquids. It has been shown that even though an ideal gas with a constant heat capacity
ratio value has a critical pressure ratio, liquid with a constant bulk modulus value experiences a critical pressure
difference. As the outlet pressure gradually decreases, the liquid reaches the local speed of sound, and further
reduction of this pressure does not lead to an increase in mass flow. This phenomenon occurs in liquids without
considering the change from a liquid to a gaseous phase. Behavior is confirmed analytically for different bulk
modulus models, and for a constant bulk modulus value, the phenomenon is verified by numerical simulation
using computational fluid dynamics. The conclusions published in this work point to striking analogies between
the behavior of liquids and ideal gas. The equations governing the motion of liquids derived in this work, thus
complete the fundamental description of the critical flow of fluids.
DOI: 10.1103/PhysRevE.105.045107
I. INTRODUCTION
It is well known that the velocity of gas leaving the
reservoir of pressure p
0
and velocity v
0
≈ 0 through a duct
increases due to a gradual reduction of the outlet pressure until
the local speed of sound is reached. Mass flow increases until
the ratio p/ p
0
acquires the so-called critical pressure ratio,
where p is local pressure in the duct and further reduction of
the outlet pressure no longer leads to an increase in mass flow.
This phenomenon is called gas-dynamic choking and can
occur even in the case of flow with no energy losses. The crit-
ical pressure ratio represents a limitation of the mass flow, not
in terms of energy losses in the gas but as a physical limitation
resulting from the nature of the change in density and velocity
of the expanding gas. Therefore, critical flow is a limiting
factor that causes the flow to choke, even in the case of zero
gas viscosity. To prevent flow choking when the pressure ratio
is smaller than critical, a convergent-divergent pipe must be
used. Detailed analyses of one-dimensional flow can be found
in almost any literature dealing with compressible flows [1–4].
The fundamental steps of the derivation of critical pressure
are shown here to develop the problem for liquids. To deter-
mine the critical pressure ratio of an ideal gas, it is necessary
to take into account the equation of the adiabatic process of
the ideal gas
pρ
−κ
= p
0
ρ
−κ
0
= constant , (1)
where ρ is density, κ is heat capacity ratio, and subscript
0 indicates the location inside the reservoir. Conservation of
*
jiri.vacula@vutbr.cz
momentum is regarded in the form
v d v =−
dp
ρ
. (2)
Note that Eq. (2) assumes a one-dimensional stationary
flow, neglecting the influence of body and friction forces.
Substituting density ρ from Eq. (1) into the conservation of
momentum Eq. (2) and its subsequent integration, the velocity
of the gas leaving a convergent nozzle, taking the adiabatic
process into account, can be determined as
v =
2κ
κ − 1
p
0
ρ
0
1 −
p
p
0
(κ −1)/κ
(3)
and assumes negligible velocity of the gas in the reservoir,
v
0
= 0. The integral form of the continuity equation for one-
dimensional stationary flow has the form
˙ m = ρ vA = constant , (4)
where ρ is determined by Eq. (1), v is determined by Eq. (3),
and A denotes the size of the cross-sectional area through
which the fluid flows. At this point, it would be appropriate to
investigate the product, ρ v; however, function ψ is sometimes
introduced in the literature [5]. The critical pressure ratio can
be derived both from the product ρ v and the form of the
continuity equation considering function ψ . For the purposes
of this work, function ψ is introduced and shown here. The
continuity equation with the introduction of function ψ takes
the form
˙ m = Aψ
2 p
0
ρ
0
, (5)
2470-0045/2022/105(4)/045107(10) 045107-1 ©2022 American Physical Society