Dimensionless Groups by Entropic Similarity: II—Wave Phenomena and Information-Theoretic Flow Regimes

Abstract

The aim of this study is to explore the insights of the information-theoretic definition of similarity for a multitude of flow systems with wave propagation. This provides dimensionless groups of the form Πinfo=U/c, where U is a characteristic flow velocity and c is a signal velocity or wave celerity, to distinguish different information-theoretic flow regimes. Traditionally, dimensionless groups in science and engineering are defined by geometric similarity, based on ratios of length scales; kinematic similarity, based on ratios of velocities or accelerations; and dynamic similarity, based on ratios of forces. In Part I, an additional category of entropic similarity was proposed based on ratios of (i) entropy production terms; (ii) entropy flow rates or fluxes; or (iii) information flow rates or fluxes. In this Part II, the information-theoretic definition is applied to a number of flow systems with wave phenomena, including acoustic waves, blast waves, pressure waves, surface or internal gravity waves, capillary waves, inertial waves and electromagnetic waves. These are used to define the appropriate Mach, Euler, Froude, Rossby or other dimensionless number(s)—including new groups for internal gravity, inertial and electromagnetic waves—to classify their flow regimes. For flows with wave dispersion, the coexistence of different celerities for individual waves and wave groups—each with a distinct information-theoretic group—is shown to imply the existence of more than two information-theoretic flow regimes, including for some acoustic wave systems (subsonic/mesosonic/supersonic flow) and most systems with gravity, capillary or inertial waves (subcritical/mesocritical/supercritical flow). For electromagnetic wave systems, the additional vacuum celerity implies the existence of four regimes (subluminal/mesoluminal/transluminal/superluminal flow). In addition, entropic analyses are shown to provide a more complete understanding of frictional behavior and sharp transitions in compressible and open channel flows, as well as the transport of entropy by electromagnetic radiation. The analyses significantly extend the applications of entropic similarity for the analysis of flow systems with wave propagation.

Full text

Citation: Niven, R.K. Dimensionless
Groups by Entropic Similarity:
II—Wave Phenomena and
Information-Theoretic Flow Regimes.
Entropy 2023,25, 1538. https://
doi.org/10.3390/e25111538
Academic Editor: Jean-Noël Jaubert
Received: 23 October 2023
Revised: 8 November 2023
Accepted: 9 November 2023
Published: 11 November 2023
Copyright: © 2023 by the author.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
entropy
Article
Dimensionless Groups by Entropic Similarity: II—Wave
Phenomena and Information-Theoretic Flow Regimes
Robert K. Niven
School of Engineering and Technology, The University of New South Wales, Canberra, ACT 2600, Australia;
r.niven@adfa.edu.au
Abstract:
The aim of this study is to explore the insights of the information-theoretic definition of
similarity for a multitude of flow systems with wave propagation. This provides dimensionless
groups of the form
Πinfo =U/c
, where
U
is a characteristic flow velocity and
c
is a signal velocity or
wave celerity, to distinguish different information-theoretic flow regimes. Traditionally, dimensionless
groups in science and engineering are defined by geometric similarity, based on ratios of length scales;
kinematic similarity, based on ratios of velocities or accelerations; and dynamic similarity, based on
ratios of forces. In Part I, an additional category of entropic similarity was proposed based on ratios of
(i) entropy production terms; (ii) entropy flow rates or fluxes; or (iii) information flow rates or fluxes.
In this Part II, the information-theoretic definition is applied to a number of flow systems with wave
phenomena, including acoustic waves, blast waves, pressure waves, surface or internal gravity waves,
capillary waves, inertial waves and electromagnetic waves. These are used to define the appropriate
Mach, Euler, Froude, Rossby or other dimensionless number(s)—including new groups for internal
gravity, inertial and electromagnetic waves—to classify their flow regimes. For flows with wave
dispersion, the coexistence of different celerities for individual waves and wave groups—each with a
distinct information-theoretic group—is shown to imply the existence of more than two information-
theoretic flow regimes, including for some acoustic wave systems (subsonic/mesosonic/supersonic
flow) and most systems with gravity, capillary or inertial waves (subcritical/mesocritical/supercritical
flow). For electromagnetic wave systems, the additional vacuum celerity implies the existence of four
regimes (subluminal/mesoluminal/transluminal/superluminal flow). In addition, entropic analyses
are shown to provide a more complete understanding of frictional behavior and sharp transitions
in compressible and open channel flows, as well as the transport of entropy by electromagnetic
radiation. The analyses significantly extend the applications of entropic similarity for the analysis of
flow systems with wave propagation.
Keywords:
dimensional analysis; entropic similarity; compressible flow; open channel flow; gravity
flow; electromagnetic radiation
1. Introduction
Since the seminal work of Buckingham [
1
], built on the insights of many predeces-
sors [
2
–
9
], dimensional analysis and similarity arguments based on dimensionless groups
have provided a powerful tool—and in many cases, the most important tool—for the anal-
ysis of physical, chemical, biological, geological, environmental, astronomical, mechanical
and thermodynamic systems, especially those involving fluid flow. The dimensionless
groups obtained are usually classified into those arising from geometric similarity, based on
ratios of length scales (or areas or volumes); kinematic similarity, based on ratios of velocities
or accelerations; and dynamic similarity, based on ratios of forces [
10
–
17
]. Thus, for example,
the Froude number [18] is often interpreted by dynamic similarity as [10,12,19]:
Fr =inertial force
gravity force =FI
Fg∼ρ`3(U2/`)
ρg`3=U2
g`(1)
Entropy 2023,25, 1538. https://doi.org/10.3390/e25111538 https://www.mdpi.com/journal/entropy

Entropy 2023,25, 1538 2 of 36
(or as the square root), where
∼
indicates “of the order of” (discarding numerical constants),
ρ
is the fluid density [SI units: kg m
−3
],
`
is an applicable length scale [m],
U
is a velocity
scale [m s
−1
] and
g
is the acceleration due to gravity [m s
−2
]. In common with many
dimensionless groups, Equation
(1)
provides an identifier of the flow regime, in this case
the dominance of gravity forces at low Froude numbers, associated with subcritical flow;
and the dominance of inertial forces at high Froude numbers, producing supercritical flow.
Part I of this study [
20
] proposes a new interpretation for a large class of dimensionless
groups based on the principle of entropic similarity, involving ratios of (i) entropy produc-
tion terms; (ii) entropy flow rates or fluxes; or (iii) information flow rates or fluxes. Since
all processes involving work against friction, dissipation, diffusion, dispersion, mixing,
separation, chemical reaction, gain of information or other irreversible changes are driven
by (or must overcome) the second law of thermodynamics, it is appropriate to analyze
these processes directly in terms of competing entropy-producing and transporting phe-
nomena and the dominant entropic regime, rather than indirectly in terms of forces. The
entropic perspective is shown to provide a new entropic interpretation of many known
dimensionless groups, as well as a number of new groups [
20
], significantly expanding the
scope of dimensional arguments for the resolution of new and existing problems.
The aim of this Part II study is to explore the insights of the information-theoretic
definition of similarity, by application to a multitude of fluid flow systems subject to
various wave phenomena. This has application to a wide range of fluid flow systems,
including compressible flows (acoustic waves), enclosed flows (pressure waves), open
channel flows, lakes and oceans (surface gravity and capillary waves), oceanographic and
meteorological flows (internal gravity waves and internal waves), and subatomic particle
flows (electromagnetic waves). This work is set out as follows. In Section 3, we provide a
recap of entropy concepts, dimensionless groups and the principle of entropic similarity,
including the information-theoretic definition. In Section 4, we then examine a number of
flow systems with wave phenomena. For each wave type, the information-theoretic defi-
nition of similarity and its role as a discriminator between different information-theoretic
flow regimes is examined in detail, leading to a revised interpretation of several known
dimensionless groups and a number of new groups. For systems with wave dispersion, the
analyses also suggest the possibility of more than two information-theoretic flow regimes.
For several flow types, the information-theoretic flow regimes, combined with a direct
entropic analysis (applying the second law of thermodynamics), are also shown to provide
a more complete understanding of the observed frictional flow regimes and the occurrence
of sharp transitions. The findings are summarized in the conclusions in Section 5.
2. Theoretical Foundations
The (dimensionless) discrete entropy and relative entropy functions are [21–26]:
HSh =−
n
∑
i=1
piln piand H=−
n
∑
i=1
piln pi
qi(2)
where
pi
is the probability of the
i
th outcome, from
n
such outcomes, and
qi
is the prior
probability of the
i
th outcome. The second form reduces to the first, plus a constant, in the
case of equal prior probabilities. Both functions and their continuous form can be derived
from the axioms of a measure of uncertainty [
21
,
26
,
27
] or from the combinatorial definition
of entropy [28,29].
The dimensionless entropy concept
(2)
provides the foundation for the thermodynamic
entropy
S=kBH∗
[J K
−1
], where
kB
is the Boltzmann constant [J K
−1
] and
H∗
is the
maximum entropy [
23
–
25
,
30
,
31
]. By analysis of the thermodynamic entropy balance in an
open system, it is possible to derive expressions for the global and local entropy production,

Entropy 2023,25, 1538 3 of 36
respectively, within an integral control volume or an infinitesimal fluid element [32–37]:
˙
σ=˚
CV
ˆ
˙
σdV =˚
CV ∂
∂tρs+∇·(jS+ρsu)dV ≥0 (3)
ˆ
˙
σ=∂
∂tρs+∇·(jS+ρsu)≥0 (4)
where
CV
is the control volume,
t
is time [s],
s
is the specific entropy (per unit mass of fluid)
[J K
−1
kg
−1
],
jS
is the non-fluid entropy flux [J K
−1
m
−2
s
−1
],
u
is the fluid velocity [m
s
−1
],
dV
is a volume element [m
3
],
n
is an outwardly directed unit normal [-] and
∇
is the
Cartesian nabla operator [m
−1
]. The inequalities in Equations
(3)
and
(4)
express the global
and local definitions of the second law of thermodynamics, applicable to an open system.
In information theory, it is usual to rewrite Equation
(2)
as the binary entropy and
relative entropy, expressed in binary digits or “bits” [38]:
BSh =−
n
∑
i=1
pilog2piand B=−
n
∑
i=1
pilog2
pi
qi(5)
Comparing Equation
(2)
, we obtain
HSh =BSh ln
2 and
H=Bln
2. The change in informa-
tion about a system can then be defined as the negative change in its binary entropy [
39
–
44
]:
∆I=−∆BSh or ∆I=−∆B(6)
For example, consider a coin toss with equiprobable outcomes H and T. Without knowledge
of the outcome, an observer will assign
pH=pT=1
2
; hence, the entropy in Equation
(5)
gives
Binit
Sh =
1 bit. Once informed of the outcome, the observer must assign one probability
to zero and the other to unity, hence
Bfinal
Sh =
0 bits and
∆BSh =
0
−
1
=−
1 bit. From
Equation
(6)
, the information gained in the binary decision is
∆I=−∆BSh =
1 bit [
21
,
38
,
45
].
If instead we use the relative entropy in Equation
(5)
with equal priors
qH=qT=1
2
, we
obtain Binit =0 bits, Bfinal =−1 bits and ∆B=−1−0=−1 bits, so again ∆I=1 bit.
This idea was taken further by Szilard [
45
] and later authors based on analyses of
Maxwell’s demon [
40
–
43
,
46
,
47
], to establish a fundamental relationship between changes in
information, thermodynamic entropy and energy. From the second law of thermodynamics:
∆Suniv =∆Ssys +∆SROU ≥0 (7)
where
∆Suniv
is the change in thermodynamic entropy of the universe [J K
−1
], which
can be partitioned into the changes
∆Ssys
within a system and
∆SROU
in the rest of the
universe [48]. Substituting for ∆Ssys using Equations (5) and (6) gives:
kBln 2 ∆B+∆SROU =−kBln 2 ∆I+∆SROU ≥0 (8)
In general, a system can only influence the entropy of the rest of the universe by a transfer
of disordered energy
∆E=T∆SROU
[J], where
T
is the absolute temperature [K], such as
that carried by heat or chemical species. Rearranging Equation
(8)
and substituting for
∆E
gives:
kBln 2 ∆I≤∆SROU or kBTln 2 ∆I≤∆E(9)
or in time rate form:
kBln 2 ∂I
∂t≤∂SROU
∂tor kBTln 2 ∂I
∂t≤∂E
∂t(10)
Equations
(9)
and
(10)
provide an information-theoretic formulation of the second law of
thermodynamics, in which each bit of information gained by an observer about a system

Entropy 2023,25, 1538 4 of 36
must be paid for by an energy cost of at least
kBTln
2, or an entropy cost of at least
kBln
2.
This imposes a fundamental limit on processes that involve the transmission of information:
if the penalty incurred in Equations
(9)
and
(10)
is not paid, then the information will not
be transmitted.
3. Dimensionless Groups and the Principle of Entropic Similarity
As discussed in Part I [
20
], a dimensionless group is a unitless parameter used to rep-
resent an attribute of a physical system, independent of the system of units used. These
can be identified and applied in three ways. First, the matching of dimensionless groups
between a system (prototype) and its model, known as similarity or similitude, can be used
for experimental scaling [
5
,
7
]. Second, the governing dimensionless groups for a system
can be extracted from its list of parameters by the method of dimensional analysis [
1
]. Third,
the governing partial differential equation(s) for a system can be converted into dimen-
sionless form, known as non-dimensionalization, to identify its governing dimensionless
groups [
11
,
12
,
14
–
16
,
49
]. Recently, it was shown that the one-parameter Lie group of point
transformations provides a rigorous method for the non-dimensionalization of a differ-
ential equation, based on the intrinsic dimensions of the system [
50
]. For over a century,
dimensional methods have been recognized as powerful tools—and in many cases, the
primary tools—for the analysis of a wide range of systems across all branches of science
and engineering [51–57].
Dimensionless groups—commonly labeled Π—can be classified as [10–17]:
(i)
Those arising from geometric similarity, based on ratios of length scales
`i
[m] or associ-
ated areas or volumes:
Πgeom =`1
`2or Πgeom =`2
1
`2
2
or Πgeom =`3
1
`3
2
(11)
(ii)
Those arising from kinematic similarity, based on ratios of magnitudes of velocities
Ui
[m s−1] or accelerations ai[m s−2]:
Πkinem =U1
U2or Πkinem =a1
a2(12)
(iii) Those arising from dynamic similarity, based on ratios of magnitudes of forces Fi[N]:
Πdynam =F1
F2(13)
In Part I [
20
], an additional category of dimensionless groups was proposed based on
entropic similarity, based on the following definitions:
(i) Those defined by ratios of global or local entropy production terms:
Πentrop =˙
σ1
˙
σ2or ˆ
Πentrop =ˆ
˙
σ1
ˆ
˙
σ2(14)
where
Π
represents a global or summary dimensionless group,
ˆ
Π
is a local group,
˙
σi
is the global entropy production by the
i
th process
(3)
[J K
−1
s
−1
] and
ˆ
˙
σi
is the local
entropy production by the ith process (4) [J K−1m−3s−1].
(ii)
Those defined by ratios of global flow rates of thermodynamic entropy, or by compo-
nents or magnitudes of their local fluxes:
Πentrop =FS,1
FS,2 or ˆ
Πentrop(n) = jS1·n
jS2·nor ˆ
Πentrop =||jS1||
||jS2|| (15)
where
FS,i
is the entropy flow rate of the
i
th process [J K
−1
s
−1
],
jSi
is the non-fluid

Entropy 2023,25, 1538 5 of 36
entropy flux of the
i
th process [J K
−1
m
−2
s
−1
] (see
(4)
),
n
is a unit normal and
||a|| =
√a>ais the Euclidean norm for vector a.
(iii)
Those defined by an information-theoretic threshold, for example by the ratio of the
local information flux carried by the flow
jI,flow
[bits m
−2
s
−1
] to that transmitted by a
carrier of information jI,signal [bits m−2s−1]:
ˆ
Πinfo =||jI,flow||
||jI,signal|| =||ρI,flow uflow||
||ρI,signal usignal || (16)
In this perspective, flows in which the information flux of the fluid exceeds that of
a signal (
ˆ
Πinfo >
1) will experience a different information-theoretic flow regime to
those in which the signal flux dominates (
ˆ
Πinfo <
1). In Equation
(16)
, each information
flux is further reduced to the product of an information density
ρI
[bits m
−3
] and the
corresponding fluid velocity
uflow
or signal velocity
usignal
[m s
−1
]. Making the strong
assumption that the two information densities are comparable, Equation
(16)
simplifies
to give the local or summary kinematic definitions:
ˆ
Πinfo =||uflow ||
||usignal ||,Πinfo =Uflow
Usignal (17)
where Uflow and Usignal are representative flow and signal velocities [m s−1].
In Part I [
20
], the first two definitions of entropic similarity in Equations
(14)
and
(15)
were applied to a range of diffusion, chemical reaction and dispersion phenomena, to
reveal the entropic interpretation of many known dimensionless groups, and to define a
number of new groups. To continue the application of entropic similarity to flow processes,
it is necessary to examine the third definition based on information-theoretic similarity in
Equation (17), and its application to flows with wave motion.
4. Wave Motion and Information-Theoretic Flow Regimes
Awave can be defined as an oscillatory process that facilitates the transfer of energy
through a medium or free space. Generally, this is governed by the wave equation:
∂2φ
∂t2=c2∇2φ(18)
where
φ
is a displacement parameter and
c
is the characteristic wave velocity (celerity) [m
s
−1
], measured relative to the medium. Wave motion, in its own right, does not produce
entropy, although some waves are carriers of entropy, and wave interactions with materials
or boundaries can be dissipative in some situations. However, a wave is also a carrier
of information, communicating the existence and strength of a disturbance or source of
energy. Its celerity therefore provides an intrinsic velocity scale for the rate of transport
of information through the medium. For a fluid flow with local velocity
u
, the celerity
provides a threshold between two different information-theoretic flow regimes, which
respectively can (
||u|| <c
) or cannot (
||u|| >c
) be influenced by downstream disturbances.
Adopting the information-theoretic formulation of similarity
(17)
and allowing for a vector
celerity field
c
with magnitude
||c|| =c
, this can be used to define the local directional, local
vector, macroscopic vector and macroscopic scalar dimensionless groups, respectively:
ˆ
Πinfo(n) = u·n
c·n,ˆ
Πinfo =u
c,Πinfo =U
c,Πinfo =U
c(19)
where
n
is a given unit normal. The first group provides a local direction-dependent
definition based on a vector celerity, but will exhibit a singularity as
n→c
. To overcome

Entropy 2023,25, 1538 6 of 36
this, an alternative definition can be adopted based on component-wise division [20]:
e
Πinfo =uc(20)
where

is the component-wise (Hadamard) division operator. The second definition
in Equation
(19)
gives a local vector definition with a scalar celerity, the third gives a
vector definition based on a summary velocity vector
U
[m s
−1
], and the last gives a
summary criterion based on a summary velocity magnitude
U
[m s
−1
]. Each of these
groups provides a discriminator between two information-theoretic flow regimes separated
by the critical value of 1 (or for some definitions, the vector
1
), governed respectively by
downstream-controlled or upstream-controlled processes.
In the following sections, we examine several wave types from this perspective. For
some flows, a sharp junction can be formed between the two information-theoretic flow
regimes (e.g., a shock wave or hydraulic jump), with a high rate of entropy production.
Wave-carrying flows are also subject to friction, with distinct differences between the two
flow regimes. Both these features are explored for several flows, drawing on all definitions
of entropic similarity in Equations
(14)
–
(17)
as needed. Due to common practice, this study
makes some excursions from the notation used in Part I [
20
]; these are mentioned explicitly.
4.1. Acoustic Waves
4.1.1. Mach Numbers and Compressible Flow Regimes
An acoustic or sound wave carries energy through a material by longitudinal compres-
sion and decompression at the sonic velocity
a=pdp/dρ=pK/ρ
[m s
−1
], where
p
is the
pressure [Pa] and
K
is the bulk modulus of elasticity [Pa] [
10
–
12
,
58
–
60
]. For isentropic (adi-
abatic and reversible) changes in an ideal gas, this reduces to
a=pγp/ρ=√γR∗T
, where
γ
is the adiabatic index [-] and
R∗
is the specific gas constant [J K
−1
kg
−1
]. By information-
theoretic similarity
(19)
, this can be used to define the local scalar and macroscopic Mach
numbers:
ˆ
Πa=ˆ
M=||u||
a
ideal gas
−−−−−→
isentropic ||u||
√γR∗T,Πa=M∞=U∞
a∞
ideal gas
−−−−−→
isentropic
U∞
√γR∗T∞
(21)
where
U∞
is the free-stream fluid velocity [m s
−1
] and
T∞
is the free-stream temperature
[K] [10–12,58,59]. These groups discriminate between two flow regimes:
1.
Subsonic flow (locally
ˆ
M<
1 or summarily
M∞.
0.8), subject to the influence of the
downstream pressure, of lower uand often of higher p,ρ,Tand s; and
2.
Supersonic flow (locally
ˆ
M>
1 or summarily
M∞&
1.2), which cannot be influenced
by the downstream pressure, of higher uand often of lower p,ρ,Tand s.
Locally
ˆ
M=
1 is termed sonic flow, while summarily 0.8
.M∞.
1.2 indicates
transonic flow and
M∞&
5hypersonic flow [
59
]. Commonly, the Mach number
(21)
is
interpreted by dynamic similarity as the square root of the ratio of inertial to elastic
forces [
10
,
12
]. Instead of Equation
(21)
, some authors use the Cauchy number
Ca∞=
M2
∞=ρU2
∞/K.
Generally, acoustic waves are considered to have a single sonic velocity
a
under
given thermodynamic conditions. However, some acoustic waves–such as in bounded
systems [
3
,
61
,
62
]–exhibit wave dispersion, in which the angular frequency
ω
[s
−1
] is a
function of the angular wavenumber
k
[m
−1
], such that the sonic velocity (the phase celerity)
a=ω/k
is a function of frequency [
63
,
64
]. If this occurs, interference between different
waves will produce wave groups (beats), with the group celerity [
63
,
64
] and corresponding
local group Mach number:
agroup =dω
dk =d(ak)
dk =a+kda
dk ,ˆ
Πgroup
a=ˆ
Mgroup =||u||
agroup , (22)
For normal wave dispersion,
agroup <a
,
da/dk <
0 and
ˆ
Mgroup >ˆ
M
. This implies the

Entropy 2023,25, 1538 7 of 36
existence of three distinct information-theoretic flow regimes:
1.
Subsonic flow (locally
ˆ
M<ˆ
Mgroup <
1), subject to the influence of acoustic waves and
wave groups;
2.
Normal mesosonic flow (locally
ˆ
M<
1
<ˆ
Mgroup
), influenced by individual acoustic
waves but not wave groups; and
3.
Supersonic flow (locally 1
<ˆ
M<ˆ
Mgroup
), which cannot be influenced by acoustic
waves or wave groups.
We here use the Greek prefix meso- for the “middle” regime. For the transitions, we retain
the term sonic flow for
ˆ
M=
1, and describe
ˆ
Mgroup =
1 as group sonic flow. For the opposite
case of anomalous wave dispersion,
agroup >a
,
da/dk >
0 and
ˆ
Mgroup <ˆ
M
, suggesting
the existence of an anomalous mesosonic flow regime (locally
ˆ
Mgroup <
1
<ˆ
M
), influenced
by wave groups but not individual waves.
At present, the physical manifestations—if any—of the postulated normal and anoma-
lous mesosonic flow regimes for media with acoustic dispersion are not known. Such effects
may be masked by the common use of summary rather than local Mach numbers
(21)
,
giving a lumped “transonic” flow with complicated properties. Similarly, the effects of
these flow regimes on gradual or sharp flow transitions (such as shock waves) and frictional
flow properties are not understood. These phenomena warrant more detailed experimental
and theoretical investigation.
4.1.2. Shock Waves
In compressible flows with non-dispersive acoustic waves, it is possible to effect a
smooth, isentropic transition between subsonic and supersonic flow (or vice versa) using a
nozzle or diffuser, described as a choke [
10
–
12
]. However, the transition from supersonic to
subsonic flow is often manifested as a normal shock wave, a sharp boundary normal to the
flow with discontinuities in
u
,
p
,
ρ
,
T
and
s
[
59
]. From the local entropy production [
20
] at
steady state:
˙
σsteady =‹
CS
(jS+ρsu)·ndA ≥0 (23)
where
dA
is an infinitesimal area element on the control surface
CS
. Adopting a control
volume for a normal shock wave of narrow thickness, with inflow 1, outflow 2 and no
non-fluid entropy fluxes (
jS=
0), Equation
(23)
gives the entropy production per unit area
across the shock [J K−1m−2s−1] (c.f., [65]):
˘
˙
σshock =∆(ρsu)·n=ρ2s2u2−ρ1s1u1≥0 (24)
Using relations for
u
,
p
,
ρ
and
T
derived from the conservation of fluid mass, momentum
and energy for inviscid adiabatic steady-state flow across the shock
[10,11,17,58,59,66–73]
,
Equation
(24)
can be rescaled by the internal entropy flux to give the entropic dimensionless
group (see Appendix A):
˘
Πshock =˘
˙
σshock
ρ1cpu1=s2−s1
cp=ln(2+ (γ−1)ˆ
M2
1)
(γ+1)ˆ
M2
1+1
γln2γˆ
M2
1−γ+1
γ+1
with ˆ
M2=s2+ (γ−1)ˆ
M2
1
2γˆ
M2
1−γ+1
(25)
where
cp
is the specific heat capacity at constant pressure [J K
−1
kg
−1
]. From Equation
(25)
,
˘
Πshock >
0 and
˘
˙
σshock >
0 for
ˆ
M1>
1 and
ˆ
M2<
1, so the formation of an entropy-
producing normal shock in the transition from supersonic to subsonic flow is permitted
by the second law. However,
˘
Πshock <
0 and
˘
˙
σshock <
0 for
ˆ
M1<
1 and
ˆ
M2>
1, so the
formation of a normal shock in the transition from subsonic to supersonic flow (a rarefaction

Entropy 2023,25, 1538 8 of 36
shock p2<p1) is prohibited by the second law (3) (see Appendix A) [10,11,58,59,70,74].
By inwards deflection of supersonic flow (a concave corner), it is also possible to form
an oblique shock wave, a sharp transition to a different supersonic or subsonic flow with
increasing
p
,
ρ
and
T
[
17
,
59
]. This satisfies the same relations for the entropy production
in Equations
(24)
and
(25)
as a normal shock, but written in terms of velocity components
u1
and
u2
normal to the shock, thus with normal Mach components
ˆ
Mn1=ˆ
M1sin β
and
ˆ
Mn2=ˆ
M2sin(β−θ)
, where
β
is the shock wave angle and
θ
is the deflection angle. From
the second law ˘
Πshock >0, an oblique shock wave is permissible for ˆ
M1>ˆ
M2, in general
with a supersonic transition Mach number, and either two or no
β
solutions depending
on
θ
[
59
]. In contrast, outwards deflection of supersonic flow (a convex corner) creates
an expansion fan, a continuous isentropic transition
ˆ
M2>ˆ
M1
with decreasing
p
,
ρ
and
T[11,17,59].
The above analysis highlights the confusion in the aerodynamics literature between
the specific entropy and the local entropy production. For a non-equilibrium flow system,
the second law is defined exclusively by
˙
σ≥
0 in Equation
(3)
[
20
], reducing for a sudden
transition to
˘
˙
σshock ≥
0 in Equation
(24)
, while the change in specific entropy can (in
principle) take any sign
∆s=s2−s1S
0. From the above analysis,
∆s≥
0 implies
˘
˙
σshock ≥
0 only for flow transitions satisfying Equation
(24)
and local continuity
ρ1u1=ρ2u2
; these
include normal and oblique shocks. For transitions involving a change in fluid mass flux,
or if there are non-fluid entropy fluxes in Equation
(24)
,
∆s
and
˘
˙
σshock
can have different
signs. Similarly, an isentropic process
∆s=
0 need not indicate zero entropy production
˘
˙
σshock =
0. The above analyses are also complicated by fluid turbulence, which generates
additional Reynolds entropy flux terms in the local entropy production equation (see
analyses in [20,65,75]).
4.1.3. Frictional Compressible Flow
For frictional internal compressible flow with non-dispersive acoustic waves at steady
state, the local entropy production
(4)
reduces to
ˆ
˙
σ=∇·(jS+ρsu)
[
20
]. Assuming one-
dimensional adiabatic flow of an ideal gas without chemical or charge diffusion in a conduit
of constant cross sections, by the conservation of fluid mass, momentum and energy with
friction [11,17,69] and entropic scaling gives the local entropic group (see Appendix B):
ˆ
Πcompr(x) = ˆ
˙
σ(x)dH
ρacpa=dH
ρacpa
dρ(x)s(x)u(x)
dx =dH
cp
ds(x)
dx
=2dH(γ−1)(1−ˆ
M(x)2)
γ2+ (γ−1)ˆ
M(x)2ˆ
M(x)
dˆ
M(x)
dx =dHΘ(x)dˆ
M(x)
dx =1
2(γ−1)fˆ
M(x)2
(26)
where
x
is the flow coordinate [m],
dH
is the pipe hydraulic diameter [m],
f
is the Darcy
friction factor [-] and subscript
a
denotes a fluid property at the sonic point. Note the
different roles of the specific entropy and the local entropy production. Analysis of the
group Θ(x)defined in Equation (26) for γ>1 reveals the following effects of friction:
1.
The last term in Equation
(26)
is positive for all
ˆ
M>
0, hence
ˆ
Πcompr >
0 and
ˆ
˙
σ>
0,
i.e., the entropy production cannot be zero for finite flow.
2.
For subsonic flow
ˆ
M<
1 and
Θ>
0; hence, the second law
ˆ
Πcompr >
0 or
ˆ
˙
σ>
0
implies dˆ
M/dx >0, and so ˆ
Mwill increase with xtowards ˆ
M=1;
3.
For supersonic flow
ˆ
M>
1 and
Θ<
0; hence, the second law
ˆ
Πcompr >
0 or
ˆ
˙
σ>
0
implies dˆ
M/dx <0, and so ˆ
Mwill decrease with xtowards ˆ
M=1;
4.
In both cases, the second law
ˆ
Πcompr >
0 or
ˆ
˙
σ>
0 implies
ds/dx >
0, so the
specific entropy
s
will increase with
x
towards
ˆ
M=
1. Integrating Equation
(26)
, this
terminates at the maximum specific entropy sa;
5.
In the sonic limit
ˆ
M→
1
∓
,
Θ→
0 and
dˆ
M/dx → ±∞
, but these limits combine to
give lim ˆ
M→1ˆ
Πcompr =1
2(γ−1)f>0 from either direction.
These statements are supported by the plots of
dˆ
M/dx
,
Θ
, fluid properties and
ˆ
Πcompr
as

Entropy 2023,25, 1538 9 of 36
functions of ˆ
Mfor the flow of dry air, presented in Appendix B.
The sonic point
x=L∗
and
M(x)
can then be calculated numerically from the inte-
grated friction equation [10,11,17,69,73]:
f(L∗−x)
dH=γ+1
2γln (γ+1)ˆ
M(x)2
2+ (γ−1)ˆ
M(x)2+1−ˆ
M(x)2
γˆ
M(x)2(27)
Flows in conduits longer than
L∗
undergo frictional choking, producing a lower subsonic
entry Mach number or supersonic flow with a normal shock, so that the flow exits at
ˆ
M=
1 [
11
,
69
]. Clearly, such flows are controlled by their entropy production: since they
are adiabatic, they cannot export heat, so each fluid element can only achieve a positive
local entropy production
ˆ
˙
σ>
0 by increasing its specific entropy
s
in Equation
(26)
, via
permissible changes in
p
and
T
. When
s
reaches its maximum, no solution to Equation
(26)
with
ds/dx >
0 is physically realizable, to enable a positive entropy production. This
triggers unsteady flow to create the choke. For isothermal flows, flows with heat fluxes,
other non-fluid entropy fluxes or chemical reactions, extensions of Equations
(26)
and
(27)
are required [10,11,69].
For frictional external compressible flow, the entropy production due to inertial drag
and lift can be written in the vector form [20]:
˙
σcompr
ext,I=FD·U
T∞=
1
2ρ∞AsCD·U||U||2
T∞(28)
where
FD
is the drag force [N],
U
is a representative velocity of the fluid relative to the
solid [m s
−1
],
ρ∞
is the free-stream density [kg m
−3
],
As
is the cross-sectional area of the
solid [m
2
] and
CD
is a vector drag-lift coefficient [-]. Equation
(28)
can be scaled by sonic
conditions to give the entropic group:
Πcompr
ext,I=˙
σcompr
ext,I
˙
σsonic
ext,I
=ρ∞CD·M||M||2/T∞
ρaCDa/Ta∼CD·M||M||2(29)
where
˙
σsonic
ext,I
is the sonic inertial entropy production and
M
is a summary vector Mach
number. Generally, the drag coefficient increases significantly beyond a critical Mach
number
||Mc|| .
1, due to the local onset of supersonic flow and the formation of shock
waves, and then falls to an asymptotic value with increasing
||M|| >
1 [
10
,
17
,
59
]. In
contrast, the lift coefficient of an airfoil exhibits a gradual rise and sudden fall over
||M|| <
||Mc|| <1, also increasing with the angle of attack [69].
4.2. Blast Waves
For chemical combustion in a fluid or solid, the reaction is driven by a combustion wave
or blast wave that moves relative to the reactants at the explosive or detonation velocity
Uexplos
[m s
−1
]. From Equation
(19)
, this can be scaled by the acoustic velocity measured in
the reactants aR, giving the information-theoretic group [69]:
ΠaR=Mexplos =Uexplos
aR(30)
This defines an explosive Mach number, which discriminates between detonation of a high
explosive for
Mexplos >
1 (typically
Mexplos 
1
)
in a (compressive) supersonic shock
front, or deflagration of a low explosive for
Mexplos <
1 in a (rarefaction) subsonic flame
front [52,69,72].
Explosions in a compressible fluid can be modeled by the one-dimensional conserva-
tion equations used for a normal shock in Equation
(25)
, adding the reaction enthalpy and
a minimum entropy assumption [
69
,
72
,
76
]. This predicts alternative incoming velocities
corresponding to detonation or deflagration; for the former, the outgoing combustion

Entropy 2023,25, 1538 10 of 36
products are expelled at the acoustic velocity relative to the shock front. A kinetic model
of detonation (ZND theory) extends this finding, with compression of the reactants at the
shock front, causing ignition, heat release and acceleration of the combustion products to
the choke point [77–80].
A large chemical, gas or nuclear explosion in the atmosphere will generate a spherical
shock wave expanding radially from the source. This provides a famous example of the
use of dimensional scaling. Consider a point explosion with shock wave radius
R
[m]
governed only by the energy
E
[J], initial density
ρ0
[kg m
−3
] and time
t
[s]. Dimensional
reasoning gives the self-similar solution
R∝(Et2/ρ0)1/5
, which with the conservation of
mass, momentum and energy for inviscid flow yields power-law relations for
u
,
p
,
ρ
and
T
with time
t
and radius
r
[
52
,
55
,
57
,
70
,
78
,
81
–
83
]. For short times, these reveal strong heating
and near-evacuation of air from the epicenter, and its accumulation behind the shock front.
Surprisingly few authors have examined explosions from an entropic perspective
[84–90]
,
despite its role as their driving force, and the use of minimum [
91
,
92
] or maximum [
93
]
entropy closures in some analyses. From Section 4.1, we suggest the use of the local entropic
group
ˆ
Πexplos(x)
or
ˆ
Πexplos(r)
, extending Equations
(26)
and
(27)
to include shock wave,
heating and chemical reaction processes.
4.3. Pressure Waves
Also related to acoustic waves is the phenomenon of water hammer, an overpressure
(underpressure) wave in an internal flow of a liquid or gas, caused by rapid closure of
a downstream (upstream) valve or pump [
12
,
17
]. By reflection at the pipe ends, this
causes the cyclic propagation of overpressure and underpressure waves along the pipe,
commonly analyzed by the method of characteristics. For flow of an elastic liquid in a
thin-walled elastic pipe, the acoustic velocity and the magnitude of the change in pressure
are, respectively [12,17,94]:
aH=sK
ρ1+Kd
Eθ(1−ν2
p)−1
,|∆p|=ρaH|∆U|, (31)
where
|∆U|
is the magnitude of the change in mean velocity [m s
−1
],
K
is the bulk elastic
modulus of the fluid [Pa],
E
is the elastic modulus of the pipe [Pa],
d
is the pipe diameter
[m],
θ
is the pipe wall thickness [m] and
νp
is Poisson’s ratio for the pipe material [-]. An
extended relation is available for gas flows [
17
]. In liquids, the underpressure wave can
cause cavitation (the formation of vapor bubbles), leading to additional shock waves when
these collapse at higher pressures [17].
Equations (19) and (31) give the information-theoretic dimensionless group:
ΠaH=EuH=|∆U|
aH=|∆p|
ρa2
H
(32)
which can be recognized as an Euler number defined for water hammer. By frictional
damping in accordance with the Darcy–Weisbach equation [
20
], the pressure pulse
∆p
—
hence, the wave speed ∆Uand the group ΠaH—will also diminish with time.
4.4. Stress Waves
Related to acoustic and pressure waves, a variety of waves can occur in solids, liquids
and/or along phase boundaries due to the transport of compressive, shear or torsional
stresses generated by a sudden failure, expansion or impact. These can be divided into
elastic or inelastic waves, involving reversible or irreversible solid deformation [
95
–
97
], and
also classified into various types of seismic or earthquake waves. Stress waves can be analyzed
by information-theoretic constructs such as Equation
(19)
to identify the flow regime, but
generally are not associated with the mean motion of the medium, so are not examined
further here. Blast waves can also be generated in a solid by an explosion or impact, as

Entropy 2023,25, 1538 11 of 36
discussed in Section 4.2.
4.5. Surface Gravity Waves
4.5.1. Froude Numbers, Wave Types and Liquid Body Flow Regimes
On the surface of a liquid, energy can be carried by gravity waves, involving circular
or elliptical rotational oscillations of the fluid in the plane normal to the surface, reducing
in scale with depth. These can be classified as standing waves, which remain in place, or
progressive waves, which move across the surface. Using Airy (linear) wave theory, the
angular frequency and individual wave (phase) celerity of a two-dimensional progressive
surface gravity wave are given by [19,60,98–101]:
ω2=gk tanh(ky),csurf =ω
k=rg
ktanh(ky) = rλg
2πtanh 2πy
λ(33)
where
λ
is the wavelength [m] and
y
is the liquid depth [m]. For an ambient flow with
the summary horizontal velocity
U
[m s
−1
], applying information-theoretic similarity
(19)
gives the generalized summary Froude number:
Πcsurf =Frsurf =U
csurf =U
qg
ktanh(ky)
=U
qλg
2πtanh 2πy
λ
(34)
A local vector Froude number
ˆ
Frsurf
can also be defined based on the local mean velocity
¯u
.
However, due to wave dispersion, surface gravity waves generally travel in wave groups.
The group celerity–equivalent to the speed of energy transmission [
60
,
99
,
101
]–and the
corresponding summary group Froude number are:
cgroup
surf =dω
dk =d(csurfk)
dk =csurf +kdcsurf
dk =csurf
21+2ky
sinh(2ky),
Πgroup
csurf =Frgroup
surf =U
cgroup
surf
=2Frsurf1+2ky
sinh(2ky)−1(35)
Curiously, the two Froude numbers in Equations
(34)
and
(35)
are not in common use.
Variants of the celerities in Equations
(33)
–
(35)
are available for gravity waves on the
interface between two liquids [
100
,
102
]. Since 0
<x/ sinh(x)≤
1 for
x=
2
ky >
0, surface
gravity waves usually exhibit normal dispersion
cgroup
surf <csurf
and
dcsurf/dk <
0, with
individual waves advancing faster than the group. The exception in the limit
ky →
0 is
examined below.
Usually, three cases of surface gravity waves are distinguished:
1.
For deepwater (deep liquid) or short waves:
ky &π
or
λ/y.
2; thus,
tanh(ky)→
1 in
Equation (33), hence [19,60,83,98–100,103]:
cλ=rg
k=rλg
2π,Πcsurf →Frλ=U
cλ
=Usk
g=Us2π
λg(36)
Such waves move freely by circular motions of the fluid, with little net horizontal
transport. Deep waves travel in wave groups: in the deepwater limit
ky →∞
,
2
ky/ sinh(
2
ky)→
0 in Equation
(35)
, giving the group celerity
cgroup
λ=1
2cλ
and
group Froude number
Frgroup
λ=
2
Frλ
. Despite their simplicity, neither
Frgroup
λ
nor
Frλ
are in common use. For wave drag on a ship, the Froude number
Frship =U/pgL
is used, where Uis the ship velocity [m s−1] and Lis the ship length [m] [10,18].
2.
For transitional waves:
π/
10
.ky .π
or 2
.λ/y.
20, the wave motion is impeded
by contact with the bottom, producing elliptical motions of the fluid. Such waves
form in natural water bodies by the shoaling of deepwater waves as they approach
the shoreline. The generalized phase celerity and Froude number
(33)
and
(34)
, and

Entropy 2023,25, 1538 12 of 36
the generalized group celerity and Froude number
(35)
, apply. More complicated
(nonlinear) wave descriptions can also be used, including Stokesian waves for
λ/y.
10,
a superposition of cosine wave forms, and cnoidal waves for
λ/y&
10, comprising
horizontally asymmetric waveforms with pointed crests [98].
3.
For shallow or long waves:
ky .π/
10 or
λ/y&
20; thus,
tanh(ky)→ky
in Equa-
tion (33), giving [10,12,19,60,83,100,101,103,104]:
cy=√gy,Πcsurf →Fry=U
√gy (37)
In the shallow limit,
ky →
0,
sinh(
2
ky)→
2
ky
and
Frgroup
y→Fry
in Equation
(35)
, so
there is no separate group celerity (producing non-dispersive waves). Equation
(37)
is
applied to open channel flows with rectangular cross sections. For channels of low
slope and arbitrary cross sections (of low aspect ratio), Equation
(37)
is commonly
generalized as [10,19,104,105]:
cyh=√gyh,Πcsurf →Fryh=U
√gyh(38)
where
yh=A/B
is the hydraulic mean depth [m],
A
is the channel cross-sectional
area [m2] and Bis the channel top width [m]. For a rectangular channel, yh=y.
Steady incompressible open channel flows generally satisfy the conditions for shallow
waves, which communicate the occurrence of a downstream influence (such as a sudden
obstruction). The Froude number in Equation
(37)
or
(38)
then discriminates between two
flow regimes [10,12,104]:
1.
Subcritical flow (
Fry
or
Fryh<
1), subject to the influence of downstream obstructions,
of lower velocity Uand higher water height y; and
2.
Supercritical flow (
Fry
or
Fryh>
1), which cannot be influenced by downstream
obstructions, of higher velocity Uand lower water height y.
Locally Fryor Fryh=1 is termed critical flow, occurring at the critical depth yc[m].
The above flow regimes can be extended to flows with deepwater or transitional waves,
but the analysis must take into account the effect of wave dispersion, which produces
two different Froude numbers
Frsurf
and
Frgroup
surf
(Equations
(34)
and
(35)
, respectively),
for individual waves and wave groups. By normal wave dispersion,
cgroup
surf <csurf
and
Frgroup
surf >Frsurf. This creates the possibility of three information-theoretic flow regimes:
1.
Subcritical flow (
Frsurf <Frgroup
surf <
1), subject to the influence of surface gravity waves
and wave groups;
2.
Normal mesocritical flow (
Frsurf <
1
<Frgroup
surf
), influenced by individual surface
gravity waves but not wave groups; and
3.
Supercritical flow (1
<Frsurf <Frgroup
surf
), which cannot be influenced by surface gravity
waves or wave groups.
We retain the term critical flow for
Frsurf =
1, and describe
Frgroup
surf =
1 as group critical flow.
The physical manifestations of the postulated mesocritical flow regime in systems with
deepwater or transitional waves are not known, and may again be masked by the common
use of summary (Equation
(34)
) rather than local Froude numbers. These systems warrant
more detailed experimental and theoretical investigation. Deep liquid bodies will also be
influenced by internal gravity waves, examined in Section 4.7.
4.5.2. Hydraulic Jumps in Open Channel Flow
In many open channel flows, it is possible to effect a smooth transition between
subcritical and supercritical flow (or vice versa) using a pinched channel (Venturi flume) or
stepped bed, described as a choke [
19
,
98
,
104
]. However, the transition from supercritical
to subcritical flow is often manifested as an hydraulic jump, with sharp changes in
U
and
y
. For a macroscopic control volume extending across a jump in a rectangular channel,

Entropy 2023,25, 1538 13 of 36
with inflow 1, outflow 2 and no non-fluid entropy fluxes, by the conservation of mass and
momentum with energy loss [11,19,98,104,105], the total entropy production is:
˙
σjump =ρgQ∆E
T=ρgQ
T
(y2−y1)3
4y1y2≥0
with y2
y1=1
2q1+8Fr2
y1−1=2q1+8Fr2
y2−1−1(39)
where
Q
is the volumetric flow rate [m
3
s
−1
] and
∆E
is the loss in energy per unit weight [J
N−1= m]. Scaling by the entropy flow rate gives the entropic group:
Πjump =˙
σjump
ρcpQ=g∆E
cpT=g
cpT
(y2−y1)3
4y1y2≥0 (40)
From Equations
(39)
and
(40)
,
Πjump >
0 and
˙
σjump >
0 for
Fry1>
1 and
Fry2<
1, so the
formation of an entropy-producing hydraulic jump in the transition from supercritical
to subcritical flow is permitted by the second law. However,
Πjump <
0 and
˙
σjump <
0
for
Fry1<
1 and
Fry2>
1, so the formation of an hydraulic jump in the transition from
subcritical to supercritical flow (a reverse jump
y2<y1
) is prohibited by the second law
(see Appendix C).
By the inward deflection of supercritical flow or by interaction with wall boundaries, it
is also possible to form an oblique hydraulic jump, a sharp transition to a different supercritical
or subcritical flow at the angle
β∈(
0,
π)
to the flow centerline [
98
]. This satisfies the same
relations for the entropy production in Equations
(39)
and
(40)
as a normal jump, but
written in terms of the velocities
U1
and
U2
normal to the jump, thus with normal Froude
numbers
Fry,ni =Fryi sin β
for
i∈ {
1, 2
}
[
98
]. From the second law
Πjump >
0, an oblique
jump is permissible for
Fry1>(sin β)−1>Fry2
, so in general with the transitional Froude
number Fryc = (sin β)−1≥1.
4.5.3. Frictional Gradually Varied Open Channel Flow
Frictional open channel flows at a steady state can be classified as (i) uniform flows of
constant water elevation
y=y0
, due to the equilibrium between frictional and gravitational
forces in a long channel; (ii) gradually-varied flows, with a smooth flow profile
y(x)
, where
x
is the flow direction [m]; or (iii) rapidly-varied flows, with a sharp change in
y(x)
in response
to a sudden constriction [
98
,
104
,
105
]. The resistance for arbitrary cross sections is often
represented by the Manning equation
U=R2/3
H√S/n
, where
RH=A/Pw
is the hydraulic
radius [m],
A
is the channel cross-sectional area [m
2
],
Pw
is the wetted perimeter [m],
n
is
Manning’s constant for the channel type [s m
−1/3
] and S
=dHL/dx
is the hydraulic slope
[-], in which
HL
is the head loss [m] and
x
is the horizontal coordinate [m] [
19
,
98
,
104
,
105
].
The entropy production per unit channel length [J K
−1
m
−1
s
−1
] by inertial dispersion
is [20,50,106–109]:
e˙
σopen(x) = ρgQ
T
dHL
dx =ρgSQ
T=ρgQ3n2P4/3
w
A10/3T(41)
For a rectangular channel of bed slope S
0
[-], constant width
B
and constant flow rate
Q=qB
, where
q=Uy
is the flow rate per unit width [m
2
s
−1
], substituting
A=By
,
Pw=B+
2
y
,
Fry=q/(y3/2 √g)
and the energy equation into Equation
(41)
and rescaling

Entropy 2023,25, 1538 14 of 36
gives the local entropic group (see Appendix D):
e
Πopen(x) = e˙
σopen(x)
ρcpq=gBS(x)
cpT=gBS0(x)
cpT+gB(S(x)−S0(x))
cpT
=gBS0(x)
cpT+gBFry(x)2−1
cpT
dy(x)
dx
=gBS0(x)
cpT−2BFry(x)2−1(gq)2/3
3cpT Fry(x)5/3
dFry(x)
dx =q2g n2B+2y(x)4/3
cpT y(x)10/3 B1/3
(42)
Analysis of Equation (42) reveals the following effects of friction:
1.
The last term in Equation
(42)
is positive for all
Fry>
0 and
y>
0, hence
e
Πopen >
0
and e˙
σopen >0, i.e., the entropy production cannot be zero for finite flow.
2.
In contrast to frictional compressible flows (Section 4.1.3), frictional open channel
flows are subject to a larger set of upstream and downstream boundary conditions.
These, in combination with the channel slope, flow rate and flow regime—under the
constraint of a positive entropy production—determine the flow profile
y(x)
that will
be realized. Some profiles terminate or start at the critical depth
y=yc
, at which
Fryh=
1; some at the uniform depth
y=y0
, at which friction and gravity are in
equilibrium; some start from a (theoretical) zero depth
y=
0; and some terminate in a
horizontal water surface [19,98,104,105].
3.
For subcritical flow
Fry<
1 and
y>yc
, from the second law
e
Πopen >
0 or
e˙
σopen >
0
in Equation (42):
(a)
For uniform flow S
=
S
0
, Equation
(42)
implies constant
y=y0
and
Fry=Fry0
;
(b)
For S
0<
S, Equation
(42)
implies
dy/dx <
0 and
dFry/dx >
0, so
Fry
will
increase with x, while y(x)will decrease with x(a drawdown curve);
(c)
For 0
<
S
<
S
0
, Equation
(42)
implies
dy/dx >
0 and
dFry/dx <
0, so
Fry
will
decrease with x, while y(x)will increase with x(a backwater curve).
4.
For supercritical flow
Fry>
1 and
y<yc
, from the second law
e
Πopen >
0 or
e˙
σopen >
0
in Equation (42):
(a)
For uniform flow S
=
S
0
, Equation
(42)
implies constant
y=y0
and
Fry=Fry0
;
(b)
For S
0<
S, Equation
(42)
implies
dy/dx >
0 and
dFry/dx <
0; hence,
Fry
will
decrease with x, while y(x)will increase with x(a backwater curve);
(c)
For 0
<
S
<
S
0
, Equation
(42)
implies
dy/dx <
0 and
dFry/dx >
0; hence,
Fry
will increase with x, while y(x)will decrease with x(a drawdown curve);
5.
In the critical limit
Fry→
1
∓
and
y→y±
c
,
(Fr2
y−
1
)→
0 and
dy/dx → ∓∞
, but
these limits combine to give
limFry→1e
Πopen =q2g n2B+2yc4/3/cpT y10/3
cB1/3 >
0
from either direction. A special case of critical uniform flow (
y=y0=yc
and
Fry=Fry0=
1) can form, but otherwise, critical flow will occur as a limiting case at
the position x=xc.
These statements are supported by the plots of
y
and
e
Πopen
as functions of
Fry
for a worked
example of open channel flow, presented in Appendix D.
For gradually varied flows, the flow profile
y(x)
and
Fry(x)
can be calculated by nu-
merical integration of the friction equation in
(42)
[
19
,
98
,
104
,
105
]. For profiles terminating
at
yc
, if the channel is longer than th