1 The Second Law of Thermodynamics in the Carathéodory Form
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318
B.E. Pobedria and D.V. Georgievskii
1◦ . A ∼ A (reflexivity).
2◦ . A ∼ B =⇒ B ∼ A (symmetry).
3◦ . A ∼ B ∨ B ∼ C =⇒ A ∼ C (transitivity).
Then, the relation of heat equilibrium is the equivalence relation. All systems are
divided by equivalence classes so that two systems of type A will belong to the same
class if and only if they are found in mutual heat equilibrium.
Among various kinds of thermodynamic systems, there are such ones that are
characterized by only scalar thermodynamic parameter of state TE . We choose one
of these systems and will call it as ”the system E.” So, there exists the functional
connection for the systems of each type, for example
TE = ϕ A (A1 , A2 , . . . , An A ), TE = ϕ B (B1 , B2 , . . . , Bn B ), TE = ϕC (C1 , C2 , . . . , Cn C )
(16.1)
such that two systems A and B are found in heat equilibrium if and only if
TE = ϕ A (A1 , A2 , . . . , An A ) = ϕ B (B1 , B2 , . . . , Bn B )
(16.2)
Thus, zero law of thermodynamics leads to definition of the new parameter of
state TE being suitable for all thermodynamic systems. This parameter is said to be
empirical temperature; it is convenient to introduce it as an independent parameter
of state. Any scalar parameter, for example An A , may be expressed as
An A = ψ A (A1 , . . . , Am , TE ), m = n A − 1
(16.3)
The first law of thermodynamics ensures an introduction of the notions of internal
energy E and heat Q. The value δ Q is an energy transmitted from one system to
another due to a difference of its empirical temperatures. For adiabatic processes,
d E + δA(int) = 0
(16.4)
where δA(int) is the change of work of internal forces.
The second law of thermodynamics is used for introduction of the notions of
absolute temperature scale and entropy. The relation (16.3) demonstrates that every
thermodynamic parameter of state, for example the internal energy E, is expressed
in terms of thermodynamic parameters of state in the form
E = E(A1 , . . . , Am , TE )
(16.5)
The tensor values A1 , A2 , . . . , Am may be considered as generalized displacements; we denote by P j the corresponding to its generalized forces:
P j = P j (A1 , . . . , Am , TE ),
j = 1, . . . , m
(16.6)
16 Two Thermodynamic Laws as the Forth and the Fifth Integral …
Therefore
319
m
δA(int) =
Pj d A j
(16.7)
j=1
The generalized forces (16.6) are connected with the generalized displacements
by some constitutive relations (the state equations). If the process is balanced and it is
effected so slowly that every generalized force (16.6) corresponds to the state equations in any time moment then the first law of thermodynamics gives the following
relation:
m
∂E
∂E
(int)
δ Q = d E + δA
=
+ Pj d A j +
dTE
(16.8)
∂
A
∂T
j
E
j=1
So, according to (16.8) the value δ Q for balanced processes is a linear differential
form (the Pfaffian form) of independent thermodynamic parameters of state.
C. Carathéodory suggested (1909) the statement of the second law of thermodynamics in the form of following principle.
• For any state of thermodynamic system, one may produce the state with two properties:
(a) it is arbitrarily close to the original state;
(b) it is not reached from the original state by means of adiabatic balanced process.
Because δ Q = 0 for adiabatic balanced process (16.8) becomes an equation in
total differentials:
m
j=1
∂E
∂E
+ Pj d A j +
dTE = 0
∂Aj
∂TE
(16.9)
According to the Carathéodory principle, there are close states which cannot be
joined with the help of the solution of (16.9). Carathéodory established that this
means an integrability of the Pfaffian form (16.8) i. e., an existence of the integrating
factor ν(A1 , . . . , Am , TE ) and the associated function M(A1 , . . . , Am , TE ) such that
δQ
= dM
ν
(16.10)
or in detail
1
ν
m
j=1
∂E
∂E
+ Pj d A j +
dTE =
∂Aj
∂TE
m
j=1
∂M
∂M
dAj +
dTE
∂Aj
∂TE
(16.11)
It can be shown that among all possible integrating factors ν, there exists unique (to
within constant) factor depending on temperature TE only. It is denoted by T (TE )
and is said to be an absolute temperature. This is an universal function of state
320
B.E. Pobedria and D.V. Georgievskii
applicable to any thermodynamic system. The associated to it function is denoted by
S(A1 , . . . , Am , TE ) = S(μ1 , . . . , μm , T ) and is said to be an entropy of the system
under consideration. Then, the Eq. (16.10) has the following form
m
d Q = T d S = d E + dA(int) =
j=1
∂E
+ Pj
∂μ j
: dμ j +
∂E
dT
∂T
(16.12)
where symbol “:” means a full contraction of the 2nd rank tensors. It is valid for any
balanced process between adjacent states.
We see that the Carathéodory principle allows to introduce both entropy and
absolute temperature not having recourse to the model of perfect gas and the Carnot
cycle. In the case of gas with the state equation f ( p, V, T ) = 0, we receive from
(16.12)
∂E
1
1 ∂E
dT +
+ p dV
(16.13)
dS =
T ∂T V
T
∂T T
The number of independent parameters of state equals two, so any Pfaffian form has
an integrating factor and
∂E
∂V
=T
T
∂p
∂T
−p
(16.14)
V
It is easy to verify a realizability of (16.14) for both perfect gas and the van der Waals
gas.
16.2 Legendre Transforms and Thermodynamic Potentials
Side by side with the internal energy E(μ1 , . . . , μm , S) let us consider the following
thermodynamic potentials:
• the enthalpy (heat content) H (P1 , . . . , Pm , S):
m
H=E+
Pj : μ j
(16.15)
j=1
• the Helmholtz free energy F(μ1 , . . . , μm , T ):
F = E −TS
(16.16)
• the Gibbs potential G(P1 , . . . , Pm , T ):
G = H −TS
(16.17)
16 Two Thermodynamic Laws as the Forth and the Fifth Integral …
321
Using both the first and the second laws of thermodynamics, we may write (16.14)
in the following way
m
δA(int) =
P j : dμ j
(16.18)
j=1
as well as represent the thermodynamic identity:
m
dE = T dS −
P j : dμ j
(16.19)
j=1
In order to pass from one thermodynamic potential to some other, it is efficient to
use the Legendre transform of function ϕ(x1 , x2 , . . . ) with the total differential
dϕ =
∂ϕ
∂ϕ
d x1 +
d x2 + · · · ≡ X 1 d x1 + X 2 d x2 + . . .
∂x1
∂x2
The Legendre transform poses the function
the function ϕ(x1 , x2 , . . . ) such that
(X 1 , X 2 , . . . ) in correspondence with
= ϕ − X 1 x1 − X 2 x2 − . . .
d
(16.20)
(16.21)
= dϕ − X 1 d x1 − x1 d X 1 − X 2 d x2 − x2 d X 2 − . . .
(16.22)
A transition from the internal energy E to the enthalpy (16.15) is realized by
means of the following transform
∂E
,
∂μ j
− Pj =
j = 1, . . . , m
(16.23)
P j : dμ j + T d S
(16.24)
Then, repeating (16.19)
m
dE = −
j=1
m
dH = dE +
m
P j : dμ j +
j=1
m
μ j : d Pj = T d S +
j=1
∂H
= T,
∂S
μ j : d Pj
(16.25)
j=1
∂H
= μj
∂ Pj
(16.26)
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B.E. Pobedria and D.V. Georgievskii
Analogous relations take place in cases of the transition H → G:
m
dG = d H − T d S − S dT =
μ j : d P j − S dT
(16.27)
j=1
∂G
= −S,
∂T
∂G
= μj
∂ Pj
(16.28)
as well as the transition E → F:
m
d F = d E − T d S − S dT = −
P j : dμ j − S dT
(16.29)
j=1
∂F
= −S,
∂T
∂F
= −P j
∂μ j
(16.30)
It should be noted that both the thermodynamic parameters μ j and its fluxes P j must
be given by choose of the model.
16.3 Mass Densities of Thermodynamic Potentials
To describe the models in continuum mechanics, it is more convenient to use densities
of the thermodynamic functions and potentials under consideration. We remember
the statements of the first law of thermodynamics
or
d E + d K = δA(ext) + δ Q
(16.31)
d E = −δA(int) + δ Q
(16.32)
and the second law of thermodynamics
T d S = δ Q + W ∗ dt
(16.33)
where W ∗ is the dispersion function. Excepting δ Q from (16.32) and (16.33), we
receive the following thermodynamic identity
d E = T d S − δA(int) − W ∗ dt
(16.34)
Let us introduce a mass density of internal energy e(x, t), a mass density of entropy
s(x, t) and a density of dispersion w∗ (x, t) by means of the following relations for
arbitrary moving volume V :
16 Two Thermodynamic Laws as the Forth and the Fifth Integral …
E=
ρs d V, W ∗ =
ρe d V, S =
V
V
323
w∗ d V
(16.35)
V
In order to present the expression for the value δ Q in (16.31) – (16.33), we consider
an arbitrary finite volume V bounded by surface Σ with external unit normal n. Let
mass density of heat q(x, t) is given in any material point of this volume, and normal
component q (n) (y, t) of the heat flux vector q is given on each square element dΣ
(y ∈ Σ):
(16.36)
q (n) = qi n i = q · n
Then a heat influx in the volume V for some time interval dt is equal to
q (n) dΣ + dt
δ Q = −dt
Σ
ρq d V = dt
V
(ρq − div q) d V
(16.37)
V
The sign minus before the surface integral in (16.37) is explained by fact that
vector n is the external normal, whereas positive surface heat influx must be directed
from the outside toward the interior the volume V .
Physical dimensions of the introduced mass densities [e] = L2 T−2 , [s] =
2 −2 −1
L T K , [q] = L2 T−3 , [q (n) ] = MT−3 , [w∗ ] = ML−1 T−3 demonstrate that the
scalar fields e(x, t), w∗ (x, t), q(x, t), q (n) (y, t) have purely mechanical nature (in
spite of the word “heat”) and may be defined without the notion “temperature.”
It follows from the expressions (16.33), (16.35), (16.37) that in any material point
of the volume V :
ds
= ρq − qi,i + w∗
ρT
(16.38)
dt
This equation is known as the heat influx equation and it is the local consequence of
the second law of thermodynamics.
For a broad class of continuums, the constitutive relations connecting the heat
influx vector q and gradient of temperature grad T are valid. The Fourier law of heat
conduction
q = −Λ · grad T
(16.39)
represents the simplest such relation. Here Λ is a positive definite symmetric tensor
of the second rank named the tensor of heat conduction. Using (16.39) the heat influx
Eq. (16.38) may be written as
ρT
ds
= ρq + (Λkl T,l ),k + w∗
dt
(16.40)
324
B.E. Pobedria and D.V. Georgievskii
16.4 Two Thermodynamic Laws in the Form
of Integral Postulates
The introduced mass densities allow to formulate two thermodynamic laws as the
4th and 5th postulates of continuum mechanics, thus to add them to statements of
boundary-value problems. The integral statement of the first law is the following
[4, 5].
• Let Ω ∈ R 3 be a material volume in actual frame of reference, V be an arbitrary
moving volume in Ω and Σ be its boundary with unit external normal n. Then
d
dt
ρ e+
|v|2
dV =
2
V
(P(n) · v − q (n) ) dΣ
ρ(F · v + q) d V +
(16.41)
Σ
V
or taking into account the theorem of kinetic energy
d
dt
ρe d V =
V
q (n) dΣ
(ρq + P : D) d V −
(16.42)
Σ
V
Here D is strain rate tensor, F is mass force.
Differential consequence of the formulation (16.42) represents the local energy
equation
de
= ρq − div q + P : D
(16.43)
ρ
dt
Integral statement of the second law of thermodynamics may be following [4, 5].
• Let Ω ∈ R 3 be a material volume in actual frame of reference, V be an arbitrary
moving volume in Ω and Σ be its boundary with unit external normal n. Then
d
dt
ρq
dV −
T
ρs d V =
V
V
Σ
q (n)
dΣ +
T
w∗
q · grad T
dV
−
T
T2
(16.44)
V
The last integral in the right hand of (16.44) is said to be the production of entropy. It
is always nonnegative by virtue of positive definiteness of the tensor Λ (see (16.39))
as well as the inequalities w∗ ≥ 0, T > 0:
q · grad T
w∗
−
dV =
T
T2
S∗ =
V
1
w∗
+ 2 grad T · Λ · grad T ≥ 0
T
T
V
(16.45)
16 Two Thermodynamic Laws as the Forth and the Fifth Integral …
325
Substituting the surface integral in (16.44) on volume one:
Σ
q (n)
dΣ =
T
div
V
div q q · grad T
dV
−
T
T2
q
dV =
T
(16.46)
V
we easily receive the differential consequence of the 5th postulate, namely the equation of heat influx
ds
= ρq − div q + w∗
ρT
(16.47)
dt
The models of continuum mechanics for which w∗ = 0 are said to be reversible
ones. The inequality (16.45) demonstrates that the production of entropy may be not
equal to zero even for reversible models.
References
1. Germain, P.: Cours de Mécanique des Milieux Continus. T. 1. Théorie Générale. Masson Éditeurs, Paris (1973)
2. Sedov, L.I.: Mechanics of Continuous Media, vols. I, II. World Scientific Publ, Singapore (1997)
3. Ilyushin, A.A.: Mechanics of Continuous Media. Moscow State Univ. Publ, Moscow (1990).
[in Russian]
4. Pobedria, B.E., Georgievskii, D.V.: Foundations of Mechanics of Continuous Media. Fizmatlit,
Moscow (2006). [in Russian]
5. Pobedria, B.E., Georgievskii, D.V.: Uniform approach to construction of nonisothermal models
in the theory of constitutive relations. Continuous and Distributed Systems II. Ser. Studies in
Systems, Decision and Control, vol. 30, pp. 341–352 (2015)
Chapter 17
Flow Control Near a Square Prism
with the Help of Frontal Flat Plates
Iryna M. Gorban and Olha V. Khomenko
Abstract The case of two symmetrical flat plates fixed in front of a square prism for
passive control of a near-body flow pattern is numerically investigated at moderate
Reynolds numbers. The plates are used for generation of a pair of the frontal stable
vortices which would be able suppress flow separation in the neighbor body edges.
The improvement of body loads in this case is achieved by wake constriction and
reducing the difference between bottom and frontal pressure. The control scheme
presented was found to be sensitive to its geometrical parameters. The dynamic
system analysis is attracted for studying the flow topology in the area and deriving
optimum parameters of the control device. It was found that the plate length l ≈ 0.2d
and r ≈ 0.16d, where d is the prism side and r is the distance between the plate base
and the prism edge, is the appropriate choice which permits reduce the prism drag
approximately per 20 %. An influence of the Reynolds number on the effectiveness
of the control scheme is also investigated.
17.1 Introduction
Square prism, as a circular cylinder, is a fundamental bluff body configuration used
in many engineering applications including heat exchangers, architectural structures,
and marine equipment. Exploitation of these systems in air and fluid flows is accompanied by vortex shedding from the body that causes large unsteady forces, acoustic
noise and resonance, structural vibrations, and other dangerous effects. So, to improve
productivity of the equipment and prevent its destruction, different passive and active
methods have been proposed for control of vortex dynamics near a bluff body. The
I.M. Gorban
Institute of Hydromechanics, National Academy of Sciences of Ukraine,
Zheliabova St. 8/4, Kyiv 03680, Ukraine
e-mail: ivgorban@gmail.com
O.V. Khomenko (B)
Institute for Applied System Analysis, National Technical University of Ukraine
“Kyiv Polytechnic Institute”, Peremogy Ave. 37, Build 35, Kyiv 03056, Ukraine
e-mail: olgkhomenko@ukr.net
© Springer International Publishing Switzerland 2016
V.A. Sadovnichiy and M.Z. Zgurovsky (eds.), Advances in Dynamical Systems
and Control, Studies in Systems, Decision and Control 69,
DOI 10.1007/978-3-319-40673-2_17
327
328
I.M. Gorban and O.V. Khomenko
earliest methods for suppression of vortex shedding and fluid forces of bluff bodies
were discussed by Zdravkovich [1] who classified the schemes of flow control with
respect to the subject of exposure. He considered surface protrusions affecting separated shear layers, shrouds acting entrainment layers and near wake stabilizers. In
most cases, the methods are based on body surface modifications and are passive in
the sense that there is no power input. Further development of flow control has lead to
creation of the schemes requiring energy supplying from external sources (blowing
and suction, injection of micro-bubbles, surface heating or cooling, etc.). In paper
[2], those are classified into active open-loop and active closed-loop controls.
Majority of the both passive and active flow control researches deals with a circular
cylinder, which is the most popular bluff body configuration. At the same time, the
flow control for a square prism may differ from that for a cylinder owing to the fixed
separation points. In this case, wake modification does not require any influence on
the body boundary layer with the aim to delay the flow separation. Then the control
has to be brought to the body wake directly.
Among the most known direct-wake control methods for a square prism, are using
a downstream splitter plate [3, 4], installation of the small element, such as a flat
plate or rod, upstream of the prism [5, 6], and base blowing/suction [7–9]. Those
change vortex wake dynamics; as a result, the fluid forces acting on a square prism
are reduced. Notice that optimal control in these researches is derived by systematic changing parameters of the proposed devices, which is a very time-consuming.
To overcome this difficulty, the control theory based on the rigorous mathematical
apparatus can be applied. Researchers have shown that feedback control algorithms
based on mathematical analysis, such as optimal control approach and dynamical
systems theory, effectively control strong nonlinear flows generating in bluff body
wakes [10–12].
To achieve the desired effects to the flow, not only active but and passive methods
are in want of optimization in frames of the chosen control strategy. It is known
that one of the successful ways to control flow–body interactions at large Reynolds
numbers is connected with modification of near-body flow by creating artificial largescale vortices there. In passive control, the special surface irregularities, such as cross
groves or plates, are usually used for this purpose. This conception known as trapped
vortex approach has found its practical application in aviation, marine engineering,
and hydraulic systems [13–15].
The principal requirement to the algorithms based on the generation of large-scale
stable vortices near a body consists in possibility to forecast and control the behavior
of those. Therefore, the control scheme will be effective, if it applies information
about critical points, dynamic properties of the vortices created and other topological
features of flow field. It was stated in paper [16], a knowledge of critical-point
theory is important for interpreting and understanding flow patterns whether they are
obtained experimentally or computationally. Modern control algorithms are not only
used the above-mentioned information, but also directed to creating the necessary
topology in the flow field that includes changing the location and type of flow critical
points in accordance with the control goals.
17 Flow Control Near a Square Prism with the Help of Frontal Flat Plates
329
In this work, the trapped vortex approach is used to improve loads of a square
prism. Two stable symmetrical vortices are proposed to be generated in front of the
prism with the help of special attaching plates. The effects of the control plates on
the force coefficients, flow pattern, and vortex shedding frequency of the prism are
numerically studied at moderate Reynolds numbers, with Re based on the side of
prism.
To identify the optimum position and length of the plates, the so-called reduced
order model is applied [11]. This concept is based on the nonviscous model of
point vortices, in which the vorticity field is represented by a discrete set of isolated
circulatory elements whose axes are perpendicular to the flow plane. The flow field
in this case is reduced to the finite system of vortices moving along the trajectories of
fluid particles. Analyzing the vortex system dynamics in the considered area, one is
able to derive the main regularities of the flow pattern there. The model of the vortex
dynamics has ensured many important results in regard to the flow control [11, 12,
16–18].
We use here the model with one degree of freedom to study the dynamic behavior
of the trapped vortex clamped between the control plate and the prism front side. It
is supposed the recirculation zone formed due to flow separation in the plate edge is
replaced by a singular vortex. According to the present control strategy, the vortex
has to be immovable and prevent the flow separation in the prism leading edge. Then,
the problem is reduced to the PDE system relatively coordinates and circulation of
the vortex as well as parameters of the control plate.
Numerical modeling of flow patterns around the square prism with two frontal
plates is performed in 2D space by the vortex method [19, 20], which belongs to highresolution Lagrangian-type schemes developed as fast alternative to direct numerical
simulations (DNS) [21]. As pointed out by Liu and Kopp [22], the accuracy in the
last versions of the vortex method is compared well to nondissipative and high-order
finite-difference schemes, especially in large and intermediate scales.
The Reynolds number in the present investigation is changed in the range Re =
100 ÷ 500. The second wake instability connected with 3D transition is known to
develop in the square cylinder flow starting from Re ≈ 170 [23]. But it has been
shown from previous researches two-dimensional calculations at higher Reynolds
number simulate flow patterns, mean forces, and separation frequency for the square
cylinder quite successfully [22].
Position and length of the control plate are set using the information obtained with
the help of the simplified model of trapped, or standing, vortices [24]. The objective
of this work was to estimate an influence of the control plates on flow patterns and
fluid forces of the square prism as well as demonstrate that the reduced order model
is able to ensure optimum parameters of the control device. Wake stabilization and a
significant decrease of both drag and lateral force are observed in the flow under the
control. It follows from the present-study new successful control methods for fluid
flows can be developed on base of the dynamic models taking into account the flow
topology.