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1 The Second Law of Thermodynamics in the Carathéodory Form

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)



322



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.



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