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Proceedings of ICHIT- 06

26 February - 5 March 2006, Moscow, Russia



Mechanism of origination of turbulence


LUDMILA I. PETROVA

Moscow State University, Russia, ptr@cs.msu.ru


Abstract
In the work it is shown that the conservation laws for material media (the
balance conservation laws for energy, linear momentum, angular momentum,
and mass, that establish a balance between the variation of a physical
quantity and the corresponding external action), turn out to be
noncommutative. The noncommutativity of the conservation laws that leads to
an emergence of internal forces and an appearance of the nonequilibrium is
a cause of development of instability in material media (material
systems).
A mechanism of development of instability in gas dynamical systems is
described and there are explained such processes as emergence of waves,
vortices, turbulent pulsations and so on.
These results were obtained with the help of the mathematical
apparatus of skew-symmetric differential forms.

1. equations of the balance conservations law

To study a development of instability it is necessary to analyze equations
of the balance conservation laws. For example, we take the simplest gas
dynamical system, namely, a flow of ideal (inviscid, heat nonconductive)
gas [1].
Assume that gas is a thermodynamic system in the state of local
equilibrium, that is, it is satisfied the relation [2]
[pic] (1)
There [pic],[pic] and [pic] are the temperature, the pressure and the gas
volume, [pic], [pic] are entropy and internal energy per unit volume.
Let us introduce two frames of reference: an inertial one that is not
connected with the material system and an accompanying frame of reference
that is connected with the manifold formed by the trajectories of the
material system elements.
In the inertial frame of reference the Euler equations are the balance
conservation laws for energy, linear momentum and mass of ideal gas [1] can
be written as
[pic] (2)
where [pic] is the total derivative with recpect to time. Here [pic] and
are [pic] respectively the mass and the entalpy densities of the gas.
Expressing entalpy in terms of internal energy [pic] with the help of
formula [pic] and using relation (1) the balance conservation law equation
can be put to the form
[pic] (3)
And respectively, the equation of the balance conservation law for linear
momentum can be presented as [1,3]
[pic] (4)
where [pic]is the velocity of the gas particle, [pic], [pic] is the mass
force. The operator [pic]in this equation is defined only in the plane
normal to the trajectory. [Here it was tolerated a certain incorrectness.
Equations (3) and (4) are written in different forms. This is connected
with difficulties when deriving these equations themselves. However, this
incorrectness will not effect on results of the qualitative analysis of the
evolutionary relation obtained from these equations.]
Since the total derivative with respect to time is that along the
trajectory, in the accompanying frame of reference equations (3) and (4)
take the form:
[pic] (5)
[pic] [pic] (6)
where [pic] is the coordinate along the trajectory, [pic] is the left-hand
side of equation (4), and [pic] is obtained from the right-hand side of
relation (4).
{In the common case when gas is nonideal equation (3) can be written
in the form
[pic] (7)
where [pic] is an expression that depends on the energetic actions. In the
case of ideal gas [pic] and equation (7) transforms into (5). In the case
of the viscous heat-conductive gas described by a set of the Navier-Stokes
equations, in the inertial frame of reference the expression [pic]can be
written as [1]
[pic] (8)
Here [pic] is the heat flux, [pic] is the viscous stress tensor. In the
case of reacting gas extra terms connected with the chemical nonequilibrium
are added [1]. }
Equations (5) and (6) can be convoluted into the relation
[pic] (9)
where [pic] is the first degree differential form (here [pic]).
The entropy enters into equation (9), as well as into the
thermodynamic relation, it enters the entropy. However, in relation (1) for
thermodynamic system the entropy depends on thermodynamic variables (see
relation (1)), whereas in the evolutionary relation for gas dynamic system
the entropy dependence on the space-time variables.

2. the development of the gas dynamic instability

Relation (9) is an evolutionary relation since this relation has been
obtained from the evolutionary equations. It is just the equation that
describes a state of material system and a mechanism of evolution of the
gas dynamic instability (in the case of local thermodynamic equilibrium).
If relation (9) appears to be identical one (if the form [pic] be the
closed form, and hence it is a differential), one can obtain a differential
of entropy [pic] and find entropy as a function of space-time coordinates.
{It should underline once again that entropy as a thermodynamic function of
state is not gas dynamic function of state}. The availability of the gas
dynamic function of state would point to the equilibrium state of the gas
dynamic system.
If relation (9) be not identical, then from this relation the
differential of entropy [pic] cannot be defined. This will point to an
absence of the gas dynamic function of state and nonequilibrium state of
the system. Such nonequilibrium is a cause of the development of
instability.
The evolutionary relation is a nonidentical one as it involves an
unclosed differential form.

Let us consider the first-degree form [pic]. The differential of this
form can be written as [pic], where [pic] are the components of the
commutator of the form [pic]. The components of the commutator of a form
[pic]can be written as follows:
[pic] (10)
(here the term connected with the nondifferentiability of the manifold has
not yet been taken into account). The coefficients of the form [pic] have
been obtained either from the equation of the balance conservation law for
energy or from that for linear momentum. This means that in the first case
the coefficients depend on the energetic action and in the second case they
depend on the force action. In actual processes energetic and force actions
have different nature and appear to be inconsistent. The commutator
constructed from the derivatives of such coefficients is nonzero. This
means that the differential of the form [pic] is nonzero as well. Thus, the
form [pic] proves to be unclosed. This means that the evolutionary relation
cannot be an identical one. In the left-hand side of this relation it
stands a differential, whereas in the right-hand side it stands an unclosed
form that is not a differential.
Since the evolutionary relation is not identical, from this relation
one cannot get the state differential [pic] that may point to the
equilibrium state of the material system. This means that the material
system state is nonequilibrium.
To the nonequilibrium state it leads everything that makes a
contribution into the commutator of the form [pic]. .
From the analysis of coefficients of the form [pic] one can see
that the development of instability is caused by not a simply connectedness
of the flow domain, nonpotential external (for each local domain of the
gas dynamic system) forces, a nonstationarity of the flow, transport
phenomena. {In common case on the gas dynamic instability it will effect
the thermodynamic, chemical, oscillatory, rotational, translational
nonequilibrium}.
All these factors lead to emergence of internal forces, that is, to
nonequilibrium and to development of various types of instability.
And yet for every type of instability one can find an appropriate term
giving contribution into the evolutionary form commutator, which is
responsible for this type of instability. Thus, there is an unambiguous
connection between the type of instability and the terms that contribute
into the evolutionary form commutator in the evolutionary relation. {In the
general case one has to consider the evolutionary relations that correspond
to the balance conservation laws for angular momentum and mass as well}.

3. the transition from the nonequilibrium state of the system to the
locally equilibrium state

The locally equilibrium state corresponds the state differential that is a
closed form. The transition from evolutionary differential form [pic] to
closed form, that would correspond to the transition from the
nonequilibrium state of the system to the locally equilibrium state, is
possible only as the degenerate transform, i.e. the transform that does not
conserve the differential.
To the degenerate transform it must correspond a vanishing of some
functional expressions. Such functional expressions may be Jacobians,
determinants, the Poisson brackets, residues and others. It is obvious that
the condition of degenerate transform has to be due to the gas dynamic
system properties. This may be, for example, the availability of any
degrees of freedom in the gas dynamic system.
If the transform is degenerate, from the unclosed evolutionary form it
can be obtained a differential form closed on some structure
(pseudostructure) that is a differential. On the pseudostructure
evolutionary relation (9) transforms into the identical relation.
The identical relation obtained from the nonidentical evolutionary relation
under degenerate transform integrates the state differential and the closed
(inexact) exterior differential form.
The availability of the state differential indicates that the material
system state becomes a locally equilibrium state. The availability of the
exterior closed on the pseudostructure differential form means that the
physical structure is present.
This shows that the transition of material system into the locally
equilibrium state is accompanied by the origination of physical structures.

The gas dynamic formations that correspond to these physical structures
are waves, vortex, shock waves, turbulent pulsations and so on.
Characteristics of the formation (intensity, vorticity, absolute and
relative speeds of propagation of the formation) are determined by the
evolutionary form and its commutator, by closed forms obtained and by the
material system characteristics
Let as analyze which types of instability and what gas dynamic
formation can originate under given external action.
1). Shock, break of diaphragm and others. The instability originates
Because of nonstationarity. The last term in equation (4) gives a
contribution into the commutator. In the case of ideal gas whose flow is
described by equations of the hyperbolic type the transition to the locally
equilibrium state is possible on the characteristics and their envelopes.
The corresponding structures are weak shocks and shock waves.
2). Flow of ideal gas around bodies. The instability develops because of
the multiple connectedness of the flow domain and a nonpotentiality of the
body forces. The contribution into the commutator comes from the second and
third terms of the right-hand side of equation (4). Since the gas is ideal
one and [pic], that is, there is no contribution into the each fluid
particle, an instability of convective type develops. For [pic] ([pic] is
the velocity of the gas particle, [pic] is the speed of sound) a set of
equations of the balance conservation laws belongs to the hyperbolic type
and hence the transition to the locally equilibrium state is possible on
the characteristics and on the envelopes of characteristics as well, and
weak shocks and shock waves are the structures of the system. If [pic]
when the equations are of elliptic type, such a transition is possible only
at singular points. The structures emerged due to a convection are of the
vortex type. Under long acting the large-scale structures can be produced.
3). Boundary layer. Contributions into the commutator produce the second
term in the right-hand side of equation (4) and the second and third terms
in expression (8). The transition to the locally equilibrium state is
allowed at singular points. Because in this case [pic], that is, the
external exposure acts onto the gas particle separately, the development of
instability and the transitions to the locally equilibrium state are
allowed only in an individual fluid particle. Hence, the structures emerged
behave as pulsations. These are the turbulent pulsations.
Studying the instability on the basis of the analysis of entropy
behavior was carried out in the works by Prigogine and co-authors [4]. In
that works entropy was considered as the thermodynamic function of state
(though its behavior along the trajectory was analyzed). By means of such
state function one can trace the development (in gas fluxes) of the
hydrodynamic instability only. To investigate the gas dynamic instability
it is necessary to consider entropy as the gas dynamic state function, i.e.
as a function of the space-time coordinates. Whereas for studying the
thermodynamic instability one has to analyze the commutator constructed by
the mixed derivatives of entropy with respect to the thermodynamic
variables, for studying the gas dynamic instability it is necessary to
analyze the commutators constructed by the mixed derivatives of entropy
with respect to the space-time coordinates.
It is commonly believed that the instability is an emergence of any
structures in the gas dynamic flow. From this viewpoint the laminar
boundary layer is regarded as stable one, whereas the turbulent layer
regarded as unstable layer. However the laminar boundary layer cannot be
regarded as a stable one because of the fact that due to the not simple
connectedness of the flow domain and the transport processes the
instability already develops although any structures do not originate. In
the turbulent boundary layer the emergence of pulsations is the transition
to the locally equilibrium state, and the pulsations themselves are local
formations. The other matter, due to the global nonequilibrium the locally
equilibrium state is broken up and the pulsations weaken.
In conclusion it should be said a little about modelling instable
flows. As it is known, some authors tried to account for the development of
instability by means of improving the equations modelling the balance
conservation laws (for example, by introducing the high-order moments) or
by introducing additional equations. However, such attempts give no
satisfactory results. To describe the nonequilibrium flow and the emergence
of the gas dynamic structures (waves, vortices, turbulent pulsations) one
must add the evolutionary relation obtained from the balance conservation
law equations to the balance conservation law equations. Under numerical
modeling the gas flows one has to trace for the transition from the
evolutionary nonidentical relation to the identical relation (for the
transition from an evolutionary unclosed form to an exterior closed form),
and this will point to the emergence of a certain physical structure.

REFERENCES

1. Clark J.F., Machesney ~M., The Dynamics of Real Gases. Butterworths,
London, 1964.
2. Haywood R.W., Equilibrium Thermodynamics. Wiley Inc. 1980.
3. Liepman H.W., Roshko A., Elements of Gas Dynamics. Jonn Wiley, New York,
1957.
4. Glansdorff P., Prigogine I. Thermodynamic Theory of Structure, Stability
and Fluctuations. Wiley, N.Y., 1971.