On some mathematical model of turbulent flow with intensive self-mixing

This paper presents a new possibility of mathematical modeling of turbulent flow with intensive self-mixing. The flow is described in this model by a family of mappings Φt,to:ω→S(t, to,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Phi_{t, t^o} : \omega \rightarrow S(t,t^o, \omega)}$$\end{document} ; ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document} , S⊂R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S \subset \mathbb{R}^3}$$\end{document}, t > to such that, in general, ω1∩ω2=∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega^1 \cap \omega^2 = \emptyset}$$\end{document} does not imply S(t,to,ω1∩S(t,to,ω2=∅)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S(t,t^o, \omega^1 \cap S(t,t^o, \omega^2 = \emptyset)}$$\end{document} The model allows a new approach to certain problems of turbulent flow. For example, there are possibilities to introduce some models of turbulent chemical reactors, or description of the unpredictable explosions of turbulence in the calm laminate flow. In this model we introduce mathematical tools that allow definition of various mixing states of the fluid. It is possible to formulate, in appropriate mathematical language, the idea that the given fluid can have various mixing states. It is also possible to determine the influence of a given mixing state on the flow parameters such as velocity field and density.


General features of the model
For the balls ω ε (x) = {y ∈ R 3 : |y − x| < ε}, ε > 0, consider the sets V ε (x, t) ⊂ R 3 of velocities of fluid particles moving at time t in ω ε (x). The velocity sets may be measured. The laminar model and our model may be shortly characterized in the following way. such that for numbers Δ > 0, ε > 0 small enough the following approximation holds in the sense of measure. Based on assumptions (I) and (II), we can formulate the integral conservation laws of mass, impulse, energy and impulse momentum. Moreover, we obtain the equivalent closed integrodifferential system. In this way we obtain a model of fluid flow in which different fluid portions are mixed and loosing their identity.
To this end, we have to introduce first the notion of the physical αquantities. We begin with the definition of the α-density. This is a nonnegative function with the following properties. For the mass portion filling up ω at time t, the following equalities hold: where (x, t) is the usual density, and κ > 0 is some scaling constant.
We construct an approximation of (x, t, α) explaining its physical sense. The sets where Δ > 0, ε > 0 small enough, are the minimal sets that contain at time t + Δ the whole fluid filling up ω ε (x) at time t. This fluid in S(t + Δ, t, ω ε (x)) at t + Δ has the density g(y) for y ∈ S(t + Δ, t, ω ε (x)).
Using (1.2), we have and we obtain the following approximation of the α-density: In our model we use the general notion of the α-quantity. For example, we define the α-impulse as α (x, t, α), so that the impulse of the fluid portion m(ω, t) is equal to Given the α-quantities, we may obtain the observed mean Euler velocity v(x, t) of the flow: In almost all physical situations one may take some ball A ⊂ R 3 , for which we know that (x, t, α) = 0 for α ∈ R 3 \A, and we may integrate over A only.

The mass conservation law
We have to introduce the fluid portions m(ω, a, t), a ⊂ A, which are parts of the fluid in ω moving at time t with velocities α ∈ a ⊂ A. We have Moreover, we introduce the mass mixer : which describes the following mixing processes: In the mixing process if one of the portions m(ω, a, t) or m(ω, b, t) is underlined, then it means that we ask amount of mass (positive or negative equal to the integrals in the frames on the right) which is transported to the underlined portion in the unit time . It turns out that The following three mixing processes (2.2), (2.3), (2.4) should be taken into account in the mass conservation law for the portion m(ω, a, t), where a ⊂ A: The amount of mass transported in the unit time in this processes to the portion m(ω, a, t) is respectively equal to where n(x) is the external normal unit vector to the boundary ∂ω. In (2.2) we used the properties (2.1) of the mass mixer M (x, t, α, β). We consider the process (2.3) like in the laminar model without any mixer, so as in this process we have on both sides of the boundary ∂ω, fluids with the same kinematical characteristics. In the process (2.3) we take into account the mass transport of the diffusion type with the constant E > 0. The process (2.4) is a mixing process of two fluid portions with different kinematical characteristics across the boundary ∂ω. Therefore, we have to introduce the boundary mass mixer B: It describes the amount of mass transported in the unit time to m(ω, a, t) in the process (2.4).

Vol. 15 (2014)
Model of turbulent fl ow 463 Finally, taking together (2.5), (2.6) and (2.7), we obtain the following integral mass conservation law for the portion m(ω, a, t): where ω ⊂ R 3 , a ⊂ A. We shall use the following simple Localization Theorem.
Localization Theorem. Let D : R 7 → R and F : R 10 → R be continuous functions. Then the condition where ω ⊂ R 3 , a ⊂ A, is equivalent to the following system of equations: Applying this theorem to the integral conservation law (2.8), we obtain the following integrodifferential system equivalent to the integral law: (2.12)

The impulse conservation law
We introduce the following impulse mixer J(x, t, α, β). It is the function where αM (x, t, α, β) describes the transport of impulse caused by the mass transport, and i(x, t, α, β) describes other types of impulse changes. The physical sense of the mixer J(x, t, α, β) is the following. The mixer J(x, t, α, β) describes the amount of impulse transported in a unit time to the portion m(ω, a, t) in the process This amount is equal to On the other hand, this change is equal to the force acting on m(ω, a, t). This force is the sum of the following forces: The forces acting on m(ω, a, t) by fluid portions: m(ω, A\a, t), m(R 3 \ω, a, t), m(R 3 \ω, A\a, t)

+
The external forces acting on m(ω, a, t) Let us determine these forces. The forces acting on m(ω, a, t) are equal to the amount of impulse transported in a unit time to m(ω, a, t).
In the process m(ω, a, t) ⇐⇒ m(ω, A\a, t) where we introduced the boundary impulse mixer J B (x, t, α, β). It is a matrix J B : R 10 → R 3 ×R 3 . Finally, let us take into account the processes (2.2), (2.3), (2.4), and the external forces. Then, after changing the surface integrals into volume ones, we obtain the following integral form of the impulse conservation law for the portion m(ω, a, t):

2)
Vol. 15 (2014) Model of turbulent fl ow 465 where ω ⊂ R 3 , a ⊂ A and is the external force acting on m(ω, a, t). In the case of the earth gravitation field, we have f (x, t, α) = g (x, t, α), g = const.
Applying the Localization Theorem to (3.2), we obtain the following integrodifferential system equivalent to the integral impulse conservation law:

The energy conservation law
Let us introduce the α-inner energy of the flow. This is a function ε(x, t, α) = 0 for α ∈ A\V (x, t).
Denote by e(ω, a, t) the inner energy of the fluid portion m(ω, a, t). Then the physical sense of the α-energy is explained by the formula Denoting by e(ω, t) the inner energy of the fluid portion m(ω, t), we obtain the following equality: Hence, we get The force F (ω, a, t) acting on the portion m(ω, a, t) is equal to the increase of the impulse of m(ω, a, t) in the unit time. Hence, from the impulse conservation law we obtain The increase of the energy of the portion m(ω, a, t) in the unit time is equal to the power due to the force F (ω, a, t). Let us evaluate it. Divide ω and a into small parts Δω ⊂ ω and Δa ⊂ a, and notice that F (ω, a, t) acts on m(ω, a, t) in such a way that on the small parts m (Δω, Δa, t) ⊂ m(ω, a, t), the following forces are acting: The power due to F (ω, a, t) is the sum of the powers P (Δω, Δa, t) due to the forces F (Δω, Δa, t), where Δω ⊂ ω and Δa ⊂ a. For o x ∈ Δω, o α ∈ Δa, we obtain the following approximation for small Δt: Finally, for the whole power P (ω, a, t) due to the force F (ω, a, t), we obtain P (ω, a, t) = where ω ⊂ R 3 , a ⊂ A. Putting a = A and taking into account (4.1), we obtain the following relation: In this way, one determines the mean inner energy ε(x, t) for a given α-density (x, t, α).

Vol. 15 (2014)
Model of turbulent fl ow 467 Applying the Localization Theorem to (4.2), we obtain the following integrodifferential system equivalent to the integral energy conservation law: (4.4)

The impulse momentum conservation law
Applying the principle The derivative ∂ t of the impulse momentum = The momentum of acting forces and our impulse conservation law, we can formulate the impulse momentum conservation law for the fluid portion m(ω, a, t).
If we denote F ∧H = ((F ∧H) 1 , (F ∧H) 2 , (F ∧H) 3 ) for F, H ∈ R 3 , then we obtain the following integrodifferential system equivalent to the integral impulse momentum conservation law: where the matrix J B is denoted as , i,j = 1, 2, 3.

The constitutive relation and the full closed system of the model
In order to formulate the constitutive relation, let us express approximately the boundary mixer B(x, t, α, β) as a function of the mixer M (x, t, α, β). To