**Statistical Modeling for the Energy-Containing Structure of Turbulent Flows**

Yuriy Nuzhnov

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67844

## Abstract

The development of statistical theory for the energy-containing structure of turbulent flows, taking the phenomenon of internal intermittency into account, is proposed, and new differential equations for conditional means of turbulent and nonturbulent fluid flow are established. Based on this fact, a new principle of constructing mathematical models is formulated as the method of autonomous statistical modeling of turbulent flows, ASMTurb method. Testing of the method is attained on the example of constructing a mathematical model for the conditional means of turbulent fluid flow in a turbulent mixing layer of co-current streams. Test results showed excellent agreements between the predictions of the ASMTurb model and known experimental data.

Keywords: turbulence, statistical modeling, intermittency, ASMTurb method

## 1. Introduction

The Reynolds-averaged Navier-Stokes equations (RANS) method does not take the intermittency of turbulent and nonturbulent fluid into consideration. As a result, this method allows us to model only the unconditional averages of a turbulent flow and does not provide a description of the conditional averages for each of the intermittent region, taking place in a turbulent stream. At the same time, the intermittency is an inherent property of such flows and that is why the conditional average modeling is necessary, for example see [1–4]. The phenomenon of intermittency (hydrodynamic intermittency) represents an interleaving process of the spacetime domains of the flow, hydrodynamic structures of which are different. As is known, such domains contain so-called "turbulent" and "nonturbulent" fluid [1]. In this connection, the turbulent fluid contains a hierarchy of all possible scales and amplitudes of the fluctuations (pulsations) of hydrodynamic values, i.e., the whole spectrum of wavenumbers, while the

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nonturbulent fluid may contain only the large-scale fluctuations or absolutely does not contain any ones (when the nonturbulent fluid is far away from the mixing layer). The main purpose of this chapter is to justify a new method of statistical modeling of turbulent flows as the ASMTurb method, which enables to construct mathematical models of such flows with a high efficiency. The presented ASMTurb method, declared in [5], fundamentally differs from the previously proposed (for example, see Refs. [6–8]) in that it is based on the conditional statistical averaging of the Navier-Stokes equations, as applied to each of the intermittent region of turbulent flow, while the generating process of the turbulent fluid begins in a thin superlayer between turbulent and nonturbulent fluid and finishes in separate small areas, involved inside the turbulent flow. The first attempts to substantiate such an approach [5] have been presented previously [9, 10]. However, the deficiency of the mathematical body of statistical hydrodynamics under the intermittency conditions makes such an approach vulnerable. In this regard, we need primarily to develop a mathematical body for statistical modeling of turbulent flows.

#### 2. Development of the statistical modeling theory

A spectacular example of the intermittent turbulent flow is the flow in the mixing layer of cocurrent streams, Figure 1. With that at point x = x0, with the course of time, will be observed an interleaving of the turbulent and nonturbulent fluid. The behavior of the instantaneous longitudinal velocity u(x, t) in the flow range with strong intermittency at the point x<sup>0</sup> is shown in Figure 2. As it seen, the structure of the turbulent fluid flow is fundamentally different from the structure of the nonturbulent fluid flow (the nonturbulent fluid involvement is shown with the arrows in Figure 1). It is evident that the behavior of any other hydrodynamic variable f(x, t) will be the same. It is important to note that the conditional averaging of variable f(x, t) is interpreted as the result of the averaging operation, referring only to as the turbulent (r = t) or nonturbulent (r = n) fluid, i.e., for the conditional time averaging

$$\overline{f(\mathbf{x},t)}^r = \lim\_{\mathbf{t}\_0 \to \mathbf{s}} \frac{1}{\mathbf{t}\_r} \Big| f(\mathbf{x}, t; \tau\_0) dt, \ r = t, \ n \tag{1}$$

Figure 1. A sketch of the turbulent and nonturbulent fluid in the mixing layer of co-current streams. Here Dt is the region with the turbulent fluid and Dn– is the region with the nonturbulent fluid.

nonturbulent fluid may contain only the large-scale fluctuations or absolutely does not contain any ones (when the nonturbulent fluid is far away from the mixing layer). The main purpose of this chapter is to justify a new method of statistical modeling of turbulent flows as the ASMTurb method, which enables to construct mathematical models of such flows with a high efficiency. The presented ASMTurb method, declared in [5], fundamentally differs from the previously proposed (for example, see Refs. [6–8]) in that it is based on the conditional statistical averaging of the Navier-Stokes equations, as applied to each of the intermittent region of turbulent flow, while the generating process of the turbulent fluid begins in a thin superlayer between turbulent and nonturbulent fluid and finishes in separate small areas, involved inside the turbulent flow. The first attempts to substantiate such an approach [5] have been presented previously [9, 10]. However, the deficiency of the mathematical body of statistical hydrodynamics under the intermittency conditions makes such an approach vulnerable. In this regard, we need primarily to develop a mathematical body for statistical modeling of turbulent flows.

A spectacular example of the intermittent turbulent flow is the flow in the mixing layer of cocurrent streams, Figure 1. With that at point x = x0, with the course of time, will be observed an interleaving of the turbulent and nonturbulent fluid. The behavior of the instantaneous longitudinal velocity u(x, t) in the flow range with strong intermittency at the point x<sup>0</sup> is shown in Figure 2. As it seen, the structure of the turbulent fluid flow is fundamentally different from the structure of the nonturbulent fluid flow (the nonturbulent fluid involvement is shown with the arrows in Figure 1). It is evident that the behavior of any other hydrodynamic variable f(x, t) will be the same. It is important to note that the conditional averaging of variable f(x, t) is interpreted as the result of the averaging operation, referring only to as the turbulent (r = t) or

> 1 τr τðr

0

Figure 1. A sketch of the turbulent and nonturbulent fluid in the mixing layer of co-current streams. Here Dt is the region

fðx, t; τ0Þdt, r ¼ t, n ð1Þ

2. Development of the statistical modeling theory

100 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

nonturbulent (r = n) fluid, i.e., for the conditional time averaging

fðx,tÞ r ¼ lim τ0!∞

with the turbulent fluid and Dn– is the region with the nonturbulent fluid.

Figure 2. Behavior of the instantaneous longitudinal velocity in different regions, interleaving at the preset point x = x<sup>0</sup> in Figure 1: (a) unconditional velocity u = u(x, t), (x, t) ∈ G; (b) "cross-linking" of the velocity over the turbulent fluid domain, ut <sup>¼</sup> utðx, tÞj<sup>I</sup>ðx,tÞ¼<sup>1</sup>, <sup>ð</sup>x, tÞ<sup>∈</sup> Gt; (c) "cross-linking" of the velocity over the nonturbulent fluid domain, un <sup>¼</sup> unðx, tÞj<sup>I</sup>ðx,tÞ¼<sup>0</sup>, <sup>ð</sup>x, tÞ<sup>∈</sup> Gn. Here, <sup>G</sup> <sup>=</sup> Gt <sup>+</sup> Gn, <sup>u</sup> is the total time average, <sup>u</sup><sup>t</sup> and <sup>u</sup><sup>n</sup> is the conditional time means, ts is the time point of observing the interfacial joint between the turbulent and nonturbulent flow domain in which cross-linking is carried out.

where fðx,tÞ r � f <sup>r</sup>ðx,tÞ r , f t <sup>ð</sup>x, tÞ ¼ <sup>f</sup>ðx, tÞj<sup>I</sup>¼1, <sup>f</sup> <sup>n</sup>ðx, tÞ ¼ <sup>f</sup>ðx, tÞj<sup>I</sup>¼0, <sup>I</sup> <sup>¼</sup> <sup>I</sup>ðx, t<sup>Þ</sup> is the intermittency function and τ<sup>0</sup> ¼ τ<sup>t</sup> þ τn. At that the total average is

$$\overline{f(\mathbf{x},t)} = \gamma(\mathbf{x})\overline{f(\mathbf{x},t)}^t + \left(1 - \gamma(\mathbf{x})\right)\overline{f(\mathbf{x},t)}^n \tag{2}$$

and γðxÞ ¼ Iðx, tÞ is an intermittency factor. At the same time in the theory of statistical modeling are used the statistical characteristics, i.e., instead of the averaging operation of Eq. (1) is required the operation of statistical averaging.

To construct the mathematical model, first of all, it is necessary to determine what kind of statistical characteristics are the most suitable for modeling. In the classical RANS method, such characteristics are the unconditional means. In the methods, taking the intermittency into consideration, such characteristics are the conditional means of each intermittent region of turbulent flow. But in this case, it requires the development of a theory of statistical hydrodynamics under the conditions of intermittency.

The mathematical body of this theory may be developed from both the theoreticalprobabilistic approach, based on the N-th repetition of the turbulent flow experiment [4, 11], and the theoretical-set approach [12, 13], which elementary events can be represented as a some set in the generalized space of the specifically considered turbulent flow. The advantage of the theoretical-set approach is that it can be implemented in the experimental research.

#### 2.1. The mathematical body of statistical hydrodynamics

Getting started to the development of the mathematical body of statistical hydrodynamics in terms of intermittency, first of all we need to create a probability space ðΩ, F, PÞ of a random field of any hydrodynamic value as a random process in the generalized physical space of turbulent flow, where Ω – is the sample space, F – is σ-algebra of subsets, P– is the probability measure in F.

#### 2.1.1. The introduction of the sample space into the body of statistics

For the introduction of the sample space Ω, we consider the behavior of the value of f(x, t), measured by the sensor at the point x = x<sup>0</sup> of statistically stationary turbulent flow with strong intermittency, i.e., when γðxÞ ffi 0:5, see Figure 3. According to Figure 2, function f(x, t) forms a random continuous field in the space G ¼ D � ½0, τ<sup>0</sup> �. Hence it follows that at the point x<sup>0</sup> ∈ D we have a continuous random varying function of time f(x0, t). Let the measurements of f = f(x0, t) were carried out in a fairly wide range (order to the averaged statistical value of this function was stable) of the observation time t = [0, τ0]. It allows us to form an ensemble of values Ω as the onepoint countable set of elementary events f, if we split the range of values of function f(x0, t) at

Figure 3. Illustration of statistical averaging of the instantaneous hydrodynamic variable. Here f = {f(x, t)} is the range of function f(x, t) at the point x = x0; 〈f〉 is the total statistical average; 〈f〉<sup>t</sup> and 〈f〉<sup>n</sup> is the conditional statistical mean in each of the intermittent media of the turbulent flow; f <sup>0</sup> ¼ f � 〈f〉, f 0 <sup>t</sup> ¼ f <sup>t</sup> � 〈f〉t, and f 0 <sup>n</sup> ¼ f <sup>n</sup> � 〈f〉<sup>n</sup> are fluctuations (pulsations), measured from its own statistical means; τ<sup>0</sup> is the period of averaging time, sufficient to ensure sustainable statistical mean of values f; Δτ\* is the characteristic time of the superlayer observation, I = I(x, t) is the intermittency function of the turbulent fluid domain.

sufficiently small intervals Δf, and the range [0, τ0] at sufficiently small intervals Δt, Figure 2. Having fixed a certain value of function fi in each of the selected intervals Δf we come to the Lebesgue integral in terms of the set theory, formed in the physical space. Indeed, in the i-th layer there are Ni sampled values fi in the form of shaded elementary cells Δf Δt, Figure 2 (the selection of one particular value fi from these cells plays no special role due to their small value). The total number of cells N = τ<sup>0</sup> / Δt and represented as the ensemble of values f, and also in the limit Δf ! 0 and Δt ! 0 this set will be dense, and the numerical value of f will be an element of this set, i.e., an elementary event.

In other words, for every fixed point x = x0, the total number of all sample values fi forms a random continuous field of values f ∈ Ω in the physical space G = D � [0, τ0]. As a result, we come to a random process in the Borel space, in which a random variable f(x, t) takes all values of f = {f(x, t)}, which are the elements of continuous set

$$\mathcal{Q} = \{ f : f\_{\text{min}} \le f(\mathbf{x}, t) < f\_{\text{max}'} N \big|\_{\mathbf{x} = \mathbf{x}\_0} = \lim\_{\tau\_0 \to \infty} \tau\_0 / \Delta t, \ (\mathbf{x}, t) \in \mathcal{G} \ \} \tag{3}$$

So, from the physical space G with the hydrodynamic quantity f(x, t) we went to sample space Ω, elements of which are a set of values of f = {f(x, t)}, i.e.

$$f(\mathbf{x},t) \to f = \{ f(\mathbf{x},t) \}; \\ f(\mathbf{x},t)|\_{l=1} \to f\_t = \{ f(\mathbf{x},t)|\_{l=1} \}; \\ f(\mathbf{x},t)|\_{l=0} \to f\_n = \{ f(\mathbf{x},t)|\_{l=0} \} \tag{4}$$

$$
\overline{f(\mathbf{x},t)} \to \langle f \rangle; \, \overline{f(\mathbf{x},t)}^t \to \langle f \rangle\_{\nu}, \, \overline{f(\mathbf{x},t)}^n \to \langle f \rangle\_n \tag{5}
$$

$$f'(\mathbf{x},t) \to f' = \{f(\mathbf{x},t) - \overline{f(\mathbf{x},t)}\}; \\ f'\_r(\mathbf{x},t) \to f'\_r = \{f\_r(\mathbf{x},t) - \overline{f(\mathbf{x},t)}'\} \tag{6}$$

Now we need develop the apparatus of statistics together with the operations of statistical averaging of the hydrodynamic quantities. For this we represent the apparatus of statistics based on a formal using of the probability density function (one-point PDF) of some hydrodynamic quantity f = {f(x, t)}. At that the intermittency function I = I(x, t) will be used to obtain conditional one-point statistics.

#### 2.1.2. The introduction of the algebra of events and PDFs

turbulent flow. But in this case, it requires the development of a theory of statistical hydrody-

The mathematical body of this theory may be developed from both the theoreticalprobabilistic approach, based on the N-th repetition of the turbulent flow experiment [4, 11], and the theoretical-set approach [12, 13], which elementary events can be represented as a some set in the generalized space of the specifically considered turbulent flow. The advantage of the theoretical-set approach is that it can be implemented in the experimental research.

Getting started to the development of the mathematical body of statistical hydrodynamics in terms of intermittency, first of all we need to create a probability space ðΩ, F, PÞ of a random field of any hydrodynamic value as a random process in the generalized physical space of turbulent flow, where Ω – is the sample space, F – is σ-algebra of subsets, P– is the probability

For the introduction of the sample space Ω, we consider the behavior of the value of f(x, t), measured by the sensor at the point x = x<sup>0</sup> of statistically stationary turbulent flow with strong intermittency, i.e., when γðxÞ ffi 0:5, see Figure 3. According to Figure 2, function f(x, t) forms a random continuous field in the space G ¼ D � ½0, τ<sup>0</sup> �. Hence it follows that at the point x<sup>0</sup> ∈ D we have a continuous random varying function of time f(x0, t). Let the measurements of f = f(x0, t) were carried out in a fairly wide range (order to the averaged statistical value of this function was stable) of the observation time t = [0, τ0]. It allows us to form an ensemble of values Ω as the onepoint countable set of elementary events f, if we split the range of values of function f(x0, t) at

Figure 3. Illustration of statistical averaging of the instantaneous hydrodynamic variable. Here f = {f(x, t)} is the range of function f(x, t) at the point x = x0; 〈f〉 is the total statistical average; 〈f〉<sup>t</sup> and 〈f〉<sup>n</sup> is the conditional statistical mean in each of

0

measured from its own statistical means; τ<sup>0</sup> is the period of averaging time, sufficient to ensure sustainable statistical mean of values f; Δτ\* is the characteristic time of the superlayer observation, I = I(x, t) is the intermittency function of the

<sup>t</sup> ¼ f <sup>t</sup> � 〈f〉t, and f

0

<sup>n</sup> ¼ f <sup>n</sup> � 〈f〉<sup>n</sup> are fluctuations (pulsations),

<sup>0</sup> ¼ f � 〈f〉, f

namics under the conditions of intermittency.

measure in F.

the intermittent media of the turbulent flow; f

turbulent fluid domain.

2.1. The mathematical body of statistical hydrodynamics

102 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

2.1.1. The introduction of the sample space into the body of statistics

Let us introduce a one-point probability density function p(f) = p(f; x, t) into the body of statistics. According to the Kolmogorov axioms [12], it can be carried out via the Lebesgue-Stieltjes integral:

$$P(\Omega) = \int\_{\Omega} p(f) df \tag{7}$$

where pðfÞ ¼ lim N!∞ <sup>p</sup>ðf; f <sup>∈</sup> <sup>Ω</sup>, N <sup>Þ</sup> and <sup>ð</sup> Ω pðfÞdf ¼ 1.

For introduction of the algebra of events, we suppose that the space Ω, defined by Eq. (3), contains two independent subspaces (subsets)

$$
\left.\Omega\_1 = \Omega\right|\_{\mathcal{I}\_1 = 1^\nu} \quad \Omega\_2 = \Omega|\_{\mathcal{I}\_2 = 1} \tag{8}
$$

i.e., the generalized set Ω = Ω<sup>1</sup> + Ω2, and we have F as σ-algebra of the subsets. The indicators of these subsets are the characteristic functions

$$\mathcal{I}\_1 = \begin{cases} 1 \text{ if } f \in \Omega\_1 \\ 0 \text{ otherwise} \end{cases}, \mathcal{I}\_2 = \begin{cases} 1 \text{ if } f \in \Omega\_2 \\ 0 \text{ otherwise} \end{cases} \tag{9}$$

at that the set of values f, belonging to the super-layer, is excluded. In the results, we have the Borel algebra subsets of the set Ω with the Kolmogorov axioms, which according to the total probability formula gives

$$\mathcal{P}(\mathcal{Q}) = \sum\_{k=1}^{2} \mathcal{P}\{\mathcal{Q}|\_{\mathcal{I}\_k=1}\} \mathcal{P}\{\mathcal{J}\_k = 1\} \tag{10}$$

where PfJ <sup>k</sup> ¼ 0g ¼ 0 as an impossible event.

#### 2.2. Statistical averaging of hydrodynamic quantities

#### 2.2.1. Applied to the intermittent turbulent flows

For the intermittent turbulent flows, the sample sets, which we designated as Ω<sup>t</sup> ¼ Ωj <sup>J</sup> <sup>1</sup>¼<sup>1</sup> and Ω<sup>n</sup> ¼ Ωj <sup>J</sup> <sup>2</sup><sup>¼</sup>1, are the set of values of hydrodynamic variable, belonging to the turbulent and nonturbulent fluid of turbulent flow. Indicators of these sets are the functions J <sup>1</sup> ¼ J <sup>t</sup> and J <sup>2</sup> ¼ J <sup>n</sup>, while PfJ <sup>t</sup> ¼ 1g ¼ γ<sup>t</sup> and PfJ <sup>n</sup> ¼ 1g ¼ γ<sup>n</sup> are the measures of these sets with the condition γ<sup>t</sup> þ γ<sup>n</sup> ¼ 1, and represent the intermittency factors as the probability of observing the turbulent and nonturbulent fluid at the point x of turbulent flow, i.e., γ<sup>t</sup> ¼ γ<sup>t</sup> ðxÞ and γ<sup>n</sup> ¼ γnðxÞ. Now, according to Eq. (10),

$$\mathbf{P}(\Omega) = \boldsymbol{\gamma}\_t \mathbf{P}(\Omega\_t) + \boldsymbol{\gamma}\_n \mathbf{P}(\Omega\_n) \tag{11}$$

where P(Ωt) и P(Ωn) – conditional random set of value f, belonging to the turbulent Ω<sup>t</sup> ¼ ff <sup>t</sup> g and nonturbulent Ω<sup>n</sup> ¼ ff <sup>n</sup>g fluid at the point x; the values γ<sup>t</sup> ¼ γ<sup>t</sup> ðxÞ and γ<sup>n</sup> ¼ γnðxÞ, while the one-point PDF

$$p(f) = \gamma\_t p\_t(f) + \gamma\_n p\_n(f) \tag{12}$$

where pt ðfÞ ¼ pt ðf ; x, tÞ, ðx, tÞ∈ Gt and pnðfÞ ¼ pnðf ; x, tÞ, ðx, tÞ∈ Gn represent the conditional one-point PDFs. As it turns out, a PDF may have or not to have an explicit dependence on x. In actual fact, if the flow is intermittent, it has a dual structure [1] and in the generalized set we have Ω = Ω<sup>t</sup> + Ω<sup>n</sup> so that the measures of sample sets γ<sup>t</sup> and γ<sup>n</sup> are depend on x; if the flow is not intermittent (when the phenomenon of intermittency is not considered) it occurs in a "single" space as a set of elementary events Ω = ΩR, the measure of which does not depend on x. In the case of the explicit dependence, we denote the PDF p(f) in Eq. (7) as

$$P(f) = \lim\_{N \to \infty} p\left(f; \ f \in (\Omega\_t + \Omega\_n)\_\prime \ N, \ \mathbf{x}\right) \tag{13}$$

and we call this function as the "total" PDF, and the flow—flow of the "intermittent" continuous media with turbulent and nonturbulent fluid. In the absence of such dependence, we denote it as

$$p\_{\mathcal{R}}(f) = \lim\_{N \to \infty} p(f; f \in \mathcal{Q}\_{\mathcal{R}} \text{ N}) \tag{14}$$

and called the "unconditional" PDF pR(f), and the flow—flow of the "nonintermittent" continuous medium, which is modeled by the RANS method. The explicit dependence of the PDF P(f) Eq. (13) on the coordinates creates certain difficulties in its using in the statistical modeling and also leads to the necessity of introducing in the theory of statistical hydrodynamics the conditional PDF for the hydrodynamic characteristics of turbulent and nonturbulent media.

So, to perform the conditional averaging of the instantaneous characteristics of the flow, we introduce into statistical body the conditional PDF, i.e., the CPDFs:

$$p\_t(f) = p(f|\_{l=1}), \ p\_n(f) = p(f|\_{l=0}) \tag{15}$$

with the indicator (characteristic) function of the turbulent fluid

Ω<sup>1</sup> ¼ Ωj

104 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

<sup>J</sup> <sup>1</sup> <sup>¼</sup> <sup>1</sup> if f <sup>∈</sup> <sup>Ω</sup><sup>1</sup>

<sup>P</sup>ðΩÞ ¼ <sup>X</sup> 2

k¼1

For the intermittent turbulent flows, the sample sets, which we designated as Ω<sup>t</sup> ¼ Ωj

ing the turbulent and nonturbulent fluid at the point x of turbulent flow, i.e., γ<sup>t</sup> ¼ γ<sup>t</sup>

PðΩÞ ¼ γ<sup>t</sup>

pðfÞ ¼ γ<sup>t</sup>

on x. In the case of the explicit dependence, we denote the PDF p(f) in Eq. (7) as

and nonturbulent Ω<sup>n</sup> ¼ ff <sup>n</sup>g fluid at the point x; the values γ<sup>t</sup> ¼ γ<sup>t</sup>

<sup>J</sup> <sup>2</sup><sup>¼</sup>1, are the set of values of hydrodynamic variable, belonging to the turbulent

and nonturbulent fluid of turbulent flow. Indicators of these sets are the functions J <sup>1</sup> ¼ J <sup>t</sup> and J <sup>2</sup> ¼ J <sup>n</sup>, while PfJ <sup>t</sup> ¼ 1g ¼ γ<sup>t</sup> and PfJ <sup>n</sup> ¼ 1g ¼ γ<sup>n</sup> are the measures of these sets with the condition γ<sup>t</sup> þ γ<sup>n</sup> ¼ 1, and represent the intermittency factors as the probability of observ-

where P(Ωt) и P(Ωn) – conditional random set of value f, belonging to the turbulent Ω<sup>t</sup> ¼ ff <sup>t</sup>

pt

one-point PDFs. As it turns out, a PDF may have or not to have an explicit dependence on x. In actual fact, if the flow is intermittent, it has a dual structure [1] and in the generalized set we have Ω = Ω<sup>t</sup> + Ω<sup>n</sup> so that the measures of sample sets γ<sup>t</sup> and γ<sup>n</sup> are depend on x; if the flow is not intermittent (when the phenomenon of intermittency is not considered) it occurs in a "single" space as a set of elementary events Ω = ΩR, the measure of which does not depend

ðf ; x, tÞ, ðx, tÞ∈ Gt and pnðfÞ ¼ pnðf ; x, tÞ, ðx, tÞ∈ Gn represent the conditional

PfΩj

of these subsets are the characteristic functions

where PfJ <sup>k</sup> ¼ 0g ¼ 0 as an impossible event.

2.2.1. Applied to the intermittent turbulent flows

γ<sup>n</sup> ¼ γnðxÞ. Now, according to Eq. (10),

2.2. Statistical averaging of hydrodynamic quantities

probability formula gives

Ω<sup>n</sup> ¼ Ωj

the one-point PDF

ðfÞ ¼ pt

where pt

<sup>J</sup> <sup>1</sup>¼<sup>1</sup>, <sup>Ω</sup><sup>2</sup> <sup>¼</sup> <sup>Ω</sup><sup>j</sup>

<sup>0</sup> otherwise , <sup>J</sup> <sup>2</sup> <sup>¼</sup> <sup>1</sup> if f <sup>∈</sup> <sup>Ω</sup><sup>2</sup>

0 otherwise

i.e., the generalized set Ω = Ω<sup>1</sup> + Ω2, and we have F as σ-algebra of the subsets. The indicators

at that the set of values f, belonging to the super-layer, is excluded. In the results, we have the Borel algebra subsets of the set Ω with the Kolmogorov axioms, which according to the total

( (

<sup>J</sup> <sup>2</sup>¼<sup>1</sup> <sup>ð</sup>8<sup>Þ</sup>

<sup>J</sup> <sup>k</sup>¼<sup>1</sup>gPf<sup>J</sup> <sup>k</sup> <sup>¼</sup> <sup>1</sup>g ð10<sup>Þ</sup>

PðΩtÞ þ γnPðΩnÞ ð11Þ

ðfÞ þ γnpnðfÞ ð12Þ

ð9Þ

<sup>J</sup> <sup>1</sup>¼<sup>1</sup> and

ðxÞ and

ðxÞ and γ<sup>n</sup> ¼ γnðxÞ, while

g

$$I = \begin{cases} 1 \text{ if } f \in \Omega\_t \\ 0 \text{ if } f \in \Omega\_n \end{cases} \tag{16}$$

represents a probability of observing the turbulent flow at the given point x, i.e., it is the intermittency factor γ = γ(x). Now the expression for the "total" PDF in Eq. (12), by virtue of the fact that γ<sup>n</sup> = 1 � γ, is transformed into

$$P(f) = \gamma p\_t(f) + (1 - \gamma)p\_n(f) \tag{17}$$

with the explicit dependence on x, while the CPDF pt(f) and pn(f) obviously do not depend on x.

Now conduct the operations of statistical averaging of hydrodynamic quantities. These operations we will conduct with the help of a formal using of the PDFs, i.e., when a particular form of this function does not necessarily need to know.

#### 2.2.2. Operations of statistical averaging of the hydrodynamic quantity

The statistical averaging of the hydrodynamic quantity f(x, t) can be performed by a formal using of the PDF. The results of statistical averaging operation are the conditional statistical means when r = t for turbulent and r = n for nonturbulent fluid

$$\begin{aligned} \langle f \rangle\_r &= \int\_{\Omega\_r} f p\_r(f) df, \\ r &= t, \ n \end{aligned} \tag{18}$$

and also the total statistical average

$$
\langle f \rangle = \int\_{\Omega} f P(f) df \tag{19}
$$

which by virtue of the expression in Eq. (17) gives

$$
\langle f \rangle = \gamma \langle f \rangle\_t + (1 - \gamma) \langle f \rangle\_n \tag{20}
$$

At that by definition the value f <sup>t</sup> ¼ f j <sup>I</sup>¼<sup>1</sup> and <sup>f</sup> <sup>n</sup> <sup>¼</sup> <sup>f</sup> <sup>j</sup> <sup>I</sup>¼<sup>0</sup> and for the "pulsations" we have

$$f' = f - \langle f \rangle ; f'\_r = f\_r - \langle f\_r \rangle\_r ; f'|\_{l=1} = f'\_t + \langle f \rangle\_t - \langle f \rangle ; f'|\_{l=0} = f'\_n + \langle f \rangle\_n - \langle f \rangle \tag{21}$$

whence it follows that

$$
\langle f\_r \rangle\_r = \langle f \rangle\_r; \langle f\_r' \mathbf{g}\_r' \rangle\_r \neq \langle f' \mathbf{g}' \rangle\_r \tag{22}
$$

As is evident that the total average in Eq. (20) represents the statistical characteristic of a rather complex structure, while the unconditional mean, when in Eq. (18) we have r = R, is a characteristic of the "simplified" flow without considering effects of intermittency. At that 〈f〉<sup>R</sup> ffi 〈f〉 because the total average 〈f〉 does not contain the values of f belonging to the superlayer [16].

#### 2.2.3. Operations of statistical averaging of derivative of the hydrodynamic quantity

The statistical averaging of the derivative of hydrodynamic quantity ξ ¼ ∂f =∂x gives the following. In point of fact, on the one side using the joint CPDF prðf , ξÞ we have

$$\iint \frac{\partial f}{\partial \mathbf{x}} p\_r(f, \xi) d\xi d\xi = \frac{\partial}{\partial \mathbf{x}} \int \langle f|\_{\xi} \rangle\_r p\_r(\xi) d\xi = \frac{\partial \langle f \rangle\_r}{\partial \mathbf{x}} \tag{23}$$

because that in accordance with [14]

$$p\_r(f, \xi) = p\_r(f|\_{\xi}) p\_r(\xi)$$

and

$$\iint \left( \partial f p\_r(f,\xi) / \partial \mathbf{x} - f \partial p\_r(f,\xi) / \partial \mathbf{x} \right) d\xi d\xi = \int \left( \partial \left( \int p\_r(f|\_{\xi}) d\xi \right) / \partial \mathbf{x} \right) p\_r(\xi) d\xi \tag{24}$$

where 〈f j <sup>ξ</sup>〉<sup>r</sup> is the conditional mean values of f in turbulent (r = t) or nonturbulent (r = n) medium for all possible fixed values of ξ. At that ∂prðf , ξÞ=∂x ¼ 0 due to the fact that the function (f, ξ) does not depend obviously on the coordinate x. On the flip side, we have

$$\iiint \frac{\partial f}{\partial x} p\_r(f, \xi) df d\xi = \left< \frac{\partial f}{\partial x} \right>\_r \tag{25}$$

because that

$$\iint \xi p\_r(\xi|\_f) p\_r(f) d\xi df = \int \langle \xi|\_f \rangle\_r p\_r(f) df \tag{26}$$

where 〈ξj <sup>f</sup>〉<sup>r</sup> is the conditional mean of the gradient ξ ¼ ∂f =∂x in turbulent or nonturbulent medium, given for all possible fixed values of f. As a result, we have

$$
\left< \frac{\partial f}{\partial x} \right>\_r = \frac{\partial \langle f \rangle\_r}{\partial x}, \; r = t, n \tag{27}
$$

Thus, the operation of conditional statistical averaging of derivatives is permutational. So, we have proved that the permutation of conditional averaging operation has a strict mathematical justification.

It is appropriate to note that in the classical method of RANS, the operation of unconditional statistical averaging of derivatives gives the same result. Actually, the unconditional joint PDF pRðf , ξÞ of Eq. (14) does not depend on the coordinates obviously and therefore it is correctly Eqs. (23)–(27) with index r = R, i.e.,

$$
\left< \frac{\partial f}{\partial \mathbf{x}} \right>\_{\mathbb{R}} = \frac{\partial \langle f \rangle\_{\mathbb{R}}}{\partial \mathbf{x}} \tag{28}
$$

that proves the rule of permutation of the operation of unconditional averaging of derivatives in the method of RANS.

About the permutation of the operation of derivatives total averaging I must say the following. The operation of total statistical averaging of partial derivatives of type ξ ¼ ∂f =∂x by Eq. (19) for intermittent continuous media with turbulent and nonturbulent fluid cannot be a permutational. This operation is carried out similarly in Eqs. (23)–(27). Here, however, must keep in mind that now the total PDF P(f) in Eq. (17) obviously depend on the coordinates due to γ = γ(x). The legitimacy of such a permutation of the operation is easy to establish if we attract Eq. (20) as applied to the partial derivatives. In this case

$$
\left\langle \frac{\partial \mathbf{\hat{f}}}{\partial \mathbf{x}} \right\rangle = \gamma \left\langle \frac{\partial \mathbf{\hat{f}}}{\partial \mathbf{x}} \right\rangle\_t + (1 - \gamma) \left\langle \frac{\partial \mathbf{\hat{f}}}{\partial \mathbf{x}} \right\rangle\_n \tag{29}
$$

and

〈f〉 ¼ ð

which by virtue of the expression in Eq. (17) gives

<sup>r</sup> ¼ f <sup>r</sup> � 〈f <sup>r</sup>〉r; f <sup>0</sup>

106 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

At that by definition the value f <sup>t</sup> ¼ f j

<sup>0</sup> ¼ f � 〈f〉 ; f <sup>0</sup>

f

superlayer [16].

and

where 〈f j

because that

whence it follows that

Ω

<sup>I</sup>¼<sup>1</sup> and <sup>f</sup> <sup>n</sup> <sup>¼</sup> <sup>f</sup> <sup>j</sup>

0 rg0 <sup>r</sup>〉<sup>r</sup> 6¼ 〈f 0 g0

As is evident that the total average in Eq. (20) represents the statistical characteristic of a rather complex structure, while the unconditional mean, when in Eq. (18) we have r = R, is a characteristic of the "simplified" flow without considering effects of intermittency. At that 〈f〉<sup>R</sup> ffi 〈f〉 because the total average 〈f〉 does not contain the values of f belonging to the

The statistical averaging of the derivative of hydrodynamic quantity ξ ¼ ∂f =∂x gives the

∂x ð

prðf , ξÞ ¼ prðf j

� df dξ ¼

〈<sup>f</sup> <sup>j</sup>ξ〉rprðξÞd<sup>ξ</sup> <sup>¼</sup> <sup>∂</sup>〈f〉<sup>r</sup>

f prðf j <sup>ξ</sup>Þdf � =∂x �

<sup>ξ</sup>ÞprðξÞ

ð � ∂ �ð

<sup>ξ</sup>〉<sup>r</sup> is the conditional mean values of f in turbulent (r = t) or nonturbulent (r = n)

∂x � �

r

medium for all possible fixed values of ξ. At that ∂prðf , ξÞ=∂x ¼ 0 due to the fact that the function (f, ξ) does not depend obviously on the coordinate x. On the flip side, we have

prðf , <sup>ξ</sup>Þdf d<sup>ξ</sup> <sup>¼</sup> <sup>∂</sup><sup>f</sup>

j <sup>I</sup>¼<sup>1</sup> <sup>¼</sup> <sup>f</sup> 0

〈f <sup>r</sup>〉<sup>r</sup> ¼ 〈f〉r;〈 f

2.2.3. Operations of statistical averaging of derivative of the hydrodynamic quantity

following. In point of fact, on the one side using the joint CPDF prðf , ξÞ we have

prðf , <sup>ξ</sup>Þdf d<sup>ξ</sup> <sup>¼</sup> <sup>∂</sup>

ðð ∂f ∂x

∂f prðf , ξÞ=∂x � f ∂prðf , ξÞ=∂x

ðð ∂f ∂x

because that in accordance with [14]

ðð �

f PðfÞdf ð19Þ

<sup>I</sup>¼<sup>0</sup> and for the "pulsations" we have

<sup>n</sup> þ 〈f〉<sup>n</sup> � 〈f〉 ð21Þ

〉<sup>r</sup> ð22Þ

<sup>∂</sup><sup>x</sup> <sup>ð</sup>23<sup>Þ</sup>

prðξÞdξ ð24Þ

ð25Þ

<sup>j</sup><sup>I</sup>¼<sup>0</sup> <sup>¼</sup> <sup>f</sup> 0

〈f〉 ¼ γ〈f〉<sup>t</sup> þ ð1 � γÞ〈f〉<sup>n</sup> ð20Þ

<sup>t</sup> þ 〈f〉<sup>t</sup> � 〈f〉 ; f <sup>0</sup>

$$\frac{\partial \langle f \rangle}{\partial \mathbf{x}} = \gamma \frac{\partial \langle f \rangle\_t}{\partial \mathbf{x}} + (1 - \gamma) \frac{\partial \langle f \rangle\_n}{\partial \mathbf{x}} + (\langle f \rangle\_t - \langle f \rangle\_n) \frac{\partial \gamma}{\partial \mathbf{x}} \tag{30}$$

after comparing of which, with regard to Eq. (27) we get

$$
\left\langle \frac{\partial f}{\partial \mathbf{x}} \right\rangle = \frac{\partial \langle f \rangle}{\partial \mathbf{x}} - \left( \langle f \rangle\_t - \langle f \rangle\_n \right) \frac{\partial \mathcal{V}}{\partial \mathbf{x}} \tag{31}
$$

It follows that the permutation of the operation of total statistical averaging of derivatives is not legitimate, i.e.,

$$
\left<\frac{\partial f}{\partial \mathbf{x}}\right> \neq \frac{\partial \langle f \rangle}{\partial \mathbf{x}}\tag{32}
$$

With regard to the total statistical averaging of time derivatives, instead of expression (31) we have 〈∂f =∂t〉 ¼ ∂〈f〉=∂t because of ∂γ=∂t ¼ 0, i.e., for statistically stationary turbulent flows such a permutation is possible. The same applies to the conditional averaging of derivatives. So, we showed that the statistical modeling of turbulent flows, in the case of taking into account the effects of intermittency, should be based on Eqs. (20), (27), and (32).

#### 2.2.4. The statistical averaging of hydrodynamics equations

The Navier-Stokes equations for an incompressible fluid together with the continuity equation are accepted as the basis of the hydrodynamic equations system [11]. When the external forces are absent, this system has the following form:

$$\begin{cases} \frac{\partial u\_i}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_k} \left( u\_i u\_k + \frac{p \delta\_{ik}}{\rho} - \sigma\_{ik} \right) = 0, & i = 1, 2, 3 \\ & \frac{\partial u\_k}{\partial \mathbf{x}\_k} = 0 \end{cases} \tag{33}$$

where σik ¼ μð∂ui=∂xk þ ∂uk=∂xiÞ is the tensor of viscous stress, μ is the dynamic factor of viscosity, p is the pressure, and ρ is the density. Our primary goal is to conduct an operation of statistical averaging of the SE (33) so as to obtain a system of equations for the conditional mean 〈ui〉t. At the beginning, we will conduct an operation of conditional statistical averaging of the continuity equation in SE (33). For this, we introduce the joint CPDF

$$p\_r(\mu\_i, \xi\_i) = p\_r(\mu\_1, \mu\_2, \mu\_3, \xi\_1, \xi\_2, \xi\_3) \tag{34}$$

with index r = t for turbulent and r = n for nonturbulent fluid, ξ<sup>1</sup> ¼ ∂u1=∂x<sup>1</sup> , ξ<sup>2</sup> ¼ ∂u2=∂x2, ξ<sup>3</sup> ¼ ∂u3=∂x3, and use the procedure of conditional averaging (23). As a result, we reach the following averaging procedure:

$$\int \dots \int \frac{\partial u\_k}{\partial \mathbf{x}\_k} p\_r(\mathbf{u}\_\nu, \xi\_i) d\mathbf{u}\_1 \dots, d\xi\_3 = \int \dots \int \frac{\partial u\_k p\_r(\mathbf{u}\_\nu, \xi\_i)}{\partial \mathbf{x}\_k} d\mathbf{u}\_1 \dots, d\xi\_3 = \frac{\partial}{\partial \mathbf{x}\_k} \int \dots \int u\_k p\_r(\mathbf{u}\_\nu, \xi\_i) d\mathbf{u}\_1 \dots, d\xi\_3 = 0 \tag{35}$$

because that the function prðui, ξÞ does not depend on xk, i.e., ∂prðui, ξiÞ=∂xk ¼ 0 and prðui, ξiÞ∂uk=∂xk ¼ ∂ukprðui, ξiÞ=∂xk. From here toward k = 1 in Eq. (35), we deduce

$$\frac{\partial}{\partial \mathbf{x}\_1} \int u\_1 p\_r(u\_1) du\_1 = \frac{\partial \langle u\_1 \rangle\_r}{\partial \mathbf{x}\_1} \tag{36}$$

as prðuiÞ ¼ <sup>Ð</sup> …Ð prðu1, u2, u3, <sup>ξ</sup>1, <sup>ξ</sup>2, <sup>ξ</sup>3Þdui�<sup>1</sup>, duiþ<sup>1</sup>::dξ3, du<sup>0</sup> = 1, for example, prðu1Þ ¼ <sup>Ð</sup> …Ð prðu1, u2, u3, ξ1, ξ2, ξ3Þdu2…dξ3. The same operation is carried out for k = 2, 3 using pr(u2) and pr(u3). Now then, the conditionally averaged continuity equation for each of the intermittent media of turbulent flow has the form

∂f ∂x � �

account the effects of intermittency, should be based on Eqs. (20), (27), and (32).

2.2.4. The statistical averaging of hydrodynamics equations

∂ui ∂t þ

8 >><

>>:

following averaging procedure:

…Ð

prðui, ξiÞdu1, …, dξ<sup>3</sup> ¼

ð :: ð ∂uk ∂xk

as prðuiÞ ¼ <sup>Ð</sup>

∂ ∂xk

108 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

of the continuity equation in SE (33). For this, we introduce the joint CPDF

<sup>ð</sup> <sup>∂</sup>ukprðui, <sup>ξ</sup>i<sup>Þ</sup> ∂xk

prðui, ξiÞ∂uk=∂xk ¼ ∂ukprðui, ξiÞ=∂xk. From here toward k = 1 in Eq. (35), we deduce

∂ ∂x<sup>1</sup> ð

ð ::

are absent, this system has the following form:

6¼ <sup>∂</sup>〈f〉

With regard to the total statistical averaging of time derivatives, instead of expression (31) we have 〈∂f =∂t〉 ¼ ∂〈f〉=∂t because of ∂γ=∂t ¼ 0, i.e., for statistically stationary turbulent flows such a permutation is possible. The same applies to the conditional averaging of derivatives. So, we showed that the statistical modeling of turbulent flows, in the case of taking into

The Navier-Stokes equations for an incompressible fluid together with the continuity equation are accepted as the basis of the hydrodynamic equations system [11]. When the external forces

> <sup>ρ</sup> � <sup>σ</sup>ik � �

> > ∂uk ∂xk ¼ 0

where σik ¼ μð∂ui=∂xk þ ∂uk=∂xiÞ is the tensor of viscous stress, μ is the dynamic factor of viscosity, p is the pressure, and ρ is the density. Our primary goal is to conduct an operation of statistical averaging of the SE (33) so as to obtain a system of equations for the conditional mean 〈ui〉t. At the beginning, we will conduct an operation of conditional statistical averaging

with index r = t for turbulent and r = n for nonturbulent fluid, ξ<sup>1</sup> ¼ ∂u1=∂x<sup>1</sup> , ξ<sup>2</sup> ¼ ∂u2=∂x2, ξ<sup>3</sup> ¼ ∂u3=∂x3, and use the procedure of conditional averaging (23). As a result, we reach the

because that the function prðui, ξÞ does not depend on xk, i.e., ∂prðui, ξiÞ=∂xk ¼ 0 and

prðu1, u2, u3, ξ1, ξ2, ξ3Þdu2…dξ3. The same operation is carried out for k = 2, 3 using pr(u2) and

<sup>u</sup>1prðu1Þdu<sup>1</sup> <sup>¼</sup> <sup>∂</sup>〈u1〉<sup>r</sup>

¼ 0, i ¼ 1, 2, 3

prðui, ξiÞ ¼ prðu1, u2, u3, ξ1, ξ2, ξ3Þ ð34Þ

∂xk ð :: ð

ukprðui, ξiÞdu1…, dξ<sup>3</sup> ¼ 0

du1, …, dξ<sup>3</sup> <sup>¼</sup> <sup>∂</sup>

∂x<sup>1</sup>

prðu1, u2, u3, <sup>ξ</sup>1, <sup>ξ</sup>2, <sup>ξ</sup>3Þdui�<sup>1</sup>, duiþ<sup>1</sup>::dξ3, du<sup>0</sup> = 1, for example, prðu1Þ ¼ <sup>Ð</sup>

uiuk <sup>þ</sup> <sup>p</sup>δik

<sup>∂</sup><sup>x</sup> <sup>ð</sup>32<sup>Þ</sup>

ð33Þ

ð35Þ

ð36Þ

…Ð

$$\frac{\partial \langle \mu\_k \rangle\_r}{\partial x\_k} = 0, \ r = t, n \tag{37}$$

To conduct the operation of conditional statistical averaging of the Navier-Stokes SE (33), we use the CPDF pr <sup>¼</sup> pr<sup>ð</sup> <sup>ξ</sup>1, <sup>ξ</sup>2Þ, where <sup>ξ</sup><sup>1</sup> <sup>¼</sup> <sup>∂</sup>ui <sup>∂</sup><sup>t</sup> , <sup>ξ</sup><sup>2</sup> <sup>¼</sup> <sup>∂</sup> <sup>∂</sup>xk uiuk <sup>þ</sup> <sup>p</sup>δik <sup>ρ</sup> � <sup>σ</sup>ik � � with the summation over k = 1, 2, 3. Then, according to Eq. (25) for the momentum equation in SE (33) we obtain ðð <sup>ξ</sup>1pr<sup>ð</sup> <sup>ξ</sup>1, <sup>ξ</sup>2Þdξ1dξ<sup>2</sup> <sup>¼</sup> 〈∂ui=∂t〉r. Similarly, it conducted the averaging operation of the value <sup>ξ</sup>2:

$$\iint \xi\_2 p\_r(\ \xi\_1, \ \xi\_2) d\xi\_1 d\xi\_2 = \left\langle \frac{\partial}{\partial x\_k} \left( \mu\_i \mu\_k + \frac{p\delta\_{ik}}{\rho} - \sigma\_{ik} \right) \right\rangle\_r \tag{38}$$

Applying the rule of permutation (27) and using the Reynolds development f <sup>r</sup> ¼ 〈f〉<sup>r</sup> þ f <sup>r</sup> 0 , we deduce 〈uiuk〉<sup>r</sup> ¼ 〈ui〉r〈uk〉<sup>r</sup> þ 〈u<sup>0</sup> iru<sup>0</sup> kr〉<sup>r</sup> in view of Eq. (22). As a result of the above-performed operation of statistical averaging of SE (33) now for the statistically stationary turbulent flow, we have the system of equations with two autonomous subsystems for the flow's conditional means of each of the intermittent media with turbulent and nonturbulent fluid:

$$\begin{cases} \frac{\partial \langle u\_{i} \rangle\_{r}}{\partial t} + \frac{\partial \langle u\_{i} \rangle\_{r} \langle u\_{k} \rangle\_{r}}{\partial \mathbf{x}\_{k}} + \frac{\langle \mu\_{\text{ir}}' \mu\_{\text{kr}}' \rangle\_{r}}{\partial \mathbf{x}\_{k}} + \frac{\partial \langle (\!(\!p\delta\_{\text{ik}} - \sigma\_{\text{ik}}) \!) / \rho \rangle\_{r}}{\partial \mathbf{x}\_{k}} = \mathbf{0} \\ \qquad \qquad \frac{\partial \langle u\_{k} \rangle\_{r}}{\partial \mathbf{x}\_{k}} = \mathbf{0}, \ r = t, \ n \end{cases} \tag{39}$$

where the fluctuating velocity of the turbulent or nonturbulent fluid flow u<sup>0</sup> ir ¼ uir � 〈uir〉<sup>r</sup> and 〈ui〉<sup>r</sup> � 〈uir〉r, but the one-point covariances 〈u<sup>0</sup> iru<sup>0</sup> kr〉<sup>r</sup> 6¼ 〈u<sup>0</sup> i u0 <sup>k</sup>〉<sup>r</sup> according to Eq. (22). Besides, ∂〈ui〉r=∂t ¼ 0 for statistically stationary turbulent flows. Each SS (39) with index r = t or r = n is statistically independent and is determined by the fact that the one-point correlation of the hydrodynamic quantities of turbulent and nonturbulent media is absent, i.e., 〈f <sup>t</sup> f <sup>n</sup>〉 ¼ 0. These subsystems allow the conditional means modeling of each of the intermittent media with turbulent and nonturbulent fluid independently from the each other.

The derivation of the turbulent kinetic energy budget equation by the RANS method is well known [1, 4]. The procedure of the budget equations derivation for conditional means of kinetic energy fluctuations in each of the intermittent medium of the turbulent flow will be the same. In the approximation of a free boundary layer, these equations have the following form:

$$\frac{\partial \langle E\_r \rangle\_r}{\partial t} + \langle u\_k \rangle\_r \frac{\partial \langle E\_r \rangle\_r}{\partial \mathbf{x}\_k} + \frac{\partial \langle (E\_r + p'\_r/\rho) \mathbf{v}'\_r \rangle\_r}{\partial \mathbf{x}\_k} + \langle u'\_{\dot{n}} u'\_{\dot{k}r} \rangle\_r \frac{\partial \langle u\_i \rangle\_r}{\partial \mathbf{x}\_k} + \langle \mathbf{c}\_r \rangle\_r = 0 \tag{40}$$

Hereinafter Er ¼ 0:5ðu 0 2 <sup>r</sup> þ v 0 2 <sup>r</sup> þ w<sup>0</sup> 2 <sup>r</sup> Þ and 〈Er〉<sup>r</sup> ¼ 0:5ð〈u 0 2 <sup>r</sup> 〉<sup>r</sup> þ 〈v 0 2 <sup>r</sup> 〉<sup>r</sup> þ 〈w<sup>0</sup> 2 <sup>r</sup> 〉rÞ.

## 3. The ASMTurb method

The new principle of constructing mathematical models of the energy-containing structure of turbulent flows (the large-scale turbulent motion) is as follows: (1) as the main statistical characteristics of modeling are chosen the conditional averages of hydrodynamic quantities of the turbulent and nonturbulent fluid; (2) to describe the conditional means of hydrodynamic quantities are used two statistically independent (autonomous) systems of differential equations; (3) each of the autonomous systems for the conditional averages is closed by its own closure hypothesis; and (4) the total average of hydrodynamic quantities is obtained by the algebraic relations of statistical hydrodynamics, which bind the total and conditional means through the mediation of the intermittency factor. To realize this principle, the mechanism of the turbulent fluid formation in a turbulent flow is proposed. This is achieved by the introduction of the "superlayer" between turbulent and nonturbulent fluid, where shear rate and pressure fluctuations in the turbulent fluid generate the pressure fluctuations in the nonturbulent fluid. This process leads to the so-called "nonlocal" transfer of the impulse and initiates the occurrence of velocity fluctuations (for particulars see in [15, 16]). The formulated principle of constructing mathematical models is called the ASMTurb method [5].

#### 3.1. Mathematical tools of the ASMTurb method

#### 3.1.1. General system of equations for conditional means

According to the ASMTurb method, we have two autonomous subsystems (SS) of the difference equations corresponding to Eqs. (39) and (40) in the form of

$$\begin{cases} \begin{aligned} \frac{\partial \langle u\_{i} \rangle\_{t}}{\partial t} + \frac{\partial \langle u\_{i} \rangle\_{t} \langle u\_{k} \rangle\_{t}}{\partial \mathbf{x}\_{k}} + \frac{\partial \langle u'\_{i\mathbf{i}} \mu'\_{k\mathbf{i}} \rangle\_{t}}{\partial \mathbf{x}\_{k}} + \frac{\partial \langle (p\_{i}\delta\_{k} - \sigma\_{t\mathbf{i} k})/\rho \rangle\_{t}}{\partial \mathbf{x}\_{k}} = 0 \\\\ \frac{\partial \langle u\_{k} \rangle\_{t}}{\partial \mathbf{x}\_{k}} + \langle u\_{k} \rangle\_{t} \frac{\partial \langle E\_{t} \rangle\_{t}}{\partial \mathbf{x}\_{k}} + \frac{\partial \langle (E\_{t} + p'\_{t}/\rho)\mathbf{v}'\_{t} \rangle\_{t}}{\partial \mathbf{x}\_{k}} + \langle u'\_{k} \mu'\_{k\mathbf{i}} \rangle\_{t} \frac{\partial \langle u\_{i} \rangle\_{t}}{\partial \mathbf{x}\_{k}} + \langle \mathbf{c}\_{t} \rangle\_{t} = 0 \end{aligned} \end{cases} (41)$$

and

$$\begin{cases} \frac{\partial \langle u\_{i} \rangle\_{n}}{\partial t} + \frac{\partial \langle u\_{i} \rangle\_{n} \langle u\_{k} \rangle\_{n}}{\partial \mathbf{x}\_{k}} + \frac{\partial \langle u'\_{n} \boldsymbol{u}'\_{kn} \rangle\_{n}}{\partial \mathbf{x}\_{k}} + \frac{\partial \langle (\boldsymbol{p}\_{n} \delta \boldsymbol{\delta}\_{k} - \boldsymbol{\sigma}\_{nk})/\rho \rangle\_{n}}{\partial \mathbf{x}\_{k}} = \mathbf{0} \\\\ \frac{\partial \langle u\_{k} \rangle\_{n}}{\partial \mathbf{x}\_{k}} = \mathbf{0} \\\\ \frac{\partial \langle E\_{n} \rangle\_{n}}{\partial t} + \langle u\_{k} \rangle\_{n} \frac{\partial \langle E\_{n} \rangle\_{n}}{\partial \mathbf{x}\_{k}} + \frac{\partial \langle (E\_{n} + p'\_{n}/\rho) \mathbf{v}'\_{n} \rangle\_{n}}{\partial \mathbf{x}\_{k}} + \langle u'\_{in} \boldsymbol{u}'\_{kn} \rangle\_{n} \frac{\partial \langle u\_{i} \rangle\_{n}}{\partial \mathbf{x}\_{k}} + \langle \boldsymbol{\epsilon}\_{n} \rangle\_{n} = 0 \end{cases} (42)$$

that describe the conditional mean flow characteristics of each of the intermittent media with the turbulent (r = t) and nonturbulent (r = n) fluid. Let us note that each of the SS (41) and SS (42) is statistically independent, in terms of the one-point correlations 〈f <sup>t</sup> f <sup>n</sup>〉 ¼ 0, so after the completion of these subsystems using the corresponding expressions for 〈u<sup>0</sup> iru<sup>0</sup> kr〉r, 〈ðErþ p0 r=ρÞv<sup>0</sup> <sup>r</sup>〉<sup>r</sup> and 〈εr〉<sup>r</sup> as the closure hypothesis we obtain mathematical models for the flow of the turbulent and nonturbulent fluid.

#### 3.1.2. The closure hypothesis

3. The ASMTurb method

The new principle of constructing mathematical models of the energy-containing structure of turbulent flows (the large-scale turbulent motion) is as follows: (1) as the main statistical characteristics of modeling are chosen the conditional averages of hydrodynamic quantities of the turbulent and nonturbulent fluid; (2) to describe the conditional means of hydrodynamic quantities are used two statistically independent (autonomous) systems of differential equations; (3) each of the autonomous systems for the conditional averages is closed by its own closure hypothesis; and (4) the total average of hydrodynamic quantities is obtained by the algebraic relations of statistical hydrodynamics, which bind the total and conditional means through the mediation of the intermittency factor. To realize this principle, the mechanism of the turbulent fluid formation in a turbulent flow is proposed. This is achieved by the introduction of the "superlayer" between turbulent and nonturbulent fluid, where shear rate and pressure fluctuations in the turbulent fluid generate the pressure fluctuations in the nonturbulent fluid. This process leads to the so-called "nonlocal" transfer of the impulse and initiates the occurrence of velocity fluctuations (for particulars see in [15, 16]). The formulated

principle of constructing mathematical models is called the ASMTurb method [5].

þ ∂〈u<sup>0</sup> itu0 kt〉t ∂xk

∂〈ðEt þ p<sup>0</sup> t =ρÞv<sup>0</sup> t〉t

þ ∂〈u<sup>0</sup> inu<sup>0</sup> kn〉<sup>n</sup> ∂xk

∂〈ðEn þ p<sup>0</sup>

(42) is statistically independent, in terms of the one-point correlations 〈f <sup>t</sup>

According to the ASMTurb method, we have two autonomous subsystems (SS) of the differ-

∂〈uk〉<sup>t</sup> ∂xk

∂xk

∂〈uk〉<sup>n</sup> ∂xk

∂xk

that describe the conditional mean flow characteristics of each of the intermittent media with the turbulent (r = t) and nonturbulent (r = n) fluid. Let us note that each of the SS (41) and SS

n=ρÞv<sup>0</sup> n〉n

þ ∂〈ð pt

¼ 0

þ

¼ 0

þ 〈u<sup>0</sup> itu0 kt〉t

þ 〈u<sup>0</sup> inu<sup>0</sup> kn〉<sup>n</sup>

δik � σtikÞ=ρ〉<sup>t</sup> ∂xk

> ∂〈ui〉<sup>t</sup> ∂xk

> > ∂〈ui〉<sup>n</sup> ∂xk

∂〈ð pnδik � σnikÞ=ρ〉<sup>n</sup> ∂xk

¼ 0

þ 〈εt〉<sup>t</sup> ¼ 0

¼ 0

þ 〈εn〉<sup>n</sup> ¼ 0

f <sup>n</sup>〉 ¼ 0, so after the

ð41Þ

ð42Þ

3.1. Mathematical tools of the ASMTurb method

∂〈ui〉<sup>t</sup> ∂t þ

<sup>∂</sup><sup>t</sup> <sup>þ</sup> 〈uk〉<sup>t</sup>

∂〈ui〉<sup>n</sup> ∂t þ

<sup>∂</sup><sup>t</sup> <sup>þ</sup> 〈uk〉<sup>n</sup>

∂〈Et〉<sup>t</sup>

8

>>>>>>>><

>>>>>>>>:

8

>>>>>>>><

>>>>>>>>:

∂〈En〉<sup>n</sup>

and

3.1.1. General system of equations for conditional means

ence equations corresponding to Eqs. (39) and (40) in the form of

110 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

∂〈Et〉<sup>t</sup> ∂xk þ

∂〈ui〉n〈uk〉<sup>n</sup> ∂xk

∂〈En〉<sup>n</sup> ∂xk þ

∂〈ui〉t〈uk〉<sup>t</sup> ∂xk

The closure hypothesis for SS (41) and SS (42) we will choose in the form of a simple expression gradient relation [16]:

$$-\langle \mu\_r' \upsilon\_r' \rangle\_r = \nu\_r \frac{\partial \langle \mu \rangle\_r}{\partial y} \tag{43}$$

$$\left\langle \left( E\_r + \frac{p\_r'}{\rho} \right) v\_r' \right\rangle\_r = -\nu\_r \frac{\partial \langle E\_r \rangle\_r}{\partial y} \text{, } \langle \mathfrak{e}\_r \rangle\_r = \mathfrak{e}\_\* \frac{\nu\_r \langle E\_r \rangle\_r}{L\_r^2} \tag{44}$$

where ν<sup>r</sup> is the coefficient of turbulent viscosity, expressed by the "second" Prandtl formula

$$
\Delta \nu\_r = k\_r (\mu\_1 - \mu\_2) \mathbf{x} \tag{45}
$$

It is clear that the use of Eq. (45) allows us to solve our "dynamic task" (i.e., the continuity and momentum equations in SS (41) and (42)) without distinction of "fluctuating task" (i.e., turbulent-kinetic-energy budget equations in SS (41) and (42)). This approach greatly simplifies the modeling process.

#### 3.1.3. Modeling of the total averages

To calculate the total statistical averages, we will use the statistical ratio (20). For example, for the velocity components

$$<\langle \mu\_i \rangle = \gamma \langle \mu\_i \rangle\_t + (1 - \gamma) \langle \mu\_i \rangle\_n \tag{46}$$

To determine the total averages for correlations of velocity pulsations (the covariances), we will use the ratios of the type

$$
\langle \mu' \boldsymbol{\upsilon}' \rangle = \gamma \langle \mu'\_t \boldsymbol{\upsilon}'\_t \rangle\_t + (1 - \gamma) \langle \mu'\_n \boldsymbol{\upsilon}'\_n \rangle\_n + \gamma (1 - \gamma) (\langle \mu \rangle\_t - \langle \mu \rangle\_n)(\langle \boldsymbol{\upsilon} \rangle\_t - \langle \boldsymbol{\upsilon} \rangle\_n) \tag{47}
$$

This equation can be obtained according to our theory. In actual fact, for the velocity pulsations we have ui 0 j <sup>I</sup>¼<sup>1</sup> <sup>¼</sup> 〈uit〉<sup>t</sup> <sup>þ</sup> ui 0 <sup>t</sup> � 〈ui〉 and ui 0 j <sup>I</sup>¼<sup>0</sup> <sup>¼</sup> 〈uin 〉<sup>n</sup> <sup>þ</sup> ui 0 <sup>n</sup> � 〈ui〉 according to Eqs. (21) and (22) whence it follows from Eq. (47), since 〈u<sup>0</sup> v0 〉 ¼ γ〈u<sup>0</sup> v0 〉<sup>t</sup> þ ð1 � γÞ〈u<sup>0</sup> v0 〉<sup>n</sup> and 〈u<sup>0</sup> <sup>r</sup>〈vr〉r〉<sup>r</sup> ¼ 0, 〈〈u〉〉<sup>t</sup> ¼ 〈u〉 and so on. Eq. (47) aligns with the expression in [4, 17].

The fluctuating structure modeling is determined by the separate terms of equations for kinetic energy of the velocity fluctuations in each of the intermittent media, i.e., the turbulent kinetic energy budget equations in SS (41) and (42). In addition, the expression for the total average of turbulent energy is the same as Eq. (47), viz.,

$$
\langle E \rangle = \gamma \langle E\_t \rangle\_t + (1 - \gamma) \langle E\_n \rangle\_n + Ed \tag{48}
$$

where

$$Ed = 0.5\gamma (1 - \gamma) [(\langle u \rangle\_t - \langle u \rangle\_n)^2 + (\langle v \rangle\_t - \langle v \rangle\_n)^2 + (\langle w \rangle\_t - \langle w \rangle\_n)^2] \tag{49}$$

Let us note that Eq. (48) can also be obtained as Eq. (47). According to Eq. (47), wherein u<sup>0</sup> ¼ v<sup>0</sup> , we have

$$
\langle \mathfrak{u}'^2 \rangle = \gamma \langle \mathfrak{u}'^2 \rangle\_t + (1 - \gamma) \langle \mathfrak{u}'^2\_n \rangle\_n + \mathfrak{u}\_{d\nu} \ \mathfrak{u}\_d = \left. \gamma (1 - \gamma) (\langle \mathfrak{u} \rangle\_t - \langle \mathfrak{u} \rangle\_n)^2 \right. \tag{50}
$$

To calculate the total averages, as is evident from the foregoing, distribution of the intermittency factor γ is required. To model the intermittency factor γ we will use the expression in [16]:

$$\gamma \cong \langle \mathfrak{e}\_t \rangle\_t / \langle \mathfrak{e} \rangle \tag{51}$$

## 4. Testing of the ASMTurb method

The ASMTurb method has been tested in [15, 16] on the example of constructing the mathematical models for self-similar turbulent shear flows such as: I, the two-stream plane mixing layer; II, the outer region of the boundary layer on the wall; III, the far wake behind a crossstreamlined cylinder; and IV, the axisymmetric submerged jet. Test results were presented in the form calculating the main conditional and total statistical averages applied to a self-similar region of turbulent flows. A comparison was performed between the predictions and known experimental data for the energy-containing structure of turbulent flow, and excellent agreements were noted. By this means, it was shown that these ASMTurb models are more accurate and more detailed than the RANS models.

In view of the fact that construction of each mathematical model requires a significant volume, here we will present without details only testing results the ASMTurb method on the example of constructing a mathematical model for the turbulent fluid flow in a self-similar mixing layer. It is doing because all turbulence processes existing only into turbulent fluid. Calculations of the main "dynamic" and "fluctuating" characteristics we will compare with known experimental data. More detailed of this model see in [16].

#### 4.1. Construction of the model for two-stream plane mixing layer

The mathematical ASMTurb model for two-stream mixing layer (see [18, 19], etc.), formed as a result of turbulent mixing of two co-current streams with identical fluid and ρ = Const, moving with different velocities u<sup>1</sup> = umax and u<sup>1</sup> = umin, includes two subsystems SS (41) and SS (42) for conditional means of each of the intermittent media of the turbulent and nonturbulent fluid. In this case, first of all, we use the SS (41) that was written in approximation of a free boundary layer and reduced to a nondimensional form after the introduction of nondimensional variables. The task of modeling only the velocity field of turbulent flow has been called as "dynamic task."

#### 4.1.1. Modeling of the turbulent fluid flow

〈E〉 ¼ γ〈Et〉<sup>t</sup> þ ð1 � γÞ〈En〉<sup>n</sup> þ Ed ð48Þ

<sup>n</sup> 〉<sup>n</sup> þ ud, ud ¼ γð1 � γÞð〈u〉<sup>t</sup> � 〈u〉nÞ

<sup>2</sup> þ ð〈w〉<sup>t</sup> � 〈w〉n<sup>Þ</sup>

γ ffi 〈εt〉t=〈ε〉 ð51Þ

2

� ð49Þ

<sup>2</sup> <sup>ð</sup>50<sup>Þ</sup>

,

<sup>2</sup> þ ð〈v〉<sup>t</sup> � 〈v〉n<sup>Þ</sup>

Let us note that Eq. (48) can also be obtained as Eq. (47). According to Eq. (47), wherein u<sup>0</sup> ¼ v<sup>0</sup>

To calculate the total averages, as is evident from the foregoing, distribution of the intermittency factor γ is required. To model the intermittency factor γ we will use the expression in [16]:

The ASMTurb method has been tested in [15, 16] on the example of constructing the mathematical models for self-similar turbulent shear flows such as: I, the two-stream plane mixing layer; II, the outer region of the boundary layer on the wall; III, the far wake behind a crossstreamlined cylinder; and IV, the axisymmetric submerged jet. Test results were presented in the form calculating the main conditional and total statistical averages applied to a self-similar region of turbulent flows. A comparison was performed between the predictions and known experimental data for the energy-containing structure of turbulent flow, and excellent agreements were noted. By this means, it was shown that these ASMTurb models are more accurate

In view of the fact that construction of each mathematical model requires a significant volume, here we will present without details only testing results the ASMTurb method on the example of constructing a mathematical model for the turbulent fluid flow in a self-similar mixing layer. It is doing because all turbulence processes existing only into turbulent fluid. Calculations of the main "dynamic" and "fluctuating" characteristics we will compare with known experi-

The mathematical ASMTurb model for two-stream mixing layer (see [18, 19], etc.), formed as a result of turbulent mixing of two co-current streams with identical fluid and ρ = Const, moving with different velocities u<sup>1</sup> = umax and u<sup>1</sup> = umin, includes two subsystems SS (41) and SS (42) for conditional means of each of the intermittent media of the turbulent and nonturbulent fluid. In this case, first of all, we use the SS (41) that was written in approximation of a free boundary layer and reduced to a nondimensional form after the introduction of nondimensional variables. The task of modeling only the velocity field of turbulent flow has been called as

where

we have

〈u0<sup>2</sup>

〉 <sup>¼</sup> <sup>γ</sup>〈u0<sup>2</sup>

4. Testing of the ASMTurb method

and more detailed than the RANS models.

"dynamic task."

mental data. More detailed of this model see in [16].

4.1. Construction of the model for two-stream plane mixing layer

Ed ¼ 0:5γð1 � γÞ½ð〈u〉<sup>t</sup> � 〈u〉nÞ

112 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

<sup>t</sup> 〉<sup>t</sup> þ ð<sup>1</sup> � <sup>γ</sup>Þ〈u0<sup>2</sup>

So, the dynamic task for modeling the velocity field of the turbulent fluid is reduced to solving the following system of equations

$$\begin{cases} \langle u \rangle\_t \frac{\partial \langle u \rangle\_t}{\partial x} + \langle v \rangle\_t \frac{\partial \langle u \rangle\_t}{\partial y} + \frac{\partial \langle u'\_t v'\_t \rangle\_t}{\partial y} = 0 \\\\ \frac{\partial \langle u \rangle\_t}{\partial x} + \frac{\partial \langle v \rangle\_t}{\partial y} = 0 \end{cases} \tag{52}$$

with boundary conditions, which initially assuming as asymptotical, namely

$$\langle u \rangle\_t = \begin{cases} \ \mu\_{1\prime} \ y \to \infty \\ \ \mu\_{2\prime} \ y \to -\infty \end{cases}, \ (\mathfrak{x}, t) \in \mathcal{G}\_t \tag{53}$$

At that the closure hypothesis in Eqs. (43) and (45) take the form of

$$-\langle u'\_t v'\_t \rangle\_t = \nu\_t \frac{\partial \langle u \rangle\_t}{\partial y}, \ \nu\_t = k\_t (u\_1 - u\_2) \mathbf{x} \tag{54}$$

where ν<sup>t</sup> is the coefficient of turbulent viscosity, kt ¼ ktðmÞ, m ¼ u2=u1). For transformation of the SE (52) to the self-similar mode in order to deduce the self-similar solution of our task, let us introduce dimensionless variables

$$\frac{\langle u \rangle\_t}{u\_1} = F'\_t(\eta)\_\prime \ \eta = \frac{y}{x} \tag{55}$$

where F<sup>0</sup> <sup>t</sup> ¼ ∂Ft=∂η with transformation ∂=∂x ¼ �η=xd=dη, ∂=∂y ¼ 1=xd=dη. The boundary conditions (53) take the form of

$$F\_t'(\eta) \cong \begin{cases} -1, \ \eta \to \infty \\ m, \ \eta \to -\infty \end{cases} \tag{56}$$

The nondimensional transverse velocity is defined after integrating the continuity equation in SE (52):

$$\frac{\langle v \rangle\_t}{u\_1} = \eta F'\_t - F\_t \tag{57}$$

the while correlation in Eq. (52) is

$$-\frac{\langle u\_t' v\_t' \rangle\_t}{u\_1^2} = k\_t (1 - m) F\_t' \tag{58}$$

As a consequence, the momentum equation in SE (52) takes the form of ordinary differential equation

$$
\sigma\_t''' + 2\sigma\_t^2 F\_t F\_t' = 0 \tag{59}
$$

where σ<sup>t</sup> is a first empirical parameter of the model, the value of which is determined by the condition of the best agreement of calculated and measurements of the longitudinal velocity. We now represent a function Ft(η) as a power series in the small parameter (m � 1):

$$F\_t = \sum\_{i=0}^{\infty} (m-1)^i F\_{it} = F\_{0t} + (m-1)F\_{1t} + (m-1)^2 F\_{2t} + \dots \tag{60}$$

If we substitute this expression into Eq. (59) and compare the components at the same powers of parametric value (<sup>m</sup> � 1)<sup>i</sup> , we obtain a system of sequentially interconnected ordinary differential equations (here we confine ourselves to the second approximation of our task):

$$\begin{cases} & F\_{0t}^{\prime\prime} + 2\sigma\_t^2 F\_{0t} F\_{0t}^{\prime} = 0, \; i = 0 \\ & F\_{1t}^{\prime\prime} + 2\sigma\_t^2 \left( F\_{0t} F\_{1t}^{\prime} + F\_{1t} F\_{0t}^{\prime} \right) = 0, \; i = 1 \\ & F\_{2t}^{\prime\prime} + 2\sigma\_t^2 \left( F\_{0t} F\_{2t}^{\prime\prime} + F\_{1t} F\_{1t}^{\prime\prime} + F\_{2t} F\_{0t}^{\prime} \right) = 0, \; i = 2 \end{cases} \tag{61}$$

From the boundary conditions (56) it follows that

$$\begin{cases} \quad F\_{0t}' = 1 \; \; \; \; \; F\_{1t}' = 0 \; \; \; \; \; F\_{2t}' = 0 \; \; as \; \; \eta \to \infty \\\ \; \; \; \; F\_{0t}' = 1 \; \; \; \; \; \; F\_{1t}' = 1 \; \; \; \; \; F\_{2t}' = 0 \; as \; \; \eta \to -\infty \end{cases} \tag{62}$$

where we get after integration

$$F\_{0t}' = 1, \; F\_{0t} = \; \eta - \eta\_{0t} \tag{63}$$

where η0<sup>t</sup> ¼ η0<sup>t</sup> ðmÞ is the displacement of the symmetry plane of the mixing layer η = 0. Now the SE (61) takes the form of

$$\begin{cases} \begin{aligned} \left. \overset{\text{''}'''}{\tilde{F}}\_{1t}^{\prime\prime} + 2q \tilde{F}\_{1t}^{\prime\prime} = 0 \\ \left. \overset{\text{''}'''}{\tilde{F}}\_{2t}^{\prime\prime} + 2q \tilde{F}\_{2t}^{\prime\prime} = -2 \tilde{F}\_{1t} \tilde{F}\_{1t}^{\prime\prime} \end{aligned} \right. \\ \end{cases} \tag{64}$$

where F~<sup>0</sup> <sup>t</sup> <sup>¼</sup> <sup>∂</sup>F~=∂<sup>ϕ</sup> , <sup>F</sup><sup>~</sup> 00 <sup>t</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup>F~=∂ϕ2, etc. and

$$
\tilde{F}\_t(\varphi) = \sigma\_t F\_t(\eta), \ \varphi = \ \sigma\_t(\eta - \eta\_{0t}) \tag{65}
$$

To determine the value η0<sup>t</sup> will use the known Karman's condition, namely

$$
\langle \langle \boldsymbol{\upsilon}(\circ) \rangle\_t + m \langle \boldsymbol{\upsilon}(\cdot \circ) \rangle\_t = 0 \tag{66}
$$

The boundary conditions (56) because of 〈u〉t=u<sup>1</sup> ¼ F<sup>0</sup> <sup>t</sup>ðηÞ ¼ <sup>F</sup>~<sup>t</sup> 0 ðϕÞ are converted in accordance with the boundary conditions (62) to the form

$$\begin{cases} \begin{array}{ll} \tilde{F}'\_{0t} = 1, \ \tilde{F}'\_{1t} = 0, \ \tilde{F}'\_{2t} = 0 \text{ as } \boldsymbol{\varrho} \to \boldsymbol{\Leftrightarrow} \\ \tilde{F}'\_{0t} = 1, \ \tilde{F}'\_{1t} = 1, \ \tilde{F}'\_{2t} = 0 \text{ as } \boldsymbol{\varrho} \to -\boldsymbol{\Leftrightarrow} \end{array} \end{cases} \tag{67}$$

The solution of the dynamic task in the first approximation is easy to obtain in an analytical form [16]. At that according to momentum equation in SE (64) we have

$$\frac{\langle u \rangle\_t}{u\_1} = \tilde{F}\_t' = 1 + \frac{m-1}{2}(1 - \text{erf}\varphi) \tag{68}$$

while the transverse velocity in Eq. (57) and correlation in Eq. (58) take the following form

$$\frac{\langle \boldsymbol{\sigma} \rangle\_t}{\boldsymbol{\mu}\_1} = \frac{(\boldsymbol{\varrho} + \boldsymbol{\sigma} \boldsymbol{\eta}\_{0t}) \tilde{\boldsymbol{F}}\_t^{\prime} - \tilde{\boldsymbol{F}}\_t}{\boldsymbol{\sigma}\_t} \tag{69}$$

$$-\frac{\langle u'\_t v'\_t \rangle\_t}{\Delta U^2} = \frac{\tilde{F}''\_t}{2\sigma\_t (1 - m)^2} \tag{70}$$

where ΔU ¼ u<sup>1</sup> � u<sup>2</sup> and the flow function (65) is

F<sup>000</sup> <sup>t</sup> þ 2σ<sup>t</sup> 2 FtF<sup>00</sup>

We now represent a function Ft(η) as a power series in the small parameter (m � 1):

Ft <sup>¼</sup> <sup>X</sup><sup>∞</sup> i¼0

of parametric value (<sup>m</sup> � 1)<sup>i</sup>

where we get after integration

the SE (61) takes the form of

<sup>t</sup> <sup>¼</sup> <sup>∂</sup>F~=∂<sup>ϕ</sup> , <sup>F</sup><sup>~</sup>

00

The boundary conditions (56) because of 〈u〉t=u<sup>1</sup> ¼ F<sup>0</sup>

with the boundary conditions (62) to the form

where η0<sup>t</sup> ¼ η0<sup>t</sup>

where F~<sup>0</sup>

ðm � 1Þ i

114 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

F 000 <sup>1</sup><sup>t</sup> þ 2σ<sup>t</sup>

F0

(

F0

F 000 <sup>2</sup><sup>t</sup> þ 2σ<sup>t</sup>

8 >>><

>>>:

From the boundary conditions (56) it follows that

F 000 <sup>0</sup><sup>t</sup> þ 2σ<sup>t</sup>

<sup>2</sup>ðF0tF 00 <sup>2</sup><sup>t</sup> þ F1tF 00 <sup>1</sup><sup>t</sup> þ F2tF 00

<sup>0</sup><sup>t</sup> ¼ 1 , F<sup>0</sup>

<sup>0</sup><sup>t</sup> ¼ 1 , F<sup>0</sup>

F0

F~ 000 <sup>2</sup><sup>t</sup> <sup>þ</sup> <sup>2</sup>ϕF<sup>~</sup>

To determine the value η0<sup>t</sup> will use the known Karman's condition, namely

8 < :

<sup>t</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup>F~=∂ϕ2, etc. and

F~ 000 <sup>1</sup><sup>t</sup> <sup>þ</sup> <sup>2</sup>ϕF<sup>~</sup>

where σ<sup>t</sup> is a first empirical parameter of the model, the value of which is determined by the condition of the best agreement of calculated and measurements of the longitudinal velocity.

If we substitute this expression into Eq. (59) and compare the components at the same powers

differential equations (here we confine ourselves to the second approximation of our task):

<sup>2</sup>ðF0tF 00 <sup>1</sup><sup>t</sup> þ F1tF 00

<sup>2</sup>F0tF 00

<sup>1</sup><sup>t</sup> ¼ 0, F<sup>0</sup>

<sup>1</sup><sup>t</sup> ¼ 1, F<sup>0</sup>

Fit ¼ F0<sup>t</sup> þ ðm � 1ÞF1<sup>t</sup> þ ðm � 1Þ

<sup>0</sup><sup>t</sup> ¼ 0, i ¼ 0

<sup>0</sup>tÞ ¼ 0, i ¼ 1

<sup>2</sup><sup>t</sup> ¼ 0 as η ! ∞

<sup>2</sup><sup>t</sup> ¼ 0 as η ! �∞

ðmÞ is the displacement of the symmetry plane of the mixing layer η = 0. Now

00 1t

<sup>t</sup>ðηÞ ¼ <sup>F</sup>~<sup>t</sup> 0

00 <sup>1</sup><sup>t</sup> ¼ 0

<sup>2</sup><sup>t</sup> ¼ �2F~1tF<sup>~</sup>

00

<sup>F</sup>~tðϕÞ ¼ <sup>σ</sup>tFtðηÞ, <sup>ϕ</sup> <sup>¼</sup> <sup>σ</sup>tð<sup>η</sup> � <sup>η</sup>0<sup>t</sup>

<sup>t</sup> ¼ 0 ð59Þ

F2<sup>t</sup> þ … ð60Þ

ð61Þ

ð62Þ

ð64Þ

Þ ð65Þ

ðϕÞ are converted in accordance

2

, we obtain a system of sequentially interconnected ordinary

<sup>0</sup>tÞ ¼ 0, i ¼ 2

<sup>0</sup><sup>t</sup> ¼ 1, F0<sup>t</sup> ¼ η � η0<sup>t</sup> ð63Þ

〈vð∞Þ〉<sup>t</sup> þ m〈vð�∞Þ〉<sup>t</sup> ¼ 0 ð66Þ

$$\tilde{F}\_t = \varphi + \frac{m-1}{2} \left( \varphi - \varrho \text{erf}\varphi - \frac{1}{\sqrt{\pi}} e^{-\varphi^2} + 2c\_{1t} \right) \tag{71}$$

Now, we can calculate both the longitudinal velocity profile by Eq. (68) and the correlation profile by Eq. (70) to evaluate the accuracy of our model in the first approximation. These calculations showed that the velocity profile in Eq. (68) coincides with the known experimental data at σ<sup>t</sup> = 18.0, while the correlation profile of fluctuating velocities in Eq. (70) greatly overestimated (see Figure 4b where according to (47) we have to have 〈u<sup>0</sup> v0 〉 ¼ 〈u<sup>0</sup> tv0 <sup>t</sup>〉<sup>t</sup> as γ = 1). Therefore, for specification of our model, we must consider the second approximation of our task.

Figure 4. (a) The self-similar profile of the normalized conditional mean longitudinal velocity 〈U〉<sup>t</sup> ¼ 〈u〉t=u<sup>1</sup> over the turbulent fluid. (b) Profiles of the normalized conditional mean shear stress 〈τ〉<sup>t</sup> ¼ �〈u<sup>0</sup> tv0 t〉t=ΔU<sup>2</sup> : 1, calculation 〈τ〉<sup>t</sup> corresponding to the solution of the dynamic task in the first approximation, σt<sup>1</sup> ¼ 18:0,; and 2, calculation 〈τ〉<sup>t</sup> corresponding to the solution of the dynamic task in the second approximation, σ<sup>t</sup> ¼ 21:5. Symbol o is the measurements of the total average 〈τ〉 ¼ �〈u<sup>0</sup> v0 〉=ΔU<sup>2</sup> (measurements 〈τ〉<sup>t</sup> of [20] are absent). From now on the curves—our calculations, symbols—experimental data [20] in the self-similar mixing layer at the parameter m = 0.305.

The solution of the dynamic task in the second approximation was found in such a manner. The solution of the second equation in SE (64)

$$\tilde{F}\_{2t}^{\prime\prime} = e^{-\varphi^2} (c\_0 - \int \tilde{F}\_{1t} \tilde{F}\_{2t}^{\prime\prime} e^{-\varphi^2} d\varphi) \tag{72}$$

was found by numerical calculation. At that according to Eq. (60), function F~<sup>t</sup> in the second approximation contains two constants of integration c1<sup>t</sup> и c2t. To determine these constants, the integral relation was involved (for example, see [16]):

$$\lim\_{\substack{\varphi\_1,\varphi\_2 \to \infty \\ -\varphi\_2}} \int\_{-\varphi\_2}^{\varphi\_1} (\,\tilde{F}\_t^{\prime\prime\prime} + 2\,\,\tilde{F}\_t\,\,\tilde{F}\_t^{\prime\prime}) d\varphi = 0 \tag{73}$$

Hence, the values of the constants are defined with the help of numerical calculation, c1<sup>t</sup> ffi 0:4, c2<sup>t</sup> ffi �0:1. To determine the value η0<sup>t</sup> ¼ η0<sup>t</sup> ðmÞ in the expression of the dimensionless coordinates ϕ ¼ σtðη � η0<sup>t</sup> Þ we will use Eqs. (69) and (66).

The results of calculations of conditional means of this dynamic task for the mixing layer with the parameter m = 0.305 in comparison with the experimental data of [20] are shown in Figure 4. In this case, according to our model the calculated value η0<sup>t</sup> = �0.0181 when σ<sup>t</sup> = 21.5 (in [20] empirical value η0<sup>S</sup> = �0.02, i.e., we have a good accuracy for η0<sup>t</sup> ffi η0S). Hereinafter curves – our calculations, symbols —measurements are mentioned [20].

Solution of the "fluctuating" task was found in such a manner. The equation of kinetic energy of the velocity fluctuations in SS (41) for the statistically stationary flow of the turbulent fluid, now in the approximation of a free boundary layer, has the following form:

$$\underbrace{\langle u \rangle\_{t} \frac{\partial \langle E\_{t} \rangle\_{t}}{\partial x} + \langle v \rangle\_{t} \frac{\partial \langle E\_{t} \rangle\_{t}}{\partial y}}\_{\text{Conv}\_{l}} + \underbrace{\frac{\partial \langle (E\_{t} + p\_{t}^{\prime}/\rho) v\_{t}^{\prime} \rangle\_{t}}{\partial y}}\_{\text{Turb } D\_{t}} + \underbrace{\langle u\_{\text{il}}^{\prime} u\_{\text{il}}^{\prime} \rangle\_{t} \frac{\partial \langle u \rangle\_{t}}{\partial y}}\_{\text{Prod}\_{l}} + \underbrace{\langle \varepsilon\_{t} \rangle\_{t}}\_{\text{Dis}\_{l}} = 0 \tag{74}$$

For completion of Eq. (74) was used the known expressions (43)–(45) with index r = t. Transformation of Eq. (74) taking into account to an automodel form gives

$$\frac{d^2 \langle \mathcal{E}\_t \rangle\_{\ast t}}{d\rho^2} + 2\,\,\tilde{\mathcal{F}}\_t \frac{d \langle \mathcal{E}\_t \rangle\_{\ast t}}{d\rho} - 2\nu\_{\text{Et}} \langle \mathcal{E}\_t \rangle\_{\ast t} = -\frac{\tilde{\mathcal{F}}\_t^{\ast 2}}{\left(1 - m\right)^2} \tag{75}$$

Here 〈Et〉�<sup>t</sup> � 〈Et〉t=ΔU<sup>2</sup> , ΔU ¼ u<sup>1</sup> � u2, Lt ¼ a0tx; the second empirical parameter of the model is

$$\nu\_{\rm Ef} = \frac{c\_\*}{2(a\_{0t}\sigma\_t)^2} \tag{76}$$

and is determined by the condition of the best agreement of calculated and experimental data of turbulent kinetic energy. The separate components in Eq. (75) correspond to Eq. (74) and have a definite physical meaning:

$$\mathsf{Conv}\_{l} = \bar{\mathsf{F}}\_{l} d\langle E\_{l} \rangle\_{st} / d\mathsf{q} \to \langle \mathsf{u} \rangle\_{l} \mathfrak{d} \langle E\_{l} \rangle\_{l} / \mathsf{\partial x} + \langle \mathsf{v} \rangle\_{l} \mathfrak{d} \langle E\_{l} \rangle\_{l} / \mathsf{\partial y} - \text{convective transfer} \qquad (77)$$

TurbDt <sup>¼</sup> <sup>0</sup>:5d<sup>2</sup> 〈Et〉�<sup>t</sup>=dϕ<sup>2</sup> ! <sup>∂</sup>〈ðEt <sup>þ</sup> <sup>p</sup><sup>0</sup> t =ρÞv<sup>0</sup> t 〉t=∂y—diffusion through the velocity fluctuations ð78Þ

$$\text{Prod}\_{t} = \tilde{\mathcal{F}}\_{t}^{\ast 2} / 2(1 - m)^{2} \rightarrow \langle \mathfrak{u}\_{it}^{\prime} \mathfrak{u}\_{kt}^{\prime} \rangle\_{t} \mathfrak{d}\langle \mathfrak{u} \rangle\_{t} / \mathfrak{d}y - \text{production of the energy fluctuations} \tag{79}$$

$$D\mathbf{is}\_{l} = -\nu\_{\rm Ef} \langle E\_{l} \rangle\_{\ast l} \to \langle \varepsilon\_{l} \rangle\_{\ast l} - \text{dissipation rate of the energy fluctuations} \tag{80}$$

Eq. (75) was solved with boundary conditions in the form

$$\frac{d\langle E \rangle\_{\ast t}}{d\varphi} = 0 \,, \ \varphi = \begin{cases} \ 1.65\\ -1.65 \end{cases} \tag{81}$$

To calculate separate components of intensity (variance) of fluctuating velocity, we will use approximate ratios:

$$
\langle \langle u\_t^{\prime 2} \rangle\_t \cong \langle E\_t \rangle\_t, \langle v\_t^{\prime 2} \rangle\_t \cong \langle w\_t^{\prime 2} \rangle\_t \tag{82}
$$

Eq. (75) was solved by the numerical method (mathematical package MathCad was used). Interestingly, the solution of Eq. (75) with using asymptotic boundary conditions 〈Et〉�<sup>t</sup> ! 0 as ϕ ! �∞ gives the bad calculation data. In this regard for the flow of the turbulent fluid have been used the hard boundary conditions in the form (for m = 0.305)

$$\frac{\langle u \rangle\_t}{u\_1} = \begin{cases} 0.99, \ q\_1 = 1.65\\ 0.32, \ q\_2 = -1.65 \end{cases} \tag{83}$$

and

The solution of the dynamic task in the second approximation was found in such a manner. The

was found by numerical calculation. At that according to Eq. (60), function F~<sup>t</sup> in the second approximation contains two constants of integration c1<sup>t</sup> и c2t. To determine these constants, the

Hence, the values of the constants are defined with the help of numerical calculation,

Þ we will use Eqs. (69) and (66). The results of calculations of conditional means of this dynamic task for the mixing layer with the parameter m = 0.305 in comparison with the experimental data of [20] are shown in Figure 4. In this case, according to our model the calculated value η0<sup>t</sup> = �0.0181 when σ<sup>t</sup> = 21.5 (in [20] empirical value η0<sup>S</sup> = �0.02, i.e., we have a good accuracy for η0<sup>t</sup> ffi η0S). Hereinafter curves –

Solution of the "fluctuating" task was found in such a manner. The equation of kinetic energy of the velocity fluctuations in SS (41) for the statistically stationary flow of the turbulent fluid, now in

<sup>þ</sup> <sup>2</sup> <sup>F</sup>~<sup>t</sup> <sup>F</sup>~<sup>t</sup>

00

dϕÞ ð72Þ

Þdϕ ¼ 0 ð73Þ

ðmÞ in the expression of the dimension-

<sup>þ</sup> 〈εt〉<sup>t</sup> |ffl{zffl} Disst

<sup>2</sup> ð76Þ

¼ 0 ð74Þ

<sup>2</sup> ð75Þ

solution of the second equation in SE (64)

F~000 <sup>2</sup><sup>t</sup> ¼ e �ϕ<sup>2</sup> ðc<sup>0</sup> � ð F~1tF~ 00 2t e �ϕ<sup>2</sup>

116 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

lim ϕ1, ϕ2!∞

ð ϕ1

ð F~t 000

�ϕ<sup>2</sup>

integral relation was involved (for example, see [16]):

c1<sup>t</sup> ffi 0:4, c2<sup>t</sup> ffi �0:1. To determine the value η0<sup>t</sup> ¼ η0<sup>t</sup>

our calculations, symbols —measurements are mentioned [20].

the approximation of a free boundary layer, has the following form:

þ

formation of Eq. (74) taking into account to an automodel form gives

∂〈ðEt þ p<sup>0</sup> t =ρÞv<sup>0</sup> t〉t

d〈Et〉�<sup>t</sup>

∂y |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} Turb Dt

For completion of Eq. (74) was used the known expressions (43)–(45) with index r = t. Trans-

<sup>ν</sup>Et <sup>¼</sup> <sup>c</sup>�

and is determined by the condition of the best agreement of calculated and experimental data of turbulent kinetic energy. The separate components in Eq. (75) correspond to Eq. (74) and

2ða0tσtÞ

þ 〈u<sup>0</sup> itu0 kt〉t ∂〈u〉<sup>t</sup> ∂y

<sup>d</sup><sup>ϕ</sup> � <sup>2</sup>νEt〈Et〉�<sup>t</sup> ¼ � <sup>F</sup>~<sup>t</sup>

, ΔU ¼ u<sup>1</sup> � u2, Lt ¼ a0tx; the second empirical parameter of the model is


00 2

ð1 � mÞ

∂〈Et〉<sup>t</sup> ∂y

less coordinates ϕ ¼ σtðη � η0<sup>t</sup>

〈u〉<sup>t</sup>

Here 〈Et〉�<sup>t</sup> � 〈Et〉t=ΔU<sup>2</sup>

have a definite physical meaning:

∂〈Et〉<sup>t</sup> ∂x

þ 〈v〉<sup>t</sup>


> d2 〈Et〉�<sup>t</sup> <sup>d</sup>ϕ<sup>2</sup> <sup>þ</sup> <sup>2</sup> <sup>F</sup>~<sup>t</sup>

$$\frac{d\langle E \rangle\_{\ast t}}{d\boldsymbol{\rho}} = \begin{cases} 0, \ \boldsymbol{\rho}\_1 = 1.65\\ 0, \ \boldsymbol{\rho}\_2 = -1.65 \end{cases} \tag{84}$$

The results of our calculations of conditional means of this "fluctuating" task are presented in Figure 5. Figure 5a shows the calculation 〈u0<sup>2</sup> <sup>t</sup> 〉<sup>t</sup> <sup>⁄</sup>ΔU<sup>2</sup> corresponding to Eqs. (75) and (82). Figure 5b shows the turbulent kinetic energy budget according to Eqs. (77)–(80). At that value of the parameter νEt ¼ 2. It is worth noting that only Eq. (75) gives the hard edges �0:075 ≤ η � η0<sup>S</sup> ≤ 0:079 (the same (84)) for the flow of the turbulent fluid due to the fact that the solution of Eq. (75) loses its physical sense outside these boundaries (see Figure 5a). So, we got the hard edges only to the flow of the turbulent fluid.

#### 4.1.2. Modeling of the nonturbulent fluid flow

Solution of the dynamic task for the flow of a nonturbulent fluid was defined in such a manner. Modeling of this flow was carried out according to the SS (42) and was related to modeling of the flow of a turbulent fluid by means of statistical ratios in the central field of the mixing layer. It appeared that division of this subsystem into two with high velocity and low velocity

Figure 5. (a) The profile of the normalized conditional mean intensity of longitudinal velocity fluctuations 〈u0<sup>2</sup> <sup>t</sup> 〉�<sup>t</sup> <sup>¼</sup> 〈u0<sup>2</sup> <sup>t</sup> 〉t=ΔU<sup>2</sup> . (b) The turbulent kinetic energy budget over the turbulent fluid: 1, Convt; 2, TurbDt, 3, Prodt,; 4, Disst. The calculated parameter νEt ¼ 2. Measurements in [20] are absent.

regions Gn<sup>1</sup> þ Gn<sup>2</sup> ¼ Gn gives more precise results of modeling. The flow of a nonturbulent fluid in one of these regions has not only its own parameters (σn<sup>1</sup> ¼ 51:44, η0n<sup>1</sup> ¼ �0:015; σn<sup>2</sup> ¼ 36:4, η0n<sup>2</sup> ¼ �0:016) but also boundary conditions: asymptotic ones in external regions and hard ones inside the mixing layer. A butting of the obtained solutions was carried out on the line η � η0<sup>s</sup> ffi 0:009 where the condition 〈u〉<sup>t</sup> ffi 〈u〉n<sup>1</sup> ffi 〈u〉n<sup>2</sup> is satisfied. Solution of the fluctuating task for the flow of a nonturbulent fluid was defined in such a manner. In this case, the solution of the fluctuating kinetic-energy budget equation in SS (42) was found the same as task for the flow of turbulent fluid. Here, however, boundary conditions were given as asymptotic ones. The results of the modeling are presented in Figures 7 and 8a.

#### 4.1.3. Modeling of the total averages

The total averages calculation is required a distribution of the intermittency factor γ. Modeling of this factor can be performed with the help of the statistical ratio (51) in view of the dissipation rate 〈ε〉 ¼ γ〈εt〉<sup>t</sup> þ ð1 � γÞ〈εn〉<sup>n</sup> and 〈ε〉<sup>n</sup> ffi 0. In this case, the value 〈εt〉<sup>t</sup> is in the process of modeling of turbulent fluid by Eq. (74). To calculate the total average of the dissipation rate 〈ε〉, we propose to use the assumption on its equality to the unconditional mean, which is found from the RANS model constructed for the mixing layer. At that the empirical constants σ<sup>R</sup> ¼ 29:0 and η0<sup>R</sup> ¼ �0:0134 are chosen only from the condition of agreement of the intermittency factor γ calculation with the experimental data. Figure 6 presents the

Figure 6. (a) Profiles of the dissipation rate of the energy fluctuations ε<sup>r</sup> ¼ Dissr: 1, Diss<sup>R</sup> ¼ �νER〈ER〉�<sup>R</sup> and 2, Diss<sup>t</sup> ¼ �νEt〈Et〉�<sup>t</sup> at the calculated parameter νER ¼ νEt ¼ 2. (b) The profile of the intermittency factor γ of the turbulent fluid.

Figure 7. (a) Profiles of normalized conditional and total average longitudinal velocity 〈U〉<sup>r</sup> <sup>¼</sup> 〈u〉r=u1: 1-Δ-〈U〉t; 2-□- 〈U〉n1; 3-□-〈U〉n2; 4-o-〈U〉. (b) Profiles of normalized conditional and total average shear stress 〈τ〉<sup>r</sup> ¼ �〈<sup>u</sup> 0 rv 0 r〉r=ΔU<sup>2</sup> : 1–〈τ〉t; 2–〈τ〉n; 3–o–〈τ〉 (measurements of 〈τ〉<sup>t</sup> and 〈τ〉<sup>n</sup> in [20] are absent).

Figure 8. (a) Profiles of normalized conditional and the total average intensity of longitudinal velocity fluctuations 〈u 0 <sup>2</sup>〉�<sup>r</sup> <sup>¼</sup> 〈u0<sup>2</sup> <sup>r</sup> 〉r=ΔU<sup>2</sup> : 1-Δ-〈u 0 <sup>2</sup>〉�<sup>t</sup>; 2-□-〈<sup>u</sup> 0 <sup>2</sup>〉�<sup>n</sup>1, 3-□-〈<sup>u</sup> 0 <sup>2</sup>〉�<sup>n</sup>2; 4-o-〈<sup>u</sup> 0 <sup>2</sup>〉�; 5 – ud=ð<sup>1</sup> � <sup>m</sup><sup>Þ</sup> 2 . (b) Profiles of normalized total average turbulent-kinetic-energy 〈E〉� <sup>¼</sup> 〈E〉=ΔU<sup>2</sup> and intensity of longitudinal velocity fluctuations 〈<sup>u</sup> 0 <sup>2</sup>〉� <sup>¼</sup> 〈<sup>u</sup> 0 <sup>2</sup>〉=ΔU<sup>2</sup> : 1-o-〈E〉�; 2–Δ–〈u 0 <sup>2</sup>〉�; 3 – ud=ð<sup>1</sup> � <sup>m</sup><sup>Þ</sup> 2 . 4 – Ed=ð1 � mÞ 2 .

calculation. As RANS models give a good result only in the regions with insignificant intermittency, such a method for determination of the intermittency factor should be considered only as an approximate one. The results of the modeling of the total averages are presented in Figures 7 and 8. The some results of the unconditional means, obtained by the RANS model, are presented in Figure 9. As it seen that the RANS model does not give good results.

#### 5. Conclusion

regions Gn<sup>1</sup> þ Gn<sup>2</sup> ¼ Gn gives more precise results of modeling. The flow of a nonturbulent fluid in one of these regions has not only its own parameters (σn<sup>1</sup> ¼ 51:44, η0n<sup>1</sup> ¼ �0:015; σn<sup>2</sup> ¼ 36:4, η0n<sup>2</sup> ¼ �0:016) but also boundary conditions: asymptotic ones in external regions and hard ones inside the mixing layer. A butting of the obtained solutions was carried out on the line η � η0<sup>s</sup> ffi 0:009 where the condition 〈u〉<sup>t</sup> ffi 〈u〉n<sup>1</sup> ffi 〈u〉n<sup>2</sup> is satisfied. Solution of the fluctuating task for the flow of a nonturbulent fluid was defined in such a manner. In this case, the solution of the fluctuating kinetic-energy budget equation in SS (42) was found the same as task for the flow of turbulent fluid. Here, however, boundary conditions were given as asymp-

Figure 5. (a) The profile of the normalized conditional mean intensity of longitudinal velocity fluctuations

. (b) The turbulent kinetic energy budget over the turbulent fluid: 1, Convt; 2, TurbDt, 3, Prodt,; 4,

The total averages calculation is required a distribution of the intermittency factor γ. Modeling of this factor can be performed with the help of the statistical ratio (51) in view of the dissipation rate 〈ε〉 ¼ γ〈εt〉<sup>t</sup> þ ð1 � γÞ〈εn〉<sup>n</sup> and 〈ε〉<sup>n</sup> ffi 0. In this case, the value 〈εt〉<sup>t</sup> is in the process of modeling of turbulent fluid by Eq. (74). To calculate the total average of the dissipation rate 〈ε〉, we propose to use the assumption on its equality to the unconditional mean, which is found from the RANS model constructed for the mixing layer. At that the empirical constants σ<sup>R</sup> ¼ 29:0 and η0<sup>R</sup> ¼ �0:0134 are chosen only from the condition of agreement of the intermittency factor γ calculation with the experimental data. Figure 6 presents the

Figure 6. (a) Profiles of the dissipation rate of the energy fluctuations ε<sup>r</sup> ¼ Dissr: 1, Diss<sup>R</sup> ¼ �νER〈ER〉�<sup>R</sup> and 2, Diss<sup>t</sup> ¼ �νEt〈Et〉�<sup>t</sup> at the calculated parameter νER ¼ νEt ¼ 2. (b) The profile of the intermittency factor γ of the turbulent fluid.

totic ones. The results of the modeling are presented in Figures 7 and 8a.

Disst. The calculated parameter νEt ¼ 2. Measurements in [20] are absent.

118 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

4.1.3. Modeling of the total averages

〈u0<sup>2</sup> <sup>t</sup> 〉�<sup>t</sup> <sup>¼</sup> 〈u0<sup>2</sup>

<sup>t</sup> 〉t=ΔU<sup>2</sup>

The new differential equations for the conditional means of turbulent flow are the theory result of this chapter. On the basis of these equations, the method of autonomous statistical modeling ASMTurb of such flow was justified. The main feature of this method is that it allows us to construct the mathematical models for the conditional means of each of the intermittent media taking place into a turbulent stream autonomously, i.e., independently. The main advantage of this method is that the system of differential equations for the conditional means does not contain the source terms. According to this method, the process of transformation the nonturbulent fluid in the turbulent fluid (as a generator of the turbulent fluid) occurs in the superlayer. Even more, the ASMTurb method allows us to construct the model only for the turbulent fluid flow, without considering the nonturbulent fluid flow. As far as all the mixing turbulent processes (and, as consequence, the processing modeling of turbulent heat and mass transfer) take place only into the turbulent fluid, this peculiarity essentially simplifies the modeling of such processes. Especially it refers to the turbulent combustion processes, in which modeling is attended by difficulties. It is important to note that ASMTurb SS (41) and (42) for conditional means of the turbulent and nonturbulent fluid differ from the known ones (for example [7]). It should be emphasized that the presented model contains only two empirical parameters σ<sup>t</sup> and νEt. With regard to these parameters, it must be said that their appearance is due to the fact that we do not know neither the expansion rate of the turbulent fluid downstream nor the maximum value of the turbulent energy generated by the shear rate.

We now make several important remarks.

On the operation of conditional statistical averaging. Sometimes the value 〈Y1jY2〉 is also called "conditional" mean that makes some confusion in comparison with the conditional means 〈Y1jY2〉r, r = t or r = n. Indeed, variable 〈Y1jY2〉 ¼ γ〈Y1jY2〉<sup>t</sup> þ ð1 � γÞ〈Y1jY2〉<sup>n</sup> where 〈Y1jY2〉<sup>t</sup> and 〈Y1jY2〉<sup>n</sup> are the conditional means of the characteristics for the turbulent and nonturbulent fluid, respectively. So, the value of 〈Y1jY2〉 actually is the total average of the random variable Y1, obtained under the condition of the variable Y2.

On the source terms. The known equations for conditional means contain the source terms, which are intended to describe the increase in volume of the turbulent fluid downstream. Here, it is interesting to discover the reasons of such source terms appearance. For this, we consider the procedure of statistical "unconditional" averaging of the continuous equation, premultiplied by the intermittency function

$$I\frac{\partial u\_k}{\partial \mathbf{x}\_k} = 0 \to I\frac{\partial u\_k}{\partial \mathbf{x}\_k} \to \left\langle I\frac{\partial u\_k}{\partial \mathbf{x}\_k} \right\rangle = \gamma \left\langle \frac{\partial u\_k}{\partial \mathbf{x}\_k} \right\rangle\_t = 0 \tag{85}$$

However, the permutation of averaging and differentiation operations, used in the approach, gives

$$
\left\langle I \frac{\partial \boldsymbol{u}\_k}{\partial \mathbf{x}\_k} \right\rangle = \left\langle \frac{\partial \boldsymbol{I} \boldsymbol{u}\_k}{\partial \mathbf{x}\_k} - \boldsymbol{u}\_k \frac{\partial I}{\partial \mathbf{x}\_k} \right\rangle \to \frac{\partial \langle \boldsymbol{I} \boldsymbol{u}\_k \rangle}{\partial \mathbf{x}\_k} - \left\langle \boldsymbol{u}\_k \frac{\partial I}{\partial \mathbf{x}\_k} \right\rangle \to \frac{\partial \gamma \langle \boldsymbol{u}\_k \rangle\_t}{\partial \mathbf{x}\_k} - \left\langle \boldsymbol{u}\_k \frac{\partial I}{\partial \mathbf{x}\_k} \right\rangle = \mathbf{0} \tag{86}
$$

i.e., gives rise to the appearance of the source terms of a singular type. It stands to reason that the appearance of such source term is only due to the accepted commutation of the averaging operation of the partial derivates and has no physical justification.

On the mathematical model for the turbulent fluid flow. The ASMTurb method allows us to construct a model for the turbulent fluid flow without considering the nonturbulent fluid flow. As far as all mixing turbulent processes take place only in the turbulent fluid, this peculiarity essentially simplifies the modeling. Even more, this approach allows us to take into account the source term, using one of the semi-empirical parameters of the mathematical model. To solve the "pulsation" task we use the turbulent-kinetic-energy budget equation. To distribute the intensity of the longitudinal velocity pulsations we use the ratio 〈u0<sup>2</sup> <sup>r</sup> 〉<sup>r</sup> ffi 〈Er〉r.

Figure 9. (a) Unconditional mean longitudinal velocity RANS 〈U〉<sup>R</sup> ¼ 〈u〉R=u1. (b) Profiles of the normalized unconditional mean RANS turbulent kinetic energy 〈E〉�<sup>R</sup> <sup>¼</sup> 〈E〉R=ΔU<sup>2</sup> .

What gives the ASMTurb method. The results of testing the ASMTurb method showed a "surprising" precision for the turbulent flows modeling—calculations of the conditional and total averages of statistical characteristics practically completely agreed with the known measurements [20] (see Figures 7 and 8 where curves—our calculation, symbols—experimental data are mentioned[20]).

What gives the RANS method. The some results of the unconditional means, obtained by the RANS model, are presented in Figure 9. As can be seen, the RANS model does not gives good results.

So, the ASMTurb differential equations for the conditional averaged characteristics of the turbulent and nonturbulent fluid flows coincide with each other in external view. Moreover, the RANS differential equations have the same external view. However, the boundary conditions and closure hypothesis for the turbulent and nonturbulent fluid flows in the ASMTurb models may be different. It is this circumstance allows us to construct highly efficient ASMTurb models of turbulent flows. The RANS method does not have this property and thus a searching for the "satisfactory" closure hypotheses for the RANS models will not give good results.

## Author details

Yuriy Nuzhnov

nonturbulent fluid in the turbulent fluid (as a generator of the turbulent fluid) occurs in the superlayer. Even more, the ASMTurb method allows us to construct the model only for the turbulent fluid flow, without considering the nonturbulent fluid flow. As far as all the mixing turbulent processes (and, as consequence, the processing modeling of turbulent heat and mass transfer) take place only into the turbulent fluid, this peculiarity essentially simplifies the modeling of such processes. Especially it refers to the turbulent combustion processes, in which modeling is attended by difficulties. It is important to note that ASMTurb SS (41) and (42) for conditional means of the turbulent and nonturbulent fluid differ from the known ones (for example [7]). It should be emphasized that the presented model contains only two empirical parameters σ<sup>t</sup> and νEt. With regard to these parameters, it must be said that their appearance is due to the fact that we do not know neither the expansion rate of the turbulent fluid downstream nor the maximum value of the turbulent energy generated by the shear rate.

On the operation of conditional statistical averaging. Sometimes the value 〈Y1jY2〉 is also called "conditional" mean that makes some confusion in comparison with the conditional means 〈Y1jY2〉r, r = t or r = n. Indeed, variable 〈Y1jY2〉 ¼ γ〈Y1jY2〉<sup>t</sup> þ ð1 � γÞ〈Y1jY2〉<sup>n</sup> where 〈Y1jY2〉<sup>t</sup> and 〈Y1jY2〉<sup>n</sup> are the conditional means of the characteristics for the turbulent and nonturbulent fluid, respectively. So, the value of 〈Y1jY2〉 actually is the total average of the

On the source terms. The known equations for conditional means contain the source terms, which are intended to describe the increase in volume of the turbulent fluid downstream. Here, it is interesting to discover the reasons of such source terms appearance. For this, we consider the procedure of statistical "unconditional" averaging of the continuous equation,

random variable Y1, obtained under the condition of the variable Y2.

120 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

¼ 0 ! I

operation of the partial derivates and has no physical justification.

the intensity of the longitudinal velocity pulsations we use the ratio 〈u0<sup>2</sup>

∂uk ∂xk

! <sup>∂</sup>〈Iuk〉 ∂xk

! I

∂uk ∂xk 

� uk

However, the permutation of averaging and differentiation operations, used in the approach,

i.e., gives rise to the appearance of the source terms of a singular type. It stands to reason that the appearance of such source term is only due to the accepted commutation of the averaging

On the mathematical model for the turbulent fluid flow. The ASMTurb method allows us to construct a model for the turbulent fluid flow without considering the nonturbulent fluid flow. As far as all mixing turbulent processes take place only in the turbulent fluid, this peculiarity essentially simplifies the modeling. Even more, this approach allows us to take into account the source term, using one of the semi-empirical parameters of the mathematical model. To solve the "pulsation" task we use the turbulent-kinetic-energy budget equation. To distribute

¼ γ

∂I ∂xk 

∂uk ∂xk 

t

! <sup>∂</sup>γ〈uk〉<sup>t</sup> ∂xk

� uk

∂I ∂xk 

<sup>r</sup> 〉<sup>r</sup> ffi 〈Er〉r.

¼ 0 ð86Þ

¼ 0 ð85Þ

We now make several important remarks.

premultiplied by the intermittency function

<sup>¼</sup> <sup>∂</sup>Iuk ∂xk

gives

I ∂uk ∂xk  ∂uk ∂xk

> � uk ∂I ∂xk

Address all correspondence to: ynuzhnov@gmail.com

Institute of Combustion Problems, Almaty, Kazakhstan

## References

