## 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 space-time 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 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 co-current streams, Figure 1. With that at point **x** = **x**_{0}, 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**_{0} 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

where

and

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 theoretical-probabilistic 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 *Ω* – is the sample space, *is σ*-algebra of subsets, *P*– is the probability measure in

#### 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**_{0} of statistically stationary turbulent flow with strong intermittency, i.e., when *f*(**x**, *t*) forms a random continuous field in the space **x**_{0} ∈ *D* we have a continuous random varying function of time *f*(**x**_{0}, *t*). Let the measurements of *f* = *f*(**x**_{0}, *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, *τ*_{0}]. It allows us to form an ensemble of values *Ω* as the one-point countable set of elementary events *f*, if we split the range of values of function *f*(**x**_{0}, *t*) at sufficiently small intervals Δ*f*, and the range [0, *τ*_{0}] at sufficiently small intervals Δ*t*, Figure 2. Having fixed a certain value of function *f*_{i} in each of the selected intervals Δ*f* 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 *N*_{i} sampled values *f*_{i} in the form of shaded elementary cells Δ*f* Δ*t*, Figure 2 (the selection of one particular value *f*_{i} from these cells plays no special role due to their small value). The total number of cells *N* = *τ*_{0} / Δ*t* and represented as the ensemble of values *f*, and also in the limit Δ*f* → 0 and Δ*t* → 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** = **x**_{0}, the total number of all sample values *f*_{i} forms a random continuous field of values *f* ∈ *Ω* in the physical space *G* = *D* × [0, *τ*_{0}]. 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

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.

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

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:

where

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

i.e., the generalized set *Ω* = *Ω*_{1} + *Ω*_{2}, and we have *σ*-algebra of the subsets. The indicators of these subsets are the characteristic functions

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

where

### 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

where **P**(*Ω*_{t}) и **P**(*Ω*_{n}) – conditional random set of value *f*, belonging to the turbulent **x**; the values

where **x**. In actual fact, if the flow is intermittent, it has a dual structure [1] and in the generalized set we have *Ω* = *Ω*_{t} + *Ω*_{n} so that the measures of sample sets *γ*_{t} and *γ*_{n} 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* = *Ω*_{R}, 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

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

and called the “*unconditional*” PDF *p*_{R}(*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:

with the indicator (characteristic) function of the turbulent fluid

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*_{n} = 1 − *γ*, is transformed into

with the explicit dependence on **x**, while the CPDF *p*_{t}(*f*) and *p*_{n}(*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

and also the *total* statistical average

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

At that by definition the value

whence it follows that

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* 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

because that in accordance with [14]

and

where *f* in turbulent (*r* = *t*) or nonturbulent (*r* = *n*) medium for all possible fixed values of *ξ*. At that *f, ξ*) does not depend obviously on the coordinate *x*. On the flip side, we have

because that

where *f*. As a result, we have

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 *r* = *R*, i.e.,

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 *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

and

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

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

With regard to the total statistical averaging of time derivatives, instead of expression (31) we have

#### 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:

where *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

with index *r* = *t* for turbulent and *r* = *n* for nonturbulent fluid,

because that the function *x*_{k}, i.e., *k* = 1 in Eq. (35), we deduce

as *du*_{0} = 1, for example, *k* = 2, 3 using *p*_{r}(*u*_{2}) and *p*_{r}(*u*_{3}). Now then, the conditionally averaged continuity equation for each of the intermittent media of turbulent flow has the form

To conduct the operation of conditional statistical averaging of the Navier-Stokes SE (33), we use the CPDF *k* = 1, 2, 3. Then, according to Eq. (25) for the momentum equation in SE (33) we obtain *ξ*_{2}:

Applying the rule of permutation (27) and using the Reynolds development

where the fluctuating velocity of the turbulent or nonturbulent fluid flow *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.,

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:

Hereinafter

## 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

and

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

#### 3.1.2. The closure hypothesis

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

where

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

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

This equation can be obtained according to our theory. In actual fact, for the velocity pulsations we have

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.,

where

Let us note that Eq. (48) can also be obtained as Eq. (47). According to Eq. (47), wherein

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]:

## 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 cross-streamlined 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*_{1} = *u*_{max} and *u*_{1} = *u*_{min}, 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

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

with boundary conditions, which initially assuming as asymptotical, namely

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

where

where

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

the while correlation in Eq. (52) is

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

where *σ*_{t} 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 *F*_{t}(*η*) as a power series in the small parameter (*m* − 1):

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

From the boundary conditions (56) it follows that

where we get after integration

where *η* = 0. Now the SE (61) takes the form of

where

To determine the value *η*_{0t} will use the known Karman’s condition, namely

The boundary conditions (56) because of

*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

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

where

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*_{t} = 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 *γ* = 1). Therefore, for specification of our model, we must consider the second approximation of our task.

*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)

was found by numerical calculation. At that according to Eq. (60), function *c*_{1t} и *c*_{2t}. To determine these constants, the integral relation was involved (for example, see [16]):

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

*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 *η*_{0t} = −0.0181 when *σ*_{t} = 21.5 (in [20] empirical value *η*_{0S} = −0.02, i.e., we have a good accuracy for

*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:

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

Here

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:

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

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

Eq. (75) was solved by the numerical method (mathematical package *MathCad* was used). Interestingly, the solution of Eq. (75) with using asymptotic boundary conditions *m* = 0.305)

and

*The results of our calculations of conditional means* of this *“fluctuating” task* are presented in Figure 5. Figure 5a shows the calculation

#### 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 regions

#### 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 *RANS* model constructed for the mixing layer. At that the empirical constants *γ* calculation with the experimental data. Figure 6 presents the 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

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*_{t} and

We now make several important remarks.

*On the operation of conditional statistical averaging*. Sometimes the value *r* = *t* or *r* = *n*. Indeed, variable *total* average of the random variable *Y*_{1}, obtained under the condition of the variable *Y*_{2}.

*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

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

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

*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.