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 (
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
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
The mathematical body of this theory may be developed from both the theoretical-probabilistic approach, based on the
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
2.1.1. The introduction of the sample space into the body of statistics
For the introduction of the sample space
In other words, for every fixed point
So, from the physical space
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
2.1.2. The introduction of the algebra of events and PDFs
Let us introduce a one-point probability density function
where
For introduction of the algebra of events, we suppose that the space
i.e., the generalized set
at that the set of values
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
where
and we call this function as the “
and called the “
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
with the explicit dependence on
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
and also the
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
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
because that
where
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
that proves the rule of permutation of the operation of unconditional averaging of derivatives in the method of
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
with index
because that the function
as
To conduct the operation of conditional statistical averaging of the Navier-Stokes SE (33), we use the CPDF
Applying the rule of permutation (27) and using the Reynolds development
where the fluctuating velocity of the turbulent or nonturbulent fluid flow
The derivation of the turbulent kinetic energy budget equation by the
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
3.1. Mathematical tools of the ASMTurb method
3.1.1. General system of equations for conditional means
According to the
and
that describe the conditional mean flow characteristics of each of the intermittent media with the turbulent (
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
4. Testing of the ASMTurb method
The
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
4.1. Construction of the model for two-stream plane mixing layer
The mathematical
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
If we substitute this expression into Eq. (59) and compare the components at the same powers of parametric value (
From the boundary conditions (56) it follows that
where we get after integration
where
where
To determine the value
The boundary conditions (56) because of
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
was found by numerical calculation. At that according to Eq. (60), function
Hence, the values of the constants are defined with the help of numerical calculation,
For completion of Eq. (74) was used the known expressions (43)–(45) with index
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
and
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
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
We now make several important remarks.
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.
So, the
References
- 1.
Townsend AA. The Structure of Turbulent Shear Flow. Cambridge University Press; 1956 - 2.
Libby PA, Williams FA. Turbulent Reacting Flows. Springer-Verlag; 1980 - 3.
Kuznetsov VR, Sabel’nikov VA. Turbulence and Combustion. Hemisphere Publishing Corporation; 1990 - 4.
Pope SB. Turbulent Flows. Cambridge University Press; 2000 - 5.
Nuzhnov YV. Method of autonomous modeling of turbulent flows (ASMTurb). IP 0010816, Bulletin No. 1392, issued on 21 October 2013. (in Russian) - 6.
Libby P. On the prediction of intermittent turbulent flows. Journal of Fluid Mechanics. 1975; 68 (2):273–295 - 7.
Dopazo C. On conditioned averages for intermittent turbulent flows. Journal of Fluid Mechanics. 1977; 81 (3):433–438 - 8.
Dopazo C, O’Brien EE. Intermittency in free turbulent shear flows. In: Durst F, Launder BW, Schmidt FW, Whitelaw JH, editors. Turbulent Shear Flows. 1st ed. New York: Springer-Verlag; 1979. pp. 6–23 - 9.
Nuzhnov YV. Conditional averaging of the Navier-Stokes equations and a new approach to modeling intermittent turbulent flows. Journal of Fluid Dynamics. 1997; 32 :489–494 - 10.
Nuzhnov YV. On the theory of turbulent heat and mass transfer with allowance for intermittence effects. Journal of Engineering Physics and Thermophysics. 2011; 84 :160–170 - 11.
Monin AS, Yaglom AM. Statistical Fluid Mechanics: The Mechanics of Turbulence. MIT Press; 1971 - 12.
Kolmogorov AN. Foundations of the Theory of Probability. Chelsea Publishing Company; 1956 - 13.
Feller W. Introduction to Probability Theory and its Applications. Wiley; 1957 - 14.
Rozanov YA. Probability Theory: A Concise Course. Dover Publications Incorporated; 1977 - 15.
Nuzhnov YV. The Method of Autonomous Statistical Modeling ASMTurb and its Testing on the Example of Classical Turbulent Flows. In: ASME Congress (IMECE), Volume 7: Fluids Engineering Systems and Technologies. Quebec, Montreal; 2014. - 16.
Nuzhnov YV. Statistical modeling of the intermittent turbulent flows (Қaзaқ yнивepcитeтi, Aлмaты, 2015 (in Russian)). - 17.
Pope SB. Calculations of a plane turbulent jet. AIAA Journal. 1984; 22 (7):896–904 - 18.
Wygnanski G, Fiedler HE. The two-dimensional mixing region. Journal of Fluid Mechanics. 1970; 41 :327-361 - 19.
Dimotakis PE. Two-dimensional shear-layer entrainment. AIAA Journal. 1986; 11 :1791–1796 - 20.
Spenser BW, Jones BG. Statistical investigation of pressure and velocity fields in the turbulence two-stream mixing layer. AIAA Journal. 1971. Paper No. 71–613