Theory of turbulence and modeling of turbulent transport in the atmosphere

The Reynolds number calculated on the dynamic roughness parameters is given by

.

This statistical relationship between the dynamic roughness parameters corresponds to the special type of the Nave-Stocks equation solution transformation (2.4). Let's consider small variation of the second velocity scale around the mean value . In this case in the first equation (2.16) and this parameter can be written as follows

where is the parameter characterising the dynamic roughness structure. If , then . This case corresponds to the special type of the dynamic roughness composed of furrows elongated along the streamlines of the mean flow. If , then . It can be considered as the relationship between the scales of the dynamic roughness elements in the X- and Y-directions. In this case fluctuates around the mean value, , and produces the velocity and pressure fluctuation. The velocity gradient on the wall fluctuates with as follows

For we have . Therefore in this case the velocity gradient on the wall is given by

(2.51)

The pressure gradient on the wall also depends on the second velocity scale as

(2.52)

The velocity components near the smooth wall for can be written as follows (see Kutateladze [62])

,

where are the viscous sublayer functions. Substituting the velocity approximation formulas in the x-component of momentum equation and supposing that one can derive

where is the restricted value for . Using equations (2.51)-(2.52) finally we have the dynamic roughness surface equation in case of the steady turbulent boundary layer over a smooth surface

(2.53)

where .

The steady turbulent flow dynamic roughness is realised for . In this case equation (2.53) can be transformed into the quasi-linear differential equation

(2.54)

where .

The point in which is the singular point of the equation (2.54). In this point . As it has been estimated in the numerical experiments for the mean flow, therefore . For the stationary case, i.e. for , the equation (2.54) can be written in the quasi-elliptical form

, (2.55)

were , .

Using the function one can calculate the dynamic roughness surface parameters as follows

 (2.56)

In the special case when , equation (2.55) has the periodical solution

where is the amplitude, is the wave number in the y-direction. Therefore the transversal length scale of coherent structures can be estimated as

The predicted length scale is in a good agreement with the experimental value, , obtained by Kline et al. [70]. This type of coherent structures corresponds to the furrows considered above.

In the special case when , the periodical solution of the equation (2.55) is given by

where . Therefore in this case depends on the amplitude . The periodical solution of the first equation (2.56) can be written as

, (2.57)

.

The transversal phase velocity of the dynamic roughness surface disturbances can be determined as . The dynamic roughness length scale depends on the amplitude as follows

.

For the estimated streamwise length scale of coherent structures agrees with the experimental value obtained by Blackwelder & Eckelmann [74], and discussed by Cantwell [52].

In this paper the problems of non-linear theory of turbulent boundary layer have been studied. The algorithm of numerical solution of the problem has been considered. Equation is deduced, connecting constants of non-linear theory.

A fundamental parameter of the turbulent boundary layer length has been determined, which coincides with the position of velocity peak of turbulence energy generation according to Klebanoff [49] and Laufer's [50] data. It is shown that the profile of an average velocity in the boundary layer can be described satisfactory, using only one constant. The Karman constant can be used for this purpose. The second constant of the logarithmic profile can be estimated within this theory. Velocity profile, calculated according to the model suggested, conforms well to the data of direct numerical modelling, to experimental data and models of other authors. Results of velocity intensity pulsation modelling have been presented, as well as their compliance with the results of direct numerical modelling and experimental data. A model of dynamic roughness in turbulent boundary layer has been suggested. It has also been shown that in a stationary case there are two types of periodic solutions. One solution corresponds to dynamic roughness in a kind of furrows, stretched along the main flow. The viscous flow over the structures is a physical mechanism of formation of logarithmic profile of velocity. The second solution corresponds to the perturbations of limited amplitude which have a limited length in the direction of the mean flow. It is shown that the parameters of dynamic roughness, having been calculated on the base of this model, coincide with the data of experiments.

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