 # Semiempirical Hypotheses of the Theory of Turbulence in the Anisotropic Boundary Layer

## Main Principles of the Semiempirical Theory of Turbulence

As is well-known, the theory of turbulence is based on the description of fluid and gas flows by equations of flow dynamics. The complete statistical description of random hydrodynamic fields is given by the characteristic functional [1,2,4,5]. The characteristic functional contains information about the infinite set of field moments and satisfies the dynamic equations with functional derivatives. Now there are well-known acceptable methods for solution of such equations (see, e.g., Chapter 2). At the same time, for many practical problems, it is sufficient to determine only lowest-order statistical moments. That is why the studies in the theory of turbulence are traditionally based on the Reynolds system of equations being a result of averaging of the flow dynamics equations [1—7]. However, in the Reynolds system of equations, the number of unknowns exceeds the number of equations. This system is usually closed by specifying some relationships between moments of hydrodynamic fields. These relationships found from experiments or derived from physical reasoning (e.g., from dimensional considerations) are called semiempirical hypotheses of the theory of turbulence.

The main semiempirical hypotheses can usually be reduced to specification of the relationship between the second moments of pulsations (i.e., deviations from the average) of the velocity (v/ • v'j) and temperature (v'j -T') and the averaged fields of velocity Щ and temperature T. These hypotheses are usually based on the analogy between the turbulent and molecular motions. Thus, the terms v -dvi Idxj and % -dTIdxj, presented in the averaged equations, are proportional to the components of the momentum and heat fluxes. Here, v is the kinematic viscosity and X is the temperature conductivity. They describe a medium without turbulence and are caused by molecular diffusion. In the turbulent medium, these components are supplemented with the terms — (vjvj), and — (vj -T'). Therefore, these characteristics can be considered as components of the turbulent momentum and heat fluxes. Within the scope of the semiempirical theory, the structure of the dependence of turbulent momentum and heat fluxes on Щ and T is the same as in the case of purely molecular diffusion. In the general case of anisotropic turbulence, it was assumed that where the repeating subscripts imply summation. The components Ky of the symmetric tensor K in definitions (3.1) are called the coefficients of turbulent viscosity, while the components KTy of the tensor KT have the meaning of the coefficients of turbulent temperature conductivity or coefficients of turbulent diffusion for a passive admixture, which is the potential temperature T (in the boundary layer, the ordinary and potential temperature may be considered identical). Hypotheses (3.1) replace 12 components of turbulent fluxes of momentum and heat with 27 new characteristics (by six components in symmetric tensors Kj and Фу nine in tensor KTy three derivatives dT/dxj and three components in the sum (v'n • vn)).

It was shown in References 1, 2, 4—8 that in the plane-parallel flows (between separated planes and in pipes) turbulent phenomena in the boundary layer are well described by the semiempirical hypotheses with application of only two scalar parameters K and KT (which are called the coefficients of turbulent viscosity (turbulent exchange) and turbulent temperature conductivity, respectively). Turbulence in the boundary layer of the Earth’s atmosphere can be considered as a particular case of the plane-parallel flow, if only we consider flows over extended surface area having a smooth, homogeneous (identical in the structure), and uniformly heated surface. As can be seen from Equation 3.1, for the plane-parallel flows the tensors K and KT are isotropic (Kj = K • 6у KTy = KT- 6ij). In this connection, the boundary layer with the isotropic tensors K and KT will be referred to as the isotropic boundary layer for brevity. If at least one of the tensors K and KT is anisotropic, then the boundary layer is referred to as anisotropic .

In practice, however, it is necessary to specify these definitions. Thus, the isotropic boundary layer appears to be the more general concept than the boundary layer in plane-parallel flows. In contrast to plane-parallel flows, in the isotropic layer, the conditions that the horizontal derivatives and the vertical component

of the average velocity are nonzero can take place in the general case. It is natural to call this case weakly isotropic, leaving the concept of the isotropic (or strongly isotropic) boundary layer only for plane-parallel flows. The analogous separation can also be done for the anisotropic boundary layer. Thus, if one of the tensors K or KT is anisotropic, then the boundary layer can be called weakly anisotropic. If the both tensors K and KT are anisotropic, then the layer can be called strongly anisotropic.

The concept of the isotropic boundary layer (for plane-parallel flows) is not connected with the isotropy of hydrodynamic fields themselves. In the isotropic layer, there is a preferred direction (distance from the boundary plane); therefore, the fields are not isotropic.

The components of the tensors K and KT can be represented in the form of the products of the root-mean-square value of velocity pulsations by the components of the tensors of turbulence scales lj and lTj (i.e., the scales which are the average distances, turbulent formations can move to, keeping their individuality): For isotropic tensors K and KT the ellipsoids of the scales lj and lTij transform into spheres. In the general case, the tensors Kj and KTj do not coincide. The variability of the temperature scales lTj is usually higher than that of the velocity scales 1{. This can be seen in the case of free convection when there are no wind and no friction. Then turbulence receives energy from the energy of temperature instability, rather than from the energy of the averaged motion, and has the character of vertical thermal flows. Therefore, in the first approximation, the tensor K in Equation 3.1 can be believed isotropic (the tensor (vjvj) still remains anisotropic). Then, according to References 1, 7, and the number of unknowns decreases down to 22.

The semiempirical hypotheses are actively used in the studies on the turbulent diffusion of passive admixtures, including the temperature diffusion. Hypotheses (3.2) are usually taken as a basis. The equations for the coefficients K and KTij, accepted now, are the results of generalization of experimental data obtained over approximately smooth surface (i.e., not in mountain regions). They take into account the action of thermal stratification. For the average wind velocity directed along the axis x1, we have Atmospheric Turbulence in the Anisotropic Boundary Layer ? 81 Here, ж = 0.4 is the Karman constant; z is the height above the underlying surface (x1 = x, x2 = y, x3 = z); p(Q is the universal similarity function specifying the type of stratification (see Figure 3.1); Pr is the turbulent Prandtl number. Here also the error of determination of the coefficients does not exceed 30% .

The similarity function depends on the stratification parameter Z = z/L, in which the scale of the length L is called the Monin—Obukhov scale (or the thickness of sublayer of dynamic turbulence). The scale L has the essential significance in the theory of thermally stratified atmosphere. It was introduced by Monin and Obukhov  from dimensional considerations and is described by the equation where g is the gravity acceleration, T is the average value of the absolute temperature, V* is the friction velocity (or the turbulent scale of velocity), and T* is the Figure 3.1 Universal similarity function constructed by sewing of empirical values. The top left plot: data are taken from Reference 8 with known asymptotics : (0 = 0.4 (—Z)-1/3, Z < 0, ia » Zo; ?>(0 = 7.0 Z, Z » Zo; Zo = 0.05. The error of measurement  of the coefficient of 0.4 was about 20%, while that for the coefficient of 7.0 was about 40%.

turbulent scale of the temperature field. For the neutral stratification, the parameter Z coincides with the dynamic Richardson number Rf (Rf = (7^(0, Ri = Rf/a). Therefore, by analogy with the Richardson number, the parameter Z is often called the Monin—Obukhov number. Experimental data for the ratio a(Z) = K-JK show that at the neutral (|Z| < 0.05) and unstable (Z < -0.05) stratifications, the value of a(Z) is close to constant, a и 1.17, Pr и 0.85 [1,2]. However, at the stable stratification (Z > +0.05), it can decrease markedly (at high stability).

In Equations 3.3 and 3.4, the coefficient of turbulent viscosity K(z) corresponds to the isotropic boundary layer, in which the characteristics V,, T,, and T are believed to be constant over all the layers. Therefore, the Monin—Obukhov scale L and the Monin—Obukhov number Z (at a given height z) are numerical parameters of the turbulent flow over the entire considered temperature-stratified area of the Earth’s surface. As can be seen from Equation 3.3, the tensor KT is anisotropic, that is, boundary layer (3.2)—(3.4) is anisotropic as well. Since the anisotropy of boundary layer of turbulent diffusion (3.2)—(3.4) is caused only by the anisotropy of the temperature tensor KTij, while the other characteristics correspond to the isotropic layer, boundary layer (3.2)—(3.4) is weakly anisotropic.

Equations 3.2 through 3.4 are basic in the theory of similarity of turbulent flows in the atmosphere. This theory is usually referred to as the Monin—Obukhov similarity theory. Turbulent flows in mountain regions is of particular interest. Here, we could not expect the constant Monin—Obukhov scale over the entire territory. Over the mountain terrain, stable vortices are formed. Distortions of air flows from such rotor formations are observed up to high altitudes (from a mountain, for example, from 1 km-height up to 7—9 km ). At the same time, in atmospheric- optical studies, especially, in investigations on the influence of turbulence on the quality of optical images, we often have to deal with the anisotropic boundary layer in mountains (to decrease turbulent distortions, ground-based receiving telescopes are often installed on mountain tops). However, turbulence models developed for the isotropic boundary layer are usually inapplicable in mountains.

The applicability of the model of anisotropic layer (3.2)—(3.4) for mountains was not assessed yet. Therefore, it is interesting to experimentally check semiempirical hypotheses (3.1) or (3.2) directly for mountain conditions. Earlier this check was not carried out in the needed volume. This is connected with the need of recording (at every point of the mountain area of the surface) experimental data simultaneously for the large number of parameters.

In this subsection we established, following Reference 9, that the similarity theory of turbulent flows can be extended to an arbitrary anisotropic boundary layer. With the use of semiempirical hypotheses of the theory of turbulence, it was shown theoretically and experimentally that the arbitrary anisotropic boundary layer can be considered as locally weakly anisotropic. Excepting for the tensor of coefficients of turbulent temperature conductivity, the statements of the theory of isotropic layer (for plane- parallel flows) become true in the vicinity of every point of the layer. In the arbitrary boundary layer, the main parameter of turbulence is the variable Monin—Obukhov number. It was found that the anisotropic boundary layer can be replaced with the effective isotropic layer. Theoretical equations have been derived for the vertical outer scale of turbulence in the anisotropic boundary layer, and the agreement between experimental and theoretical values of the outer scale has been demonstrated. 