Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1960, Volume 135, No. 2
Hydromechanics
V. G. Nevzglyadov
On the Theory of Anisotropic Turbulence
(Presented by Academician V. A. Fock, 28 March 1960)
The task of a phenomenological theory consists in constructing, in covariant form, a closed system of equations describing experimental facts over as broad a range as possible. In recent years a series of works has appeared on measuring the characteristics of turbulence in the boundary layer. The results of the measurements do not fit within the framework of the simple theory previously proposed by the author \((^{1})\). In the present paper considerations are advanced which generalize the theory and make it capable of describing the new facts.
§ 1. By an isotropic relation between the tensor of turbulent stresses
\(\Pi_{ik} \equiv - \rho \overline{u'_i u'_k}\), where \(u'_i\) are the fluctuating velocities, and the strain-rate tensor
\(\dot e_{ik} = \dfrac{1}{2}\left(\dfrac{\partial \bar u_i}{\partial x_k} + \dfrac{\partial \bar u_k}{\partial x_i}\right)\), we shall mean a relation of the form
\[ \hat{\Pi}_{ik} = 2K\dot e_{ik}, \tag{1,1} \]
where \(\hat{\Pi}_{ik}\) is the deviator, namely
\[ \hat{\Pi}_{ik} \equiv \Pi_{ik} + \Pi\delta_{ik}; \qquad \Pi \equiv -\frac{1}{3}(\Pi_{11} + \Pi_{22} + \Pi_{33}) \tag{1,2} \]
and \(K\) is a scalar coefficient of viscosity, which on general grounds should be assumed to be a function of the turbulent pressure \(\Pi\). The existence of the isotropic relation (1,1) does not mean that the turbulence is isotropic in the generally accepted sense of this term, since the diagonal components of the deviator (1,1), generally speaking, are not equal to zero. The isotropic equations of state (1,1) of the theory \((^{1})\) also describe anisotropic turbulence.
However, in a steady flow in a circular pipe the tensor \(\dot e_{ik}\) has only one nonzero component, \(e_{12} = \dfrac{1}{2}\dfrac{du}{dr}\), and therefore (1,1) gives
\[ \hat{\Pi}_{11} = \hat{\Pi}_{22} = \hat{\Pi}_{33} = 0; \qquad \Pi_{12} = K \frac{du}{dy}. \tag{1,3} \]
Laufer \((^{2})\) measured the distributions of \(\overline{u'_i u'_k}\) along the radius of a pipe of circular cross-section. It was found that the curves for \(\overline{u_1'^2}\), \(\overline{u_2'^2}\), and \(\overline{u_3'^2}\) do not coincide. They are close in the core of the flow, but diverge appreciably as the wall is approached. All of them have a sharp maximum near the wall. The turbulent pressure \(\Pi\), calculated from Laufer’s data, increases monotonically from the pipe axis to the wall, reaching a maximum at a distance of the order of \(0.01a\) (\(a\) is the radius), and then rapidly falls to zero. The maximum value of \(\Pi\) is 4–5 times greater than on the axis. Laufer’s data do not agree with (1,3). He obtained
\[ \Pi_{11} \ne \Pi_{22} \ne \Pi_{33}; \qquad \Pi_{12} - K\frac{du}{dy} \equiv F_{12} \ne 0, \tag{1,4} \]
with \(F_{12} \cong 0\) in the core of the flow.
Laufer’s data show agreement of the experiment with the isotropic relation (1.1) in the core of the flow and a deviation from it near the wall. We shall call this deviation anisotropic turbulence.
§ 2. Denote the difference of the tensors by
\[ \hat{\Pi}_{ik}-2K\dot e_{ik}\equiv \hat F_{ik};\qquad \hat F_{ik}=F_{ik}-{}^1\!/\!_3\,F_{ss}\delta_{ik}. \tag{2.1} \]
\(F_{ik}\) is an additional tensor which must describe the observed departure from the isotropic relation; \(\hat F_{ik}\) is its deviator. For \(F_{ik}\) one must formulate equations of state expressing it in terms of the basic quantities, which for an incompressible fluid are
\[ \bar u_i,\ \Pi,\ \bar p. \tag{2.2} \]
In addition, we add two unit vectors: \(\mathbf{s}\), in the direction of the flow, and \(\mathbf{v}\), along the principal normal to the trajectory; in the boundary layer this second vector may be replaced by \(\mathbf{n}\), the normal to the wall directed into the flow.
In forming \(F_{ik}\), we form derivatives from (2.2) and then use multiplication and contraction of tensors; here we assume that the equations of state depend only on the “internal state” of a fluid particle, and therefore we do not use tensors of the form \(\bar u_i A_k\). In this way a large number of tensors can be constructed. We divide them into two groups: the group of tensors that do not have the character of viscous stresses, i.e., do not depend on \(\dot e_{ik}\), we shall denote by \(P_{ik}\); they are analogous to thermal stresses in the broad sense of the word (analogous to certain tensors appearing in the known Burnett equations used in gas dynamics), and the group of tensors characterizing the viscous stresses proper, i.e., essentially dependent on \(\dot e_{ik}\), we denote by \(T_{ik}\); they may be given the form
\[ T_{ik}=2K_{ikmn}\dot e_{mn};\qquad \hat T_{ik}=2\hat K_{ikmn}\dot e_{mn}, \tag{2.3} \]
where \(K_{ikmn}\) is a fourth-rank viscosity tensor describing anisotropic viscosity, with
\[ \hat K_{ikmn}\equiv K_{ikmn}-{}^1\!/\!_3 K_{ssmn}\delta_{ik}. \tag{2.4} \]
The symmetry of the tensors \(T_{ik}\) and \(\dot e_{mn}\) requires the equalities
\[ K_{ikmn}=K_{iknm}=K_{kimn}=K_{kinm}. \tag{2.5} \]
The general form of the deviator of the stress tensor now becomes
\[ \hat{\Pi}_{ik}=2\left(K\delta_{im}\delta_{kn}+\hat K_{ikmn}\right)\dot e_{mn}+\hat P_{ik}. \tag{2.6} \]
We shall narrow the number of tensors to be discussed by subjecting them to the requirement: in the core of the flow the isotropic relation (1.1) is valid and there are no nonviscous stresses, i.e.,
\[ \hat P_{ik}\to 0;\qquad K_{ikmn}\to 0 \quad \text{in the core of the flow.} \tag{2.7} \]
The simplest tensors \(P_{ik}\) satisfying (2.7) are
\[ P^{(1)}_{ik}=a_1\frac{\partial \Pi}{\partial x_i}\frac{\partial \Pi}{\partial x_k};\qquad P^{(2)}_{ik}=a_2\left(\frac{\partial \bar p}{\partial x_i}\frac{\partial \Pi}{\partial x_k}+\frac{\partial \bar p}{\partial x_k}\frac{\partial \Pi}{\partial x_i}\right). \tag{2.8} \]
The tensor of anisotropic viscosity, in general, depends on \(\dot e_{ik}\), \(\Pi\), \(\bar p\) and on their derivatives. In forming, by multiplication, the simplest tensors satisfying (2.7), we consider, all other conditions being equal, the terms with second derivatives to be more substantial than those with products of first derivatives, i.e. li-
linear ones to nonlinear ones:
\[ K^{(1)}_{ikmn}=a_1 \dot e_{im}\delta_{kn}+\ldots;\qquad K^{(2)}_{ikmn}=a_2(\dot e_{im}\dot e_{kn}+\dot e_{km}\dot e_{in}); \tag{2,9} \]
\[ K^{(3)}_{ikmn}=a_3\,\frac{\partial^2 \dot e_{mn}}{\partial x_i\partial x_k};\qquad K^{(4)}_{ikmn}=a_4\left(\frac{\partial^2\Pi}{\partial x_i\partial x_m}\dot e_{kn}+\ldots\right); \tag{2,10} \]
we do not write out the terms needed for the symmetrization of (2,5). Using the characteristic directions important in the boundary layer, \(\mathbf{s}, \mathbf{n}\), we obtain another viscosity tensor of the form
\[ K^{(5)}_{ikmn}=A\,(s_i n_m \dot e_{kn}+\ldots). \tag{2,11} \]
For the tensor of anisotropic viscous stresses, nonlinear dependence on \(\dot e_{ik}\) is characteristic.
§ 3. The role of the tensors introduced may be estimated by considering plane flow. We place the origin of the axes \(Oxy\) on the wall and direct the \(Ox\) axis along the flow, whose width is \(2l\). Then
\[ \overline{\mathbf{u}}=u(y)\mathbf{s};\qquad \Pi=\Pi(y);\qquad \overline p=ax+f(y). \tag{3,1} \]
The constant \(a<0\). The tensors (2,8) take the form
\[ \hat P^{(1)}_{11}=\hat P^{(1)}_{33}=-\frac{1}{2}\hat P^{(1)}_{22} =-\frac{a_1}{3}\left(\frac{d\Pi}{dy}\right)^2;\qquad \hat P^{(1)}_{12}=0; \]
\[ \hat P^{(2)}_{11}=\hat P^{(2)}_{33}=-\frac{1}{2}\hat P^{(2)}_{22} =-\frac{2}{3}a_2\frac{\partial\overline p}{\partial y}\frac{d\Pi}{dy};\qquad \hat P^{(2)}_{12}=a_2\frac{\partial\overline p}{\partial x}\frac{d\Pi}{dy}. \tag{3,2} \]
The viscous-stress tensors \(T^{(1)}_{ik}, T^{(3)}_{ik}, T^{(4)}_{ik}\) describe only the anisotropy of the diagonal terms; for them \(T_{12}=0\), while the tensors \(T^{(2)}_{ik}\) and \(T^{(5)}_{ik}\) describe only shear anisotropy; for them the diagonal terms are zero:
\[ \hat T^{(2)}_{12}=a_2\left(\frac{du}{dy}\right)^3; \tag{3,3} \]
\[ \hat K^{(5)}_{ik12}=0,\quad \text{except } \hat K^{(5)}_{1212}=A\frac{du}{dy};\qquad \hat T^{(5)}_{12}=A\left(\frac{du}{dy}\right)^2. \tag{3,4} \]
Let us note that Prandtl’s mixing-length theory, which gives a coefficient of turbulent viscosity of the form
\[ \rho l^2\frac{du}{dy}, \tag{3,5} \]
is contained as a special case in the viscosity tensor \(\hat K^{(5)}_{ikmn}\), as is evident from comparing (3,4) and (3,5). Prandtl’s theory of viscosity, properly speaking, is a theory of anisotropic viscosity, and its natural generalization to spatial flows is the tensor (2,11), not the invariant \(J\), as Prandtl assumes \({}^{(3)}\).
In a plane flow only one quantity—the pressure \(\overline p\)—depends on \(x\) (see (3,1)), and a single integration of the averaged equations of motion gives the equation:
\[ a(y-l)-\eta\frac{du}{dy} =\hat P_{12}+\left[K+A\frac{du}{dy}+a_2\left(\frac{du}{dy}\right)^2\right]\frac{du}{dy}; \tag{3,6} \]
\(\eta\) is the coefficient of molecular viscosity. To formulate a closed system of equations, the coefficients \(a_i,\alpha_i,A\) must be assumed to be functions of the turbulent pressure \(\Pi\), their form being determined from experiment.
To determine \(\Pi\) as a function of the coordinates and times there is equation (1), which for a plane stationary flow has the form
\[ \hat{\Pi}_{12}\frac{du}{dy} -\frac{d}{dy}(I_2+P_2) +\eta\left(\frac{3}{2\rho}\frac{d^2\Pi}{dy^2} -\frac{\overline{\partial u_i'\,\partial u_i'}}{\partial x_k\,\partial x_k}\right)=0; \tag{3,7} \]
\(I_k\) is the vector of the flux density of turbulent energy
\[ I_k \equiv \frac{1}{2}\rho\,\overline{(u_1'^2+u_2'^2+u_3'^2)u_k'}, \tag{3,8} \]
which is expressed through \(\partial \Pi/\partial x_k\). In the work of W. Dryden (4) data are given indicating the existence of an anisotropic relation between \(I_k\) and \(\partial \Pi/\partial x_i\). This relation in the author’s work (5) is given in the form
\[ I_k=-L_{ki}\frac{\partial \Pi}{\partial x_i}; \qquad L_{ki}=L(\Pi)\delta_{ki}+\hat{L}_{ki}, \tag{3,9} \]
where \(\hat{L}_{ki}\to 0\) in the core of the flow. The vector \(P_k\) has an analogous structure. Equation (3,7) for a plane flow, using (3,6), takes the form
\[ \left[a(y-l)-\eta\frac{du}{dy}\right]\frac{du}{dy} +\frac{d}{dy}\left(L\frac{d\Pi}{dy}\right) +\frac{3}{2}\nu\frac{d^2\Pi}{dy^2} -\eta g\Pi^\alpha=0; \tag{3,10} \]
\(g\) and \(\alpha\) are constants of the theory (1). Adding here equation (3,6), we obtain a system of two equations for determining the two functions \(u=u(y)\) and \(\Pi=\Pi(y)\). The complete coefficient of anisotropic viscosity
\[ K+A\frac{du}{dy}+\alpha_2\left(\frac{du}{dy}\right)^2, \]
in which \(K\), \(A\), \(\alpha_2\) are certain functions of \(\Pi\), satisfies all qualitative requirements. Near the axis of the flow it passes into the invariant \(K\), which is determined by direct measurements in this region; after this, by measurements near the wall one can measure \(A+\alpha_2\,du/dy\).
With regard to the existence in the boundary layer of nonviscous stresses \(P_{ik}\), there is an indication in Dryden’s work (6). Additional experiments are needed here. Their task consists in determining the phenomenological coefficients in the equations of state. If this is done fully (for different Reynolds numbers) from measurements in flumes and pipes, then all other boundary-value problems can be solved by numerical integration of the equations of the theory.
Leningrad State University
named after A. A. Zhdanov
Received
16 III 1960
CITED LITERATURE
- V. G. Nevzglyadov, ZhETF, 16, 614 (1946); DAN, 58, 547 (1947); Vestn. LGU, No. 3, 3 (1948).
- J. Laufer, NACA, Report 1174 (1955).
- S. Goldstein, Modern State of Hydrodynamics, 1, IL, 1948, p. 239.
- W. A. Dryden, J. Meteorol., 13, No. 5, 433 (1956).
- V. G. Nevzglyadov, DAN, 124, No. 2, 288 (1959).
- Collection Problems of Mechanics, IL, 1955.