E. M. KHAZEN

ON THE THEORY OF TURBULENCE IN NONUNIFORM FLOWS*

(Presented by Academician A. N. Kolmogorov, 2 VIII 1962)

I. The origin, development, and maintenance of turbulence in an incompressible viscous fluid are effected at the expense of the energy of the averaged flow and of external random actions. The study of the mechanism of turbulence therefore requires consideration of nonuniform turbulent flows. In turn, turbulent fluctuations influence the mean flow: the equations of averaged motion

\[ \frac{\partial U_i}{\partial t} + U_j \frac{\partial U_i}{\partial x_j} = \nu \Delta U_i - \frac{1}{\rho}\frac{\partial p}{\partial x_i} - \frac{\partial \overline{\delta V_j(x,t)\,\delta V_i(x,t)}}{\partial x_j} \tag{1} \]

contain the correlation functions of the fluctuations
\(b_{ij}(x,t)=\overline{\delta V_j(x,t)\,\delta V_i(x,t)}\).

If the level of turbulence in the flow is regarded as small and, in averaging, terms of order \(\overline{\delta V_i \delta V_j \delta V_k}\) are neglected, then from the Navier—Stokes equations it follows that

\[ \frac{\partial b_{ij}(x,x',t)}{\partial t} + \left( U_k(x,t)\frac{\partial}{\partial x_k} + U_k(x',t)\frac{\partial}{\partial x'_k} \right)b_{ij}(x,x',t) + \]

\[ + \frac{\partial U_i}{\partial x_k} b_{kj} + \frac{\partial U_j}{\partial x'_k} b_{ik} = -\frac{1}{\rho} \left( \frac{\partial b_{pj}}{\partial x_i} + \frac{\partial b_{ip}}{\partial x'_j} \right) + \nu \Delta_x b_{ij} + \nu \Delta_{x'} b_{ij}. \tag{2} \]

The functions \(b_{pi}=\overline{\delta p\,\delta V_i}\) are eliminated with the help of the incompressibility condition. For a compressible fluid one can obtain a closed system of equations analogous to (1), (2), if one introduces the relation \(\delta p=v_s^2\delta \rho\), where \(v_s\) is the speed of sound.

We shall use the closed system of equations for the mean flow velocity \(U(x,t)\) and the second central moments
\(b_{ij}(x,x',t)=\overline{\delta V_i(x,t)\,\delta V_j(x',t)}\)
to describe the initial stage of the onset of turbulence.

II. Consider the case in which the flow is nonuniform, but the initial disturbances are such that the maximum scale of turbulence \(l\) is smaller than the characteristic scale of the averaged flow \(L\). We introduce, under these conditions, the spectral function

\[ \Phi_{ij}(k,x,t) = \int_{-\infty}^{\infty} e^{ikr}\, \overline{\delta V_i(x-r/2)\,\delta V_j(x+r/2)}\,dr, \tag{3} \]

which depends on the coordinates \(x\) not explicitly, but only through the slowly varying functions \(U(x,t)\).

* Reported on May 25, 1962 at the research seminar of the Department of Physics for the Mechanics and Mathematics Faculty of Moscow State University, and on June 6, 1962 at the research seminar of the Department of Aeromechanics of Moscow State University.

** Summation is carried out over repeated indices; \(x\) denotes the vector \((x_1,x_2,x_3)\).

Using equation (2) and retaining only the first derivatives with respect to the coordinates of the function \(\mathbf U(\mathbf x,t)\), we obtain the following system of equations:

\[ \left( \frac{\partial}{\partial t}+U_k\frac{\partial}{\partial x_k} \right)\Phi_{ij}(\mathbf k,\mathbf x,t) -\frac{\partial U_k}{\partial x_l}k_k\frac{\partial}{\partial k_l}\Phi_{ij} + \frac{\partial U_l}{\partial x_k} \left(\delta_{li}-2\frac{k_lk_i}{k^2}\right)\Phi_{kj} + \frac{\partial U_l}{\partial x_k} \left(\delta_{lj}-2\frac{k_lk_j}{k^2}\right)\Phi_{ik} = -2\nu k^2\Phi_{ij}. \tag{4} \]

Equations (4) correspond to the case \(l\ll L\).

Consider a plane-parallel flow \(\mathbf U(\mathbf x,t)\); let the axis \(x_1\) be directed along the flow, and the axis \(x_2\) across it. Then only \(U_1\) and \(\partial U_1/\partial x_2\ne 0\). We find the solution, regarding the functions \(U_k(t)\) as constant; denote \(\partial U_1/\partial x_2\) by \(C(\mathbf x)\). Then

\[ \Phi_{22}(\mathbf k,t) = \Phi_{22}^0(k_1,k_2+k_1Ct,k_3) \left[ \frac{k_1^2+(k_2+k_1Ct)^2+k_3^2}{k_1^2+k_2^2+k_3^2} \right]^2 \times \]

\[ \times \exp\{-2\nu[k^2t+k_1k_2Ct^2+k_1^2C^2t^3/3]\}; \tag{5} \]

\[ \Phi_{11}(\mathbf k,t) = \exp\{-2\nu[k^2t+k_1k_2Ct^2+k_1^2C^2t^3/3]\} \Bigl[ \Phi_{11}^0(k_1,k_2+k_1Ct,k_3)+ \]

\[ + \Phi_{12}^0(k_1,k_2+k_1Ct,k_3)\, G(k_1,k_2+k_1Ct,k_3;k_2) + \]

\[ + \Phi_{22}^0(k_1,k_2+k_1Ct,k_3) \int_0^t G(k_1,k_2+k_1Ct,k_3;k_2-k_1C\tau)\times \]

\[ \times(-2C) \left[ \frac{ \bigl(k_1^2+(k_2+k_1Ct)^2+k_3^2\bigr) \bigl(k_3^2-k_1^2+(k_2+k_1C(t-\tau))^2\bigr) }{ k_1^2+(k_2+k_1C(t-\tau))^2+k_3^2 } \right]\,d\tau. \tag{6} \]

Here \(\Phi_{ij}^0\) are the values of the functions at \(t=0\); the function \(G\) is equal to

\[ G(k_1,k_2+k_1Ct,k_3;k_2) = \operatorname{arc\,tg}\frac{k_2}{\sqrt{k_1^2+k_3^2}} \cdot \left( \frac{k_1^2+(k_2+k_1Ct)^2+k_3^2}{k_1\sqrt{k_1^2+k_3^2}} \right) - \]

\[ - \frac{ 2k_1\bigl(k_1^2+(k_2+k_1Ct)^2+k_3^2\bigr) }{ (k_1^2+k_3^2)^{3/2} } \left( \frac{k_2\sqrt{k_1^2+k_3^2}}{2k^2} + \frac12\operatorname{arc\,tg} \frac{k_2}{\sqrt{k_1^2+k_3^2}} \right). \]

For \(\mathbf U=0\), which corresponds to homogeneous and isotropic turbulence, and
\(\Phi_{11}^0=k_2^2+k_3^2,\ \Phi_{22}^0=k_1^2+k_3^2,\ \Phi_{12}^0=-k_1k_2\), the solution (5), (6) goes over into the well-known expression of M. D. Millionshchikov \((^1)\).

The energy of the pulsations is equal to

\[ B(t) = \int_{-\infty}^{\infty}\Phi_{11}(\mathbf k,t)\,d\mathbf k + \int_{-\infty}^{\infty}\Phi_{22}(\mathbf k,t)\,d\mathbf k = b_{11}(t)+b_{22}(t). \tag{7} \]

For a wave initial disturbance we have
\(\Phi_{ij}^0=A_{ij}\delta(k_1-a)\times\delta(k_2)\delta(k_3)\)—the initial correlation function is spatially periodic along the flow with period \(2\pi/a\); \(A_{ij}\) is a constant tensor. In this case it is evident from (5)—(7) that the energy of pulsations decreases monotonically with increasing \(t\). Consequently, in a flow with a constant linear profile of mean velocities any wave initial disturbance decreases monotonically. From (5)—(7), however, it is evident that there exist initial disturbances for which the energy of pulsations begins to increase with increasing \(t\) and, at a fixed local Reynolds number \(\mathrm{Re}=(\partial U_1/\partial x_2)(l^2/\nu)\), increases over a finite interval of time, while with increasing \(\mathrm{Re}\) the maximum of the ratio \(B(t)/B(0)\) becomes larger than any prescribed number. Indeed, for vortex initial disturbances consisting of vortices of the type
\(\delta V_1(\mathbf x)=-\partial e^{-a^2x_2^2}/\partial x_2\),
\(\delta V_2(\mathbf x)=\partial e^{-a^2x_1^2}/\partial x_1\), randomly scattered in the plane \(x_1,x_2\), we have

\[ \Phi_{11}^0(\mathbf k)=Ak_2^2e^{-k^2/a^2}; \qquad \Phi_{22}^0(\mathbf k)=Ak_1^2e^{-k^2/a^2}; \qquad \Phi_{12}^0(\mathbf k)=-Ak_1k_2e^{-k^2/a^2}. \tag{8} \]

Here it is natural to take \(l=1/a\). From (5), (7), (8), passing to dimensionless

variables \(k^*=k/a,\ t^*=t\,\partial U_1/\partial x_2\), we find

\[ \frac{B(t^*)}{B(0)}>\frac{b_{22}(t^*)}{B(0)} = \int_{-\infty}^{\infty} \frac{(k_1^*)^2}{2\pi} \exp\left[-k_1^{*2}+(k_2^*+k_1^*t^*)^2+k_3^{*2}\right]\times \]

\[ \times \left[ \frac{k_1^{*2}+(k_2^*+k_1^*t^*)^2+k_3^{*2}} {k_1^{*2}+k_2^{*2}+k_3^{*2}} \right]^2 \exp\left\{ -\frac{2}{\mathrm{Re}}\left(t^*\mathbf{k}^{*2}+k_1^*k_2^*t^{*2}+k_1^{*2}t^{*3}/3\right) \right\}d\mathbf{k} \]

\[ > \int_{Q(t)} \frac{(k_1^*)^2}{6\pi} \left[ \frac{k_1^{*2}+(k_2^*+k_1^*t^*)^2+k_3^{*2}} {k_1^{*2}+k_2^{*2}+k_3^{*2}} \right]^2 \exp\left\{ -\frac{2}{\mathrm{Re}}\left(t^*\mathbf{k}^{*2}+k_1^*k_2^*t^{*2}+k_1^{*2}t^{*3}/3\right) \right\}d\mathbf{k}^*, \]

where \(Q(t)\) is the domain defined by the inequalities
\(|k_1^*|<1/3,\ |k_2^*+k_1^*t^*|<1/3,\ |k_3^*|<1/3\). Let \(M\) be any prescribed number. For \(t^*<N(M)\) and Reynolds numbers \(\mathrm{Re}>N^3(M)\), we have, for all \(\mathbf{k}^*\in Q(t)\), the inequality

\[ \exp\left\{ -\frac{2}{\mathrm{Re}}\left(t^*k^{*2}+k_1^*k_2^*t^{*2}+k_1^{*2}t^{*3}/3\right) \right\}>1/3. \]

Consequently,

\[ \frac{B(t^*)}{B(0)} > \int_{Q(t)} \frac{(k_1^*)^6(t^*)^4\,dk_1^*dk_2^*dk_3^*} {18\pi\,[k_1^{*2}+k_2^{*2}+k_3^{*2}]^2} > \frac{(t^*)^4}{3^9[1+t^{*2}/2]} \]

for \(t^*<N(M),\ \mathrm{Re}>N^3(M)\). Choosing \(N(M)=2[3^9M]^{1/2}\), we find that for \([M3^9]^{1/2}<t<2[3^9M]^{1/2}\) the relation \(B(t)/B(0)>M\) is satisfied. This also explains the emergence of turbulence in flows that are stable with respect to all wave-type initial disturbances.

We shall show that the loss of stability of a laminar flow and the emergence of turbulence occur in the following way. Vortical random disturbances with small initial amplitude in a nonuniform flow begin to grow. If the values of the Reynolds number and of the initial amplitude are such that the maximum amplitude of the pulsations that is reached does not exceed a certain barrier (up to which the linear approximation for \(\delta V(\mathbf{x},t)\) is still applicable), then the pulsations decay as \(t\to\infty\). If this barrier is exceeded, the energy of the pulsations remains nondecreasing; the laminar flow is replaced by a turbulent one.

Since in an incompressible fluid any velocity field contains a vortical component, the described mechanism of the onset of turbulence from vortical small initial disturbances is universal.

Let us consider the equation for \(\Phi_{ij}(\mathbf{k},\mathbf{x},t)\), constructed with allowance for the terms nonlinear in \(\delta \mathbf{V}(\mathbf{x},t)\) in the Navier—Stokes equations. It can be shown that its solution with the initial condition (8) does not decrease as \(t\to\infty\) at large \(\mathrm{Re}\), if the initial amplitude \(A\) was sufficiently large: one must have \(A>K/\mathrm{Re}\), where \(K\) is a constant. Indeed, for \(\Phi_{ij}(\mathbf{k},\mathbf{x},t)\), retaining under averaging of the Navier—Stokes equations the terms nonlinear in \(\delta \mathbf{V}(\mathbf{x},t)\), we find (for a plane-parallel mean flow):

\[ \frac{\partial}{\partial t}\Phi_{ij}(\mathbf{k},\mathbf{x},t) -\frac{\partial U_1}{\partial x_2}k_1\frac{\partial}{\partial k_2}\Phi_{ij} +\frac{\partial U_1}{\partial x_2}\left(\delta_{il}-2\frac{k_1k_i}{k^2}\right)\Phi_{kj} + \]

\[ +\frac{\partial U_1}{\partial x_2}\left(\delta_{ij}-2\frac{k_1k_j}{k^2}\right)\Phi_{ik} +2\nu k^2\Phi_{ij} = \tag{9} \]

\[ = k_lk_s \int_0^t\int_{-\infty}^{\infty} G(\mathbf{k}-\mathbf{k}',t-\tau)\, G(k_1-k_1',t-\tau)\, \varphi_{ijls}(\mathbf{k}',k_1',\tau)\, d\mathbf{k}'\,dk_1'\,dk_1\,d\tau; \]

here the functions \(G(\mathbf{k},t)\) are defined analogously to the factor multiplying \(\Phi_{22}^0\) in formula (5); the functions \(\varphi_{ijls}\) are expressed through pairwise products of the functions \(\Phi_{ij}\) and \(\Phi_{ls}\). In deriving equation (9), the hypothesis of the Gaussian character of the relation between the fourth \((\delta V_i\delta V_j\delta V_k\delta V_l)\) and second \((\delta V_i\delta V_j)\) moments was used.

The solution with initial condition (8) is found by the method of successive approximations (in the first approximation \(\Phi'_{ij}(\mathbf{k},t)\) is determined by (5), (6)). Then

\[ \Phi_{22}(\mathbf{k},t)=\sum_{n=1}^{\infty} A^n G_n(\mathbf{k},t), \]

where, for the terms of the series, the estimate

\[ C_1\,\frac{t^{3n+2}}{n!K^n}\,e^{-t^3/\mathrm{Re}} < \int G_n(\mathbf{k},t)\,d\mathbf{k} < C_2\,\frac{t^{3n+2}}{n!K^n}\,e^{-t^3/\mathrm{Re}} . \]

Therefore

\[ C_1 t^2 \exp\{(A/K-1/\mathrm{Re})t^3\} < b_{22}(t) < C_2 t^2 \exp\{(A/K-1/\mathrm{Re})t^3\}, \]

and for \(A<K/\mathrm{Re}\) we have \(B(t)\to 0\) as \(t\to\infty\); if \(A\ge K/\mathrm{Re}\), \(B(t)\) remains greater than the positive quantity \(C_1\) for all \(t\), as was required.

Returning to the linear approximation in \(\delta \mathbf{V}(\mathbf{x},t)\), let us also consider inhomogeneous flows with allowance for the curvature of the mean velocity profile. In this case, in equation (4) one must retain the second derivatives with respect to the coordinates of \(\mathbf{U}(\mathbf{x},t)\). Then

\[ \begin{aligned} &\left(\frac{\partial}{\partial t}+U_k\frac{\partial}{\partial x_k}\right)\Phi_{ij}(\mathbf{k},\mathbf{x},t) -\frac{\partial U_n}{\partial x_l}\,k_n\frac{\partial}{\partial k_l}\Phi_{ij} -\frac{i}{4}\frac{\partial^2 U_n}{\partial x_l\partial x_m}\,k_n \frac{\partial^2\Phi_{ij}}{\partial k_l\partial k_m} \\ &\quad +\frac{\partial U_s}{\partial x_k}\left(\delta_{si}-\frac{2k_i k_s}{k^2}\right)\Phi_{kj} +\frac{\partial U_s}{\partial x_k}\left(\delta_{sj}-\frac{2k_j k_s}{k^2}\right)\Phi_{ik} \\ &\quad +\frac{i}{2}\frac{\partial^2 U_i}{\partial x_l\partial x_k} \left(\frac{\partial\Phi_{kj}}{\partial k_l}-\frac{k_i}{k^2}\Phi_{lk}\right) +\frac{i}{2}\frac{\partial^2 U_j}{\partial x_l\partial x_k} \left(\frac{\partial\Phi_{ik}}{\partial k_l}-\frac{k_i}{k^2}\Phi_{lk}\right) = -2\nu k^2\Phi_{ij}. \tag{10} \end{aligned} \]

For a plane-parallel flow \(\mathbf{U}(\mathbf{x},t)\), only \(\partial U_1/\partial x_2\ne 0\), \(\partial^2 U_1/\partial x_2^2\ne 0\), and equation (10) is simplified. The solution \(\Phi_{22}(\mathbf{k},\mathbf{x},t)\) has the form

\[ \begin{aligned} \Phi_{22}(\mathbf{k},\mathbf{x},t) &= \int_{-\infty}^{\infty} \Phi^0_{22}(k_1^0,k_2^0,k_3^0)\, \frac{1}{\sqrt{(i\pi k_1/2)(\partial^2 U_1/\partial x_2^2)}}\, \delta(k_1-k_1^0)\,\delta(k_3-k_3^0) \\ &\quad\times \exp\left\{ -\frac{\left(k_2-k_2^0+(\partial U_1/\partial x_2)k_1^0 t\right)^2} {(i k_1/4)(\partial^2 U_1/\partial x_2^2)t} \right\} \\ &\quad\times \exp\left\{ -\int_0^t 2\nu\left[ (k_1^0)^2+(k_3^0)^2+ \left(k_2^0-k_1^0\frac{\partial U_1}{\partial x_2}\tau\right)^2 \right]\,d\tau \right\} \\ &\quad\times \left[ \frac{(k_1^0)^2+(k_2^0)^2+(k_3^0)^2} {(k_1^0)^2+(k_2^0-k_1^0 t\,\partial U_1/\partial x_2)^2+(k_3^0)^2} \right]^2 \,d\mathbf{k}^0 . \end{aligned} \]

By analogous formulas one writes \(\Phi_{12}(\mathbf{k},\mathbf{x},t)\) and \(\Phi_{11}(\mathbf{k},\mathbf{x},t)\).

For wave-like initial disturbances \(\delta \mathbf{V}(\mathbf{x},t)\), periodic in \(x_1,x_2,x_3\), substituting
\[ \Phi^0_{ij}=A^0_{ij}\delta(k_1-a_1)\delta(k_2-a_2)\delta(k_3-a_3), \]
we find that \(b_{11}, b_{22}\) decrease.

Conclusion. In the present work, spectral functions \(\Phi_{ij}(\mathbf{k},\mathbf{x},t)\) (3) are introduced to describe inhomogeneous turbulence. A closed system of equations has been obtained for the mean velocities \(\mathbf{U}(\mathbf{x},t)\) and the spectral functions of turbulent pulsations \(\Phi_{ij}(\mathbf{k},\mathbf{x},t)\) ((3), (1), (4), (9), (10)) in inhomogeneous flows with a low level of turbulence. On the basis of the system obtained, the mechanism of the onset of turbulence and of the transfer of energy from large-scale to small-scale motion is revealed.

The work was carried out under the supervision of Yu. L. Klimontovich, to whom the author expresses deep gratitude for his great attention and cooperation in the work. The author also expresses deep gratitude to A. N. Kolmogorov for valuable suggestions.

Moscow State University
named after M. V. Lomonosov

Received
2 VIII 1962

REFERENCES

1. M. D. Millionshchikov, DAN, 22, No. 5, 236 (1939).