= Solution
Let $E(k)$ be the one-dimensional <turbulent energy spectrum>, normalized so that $\tfrac12\langle|\boldsymbol u'|^2\rangle=\int_0^\infty E(k)\,dk$. Large energy-containing eddies receive energy from the forcing. In the <inertial range>, nonlinear transfer passes this energy towards smaller scales at a mean rate $\epsilon$ per unit mass. Dimensional analysis of <Kolmogorov 1941 theory> gives
$$
\boxed{E(k)=C_K\epsilon^{2/3}k^{-5/3},\qquad k_0\ll k\ll k_d,\quad k_d\sim(\epsilon/\nu^3)^{1/4}.}
$$
The <dissipation range> begins near the inverse of the <Kolmogorov length scale>, where <kinematic viscosity> removes the transferred <kinetic energy>. The large-scale spectrum depends on the forcing; the rising low-$k$ branch in this sketch is illustrative, rather than a universal power law.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-79-spectrum.png]
{title=Schematic turbulence energy spectrum showing energy-containing eddies, the inertial range and viscous dissipation}
The tube has neither homogeneous nor isotropic large-scale <turbulence>: gravity selects a vertical direction, its walls constrain the motion, and its <density gradient> varies with height. The width $W$ bounds the transverse size of the energy-containing eddies. At sufficiently high <Reynolds number>, much smaller eddies may become approximately locally isotropic and exhibit an <inertial range>, even though the full tube flow does not satisfy the homogeneous-isotropic assumptions.
To obtain the <Reynolds stress>, write $u_i=U_i+u_i'$ and $p=P+p'$, with $\langle u_i'\rangle=0$. Averaging the incompressible <Navier-Stokes equations> uses $\langle u_i u_j\rangle=U_iU_j+\langle u_i'u_j'\rangle$ and gives the <Reynolds-averaged momentum equation>
$$
\partial_tU_i+U_j\partial_jU_i=-\frac{1}{\rho_0}\partial_iP+\nu\nabla^2U_i-\partial_j\langle u_i'u_j'\rangle+\langle b\rangle\delta_{iz}.
$$
Here the final term is the mean <buoyancy> force, with the hydrostatic reference removed. Thus averaging the quadratic momentum flux introduces the covariance $\langle u_i'u_j'\rangle$; multiplied by $\rho_0$, it has the dimensions of a stress. Its divergence transfers momentum between the mean flow and the fluctuations.
An <eddy viscosity> model represents this <Reynolds stress> by
$$
\langle u_i'u_j'\rangle=\frac23k_T\delta_{ij}-2\nu_T S_{ij},\qquad k_T=\frac12\langle u_i'u_i'\rangle,\qquad S_{ij}=\frac12(\partial_iU_j+\partial_jU_i).
$$
The isotropic term can be absorbed into the mean pressure, while the deviatoric term gives a diffusion of mean momentum with <turbulent viscosity> $\nu_T\sim u_*\ell$, where $u_*$ and $\ell$ are representative turbulent speed and <mixing length>. Unlike <kinematic viscosity>, this coefficient depends on the flow. The closure assumes a local relation between stress and mean strain; anisotropic, buoyancy-driven large eddies may require a more detailed <Reynolds stress> model.
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