Nonequilibrium turbulent dissipation in buoyant axisymmetric plume

We report existence of nonequilibrium turbulent dissipations in buoyant axisymmetric plumes at infinite Reynolds number limit. The plume statistical equations remain form invariant under one translation parameterized by $a_0$ and two unequal stretching transformations parameterized by $a_1$ and $a_2$ of flow variables, except the pressure-strain-rate equation. The dissipation coefficients, unlike Kolmogorov theory, in the scaling of dissipations ${\epsilon }_u$ of velocity fluctuation $u^{\prime }$ and ${\epsilon }_{\theta }$ of thermal fluctuation ${\theta }^{\prime }$ evolve linearly with spreading rate $d\delta /dz$. The spreading rate is expressed as inversely proportional to local Reynolds number ${\mathrm{Re}}_{\delta }$ raised to the exponent $m=3(a_1-a_2)/(a_1+a_2), a_1\ne -a_2$ and is varying with ${\mathrm{Re}}_{\delta }$ for $m\ne 0$. The components of Reynolds stress tensor have different streamwise (s) evolutions as ${\left(\overline{w^{\prime 2}}\right)}_s={\left(\overline{v^{\prime 2}}\right)}_s\propto {(d\delta /dz)}^2{\left(\overline{u^{\prime 2}}\right)}_s$ and ${\left(\overline{u^{\prime }{v}^{\prime }}\right)}_s\propto d\delta /dz{\left(\overline{u^{\prime 2}}\right)}_s$ unless $d\delta /dz$ is a constant, which holds only if $m=0$ giving $a_1=a_2$. This implies that Kolmogorov equilibrium theory ($m=0$) is intertwined inextricably with complete self-preservation. There is a direct universal relationship between turbulent dissipation and entrainment coefficient $\alpha$ as $d\delta /dz\propto \alpha$, the proportionality constant depends on integrals of mean axial velocity, temperature difference, and turbulent stresses. The power exponent in power-law streamwise evolutions of flow quantities are specified by the ratio $a_2/a_1$ and differs from the conventional results unless $m=0$. Both the local axial velocity and temperature difference widths are increasing with the increase of vertical distance from the source, as same power-law scaling exponents but differ due to different prefactors. However, the spreading rates and entrainment coefficient decrease in the non-Kolmogorov dissipation region, which occurs when $m\ne 0$ preferably lying in (0,1] ($a_2/a_1\in [1/2,1)$). Similar trends of spreading rate and entrainment were also measured experimentally in planar jet [Cafiero and Vassilicos, Proc. R. Soc. London A 475, 20190038 (2019)] and established theoretically in planar jet and plume [Layek and Sunita, Phys. Rev. Fluids 3, 124605 (2018); Layek and Sunita, Phys. Fluids 30, 115105 (2018)].

Figure 1

Axisymmetric plume flow definition.

Figure 2

Comparison of (a) velocity $1/e$ width and (b) temperature difference $1/e$ width and centerline evolution (c) mean axial velocity (d) mean temperature difference (solid curve) with $a=-0.5z_0$ when $m=0$ with the measured data ($\circ$) of Papanicolaou and List (PL) [12].

Figure 3

Radial profiles of (a) mean axial velocity and (b) mean temperature difference normalized by their centerline value when $m=0$ are compared with the measured data of normalized velocity and temperature/concentration profiles of Shabbir and George (SG94) [2] for both two and three wire probes and Papanicolaou and List (PL) [12]. The data of PL is obtained by digitization.

Figure 4

Radial profile of mean radial velocity when $m=0$ compared with the measured data of Shabbir and George (SG94) [2].

Figure 5

Profiles of (a) shear stress made dimensionless by ${\overline{u}}_c^2$ and (b) radial heat flux made dimensionless by ${\overline{u}}_c{\theta }_c$ when $m=0$ are compared with the measured data of Shabbir and George (SG94) [2] and Wang and Law (WL) [37]. The data of WL is obtained by digitization for comparison.

Figure 6

Spreading rate of velocity half-width (a) axisymmetric plume, (b) planar jet in Refs. [21, 22] when $m=1$.

Figure 7

Centerline evolution when $m=1$ of (a) mean axial velocity (b) mean temperature difference.

Figure 8

Profiles of (a) mean axial velocity, (b)mean temperature difference, (c) Reynolds shear stress, and (d) radial heat flux when $m=0, 1/2$, and 1.