5 results
The ultimate state of turbulent permeable-channel flow
- Shingo Motoki, Kentaro Tsugawa, Masaki Shimizu, Genta Kawahara
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- Journal:
- Journal of Fluid Mechanics / Volume 931 / 25 January 2022
- Published online by Cambridge University Press:
- 18 November 2021, R3
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Direct numerical simulations have been performed for heat and momentum transfer in internally heated turbulent shear flow with constant bulk mean velocity and temperature, $u_{b}$ and $\theta _{b}$, between parallel, isothermal, no-slip and permeable walls. The wall-normal transpiration velocity on the walls $y=\pm h$ is assumed to be proportional to the local pressure fluctuations, i.e. $v=\pm \beta p/\rho$ (Jiménez et al., J. Fluid Mech., vol. 442, 2001, pp. 89–117). The temperature is supposed to be a passive scalar, and the Prandtl number is set to unity. Turbulent heat and momentum transfer in permeable-channel flow for the dimensionless permeability parameter $\beta u_b=0.5$ has been found to exhibit distinct states depending on the Reynolds number $Re_b=2h u_b/\nu$. At $Re_{b}\lesssim 10^4$, the classical Blasius law of the friction coefficient and its similarity to the Stanton number, $St\approx c_{f}\sim Re_{b}^{-1/4}$, are observed, whereas at $Re_{b}\gtrsim 10^4$, the so-called ultimate scaling, $St\sim Re_b^0$ and $c_{f}\sim Re_b^0$, is found. The ultimate state is attributed to the appearance of large-scale intense spanwise rolls with the length scale of $O(h)$ arising from the Kelvin–Helmholtz type of shear-layer instability over the permeable walls. The large-scale rolls can induce large-amplitude velocity fluctuations of $O(u_b)$ as in free shear layers, so that the Taylor dissipation law $\epsilon \sim u_{b}^{3}/h$ (or equivalently $c_{f}\sim Re_b^0$) holds. In spite of strong turbulence promotion there is no flow separation, and thus large-amplitude temperature fluctuations of $O(\theta _b)$ can also be induced similarly. As a consequence, the ultimate heat transfer is achieved, i.e. a wall heat flux scales with $u_{b}\theta _{b}$ (or equivalently $St\sim Re_b^0$) independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction.
Multi-scale steady solution for Rayleigh–Bénard convection
- Shingo Motoki, Genta Kawahara, Masaki Shimizu
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- Journal:
- Journal of Fluid Mechanics / Volume 914 / 10 May 2021
- Published online by Cambridge University Press:
- 05 March 2021, A14
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We found a multi-scale steady solution of the Boussinesq equations for Rayleigh–Bénard convection in a three-dimensional periodic domain between horizontal plates with a constant temperature difference. This was realised using a homotopy from the wall-to-wall optimal transport solution provided by Motoki et al. (J. Fluid Mech., vol. 851, 2018, R4). A connected steady solution, which is a consequence of bifurcation from a thermal conduction state at Rayleigh number $Ra\sim 10^{3}$, is tracked up to $Ra\sim 10^{7}$ using a Newton–Krylov iteration. The three-dimensional exact coherent thermal convection exhibits a scaling of $Nu\sim Ra^{0.31}$ (where $Nu$ is the Nusselt number) as well as multi-scale thermal plume and vortex structures, which are quite similar to those in turbulent Rayleigh–Bénard convection. The mean temperature profiles and the root-mean-square of the temperature and velocity fluctuations are in good agreement with those of the turbulent states. Furthermore, the energy spectrum follows Kolmogorov's $-5/3$ scaling law with a consistent prefactor, and the energy transfer to small scales with a nearly constant flux in the wavenumber space is in accordance with the turbulent energy transfer.
Ultimate heat transfer in ‘wall-bounded’ convective turbulence
- Koki Kawano, Shingo Motoki, Masaki Shimizu, Genta Kawahara
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- Journal:
- Journal of Fluid Mechanics / Volume 914 / 10 May 2021
- Published online by Cambridge University Press:
- 05 March 2021, A13
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Direct numerical simulations have been performed for turbulent thermal convection between horizontal no-slip, permeable walls with a distance $H$ and a constant temperature difference $\Delta T$ at the Rayleigh number $Ra=3\times 10^3\text {--}10^{10}$. On the no-slip wall surfaces $z=0$, $H$, the wall-normal (vertical) transpiration velocity is assumed to be proportional to the local pressure fluctuation, i.e. $w=-\beta p'/\rho$, $+\beta p'/\rho$ (Jiménez et al., J. Fluid Mech., vol. 442, 2001, pp. 89–117). Here $\rho$ is mass density, and the property of the permeable wall is given by the permeability parameter $\beta U$ normalised with the buoyancy-induced terminal velocity $U=(g\alpha \Delta TH)^{1/2}$, where $g$ and $\alpha$ are acceleration due to gravity and volumetric thermal expansivity, respectively. The critical transition of heat transfer in convective turbulence has been found between the two $Ra$ regimes for fixed $\beta U=3$ and fixed Prandtl number $Pr=1$. In the subcritical regime at lower $Ra$ the Nusselt number $Nu$ scales with $Ra$ as $Nu\sim Ra^{1/3}$, as commonly observed in turbulent Rayleigh–Bénard convection. In the supercritical regime at higher $Ra$, on the other hand, the ultimate scaling $Nu\sim Ra^{1/2}$ is achieved, meaning that the wall-to-wall heat flux scales with $U\Delta T$ independent of the thermal diffusivity, although the heat transfer on the wall is dominated by thermal conduction. In the supercritical permeable case, large-scale motion is induced by buoyancy even in the vicinity of the wall, leading to significant transpiration velocity of the order of $U$. The ultimate heat transfer is attributed to this large-scale significant fluid motion rather than to transition to turbulence in boundary-layer flow. In such ‘wall-bounded’ convective turbulence, a thermal conduction layer still exists on the wall, but there is no near-wall layer of large change in the vertical velocity, suggesting that the effect of the viscosity is negligible even in the near-wall region. The balance between the dominant advection and buoyancy terms in the vertical Boussinesq equation gives us the velocity scale of $O(U)$ in the whole region, so that the total energy budget equation implies the Taylor dissipation law $\epsilon \sim U^3/H$ and the ultimate scaling $Nu\sim Ra^{1/2}$.
Maximal heat transfer between two parallel plates
- Shingo Motoki, Genta Kawahara, Masaki Shimizu
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- Journal:
- Journal of Fluid Mechanics / Volume 851 / 25 September 2018
- Published online by Cambridge University Press:
- 31 July 2018, R4
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The divergence-free time-independent velocity field has been determined so as to maximise heat transfer between two parallel plates with a constant temperature difference under the constraint of fixed total enstrophy. The present variational problem is the same as that first formulated by Hassanzadeh et al. (J. Fluid Mech., vol. 751, 2014, pp. 627–662); however, the search range for optimal states has been extended to a three-dimensional velocity field. A scaling of the Nusselt number $Nu$ with the Péclet number $Pe$ (i.e., the square root of the non-dimensionalised enstrophy with thermal diffusion time scale), $Nu\sim Pe^{2/3}$, has been found in the three-dimensional optimal states, corresponding to the asymptotic scaling with the Rayleigh number $Ra$, $Nu\sim Ra^{1/2}$, expected to appear in an ultimate state, and thus to the Taylor energy dissipation law in high-Reynolds-number turbulence. At $Pe\sim 10^{0}$, a two-dimensional array of large-scale convection rolls provides maximal heat transfer. A three-dimensional optimal solution emerges from bifurcation on the two-dimensional solution branch at $Pe\sim 10^{1}$, and the three-dimensional solution branch has been tracked up to $Pe\sim 10^{4}$ (corresponding to $Ra\approx 2.7\times 10^{6}$). At $Pe\gtrsim 10^{3}$, the optimised velocity fields consist of convection cells with hierarchical self-similar vortical structures, and the temperature fields exhibit a logarithmic-like mean profile near the walls.
Optimal heat transfer enhancement in plane Couette flow
- Shingo Motoki, Genta Kawahara, Masaki Shimizu
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- Journal:
- Journal of Fluid Mechanics / Volume 835 / 25 January 2018
- Published online by Cambridge University Press:
- 01 December 2017, pp. 1157-1198
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Optimal heat transfer enhancement has been explored theoretically in plane Couette flow. The vector field (referred to as the ‘velocity’) to be optimised is time independent and divergence free, and temperature is determined in terms of the velocity as a solution to an advection-diffusion equation. The Prandtl number is set to unity, and consistent boundary conditions are imposed on the velocity and the temperature fields. The excess of a wall heat flux (or equivalently total scalar dissipation) over total energy dissipation is taken as an objective functional, and by using a variational method the Euler–Lagrange equations are derived, which are solved numerically to obtain the optimal states in the sense of maximisation of the functional. The laminar conductive field is an optimal state at low Reynolds number $Re\sim 10^{0}$. At higher Reynolds number $Re\sim 10^{1}$, however, the optimal state exhibits a streamwise-independent two-dimensional velocity field. The two-dimensional field consists of large-scale circulation rolls that play a role in heat transfer enhancement with respect to the conductive state as in thermal convection. A further increase of the Reynolds number leads to a three-dimensional optimal state at $Re\gtrsim 10^{2}$. In the three-dimensional velocity field there appear smaller-scale hierarchical quasi-streamwise vortex tubes near the walls in addition to the large-scale rolls. The streamwise vortices are tilted in the spanwise direction so that they may produce the anticyclonic vorticity antiparallel to the mean-shear vorticity, bringing about significant three-dimensionality. The isotherms wrapped around the tilted anticyclonic vortices undergo the cross-axial shear of the mean flow, so that the spacing of the wrapped isotherms is narrower and so the temperature gradient is steeper than those around a purely streamwise (two-dimensional) vortex tube, intensifying scalar dissipation and so a wall heat flux. Moreover, the tilted anticyclonic vortices induce the flow towards the wall to push low- (or high-) temperature fluids on the hot (or cold) wall, enhancing a wall heat flux. The optimised three-dimensional velocity fields achieve a much higher wall heat flux and much lower energy dissipation than those of plane Couette turbulence.