scholarly journals Near wall Prandtl number effects on velocity gradient invariants and flow topologies in turbulent Rayleigh–Bénard convection

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Sahin Yigit ◽  
Josef Hasslberger ◽  
Markus Klein ◽  
Nilanjan Chakraborty
Keyword(s):  
Natural Convection ◽  
Velocity Gradient ◽  
Bulk Region ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
Turbulent Natural Convection ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Abstract The statistical behaviours of the invariants of the velocity gradient tensor and flow topologies for Rayleigh–Bénard convection of Newtonian fluids in cubic enclosures have been analysed using Direct Numerical Simulations (DNS) for a range of different values of Rayleigh (i.e. $$Ra=10^7-10^9$$ R a = 10 7 - 10 9 ) and Prandtl (i.e. $$Pr=1$$ P r = 1 and 320) numbers. The behaviours of second and third invariants of the velocity gradient tensor suggest that the bulk region of the flow at the core of the domain is vorticity-dominated whereas the regions in the vicinity of cold and hot walls, in particular in the boundary layers, are found to be strain rate-dominated and this behaviour has been found to be independent of the choice of Ra and Pr values within the range considered here. Accordingly, it has been found that the focal topologies S1 and S4 remain predominant in the bulk region of the flow and the volume fraction of nodal topologies increases in the vicinity of the active hot and cold walls for all cases considered here. However, remarkable differences in the behaviours of the joint probability density functions (PDFs) between second and third invariants of the velocity gradient tensor (i.e. Q and R) have been found in response to the variations of Pr. The classical teardrop shape of the joint PDF between Q and R has been observed away from active walls for all values of Pr, but this behavior changes close to the heated and cooled walls for high values of Pr (e.g. $$Pr=320$$ P r = 320 ) where the joint PDF exhibits a shape mirrored at the vertical Q-axis. It has been demonstrated that the junctions at the edges of convection cells are responsible for this behaviour for $$Pr=320$$ P r = 320 , which also increases the probability of finding S3 topologies with large negative magnitudes of Q and R. By contrast, this behaviour is not observed in the $$Pr=1$$ P r = 1 case and these differences between flow topology distributions in Rayleigh–Bénard convection in response to Pr suggest that the modelling strategy for turbulent natural convection of gaseous fluids may not be equally well suited for simulations of turbulent natural convection of liquids with high values of Pr.

scholarly journals Bulk scaling in wall-bounded and homogeneous vertical natural convection

2018 ◽  
Vol 841 ◽  
pp. 825-850 ◽  
Author(s):  
Chong Shen Ng ◽  
Andrew Ooi ◽  
Detlef Lohse ◽  
Daniel Chung
Keyword(s):  
Nusselt Number ◽  
Natural Convection ◽  
Temperature Gradient ◽  
Power Law ◽  
Exact Relation ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Previous numerical studies on homogeneous Rayleigh–Bénard convection, which is Rayleigh–Bénard convection (RBC) without walls, and therefore without boundary layers, have revealed a scaling regime that is consistent with theoretical predictions of bulk-dominated thermal convection. In this so-called asymptotic regime, previous studies have predicted that the Nusselt number ($\mathit{Nu}$) and the Reynolds number ($\mathit{Re}$) vary with the Rayleigh number ($\mathit{Ra}$) according to $\mathit{Nu}\sim \mathit{Ra}^{1/2}$ and $\mathit{Re}\sim \mathit{Ra}^{1/2}$ at small Prandtl numbers ($\mathit{Pr}$). In this study, we consider a flow that is similar to RBC but with the direction of temperature gradient perpendicular to gravity instead of parallel to it; we refer to this configuration as vertical natural convection (VC). Since the direction of the temperature gradient is different in VC, there is no exact relation for the average kinetic dissipation rate, which makes it necessary to explore alternative definitions for $\mathit{Nu}$, $\mathit{Re}$ and $\mathit{Ra}$ and to find physical arguments for closure, rather than making use of the exact relation between $\mathit{Nu}$ and the dissipation rates as in RBC. Once we remove the walls from VC to obtain the homogeneous set-up, we find that the aforementioned $1/2$-power-law scaling is present, similar to the case of homogeneous RBC. When focusing on the bulk, we find that the Nusselt and Reynolds numbers in the bulk of VC too exhibit the $1/2$-power-law scaling. These results suggest that the $1/2$-power-law scaling may even be found at lower Rayleigh numbers if the appropriate quantities in the turbulent bulk flow are employed for the definitions of $\mathit{Ra}$, $\mathit{Re}$ and $\mathit{Nu}$. From a stability perspective, at low- to moderate-$\mathit{Ra}$, we find that the time evolution of the Nusselt number for homogenous vertical natural convection is unsteady, which is consistent with the nature of the elevator modes reported in previous studies on homogeneous RBC.

scholarly journals Vertical natural convection: application of the unifying theory of thermal convection

2015 ◽  
Vol 764 ◽  
pp. 349-361 ◽  
Author(s):  
Chong Shen Ng ◽  
Andrew Ooi ◽  
Detlef Lohse ◽  
Daniel Chung
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Temperature Fields ◽  
Mean Flow ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Gl Theory

AbstractResults from direct numerical simulations of vertical natural convection at Rayleigh numbers $1.0\times 10^{5}$–$1.0\times 10^{9}$ and Prandtl number $0.709$ support a generalised applicability of the Grossmann–Lohse (GL) theory, which was originally developed for horizontal natural (Rayleigh–Bénard) convection. In accordance with the GL theory, it is shown that the boundary-layer thicknesses of the velocity and temperature fields in vertical natural convection obey laminar-like Prandtl–Blasius–Pohlhausen scaling. Specifically, the normalised mean boundary-layer thicknesses scale with the $-1/2$-power of a wind-based Reynolds number, where the ‘wind’ of the GL theory is interpreted as the maximum mean velocity. Away from the walls, the dissipation of the turbulent fluctuations, which can be interpreted as the ‘bulk’ or ‘background’ dissipation of the GL theory, is found to obey the Kolmogorov–Obukhov–Corrsin scaling for fully developed turbulence. In contrast to Rayleigh–Bénard convection, the direction of gravity in vertical natural convection is parallel to the mean flow. The orientation of this flow presents an added challenge because there no longer exists an exact relation that links the normalised global dissipations to the Nusselt, Rayleigh and Prandtl numbers. Nevertheless, we show that the unclosed term, namely the global-averaged buoyancy flux that produces the kinetic energy, also exhibits both laminar and turbulent scaling behaviours, consistent with the GL theory. The present results suggest that, similar to Rayleigh–Bénard convection, a pure power-law relationship between the Nusselt, Rayleigh and Prandtl numbers is not the best description for vertical natural convection and existing empirical relationships should be recalibrated to better reflect the underlying physics.

scholarly journals Natural convection in a model chamber with the Boussinesq approximation utilization

2017 ◽  
Vol 66 (3) ◽  
pp. 75-82
Author(s):  
Weronika Radzikowska-Juś ◽  
Stefan Owczarek
Keyword(s):  
Natural Convection ◽  
Boussinesq Approximation ◽  
The Finite Element Method ◽  
Bénard Convection ◽  
Benard Convection ◽  
Starting Point ◽  
Model Chamber ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Model Room

The paper presents numerical analysis of heat transfer inside a model chamber in time after entering an object of elevated temperature therein. As a result of the temperature difference between the object and the environment in the room, natural convection has occurred. Numerical simulations were performed using ELMER software based on the Finite Element Method. In order to reduce the complexity of the simulation, the Bussinesq approximation was used. The relevant case is similar to Rayleigh-Benard convection, which was starting point to the conducted analysis. Keywords: temperature distribution in the model room, Rayleigh-Benard convection, Bussinesq approximation

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