Theory of the steady laminar natural convectiol above a horizontal line heat source and a point heat source
TL;DR: In this paper, the velocity and temperature distributions for the case of Prandtl number equal to 0.01, 0.7 and 10 are computed with an electronic computer, and differences, caused by different prandtl numbers, among the velocity distributions or the temperature distributions are described in detail.
Abstract: Steady laminar natural convection above a horizontal line heat source and a point heat source are analysed mathematically. The solutions of elementary functions are given for Pr = 2, and also for the flow above a point source for Pr = 1. The velocity and temperature distributions for the case of Prandtl number equal to 0.01, 0.7 and 10 are computed with an electronic computer, and differences, caused by different Prandtl number, among the velocity distributions or the temperature distributions are described in detail.
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TL;DR: In this paper, a simple formulation of the boundary value problem for a wide range of values of the Prandtl number has been proposed, and results of experiments with plumes are included to emphasize the various properties of plume flow and to indicate that the large thickness of boundary region in the range of stable laminar plumes (i.e. at relatively low local Grashof numbers) should encourage the calculation of higher order approximate of the flow.
Abstract: The extensive past publication concerning the calculation of the behavior of laminar natural convection plumes above energy sources contains numerous elements of uncertainty and confusion concerning proper variables, boundary conditions, and governing equations. This paper, for the first time, states the problem in simplest variables, resolves various apparent redundancies in boundary conditions, and indicates the optimum way to formulate this boundary value problem. Results of numerical calculations are given in terms of the present simple formulation for a wide range of values of Prandtl number, some not having been treated before. Results of experiments with plumes are included to emphasize the various properties of plume flow and to indicate that the large thickness of the boundary region in the range of stable laminar plumes (i.e. at relatively low local Grashof numbers) should encourage the calculation of higher order approximations of the flow.
129 citations
TL;DR: In this article, the scaling laws for the front of an isolated laminar starting plume are studied and the scaling relations are formulated and measured experimentally over a range of power, fluids, and heaters.
Abstract: We present an experimental study of the scaling laws for the front (or cap) of an isolated, laminar starting plume. The scaling relations are formulated and measured experimentally over a range of power, fluids, and heaters. The results are that the cap rises at constant velocity, grows diffusively in width, and its temperature depends inversely on height. This extends analytic results by Batchelor (1954) for the column (stem) below the front. The source size determines initial conditions for the cap, but does not affect it in the far field. The shape of the front is fitted by a model of potential flow. The interaction between plume caps is complex, but with simple underlying dynamics. We conjecture that some of our conclusions can be applied to a distribution of plumes, as in soft turbulent convection.
105 citations
TL;DR: In this article, the structure of the boundary layers in turbulent Rayleigh-B´ enard convection is studied by means of three-dimensional direct numerical simulations, and the results demonstrate that important assumptions of existing classical laminar boundary layer theories for forced and natural convection are violated, such as the strict two-dimensionalality of the dynamics or the steadiness of the fluid motion.
Abstract: The structure of the boundary layers in turbulent Rayleigh‐B´ enard convection is studied by means of three-dimensional direct numerical simulations. We consider convection in a cylindrical cell at aspect ratio one for Rayleigh numbers of RaD 3 10 9 and 3 10 10 at fixed Prandtl number PrD 0:7. Similar to the experimental results in the same setup and for the same Prandtl number, the structure of the laminar boundary layers of the velocity and temperature fields is found to deviate from the prediction of Prandtl‐Blasius‐Pohlhausen theory. Deviations decrease when a dynamical rescaling of the data with an instantaneously defined boundary layer thickness is performed and the analysis plane is aligned with the instantaneous direction of the large-scale circulation in the closed cell. Our numerical results demonstrate that important assumptions of existing classical laminar boundary layer theories for forced and natural convection are violated, such as the strict twodimensionality of the dynamics or the steadiness of the fluid motion. The boundary layer dynamics consists of two essential local dynamical building blocks, a plume detachment and a post-plume phase. The former is associated with larger variations of the instantaneous thickness of velocity and temperature boundary layer and a fully three-dimensional local flow. The post-plume dynamics is connected with the large-scale circulation in the cell that penetrates the boundary region from above. The mean turbulence profiles taken in localized sections of the boundary layer for each dynamical phase are also compared with solutions of perturbation expansions of the boundary layer equations of forced or natural convection towards mixed convection. Our analysis of both boundary layers shows that the near-wall dynamics combines elements of forced Blasius-type and natural convection.
83 citations
TL;DR: In this paper, the dependence of the flow on the Prandtl number was investigated in laminar axisymmetric starting plumes with a small electric heater in a glass tank with viscous oils.
Abstract: Experimental studies of laminar axisymmetric starting plumes are performed to investigate the dependence of the flow on the Prandtl number, focusing on large Prandtl numbers. Thermal plumes are generated by a small electric heater in a glass tank filled with viscous oils. Prandtl numbers in the range of 7–104 were investigated. Experimental conditions are such that viscosity variations due to temperature differences are negligible. Plumes ascend in two different regimes as a function of distance to source. At short distances, the plumes accelerate owing to the development of the viscous boundary layer. At distances larger than about five times the heater size, the ascent velocity is constant and increases as a function of the Prandtl number, as predicted by theory for steady plumes. This velocity is, within experimental error, proportional to the steady plume centreline velocity.
83 citations
Cites background or methods from "Theory of the steady laminar natura..."
...Numerical results are available up to a Prandtl number of 10 (e.g. Fujii 1963; Brand & Lahey 1967; Worster 1986) and an asymptotic analysis for large Prandtl numbers may be found in Worster (1986)....
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...We used the same boundary-layer approximation and the same equations as Fujii (1963) and Worster (1986). However, we are dealing with large Prandtl numbers, and hence we have rescaled variables using the thickness of the thermal boundary layer, δκ , instead of the viscous boundary layer. With our scheme, we reproduce the results quoted by Worster (1986) for σ = 10....
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...We used the same boundary-layer approximation and the same equations as Fujii (1963) and Worster (1986)....
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...We used the same boundary-layer approximation and the same equations as Fujii (1963) and Worster (1986). However, we are dealing with large Prandtl numbers, and hence we have rescaled variables using the thickness of the thermal boundary layer, δκ , instead of the viscous boundary layer....
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TL;DR: In this article, two-dimensional time-dependent buoyancy-induced flows above a horizontal line heat source inside rectangular vessels, with adiabatic sidewalls and top and bottom walls maintained at uniform temperature, are studied numerically.
Abstract: Two-dimensional time-dependent buoyancy-induced flows above a horizontal line heat source inside rectangular vessels, with adiabatic sidewalls and top and bottom walls maintained at uniform temperature, are studied numerically. Transitions to unsteady flows are performed by direct simulations for various depths of immersion of the source in the central vertical plane of air-filled vessels. For a square vessel and a line source near the bottom wall, the numerical solutions exhibit a sequence of instabilities, called natural swaying motion of confined plumes, beginning with a periodic regime having a high fundamental frequency followed by a two-frequency locked regime. Then, broadband components appearing in the spectra indicate chaotic behaviour and a weakly turbulent motion arises via an intermittent route to chaos. For rectangular vessels of aspect ratio greater than 2 and depths of immersion greater than the width, the flow undergoes a pitchfork bifurcation. This symmetry breaking is driven by the destabilization of an upper unstable layer of stagnant fluid above the plume. Then a subcritical Hopf bifurcation occurs. On the other hand, if the depth of immersion is lower than the width of the vessel, a stable layer of fluid is at rest below the line source. Then penetrative convection sets the whole air-filled vessel in motion and an oscillatory motion of very low frequency arises through supercritical Hopf bifurcation followed by a two-frequency locked state.
72 citations
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TL;DR: In this article, der Prandtlsche Mischungswegansatz, dessen Verwendung bei der Berechnung einer Reihe turbulenter Ausbreitungsvorgange gute Ubereinstimmung mit den Experimenten ergab, wird auf die Berechnings der Ausbreits eines ebenen and eines raumlichen Strahls erhitzter Luft angewandt.
Abstract: Der Prandtlsche Mischungswegansatz, dessen Verwendung bei der Berechnung einer Reihe turbulenter Ausbreitungsvorgange gute Ubereinstimmung mit den Experimenten ergab, wird auf die Berechnung der Ausbreitung eines ebenen und eines raumlichen Strahls erhitzter Luft angewandt. Die Ausbreitung des ebenen Strahls wurde auch nach dem Ansatz von Taylor berechnet. Uber die Ausbreitung des runden Strahls und seine experimentelle Untersuchung wird in dem im nachsten Heft erscheinenden II. Teil der Arbeit berichtet.
104 citations
TL;DR: Schuh's theory of steady laminar flow above a line heat source in a fluid with Pr = 0.7 is extended to fluids such as heavy oils in this article, and the solution is valid for all fluids of high Prandtl number, regardless of whether the viscosity is temperature dependent or not.
Abstract: Schuh's [1] theory of steady laminar flow above a line heat source in a fluid with Pr = 0.7 is extended to fluids such as heavy oils. The solution is valid for all fluids of high Prandtl number, regardless of whether the viscosity is temperature-dependent or not.
41 citations