scholarly journals Very viscous horizontal convection

2008 ◽  
Vol 611 ◽  
pp. 395-426 ◽  
Author(s):  
S. CHIU-WEBSTER ◽  
E. J. HINCH ◽  
J. R. LISTER
Keyword(s):  
Boundary Layer ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Initial Conditions ◽  
Analytic Solutions ◽  
Rectangular Domain ◽  
Return Flow ◽  
Linear Temperature ◽  
Linear Temperature Gradient ◽  
Horizontal Convection

‘Horizontal convection’ arises when a temperature variation is imposed along a horizontal boundary of a finite fluid volume. Here we study the infinite-Prandtl-number limit relevant to very viscous fluids, motivated by the study of convection in glass furnaces. We consider a rectangular domain with insulating conditions on the sides and bottom, and a linear temperature gradient on the top. We describe steady states for a large range of aspect ratio A and Rayleigh number Ra, and find universal scalings for the transition from small to large Rayleigh numbers. At large Rayleigh number, the top boundary-layer thickness scales as Ra−1/5, with the circulation and heat flux scaling as Ra1/5. These scalings hold for both rigid and shear-free boundary conditions on the top or on the other boundaries, which is initially surprising, but is because the return flow is dominated by a horizontal intrusion immediately beneath the top boundary layer. A downwelling plume also forms on one side, but because of strong stratification in the interior, the volume flux it carries is much smaller than that of the horizontal intrusion, decaying as the inverse of the depth below the top boundary. The fluid in the plume detrains into the interior and then returns to the top boundary, thus forming a ‘filling box’. We find analytic solutions for the interior temperature and streamfunction and match them to a similarity solution for the plume. At depths comparable to the length of the top boundary the streamfunction has O(1) values and the temperature variations scale as 1/Ra. Transient calculations with a large, but finite, Prandtl number, show how the steady state is reached from hot and cold initial conditions.

scholarly journals Horizontal convection in a rectangular enclosure driven by a linear temperature profile

2021 ◽  
Author(s):  
Tianyong Yang ◽  
Bofu Wang ◽  
Jianzhao Wu ◽  
Zhiming Lu ◽  
Quan Zhou
Keyword(s):  
Rayleigh Number ◽  
Temperature Profile ◽  
Scaling Law ◽  
Thermal Plume ◽  
Steady Regime ◽  
Linear Temperature ◽  
Square Enclosure ◽  
Horizontal Convection ◽  
Kinetic Boundary Layer ◽  
The Mean

AbstractThe horizontal convection in a square enclosure driven by a linear temperature profile along the bottom boundary is investigated numerically by using a finite difference method. The Prandtl number is fixed at 4.38, and the Rayleigh number Ra ranges from 107 to 1011. The convective flow is steady at a relatively low Rayleigh number, and no thermal plume is observed, whereas it transits to be unsteady when the Rayleigh number increases beyond the critical value. The scaling law for the Nusselt number Nu changes from Rossby’s scaling Nu ∼ Ra1/5 in a steady regime to Nu ∼ Ra1/4 in an unsteady regime, which agrees well with the theoretically predicted results. Accordingly, the Reynolds number Re scaling varies from Re ∼ Ra3/11 to Re ∼ Ra2/5. The investigation on the mean flows shows that the thermal and kinetic boundary layer thickness and the mean temperature in the bulk zone decrease with the increasing Ra. The intensity of fluctuating velocity increases with the increasing Ra.

scholarly journals On the boundary-layer structure of high-Prandtl-number horizontal convection

2010 ◽  
Vol 652 ◽  
pp. 299-331
Author(s):  
P. G. DANIELS
Keyword(s):  
Boundary Layer ◽  
Temperature Distribution ◽  
Prandtl Number ◽  
Layer Structure ◽  
Lower Surface ◽  
Horizontal Layer ◽  
Linear Temperature ◽  
Boundary Layer Problem ◽  
Boundary Layer Structure ◽  
Horizontal Boundary

This paper describes the boundary-layer structure of the steady flow of an infinite Prandtl number fluid in a two-dimensional rectangular cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated and the upper surface is assumed to be either shear-free or rigid. In the limit of large Rayleigh number (R → ∞), the solution involves a horizontal boundary layer at the upper surface of depth of order R−1/5 where the main variation in the temperature field occurs. For a monotonic temperature distribution at the upper surface, fluid is driven to the colder end of the cavity where it descends within a narrow convection-dominated vertical layer before returning to the horizontal layer. A numerical solution of the horizontal boundary-layer problem is found for the case of a linear temperature distribution at the upper surface. At greater depths, of order R−2/15 for a shear-free surface and order R−9/65 for a rigid upper surface, a descending plume near the cold sidewall, together with a vertically stratified interior flow, allow the temperature to attain an approximately constant value throughout the remainder of the cavity. For a shear-free upper surface, this constant temperature is predicted to be of order R−1/15 higher than the minimum temperature of the upper surface, whereas for a rigid upper surface it is predicted to be of order R−2/65 higher.

A study of Bénard convection with and without rotation

1969 ◽  
Vol 36 (2) ◽  
pp. 309-335 ◽  
Author(s):  
H. T. Rossby
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Linear Theory ◽  
Silicone Oil ◽  
Taylor Number ◽  
Subcritical Instability ◽  
Two Fluids ◽  
Wide Range

An experimental study of the response of a thin uniformly heated rotating layer of fluid is presented. It is shown that the stability of the fluid depends strongly upon the three parameters that described its state, namely the Rayleigh number, the Taylor number and the Prandtl number. For the two Prandtl numbers considered, 6·8 and 0·025 corresponding to water and mercury, linear theory is insufficient to fully describe their stability properties. For water, subcritical instability will occur for all Taylor numbers greater than 5 × 104, whereas mercury exhibits a subcritical instability only for finite Taylor numbers less than 105. At all other Taylor numbers there is good agreement between linear theory and experiment.The heat flux in these two fluids has been measured over a wide range of Rayleigh and Taylor numbers. Generally, much higher Nusselt numbers are found with water than with mercury. In water, at any Rayleigh number greater than 104, it is found that the Nusselt number will increase by about 10% as the Taylor number is increased from zero to a certain value, which depends on the Rayleigh number. It is suggested that this increase in the heat flux results from a perturbation of the velocity boundary layer with an ‘Ekman-layer-like’ profile in such a way that the scale of boundary layer is reduced. In mercury, on the other hand, the heat flux decreases monotonically with increasing Taylor number. Over a range of Rayleigh numbers (at large Taylor numbers) oscillatory convection is preferred although it is inefficient at transporting heat. Above a certain Rayleigh number, less than the critical value for steady convection according to linear theory, the heat flux increases more rapidly and the convection becomes increasingly irregular as is shown by the temperature fluctuations at a point in the fluid.Photographs of the convective flow in a silicone oil (Prandtl number = 100) at various rotation rates are shown. From these a rough estimate is obtained of the dominant horizontal convective scale as a function of the Rayleigh and Taylor numbers.

Free convection in an open thermosyphon, with special reference to turbulent flow

1955 ◽  
Vol 230 (1183) ◽  
pp. 502-530 ◽  
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Boundary Layer Flow ◽  
Radius Ratio ◽  
Tube Length ◽  
Prandtl Numbers ◽  
Layer Flow ◽  
Rayleigh Numbers

This paper describes an experimental investigation of heat transfer by free convection of a fluid in a heated vertical tube, sealed at its lower end. Heated fluid adjacent to the wall is discharged from the open end into a suitably cooled large reservoir, while a central core of cool fluid is continuously drawn into the tube by way of replacement. The system constitutes an unusual case of natural convection because the two streams of fluid, moving in opposite directions, are compelled to create their own internal boundary. Such an arrangement forms a static simulation of the Schmidt system (1951) for cooling high-temperature gas turbine blades, where sealed radial passages in the blades communicate with a reservoir in the rotor drum, and large centrifugal accelerations replace that due to gravity in the static system. The use of a scaled-up static tube in large measure compensates for the relatively small gravitational acceleration, when determining the working range of Rayleigh numbers, in this case from 10 7 to 10 13 . These are based on tube length, the fluid property values being referred to tube-wall temperature. Separate assessments are made of the effect of fluid Prandtl number (covering values from 7600 to 0·69) and tube length radius ratio (ranging from 7·5 to 47·5). In laminar flow the former is not found to be significant, but the quotient of the Rayleigh number (based on radius) and tube length-radius ratio determines the ranges of three laminar flow régimes. High values of the quotient correspond to 'boundary-layer flow’ and greatest heat transfer. This is followed first by ‘impeded non-similarity flow’ and then by ‘impeded similarity flow’ as the quotient becomes smaller, where the two streams of fluid mingle. These findings are in close agreement with theoretical prediction (Lighthill 1953). Turbulence arises in two ways. For Prandtl numbers near unity, transition occurs during the laminar impeded-flow régimes, resulting in a mixing effect and reduced heat transfer. This is predicted by Lighthill, but his discussion of turbulent flow is restricted to a Prandtl number of unity. For larger Prandtl numbers, transition takes place during laminar boundary-layer flow, yielding a conventional turbulent boundary-layer régime with increased heat transfer. The mean transitional Grashof numbers (based on radius) are in the range 10 4.4 to 10 4.6 ; they compare favourably with a pre­dicted range of from 10 4.0 to 10 4.3 . The tendency for the cool entering fluid to become turbulent renders turbulent boundary-layer flow potentially unstable. Both modes of transition eventually lead to a stable ‘fully mixed' régime where the two turbulent streams mix. This causes reduced circulation and heat transfer, the extent of the reduction varying directly with length-radius ratio and inversely with Prandtl number. The régime was predicted by Lighthill, but there are considerable dis­crepancies between estimated and experimental heat-transfer rates, and in the duration of the régime. In practice it appears to persist indefinitely, whereas Lighthill forecasts its replace­ment at high Rayleigh numbers by a stable boundary-layer flow. Empirical correlations show that fully mixed flow yields optimum heat transfer at a length-radius ratio, which is determined by the Rayleigh number. The suitability of the Schmidt system for blade cooling is briefly discussed in the light of the investigation.

Horizontal convection is non-turbulent

2002 ◽  
Vol 466 ◽  
pp. 205-214 ◽  
Author(s):  
F. PAPARELLA ◽  
W. R. YOUNG
Keyword(s):  
Thermal Diffusivity ◽  
Energy Dissipation ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Numerical Solutions ◽  
Inviscid Limit ◽  
Two Dimensional ◽  
Uniform Temperature ◽  
Horizontal Convection ◽  
Boussinesq Fluid

Consider the problem of horizontal convection: a Boussinesq fluid, forced by applying a non-uniform temperature at its top surface, with all other boundaries insulating. We prove that if the viscosity, ν, and thermal diffusivity, κ, are lowered to zero, with σ ≡ ν/κ fixed, then the energy dissipation per unit mass, κ, also vanishes in this limit. Numerical solutions of the two-dimensional case show that despite this anti-turbulence theorem, horizontal convection exhibits a transition to eddying flow, provided that the Rayleigh number is sufficiently high, or the Prandtl number σ sufficiently small. We speculate that horizontal convection is an example of a flow with a large number of active modes which is nonetheless not ‘truly turbulent’ because ε→0 in the inviscid limit.

Boundary-layer solutions of single-mode convection equations

1981 ◽  
Vol 102 ◽  
pp. 211-219 ◽  
Author(s):  
N. Riahi
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Thermal Convection ◽  
Single Mode ◽  
Layer Method ◽  
Boundary Layer Method ◽  
Inertial Terms ◽  
Large Rayleigh Number

Nonlinear thermal convection between two stress-free horizontal boundaries is studied using the modal equations for cellular convection. Assuming a large Rayleigh number R the boundary-layer method is used for different ranges of the Prandtl number σ. The heat flux F is determined for the values of the horizontal wavenumber a which maximizes F. For a large Prandtl number, σ [Gt ] R⅙(log R)−1, inertial terms are insignificant, a is either of order one (for $\sigma \geqslant R^{\frac{2}{3}}$) or proportional to $R^{\frac{1}{3}}\sigma^{-\frac{1}{2}}$ (for $\sigma \ll R^{\frac{2}{3}}$) and F is proportional to $R^{\frac{1}{3}}$. For a moderate Prandtl number, \[ (R^{-1}\log R)^{\frac{1}{9}} \ll \sigma \ll R^{\frac{1}{6}}(\log R)^{-1}, \] inertial terms first become significant in an inertial layer adjacent to the viscous buoyancy-dominated interior, and a and F are proportional to R¼ and \[ R^{\frac{3}{10}}\sigma^{\frac{1}{5}} (\log\sigma R^{\frac{1}{4}})^{\frac{1}{10}}, \] respectively. For a small Prandtl number, $R^{-1} \ll \sigma \ll (R^{-1} \log R)^{\frac{1}{9}}$, inertial terms are significant both in the interior and the boundary layers, and a and F are proportional to ($R \sigma)^{\frac{9}{32}} (\log R\sigma)^{-\frac{1}{32}}$ and ($R \sigma)^{\frac{5}{16}} (\log R \sigma)^{\frac{3}{16}}$, respectively.

scholarly journals Nonlinear Double-diffusive Convection in a Low Prandtl Number Fluid

1980 ◽  
Vol 33 (1) ◽  
pp. 59 ◽  
Author(s):  
N Riahi
Keyword(s):  
Boundary Layer ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Layer Structure ◽  
Solute Concentration ◽  
Double Diffusive Convection ◽  
Diffusive Convection ◽  
Solute Fluxes ◽  
Boundary Layer Method ◽  
Double Diffusive

Nonlinear double-diffusive convection is studied using the modal equations of cellular convection. The boundary layer method is used by assuming a large Rayleigh number R for a fluid of low Prandtl number (J, and different ranges of the diffusivity ratio 7: and. the solute Rayleigh number Rs. The heat and solute fluxes are found to increase with R(J and decrease with Rs. The effect of the solute is stabilizing, although the convection in a fluid with large (J is less affected by the solute concentration. The flow is shown to have a solute layer which thickens as (J, R, .-1 or R;l decreases. It is proved that it is only for this layer that the solute affects the boundary layer structure.

scholarly journals Global and local statistics in turbulent convection at low Prandtl numbers

2016 ◽  
Vol 802 ◽  
pp. 147-173 ◽  
Author(s):  
Janet D. Scheel ◽  
Jörg Schumacher
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Boundary Layer Thickness ◽  
Thermal Boundary Layer ◽  
Turbulent Convection ◽  
Scaling Exponent ◽  
Velocity Boundary Layer ◽  
Prandtl Numbers

Statistical properties of turbulent Rayleigh–Bénard convection at low Prandtl numbers $Pr$, which are typical for liquid metals such as mercury or gallium ($Pr\simeq 0.021$) or liquid sodium ($Pr\simeq 0.005$), are investigated in high-resolution three-dimensional spectral element simulations in a closed cylindrical cell with an aspect ratio of one and are compared to previous turbulent convection simulations in air for $Pr=0.7$. We compare the scaling of global momentum and heat transfer. The scaling exponent $\unicode[STIX]{x1D6FD}$ of the power law $Nu=\unicode[STIX]{x1D6FC}Ra^{\unicode[STIX]{x1D6FD}}$ is $\unicode[STIX]{x1D6FD}=0.265\pm 0.01$ for $Pr=0.005$ and $\unicode[STIX]{x1D6FD}=0.26\pm 0.01$ for $Pr=0.021$, which are smaller than that for convection in air ($Pr=0.7$, $\unicode[STIX]{x1D6FD}=0.29\pm 0.01$). These exponents are in agreement with experiments. Mean profiles of the root-mean-square velocity as well as the thermal and kinetic energy dissipation rates have growing amplitudes with decreasing Prandtl number, which underlies a more vigorous bulk turbulence in the low-$Pr$ regime. The skin-friction coefficient displays a Reynolds number dependence that is close to that of an isothermal, intermittently turbulent velocity boundary layer. The thermal boundary layer thicknesses are larger as $Pr$ decreases and conversely the velocity boundary layer thicknesses become smaller. We investigate the scaling exponents and find a slight decrease in exponent magnitude for the thermal boundary layer thickness as $Pr$ decreases, but find the opposite case for the velocity boundary layer thickness scaling. A growing area fraction of turbulent patches close to the heating and cooling plates can be detected by exceeding a locally defined shear Reynolds number threshold. This area fraction is larger for lower $Pr$ at the same $Ra$, but the scaling exponent of its growth with Rayleigh number is reduced. Our analysis of the kurtosis of the locally defined shear Reynolds number demonstrates that the intermittency in the boundary layer is significantly increased for the lower Prandtl number and for sufficiently high Rayleigh number compared to convection in air. This complements our previous findings of enhanced bulk intermittency in low-Prandtl-number convection.

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