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.

Experimental studies of the viscous boundary layer properties in turbulent Rayleigh–Bénard convection

2008 ◽  
Vol 605 ◽  
pp. 79-113 ◽  
Author(s):  
CHAO SUN ◽  
YIN-HAR CHEUNG ◽  
KE-QING XIA
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Shear Stress ◽  
Rayleigh Number ◽  
Power Law ◽  
Thermal Boundary Layer ◽  
Viscous Sublayer ◽  
Viscous Boundary Layer ◽  
Velocity Boundary Layer ◽  
Viscous Boundary

We report high-resolution measurements of the properties of the velocity boundary layer in turbulent thermal convection using the particle image velocimetry (PIV) technique and measurements of the temperature profiles and the thermal boundary layer. Both velocity and temperature measurements were made near the lower conducting plate of a rectangular convection cell using water as the convecting fluid, with the Rayleigh number Ra varying from 109 to 1010 and the Prandtl number Pr fixed at 4.3. From the measured profiles of the horizontal velocity we obtain the viscous boundary layer thickness δυ. It is found that δυ follows the classical Blasius-like laminar boundary layer in the present range of Ra, and it scales with the Reynolds number Re as δυ/H = 0.64Re−0.50±0.03 (where H is the cell height). While the measured viscous shear stress and Reynolds shear stress show that the boundary layer is laminar for Ra < 2.0 × 1010, two independent extrapolations, one based on velocity measurements and the other on velocity and temperature measurements, both indicate that the boundary layer will become turbulent at Ra ~ 1013. Just above the thermal boundary layer but within the mixing zone, the measured temperature r.m.s. profiles σT(z) are found to follow either a power law or a logarithmic behaviour. The power-law fitting may be slightly favoured and its exponent is found to depend on Ra and varies from −0.6 to −0.77, which is much larger than the classical value of −1/3. In the same region, the measured profiles of the r.m.s. vertical velocity σw(z) exhibit a much smaller scaling range and are also consistent with either a power-law or a logarithmic behaviour. The Reynolds number dependence of several wall quantities is also measured directly. These are the wall shear stress τw ~ Re1.55, the viscous sublayer δw ~ Re−0.91, the friction velocity uτ ~ Re0.80, and the skin-friction coefficient cf ~ Re−0.34. All of these scaling properties are very close to those predicted for a classical Blasius-type laminar boundary layer, except that of cf. Similar to classical shear flows, a viscous sublayer is also found to exist in the present system despite the presence of a nested thermal boundary layer. However, velocity profiles normalized by wall units exhibit no obvious logarithmic region, which is likely to be a result of the very limited distance between the edge of the viscous sublayer and the position of the maximum velocity. Compared to traditional shear flows, the peak position of the wall-unit-normalized r.m.s. profiles is found to be closer to the plate (at z+ = z/δw ≃ 5). Our overall conclusion is that a Blasius-type laminar boundary condition is a good approximation for the velocity boundary layer in turbulent thermal convection for the present range of Rayleigh number and Prandtl number.

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.

An integral approach to the calculation of skin friction and heat transfer in the laminar flow of a high-Prandtl-number power-law non-Newtonian fluid past a wedge

1991 ◽  
Vol 69 (2) ◽  
pp. 83-89 ◽  
Author(s):  
G. Ramamurty ◽  
K. Narasimha Rao ◽  
K. N. Seetharamu
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Prandtl Number ◽  
Skin Friction ◽  
Layer Thickness ◽  
Power Law ◽  
Boundary Layer Thickness ◽  
Thermal Boundary Layer ◽  
Integral Approach ◽  
High Prandtl Number

An integral approach to the theoretical analysis for the skin friction of a non-Newtonian, power-law-fluid flow over a wedge is presented, when the inertia terms in the boundary-layer equations are small but need consideration. The method adopted for the solution of the equations considers an integrated average value of the inertia terms in the momentum equation. The values of the velocities and the boundary-layer thickness obtained from the hydrodynamic analysis are used for the calculation of the thermal-boundary-layer thickness. A linear velocity profile is assumed for the flow field within the thermal boundary layer as the fluids chosen for the analysis are high-Prandtl-number fluids. The results of the skin friction and the rates of the heat transfer are tabulated for a number of values of the flow behaviour index, n, varying from 0.05 to 5.0. This analysis is applicable to viscous polymer solutions having high Prandtl numbers.

A CFD Evaluation of Multiple RANS Turbulence Models for Prediction of Boundary Layer Flows on a Turbine Vane

2013 ◽  
Author(s):  
Thomas E. Dyson ◽  
David G. Bogard ◽  
Sean D. Bradshaw
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Layer Thickness ◽  
Boundary Layer Thickness ◽  
Thermal Boundary Layer ◽  
Turbulence Models ◽  
Experimental Measurements ◽  
Thermal Boundary Layer Thickness ◽  
Mean Velocity ◽  
High Turbulence

There is a growing trend toward the use of conjugate CFD for use in prediction of turbine cooling performance. While many studies have evaluated the performance of RANS simulations relative to experimental measurements of the momentum boundary layer, no studies have evaluated their performance in prediction of the accompanying thermal boundary layer. This is largely due to the fact that, until recently, no appropriate experimental data existed to validate these models. This study compares several popular RANS models — including the realizable k-ε and k-ω SST models — with a four equation k-ω model (“Transition SST”) and experimental measurements at selected positions on the pressure and suction sides of a model C3X vane. Comparisons were made using mean velocity and temperature in the boundary layer without film cooling under conditions of high and low mainstream turbulence. The best performing model was evaluated using modification of the turbulent Prandtl number to attempt to better match the data for the high turbulence case. Overall, the models did not perform well for the low turbulence case; they greatly over-predicted the thermal boundary layer thickness. For the high turbulence case, their performance was better. The Transition SST model performed the best with an average thermal boundary layer thickness within 15% of the experimentally measured values. Prandtl number variation proved to be an inadequate means of improving the thermal boundary layer predictions.

Effects of temperature-dependent viscosity and thermal conductivity on mixed convection flow along a magnetized vertical surface

2016 ◽  
Vol 26 (5) ◽  
pp. 1580-1592 ◽  
Author(s):  
Ashraf Muhammad ◽  
Ali J Chamkha ◽  
S Iqbal ◽  
Masud Ahmad
Keyword(s):  
Magnetic Field ◽  
Thermal Conductivity ◽  
Boundary Layer ◽  
Prandtl Number ◽  
Layer Thickness ◽  
Boundary Layer Thickness ◽  
Thermal Boundary Layer ◽  
Content Type ◽  
Variation Parameter ◽  
Momentum Boundary Layer

Purpose – The purpose of this paper is to report a numerical solution for the problem of steady, two dimensional boundary layer buoyant flow on a vertical magnetized surface, when both the viscosity and thermal conductivity are assumed to be temperature-dependent. In this case, the motion is governed by a coupled set of three nonlinear partial differential equations, which are solved numerically by using the finite difference method (FDM) by introducing the primitive variable formulation. Calculations of the coupled equations are performed to investigate the effects of the different governing parameters on the profiles of velocity, temperature and the transverse component of magnetic field. The effects of the thermal conductivity variation parameter, viscosity variation parameter, magnetic Prandtl number Pmr, magnetic force parameter S, mixed convection parameter Ri and the Prandtl number Pr on the flow structure and heat transfer characteristics are also examined. Design/methodology/approach – FDM. Findings – It is noted that when the Prandtl number Pr is sufficiently large, i.e. Pr=100, the buoyancy force that driven the fluid motion is decreased that decrease the momentum boundary layer and there is no change in thermal boundary layer is noticed. It is also noted that due to slow motion of the fluid the magnetic current generates which increase the magnetic boundary layer thickness at the surface. It is observed that the momentum boundary layer thickness is increased, thermal and magnetic field boundary layers are decreased with the increase of thermal conductivity variation parameter =100. The maximum boundary layer thickness is increased for =100 and there is no change seen in the case of thermal boundary layer thickness but magnetic field boundary layer is deceased. The momentum boundary layer thickness shoot quickly for =40 but is very smooth for =50.There is no change is seen for the case of thermal boundary layer and very clear decay for =40 is noted. Originality/value – This work is original research work.

Stability of thermocapillary flows in non-cylindrical liquid bridges

2002 ◽  
Vol 458 ◽  
pp. 35-73 ◽  
Author(s):  
CH. NIENHÜSER ◽  
H. C. KUHLMANN
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Critical Reynolds Number ◽  
Volume Fraction ◽  
Thermocapillary Flow ◽  
Surface Shape ◽  
Liquid Bridges ◽  
Gravity Level ◽  
Neutral Curves ◽  
Prandtl Numbers

The thermocapillary flow in liquid bridges is investigated numerically. In the limit of large mean surface tension the free-surface shape is independent of the flow and temperature fields and depends only on the volume of liquid and the hydrostatic pressure difference. When gravity acts parallel to the axis of the liquid bridge the shape is axisymmetric. A differential heating of the bounding circular disks then causes a steady two-dimensional thermocapillary flow which is calculated by a finite-difference method on body-fitted coordinates. The linear-stability problem for the basic flow is solved using azimuthal normal modes computed with the same discretization method. The dependence of the critical Reynolds number on the volume fraction, gravity level, Prandtl number, and aspect ratio is explained by analysing the energy budgets of the neutral modes. For small Prandtl numbers (Pr = 0.02) the critical Reynolds number exhibits a smooth minimum near volume fractions which approximately correspond to the volume of a cylindrical bridge. When the Prandtl number is large (Pr = 4) the intersection of two neutral curves results in a sharp peak of the critical Reynolds number. Since the instabilities for low and high Prandtl numbers are markedly different, the influence of gravity leads to a distinctly different behaviour. While the hydrostatic shape of the bridge is the most important effect of gravity on the critical point for low-Prandtl-number flows, buoyancy is the dominating factor for the stability of the flow in a gravity field when the Prandtl number is high.

Numerical simulations of Rayleigh–Bénard convection for Prandtl numbers between 10−1 and 104 and Rayleigh numbers between 105 and 109

2010 ◽  
Vol 662 ◽  
pp. 409-446 ◽  
Author(s):  
G. SILANO ◽  
K. R. SREENIVASAN ◽  
R. VERZICCO
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Nusselt Number ◽  
Numerical Simulations ◽  
Multiple Solutions ◽  
Large Scale ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

We summarize the results of an extensive campaign of direct numerical simulations of Rayleigh–Bénard convection at moderate and high Prandtl numbers (10−1 ≤ Pr ≤ 104) and moderate Rayleigh numbers (105 ≤ Ra ≤ 109). The computational domain is a cylindrical cell of aspect ratio Γ = 1/2, with the no-slip condition imposed on all boundaries. By scaling the numerical results, we find that the free-fall velocity should be multiplied by $1/\sqrt{{\it Pr}}$ in order to obtain a more appropriate representation of the large-scale velocity at high Pr. We investigate the Nusselt and the Reynolds number dependences on Ra and Pr, comparing the outcome with previous numerical and experimental results. Depending on Pr, we obtain different power laws of the Nusselt number with respect to Ra, ranging from Ra2/7 for Pr = 1 up to Ra0.31 for Pr = 103. The Nusselt number is independent of Pr. The Reynolds number scales as ${\it Re}\,{\sim}\,\sqrt{{\it Ra}}/{\it Pr}$, neglecting logarithmic corrections. We analyse the global and local features of viscous and thermal boundary layers and their scaling behaviours with respect to Ra and Pr, and with respect to the Reynolds and Péclet numbers. We find that the flow approaches a saturation state when Reynolds number decreases below the critical value, Res ≃ 40. The thermal-boundary-layer thickness increases slightly (instead of decreasing) when the Péclet number increases, because of the moderating influence of the viscous boundary layer. The simulated ranges of Ra and Pr contain steady, periodic and turbulent solutions. A rough estimate of the transition from the steady to the unsteady state is obtained by monitoring the time evolution of the system until it reaches stationary solutions. We find multiple solutions as long-term phenomena at Ra = 108 and Pr = 103, which, however, do not result in significantly different Nusselt numbers. One of these multiple solutions, even if stable over a long time interval, shows a break in the mid-plane symmetry of the temperature profile. We analyse the flow structures through the transitional phases by direct visualizations of the temperature and velocity fields. A wide variety of large-scale circulation and plume structures has been found. The single-roll circulation is characteristic only of the steady and periodic solutions. For other regimes at lower Pr, the mean flow generally consists of two opposite toroidal structures; at higher Pr, the flow is organized in the form of multi-jet structures, extending mostly in the vertical direction. At high Pr, plumes mainly detach from sheet-like structures. The signatures of different large-scale structures are generally well reflected in the data trends with respect to Ra, less in those with respect to Pr.

Annulus Wall Boundary-Layer Measurements in a Four-Stage Compressor

1986 ◽  
Vol 108 (1) ◽  
pp. 2-6 ◽  
Author(s):  
N. A. Cumpsty
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Layer Thickness ◽  
Boundary Layers ◽  
Boundary Layer Thickness ◽  
Tip Clearance ◽  
The Other ◽  
Wall Boundary Layer ◽  
Multistage Compressors ◽  
Full Design

There are few available measurements of the boundary layers in multistage compressors when the repeating-stage condition is reached. These tests were performed in a small four-stage compressor; the flow was essentially incompressible and the Reynolds number based on blade chord was about 5 • 104. Two series of tests were performed; in one series the full design number of blades were installed, in the other series half the blades were removed to reduce the solidity and double the staggered spacing. Initially it was wished to examine the hypothesis proposed by Smith [1] that staggered spacing is a particularly important scaling parameter for boundary layer thickness; the results of these tests and those of Hunter and Cumpsty [2] tend to suggest that it is tip clearance which is most potent in determining boundary-layer integral thicknesses. The integral thicknesses agree quite well with those published by Smith.

Sign in / Sign up

Export Citation Format

Share Document