Free convection beneath a heated horizontal plate in a rapidly rotating system

2007 ◽  
Vol 586 ◽  
pp. 491-506
Author(s):  
ROBERT J. WHITTAKER ◽  
JOHN R. LISTER
Keyword(s):  
Boundary Layer ◽  
Thermal Boundary Layer ◽  
Ekman Layer ◽  
Outer Edge ◽  
Horizontal Plate ◽  
Rotating System ◽  
Plate Temperature ◽  
System Equation ◽  
Simple Boundary ◽  
Simple Boundary Condition

Laminar flow beneath a finite heated horizontal plate in a rapidly rotating system is considered in both axisymmetric and planar geometries. In particular, we examine the case where the Ekman layer is confined well within a much deeper (yet still thin) thermal boundary layer. This situation corresponds to the regime E−3/2 ≪ Ra ≪ E−5/2, where E and Ra are the natural Ekman and Rayleigh numbers for the system (equation (2.6)). The outward flux of buoyant fluid from beneath the plate occurs primarily in the Ekman layer, while outward flow in the thicker thermal boundary layer is inhibited by a dominant thermal-wind balance. The O(Ra−1/2E−3/4 thickness of the thermal boundary layer is determined by a balance between Ekman suction and diffusion. There are several possible asymptotic regimes near the outer edge of the plate, differing only by logarithmic factors, but in all cases the edge corresponds to a simple boundary condition on the interior flow. With a uniform plate temperature, the dimensionless heat transfer (equation (7.6)) is given by a Nusselt number $\Nu\,{\sim} \tfrac{1}{2}\Ra^{1/2}\Ek^{3/4}[\ln (\Ra^{-1} \Ek ^{-5/2})]^{1/2}$. The solution for a uniform plate heat flux is also presented.

Nonlinear stratified spin-up

2002 ◽  
Vol 473 ◽  
pp. 211-244 ◽  
Author(s):  
LEIF N. THOMAS ◽  
PETER B. RHINES
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Wind Stress ◽  
Thermal Boundary Layer ◽  
Ekman Layer ◽  
Ekman Transport ◽  
Ekman Pumping ◽  
Vertical Diffusion ◽  
Vertical Vorticity ◽  
Horizontal Advection

Both a weakly nonlinear analytic theory and direct numerical simulation are used to document processes involved during the spin-up of a rotating stratified fluid driven by wind-stress forcing for time periods less than a homogeneous spin-up time. The strength of the wind forcing, characterized by the Rossby number ε, is small enough (i.e. ε[Lt ]1) that a regular perturbation expansion in ε can be performed yet large enough (more specifically, ε∝E1/2, where E is the Ekman number) that higher-order effects of vertical diffusion and horizontal advection of momentum/density are comparable in magnitude. Cases of strong stratification, where the Burger number S is equal to one, with zero heat flux at the upper boundary are considered. The Ekman transport calculated to O(ε) decreases with increasing absolute vorticity. In contrast to nonlinear barotropic spin-up, vortex stretching in the interior is predominantly linear, as vertical advection negates stretching of interior relative vorticity, yet is driven by Ekman pumping modified by nonlinearity. As vertical vorticity is generated during the spin-up of the fluid, the vertical vorticity feeds back on the Ekman pumping/suction, enhancing pumping and vortex squashing while reducing suction and vortex stretching. This feedback mechanism causes anticyclonic vorticity to grow more rapidly than cyclonic vorticity. Strict application of the zero-heat-flux boundary condition leads to the growth of a diffusive thermal boundary layer E−1/4 times thicker than the Ekman layer embedded within it. In the Ekman layer, vertical diffusion of heat balances horizontal advection of temperature by extracting heat from the thermal boundary layer beneath. The flux of heat extracted from the top of the thermal boundary layer by this mechanism is proportional to the product of the Ekman transport and the horizontal gradient of the temperature at the surface. The cooling caused by this heat flux generates density inversions and intensifies lateral density gradients where the wind-stress curl is negative. These thermal gradients make the potential vorticity strongly negative, conditioning the fluid for ensuing symmetric instability which greatly modifies the spin-up process.

Buoyancy effects in a wall jet over a heated horizontal plate

2016 ◽  
Vol 793 ◽  
pp. 21-40
Author(s):  
R. Fernandez-Feria ◽  
F. Castillo-Carrasco
Keyword(s):  
Boundary Layer ◽  
Similarity Solution ◽  
Horizontal Velocity ◽  
Wall Jet ◽  
Horizontal Distance ◽  
Horizontal Plate ◽  
Plate Temperature ◽  
Boundary Layer Equations ◽  
Dimensional Parameters ◽  
Temperature And Velocity Fields

A similarity solution of the boundary layer equations for a wall jet on a heated horizontal surface at constant temperature taking into account the coupling of the temperature and velocity fields by buoyancy is described. This similarity solution exists for any value of ${\it\Lambda}=Gr/Re^{2}$, characterizing this coupling between natural and forced convection over the horizontal plate, where $Gr$ is a Grashof number and $Re$ is a Reynolds number, provided that the plate temperature is higher than the ambient temperature (${\it\Lambda}>0$, say). Two main qualitative differences are found in the flow structure in relation to the well-known Glauert’s similarity solution for a wall jet without natural convection effects (i.e. when ${\it\Lambda}=0$): the first is that the similarity variable and structure of the horizontal velocity and temperature have the same functional form for both a radially spreading jet and a two-dimensional jet; the second is that the maximum of the horizontal velocity increases as the jet spreads over the surface, instead of decreasing like in Glauert’s solution, as the radial or horizontal distance to the power $1/5$. To check this similarity solution we solve numerically the boundary layer equations for the particular case of a jet with constant velocity and temperature emerging from a slot of height ${\it\delta}$ and radius $r_{0}$ (in the radially spreading case). An approximate, analytical similarity solution near the jet exit is also found that helps to start the numerical integration. Far from the jet exit the numerical solution tends to the similarity solution for any set of values of the non-dimensional parameters governing the problem, provided that the plate is heated (${\it\Lambda}>0$). No similarity solution is found numerically for the case of a cooled plate (${\it\Lambda}<0$). For ${\it\Lambda}=0$ Glauert’s similarity solution is recovered.

Effect of Nanofluid on Heat Transfer Enhancement for Mixed Convection Flow Over a Corrugated Surface

2020 ◽  
Vol 45 (4) ◽  
pp. 373-383
Author(s):  
Nepal Chandra Roy ◽  
Sadia Siddiqa
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Nusselt Number ◽  
Mixed Convection ◽  
Local Nusselt Number ◽  
Richardson Number ◽  
Thermal Boundary Layer ◽  
Volume Fraction ◽  
Convection Flow ◽  
Mixed Convection Flow

AbstractA mathematical model for mixed convection flow of a nanofluid along a vertical wavy surface has been studied. Numerical results reveal the effects of the volume fraction of nanoparticles, the axial distribution, the Richardson number, and the amplitude/wavelength ratio on the heat transfer of Al2O3-water nanofluid. By increasing the volume fraction of nanoparticles, the local Nusselt number and the thermal boundary layer increases significantly. In case of \mathrm{Ri}=1.0, the inclusion of 2 % and 5 % nanoparticles in the pure fluid augments the local Nusselt number, measured at the axial position 6.0, by 6.6 % and 16.3 % for a flat plate and by 5.9 % and 14.5 %, and 5.4 % and 13.3 % for the wavy surfaces with an amplitude/wavelength ratio of 0.1 and 0.2, respectively. However, when the Richardson number is increased, the local Nusselt number is found to increase but the thermal boundary layer decreases. For small values of the amplitude/wavelength ratio, the two harmonics pattern of the energy field cannot be detected by the local Nusselt number curve, however the isotherms clearly demonstrate this characteristic. The pressure leads to the first harmonic, and the buoyancy, diffusion, and inertia forces produce the second harmonic.

scholarly journals Thermal boundary layer structure in convection with and without rotation

2020 ◽  
Vol 5 (11) ◽  
Author(s):  
Robert S. Long ◽  
Jon E. Mound ◽  
Christopher J. Davies ◽  
Steven M. Tobias
Keyword(s):  
Boundary Layer ◽  
Thermal Boundary Layer ◽  
Layer Structure ◽  
Boundary Layer Structure
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