ABSTRACT
It is of fundamental significance, especially with regard to application, to fully understand the flow behavior of unsteady natural convection boundary layers on a vertical plate heated by a time-dependent heat flux. Such an understanding is currently scarce. In this paper, the scaling analysis by Lin et al. [Phys. Rev. E 79, 066313 (2009)] using a simple three-region structure for the unsteady natural convection boundary layer of a homogeneous Newtonian fluid with Pr>1 under isothermal heating was substantially extended for the case when the heating is due to a time-varying sinusoidal heat flux. A series of scalings was developed for the thermal boundary thickness, the plate temperature, the viscous boundary thicknesses, and the maximum vertical velocity within the boundary layer, which are the major parameters representing the flow behavior, in terms of the governing parameters of the flow, i.e., the Rayleigh number Ra, the Prandtl number Pr, and the dimensionless natural frequency f(n) of the time-varying sinusoidal heat flux, at the start-up stage, at the transition time scale which represents the ending of the start-up stage and the beginning of the transitional stage of the boundary-layer development, and at the quasi-steady stage. These scalings were validated by comparison to 10 full numerical solutions of the governing equations with Ra, Pr, and f(n) in the ranges 10(6)≤Ra≤10(9), 3≤Pr≤100, and 0.01≤f_{n}≤0.1 and were shown in general to provide an accurate description of the flow at different development stages, except for high-Pr runs in which a further, although weak, Pr dependence is present, which cannot be accurately predicted by the current scaling analysis using the simple three-region structure, attributed to the non-boundary-layer nature of the velocity field with high-Pr fluids. Some scalings at the transition time scale and at the quasi-steady stage also produce noticeable deviations from the numerical results when f(n) is reduced, indicating that there may be a further f(n) dependence of the scalings which also cannot be accurately predicted by the current scaling analysis.
ABSTRACT
Recent studies have used scaling analysis to obtain simple power-law relations that accurately predict the Prandtl (Pr) number dependency of natural-convection boundary layers subjected to both isothermal and ramped heating conditions, when Pr>1. The analysis used in those studies cannot be extended to Pr<1 fluids, and it is not clear at present whether such simple scaling relations can be developed for Pr<1 fluids. In the present study, the Pr>1 scalings are shown to perform well for the start-up stage of the Pr<1 flow, but not for the fully developed flow. The Pr>1 scalings are modified to provide unified Prandtl number scalings for fully developed natural-convection boundary layers for both Prâ³1 and Prâ²1, with the unknown powers obtained empirically via direct numerical simulation. The modified scalings are shown to perform well for the fully developed flow, with the exception being the prediction of the inner viscous boundary-layer thickness.
ABSTRACT
In this paper, the scalings incorporating the Prandtl number (Pr) dependence have been obtained by a scaling analysis for the unsteady natural convection boundary layer of an initially quiescent isothermal Newtonian fluid of Pr>1 produced by the sudden imposition of a higher temperature on a vertical plate. It is shown that the transient flow behavior of the resulting boundary layer can be described by a three-region structure and at the start-up stage the boundary layer development is one dimensional and independent of height due to the dominance of pure conduction; however, at steady state it becomes two dimensional and height dependent as the flow becomes dominated by convection. Numerical results demonstrate that the scalings representing the thermal boundary layer development accurately represent their Pr dependence over the whole stage of flow development. The scalings representing the viscous boundary layer development are generally in good agreement with the numerical results with the Pr variation over the whole stage of flow development, although there are small deviations from the numerical results with the Pr variation that are within acceptable limits for scaling.
ABSTRACT
The flow behavior associated with cooling an initially quiescent isothermal Newtonian fluid with Prandtl number (Pr) less than one in a vertical cylinder by unsteady natural convection with an imposed lower temperature on vertical sidewalls is investigated by scaling analysis and direct numerical simulation. The flow is dominated by three distinct stages of development, i.e., the boundary-layer development stage adjacent to the sidewall, the stratification stage, and the cooling-down stage, respectively. The first stage can be further divided into three distinct substages, i.e., the start-up stage, the transitional stage, and the boundary-layer steady-state stage, respectively. A scaling analysis is carried out to obtain scaling laws for the basic flow features in terms of the flow control parameters, i.e., the Rayleigh number Ra, Pr, and the aspect ratio of the cylinder A , respectively. A series of direct numerical simulations with selected values of A , Ra, and Pr in the ranges of 1/3< or = A< or =3, 10(6) < or =Ra < or = 10(10) , and 0.01< or =Pr< or =0.5 are carried out, and it is found that the numerical results agree well with the scaling laws. These numerical results are further used to quantify these scaling laws for Ra, A , and Pr in the above-mentioned ranges.
ABSTRACT
The transient behavior of the natural convection boundary-layer flow adjacent to a vertical plate heated with a uniform flux in a quiescent homogeneous ambient fluid with Prandtl number Pr<1 is investigated by scaling analysis and direct numerical simulation (DNS). The flow is characterized by a startup stage, a short transitional stage and a steady state. The flow is parametrized by the thermal and velocity boundary-layer thickness scales, the vertical velocity scale, the time scale for the boundary layer to reach the steady state and the plate temperature scale. Scaling analysis is used to obtain laws relating these quantities to the flow governing parameters, the Rayleigh number Ra, the Prandtl number, and the Boussinesq number Bo=RaPr which is a much more important control parameter than Ra for small Pr fluids. A series of DNS with selected values of Ra and Pr in the ranges of 10(6)< or =R< or =10(10) and 0.01< or =Pr < or =0.5 are used to validate the scaling laws and obtain scaling constants.
ABSTRACT
In this study, the long-term behavior of cooling an initially quiescent isothermal Newtonian fluid in a rectangular container with an infinite length by unsteady natural convection due to a fixed wall temperature has been investigated by scaling analysis and direct numerical simulation. Two specific cases are considered. Case 1 assumes that the cooling of the fluid is caused by the imposed fixed temperature on the vertical sidewall while the top and bottom boundaries are adiabatic. Case 2 assumes that the cooling is caused by the imposed fixed temperature on both the vertical sidewall and the bottom boundary while the top boundary is adiabatic. The appropriate parameters to represent the long-term behavior of the fluid cooling in the container are the transient average fluid temperature T(a)(t) over the whole volume of the container per unit length (i.e., the transient area average fluid temperature, as used in the subsequent numerical simulations) at time t and the average Nusselt number on the cooling boundary. A scaling analysis has been carried out which shows that for both cases theta(a)(tau) scales as e(-C(ARa)(-1/4) tau), where theta(a)(tau) is the dimensionless form of T(a)(t), tau is the dimensionless time, A is the aspect ratio of the container, Ra is the Rayleigh number, and C is a proportionality constant. A series of direct numerical simulations with the selected values of A, Ra, and Pr (Pr is the Prandtl number) in the ranges of 1/3< or =A< or =3, 6 x 10(6) < or =Ra< or =6 x 10(10), and 1< or =Pr< or =1000 have been carried out for both cases to validate the developed scaling relations. It is found that these numerical results agree well with the scaling relations. The numerical results have also been used to quantify the scaling relations and it is found that C=0.645 and 0.705 respectively for Cases 1 and 2 with Ra, A and Pr in the above-mentioned ranges.
Subject(s)
Biophysics/methods , Cold Temperature , Models, Theoretical , Temperature , Time FactorsABSTRACT
The behavior of weak axisymmetric and plane fountains resulting from the injection of denser fluid upwards into large containers containing a stably stratified fluid has been explored using dimensional analysis, scaling analysis, and direct numerical simulation. For weak fountains, with Froude number Fr approximately 1.0, dimensional and scaling analyses have been used to derive scaling relations for the dimensionless fountain height, width, thickness of the temperature layer, and development times in terms of the Froude number Fr, Reynolds number Re, Prandtl number Pr, and ambient stratification number s. Numerical simulations have been carried out for a series of Fr, Re, Pr, and s for both axisymmetric and plane fountains to validate and quantify the scaling relations obtained by the dimensional and scaling analyses. The numerical results have been found to agree well with the analytical scaling relations.
