Range Of Prandtl Numbers Research Articles

Prandtl Number Effect on the Laminar Convective Heat Transfer From a Rotating Disk

Convective heat transfer in rotating disk systems is of great importance in many engineering applications. Despite the high practical relevance, there have been only a small number of experimental investigations regarding the influence of the Prandtl number larger than unity. Ever since Dorfman's pioneering work more than 50 years ago, various analytical works about the heat transfer of a rotating disk have been published. However, this study is a novelty because measurements of the laminar convective heat transfer over a free rotating disk for a wide range of Prandtl number up to Pr=5000 are presented. The accuracy of the employed experimental apparatus was assessed by heat transfer measurements in air, for which reliable literature data are widely available. Natural convection effects and temperature-dependent physical properties have been taken into consideration using the property-ratio method. The experimental results are in excellent agreement with analytical self-similar solutions and the theoretical correlation of Lin and Lin. The applicability of frequently used heat transfer correlations is assessed by the means of the new experimental data.

Natural convection flow of a two-phase dusty non-Newtonian fluid along a vertical surface

The aim of this paper is to present a boundary-layer analysis of two-phase dusty non-Newtonian fluid flow along a vertical surface by using a modified power-law viscosity model. This investigation particularly reports the flow behavior of spherical particles suspended in the non-Newtonian fluid. The governing equations are transformed into non-conserved form and then solved straightforwardly by implicit finite difference method. The numerical results of rate of heat transfer, rate of shear stress, velocity and temperature profiles and streamlines and isotherms are presented for wide range of Prandtl number, i.e., 0.7⩽Pr≤1000.0, with the representative values of the power-law index n. A good agreement is found between the present and the previous results when compared for some special cases. The key observation from the present study is that the power-law fluids with (n>1) are more likely to promote the rate of heat transfer near the leading edge.

Self-similar mixing in stratified plane Couette flow for varying Prandtl number

We investigate fully developed turbulence in stratified plane Couette flows using direct numerical simulations similar to those reported by Deusebioet al.(J. Fluid Mech., vol. 781, 2015, pp. 298–329) expanding the range of Prandtl number$Pr$examined by two orders of magnitude from 0.7 up to 70. Significant effects of$Pr$on the heat and momentum fluxes across the channel gap and on the mean temperature and velocity profile are observed. These effects can be described through a mixing length model coupling Monin–Obukhov (M–O) similarity theory and van Driest damping functions. We then employ M–O theory to formulate similarity scalings for various flow diagnostics for the stratified turbulence in the gap interior. The midchannel gap gradient Richardson number$Ri_{g}$is determined by the length scale ratio$h/L$, where$h$is the half-channel gap depth and$L$is the Obukhov length scale. As$h/L$approaches very large values,$Ri_{g}$asymptotes to a maximum characteristic value of approximately 0.2. The buoyancy Reynolds number$Re_{b}\equiv \unicode[STIX]{x1D700}/(\unicode[STIX]{x1D708}N^{2})$, where$\unicode[STIX]{x1D700}$is the dissipation,$\unicode[STIX]{x1D708}$is the kinematic viscosity and$N$is the buoyancy frequency defined in terms of the local mean density gradient, scales linearly with the length scale ratio$L^{+}\equiv L/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$, where$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$is the near-wall viscous scale. The flux Richardson number$Ri_{f}\equiv -B/P$, where$B$is the buoyancy flux and$P$is the shear production, is found to be proportional to$Ri_{g}$. This then leads to a turbulent Prandtl number$Pr_{t}\equiv \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D705}_{t}$of order unity, where$\unicode[STIX]{x1D708}_{t}$and$\unicode[STIX]{x1D705}_{t}$are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number$Fr_{h}\equiv \unicode[STIX]{x1D700}/(NU^{\prime 2})$, where$U^{\prime }$is a turbulent horizontal velocity scale, is found to vary like$Ri_{g}^{-1/2}$. All these scalings are consistent with our numerical data and appear to be independent of$Pr$. The classical Osborn model based on turbulent kinetic energy balance in statistically stationary stratified sheared turbulence (Osborn,J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89), together with M–O scalings, results in a parameterization of$\unicode[STIX]{x1D705}_{t}/\unicode[STIX]{x1D708}\sim \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D708}\sim Re_{b}Ri_{g}/(1-Ri_{g})$. With this parameterization validated through direct numerical simulation data, we provide physical interpretations of these results in the context of M–O similarity theory. These results are also discussed and rationalized with respect to other parameterizations in the literature. This paper demonstrates the role of M–O similarity in setting the mixing efficiency of equilibrated constant-flux layers, and the effects of Prandtl number on mixing in wall-bounded stratified turbulent flows.

Analysis of mixed convection in the Czochralski model in a wide range of Prandtl numbers

The results of the calculations and analysis of the effect of separate and joint rotation of the crystal and the crucible on the flow stability are presented for a wide range of Prandtl numbers (from 0.01 to 10). The regimes with a high stability threshold are determined for different combinations of the rotation velocities. It is shown that for high Prandtl numbers, simultaneous rotation of the crystal and the crucible makes it possible to increase the critical Grashof number in 9–12 times. A resultant diagram (map) of the limiting regimes of natural and mixed convection is constructed. Themethodology of control and analysis of 2D and 3D instability modes is discussed.

Excitation of Surface Waves Due to Thermocapillary Effects on a Stably Stratified Fluid Layer

In chemical engineering applications, the operation of condensers and evaporators can be made more efficient by exploiting the transport properties of interfacial waves excited on the interface between a hot vapor overlying a colder liquid. Linear theory for the onset of instabilities due to heating a thin layer from above is computed for the Marangoni–Bénard problem. Symbolic computation in the long wave asymptotic limit shows three stationary, non-growing modes. Intersection of two decaying branches occurs at a crossover long wavelength; two other modes co-exist at the crossover point—propagating modes on nascent, shorter wavelength branches. The dispersion relation is then mapped numerically by Newton continuation methods. A neutral stability method is used to map the space of critical stability for a physically meaningful range of capillary, Prandtl, and Galileo numbers. The existence of a cut-off wavenumber for the long wave instability was verified. It was found that the effect of applying a no-slip lower boundary condition was to render all long waves stationary. This has the implication that any propagating modes, if they exist, must occur at finite wavelengths. The computation of 8000 different parameter sets shows that the group velocity always lies within 1 2 to 2 3 of the longwave phase velocity.

Thermal transport due to liquid jet impingement on superhydrophobic surfaces with isotropic slip

This paper presents an analytical investigation of thermal transport due to a steady, laminar, axisymmetric liquid jet impinging normally on a superhydrophobic (SHPo) surface maintained at constant surface temperature. At the liquid-surface boundary of the spreading thin film, an isotropic hydrodynamic slip and temperature jump are imposed to approximate the SHPo surface boundary condition. Applying an integral analysis within the thin film results in a system of differential equations which are solved numerically to obtain local hydrodynamic and thermal boundary layer thicknesses, thin film height, and local and radially averaged heat flux. The classical smooth hydrophobic scenario with no-slip and no-temperature jump showed excellent agreement with previous differential analysis of the same problem. The influence of varying temperature jump length on the local Nusselt number was obtained over a range of Reynolds and Prandtl numbers. Increasing temperature jump length results in a dramatic decrease in the local thermal transport near the impingement point. The greatest decrease occurs at small temperature jump lengths. Further, local and average Nusselt numbers are less influenced by the Reynolds and Prandtl numbers as temperature jump length increases. Overall, variations in the temperature jump length exert much more influence than variations in the hydrodynamic slip length.

Stability of advective flow in an inclined plane fluid layer bounded by solid planes with a longitudinal temperature gradient. 1. Unstable stratification

The stability of steady convective flow in an inclined plane fluid layer bounded by ideally heat conducting solid planes is studied in the presence of a homogeneous longitudinal temperature gradient under unstable stratification conditions where the layer is inclined so that the temperature is higher in the lower part than in the upper part. It is shown that the inclination leads to the transition from critical perturbations to long-wavelength helical perturbations. Flow stability maps are given for the entire range of Prandtl numbers and inclination angles corresponding to unstable stratification.

Effects of viscosity and conductivity stratification on the linear stability and transient growth within compressible Couette flow

Accurate prediction of laminar to turbulent transition in compressible flows is a challenging task, as it can be affected by a combination of factors. Compressibility causes large variations in thermodynamic as well as transport properties of a gas, which in turn are known to affect flow stability. We study the stratification of individual transport properties and their combined behavior. We also examine the effect of a change in the magnitude of viscosity and conductivity on flow stability. The Couette flow of a perfect gas is our model problem and both modal and non-modal analyses are carried out. We notice a large destabilizing role of the increase in the conductivity value and a dramatic stabilizing effect of mean viscosity stratification, over a range of free-stream Mach number, Reynolds number, Prandtl number, and disturbance wavenumber. In the combined case, viscosity stratification plays a dominant role. We find this to be the case for finite-time transient growth in the parameter regime below linear instability as well as asymptotically at large time. A budget of the transient growth energy amplification is also shown to identify the effects of transport properties on the constituents of perturbation energy. The extensive results presented in this paper, we believe should motivate those studying more realistic flows to examine how these contrasting effects of stratification come together.

Asymptotic theory for torsional convection in rotating fluid spheres

This paper is concerned with the classical, well-studied problem of convective instabilities in rapidly rotating, self-gravitating, internally heated Boussinesq fluid spheres. Sanchez et al. (J. Fluid Mech., vol. 791, 2016, R1) recently found, unexpectedly via careful numerical simulation, that non-magnetic convection in the form of axially symmetric, equatorially antisymmetric torsional oscillation is physically preferred in a special range of small Prandtl number for rapidly rotating fluid spheres with the stress-free boundary condition. We derive an asymptotic solution describing convection-driven torsional oscillation – whose flow velocity and pressure are fully analytical and in closed form – that confirms the result of the numerical analysis and is in quantitative agreement with the numerical solution. We also demonstrate, through the derivation of a different asymptotic solution, that convection-driven torsional oscillation cannot occur in rapidly rotating fluid spheres with the no-slip boundary condition.

Modeling the effect of the inclination angle on natural convection from a flat plate: The case of a photovoltaic module

The main purpose of this paper is to show how the inclination angle affects natural convection from a flat-plate photovoltaic module which is mounted on the ground surface. In order to model this effect, novel correlations for natural convection from isothermal flat plates are developed by using the fundamental dimensionless number. On the basis of the available experimental and numerical results, it is shown that the natural convection correlations correspond well with the existing empirical correlations for vertical, inclined, and horizontal plates. Five additional correlations for the critical Grashof number are derived from the available data, three indicating the onset of transitional flow regime and two indicating the onset of flow separation. The proposed correlations cover the entire range of inclination angles and the entire range of Prandtl numbers. This paper also provides two worked examples, one for natural convection combined with radiation and one for natural convection combined with forced convection and radiation.

Experimental investigation on the convective heat transfer enhancement in tubes with cross-helix profile wall corrugation

Wall corrugation is a popular heat transfer enhancement technique since it acts as a disturbance source in the flow that significantly enhances the thermal performance of the tube section with a limited pressure drop augmentation if compared to other passive techniques, such as the ones based on insert devices. In the hereby presented study nine pipes characterised by a cross-helix type corrugation were tested: this kind of corrugation, that was obtained by rolling twice the same tube with two helical corrugations evolving along opposite directions, showed a performance that exceeded the single helix-type corrugation behaviour and moreover presented an earlier transition to unstable regime. In the analysis the effect of the cross-helix type corrugation profile on the forced convection heat transfer mechanism was experimentally investigated in the Reynolds and Prandtl number range 25÷1000 and 115÷150 respectively. In particular, the effect of both the corrugation depth and the corrugation pitch were analysed. The results were compared with other types of wall corrugation and with the predictions for the smooth tube in order to point out the achieved heat transfer enhancement.

Study on CO2 – water printed circuit heat exchanger performance operating under various CO₂ phases for S-CO₂ power cycle application

The supercritical CO2 (S-CO2) power cycle is receiving worldwide attention as one of the promising advanced future electricity generation technologies. Since the S-CO2 power cycle can achieve high efficiency with simple system configuration, the role of compact heat exchanger becomes more important to achieve smaller system footprint. As an example of successful compact heat exchanger, the printed circuit heat exchanger (PCHE) was recently suggested for the S-CO2 power cycle application due to the capability of enduring high pressure difference while providing large heat transfer area within a small volume. However, the S-CO2 precooler operates very close to the critical point of CO2 where the conventional design methodology may not be suitable due to substantial variation of the thermo-physical properties near the critical point. Thus, in this paper the design and operation issues of PCHE as the precooler are addressed. In this study, the verification of developed PCHE core design code and experimental test results are presented. The test conditions were 26–43°C and 7.3–8.6MPa in temperature and pressure, respectively. In terms of non-dimensional numbers, the Reynolds number range is 15,000–100,000 and the Prandtl number range is 2–33. Friction factor and heat transfer correlations were developed with the experimental data and computational analysis for the future PCHE design as a precooler in the S-CO2 power cycle.

Criteria for Predicting Transitions in Free Convection Heat Transfer From Isothermal Convex Bodies in Fluids With Any Prandtl Number: A New Analytical Model

ABSTRACTThis article presents criteria for predicting transitions in free convection from isothermal convex bodies in fluids with any Prandtl number (conduction–laminar and laminar–turbulent transitions). Laminar–turbulent transition results for vertical plates happening in the vicinity of Grashof number 109 in fluids with any Prandtl number and other geometries show that this criterion based on Grashof number is a better choice compared to that of the Rayleigh number, as the Prandtl number yields a large change in the transition Rayleigh number. Besides, for convex geometries other than vertical plates, the laminar–turbulent transition occurs at values slightly less than Grashof number 109 owing to longitudinal vortices bringing about separation over their surface and helping laminar–turbulent transition to arise sooner. On the other side, the results of the conduction–laminar transition criterion for different convex bodies disclose that the Rayleigh number is superior to the Grashof number. Likewise, the present results for five different geometries over a wide range of Prandtl numbers are compared with the available experimental data, and excellent agreement is found. This reveals that the proposed criteria is powerful enough to predict the two transition phenomena in free convection heat transfer from isothermal convex bodies for a wide range of convex geometries in fluids with any Prandtl number.

On the role of discrete mass conservation for non-Boussinesq flow simulations in enclosures

We discuss the role of discrete mass conservation on the simulations of thermobuoyant flows in enclosures through a detailed numerical investigation in the Boussinesq and non-Boussinesq regimes. Employing a low Mach number formulation in conjunction with a pressure-based approach, we devise two unified algorithms that are capable of handling a wide range of temperature differences. These algorithms can be broadly categorized as discretely conservative and discretely consistent algorithms based on their implementation. While the discretely conservative algorithm computes the density by solving the mass conservation equation in each control volume (along with other conservation laws), the discretely consistent algorithm employs the algebraic equation of state to evaluate the local density. We show that the discretely consistent algorithm is prone to a discrete mass conservation error which can be detrimental to numerical stability in steady non-Boussinesq flows and can significantly influence the computed flowfield in case of unsteady non-Boussinesq convection. We also show that although the discretely conservative algorithm does not satisfy the discrete equation of state it introduces no significant errors in thermodynamic pressure for closed systems. Several numerical experiments over a range of temperature differences, Rayleigh and Prandtl numbers are carried out and the results whilst supporting the theoretical arguments also indicate that the discrete mass conservation errors are strongly linked respectively to numerical stability and solution accuracy in steady and unsteady non-Boussinesq flows.

Buoyancy-affected backward-facing step flow with heat transfer at low Prandtl number

Motivated by applications in concentrating solar power plants, the paper presents direct numerical simulations of the turbulent upward-directed flow of a liquid metal in a vertical channel. One of the walls features a backward-facing step with the vertical wall behind the step being heated. The paper addresses the impact of buoyancy effects by comparing two simulations, one with the buoyancy term removed from the equations, the other one with a Richardson number of 0.338. After a thorough validation detailed statistical data for both cases are provided. Due to the buoyancy forces acting on the fluid at the heated wall, the flow field is substantially altered and recirculation is considerably diminished. The altered flow field and the increased advection of temperature increase heat transfer at the heated wall. The differences in the turbulent heat fluxes are thoroughly discussed and related to the altered flow conditions. The results contribute to the physical understanding of buoyancy effects on turbulent heat transfer in the considered regime. Furthermore, the generated data allows the validation and improvement of turbulence models for turbulent heat transfer in the range of low Prandtl numbers.

Towards numerical computation of double-diffusive natural convection within an eccentric horizontal cylindrical annulus

Purpose– The purpose of this paper is to numerically study the double-diffusive natural convection within an eccentric horizontal cylindrical annulus filled with a Newtonian fluid. The annulus walls are maintained at uniform temperatures and concentrations so as to induce aiding thermal and mass buoyancy forces within the fluid. For that, this simulation span a moderate range of thermal Rayleigh number (100RaT100,000), Lewis (0.1Le10), buoyancy ratio (0N5) and Prandtl number (Pr=0.71) to examine their effects on flow motion and heat and mass transfers.Design/methodology/approach– A finite volume method in conjunction with the successive under-relaxation algorithm has been developed to solve the bipolar equations. These are written in dimensionless form in terms of vorticity, stream function, temperature and concentration. Beforehand, the implemented computer code has been validated through already published findings in the literature. The isotherms, streamlines and iso-concentrations are exhibited for various values of Rayleigh and Lewis numbers, and buoyancy ratio. In addition, heat and mass transfer rates in the annulus are translated in terms of Nusslet and Sherwood numbers along the enclosure’s sides.Findings– It is observed that, for the range of parameters considered here, the results show that the average Sherwood number increases with, while the average Nusselt number slightly dips as the Lewis number increases. It is also found that, under the convective mode, the local Nusselt number (or Sherwood) increases with the buoyancy ratio. Likewise, according to Lewis number’s value, the flow pattern is either symmetric and stable or asymmetric and random. Besides that, the heat transfer is transiting from a conductive mode to a convective mode with increasing the thermal Rayleigh number, and the flow structure and the rates of heat and mass transfer are significantly influenced by this parameter.Research limitations/implications– The range of the Rayleigh number considered here covers only the laminar case, with some constant parameters, namely the Prandtl number (Pr = 0.71), and the tilt angle (α=90°). The analysis here is only valid for steady, two-dimensional, laminar and aiding flow within an eccentric horizontal cylindrical annulus. This motivates further investigations involving other relevant parameters as N (opposite flows), Ra, Pr, Le, the eccentricity, the tilt angle, etc.Practical implications– An original framework for handling the double-diffusive natural convection within annuli is available, based on the bipolar equations. In addition, the achievement of this work could help researchers design thermal systems supported by annulus passages. Applications of the results can be of value in various arrangements such as storage of liquefied gases, electronic cable cooling systems, nuclear reactors, underground disposal of nuclear wastes, manifolds of solar energy collectors, etc.Originality/value– Given the geometry concerned, the bipolar coordinates have been used to set the inner and outer walls boundary conditions properly without interpolation. In addition, since studies on double-diffusive natural convection in annuli are lacking, the obtained results may be of interest to handle other configurations (e.g., elliptical-shaped speakers) with other boundary conditions.

A thermal non-equilibrium perspective on instability mechanism of non-isothermal Poiseuille flow in a vertical porous-medium channel

We examine the influence of local thermal non-equilibrium state (LTNE) on the instability mechanism of non-isothermal Poiseuille flow in a vertical channel filled with porous medium. The temperature field of the solid porous matrix and the saturating fluid are governed by separate, but coupled, energy equations supplemented with three dimensionless LTNE parameters: inter-phase heat transfer coefficient (H), porosity scaled conductivity ratio (γ) and diffusivity ratio (Γ). The linear stability results show that the critical Rayleigh number (Rac) increases (decreases) with H (γ), which indicates that the effect of H (γ) is to stabilize (destabilize) the flow. The stabilizing impact of H for fluid with low Prandtl number (Pr) becomes high when kinetic energy due to non-isothermal effect is lost to the basic flow. The energy loss to the basic flow increases on increasing (decreasing) H (γ). Depending on the magnitude of all the studied parameters, three types of instabilities, namely: shear, mixed and buoyant have been observed. For relatively low permeable medium buoyant instability becomes the most dominant instability in the range of Prandtl number considered in this study.

Thermo-dynamic irreversibility induced by natural convection in square enclosure with inner cylinder. Part-II: Effect of vertical position of inner cylinder

Two-dimensional numerical simulations were conducted to investigate the natural convection heat transfer induced by the temperature difference between cold walls of the square enclosure and a hot inner circular cylinder in the Prandtl number range of 10−2⩽Pr⩽102 and Rayleigh number range of 103⩽Ra⩽106. In this paper as a sequent research of Mun et al. (2015), the additional geometrical configuration of the system, which is the variation in the vertical position of the inner cylinder, was considered. The change in the structure of convection cells and corresponding heat transfer characteristics were analyzed, which are induced by the variation in the vertical position of the inner cylinder located in the enclosure. And the characteristics of the entropy generation due to heat transfer and fluid friction associated with the flow and thermal structures were addressed in this study.

Thermal convection in inclined cylindrical containers

By means of direct numerical simulations (DNS) we investigate the effect of a tilt angle ${\it\beta}$, $0\leqslant {\it\beta}\leqslant {\rm\pi}/2$, of a Rayleigh–Bénard convection (RBC) cell of aspect ratio 1, on the Nusselt number $\mathit{Nu}$ and Reynolds number $\mathit{Re}$. The considered Rayleigh numbers $\mathit{Ra}$ range from $10^{6}$ to $10^{8}$, the Prandtl numbers range from 0.1 to 100 and the total number of the studied cases is 108. We show that the $\mathit{Nu}\,({\it\beta})/\mathit{Nu}(0)$ dependence is not universal and is strongly influenced by a combination of $\mathit{Ra}$ and $\mathit{Pr}$. Thus, with a small inclination ${\it\beta}$ of the RBC cell, the Nusselt number can decrease or increase, compared to that in the RBC case, for large and small $\mathit{Pr}$, respectively. A slight cell tilt may not only stabilize the plane of the large-scale circulation (LSC) but can also enforce an LSC for cases when the preferred state in the perfect RBC case is not an LSC but a more complicated multiple-roll state. Close to ${\it\beta}={\rm\pi}/2$, $\mathit{Nu}$ and $\mathit{Re}$ decrease with increasing ${\it\beta}$ in all considered cases. Generally, the $\mathit{Nu}({\it\beta})/\mathit{Nu}(0)$ dependence is a complicated, non-monotonic function of ${\it\beta}$.

3D natural convection in a cubical cavity with a thermally active heater under the presence of an external magnetic field

A numerical study of three dimensional natural convection heat transfer and fluid flow in a cubical cavity induced by a thermally active plate is described under the presence of external magnetic field. The active plate is attached to the center of one vertical wall of the cavity filled with different fluids as mercury, air and dielectric liquid FC-77. It is located at two different positions as vertical or horizontal. The governing equations are solved by using finite volume method. Computations are performed for a wide range of Hartmann number (0 ≤ Ha ≤ 300), Prandtl number (0.025 ≤ Pr ≤ 25) and different aspect ratio of the plate (Ac = 0.5 & 1.0) when the Rayleigh number Ra is fixed to be 107. It is found that the heat transfer rate is strongly suppressed by the X and Y directional magnetic fields and it is weakly suppressed by the Z directional magnetic field.