Moderate Prandtl Number Research Articles

Averaging theory for heat transfer in circular hydraulic jumps with a separation bubble

Analytical investigations of heat transfer during the vertical impingement of an unsubmerged axisymmetric liquid jet on a horizontal plate have been limited to the regions ahead of the jump. This limitation is due to the complex flow physics in the jump region arising from sudden changes in the flow field. This is addressed in here by extending the averaging theory (AT) introduced by Bohr et al. (Phys. Rev. Lett., vol. 79, issue 6, 1997, pp. 1038–1041) which was further developed by Watanabe et al. (J. Fluid Mech., vol. 480, 2003, pp. 233–265), to describe the heat transfer problem in circular hydraulic jumps including separation. The applicability of the resulting theory to determine the temperature field in the jump region is evaluated using the data available in the literature and also by means of fully resolved numerical solutions. Good agreement is observed for moderate Prandtl numbers. However, for sufficiently high Prandtl numbers, deviations become notable. The reasons for the deviations according to their relevance are (i) monotonically decreasing temperature profile inherent to the AT, whereas the fully resolved numerical solutions exhibit a local maximum in the temperature profile away from the plate; and (ii) inapplicability of the concept of dividing the flow field into a region affected and a region unaffected by heat transfer according to the thermal boundary layer thickness. This concept leads to the overestimation of the temperature close to the wall and to the existence of a threshold Prandtl number, for which the thermal boundary layer thickness does not meet the free surface anymore. Around this threshold Prandtl number, the temperature field shows a discontinuous behaviour.

Thermocapillary convection in shallow liquid layers for moderate Prandtl number liquids using different cylinder configurations

In this paper, the influence of thermocapillary convection in a shallow annular pool using different cylinder configurations is numerically investigated. An axisymmetric, two-dimensional, cylindrical geometry with two concentric cylinders kept at two different temperatures constitutes the model. The main cylinder has a radius of 40 mm and the secondary cylinder has a radius of 20 mm. The effects of Prandtl numbers (Pr = 6.7, 16.1, and 25.2) and layer depths (d = 3.0 mm and 11.0 mm) on the streamlines, isotherms, velocity, and temperature profiles at various locations of the flow are investigated. Depending on the position of the secondary cylinder, three different configurations: 1) an internal cylinder 2) an external cylinder, and a 3) central cylinder is analysed. The flow is characterized by a rapid inward flow along the hot outer wall towards the colder inner wall. In deeper pool at lower Prandtl numbers, flow patterns are characterized by a single rolling cell structure. Conversely, secondary cell formations appear in shallow pool regardless of the Prandtl number. Based on the comparative analysis of various cylinder configurations, it is clear that the external cylinder generates greater velocities and temperatures along its surface. The maximum disparity in surface velocity between an external cylinder and other configurations is 7%, 12%, and 13% for Prandtl numbers of 6.7, 16.1, and 25.2 respectively. Therefore, positioning the cylinder externally promotes swifter convection and more effective mixing of the fluid.

CONTROLLING CHAOS IN 3D POROUS MEDIA IN THE PRESENCE OF ORIENTATION EFFECT

The influence of orientation on the control of chaotic behaviors of the system in a porous medium is studied in this article based on the theory of dynamical systems. A three-dimensional system model is obtained using the spectral method. The results showed that chaos can be suppressed in the case of a low or moderate Prandtl number by choosing appropriate tilt angles.

Investigating to chemically reactive and radiative Darcy/non-Darcy stagnation point flow of ternary composite nanofluids with moderate Prandtl numbers

ABSTRACT In this study, the Darcy or non-Darcy stagnation point flow and thermal analysis of ternary composite nanofluids with modest Prandtl numbers across stretched surfaces that are susceptible to slip or no-slip environments are examined. The concept of non-linear thermal radiation is brought into play. It has been decided to apply non-linear slip. Homogeneous and heterogeneous chemical reactions, or HHCRs, make the flow problem harder to solve. Along with the shooting method, the Runge–Kutta method of the fourth order is an important part of the numerical solution needed for the non-dimensional and non-linear system of partial differential equations. In slip and no slide, both the amplified porosity parameter and the Forchheimer number work to slow down the speed of the fluid. In a medium that may be Darcy or non-Darcy, an increase in the slip parameter causes the fluid velocity and temperature to sputter and stop. When Darcy and non-Darcy media and slip and no-slip conditions are taken into account, surface viscous drag goes down as the strength of the porous matrix goes up, heat transfer rate goes down as the Brinkman number goes up, and mass transfer rate goes up as the homogeneous reaction rate parameter goes up.

Heat-transfer scaling at moderate Prandtl numbers in the fully rough regime

In the fully rough regime, proposed models predict a scaling for a roughness heat-transfer coefficient, e.g. the roughness Stanton number ${St}_k \sim (k^+)^{-p} {Pr}^{-m}$ where the exponent values $p$ and $m$ are model dependent, giving diverse predictions. Here, $k^+$ is the roughness Reynolds number and ${Pr}$ is the Prandtl number. To clarify this ambiguity, we conduct direct numerical simulations of forced convection over a three-dimensional sinusoidal surface spanning $k^+ = 5.5$ – $111$ for Prandtl numbers ${Pr} = 0.5$ , 1.0 and 2.0. These unprecedented parameter ranges are reached by employing minimal channels, which resolve the roughness sublayer at an affordable cost. We focus on the fully rough phenomenologies, which fall into two groups: $p=1/2$ (Owen & Thomson, J. Fluid Mech., vol. 15, issue 3, 1963, pp. 321–334; Yaglom & Kader, J. Fluid Mech., vol. 62, issue 3, 1974, pp. 601–623) and $p=1/4$ (Brutsaert, Water Resour. Res., vol. 11, issue 4, 1975b, pp. 543–550). Although we find the mean heat transfer favours the $p=1/4$ scaling, the Prandtl–Blasius boundary-layer ideas associated with the Reynolds–Chilton–Colburn analogy that underpin the $p=1/2$ can remain an apt description of the flow locally in regions exposed to high shear. Sheltered regions, meanwhile, violate this behaviour and are instead dominated by reversed flow, where no clear correlation between heat and momentum transfer is evident. The overall picture of fully rough heat transfer is then not encapsulated by one singular mechanism or phenomenology, but rather an ensemble of different behaviours locally. The implications of the approach to a Reynolds-analogy-like behaviour locally on bulk measures of the Nusselt and Stanton numbers are also examined, with evidence pointing to the onset of a regime transition at even-higher Reynolds numbers.

Hydro-thermo-fluidic aspects of Oldroyd B fluid with hybrid nanostructure subject to low and moderate Prandtl numbers

PurposeThe major goal of this article is to investigate flow and thermal aspects of Oldroyd B with hybrid nanostructure subject to a radially stretched surface under the influence of low and moderate Prandtl numbers.Design/methodology/approachThe non-dimensional governing equations are solved considering BVP4C in MATLAB as instrumental.FindingsEntropy generation effect is analyzed. Radial velocity and entropy generation exhibit opposite effect in response to amplified relaxation and retardation time parameters in case of both low and moderate Prandtl numbers. Augmented relaxation and retardation time parameters controls heat transfer rate.The results show that increasing the aspect ratio increases both the average Nusselt and entropy generation numbers for each value of the Prandtl number, while increasing the prandtl number decreases both. There is also a minimum value for the entropy generation number at a given relaxation and retardation parameter.Research limitations/implicationsAssume that the Oldroyd B fluid is dispersed with hybrid nanostructure in order to ameliorate thermal conductivity of Oldroyd B fluid so as to make it as best coolant.Practical implicationsThe low range of Prandtl number comprising particles of air, gas, etc. and moderate range of Prandtl number comprising particles of honey, thin motor oil, or any non-Newtonian liquid. The hybrid nanofluid is radiative in nature. Also, the effects of significant physical parameters on entropy generation are highlighted. The entropy generation number intensifies due to the rise in temperature difference parameter at low/moderate Prandtl number effectively. Entropy minimization can lead to the amelioration of available energy thereby enhances the efficiency of several thermal systems.Originality/valueThis article's primary goal is to investigate the flow and thermal aspects of Oldroyd B with a hybrid nanostructure subject to a radially stretched surface under the influence of low and moderate Prandtl numbers.

Thermal heat storage and convective melting dynamics of phase change materials in inclined square enclosures: Effect of the Prandtl number

We study numerically the effect of the Prandtl number on the thermal heat storage, heat flux, and melting dynamics in phase change materials enclosed in square geometries inclined from 0° (bottom horizontal heating) to 180° (top horizontal heating) at increments of at least 5°. Chosen phase change materials are representative of very low, moderate, and high Prandtl numbers: gallium ( P r = 0 . 021 ), water ( P r = 13 . 5 ), and erythritol ( P r = 201 ). The Stefan number is fixed to 0.5, and the Rayleigh number to R a ∼ 1 0 8 . Water and erythritol exhibit a global bifurcation angle at θ c = 3 0 ∘ , separating the liquid phase dynamics from turbulent to laminar. Gallium exhibits stages of turbulent behavior up to 165°. The optimum tilt for harvested thermal energy corresponds to 95° for water and erythritol, and 100° for gallium. The maximum energy efficiency in harvesting thermal energy corresponds to small inclinations within the turbulent region, and 50° for water and erythritol, and the region about 45° for gallium. The maxima at 50° and 95° are very close to those values of the most frequent inclinations 45° and 90° investigated in the experimental literature. There is a power-law dependence between the Nusselt and Rayleigh number N u ∼ R a α for all the Prandtl numbers. Power-law exponents for water and erythritol exhibit a global upward trend with the inclination and undefined for gallium. • Simulations on erythritol (Pr=201), water (Pr=13.5), gallium (Pr=0.021) for Rã10̂8. • Power-law with inclination between Nusselt and Rayleigh numbers for all the Prandtl. • Similar melting time, energy storage and efficiency curves for all Prandtl. • Exists a critical inclination splitting laminar from turbulent flow for all Prandtl. • Optimal inclination for energy storage: 95° water and erythritol, 100° for gallium.

Gravity-driven film flow down a uniformly heated smoothly corrugated rigid substrate

Gravity induced film flow over a rigid smoothly corrugated substrate heated uniformly from below, is explored. This is achieved by reducing the governing equations of motion and energy conservation to a manageable form within the mathematical framework of the well-known long-wave approximation; leading to an asymptotic model of reduced dimensionality. A key feature of the approach and to solving the problem of interest, is proof that the leading approximation of the temperature field inside the film must be nonlinear to accurately resolve the thermodynamics beyond the trivial case of ‘a flat film flowing down a planar uniformly heated incline.’ Superior predictions are obtained compared with earlier work and reinforced via a series of corresponding solutions to the full governing equations using a purpose written finite element analogue, enabling comparisons to be made between free-surface disturbance and temperature predictions, as well as the streamline pattern and temperature contours inside the film. In particular, the free-surface temperature is captured extremely well at moderate Prandtl numbers. The stability characteristics of the problem are examined using Floquet theory, with the interaction between the substrate topography and thermo-capillary instability modes investigated as a set of neutral stability curves. Although there are no relevant experimental data currently available for the heated film problem, recent existing predictions and experimental data concerning the behaviour of corresponding isothermal flow cases are taken as a reference point from which to explore the effect of both heating and cooling.

Zonal flow reversals in two-dimensional Rayleigh-Bénard convection

We analyse the nonlinear dynamics of the large scale flow in Rayleigh-B\'enard convection in a two-dimensional, rectangular geometry of aspect ratio $\Gamma$. We impose periodic and free-slip boundary conditions in the streamwise and spanwise directions, respectively. As Rayleigh number Ra increases, a large scale zonal flow dominates the dynamics of a moderate Prandtl number fluid. At high Ra, in the turbulent regime, transitions are seen in the probability density function (PDF) of the largest scale mode. For $\Gamma = 2$, the PDF first transitions from a Gaussian to a trimodal behaviour, signifying the emergence of reversals of the zonal flow where the flow fluctuates between three distinct turbulent states: two states in which the zonal flow travels in opposite directions and one state with no zonal mean flow. Further increase in Ra leads to a transition from a trimodal to a unimodal PDF which demonstrates the disappearance of the zonal flow reversals. On the other hand, for $\Gamma = 1$ the zonal flow reversals are characterised by a bimodal PDF of the largest scale mode, where the flow fluctuates only between two distinct turbulent states with zonal flow travelling in opposite directions.

Structure and Dynamics of Large-Scale Circulation in Turbulent Convection at High Prandtl Numbers

The structure and dynamics of the large-scale circulation in turbulent Rayleigh—Benard convection in a cubic cell are experimentally investigated for fluids with moderate and high Prandtl numbers (from 3.5 to 64). The decomposition of two-dimensional velocity fields into Fourier modes shows that with variation of the Prandtl number from 3.5 to 24 one large-scale mode predominates in the flow. With further increase in the Prandtl number the flow structure considerably changes and the energy is redistributed between several lower Fourier modes. The temporal dynamics of the predominant mode is also considerably changed.

Analytical solution for heat transfer problem in a cross-flow plate heat exchanger

Cross-flow plate heat exchangers are used for plenty of applications in industrial and domestic sectors, and the analysis of heat transfer is a key for the evaluation of their performance. There are challenges for an analytical study because heat transfer in each channel is governed by a partial differential equation coupled with temperature fields in the adjacent channels representing a three-dimensional problem. The problem is even more complicated in the turbulent regime as the effective thermal diffusivity varies within the channel cross-section. In the present study, a separate set of governing equations and boundary conditions are considered for each part of the heat exchanger. Appropriate profiles for the flow velocity and thermal diffusivity are substituted into the governing equations of the channels. The resulting partial differential equations in the channels are solved using the separation of variables method. There remains an unknown boundary condition linked to the temperature field on the plate surface which is considered to be in the form of a two-variable series function whose coefficients are calculated by applying energy balance between the two sides of the plate. The obtained solution provides explicit expressions for the temperature fields in the plate and channels. A scaling analysis is conducted showing that the model is valid when Peclet numbers in both hot and cold channels are not small. The results are compared with empirical data and a numerical model, and the accuracy of the derived heat transfer coefficients is investigated. In harmony with the scaling analysis, a close agreement is observed in both laminar and turbulent flows for moderate and high Prandtl numbers and when Peclet number is greater than 1000 in the case of turbulent flow.

Scaling and spatial intermittency of thermal dissipation in turbulent convection

We derive scaling relations for the thermal dissipation rate in the bulk and in the boundary layers for moderate and large Prandtl number (Pr) convection. Using direct numerical simulations of Rayleigh-Bénard convection, we show that the thermal dissipation in the bulk is suppressed compared to passive scalar dissipation. The suppression is stronger for large Pr. We further show that the dissipation in the boundary layers dominates that in the bulk for both moderate and large Pr. The probability distribution functions of thermal dissipation rate, both in the bulk and in the boundary layers, are stretched exponential, similar to passive scalar dissipation.

Thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool heated from inner cylinder with surface heat dissipation

This paper presents a series of three-dimensional numerical simulations on thermocapillary convection of moderate Prandtl number (Pr = 6.7) fluid in a shallow annular pool heated from the inner cylinder with surface heat dissipation. The aspect ratio and radius ratio of the annular pool are respectively fixed at ε = 0.05 and η = 0.5. Biot number varies from 0 to 40 and Marangoni number varies from 0.1 × 105 to 6.0 × 105. Results show that there is a large temperature drop near the heated cylinder and the flow becomes more intense than those when the outer cylinder is heated at adiabatic free surface. With the increase of Marangoni number, the flow will transit from the axisymmetric steady flow to the three-dimensional oscillatory flow. The critical Marangoni number decreases first, and then increases with the increase of Biot number. After the flow destabilization, thermocapillary convection with adiabatic free surface exhibits the multicellular wave pattern when the inner cylinder is heated, which is different from that when the outer cylinder is heated. The temperature fluctuation pattern depends on the coupling effect of thermocapillary convection and heat exchange on the free surface. The wave number increases first, and then decreases with the increase of Marangoni number at a fixed Biot number. Furthermore, as Marangoni number increases, the temperature fluctuation on the free surface will radially extend from the inner cylinder to the outer cylinder gradually, and the amplitude of temperature fluctuation increases obviously.

Three-Dimensional Numerical Simulation on Thermocapillary Convection of Moderate Prandtl Number Fluid in an Annular Shallow Pool with Surface Heat Dissipation

In order to understand the effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in an annular shallow pool, we performed a series of three-dimensional numerical simulations by using the finite volume method. An annular shallow pool is full of 0.65cSt silicone oil with Prandtl number of 6.7. The radius ratio and the aspect ratio of the pool are respectively fixed at 0.5 and 0.1. Surface heat dissipation Biot number is varied from 0 to 50. Results indicate that the critical Marangoni number of flow destabilization mainly depends on the coupling effect of the thermocapillary force and surface heat dissipation on the free surface, which decreases first, and then increases gradually. When Biot number is small, the steady axisymmetric flow after the flow destabilization will bifurcate to the hydrothermal wave. With the increase of Marangoni number, the flow pattern evolution at a small Biot number is similar to that of an adiabatic free surface, and the fundamental oscillatory frequency increases gradually. With the increase of Biot number, the radial roll cells near the outer cylindrical wall gradually extend to the inner wall, and eventually occupy the whole liquid pool. When Biot number is large, the flow pattern after the flow destabilization is a radial rolling cell pattern with alternating azimuthal direction. Then it gradually evolves into azimuthal waves, and the fundamental oscillatory frequency has a slight decrease.

Effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool

This paper presents a series of three-dimensional numerical simulations on the effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool. The annular pool with a fixed aspect ratio of 0.05 and radius ratio of 0.5 is filled with 0.65 cSt silicone oil. Its Prandtl number is 6.7. Biot number of surface heat dissipation ranges from 0 to 50. The results show that when Marangoni number is small, thermocapillary convection is the axisymmetric steady flow. When Marangoni number exceeds the critical value, the basic flow will destabilize and transit to the hydrothermal waves that propagate along the clockwise or anticlockwise direction. The critical Marangoni number of the flow destabilization increases with the increase of Biot number. Furthermore, when Biot number increases, the radial temperature gradient on the free surface decreases near the inner cylindrical wall, but increases near the outer cylindrical wall, which results in the thermocapillary convective cell moving gradually from the inner to the outer cylindrical walls. After the flow destabilizes, the temperature fluctuation on the free surface mainly appears near the outer cylinder at a large Biot number.

Surface Heat Dissipation Dependence of Thermocapillary Convection of Moderate Prandtl Number Fluid in an Annular Pool

In order to understand surface heat dissipation dependence of thermocapillary convection for moderate Prandtl number fluid in a deep annular pool, a series of three-dimensional numerical simulations have been carried out by using the finite volume method. The radius ratio and the aspect ratio of an annular pool are fixed at 0.5 and 1.0, respectively. The working fluid is 0.65cSt silicone oil with Prandtl number of 6.7. Surface heat dissipation Biot (Bi) number is varied from 0 to 50. Results indicate that with the increase of Biot number, the radial temperature gradient near the inner cylindrical wall decreases, and near the outer cylindrical wall it increases, so the flow is enhanced. When 0 < Bi < 10, with the increase of Marangoni number, the axisymmetric steady flow first transits to the standing wave, and then to the azimuthal waves. The standing wave should be attributed to Marangoni-Benard instability. However, the azimuthal waves should be corresponded to hydraulic instability, which is mainly driven by the azimuthal motion of temperature fluctuation from the sudden change of flow direction near the bottom and the inner cylindrical wall. When Bi ≥ 10, when the flow destabilizes, the axisymmetric steady flow transits directly to the azimuthal waves. With the increase of Biot number, the critical Marangoni number of the flow destabilization increases. Furthermore, the fundamental frequency and the wave number of three-dimensional oscillatory flow increase gradually with the increase of Biot number.

Modelling of a planar impinging jet at unity, moderate and low Prandtl number: Assessment of advanced RANS closures

The numerical simulation of turbulent heat transfer in complex industrial flows represents a major challenge for RANS models. The impinging jet configuration is a popular test case for turbulence models, due to both the challenging nature of the flow field for the RANS approach and its widespread use in industrial applications. As a consequence, a large number of studies in the literature is dedicated to the assessment of a broad variety of turbulence models in this configuration. In particular, a significant effort has been put into the assessment of different closures for the turbulent momentum flux. In contrast, relatively little attention has been paid to the modelling of the turbulent heat flux term, and the classical Reynolds analogy is almost universally employed for this purpose despite its well-known limitations. Nevertheless, recently there has been a growing interest towards the development and assessment of more advanced closures for the turbulent heat flux. In this context, in the present work six turbulence models relying on different closures for both the turbulent momentum and heat fluxes have been used to simulate a planar impinging jet at unity, moderate and low Prandtl number and thoroughly assessed against a recently published reference DNS database. It is shown that the use of an advanced differential closure for the Reynolds stresses can result in an improvement in the prediction of the flow field. This, in turn, directly results in a more accurate prediction of the mean temperature at unity Prandtl number when the Reynolds analogy is employed for the closure of the turbulent heat flux. On the other hand, the inadequacy of the latter approach appears evident at low Prandtl values, and the necessity to account at least for the variability of the turbulent Prandtl number is demonstrated. It is then inferred that the most promising approach for such complex cases is represented by the combined use of anisotropic models for both the turbulent flow and the thermal fields.

Influence of aspect ratio on the onset of thermocapillary flow instability in annular pool heated from inner wall

Thermocapillary flow instabilities in annular pool with different aspect ratios were investigated by the linear stability analysis based on the Legendre spectral element method. The energy analysis was applied to explore the underlying physical mechanism for the instability of thermocapillary flow. The results reveal two types of instability for different aspect ratios. The onset of instability is an oscillatory bifurcation in the cases of small aspect ratio whereas a stationary bifurcation in the cases of large aspect ratio. Moreover, the energy analysis indicates that the bifurcation of the flow is caused by the hydrodynamic (inertial) instability mechanism for low Prandtl number (Pr = 0.011), while by the coupled hydrodynamic effect and Marangoni effect for moderate Prandtl number (Pr = 1.4).

Instabilities and small-scale waves within the Stewartson layers of a thermally driven rotating annulus

We report on small-scale instabilities in a thermally driven rotating annulus filled with a liquid with moderate Prandtl number. The study is based on direct numerical simulations and an accompanying laboratory experiment. The computations are performed independently with two different flow solvers, that is, first, the non-oscillatory forward-in-time differencing flow solver EULAG and, second, a higher-order finite-difference compact scheme (HOC). Both branches consistently show the occurrence of small-scale patterns at both vertical sidewalls in the Stewartson layers of the annulus. Small-scale flow structures are known to exist at the inner sidewall. In contrast, short-period waves at the outer sidewall have not yet been reported. The physical mechanisms that possibly trigger these patterns are discussed. We also debate whether these small-scale structures are a gravity wave signal.

On the stability of thermocapillary convection of a Bingham fluid in an infinite liquid layer

The linear stability of thermocapillary liquid layer of a Bingham-plastic fluid is studied. Due to the yield stress of Bingham fluid, there is a plug region in the flow, which divides the yielded flow into two regions. When the flow is subjected to a small perturbation, the velocity perturbation below the upper surface of plug region is negligible, while the temperature perturbation can be found in all flow regions at moderate and small Prandtl numbers (Pr). The perturbation amplitude of the upper surface of plug region decreases rapidly with the increase of Pr. The preferred modes are the upstream oblique wave and the downstream streamwise wave at small and large Pr, respectively. The effects of the yield stress, gravity and the interfacial heat transfer on the flow stability are discussed. The perturbation amplitude only appears above the plug region, which differs from the cases in the plane Bingham–Poiseuille flow and the thermocapillary liquid layer of a Carreau fluid.