Fluid Pr Research Articles

Experimental investigation of the Reynolds analogy for high Prandtl number fluid with repeated rib roughness

Reynolds analogy describes the relationship between friction (f/2) and heat transfer (St) owing to their superficial resemblance. However, Reynolds analogy is limited to simple geometries with a Prandtl number (Pr) close to unity. To extend Reynolds analogy, we measured the St and f/2 simultaneously using a high Pr fluid in duct flow in vertical duct flow with repeated rib roughness. The experiments were performed at a fixed value of Pr, 2014, Reynolds number (Re) ranging from 9.1 × 103 to 9.6 × 104, and relative roughness (e/D) ranging from 0.01 to 0.04. In the high Pr fluid, the St was much lower than f/2, as most of heat exchange occurred in the conduction sublayer, resulting from a thin thermal boundary layer. On the rough surface, both f/2 and St were higher than on the smooth surface. However, the influence of Re and e/D was different. The differences stem from that the corresponding phenomenon to pressure drag in friction does not exist in heat transfer. The extended Reynolds analogy correlation was developed. In addition, the efficiency factor was defined as the ratio of the enhancement rates of St and f/2 on the rough surface to those on the smooth surface. The efficiency factor decreased with Re and e/D, while it increased with Pr.

Free convection in a differentially heated square cavity filled with low-Prandtl-number materials: Numerical studies using transition shear stress transport model

Computing the flow field under free convection in a cavity becomes particularly challenging for low-Prandtl-number (Pr) fluids typically encountered for liquid metals. The objective of the present study is to investigate the natural convection process in a differentially heated square cavity employing the transition shear stress transport (SST) model for the Prandtl number Pr∈[0.001,0.1] and the Rayleigh number Ra∈[104,1010]. The mean flow field is visualized through streamlines, isotherms, non-dimensional velocity, and temperature profiles, turbulence intensity, contours of intermittency (γ) times, the production of turbulent kinetic energy (Pκ), and distribution of skin friction coefficient and the Nusselt number (Nu). The transition SST model can capture the mean flow field and thermal transport over the entire parametric regime successfully. An average Nusselt number (Nu¯) on the hot wall is found to scale with a certain power (n) of the Boussinesq number (Bo), the product of Ra and Pr. The value of n is 0.18 for Ra up to 106 and 0.25 for higher Ra.

Towards the accurate prediction of axial flow and heat transfer in a tightly spaced bare rod bundle configuration

The understanding of flow behaviour and temperature distribution in rod bundles during operation is of great importance for the safety and economical design of existing nuclear reactors and as well as new nuclear technologies. In order to study the thermal-hydraulics phenomena in the fuel assembly geometries, researchers all over the world are increasingly using Computational Fluid Dynamics (CFD) as a reliable research tool. Nevertheless, the prediction capabilities of the most commonly used Reynolds Average Navier-Stokes (RANS) models must be assessed. In this regard, in the present research work, an extensive effort has been put forward and is described in three main steps. As a first step, a wide range of RANS and unsteady RANS computations have been performed to design a numerical experiment for a closely-spaced bare rod bundle in order to perform a direct numerical simulation (DNS), which will serve as a reference for the validation purpose. As a second step, the DNS of this bare rod bundle has been performed for three different Prandtl number (Pr) fluids, i.e. Pr = 2, 1 and 0.025. This DNS has been performed using a higher-order spectral element code and consists of 660 million grid points. Accordingly, an extensive database has been generated for validation purposes. Finally, this database is used to assess the prediction capabilities of different linear and non-linear RANS models. This article will provide a good overview of all three steps and will also highlight the main lessons learned from this research.

Magneto laminar mixed convection and entropy generation analyses of an impinging slot jet of Al2O3-water and Novec-649

In this study, the cooling capability and heat transfer characteristics in the mixed convective regime of a hot surface by a confined impinging slot jet in the presence of a uniform magnetic field have been investigated numerically. Besides, the second law of thermodynamics is applied to analyze the entropy generation inside the simulation medium. The simulations are for a two-dimensional slot jet that is directed vertically downward and impinges on a hot isothermal surface at the bottom. The set of non-dimensional governing equations is solved using ANSYS Fluent, which employs a finite volume approach in combination with a non-dimensionalization scheme. Flow patterns, isotherms, and entropy generation rates were calculated for various fluids (Air, Water, Novec-649, Al2O3-Water (3 %) nanofluid), that cover a wide range of Prandtl number values (Pr = 0.71–12.64), in the presence of a uniform magnetic field with Hartmann number values ranging from 0 (no magnetic field) to 100 (strong magnetic field). We prove that the Prandtl number plays a key role in determining heat transfer efficiency and entropy generation. Fluids with a high Pr number are more efficient for surface cooling, although they generate more entropy than low Pr fluids. Nanofluid with a high-volume fraction can be a better choice rather than air and water. On the other hand, we show that applying the magnetic field result in a very efficient method to reduce entropy generation although they degrade heat transfer efficiency. It is interesting that at a higher magnetic field, a high Prandtl number plays a better role. The percentage of reduction of Nuave by applying magnetic field is 28.5 % for air (Pr = 0.71), 45.2 % for water (Pr = 6.2), 43.4 % for Al2O3-water nanofluid (Pr = 4.91) and 48.3 % for Novec-649 (Pr = 12.64).We postulate that applying magnetic fields combined with high Pr number fluids, could provide an excellent balance between heat transfer efficiency and exergy destruction, thus providing efficient control options for thermal management and heat transfer.

Numerical Study of Low Pr Flow in a Bare 19-Rod Bundle Based on an Advanced Turbulent Heat Transfer Model

Compared to assuming a constant turbulent Prandtl number model, an advanced four-equation model has the potential to improve the numerical heat transfer calculation accuracy of low–Prandtl number (Pr) fluids. Generally, a four-equation model consists of a two-equation k−ε turbulence model and a two-equation kθ−εθ heat transfer model. It is essential to analyze the influence of dissimilar turbulence models on the overall calculation accuracy of the four-equation model. The present study aims to study the effect of using different turbulence models on the same kθ−εθ heat transfer model. First, based on the open-source computational fluid dynamics software OpenFOAM, an advanced two-equation kθ−εθ heat transfer model was introduced into the solver buoyant2eqnFoam, which was developed based on the self-solver buoyantSimpleFoam of OpenFOAM. In the solver buoyant2eqnFoam, various turbulence models built into OpenFOAM can be conveniently called to close the Reynolds stress and an advanced two-equation heat transfer model can be utilized to calculate the Reynolds heat flux of low-Pr fluids. Subsequently, the solver buoyant2eqnFoam was employed to study the fully developed flow heat transfer of low-Pr fluids in a bare 19-rod bundle. The numerical results were compared and analyzed with the experimental correlations and other simulation results to validate the effectiveness and feasibility of the solver buoyant2eqnFoam. Furthermore, the influence of combining different turbulence models with the same two-equation kθ−εθ heat transfer model was also presented in this study. The results show that the turbulence model has a considerable influence on the prediction of turbulent heat transfer in the high Peclet number range, suggesting that it should be prudent when picking a turbulence model in the simulations of low-Pr fluids.

Development and Assessment of an Isotropic Four-Equation Model for Heat Transfer of Low Prandtl Number Fluids

In the simple gradient diffusion hypothesis, the turbulent Prandtl number (Prt) with a constant of 0.85 is difficult to accurately predict for liquid metals having low Prandtl numbers (Pr), while a four-equation model can improve this solution by introducing the turbulence time-scale into the calculation of turbulent thermal diffusivity. However, the four-equation model’s transport form and numerical stability are so complex that suitable commercial code is lacking. Therefore, an isotropic four-equation model with simple Dirichlet wall boundary conditions is built in the present work. Based on the open-source computational fluid dynamics program OpenFOAM, the fully developed velocity, temperature, Reynolds stress, and heat flux of low Pr fluids (Pr = 0.01–0.05) in the parallel plane are obtained by numerical simulation. The results show that the time-average statistics predicted using the present four-equation model are in good agreement with the direct numerical simulation data. Then, the isotropic four-equation model is used to analyze the flow and heat of liquid metal (Pr = 0.01) in a quadrilateral infinite rod bundle. The numerical results are compared with the various and available experimental relationships. The Nusselt numbers calculated using the isotropic four-equation model are betweenness the available correlations, while the turbulent Prandtl number model using a constant of 0.85 over predicts heat transfer. More detailed local heat transfer phenomena and distribution of low Pr fluids are obtained using the present isotropic four-equation model.

Thermal/frictional performance of spiral pipe with ring-shape depression used as in-pond heat exchanger

In the present numerical study the ring-shape depressions were created on the wall of spiral pipe that can be employed as a heat exchanger in the solar ponds. Unlike available literature that focused on the optimization of solar ponds through its nature recognition, present study tried to propose an efficient passive method for maximizing heat extraction. This is a simple and cost-efficient technique for enhancing the overall performance of salinity-gradient solar ponds through augmentation of turbulence inside the modified piping system and yielding higher heat extraction accordingly. Indeed, the introduced practical technique not only prevents the increment of size and volume of the piping system but also makes it needless to use additional supplementary equipment like turbulator or even air injectors. Moreover, this is a good way to prevent fouling inside piping system which bears huge costs. 3D numerical simulation was done based on Finite Volume Method (FVM) and turbulent flow simulated by RNG k-epsilon model. Due to the steady-state nature of this problem pressure–velocity coupling algorithm was adopted. The effect of several decisive parameters including Dean, Pr (fluid type and inlet temperature), groove distance, and groove depth on Nu, f, j-factor, and TPF was studied. Interestingly, it was turned out that the values of thermal performance factor (TPF) for all cases were above 1 which shows the justifiability of made modifications. For all flow rates water has the highest Nu and the lowest friction factor compared to Theminol-55 and Ethylene-glycol. Groove pitch of 20 mm makes the highest values of Nu so that 168, 195, and 216 were the maximum obtained Nu for cases with groove depth of 0.7 mm, 1 mm, and 1.3 mm, respectively (working fluid was water and Dean = 1615). Said statement is true for f where its maximum values were 0.1, 0.14, and 0.2, respectively, for cases with groove depth of 0.7 mm, 1 mm, and 1.3 mm (working fluid was water and Dean = 269). The highest Nu of Therminol-55, Ethylene-glycol, and water were 100, 114, and 216, respectively for groove pitch of 20 mm.

Critical role of Prandtl number on multiple steady states during natural convection in square enclosures: Analysis of heat transfer rates, flow and thermal maps

The effect of Prandtl number (Pr) on the multiplicity of the steady states (flow and temperature maps) is investigated for the natural convection of Newtonian fluids within a square enclosure involving hot bottom wall, cold side walls and adiabatic top wall. The analysis is based on the steady state governing equations, which are solved by penalty Galerkin finite element method and Newton–Raphson solution scheme. A novel perturbation technique has been developed, which is applied in conjunction with an in-house continuation scheme for the initiation of various symmetric as well as asymmetric steady states. The current computation strategy involving perturbation and continuation schemes leads to a rich bifurcation diagram in the parameter space of Pr (0.01⩽Pr⩽10) for Ra (Rayleigh number) =106. It is found that the occurrence of multiple steady states depends on Pr. Only one symmetric solution is obtained in the high–Pr regime (Pr>0.42), while a highly populated bifurcation diagram consisting of 6 symmetric branches, 18 asymmetric branches and 4 isolas is obtained in the low–Pr regime (especially for Pr≲0.05). The solution branches correspond to a wide spectrum of flow structures involving single cell, multiple cells, reverse-cells, corner-cells, horizontally stacked cells, vertically stacked cells etc., which are reported for the first time. These flow structures result in a wide range of Nusselt numbers (average heat transfer rates) at a given Pr, especially for the low–Pr fluids (such as liquid metals). The flow structures are critically analyzed in terms of the location of the flow intensification and its impact on the overall heat transfer rates. It has been shown that the overall heat transfer rates for the low–Pr fluids can be significantly enhanced (80 –100% in few cases) based on corner-cells or single cell driven flow.

Thermo-magneto-convection and mechanism of multiple eddy formation due to applied magnetic field in a lid-driven cavity

Magnetohydrodynamic study of eddy formation and heat transfer in a bottom heated two-dimensional lid-driven cavity with the insulated sidewall is reported in this paper. We consider the classic problem of a two-dimensional lid driven cavity in the presence of an applied external magnetic field. In the standard nonlinear Navier–Stokes equation and the energy equation, appropriate terms are added to model the applied magnetic field and thermo-magnetic convection. And we have derived a higher-order numerical scheme that can provide solutions that are fourth-order accurate. The scheme is implemented with suitable fourth-order accurate boundary conditions. The square LDC enclosure is filled with a fluid of Pr = 6.2. The top lid of the cavity is kept moving with constant velocity. The effect of magnetic field on heat transfer and eddy formation is discussed for 0 ≤ Ha ≤ 250, and 103 ≤ Gr ≤ 104 and the results are presented in terms of contour plots of streamlines, isotherms, Lorentz force, and the kinetic energy. The physical mechanism for forming multiple recirculation zones is understood and attributed to two opposite forces acting in the fluid about the center-line of the cavity, which acts like a couple (and hence rotational flow). This is discovered by computing the Lorentz force acting in the entire domain of fluid flow. Similarly, from the computation of kinetic energy of the fluid at all the grid points in the fluid region, it is revealed that conduction is the dominant heat transfer mechanism at higher magnetic fields. At the same time, convection is the prime mode at low magnetic fields. It is ascertained that with an increase in magnetic field intensity, velocity profiles are suppressed, and heat transfer stabilizes, giving rise to the formation of recirculation zones in the cavity.

Influence of Prandtl number on bifurcation and pattern variation of non-isothermal annular Poiseuille flow

The relative influence of momentum diffusivity and thermal diffusivity, in terms of the Prandtl number (Pr), on the finite-amplitude instability of a non-isothermal annular Poiseuille flow (NAPF) is analyzed. The limiting value of the growth of instabilities under nonlinear effects is studied by deriving a cubic Landau equation. Emphasis is given especially on studying the impact of the low Prandtl number and the curvature parameter (C) on the bifurcation and the pattern variation of the secondary flow for both axisymmetric and non-axisymmetric disturbances. The finite-amplitude analysis predicts that in contrast to NAPF of water or fluid with Pr ≥ O(1) where the flow is supercritically unstable, the NAPF of low Pr fluids, particularly liquid metals, has shown both supercritical and subcritical bifurcation in the vicinity as well as away from the critical point. The nonlinear interaction of different harmonics for the liquid metal predicts a lower heat transfer rate than those by the laminar flow model, whereas for a fluid with Pr > 2, it is the other way. The maximum heat transfer takes place for the considered minimum value of C. For fluids with low Pr, a probable lower critical Rayleigh number is obtained. The corresponding variation in neutral stability curves as a function of wavenumber reveals that the instability that is supercritical for some wavenumber may be subcritical or vice versa at other nearby wavenumbers. The structural feature of the pattern of the secondary flow under the linear theory differs significantly from those of the secondary flow under nonlinear theory away from the bifurcation point. This is a consequence of the intrinsic interaction of different harmonics that are responsible for the stabilizing or the destabilizing nature of different components in the disturbance kinetic energy balance.

Linear instability of concentric annular flow: Effect of Prandtl number and gap between cylinders

The stability of non-isothermal Poiseuille flow in a coaxial annular domain is studied using normal mode analysis. The stably stratified flow (i.e., when buoyant force is in the direction of forced flow) is induced due to external pressure gradient and maintenance of linear variation of the temperature of the inner cylinder. We have emphasized the impact of the gap between cylinders in terms of curvature parameter (C) and Prandtl number (Pr) on the instability of the flow under axisymmetric as well as non-axisymmetric disturbances. In general, it has been found that the flow of low Pr fluids under axisymmetric disturbance is more stable in comparison with the flow under non-axisymmetric disturbance. When the gap between cylinders is relatively small (i.e. C is large), the flow under non-axisymmetric is most stable for the fluids having high Pr. It is known that the Newtonian pipe flow is linearly stable for all values of Reynolds number (Re). However, in the present study, we have seen the instability of the pipe flow with a thin rod placed at the center even for a very small value of heat source intensity (in terms of Rayleigh number) at Re=2000. Here the disturbance velocity is concentrated in the vicinity of the inner wall and instability is due to transfer of kinetic energy from the basic flow through Reynold’s stress. Depending on the value of controlling parameters, three types of instability: thermal-shear, interactive, thermal-buoyant have been observed. In general, for a fixed value of Pr, the type of instability is not affected by C and Re. The non-isothermal Poiseuille flow of low Pr (including mercury and gases) is governed by thermal-shear instability and the same of high Pr (including liquid and oil) is governed by thermal-buoyant instability.

Transitions near the onset of low Prandtl-number rotating convection in presence of horizontal magnetic field

We investigate the transitions near the onset of thermal convection in electrically conducting low Prandtl-number (Pr) fluids in the presence of rotation about a vertical axis and external horizontal magnetic field. Three-dimensional direct numerical simulations (DNSs) and low dimensional modeling are performed with the Rayleigh–Bénard convection system in the ranges 0 < Q ≤ 1000 and 0 < Ta ≤ 500 of the Chandrasekhar number (Q) and the Taylor number (Ta), respectively, for that purpose. For larger Q(≥32.7), DNSs show substantial enhancement of convective heat transport and only finite amplitude steady two dimensional roll patterns at the onset. On the other hand, for smaller Q(<32.7), very rich dynamics involving different stationary as well as time dependent patterns, including stationary two-dimensional rolls, cross rolls, and oscillatory cross rolls, are observed at the onset of convection. Our investigation uncovers the cause of enhancement of heat transport and the origin of different flow patterns at the onset. We establish that a first order transition to convection occurring at the onset is responsible for the enhancement of the heat transport there. Furthermore, as the Rayleigh number (Ra) is increased beyond the onset, subsequent transitions near it are also explored in detail for smaller Q, and these are found to be associated with a variety of bifurcations including subcritical/supercritical pitchfork, Hopf, imperfect pitchfork, imperfect gluing, and Neimark–Sacker.

Natural convection in a differentially heated enclosure filled with low Prandtl number fluids with modified lattice Boltzmann method

The convective motion in a three- and two- dimensional square cavity driven by a temperature gradient is analyzed. The cavity is filled with a low-Prandtl-number (Pr) fluid, typical for liquid metals. The vertical walls have constant but different temperatures, while the horizontal boundaries are adiabatic. Low Prandtl number exhibits strong non-linearity, where the advection term dominates the diffusion term in the momentum equations. Therefore, a Multi-Relaxation-Time, modified Lattice Boltzmann (MRT-LBM) approach is used to overcome the stability of the numerical solutions. Also, by using the mentioned method a wide range of Prandtl numbers, Pr, (0.01–0.5) and a wide range of Rayleigh numbers, Ra, (104 to 108) are investigated. It was found that the flow field exhibits periodic oscillations at the critical Rayleigh numbers, which are dependent on the Prandtl number. The periodic flow becomes aperiodic as the Prandtl number decreases and/or the Rayleigh number decreases. Also, the average Nusselt number are presented and correlated for the investigated range of the controlling parameters.

Prandtl number effect on thermal behavior in volumetrically heated pool in the high Rayleigh number region

The in-vessel retention by external reactor vessel cooling (IVR-ERVC) is a severe accident management strategy to hold or mitigate the molten radioactive material in the reactor pressure vessel (RPV). Highly turbulent natural convection is one of the most important phenomenon determining the thermal behavior of the molten pool. In the safety evaluation of IVR-ERVC, thermal behavior of the corium pool has been determined using experimental correlations. Property differences between the experimental simulant and the corium have been considered negligible. The effect of property differences represented by Prandtl number (Pr) on the thermal behavior of oxide pool is numerically investigated. The algebraic heat flux model (AFM) is used to simulate the turbulent natural convection and is validated with the BALI experiment. The Pr effect on the flow characteristics and the thermal behavior is investigated in the high Rayleigh number region. The present results show that the upward heat transfer tends to increase at the same Ra′ in the case of the molten pool which is lower Pr fluid than the simulant material.

Numerical Analysis of Plume Structures of Fluids on a Horizontal Heated Plate

Numerical investigations have been carried out to predict the near-wall dynamics in indirect natural convection for air (Pr = 0.7) and water (Pr = 5.2). Near-wall flow structures appear to be line plumes. Three-dimensional laminar, steady-state model was used to model the problem. Density was formulated using the Boussinesq approximation. Flux scaling, plume spacing and plume lengths obtained numerically are found to have the same trend with the results available in the literature. Plume length and Nusselt number, Nu exhibits an increasing trend with an increase in Rayleigh number, RaH for both Pr fluids. The plume spacing is found to have an inverse relationship with RaH. The cube root of Rayleigh number based on plume spacing, Raλ1/3 is found to have a slight dependence on the dimensionless plume spacing, λ/H. Nu scales as Nu∼CRaHn, n = 0.26 for air and n = 0.3 for water. Heat transfer is thus found to be dominated by near-wall phenomenon. Nu shows a nonlinear relationship with LpH/A and is found to be an accurate representation of heat transfer.

Natural convection of Bingham fluids in rectangular cross-sectional cylindrical annuli with differentially heated vertical walls

PurposeThis paper aims to numerically analyse natural convection of yield stress fluids in rectangular cross-sectional cylindrical annular enclosures. The laminar steady-state simulations have been conducted for a range of different values of normalised internal radius (ri/L1/8 to 16, whereLis the difference between outer and inner radii); aspect ratio (AR=H/Lfrom 1/8 to 8 whereHis the enclosure height); and nominal Rayleigh number (Rafrom 103to 106) for a single representative value of Prandtl number (Pris 500).Design/methodology/approachThe Bingham model has been used to mimic the yield stress fluid motion, and numerical simulations have been conducted for both constant wall temperature (CWT) and constant wall heat flux (CWHF) boundary conditions for the vertical side walls. The conservation equations of mass, momentum and energy have been solved in a coupled manner using the finite volume method where a second-order central differencing scheme is used for the diffusive terms and a second-order up-wind scheme is used for the convective terms. The well-known semi-implicit method for pressure-linked equations algorithm is used for the coupling of the pressure and velocity.FindingsIt is found that the mean Nusselt number based on the inner peripheryNu¯iincreases (decreases) with an increase inRa(Bn) due to augmented buoyancy (viscous) forces irrespective of the boundary condition. The ratio of convective to diffusive thermal transport increases with increasingri/Lfor both Newtonian (i.e.Bn= 0) and Bingham fluids regardless of the boundary condition. Moreover, the mean Nusselt numberNu¯inormalised by the corresponding Nusselt number due to pure conductive transport (i.e.Nu¯i/(Nu¯i)cond) shows a non-monotonic trend with increasingARin the CWT configuration for a given set of values ofRa,Pr,Lifor both Newtonian (i.e.Bn= 0) and Bingham fluids, whereasNu¯i/(Nu¯i)condincreases monotonically with increasingARin the CWHF configuration. The influences of convective thermal transport strengthen while thermal diffusive transport weakens with increasingAR, and these competing effects are responsible for the non-monotonicNu¯i/(Nu¯i)condvariation withARin the CWT configuration.Originality/valueDetailed scaling analysis is utilised to explain the observed influences ofRa,BN,ri/LandAR, which along with the simulation data has been used to propose correlations forNu¯i.

CFD SIMULATIONS OF THE HELIUM-XENON MIXTURE FLOW AND HEAT TRANSFER

Helium-xenon gas mixtures are generally proposed for the working fluid of gas-cooled space nuclear reactors with Brayton cycle to achieve a compact design. The Prandtl number (Pr) of helium-xenon mixtures at certain mixing ratios can be significantly lower than that of pure gases, inducing problems in describing the turbulent heat transfer. In this paper, investigations are performed to study the feasibilities of different CFD methods which were used to simulate the flow and heat transfer of helium-xenon mixtures. The thermophysical properties that are used in the simulation are firstly re-examined. Then, various turbulent models are employed to test their applicability by comparing the simulations with experimental data. It shows that all calculations have a general consistent tendency with, but lower than, experimental data. The SST κ-ω model has a better prediction ability. Considering the fact that the turbulent Prandtl number (Prt) varies perpendicular to the wall surface for the low Pr fluids, the Kays-Crawford Prt model is introduced into the SST κ-ω model, and the simulation results prove to be improved significantly in various cases.

Laminar flow and heat transfer in U-bends: The effect of secondary flows in ducts with partial and full curvature

The present study numerically explores the flow and heat transfer in “U-type” curved pipes of either partial or full curvature, which are often used as integral parts of heat exchangers and heat sinks in industrial applications such as solar thermal systems, among many others. Specifically, the case of a high-Prandtl-number Newtonian fluid (thermal oil) in laminar forced convection inside partially and fully curved “U-bends” is investigated, in order to illustrate the effect of the generated secondary flows on the various local and global scales. The 3D steady-state simulations are performed using the free and open source computational fluid dynamics software “OpenFOAM”. The analysis of the computational results is initially based on data at selected characteristic locations of interest; they are properly established and further explored by extracting internal field variables, in the form of 2D contours of quantities of interest, such as temperature and vorticity. A parametric study on the flow rate and curvature impact, namely in terms of increasing Re and De numbers (Re = 100, 1000, 2000), on the heat transfer process inside the U-bends is conducted comparatively for both geometries. The presence of secondary flows is confirmed and the differences between the geometries under investigation are visually illustrated. For moderate and high Re numbers, the presence of multiple counter-rotating secondary cells is generally observed, while the partially curved U-bend (“composite”) displays a more complex flow topology. This behavior is projected on heat transfer through the examination of temperature distribution on inner and outer arcs of both geometries. It is shown that the composite bend causes an abrupt decrease and oscillations on temperature distributions, which are found to be related with phenomena like flow impingement, separation and re-attachment.The Pr number effect is illustrated through a comparison with a lower Pr fluid and its impact is demonstrated both inside and downstream of the U-bends. The averaged Nusselt distribution along the length of both curved ducts further highlights the impact of these phenomena on local heat transfer and provides initial evidence in favor of the composite bend. The overall performance of the investigated curved geometries is compared based on a “performance factor” criterion and a superior performance of the composite U-bend is further established, attaining a value of almost 45% for higher flow rates. Finally, the effect of each U-bend on the straight pipe downstream of their respective exit is quantified. High divergence from the regular “combined hydrodynamic and thermal entrance flow” is found for the forced convection inside the downstream pipe. A characteristic overshoot of the downstream-to-upstream ratio Nusselt number is found for moderate and higher flow rates at small lengths from entrance (z*). The partially curved U-bend downstream effect seems to be greater for the investigated cases of Re = 100, 2000, while the fully curved outperforms the composite for the moderate Re case (Re = 1000). The source of this discrepancy cannot be clearly identified through this work and further investigation is needed on this matter. All in all, the simulations performed here show that partially curved U-bends can potentially be advantageous, in terms of performance, compared to the standard fully-curved pipes for laminar forced convection applications. These results can provide a good starting point for the optimal design of such ducts for several heat transfer operations.

Time Step Validation Method Research for Low-Prandtl Number Fluid Numerical Simulation

The stability condition of Courant number and diffusion number is proved for an SGSD (stability guaranteed second-order difference) scheme by von Neumann method in implicit and explicit discretization of the one-dimensional convection and diffusion terms. Then, a series of numerical simulations of fluid flow and heat transfer based on two-dimensional unsteady state model is used to study the combined natural and MHD (magnetohydrodynamics) convection in a Joule-heated cavity using the finite volume methods, for the fluid of Pr = 0.01, also we use an SGSD scheme and IDEAL (inner doubly iterative efficient algorithm for linked equations) algorithm. It is found that periodic oscillation flow evolves.We propose a new convergence concept for the simulation oscillation results; the results of the numerical experiments are presented and they confirm our theoretical conclusions. The convergence result is checked in another way. It is found that the two approaches have the same results and can judge the validity of the time step. The proposed method is helpful to get reliable results efficiently.

Flow bifurcation routes to chaos of thermocapillary convection for low Prandtl number fluid in shallow annular pool with surface heat dissipation

In order to understand clearly the transition characteristics of thermocapillary convection of low Prandtl number fluid in shallow annular pool with surface heat dissipation, some bifurcation routes to chaos of thermocapillary convection have been numerically investigated by using the finite volume method in this paper. The annular pool was filled with the low Prandtl (Pr) number fluid of Pr = 0.011. The range of Biot number is from 0 to 1. Results indicate that the flow bifurcation route of thermocapillary convection depends intensively on surface heat dissipation. When surface heat dissipation is smaller, with the increase of Marangoni number the flow bifurcation sequence of thermocapillary convection is two-dimensional axisymmetric steady flow → three-dimensional steady flow → coexisting hydrothermal waves and radial moving waves → radial moving waves → chaos. When surface heat dissipation is larger, the bifurcation sequence becomes simple as two-dimensional axisymmetric steady flow → hydrothermal waves → chaos. Furthermore, every bifurcation of thermocapillary convection is always accompanied by the variations of the wave number and the oscillatory frequency. Furthermore, the temperature fluctuation amplitude on the free surface increases gradually with the increase of Marangoni number.