Small Values Of Re Research Articles

Mixed convective and radiative heat transfer of LiF-BeF2-ThF4-UF4 in a vertical circular tube

Molten salts are promising fluids for applications involving heat transfer and thermal energy storage. The buoyancy effect and radiative heat transfer effect are frequently ignored in previous studies that concentrate on forced convective heat transfer of molten salts. However, these effects might not be negligible, particularly for laminar flows of optically semi-transparent fluids, such as molten salts. Therefore, this study tries to understand the heat transfer characteristics of molten salts, including mixed (forced and natural) convective heat transfer and radiative heat transfer. A two-dimensional axisymmetric model is created in STAR-CCM+ for assisting (buoyancy-assisted) laminar flows of LiF-BeF2-ThF4-UF4 (67.5–20.0–12.0–0.5 mol%) in a 5-m long, 15-mm-inner-diameter vertical circular tube. The effects of several non-dimensional parameters on the velocity, temperature, and Nusselt number are examined, including the Reynolds number (Re¯=2.0×102to1.8×103), Grashof number (Gr¯=3.1×103to5.4×104), optical thickness (τ=0.1τ0to10τ0, τ0 = 0.27, 2.11, and 446.47 for the three bands λI = 0.10 to 3.36 µm, λII = 3.36 to 5.42 µm, and λIII = 5.42 to 100.00 µm, respectively), and emissivity (ε=0.0to1.0). A flow regime map is also suggested to assess the radiative heat transfer effect in molten salts. Our analysis demonstrates that: (1) the predicted results in this study deviate from published experimental data by 14.5 to 45.1 % without taking the buoyancy effect and radiative heat transfer effect into account, however, these differences are dramatically decreased to 10.1 to 17.4 % while considering the buoyancy effect or 7.5 to 14.7 % while considering the buoyancy effect and radiative heat transfer effect; (2) the radiative heat transfer effect tends to mitigate the buoyancy effect, but their net effect enhances heat transfer; (3) the buoyancy effect factor fbuoy and thermal radiation effect factor frad are, respectively, up to 38.5 % and 19.5 % under the conditions investigated, illustrating the need to evaluate the two effects for molten salts; (4)fbuoy becomes larger for smaller values of Re¯, τ, and ε, and larger values of Gr¯ and non-dimensional axial location x/Di. frad becomes larger for smaller values of Re¯ and larger values of Gr¯, τ, ε, and x/Di; and (5) the flow regime map provided in this study aids in determining whether the radiative heat transfer effect in molten salts is negligible.

Bifurcation and instability of annular Poiseuille flow in the presence of stable thermal stratification: Dependence on curvature parameter

The bifurcation and instability of nonisothermal annular Poiseuille flow (NAPF) of air as well as water is studied. We have emphasized the impact of a gap between cylinders in terms of curvature parameter (C) for axisymmetric as well as nonaxisymmetric disturbances. The results from the linear stability analysis reveal that the first azimuthal mode acts as a least stable mode of the NAPF of air for relatively small values of C. In this situation, even though for some values of C, the NAPF has supercritical bifurcation, but the same flow may experience subcritical bifurcation under zero azimuthal mode. It has also been observed that for relatively larger values of the Reynolds number (Re) and lower values of C, the NAPF under axisymmetric disturbance always exhibits subcritical bifurcation. However, for small values of Re, the NAPF exhibits only supercritical bifurcation. The finite amplitude analysis predicts only supercritical bifurcation of NAPF of water. The influence of nonlinear interaction of different harmonics on the amplitude profile as well as kinetic energy spectrum is investigated. The amplitude profile possesses a jump in the vicinity of a point where the type of bifurcation is changed. In the subcritical regime, the induced shear production due to modification of the gradient production acts as a main destabilizing factor balanced by the gradient production of kinetic energy.

Natural convection effects on heat and mass transfer of slip flow of time-dependent Prandtl fluid

Abstract We proposed a mathematical model for an incompressible, viscous, natural convection, and stagnation point slip flow of MHD Prandtl fluid over an infinite plate. The governing flow equations are constructed using the Prandtl rheological model. In account of physical relevance, we investigated the Soret and Dufour effects on the flow field. The complex natured flow equations are transformed to a set of PDEs using a suitable similarity variables. The non-dimensionalized ruling problem together with physical boundary conditions is numerically analyzed via Crank-Nicolson scheme. The velocity, temperature and concentration of the diffusing species distributions are enhanced for higher values of unsteadiness parameter. It is noted that velocity is slightly decreasing for higher values of Reynolds number while smaller values of Re providing more dominant effects on the velocity, temperature and concentration of the diffusing species profiles and enhanced heat and mass transfer rates is noticed. The physical behavior of reduced Nusselt and Sherwood numbers, friction factor, for distinct values of emerging parameters is examined and representative set of graphs are presented. Highlights Flow model is presented for MHD Prandtl fluid flow over an infinite plate. Mathematical model is performed for unsteady flow with Soret and Dufour effects. The proposed model is solved via crank Nicolson finite difference scheme. Simulations are performed for skin friction, Nusselt and Sherwood numbers.

Cloud parameter retrievals from Meteosat and their effects on the shortwave radiation at the surface

ABSTRACTA method based on Spinning Enhanced Visible and Infrared Imager (SEVIRI) measured reflectance at 0.6 and 3.9 µm is used to retrieve the cloud optical thickness (COT) and cloud effective radius (re) over the Iberian Peninsula. A sensitivity analysis of simulated retrievals to the input parameters demonstrates that the cloud top height is an important factor in satellite retrievals of COT and re with uncertainties around 10% for small values of COT and re; for water clouds these uncertainties can be greater than 10% for small values of re. The uncertainties found related with geometries are around 3%. The COT and re are assessed using well-known satellite cloud products, showing that the method used characterize the cloud field with more than 80% (82%) of the absolute differences between COT (re) mean values of all clouds (water plus ice clouds) centred in the range from ±10 (±10 µm), with absolute bias lower than 2 (2 μm) for COT (re) and root mean square error values lower than 10 (8 μm) for COT (re). The cloud water path (CWP), derived from satellite retrievals, and the shortwave cloud radiative effect at the surface (CRESW) are related for high fractional sky covers (Fsc >0.8), showing that water clouds produce more negative CRESW than ice clouds. The COT retrieved was also related to the cloud modification factor, which exhibits reductions and enhancements of the surface SW radiation of the order of 80% and 30%, respectively, for COT values lower than 10. A selected case study shows, using a ground-based sky camera that some situations classified by the satellite with high Fsc values correspond to situations of broken clouds where the enhancements actually occur. For this case study, a closure between the liquid water path (LWP) obtained from the satellite retrievals and the same cloud quantity obtained from ground-based microwave measurements was performed showing a good agreement between both LWP data set values.

Local Polynomial Regression Estimator of the Finite Population Total under Stratified Random Sampling: A Model-Based Approach

In this paper, auxiliary information is used to determine an estimator of finite population total using nonparametric regression under stratified random sampling. To achieve this, a model-based approach is adopted by making use of the local polynomial regression estimation to predict the nonsampled values of the survey variable y. The performance of the proposed estimator is investigated against some design-based and model-based regression estimators. The simulation experiments show that the resulting estimator exhibits good properties. Generally, good confidence intervals are seen for the nonparametric regression estimators, and use of the proposed estimator leads to relatively smaller values of RE compared to other estimators.

Effects of Turbulent Reynolds Number on Turbulent Scalar Flux Modeling in Premixed Flames Using Reynolds-Averaged Navier–Stokes Simulations

The influence of turbulent Reynolds number (Re t ) on the statistical behavior and modeling of turbulent scalar flux (TSF) has been analyzed using a direct numerical simulation database of freely propagating turbulent premixed flames. A range of different values of Re t is considered in which the Damköhler and Karlovitz numbers are modified independent of each other to bring about the variation of Re t . It has been found that the qualitative behavior of the various terms of the TSF transport equation does not change by the variation of Re t , but their relative contributions to the transport of TSF are affected to some extent. The effects of Re t on the modeling of the TSF using both algebraic and transport equation-based closures are addressed in detail. It is demonstrated that model parameters for an existing algebraic model, and the models for turbulent transport, pressure gradient and the reaction rate terms in the TSF transport equation, all exhibit Re t dependence for small values of Re t but all assume asymptotic values for Re t ≥ 50. By contrast, the model parameters for the combined molecular diffusion term are found to be insensitive to the variation of turbulent Re t . Existing models for algebraic and transport equation-based closures of TSF have been modified to account for the observed Re t dependence.

Finite element based heatline approach to study mixed convection in a porous square cavity with various wall thermal boundary conditions

A penalty finite element method based simulation is performed to analyze the influence of various walls thermal boundary conditions on mixed convection lid driven flows in a square cavity filled with porous medium. The relevant parameters in the present study are Darcy number ( Da = 10 −5 − 10 −3), Grashof number ( Gr = 10 3 − 10 5), Prandtl number ( Pr = 0.7–7.2), and Reynolds number ( Re = 1–10 2). Heatline approach of visualizing heat flow is implemented to gain a complete understanding of complex heat flow patterns. Patterns of heatlines and streamlines are qualitatively similar near the core for convection dominant flow for Da = 10 −3. Symmetric distribution in heatlines, similar to streamlines is observed irrespective of Da at higher Gr in natural convection dominant regime corresponding to smaller values of Re. A single circulation cell in heatlines, similar to streamlines is observed at Da = 10 −3 for forced convection dominance and heatlines are found to emanate from a large portion on the bottom wall illustrating enhanced heat flow for Re = 100. Multiple circulation cells in heatlines are observed at higher Da and Gr for Pr = 0.7 and 7.2. The heat transfer rates along the walls are illustrated by the local Nusselt number distribution based on gradients of heatfunctions. Wavy distribution in heat transfer rates is observed with Da ⩾ 10 −4 for non-uniformly heated walls primarily in natural convection dominant regime. In general, exponential variation of average Nusselt numbers with Grashof number is found except the cases where the side walls are linearly heated. Overall, heatlines are found to be a powerful tool to analyze heat transport within the cavity and also a suitable guideline on explaining the Nusselt number variations.

Steady axisymmetric motion of a small bubble in a tube with flowing liquid

The steady axisymmetric behaviour of a relatively small bubble moving with a flowing liquid in a straight round tube is studied by computationally solving the nonlinear Navier–Stokes equations, using a Galerkin finite-element method with boundary-fitted mesh, for wide ranges of capillary number C a and Reynolds number R e . Here a bubble is considered relatively small when its volume-equivalent radius is less than that of the tube. At small values of R e , the velocity of a bubble increases with bubble size for large values of C a but decreases with bubble size for small values of C a . At large values of R e , however, a bubble of large size appears to move at a slower velocity for any given value of C a . When R e is large (e.g. R e = 100) and C a > 0.1, a bubble of radius greater than half of the tube radius moves at a velocity that seems to be independent of bubble size. The strong inertial effect at large R e makes a small bubble of radius greater than a quarter of the tube radius to deform into a noticeable oblate shape as C a increases from very small value, and then to be elongated into a bullet shape with further increasing C a after C a reaches an intermediate value. Even very small bubbles (e.g. of radius equal to one-tenth of the tube radius) can still be significantly deformed provided that the value of C a is adequately large. Despite significant shape deformations that may still occur, bubbles of radius less than a quarter of that of the tube almost always move at the same velocity as that of the local liquid flow at the tube centreline (i.e. twice that of the average liquid velocity), regardless the values of R e and C a . This fact suggests that very small bubbles are basically carried by the local liquid flow.

Influence of Prandtl number on stability of mixed convective flow in a vertical channel filled with a porous medium

Buoyancy opposed mixed convection is considered in a vertical channel filled with an isotropic, porous medium, in which the motion of an incompressible fluid is induced by external pressure gradients and buoyancy forces. The Brinkman-Wooding-extended Darcy model has been used to study the instability mechanisms of the basic flow and its dependence on the Prandtl number (Pr) of the fluid. The stability analysis indicated that for the same Reynolds number (Re), the fully developed base flow was highly unstable for a fluid with high Pr. For a porous medium with a Darcy number (Da) of 10−6 and Pr⩾0.7, two different types of instability, Rayleigh-Taylor (R-T) and buoyant instability, are observed. The R-T instability mode is observed for relatively small values of Re. Further, the results show that for Da=10−5 and Pr<1, the spectrum of the energy profile is abrupt and sudden, whereas the same is smooth when Da=10−6. In the case of R-T instability, the critical value of Ra at low Re is given by −2.47∕Da. Though the R-T mode of instability is independent of Pr, the range of Re that sustains the R-T mode is a function of Pr. It has been found that enhancement of Pr reduces the Re range mentioned above. In contrast to the case of a purely viscous fluid, where the effect of Pr is not significant, in isotropic porous media Pr plays a significant role in characterizing the flow stability. The instability characteristics of zero temperature flux perturbation (BC-I) and zero heat flux perturbation (BC-II) on the boundaries differ significantly in the case of the R-T stability mode. However, both conditions lead to similar results for buoyant stability, except at small values of Re.

Finite-Reynolds-Number Effects in Steady, Three-Dimensional Airway Reopening

Motivated by the physiological problem of pulmonary airway reopening, we study the steady propagation of an air finger into a buckled elastic tube, initially filled with viscous fluid. The system is modeled using geometrically non-linear, Kirchhoff-Love shell theory, coupled to the free-surface Navier-Stokes equations. The resulting three-dimensional, fluid-structure-interaction problem is solved numerically by a fully coupled finite element method. Our study focuses on the effects of fluid inertia, which has been neglected in most previous studies. The importance of inertial forces is characterized by the ratio of the Reynolds and capillary numbers, ReCa, a material parameter. Fluid inertia has a significant effect on the system's behavior, even at relatively small values of ReCa. In particular, compared to the case of zero Reynolds number, fluid inertia causes a significant increase in the pressure required to drive the air finger at a given speed.

Entropy generation due to heat and fluid flow in backward facing step flow with various expansion ratios

The present research investigated second law analysis of laminar flow over a backward facing step with various expansion ratios (ER). As Reynolds number (Re) increased the value of total entropy generation (Ns) increased. For smaller values of Re as the expansion ratio increased the value of Ns decreased. However, for higher values of Re as ER increased the value of Ns decreased but at smaller rates. The total Bejan number increased as the ER increased. At the bottom wall the minimum value of Ns occurred at the step location and the maximum value of Ns occurred inside the recirculation zone.

Forced stratified turbulence: successive transitions with Reynolds number.

Numerical simulations are made for forced turbulence at a sequence of increasing values of Reynolds number Re keeping fixed a strongly stable, volume-mean density stratification. At smaller values of Re, the turbulent velocity is mainly horizontal, and the momentum balance is approximately cyclostrophic and hydrostatic. This is a regime dominated by so-called pancake vortices, with only a weak excitation of internal gravity waves and large values of the local Richardson number Ri everywhere. At higher values of Re there are successive transitions to (a) overturning motions with local reversals in the density stratification and small or negative values of Ri; (b) growth of a horizontally uniform vertical shear flow component; and (c) growth of a large-scale vertical flow component. Throughout these transitions, pancake vortices continue to dominate the large-scale part of the turbulence, and the gravity wave component remains weak except at small scales.

Numerical simulation of blood flow in an artery with two successive bends

Blood flow in an artery with two successive bends is simulated by a finite-element computation of steady flow of a Newtonian viscous fluid through a rigid tube having the same shape as a specific part of the femoral artery. Notwithstanding the fact that the bends in the model geometry are rather gentle, the axial and secondary flow patterns, computed for a range of values of the Reynolds number Re, show strong and complicated three-dimensional flow effects. In particular, the flow pattern in the second bend for relatively small values of Re ( Re < 240) turns out to be drastically different from that for larger Re-values.

PREDICTABILITY AND THE BUTTERFLY EFFECT IN TURBULENT FLOWS: A SHELL MODEL STUDY

We discuss the dynamical growth of a disturbance in turbulent flows based on numerical calculations of an approximative model of the Navier–Stokes equations, a so-called shell model. A disturbance at small length scales is observed to propagate (and increase) towards large length scales by an inverse cascade of duration T, the predictability time. At increasing Reynolds number Re, the mean predictability time Tt is found to decrease proportionally to Re−0.47. Moreover, the probability distribution of (T − Tt)/Tt changes its shape as Re increases: at relatively small values of Re it has an almost Gaussian shape, while at large Re it gets an exponential tail, indicating the possibility of large excursions for Tt.

Aerodynamic aspects of the sealing of gas-turbine rotor-stator systems: Part 2: The performance of simple seals in a quasi-axisymmetric external flow

The sealing characteristics of a shrouded rotor-stator system with an external flow of air have been studied using flow visualization, pressure, and concentration measurements. Three shroud geometries, incorporating axial clearance or radial clearance seals, have been tested for a range of clearance ratios for both rotational and axial Reynolds numbers Re θ and Re w respectively, up to 1.2 × 10 6. It was found that there were two regimes: a rotation-dominated regime at low values of Re w/Re θ, and an external-flow-dominated regime at high values. For Re w = 0, C w, min (the dimensionless flow rate of sealing air necessary to prevent the ingress of external fluid into the rotor-stator wheel space) increased with increasing Re θ. For small values of Re w/Re θ, C w, min decreased with increasing Re w; for large values of Re w/Re θ, C w, min was proportional to Re w and was independent of Re θ. The latter effect is attributed to the nonaxisymmetric pressure distribution in the external flow: fluid moved transversely across the wheel space from high-pressure to low-pressure regions of the external flow.

Experiments on sedimentation beneath downward-facing inclined walls

It has been known since Boycott (1920) that sedimentation rates can be several times higher within inclined walls than in vertical vessels. Beneath the downward-facing inclined wall a thin boundary-layer of clear liquid is formed with an upward velocity. Mass continuity brings about the observed effect. Ponder (1925), and Nakamura & Kuroda (1937) proposed a global kinematic model without any detail of the flow field to predict the sedimentation rate. Recently, two asymptotic theories for describing the flow field became available. The investigation by Acrivos & Herbolzheimer (1979) requires both Re 2/ Gr and Re 4/ Gr to be small, with Re being a sedimentation Reynolds number and Gr a sedimentation Grashof number. The analysis due to Schneider (1982) is valid for small values of Re 2/ Gr and large values of Re 4/ Gr. This paper presents experimental sedimentation data in a 2/ Gr and large values of symmetrical, roof-shaped vessel. The experiments with spherical glass beads in a variable mixture of glycerine and water cover the whole parameter range from very small to very large values of Re 4/ Gr. Therefore, both theories could be verified. In the case of small values of Re 4/ Gr, strong waves beneath the downward-facing wall were observed and graphically evaluated in order to find a basis for further theoretical approach. Furthermore, an upward flow of particles in a sublayer of the boundary-layer leads te protruding “borns” and an oscillation of the originally horizontal discontinuity separating the suspension from the clear liquid. The measured particle distribution and deviations from a monodispersed suspension seems to be a possible explanation for this effect, which is investigated theoretically in a separate paper (Schaflinger 1984). A further experimental shortcoming, the shallow depth of the settling vessel, that was necessary for an optoelectronic measuring of the actual volume fraction solids within the bulk, may affect the convective motion when the sedimentation Reynols number is small. However, in this case the experimental data are in conformity with the theoretical predictions which do not consider the influence of wall friction on the sedimentation behaviour.

Laminar Elastico-Viscous Source Flow between Parallel Porous Disks

The steady-state laminar source flow of an incompressible elastico-viscous liquid between stationary infinite porous disks is considered. The suction or injection velocities at the disks are assumed to be uniform. The flow phenomenon has been characterized by the nondimensional parameters S (elastic number), R (wall Reynolds number), and Re* (reduced Reynolds number). The solution obtained is in the form of a perturbation from the creeping flow solution and is valid for small values of R and also for small values of Re*. The effects of the above parameters on the velocity distribution, shear stress, and the radial pressure variation have been studied.