Mixed convection cooling of a heated circular cylinder by laminar upward-directed slot jet impingement

An experimental and numerical study has been carried out to investigate the heat transfer characteristics of a horizontal circular cylinder exposed to a slot jet impingement of air. A square-edged nozzle is mounted parallel with the cylinder axis and jet flow impinges on the bottom of the cylinder. The study is focused on low Reynolds numbers ranging from 120 to 1,210, Grashof numbers up to Gr = 10Re ² and slot-to-cylinder spacing from 2 to 8 of the slot width. The flow field is greatly influenced by the slot exit velocity and the buoyancy force due to density change. A Mach–Zehnder Interferometer is used for measurement of local Nusselt number around the cylinder at 10° interval. It is observed that the average Nusselt number decreases with increasing the jet spacing and increases with rising the Reynolds number. A finite volume method utilizing a curvilinear coordinate transformation is used for numerical modeling. The numerical results show good agreement with the experimental results. The flow and thermal field are seen to be stable and symmetric around the cylinder over the range of parameters studied.     

Laminar mixed convection from a continuously moving vertical surface with suction or injection

The boundary layer flow over a uniformly moving vertical surface with suction or injection is studied when the buoyancy forces assist or oppose the flow. Similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity boundary conditions. The effect is of various governing parameters, such as Prandtl number Pr, temperature exponent n, injection parameter d, and the mixed convection parameter λ=Gr/Re², which determine the velocity and temperature distributions and the heat transfer coefficient, are studied. The heat transfer coefficient increases as λ assisting the flow for all d at Pr=0.72 however, for n=−1 it decreases sharply with λ. On the other hand, increasing λ has no effect on heat transfer coefficient for Pr=10 at n=0, and 1 for almost all values of d studied. However, for n=−1 it has similar effect as for Pr=0.72. It is also found that Nusselt number increases as n increases for fixed λ and d. Received on 26 March 1997     

Transient natural convection cooling of a high Prandtl number fluid in a cubical cavity

This work presents a numerical analysis of the effects of thermal boundary conditions, fluid variable viscosity and wall conduction on transient laminar natural convection of a high Prandtl number (Pr=4×10⁴) fluid (Golden Syrup) in a cubical cavity. The simulations consider physical situations realizable at laboratory scale using a cavity with Plexiglas walls of 1 cm of thickness, and inside dimension of L=20 cm. The initial Rayleigh (Ra) number is 10⁶. The cavity is initially full of fluid at rest and at constant temperature (T _i =45°C) higher than the temperature of the walls (T _w =25°C). The time evolution of the flow patterns, the temperature contours, the mean temperature of the fluid and the Nusselt number (Nu) of eight different cases of cooling are presented and analyzed.     

Large-scale flow structures and heat transport of turbulent forced and mixed convection in a closed rectangular cavity

Results of an experimental study of flow structure formation and heat transport in turbulent forced and mixed convection are presented. The experiments were conducted in a rectangular cavity with a square cross section, which has an aspect ratio between length and height of Γ_xz = 5. Air at atmospheric pressure was used as working fluid. The air inflow was supplied through a slot below the ceiling, while exhausting was provided by another slot, which is located directly above the floor. Both vents extend over the whole length of the cell. In order to induce thermal convection the bottom of the cell is heated while the ceiling is maintained at a constant temperature. This configuration allows to generate and study mixed convection under well defined conditions. Results of forced convection at Re = 1.07 × 10⁴ as well as mixed convection at 1.01 × 10⁴ ? Re ? 3.4 × 10⁴ and Ra = 2.4 × 10⁸ (3.3 ? Ar ? 0.3), which were obtained by means of Particle Image Velocimetry and local temperature measurements, are presented. For purely forced convection a 2D mean wind, which can be approximated by a solid body rotation, is found. With increasing Archimedes number this structure becomes unstable, leading to a transition of the solid body rotation into additional smaller convection rolls. Proper orthogonal decomposition of the instantaneous velocity fields has been performed for further analysis of these coherent large-scale structures. Their fingerprint is found in the spatial temperature distribution of the out flowing air at the end of the outlet channel, which reveals a temporally stable profile with two maxima over the length of the outlet. Moreover a maximum in the global enthalpy transport by the fluid is found at Ar ≈ 0.6.     

Rayleigh's transformation for thermally buoyant flows

 Rayleigh's classical transformation t=x/U connecting an unsteady parallel boundary flow to the steady plane Blasius-flow is generalized to t=C·x ^c and applied to the case of natural convection in the vicinity of a vertical flat plate. The parameters C and c of this “chronotopic transformation” (CTT) are determined self-consistently. It is shown that the analytic expressions obtained in this way for temperature and for the main component of the steady velocity field reproduce the “numerically exact” patterns to a good accuracy. Surprisingly, the transversal component of the steady flow (which in the unsteady flow is entirely missing) can also be generated by CTT with a remarkable precision. Moreover, the CTT is also able to extrapolate unsteady buoyant flows to steady ones with a good performance, even if these belong to a different set of boundary conditions (e.g. time-dependent vs. coordinate-dependent wall temperatures). Received on 6 May 1999     

Experimental study on turbulent natural convection heat transfer in water with sub-millimeter-bubble injection

Using thermocouples and a particle tracking velocimetry technique, temperature and velocity measurements are conducted to investigate flow and heat transfer characteristics of turbulent natural convection from a vertical heated plate in water with sub-millimeter-bubble injection. Hydrogen-bubbles generated by the electrolysis of water are used as the sub-millimeter-bubbles. In the turbulent region, the heat transfer deterioration occurs for a bubble flow rate Q = 33 mm³/s, while the heat transfer enhancement occurs for Q = 56 mm³/s. Temperature and velocity measurements suggest that the former is caused by a delay of the transition due to the bubble-induced upward flow. On the other hand, the latter is mainly due to two factors: one is the enhancement of the rotation of eddies in the outer layer, and the other is the increase in the gradient of the streamwise liquid velocity at the heated wall. These are caused by bubbles, which are located in the inner layer, rising at high speed.     

Effect of the surface thermal radiation on turbulent natural convection in tall cavities of façade elements

The effect of the surface thermal radiation in tall cavities with turbulent natural convection regime was analyzed and quantified numerically. The parameters considered were: the Rayleigh number 10⁹–10¹², the aspect ratio 20, 40 and 80 and the emmisivity 0.0–1.0. The percentage contribution of the radiative surface to the total heat transfer has a maximum value of  15.19% (Ra = 10⁹, A = 20) with emissivity equal to 1.0 and a minimum of 0.5% (Ra = 10¹², A = 80) with ε* = 0.2. The average radiative Nusselt number for a fixed emissivity is independent of the Rayleigh number, but for a fixed Rayleigh number diminishes with the increase of the aspect ratio. The results indicate that the surface thermal radiation does not modify significantly the flow pattern in the cavity, just negligible effects in the bottom and top of the cavity were observed. Two different temperature patterns were observed a conductive regime Ra = 10⁹ and a boundary layer regime Ra = 10¹².     

Characterization of fluid flow patterns and heat transfer in horizontal channel mixed convection

Two mechanisms of roll initiation are highlighted in a horizontal channel flow, uniformly heated from below, at constant heat flux (Γ = 10, Pr = 7, 50 ≤ Re ≤ 100, 0 ≤ Ra ≤ 10⁶). The first mechanism is the classical one, it occurs for low Rayleigh numbers and is initiated by the lateral wall effect. The second occurs for higher Rayleigh numbers and combines the previous effect with a supercritical vertical temperature gradient in the lower boundary layer, which simultaneously triggers pairs of rolls in the whole zone in between the two lateral rolls. We have found that in the present configuration, the transition between the two roll initiation mechanisms occurs for Ra/Re ² ≈ 18. Consequently, the heat transfer is significantly enhanced compared to the pure forced convection case owing to the flow pattern responsible of the continuous flooding the heated wall with cold fluid.     

Direct numerical simulation of turbulent flow and heat transfer in a square duct with natural convection

In this paper, a direct numerical simulation of a fully developed turbulent flow and heat transfer are studied in a square duct with an imposed temperature difference between the vertical walls and the perfectly insulated horizontal walls. The natural convection is considered on the cross section in the duct. The numerical scheme employs a time-splitting method to integrate the three dimensional incompressible Navier-Stokes equation. The unsteady flow field was simulated at a Reynolds number of 400 based on the Mean friction velocity and the hydraulic diameter (Re _m = 6200), while the Prandtl number (Pr) is assumed 0.71. Four different Grashof numbers (Gr = 10⁴, 10⁵, 10⁶ and 10⁷) are considered. The results show that the secondary flow and turbulent characteristics are not affected obviously at lower Grashof number (Gr ≤ 10⁵) cases, while for the higher Grashof number cases, natural convection has an important effect, but the mean flow and mean temperature at the cross section are also affected strongly by Reynolds stresses. Compared with the laminar heat transfer at the same Grashof number, the intensity of the combined heat transfer is somewhat decreased.     

Unsteady natural convection in an enclosure with vertical wavy walls

In this paper, unsteady heat transfer and fluid flow characteristics in an enclosure are investigated. The enclosure consists of two vertical wavy and two horizontal straight walls. The top and the bottom walls are considered adiabatic. Two wavy walls are kept isothermal and their boundaries are approximated by a cosine function. Governing equations including continuity, momentum and energy were discretized using the finite-volume method and solved by SIMPLE method in curvilinear coordinate. Simulation was carried out for a range of Grashof number Gr = 10³–10⁶, Prandtl number Pr = 0.5–4.0, wave ratio A (defined by amplitude/wavelength) 0.0–0.35 and aspect ratio W (defined by average width/wavelength) 0.5–1.0. Streamlines and isothermal lines are presented to corresponding flow and thermal fields. Local and average Nusselt number distributions are presented. The obtained results are in good agreement with available numerical and experimental data.     

Laminar Mixed Convection Adjacent to Vertical Continuously Stretching Sheets With Variable Viscosity and Variable Thermal Diffusivity

The paper presents a study of the laminar mixed convection adjacent to vertical continuously stretching sheets, taking into account the effects of variable viscosity and variable thermal diffusivity. The similarity solutions are reported for isothermal sheet moving with a velocity of the form u_w=Bx^0.5 and a continuous linearly stretching sheet with a linear surface temperature distribution. The equations of conservation of mass, momentum and energy, which govern the flow and heat transfer, are solved numerically by using the shooting method. The numerical results obtained for the flow and heat transfer characteristics reveal many interesting behaviors. The numerical results show that, variable viscosity, variable thermal diffusivity, the velocity exponent parameter, the temperature exponent parameter and the buoyancy force parameter have significant influences on the velocity and temperature profiles, shear stress and Nusselt number in two cases air and water.     

Numerical study of the laminar thermocapillary flow in a square cavity

The hydrodynamic and heat-transfer processes in the problem of a laminar thermocapillary flow of a viscous incompressible fluid in a square cavity with isothermal vertical and isentropic horizontal surfaces are investigated numerically under the assumption that the gravity is absent, the free surface is flat, and the surface tension depends linearly on the temperature. Calculations were performed by a compact-difference method on irregular grids with a fifth-order accuracy for four Prandtl numbers (Pr=1, 16, 200, and 3000) as the Marangoni (Ma) number varies from 10² to 10⁴. The maximum local heat transfer versus theMa number is obtained. It is shown that, for thePr values considered, the maxima of the distribution of the horizontal velocity component on the surface is displaced to the cold boundary according to a law inversely proportional to theMa number. Kutateladze Institute of Thermal Physics, Siberian Division, Russian Academy of Sciences, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 81–89, July–August, 1999.     

Finite element analysis of vortex shedding using equal order interpolations

An operator splitting and element‐by‐element conjugated gradient solver, and equal order interpolations are applied for solving time dependent Navier–Stokes (NS) equations to simulate flow induced vortex shedding in the present study. In addition, the convection term is corrected by balanced tensor diffusivity, which can stabilize the numerical simulation and overcome the numerical oscillations. The evolution of the interested flowing properties with time is analyzed by using spectral analysis. The developed code has been validated by the application of two examples: a driven cavity flow and a flow induced vortex vibration. Results from the first example for Reynolds number Re=10³ and Re=10⁴ are compared with other numerical simulations. Results from the second example, uniform flow past a square rod over a wide range of high Reynolds numbers from Re=10³～10⁵, are compared with experimental data and other numerical studies. Copyright © 2002 John Wiley & Sons, Ltd.     

Laminar mixed convection adjacent to vertical, continuously stretching sheets

The analysis of laminar mixed convection in boundary layers adjacent to a vertical, continuously stretching sheet has been presented. The velocity and temperature of the sheet were assumed to vary in a power-law form, that is, u _w (x)=Bx ^m and T _w (x)−T _∞=Ax ⁿ . In the presence of buoyancy force effects, similarity solutions were reported for the following two cases: (a) n=0 and m=0.5, which corresponds to an isothermal sheet moving with a velocity of the form u _w =Bx ^0.5 and (b) n=1 and m=1, which corresponds to a continuous, linearly stretching sheet with a linear surface temperature distribution, i.e. T _w −T _∞=Ax. Formulation of the present problem shows that the heat transfer characteristics depends on four governing parameters, namely, the velocity exponent parameter m, the temperature exponent parameter n, the buoyancy force parameter G ^*, and Prandtl number of the fluid. Numerical solutions were generated from a finite difference method. Results for the local Nusselt number, the local friction coefficient, and temperature profiles are presented for different governing parameters. Effects of buoyancy force and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. Received on 17 July 1997     

Boundary layer structure of oscillatory open-channel shallow flows over smooth and rough beds

The boundary layer structure of oscillatory shallow open channel flows has been studied in a wide flume. Fluorescence solution was released at a porous rough bed through a diffuser covered by gravel of 0.5 cm grain size. A planar laser-induced fluorescence (PLIF) system was used to visualise the dye plumes in both vertical and horizontal planes for a qualitative understanding of the roles of large-scale flow structures in mass transport. A variety of tests were conducted for a range of oscillatory periods (30–240 s), water depths (3–16 cm) and velocity amplitudes (0.027–0.325 m/s), which cover a wide range of oscillatory flows with Reynolds numbers Re _a varied from 0.3 × 10⁴ (laminar) to 2.1 × 10⁶ (fully turbulent). For quantitative investigation, a novel technique, namely combined laser-induced fluorescence (LIF) and 2D laser Doppler velocimetry (LDV) (LIF/LDV), was developed and used to measure the velocity and solute concentration simultaneously in a vertical plane over 50 cycles. From the dye plumes revealed by the PLIF in transitional flows, there are different patterns of flow structure and solute transport with three representative stages of acceleration, deceleration and flow reversal. In the acceleration stage, turbulence was suppressed with dye layers adhering to the surface with little vertical mass transport. In the deceleration stage, flame-like turbulent structures occurred when turbulence generation was prominent. This was investigated quantitatively by recording the percentage occurrence of the adhered smooth layers per cycle. For those smooth bed cases with Re _a < 1.8 × 10⁵, the adhered smooth dye layers type of boundary layer occupied 100% of the oscillation period. Over a sufficiently high Re _a , a rough bed can generate fully turbulent oscillatory flows without the appearance of adhering dye layers. Between these two extremes, a transitional flow regime occurs in a wide range of flow conditions: Re _a > 2.7 × 10⁴ over the rough bed and Re _a > 8.3 × 10⁶ over a smooth bed.     

Finite difference analysis of laminar free convection flow past a non isothermal vertical cone

A finite-difference analysis for the transient free convection flow of an incompressible viscous fluid past a vertical cone with variable wall surface temperature T _w^′ (x) = T _∞^′ + a x ⁿ varying as power function of distance from the apex (x = 0) is presented here. The dimensionless governing equations of the flow that are unsteady, coupled and non-linear partial differential equations are solved by an efficient, accurate and unconditionally stable finite difference scheme of Crank-Nicolson type. The velocity and temperature fields have been studied for various parameters such as Prandtl number and n (exponent in power law variation in surface temperature). The local as well as average skin-friction and Nusselt number are also presented and analyzed graphically. The present results are compared with available results in literature and are found to be in good agreement.     

Transient oscillatory conjugate natural convection in a tall water cavity

A two-dimensional numerical simulation was conducted to investigate the effects of the heat capacity of the heated wall on the transient oscillatory convection in a tall cavity. Results were particularly obtained for water (Pr=6) in a tall cavity (A=6) with constant-heat-flux heating on one side and isothermal cooling on the opposing side. Significant wall heat capacity effects were found. Specifically, a heated wall of finite heat capacity can slow down the flow evolution process to steady state at low Rayleigh number and damp the flow oscillation during the initial developing period at high Rayleigh number. In periodic flow atRa* _H =4.4×10¹⁰ the wall heat capacity significantly reduces the amplitude of the temperature oscillation. A quasi-periodic flow atRa* _H =4.8×10¹⁰ was found to become periodic when the wall heat capacity was included. The wall heat capacity does not shown noticeable effect when the flow is chaotic forRa* _H =6.0×10¹⁰.In einer zweidimensionalen numerischen Simulationsrechnung wurde der Einfluß der Wärmekapazität einer beheizten Wand auf die nichtstationäre oszillatorische Konvektion in einem großen Behälter untersucht. Die Resultate beziehen sich speziell auf Wasser (Pr=6), einen tiefen Behälter (H/W=6) und konstanten Wärmezufluß auf der einen und isotherme Kühlung auf der anderen Wandseite. Es traten signifikante Effekte in Abhängigkeit von der Wandwärmekapazität auf. Insbesondere vermag eine beheizte Wand begrenzter Wärmekapazität bei niedrigen Rayleigh-Zahlen die Entwicklung des Strömungsprozesses bis zum Stationärzustand abzubremsen und bei hohen Rayleigh-Zahlen die Strömungsoszillationen während der Einschwingphase zu dämpfen. Bei periodischem Strömungszustand undRa* _H =4,4·10¹⁰ verringert die Wandkapazität die Amplitude der Temperaturschwingung beträchtlich. Es zeigte sich ferner, daß ein quasiperiodischer Strömungszustand beiRa* _H =4,8·10¹⁰ periodisch wird, wenn die Wandkapazität Einfluß nimmt. Letztere hatte keine nachweisbaren Wirkungen bei chaotischem Strömungszustand mitRa* _H =6,0·10¹⁰.     

Time-dependent natural convection in a square cavity: Application of a new finite volume method

A new finite volume (FV) approach with adaptive upwind convection is used to predict the two-dimensional unsteady flow in a square cavity. The fluid is air and natural convection is induced by differentially heated vertical walls. The formulation is made in terms of the vorticity and the integral velocity (induction) law. Biquadratic interpolation formulae are used to approximate the temperature and vorticity fields over the finite volumes, to which the conservation laws are applied in integral form. Image vorticity is used to enforce the zero-penetration condition at the cavity walls. Unsteady predictions are carried sufficiently forward in time to reach a steady state. Results are presented for a Prandtl number (Pr) of 0-71 and Rayleigh numbers equal to 10³, 10⁴ and 10⁵. Both 11 × 11 and 21 × 21 meshes are used. The steady state predictions are compared with published results obtained using a finite difference (FD) scheme for the same values of Pr and Ra and the same meshes, as well as a numerical bench-mark solution. For the most part the FV predictions are closer to the bench-mark solution than are the FD predictions.