Heat transfer measurements in fully turbulent flows: basic investigations with an advanced thin foil triple sensor

In a former article in this journal a double layer hot film with two 10 μm nickel foils, separated by a 25 μm polyimide foil was introduced as a multi-purpose sensor. Each foil can be operated as a (calibrated) temperature sensor in its passive mode by imposing an electric current small enough to avoid heating by dissipation of electrical energy. Alternatively, however, each foil can also serve as a heater in an active mode with electric currents high enough to cause Joule heating. This double foil sensor can be used as a conventional heat flux sensor in its passive mode when mounted on an externally heated surface. In fully turbulent flows it alternatively can be operated in an active mode on a cold, i.e. not externally heated surface. Then, by heating the upper foil, a local heat transfer is initiated from which the local heat transfer coefficient h can be determined, once the lower foil is heated to the same temperature as the upper one, thus acting as a counter-heater. For further investigations with respect to the underlying sensor concept a triple sensor has been built which consists of three double layer film sensors very close to each other. Various aspects of heat transfer measurements in active modes can be addressed by this sensor.     

Experimental and numerical investigation of transition to turbulent flow and heat transfer inside a horizontal smooth rectangular duct under uniform bottom surface temperature

In this study, steady-state turbulent forced flow and heat transfer in a horizontal smooth rectangular duct both experimentally and numerically investigated. The study was carried out in the transition to turbulence region where Reynolds numbers range from 2,323 to 9,899. Flow is hydrodynamically and thermally developing (simultaneously developing flow) under uniform bottom surface temperature condition. A commercial CFD program Ansys Fluent 12.1 with different turbulent models was used to carry out the numerical study. Based on the present experimental data and three-dimensional numerical solutions, new engineering correlations were presented for the heat transfer and friction coefficients in the form of $ {\text{Nu}} = {\text{C}}_{2} {\text{Re}}^{{{\text{n}}_{ 1} }} $ and $ {\text{f}} = {\text{C}}_{3} {\text{Re}}^{{{\text{n}}_{3} }} $ , respectively. The results have shown that as the Reynolds number increases heat transfer coefficient increases but Darcy friction factor decreases. It is seen that there is a good agreement between the present experimental and numerical results. Examination of heat and mass transfer in rectangular cross-sectioned duct for different duct aspect ratio (α) was also carried out in this study. Average Nusselt number and average Darcy friction factor were expressed with graphics and correlations for different duct aspect ratios.     

Fluid flow and heat transfer in microchannels with rectangular cross section

Forced convective heat transfer coefficients and friction factors for flow of water in microchannels with a rectangular cross section were measured. An integrated microsystem consisting of five microchannels on one side and a localized heater and seven polysilicon temperature sensors along the selected channels on the other side was fabricated using a double-polished-prime silicon wafer. For the microchannels tested, the friction factor constant obtained are values between 53.7 and 60.4, which are close to the theoretical value from a correlation for macroscopic dimension, 56.9 for D _h  = 100 μm. The heat transfer coefficients obtained by measuring the wall temperature along the micro channels were linearly dependent on the wall temperature, in turn, the heat transfer mechanism is strongly dependent on the fluid properties such as viscosity. The measured Nusselt number in the laminar flow regime tested could be correlated by which is quite different from the constant value obtained in macrochannels.     

Heat transfer characteristics from in-line arrays of free impinging jets

An experimental investigation of the convective heat transfer on a flat surface in a multiple-jet system is described. A thin metal sheet was heated electrically and cooled from one side. On the other black coated side the temperature field was measured using an IR camera. Varied parameters were the jet Reynolds number in the range from 1,400 to 41,400, the normalized distance nozzle to sheet H/d from 1 to 10, and the normalized nozzle spacing S/d from 2 to 10. A geometrical arrangement of nine nozzle in-line arrays was tested. The results show that the multiple-jet system enhances the local and average heat transfer in comparison with that of a single nozzle. A maximum of the heat transfer was found for the normalized spacing S/d = 6.0. The normalized distance H/d has nearly no effect on the heat transfer in the range 2 ≤ H/d ≤ 4. The maximum average Nusselt number was correlated as a function of the jet Reynolds number      

Forced convection heat transfer and friction characteristics in multi-exit -rectangular package with inclined tube bundles

An experiment was carried out to investigate the characteristics of the heat transfer and pressure drop for forced convection airflow over tube bundles that are inclined relative to the on-coming flow in a rectangular package with one outlet and two inlets. The experiments included a wide range of angles of attack and were extended over a Reynolds number range from about 250 to 12,500. Correlations for the Nusselt number and pressure drop factor are reported and discussed. As a result, it was found that at a fixed Re, for the tube bundles with attack angle of 45 ° has the best heat transfer coefficient, followed by 60, 75 and 90 °, respectively. This investigation also introduces the factors which can be used for finding the heat transfer and the pressure drop factor on the tube bundles positioned at different angles to the flow direction. Moreover, no perceptible dependence of Cand C on Re was detected. In addition, flow visualizations were explored to broaden our fundamental understanding of the heat transfer for the present study.     

Natural convective heat transfer in water enclosed between pairs of differentially heated vertical plates

This paper presents the results of an experimental study of buoyancy-driven convective heat transfer between three parallel vertical plates, symmetrically spaced with water as the intervening medium. The centre plate was electrically heated, while the other side plates were water-cooled forming two successive parallel vertical channels of dimensions 20 cm × 3.5 cm × 35 cm (length W, gap L, height H) each. Top, bottom and sides of the channels were open to water in the chamber which is the novel aspect of this study. Plate surface temperature and bath temperature at different levels of height from the bottom of channel were measured by K-type thermocouples. Experimental data have been correlated as under:

A experimental investigation of turbulent flow and heat transfer in elliptical duets

Fully developed turbulent flow and heat transfer to air and water in ducts of elliptical cross section have been investigated experimentally. For the ducts of aspect ratio 2.5 1 and larger, a reduction in the overall heat transfer rate was found in the lower turbulent Reynold's number range (Re<25,000). Similar effects have been noted by investigators of narrow triangular cross sections where flow measurements indicated the possible co-existence of laminar and turbulent flow resulting in localised increases in thermal resistance. It was found that the analogy between momentum and heat transfer could not be applied directly to the larger aspect ratio ducts where significant circumferential variations of wall temperature occurred.

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Zusammenfassung Voll entwickelte turbulente Strömung und Wärmeübertragung an Luft und Wasser in elliptischen Kanälen wurden experimentell untersucht. Für Kanäle mit Achsenverhältnissen von 2,5 1 und größer fand man eine Verringerung des Wärmedurchgangs im Bereich geringer Reynolds-Zahlen (Re < 25 000). Ähnliche Effekte waren von anderen Autoren in engen Dreieckskanälen gefunden worden, wobei man aus Strömungsmessungen das gleichzeitige Auftreten von laminarer und turbulenter Strömung mit örtlicher Zunahme des thermischen Widerstandes folgern konnte. Die Analogie zwischen Impuls- und Wärmeübertragung konnte nicht unmittelbar auf Kanäle mit großem Achsenverhältnis, bei denen die Umfangstemperatur beträchtlich variierte, angewendet werden.

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Nomenclature A cross-sectional area - b duct wall thickness - C_p specific heat at constant pressure - d_e equivalent diameter of noncircular cross-section (=4A/p) - f Fanning friction coefficient - h local heat transfer coefficient (=q_w/(T_w-T_b)) - ¯h average circumferential heat transfer coefficient - k thermal conductivity of fluid - k_w thermal conductivity of wall material - K^* wall conductivity parameter (= k_wb/kd_e) - p wetted perimeter - q_w wall heat flux - T_b bulk fluid temperature - T_w local wall temperature - absolute viscosity - kinematic viscosity (=/) - mass density - Nu Nusselt number (= h d_e/k) - Nu average circumferential Nusselt number (= ¯h d_e/k) - Pr Prandtl number (= C_p/k) - Re Reynolds number (= d_e/) - St Stanton number (= Nu/Re · Pr)     

Momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel

Unsteady momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel is numerically investigated using FLUENT for the ranges of Reynolds numbers as 10≤Re≤500, of the blockage ratio as 0.1≤β≤0.4, and of the gap ratio as 0.125≤γ≤1 for a constant value of the Prandtl number of 0.744. The transition of the flow from steady to unsteady (characterized by critical Re) is determined as a function of γ and β. The effect of γ on the mean drag and lift coefficients, Strouhal number (St), and Nusselt number (Nu _w ) is studied. Critical Re was found to increase with decreasing γ for all values of β. and St were found to increase with decreasing values of γ for fixed β and Re. The effect of decrease in γ on was found to be negligible for all blockage ratios investigated.     

Frictional heating effect on heat transfer in forced convective flows

An analytical — experimental investigation into the effect of frictional heating on heat transfer during a forced convective laminar flow over a plate has been undertaken. The temperature distribution in laminar flow past a flat plate is obtained as the superposition of the separate distributions obtained for flow past a heated flat plate with frictional effects neglected and for flow past an adiabatic wall with the presence of frictional heat generation. For the case of Prandtl number around unity analytical calculations confirmed by experimental results show that frictional heating of the fluid adjacent to the wall results in fluid temperatures in excess of the wall temperature for values of Eckert number greater than about 2.

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Der Effekt der Reibungswärme auf die Wärmeübertragung in Strömungen mit erzwungener Konvektion

Zusammenfassung Es wurde eine analytisch-experimentelle Untersuchung des Effekts der Reibungswärme auf die Wärmeübertragung einer laminaren Strömung über einer Platte mit erzwungener Konvektion durchgeführt. Die Temperaturverteilung in laminarer Strömung über einer flachen Platte wurde als eine Superposition der einzelnen Verteilungen, die sich aus einer Strömung über eine geheizte Platte mit vernachlässigten Reibungseffekten und aus einer Strömung über eine adiabate Wand mit Reibungswärmeentwicklung zusammensetzt erhalten. Für Prandtl-Zahlen von ungefähr eins zeigen analytische Berechnungen, bestätigt durch experimentelle Ergebnisse, daß die Reibungswärme des Fluids hervorgerufen durch die angrenzende Wand für Eckert-Zahlen größer als zwei zu Fluidtemperaturen führt, die über der Wandtemperatur liegen.

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Nomenclature L characteristic length of the plate, m - P pressur, Pa. - T temperature, K - K thermal conductivity, W/mk - V local velocity, m/s - thermal diffusicity, m²/s - Pr Prandtl number - Cp specific heat, Joule/kg - Ec - dynamic viscosity, kg/ms - kinematic viscosity, m²/s - density, kg/m³ - Nu - h surface heat transfer coefficient - Tw Wall temperature, K - Taw adiabatic wall temperature, K - Re - T temperature of undisturbed stream flow - V velocity of undisturbed stream flow - V _x ,V _y velocity components along thex-andy-axis - - stream function as employed in the Blasius solution for velocity field [2] - f() - f,f first and second derivatives off with respect to     

DNS Study of the Optimal Chemical Markers for Heat Release in Syngas Flames

The purpose of this study is to identify a quantitative marker of the heat release rate (HRR) distribution using experimentally measurable species. Turbulent syngas (CO/H₂/air) flames with different equivalence ratios, H₂/CO ratios, and turbulence intensities are computed by Direct Numerical Simulations (DNS) in order to obtain an indirect but accurate estimation of heat release profiles. To check the robustness of the estimation, two different kinetic mechanisms have been considered. Based on a direct image analysis of the DNS results, normalized species concentrations combined with exponents are systematically tested in an attempt to reconstruct as accurately as possible the field of heat release rate. A systematic comparison is used to identify the best possible exponents associated with each species combination. Differing from previous studies, the present analysis takes into account the local thickness of the turbulent heat release zone. As a consequence, the obtained optimal species combinations represent not only the position of peak heat release but also local changes in the topology of the reaction zone (thickness, curvature). In the end, the heat release rate of atmospheric syngas flames can, in general, be best approximated using the concentrations of HCO and OH, using \(\overline {c}_{HCO}^{1.5}\times \overline {c}_{OH}^{0.75}\), when considering only species that are measurable by Laser-Induced Fluorescence. Another excellent reconstruction would be \(\overline {c}_{CH_{2}O}^{0.32}\times \overline {c}_{OH}^{0.8}\), for cases where CH₂O is preferred to HCO.     

Transients in flow and local heat transfer due to a pressure wave in pipe flow

The effect of a pressure wave on the turbulent flow and heat transfer in a rectangular air flow channel has been experimentally studied for fast transients, occurring due to a sudden increase of the main flow by an injection of air through the wall. A fast response measuring technique using a hot film sensor for the heat flux, a hot wire for the velocities and a pressure transducer have been developed. It was found that in the initial part of the transient the heat transfer change is independent of the Reynolds number. For the second part the change in heat transfer depends on thermal boundary layer thickness and thus on the Reynolds number. Results have been compared with a simple numerical turbulent flow and heat transfer model. The main effect on the flow could be well predicted. For the heat transfer a deviation in the initial part of the transient heat transfer has been found. From the turbulence measurements it has been found that a pressure wave does not influence the absolute value of the local turbulent velocity fluctuations. They could be considered to be frozen.Nomenclature A surface area (m²) - D diameter (m) - h heat transfer coefficient (Wm^–2 K^–1) - p pressure drop (Pa) - P pressure (Pa) - Q heat flow (W) - R tube radius (m) - T bulk temperature (K) - T _s surface temperature (K) - t time (s) - u velocity (m/s) - V voltage (V) - y distance from wall (m) - viscosity (N s m^–2) - kinematic viscosity (m^–2 s^–1) - density (kg m^–3) - _w wall shear stress (N m^–2) - Nu Nusselt number - Re Reynolds number     

Nonparabolic Asymptotic Limits of Solutions of the Heat Equation on $${\mathbb{R}}^N$$

In this paper, we construct solutions u(t,x) of the heat equation on such that has nontrivial limit points in as t → ∞ for certain values of μ > 0 and β > 1/2. We also show the existence of solutions of this type for nonlinear heat equations.      

Heat transfer from a circular cylinder rotating about an orthogonal axis in quiescent air

Direct measurements of local heat flux and temperature from rotating cylinders have been carried out using Gardon type foil heat flux sensors and a power supply cum instrumentation slip ring set up. The local and average heat transfer results are presented covering a rotational Reynolds number range of 2 × 10⁴ to 6.2 x 10⁴ corresponding to the speeds varying from 400 to 1,400 rpm. A correlation has been derived for peripherally averaged values of Nusselt numbers: . The values of surface average Nusselt number for the cylinder under the present rotating conditions are found to be higher than for a stationary cylinder in crossflow and for a cylinder rotating about its own axis, in the range of present experiments.Research scholar on leave from Faculty of Engineering, Port Said, Egypt     

Predictions on momentum,heat and mass transfer in turbulent channel flow with the aid of a boundary layer growth-breakdown model

This paper deals with theoretical aspects of momentum, heat and mass transfer in turbulent channel flow and in particular with phenomena occurring close to the wall. The analysis presented involves the use of a boundary-layer growth-breakdown model. Theoretical expressions have been derived predicting heat and mass transfer at smooth surfaces in the fully developed and entrance region and at surfaces provided with ideal two-dimensional roughness elements. The analysis is restricted to fluids having Prandtl and Schmidt numbers larger than one. Good agreement appears to exist between theoretical predictions and experimental observations.

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Zusammenfassung Diese Arbeit behandelt die Theorie der Übertragungsvorgänge von Impuls, Wärme und Stoff in turbulenter Kanalströmung unter besonderer Berücksichtigung der Vorgänge in Wandnähe. Das verwendete Modell beruht auf dem Zusammenbruch der anwachsenden Grenzschicht. Für die ausgebildete Strömung und für den Einlaufbereich bei glatter Wand und bei Oberflächen mit idealen zweidimensionalen Rauhigkeitselementen werden theoretische Ausdrücke abgeleitet bei Beschränkung auf Prandtl- und Schmidt-Zahlen über Eins. Zwischen den theoretischen Voraussagen und den Versuchsergebnissen scheint gute Übereinstimmung zu herrschen.

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Nomenclature a thermal diffusivity [m²/s] - c concentration [kg/m³] - c _p specific heat [J/kg °C] - D molecular diffusivity [m²/s] - G relative increase in friction factor due to surface roughening - d pipe diameter [m] - e height (depth) of roughness element [m] - e ^p+ dimensionless roughness height (depth) - F parameter denoting the ratio - f friction factor for smooth surface and isothermal conditions - f _h friction factor for heating conditions - f _r friction factor for artificially roughened surface - n _av average frequency of fluctuations at the wall [s^–1] - q heat flux [W/m₂] - q _w heat flux at the wall [W/m²] - q _wr heat flux at roughened wall [W/m²] - q _wx wall heat flux to growing laminar boundary layer at positionx [W/m²] - R _ma longitudinal correlation coefficient for mass transfer - R _mo longitudinal correlation coefficient for momentum transfer - T temperature [°C] - T _b bulk temperature of fluid [°C] - T ₀ fluid temperature at edge of viscous boundary layer (edge of viscous region) [°C] - T _w wall temperature [°C] - T _wx wall temperature at positionx for growing laminar boundary layer [°C] - t time [s] - t ₀ characteristic time period associated with boundary layer growth [s] - u local axial fluid velocity, at wall distancey, for turbulent flow also denoting the mean velocity at that distance [m/s] - u _b bulk fluid velocity [m/s] - u ₀ fluid velocity at edge of viscous boundary layer (edge of viscous region) [m/s] - u _0r fluid velocity at edge of viscous region for the case of an artificially roughened wall [m/s] - u axial fluid velocity fluctuation [m/s] - u ⁺ dimengionless fluid velocity,u/(_w/)^1/2 - u _i ⁺ instantaneous value ofu ⁺ - u _min ⁺ minimum value ofu _i ⁺ - u _r ⁺ root mean square value of dimensionless axial velocity - u ₀ ⁺ value ofu ⁺ at edge of viscous region - v fluid velocity normal to flow direction and normal to wall [m/s] - v fluctuation of the velocityv [m/s] - x coordinate in flow direction [m] - x axial distance interval [m] - x ⁺ dimensionless distance interval - x ₀ viscous boundary layer growth length [m] - x ₀ ⁺ dimensionless boundary growth length - x _r axial dixtance between roughness elements [m] - x _r ⁺ dimensionless distance between roughness elements - x _h value of viscous boundary growth length for heating conditions [m] - y distance from wall [m] - y ⁺ dimensionless wall distance - y _v thickness of viscous region [m] - y _v ⁺ dimensionless form ofy _v - z _u unheated (zero mass transfer) part of elementary viscous boundary layer in entrance region [m] - z _h heated (mass transfer) part of elementary viscous boundary layer [m] - z _v lateral extent of elementary viscous boundary layer [m] Greek symbols heat transfer coefficient defined with respect to bulk fluid temperature [W/m² °C] - ₀ viscous region heat transfer coefficient [W/m² °C] - _0h viscous boundary layer heat transfer coefficient averaged over lengthx ₀ for conditions of heating [W/m² °C] - _0hh viscous region heat transfer coefficient averaged over lengthx _h for conditions of heating [W/m² °C] - entrance region heat transfer coefficient at position [W/m² °C - _,t viscous boundary layer heat transfer coefficient at position and timet [W/m² °C] - mass transfer coefficient [m/s] - _av average value of mass transfer coefficient [m/s] - _x mass transfer coefficient for viscous boundary layer at positionx [m/s] - entrance region mass transfer coefficient at position [m/s] - thickness of laminar (viscous) boundary layer evaluated atu=1/2u ₀ [m] - _max maximum value of boundary layer thickness [m] - _i turbulent diffusivity for momentum transfer [m²/s] - _h turbulent diffusivity for heat transfer [m²/s] - _m turbulent diffusivity for mass transfer [m²/s] - turbulent intensity - thermal conductivity [W/m °C] - kinematic viscosity [m²/s] - ₀ value ofv at edge of viscous region [m²/s] - _w value ofv at the wall [m²/s] - density [kg/m³] - shear stress [N/m²] - _tx local value of wall shear stress associated with viscous boundary layer growth [N/m²] - ₀ value of wall shear stress averaged over lengthx ₀ [N/m²] - _0r value of ₀ for the case of an artificially roughened wall [N/m²] - _0h value of ₀ for heating conditions [N/m²] - _h value of wall shear stress for heating conditions, averaged over lengthx _h [N/m²] - _w wall shear stress for conditions of turbulent flow [N/m²] - _wh value of _w for heating conditions [N/m²] - dimensionless axial distancex/x ₀ in extrance region Dimensionless numbers Nu Nusselt number (d/) - Nu _x Entrance region Nusselt number at axial positionx - Nu _h Nusselt number for heating conditions - Nu _r Nusselt number for the case of artificially roughened surface - Pr Prandtl number (v/a) - Re Reynolds number (d u _b/v) - Re _b Boundary layer Reynolds number (1/2 u ₀/v) - Re _ber Critical value ofRe _b - Sh Sherwood number (d/D) - Sh _x entrance region Sherwood number at axial positionx - Sc Schmidt number (v/D)     

Turbulent heat transfer in circular tube flows of viscoelastic fluids

The effects of thermal entrance length, polymer degradation and solvent chemistry were found to be critically important in the determination of the drag and heat transfer behavior of viscoelastic fluids in turbulent pipe flow. The minimum heat transfer asymptotic values in the thermally developing and in the fully developed regions were experimentally determined for relatively high concentration solutions of heat transfer resulting in the following correlations: $$\begin{gathered} j_H = 0.13\left( {\frac{x}{d}} \right)^{ - 0.24} \operatorname{Re} _a^{ - 0.45} thermally developing region \hfill \\ x/d< 450 \hfill \\ j_H = 0.03 \operatorname{Re} _a^{ - 0.45} thermally developed region \hfill \\ x/d< 450 \hfill \\ \end{gathered} $$ For dilute polymer solutions the heat transfer is a function ofx/d, the Reynolds number and the polymer concentration. The Reynolds analogy between momentum and heat transfer which has been widely used in the literature for Newtonian fluids is found not to apply in the case of drag-reducing viscoelastic fluids.     

Measurements of Transitional and Turbulent Flow in a Randomly Packed Bed of Spheres with Particle Image Velocimetry

Particle image velocimetry (PIV) has been used to investigate transitional and turbulent flow in a randomly packed bed of mono-sized transparent spheres at particle Reynolds number, \(20<{{ Re}}_{\mathrm{p}}< 3220\). The refractive index of the liquid is matched with the spheres to provide optical access to the flow within the bed without distortions. Integrated pressure drop data yield that Darcy law is valid at \({{ Re}}_{\mathrm{p}} \approx 80\). The PIV measurements show that the velocity fluctuations increase and that the time-averaged velocity distribution start to change at lower \({{ Re}}_{\mathrm{p}}\). The probability for relatively low and high velocities decreases with \({{ Re}}_{\mathrm{p}}\) and recirculation zones that appear in inertia dominated flows are suppressed by the turbulent flow at higher \({{ Re}}_{\mathrm{p}}\). Hence there is a maximum of recirculation at about \({{ Re}}_{\mathrm{p}} \approx 400\). Finally, statistical analysis of the spatial distribution of time-averaged velocities shows that the velocity distribution is clearly and weakly self-similar with respect to \({{ Re}}_{\mathrm{p}}\) for turbulent and laminar flow, respectively.     

The pulsed-wire anemometer

The principles and practice of pulsed-wire anemometry are reviewed. Flow velocity is deduced from the time taken for the thermal wake of a thin wire, heated by a short pulse of current, to reach a sensor wire operating as a resistance thermometer. The advantage over the hot-wire anemometer is that reversed flows can be measured by adding a second sensor wire on the upstream side of the pulsed wire: the main advantages over the laser Doppler anemometer are cheapness and simplicity of use. The pulsed-wire anemometer can now be regarded as a cost-effective instrument for measurements in turbulent separated flows.     

Natural convection heat transfer along vertical rectangular ducts

Experimental investigations have been reported on steady state natural convection from the outer surface of vertical rectangular and square ducts in air. Seven ducts have been used; three of them have a rectangular cross section and the rest have square cross section. The ducts are heated using internal constant heat flux heating elements. The temperatures along the vertical surface and the peripheral directions of the duct wall are measured. Axial (perimeter averaged) heat transfer coefficients along the side of each duct are obtained for laminar and transition to turbulent regimes of natural convection heat transfer. Axial (perimeter averaged) Nusselt numbers are evaluated and correlated using the modified Rayleigh numbers for laminar and transition regime using the vertical axial distance as a characteristic length. Critical values of the modified Rayleigh numbers are obtained for transition to turbulent. Furthermore, total overall averaged Nusselt numbers are correlated with the modified Rayleigh numbers and the area ratio for the laminar regimes. The local axial (perimeter averaged) heat transfer coefficients are observed to decrease in the laminar region and increase in the transition region. Laminar regimes are obtained at the lower half of the ducts and its chance to appear decreases as the heat flux increases.     

The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model of 3D Turbulent Flow

We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in which hyperviscous terms are added spectrally to the standard incompressible Navier–Stokes equations (NSE). Let P _m be the projection onto the first m eigenspaces of A =−Δ, let μ and α be positive constants with α ≥3/2, and let Q _m =I − P _m , then we add to the NSE operators μ A _φ in a general family such that A _φ≥Q _m A ^α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers past a cutoff λ _m0 where m ₀ ≤ m, so that for large enough m ₀ the inertial-range wavenumbers see only standard NSE viscosity. We first obtain estimates on the Hausdorff and fractal dimensions of the attractor (respectively and ). For a constant K _α on the order of unity we show if μ ≥ ν that and if μ ≤ ν that where ν is the standard viscosity coefficient, l ₀ = λ₁^−1/2 represents characteristic macroscopic length, and is the Kolmogorov length scale, i.e. where is Kolmogorov’s mean rate of dissipation of energy in turbulent flow. All bracketed constants and K _α are dimensionless and scale-invariant. The estimate grows in m due to the term λ _m /λ₁ but at a rate lower than m ^3/5, and the estimate grows in μ as the relative size of ν to μ. The exponent on is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition , the estimates become for μ ≥ ν and for μ ≤ ν. This result holds independently of α, with K _α and c _α independent of m. In an SVV example μ ≥ ν, and for μ ≤ ν aspects of SEV theory and observation suggest setting for 1/c within α orders of magnitude of unity, giving the estimate where c _α is within an order of magnitude of unity. These choices give straight-up or nearly straight-up agreement with the Landau–Lifschitz predictions for the number of degrees of freedom in 3D turbulent flow with m so large that (e.g. in the distinguished-class case for m ₀ large enough) we would expect our solutions to be very good if not virtually indistinguishable approximants to standard NSE solutions. We would expect lower choices of λ _m (e.g. with a > 1) to still give good NSE approximation with lower powers on l ₀/l _ε, showing the potential of the model to reduce the number of degrees of freedom needed in practical simulations. For the choice , motivated by the Chapman–Enskog expansion in the case m = 0, the condition becomes , giving agreement with Landau–Lifschitz for smaller values of λ _m then as above but still large enough to suggest good NSE approximation. Our final results establish the existence of a inertial manifold for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such an of dimension N > m for the general class of operators A _φ if α > 5/2. The special class of A _φ such that P _m A _φ = 0 and Q _m A _φ ≥ Q _m A ^α has a unique spectral-gap property which we can use whenever α ≥ 3/2 to show that we have an inertial manifold of dimension m if m is large enough. As a corollary, for most of the cases of the operators A _φ in the distinguished-class case that we expect will be typically used in practice we also obtain an , now of dimension m ₀ for m ₀ large enough, though under conditions requiring generally larger m ₀ than the m in the special class. In both cases, for large enough m (respectively m ₀), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics, and in particular trajectories on are controlled by essentially NSE dynamics.      

The Semigeostrophic Equations Discretized in Reference and Dual Variables

We study the evolution of a system of n particles in . That system is a conservative system with a Hamiltonian of the form , where W ₂ is the Wasserstein distance and μ is a discrete measure concentrated on the set . Typically, μ(0) is a discrete measure approximating an initial L ^∞ density and can be chosen randomly. When d  =  1, our results prove convergence of the discrete system to a variant of the semigeostrophic equations. We obtain that the limiting densities are absolutely continuous with respect to the Lebesgue measure. When converges to a measure concentrated on a special d–dimensional set, we obtain the Vlasov–Monge–Ampère (VMA) system. When, d = 1 the VMA system coincides with the standard Vlasov–Poisson system.