A minimization problem related to the estimation of a lower bound for the temperature in a free boundary value problem related to the combustion of a solid

This paper investigates the least time τ_* of the first zero of the bounded solution to an initial boundary value problem for the heat equation. The heat equation is considered in the domain $$\left\{ {(x,t)| - \infty< x< s(t),0< t \leqslant T} \right\}$$ . The initial conditionu(x, 0)=φ(x) and the boundary conditionu _x (s(t),t)=?R are specified. Let τ=τ(φ,R, s) denote the first zero ofu onx=s(t), that is,u(s(τ), τ)=0. Let τ_*=min τ, where the minimum is taken over a class of functionss=s(t). The existence of τ_* is demonstrated, and a generalization of the problem is discussed.     

Statistical analysis of instantaneous turbulent heat transfer in circular pipe flows

Turbulent heat transfer in circular pipe flow with constant heat flux on the wall is investigated numerically via Large Eddy Simulations for frictional Reynolds number Re _τ  = 180 and for Prandtl numbers in the range 0.1 ≤ Pr ≤ 1.0. In our simulations we employ a second-order finite difference scheme, combined with a projection method for the pressure, on a collocated grid in cylindrical coordinates. The predicted statistical properties of the velocity and temperature fields show good agreement with available data from direct numerical simulations. Further, we study the local thermal flow structures for different Prandtl numbers. As expected, our simulations predict that by reducing the Prandtl number, the range of variations in the local heat transfer and the Nusselt number decrease. Moreover, the thermal flow structures smear in the flow and become larger in size with less sharpness, especially in the vicinity of the wall. In order to characterize the local instantaneous heat transfer, probability density functions (PDFs) for the instantaneous Nusselt number are derived for different Prandtl number. Also, it is shown that these PDFs are actually scaled by the square root of the Prandtl number, so that a single PDF can be employed for all Prandtl numbers. The curve fits of the PDFs are presented in two forms of log-normal and skewed Gaussian distributions.     

Conjugate heat transfer in the unbounded flow of a viscoelastic fluid past a sphere

This work addresses the conjugate heat transfer of a simplified PTT fluid flowing past an unbounded sphere in the Stokes regime (Re = 0.01). The problem is numerically solved with the finite-volume method assuming axisymmetry, absence of natural convection and constant physical properties. The sphere generates heat at a constant and uniform rate, and the analysis is conducted for a range of Deborah (0 ≤ De ≤ 100), Prandtl (10⁰ ≤ Pr ≤ 10⁵) and Brinkman (0 ≤ Br ≤ 100) numbers, in the presence or absence of thermal contact resistance at the solid–fluid interface and for different conductivity ratios (0.1 ≤ κ ≤ 10). The drag coefficient shows a monotonic decrease with De, whereas the normalized stresses on the sphere surface and in the wake first increase and then decrease with De. A negative wake was observed for the two solvent viscosity ratios tested (β = 0.1 and 0.5), being more intense for the more elastic fluid. In the absence of viscous dissipation, the average Nusselt number starts to decrease with De after an initial increase. Heat transfer enhancement relative to an equivalent Newtonian fluid was observed for the whole range of conditions tested. The dimensionless temperature of the sphere decreases and becomes more homogeneous when its thermal conductivity increases in relation to the conductivity of the fluid, although small changes are observed in the Nusselt number. The thermal contact resistance at the interface increases the average temperature of the sphere, without affecting significantly the shape of the temperature profiles inside the sphere. When viscous dissipation is considered, significant changes are observed in the heat transfer process as Br increases. Overall, a simplified PTT fluid can moderately enhance heat transfer compared to a Newtonian fluid, but increasing De does not necessarily improve heat exchange.     

Turbulent heat transfer characteristics of water flow in a rotating pipe

Large-Eddy-Simulation of turbulent heat transfer for water flow in rotating pipe is performed, for various rotation ratios (0 ≤ N ≤ 14). The value of the Reynolds number, based on the bulk velocity and pipe diameter, is Re = 5,500. The aim of this study is to examine the effect of the rotating pipe on the turbulent heat transfer for water flow, as well as the reliability of the LES approach for predicting turbulent heat transfer in water flow. Some predictions for the case of non-rotating pipe are compared to the available results of literature for validation. To depict the influence of the rotation ratio on turbulent heat transfer, many statistical quantities are analyzed (distributions of mean temperature, rms of fluctuating temperature, turbulent heat fluxes, higher-order statistics). Some contours of instantaneous temperature fluctuations are examined.     

Nusselt number and friction factor in thermally stratified turbulent channel flow under Non-Oberbeck–Boussinesq conditions

In stably stratified turbulence, computations under Oberbeck–Boussinesq (OB) hypothesis of temperature-independent fluid properties may lead to inaccurate representation of the flow field and to wrong estimates of momentum/heat transfer coefficients. This is clearly assessed here comparing direct numerical simulations of stratified turbulence under OB conditions to simulations under NOB (Non-Oberbeck–Boussinesq) conditions of temperature-dependent fluid viscosity and thermal expansion coefficient. Compared to the OB case, NOB conditions may induce local flow relaminarization with significant variations (up to 30%) of heat and momentum transfer coefficients. Together with DNS results, we propose a phenomenological model (based on turbulent bursts) for heat transfer prediction in stratified turbulence under OB and NOB conditions. Implications of NOB assumptions on mixing efficiency (i.e. flux Richardson number Ri_f) and turbulent Prandtl number (Pr_t) are also discussed. These results are of specific importance in RANS modelling, where the condition Pr_t = 1 is usually assumed (Reynolds analogy). Although this assumption is valid in some situations (i.e. boundary layer, pipe flow) there is uncertainty about its validity for stably-stratified turbulence. We demonstrate that this assumption is inaccurate when NOB effects become significant.     

Correlation of turbulent flow rate-pressure drop data for non-newtonian solutions and slurries in pipes

Isothermal and non-isothermal flow rate-pressure drop data in turbulent flow through smooth pipes have been obtained for non-Newtonian fluids, including aqueous solutions of polymers and aqueous suspensions of titanium dioxide. It has been found that the friction factor, f, is a function of a new form of Reynolds number, Re_B, based on the parameters A, x and w of Bowen's correlation, viz.

$\text{τ}{}_{\mathrm{w}}\text{D}{}^{\mathrm{x}}\text{=A}\text{u}{}^{\mathrm{w}}$

where τ_w is the wall shear strees, ?u the mean velocity, D the pipe diameter; A, x and w are experimentally derived parameters which characterise the fluid.     

Turbulent channel flow of dilute polymeric solutions: Drag reduction scaling and an eddy viscosity model

Direct numerical simulation of viscoelastic turbulent channel flows up to the maximum drag reduction (MDR) limit has been performed. The simulation results in turn have been used to develop relationships between the flow and fluid rheological parameters, i.e. maximum chain extensibility, Reynolds number, Re_τ, and Weissenberg number, We_τ and percent drag reduction (%DR) as well as the slope increment of the mean velocity profile. Moreover, based on the trends observed in the mean velocity profile and the overall momentum balance three different regimes of drag reduction (DR), namely, low drag reduction (LDR; 0 ≤ %DR ≤ 20), high drag reduction (HDR; 20 ≤ %DR ≤ 52) and MDR (52 ≤ %DR ≤ 74) have been identified and mathematical expressions for the eddy viscosity in these regimes are presented. It is found that both in LDR and HDR regimes the eddy viscosity varies with the distance from the channel wall. However, in the MDR regime the ratio of the eddy viscosity to the Newtonian one tends to a very small value around 0.1 within the channel. Based on these expressions a procedure that relies on the DNS predictions of the budgets of momentum and viscoelastic shear stress is developed for evaluating the mean velocity profile.     

Heat transfer and skin-friction in a turbulent boundary layer under a non-equilibrium longitudinal adverse pressure gradient

The experimental data on the effect of weak and moderate non-equilibrium adverse pressure gradients (APG) on the parameters of dynamic and thermal boundary layers are presented. The Reynolds number based on the momentum thickness at the beginning of the APG region was Re^** = 5500. The APG region was a slot channel with upper wall expansion angles from 0 to 14°. The profiles of the mean and fluctuation velocity components were measured using a single-component hot-wire anemometer. The friction coefficients were determined using two methods, namely, the indirect Clauser method and the direct method of weighting the lower wall region on a single-component strain-gage balance. The heat transfer coefficients were determined by a transient method using an IR camera. It is noticed that in the pressure gradient range realized the universal logarithmic region in the boundary layer profile is conserved. The values of the relative (divided by the parameters in zero gradient flow at the same value of Re^**) friction and heat transfer coefficients, together with the Reynolds analogy factor, are determined as functions of the longitudinal pressure gradient. The values of the relative friction coefficient reduced to c_f/c_f0 = 0.7 and those of the heat transfer to St/St₀ = 0.9. A maximum value of the Reynolds analogy factor (St/St₀)/(c_f/c_f0) = 1.16 was reached for the pressure gradient parameter β = 2.9.     

Effect of side ratio on fluid flow and heat transfer from rectangular cylinders using the PANS method

A computational study of heat transfer from rectangular cylinders is carried out. Rectangular cylinders are distinguished based on the ratio of the length of streamwise face to the height of the cross-stream face (side ratio, R). The simulations were performed to understand the heat transfer in a flow field comprising separation, reattachment, vortex shedding and stagnation. The Partially-Averaged Navier–Stokes (PANS) modeling approach is used to solve the turbulent flow physics associated and the wall resolve approach is used for the near wall treatment because of the flow separation involved. The simulations were performed using a finite volume based opensource software, OpenFOAM, at Reynolds number (Re) = 22,000 for rectangular cylinder at constant temperature kept in an air stream. Two critical side ratios were obtained, R = 0.62 and 3.0. At R = 0.62, the maximum value of the drag coefficient (C_d) = 2.681 was observed which gradually reduced by 54% at R = 4.0. The base pressure coefficient and global Nusselt number also attained the maximum value at R = 0.62 and from R = 2.5 to 3.0 a sharp discontinuous increase by 140% in the Strouhal number was observed. At R = 0.62, it was observed that the separated flow reattaches at the trailing edge after rolling over the side face and therefore increases the overall Nusselt number. The phase averaging was also performed to analyze the unsteady behavior of heat transfer.     

Flow instability in baffled channel flow

Flow instability in baffled channel flow, where thin baffles are mounted on both channel walls periodically in the direction of the main flow, has been numerically investigated. The geometry considered here can be regarded as a simple model for finned heat exchangers. The aim of this investigation is to understand how baffle interval (L) and Reynolds number (Re) influence the flow instability. With a fixed baffle length of one quarter of channel height (H), ratios of baffle interval to channel height (R_B = L/H) between 1 and 4 are considered. The critical Reynolds number of the primary instability, a Hopf bifurcation from steady flow to time-periodic flow, turned out to be minimum when R_B = 3.08. The friction factor (f) is strongly correlated with the critical Reynolds number for R_B ⩽ 2.5. For the particular cases of R_B = 1.456 and R_B = 1.0, we performed Floquet stability analysis in order to study the secondary instability through which time-periodic two-dimensional flow bifurcates into three-dimensional flow. The results obtained in this investigation are in good agreement with those computed from full simulations, and shed light on understanding and controlling flow characteristics in a finned heat exchanger, quite beneficial to its design.     

“Magnetic-ribs” in fully developed laminar liquid–metal channel flow

This paper documents the numerical investigation of the effects of non-uniform magnetic fields, i.e. magnetic-ribs, on a liquid–metal flowing through a two-dimensional channel. The magnetic ribs are physically represented by electric currents flowing underneath the channel walls. The Lorentz forces generated by the magnetic ribs alter the flow field and, as consequence, the convective heat transfer and wall shear stress. The dimensionless numbers characterizing a liquid–metal flow through a magnetic field are the Reynolds (Re) and the Stuart (N) numbers. The latter provides the ratio of the Lorentz forces and the inertial forces. A liquid–metal flow in a laminar regime has been simulated in the absence of a magnetic field (Re_H = 1000, N = 0), and in two different magnetic ribs configurations for increasing values of the Stuart number (Re_H = 1000, N equal to 0.5, 2 and 5). The analysis of the resulting velocity, temperature and force fields has revealed the heat transport phenomena governing these magneto-hydro-dynamic flows. Moreover, it has been noticed that, by increasing the strength of the magnetic field, the convective heat transfer increases with local Nusselt numbers that are as much 27.0% larger if compared to those evaluated in the absence of the magnetic field. Such a convective heat transfer enhancement has been obtained at expenses of the pressure drop, which increases more than twice with respect to the non-magnetic case.     

The Decay Rate and Higher Approximation of Mild Solutions to the Navier�CStokes Equations

In this paper, we consider v(t) = u(t) − e ^tΔ u ₀, where u(t) is the mild solution of the Navier–Stokes equations with the initial data u₀ ? L²(\mathbb Rⁿ)?Lⁿ(\mathbb Rⁿ){u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)} . We shall show that the L ² norm of D ^β v(t) decays like t^{-\frac |b|-1 2-\frac n4}{t^{-\frac {|\beta|-1} {2}-\frac n4}} for |β| ≥ 0. Moreover, we will find the asymptotic profile u ₁(t) such that the L ² norm of D ^β (v(t) − u ₁(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions.     

Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold

We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation u_t=D(u^m/m) = div (u^m-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb R^d{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold (\mathbb R^d,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb R^d{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.     

Particle behavior in a turbulent pipe flow with a flat bed

Particle behavior in a turbulent flow in a circular pipe with a bed height h = 0.5R is studied at Re_b = 40,000 and for two sizes of particles (5 μm and 50 μm) using large eddy simulation, one-way coupled with a Lagrangian particle tracking technique. Turbulent secondary flows are found within the pipe, with the curved upper wall affecting the secondary flow formation giving rise to a pair of large upper vortices above two smaller vortices close to the pipe floor. The behavior of the two sizes of particle is found to be quite different. The 50 μm particles deposit forming irregular elongated particle streaks close to the pipe floor, particularly at the center of the flow and the pipe corners due to the impact of the secondary flows. The deposition and resuspension rate of the 5 μm particles is high near the center of the floor and at the pipe corners, while values for the 50 μm particles are greatest near the corners. Near the curved upper wall of the pipe, the deposition rate of the 5 μm particles increases in moving from the wall center to the corners, and is greater than that for the larger particles due to the effects of the secondary flow. The maximum resuspension rate of the smaller particles occurs above the pipe corners, with the 50 μm particles showing their highest resuspension rate above and at the corners of the pipe.     

Turbulent Couette–Poiseuille flow with zero wall shear

A particular pressure-driven flow in a plane channel is considered, in which one of the walls moves with a constant speed that makes the mean shear rate and the friction at the moving wall vanish. The Reynolds number considered based on the friction velocity at the stationary wall (u_τ,S) and half the channel height (h) is Re_τ,S = 180. The resulting mean velocity increases monotonically from the stationary to the moving wall and exhibits a substantial logarithmic region. Conventional near-wall streaks are observed only near the stationary wall, whereas the turbulence in the vicinity of the shear-free moving wall is qualitatively different from typical near-wall turbulence. Large-scale-structures (LSS) dominate in the center region and their spanwise spacing increases almost linearly from about 2.3 to 4.2 channel half-heights at this Re_τ,S. The presence of LSS adds to the transport of turbulent kinetic energy from the core region towards the moving wall where the energy production is negligible. Energy is supplied to this particular flow only by the driving pressure gradient and the wall motion enhances this energy input from the mean flow. About half of the supplied mechanical energy is directly lost by viscous dissipation whereas the other half is first converted from mean-flow energy to turbulent kinetic energy and thereafter dissipated.     

Effect on the flow and heat transfer characteristics for sinusoidal pulsating laminar flow in a heated square cylinder

Two-dimensional numerical simulation is performed to understand the effect of flow pulsation on the flow and heat transfer from a heated square cylinder at Re = 100. Numerical calculations are carried out by using a finite volume method based on the pressure-implicit with splitting of operators algorithm in a collocated grid. The effects of flow pulsation amplitude (0.2 ≤ A ≤ 0.8) and frequency (0 ≤ f _p  ≤ 20 Hz) on the detailed kinematics of flow (streamlines, vorticity patterns), the macroscopic parameters (drag coefficient, vortex shedding frequency) and heat transfer enhancement are presented in detail. The Strouhal number of vortices shedding, drag coefficient for non-pulsating flow are compared with the previously published data, and good agreement is found. The lock-on phenomenon is observed for a square cylinder in the present flow pulsation. When the pulsating frequency is within the lock-on regime, time averaged drag coefficient and heat transfer from the square cylinder is substantially augmented, and when the pulsating frequency in about the natural vortex shedding frequency, the heat transfer is also substantially enhanced. In addition, the influence of the pulsating amplitude on the time averaged drag coefficient, heat transfer enhancement and lock-on occurrence is discussed in detail.     

Effect of double-diffusive convection with cross gradients on heat and mass transfer in a cubical enclosure with adiabatic cylindrical obstacles

We investigate natural convection driven by a horizontal temperature gradient and a vertical concentration gradient in fluid-filled enclosures with obstructions inside it. Within the domain, nine adiabatic and impermeable cylinders are placed, occupying 30% of the domain volume. The Boussinesq approximation is used to account for density variations within the fluid and the flow is fully resolved. The solutal Rayleigh number has been fixed at $R{a}_C={10}^6$ and the Prandtl number at $Pr=5.4$. The Lewis number has been varied in the range of 1 ≤ Le ≤ 100 and the buoyancy ratio in the range of 0.1 ≤ |N| ≤ 10. The rate of heat and mass transfer are compared to those found in single-scalar natural convection, i.e solely thermal or concentration driven convection. Besides, the obtained heat and mass transfer rate in the cylinder-packed enclosure have been compared to those found in a fluid-only domain. We observe that the addition of a destabilizing concentration gradient to a side-heated enclosure results in heat transfer enhancement, which decreases with Lewis number and thermal Rayleigh number. Similarly, the temperature gradient increases the mass transfer, especially at high Lewis numbers and lower concentration buoyancy force over its thermal counterpart. Although the presence of the cylindrical obstacles reduced the flow velocity, the mass transfer was enhanced at lower buoyancy ratio.     

Characterisation of gas-liquid two-phase flow in minichannels with co-flowing fluid injection inside the channel,part I: unified mapping of flow regimes

Present research highlights the potential of apparatuses with integrated minichannel packings to intensify gas-liquid-solid contacting. Especially an operation of these devices within the Taylor flow regime gained extraordinary attention due to its excellent heat and mass transfer and the segmented flow characteristics. However, criteria for flow regime transitions are mainly developed from water-similar fluids and are contradictory which hinders uniform flow regime prediction.This work presents a systematic analysis of adiabatic gas-liquid downflow in a square minichannel of 1.0 mm hydraulic diameter. In the mixing zone located within the flow channel, gas was injected into the co-flowing liquid by so-called capillary injectors with variable inner diameter (0.184, 0.317, 0.490 mm). Experiments were conducted using water, water-glycerol, and water-ethanol mixtures to cover a broad range of material properties. The gas and liquid superficial velocities were varied between 9.81·10^-4…2.72 m/s and 1.7·10⁻⁴…0.80 m/s, respectively. Taylor flow, Taylor-annular flow, annular flow, churn flow, and bubbly flow were observed. Using the Pi-theorem, 8 significant dimensionless groups dictating the flow transition were identified, namely u_{G, s}/u_{L, s}, Re_G, Re_L, We_G, We_L, Θ*, d_{In, CI}/d_h, and d_{Ou, CI}/d_h. Based on more than 1500 experimental data, criteria for the regime transitions of Taylor flow are provided. The derived flow regime map shows good agreement for all applied liquids and for the two larger injector geometries.     

Centreline velocity decay characterisation in low-velocity jets downstream from an extended conical diffuser

The centreline velocity decay of round airflow jets issuing from extended conical diffusers with length-to-diameter ratio 1.2≤L _t /d≤20 is studied for moderate bulk Reynolds numbers 1131≤Re _b ≤9054. The centreline velocity decay varies as a function of the initial conditions. The functional correlation between the centreline velocity decay coefficient and the initial centreline turbulence level observed on convergent nozzles (Malmström et al. in J. Fluid Mech. 246:363–377, 1997) breaks down as the initial centreline turbulence level exceeds 20 %. In addition, the centreline velocity decay coefficient expressed as function of the bulk velocity U _b decreases for U _b <3 m/s instead of initial mean velocity U ₀<6 m/s as reported for convergent nozzles (Malmström et al. in J. Fluid Mech. 246:363–377, 1997). The asymptotic values of the decay coefficient for U _b >3 m/s decrease linearly when expressed as function of the initial centreline turbulence intensity u ₀/U ₀. Studied flow and geometrical conditions are relevant to flow through the human upper airways.