Heat transfer modeling for turbulent shear flows on curved surfaces

This paper examines two models for two-dimensional curved flow heat transfer. Both models are based on the assumption of local-equilibrium turbulence. Yet, one model predicts an increase for the turbulent Prandtl number with convex curvature, while the other gives the opposite behavior. Through examination of the near wall temperature profiles, it is possible to identify the correct behavior for turbulent Prandtl number. It is found that the turbulent Prandtl number for curved flows increases with convex curvature. The discrepancy between the two models is traced to the modeling terms proposed for the pressure-temperature-gradient correlation and for wall corrections. While the introduction of the rapid component and the wall corrections in the modeling of the pressure-strain correlation do not affect the behavior of the resultant shear stress, these modeling terms in the pressure-temperature-gradient correlation cause the turbulent Prandtl number to decrease with convex curvature.

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Zusammenfassung In dieser Arbeit werden zwei Modelle für zweidimensionale Wärmeübertragung in gekrümmten Strömungen untersucht. Beide Modelle basieren auf der Annahme von lokalem Gleichgewicht innerhalb der turbulenten Strömung. Trotzdem resultiert das eine Modell in einer Erhöhung der turbulenten Prandtl-Zahl für konvexe Krümmungen, während das zweite Modell genau das Gegenteil vorhersagt. Es ist möglich, durch Untersuchung des Temperaturprofiles in Wandnähe das korrekte Verhalten der turbulenten Prandtl-Zahl zu identifizieren. Es stellt sich dabei heraus, daß die turbulente Prandtl-Zahl für gekrümmte Strömungen in konvexen Kurven zunimmt. Der Widerspruch zwischen den beiden Modellen läßt sich auf die Abbildungsausdrücke zurückführen, die für die Wechselbeziehungen zwischen den Druck- und Temperaturgradienten und die für die Wandkorrekturen angesetzt worden sind. Während die Einführung der schnellen Komponente und die Wandkorrekturen in der Darstellung der Wechselbeziehung zwischen Druck und Beanspruchung keinen Einfluß auf das Verhalten des resultierenden Schubdruckes haben, verursachen diese Darstellungsausdrücke für die Wechselbeziehung zwischen Druck- und Temperaturgradienten ein Abnehmen der turbulenten Prandtl-Zahl für konvexe Kurven.

     

On the calculation of heat transfer rates in fully turbulent wall flows

A wall function for heat transfer is derived from the heat and mass transfer laws developed by Kader and Yaglom for turbulent wall flows. The wall function is used as a component of a prediction procedure to compute heat transfer rates in boundary layers, pipes, and wall jets. The results are generally in good agreement with the experimental data, but under most conditions the new function gives only a relatively minor improvement over existing functions. However, significant improvement is obtained for very large molecular Prandtl numbers.     

On the curvature/buoyancy analogy for turbulent shear flows

Similarity in the near wall region of turbulent curved shear flows is examined. It is found that the normalized mean velocity is a function only of the dimensionless distance _c =z/L _c whereL _c is a corresponding Monin-Oboukhov length for curved shear flows. Again, the universal function is found to obey the log-linear law. Therefore, this result and the earlier derivation of So affirm that there is a very close analogy between the effects of streamline curvature and buoyancy for turbulent shear flows.

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Zusammenfassung Die Ähnlichkeitsverhältnisse in turbulenten, gekrümmten Strömungen mit Schubkräften wird für das Gebiet in der Nähe einer Wand untersucht. Es ergibt sich, daß die normalisierte mittlere Geschwindigkeit nur von der dimensionslosen Entfernung _c =z/L _c abhängt.L _c ist hierbei eine zugeordnete Monin-Oboukhov-Länge für gekrümmte Strömungen mit Schubkräften. Auch in diesem Falle geohorcht die allgemeine Gleichung dem logarithmisch-linearen Gesetz. Dieses Ergebnis und die frühere Ableitung von So bestätigen, daß eine ausgeprägte Analogie zwischen den Auswirkungen der Strömungslinienkrümmung und dem Auftrieb bei turbulenten Strömungen mit Schubkräften besteht.

     

Numerical modelling of bubbly flows with and without heat and mass transfer

In this paper, modelling gas–liquid bubbly flows is achieved by the introduction of a population balance equation combined with the three-dimensional two-fluid model. For gas–liquid bubbly flows without heat and mass transfer, an average bubble number density transport equation has been incorporated in the commercial code CFX5.7 to better describe the temporal and spatial evolution of the geometrical structure of the gas bubbles. The coalescence and breakage effects of the gas bubbles are modelled according to the coalescence by the random collisions driven by turbulence and wake entrainment while for bubble breakage by the impact of turbulent eddies. Local radial distributions of the void fraction, interfacial area concentration, bubble Sauter mean diameter, and gas and liquid velocities, are compared against experimental data in a vertical pipe flow. Satisfactory agreements for the local distributions are achieved between the predictions and measurements. For gas–liquid bubbly flows with heat and mass transfer, boiling flows at subcooled conditions are considered. Based on the formulation of the MUSIG (multiple-size-group) boiling model and a model considering the forces acting on departing bubbles at the heated surface implemented in the computer code CFX4.4, comparison of model predictions against local measurements is made for the void fraction, bubble Sauter mean diameter, interfacial area concentration, and gas and liquid velocities covering a range of different mass and heat fluxes and inlet subcooling temperatures. Good agreement is achieved with the local radial void fraction, bubble Sauter mean diameter, interfacial area concentration and liquid velocity profiles against measurements. However, significant weakness of the model is evidenced in the prediction of the vapour velocity. Work is in progress through the consideration of additional momentum equations or developing an algebraic slip model to account for the effects of bubble separation.     

On a similar solution for a turbulent curved jet

A theoretical investigation of the turbulent curved jet along a curved streamline is presented. Equations of motion referring to streamline co-ordinates are simplified by boundary-layer approximations and integrated under the assumption of similarity.The forms of zero-streamlines, velocity profiles and pressure distributions are calculated, and the effects of streamline curvatures are investigated.

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Zusammenfassung Eine theoretische Untersuchung eines turbulenten gekrümmten Freistrahles entlang einer gekrümmten Stromlinie wird gegeben. Die Bewegungsgleichungen werden in Stromlinienkoordinaten angegeben und durch die Grenzschichtvernachlässigungen vereinfacht; unter der Annahme von ähnlichen Lösungen werden die Gleichungen integriert.Die Formen der Null-Stromlinien, die Geschwindigkeitsverteilungen und die Druckverteilungen werden berechnet, und die Effekte der Stromlinienkrümmung untersucht.

     

Bubbling of the heat flows for harmonic maps from surfaces

In this article we prove that any Palais-Smale sequence of the energy functional on surfaces with uniformly L²-bounded tension fields converges pointwise, by taking a subsequence if necessary, to a map from connected (possibly singular) surfaces, which consist of the original surfaces and finitely many bubble trees. We therefore get the corresponding results about how the solutions of heat flow for harmonic maps from surfaces form singularities at infinite time. © 1997 John Wiley & Sons, Inc.     

On reducing Howard's semicircle for homogeneous shear flows

Howard's semicircle bound on the range of the complex wave velocity of an arbitrary unstable mode in the stability problem of homogeneous shear flows is further reduced. The reduction depends on the curvature of the velocity profile and the depth of the fluid layer.     

On a Morse conjecture for analytic flows on compact surfaces

The aim of this paper is to prove a Morse conjecture; in particular it is shown that a topologically transitive analytic flow on a compact surface is metrically transitive. We also build smooth topologically transitive flows on surfaces which are not metrically transitive.     

Boiling heat transfer on a sphere with turbulent vapour film

The theoretical study investigates turbulent film boiling on a sphere immersed in the stagnant liquid. It begins by assuming the surface temperature of the sphere is isothermal. The result shows that the boiling heat transfer under the turbulent vapour shows that both the temperature and the velocity present a non-linear distribution. Besides, the increase of the thermal radiation, wall temperature and buoyancy effects can enhance the heat transfer efficiency. Furthermore, a comparison between the results of the present study and those reported in a previous theoretical study of laminar film boiling is provided. It is found that turbulent film boiling with higher Rayleigh number will have higher Nusselt number.     

Investigation of structure and non-gradient turbulent transfer in swirling flows

Turbulence structure of swirling flows is being investigated in this paper, on the basis of experimental, self conducted measurements, and theoretical and numerical results. Turbulent swirling flows are extremely inhomogeneous, three dimensional and anisotropic. The aim of this paper is to investigate significant influence of swirl onto statistical parameters and non-gradient turbulent transfer. Contemporary optical measuring techniques two component laser Doppler anemometry (LDA) and stereo particle image velocimetry (SPIV) have been applied. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On free surfaces in hydrostatic flows

In the following note we investigate the occurrence of non-trivial free-surfaces for hydrostatic flows. We show that, for travelling-wave solutions of the full Euler equations where we invoke the hydrostatic approximation, no non-flat travelling wave flows occur unless the resulting flow is outside the domain of current local or global existence results for travelling water waves.     

Recurrent orbits for flows on surfaces

A flow on a surface M lifts to a flow on , the universal cover of M. For a compact surface of genus at least 2, is the interior of the unit disk and the covering transformations are a group of hyperbolic linear fractional transformations with two fixed points on the unit circle, K. The analysis in this paper will be based almost entirely on geometric properties of the lifted flow and homotopy properties of loops constructed from local cross sections and orbits. The orbits studied have at least one nonfixed point in their ω-limit set and have lifts to that limit to a point of K. The main results describe the structure of the ω-limit sets of such orbits. In Theorem 8 it is shown that all nonfixed ω-limit points are “isolated” if and only if the lifted orbit limits to a fixed point of a covering transformation, which is called a rational point. In Theorem 13 sufficient conditions on the ω-limit set are established for the existence of positively and negatively recurrent points in the omega limit set of an orbit. This result is then used in Section 6 to give new geometric proofs to two of A. Maier's classic results about the existence of positively recurrent orbits in omega limit sets.     

Direct test of a nonlinear constitutive equation for simple turbulent shear flows using DNS data

Several nonlinear constitutive equations have been proposed to overcome the limitations of the linear eddy-viscosity models to describe complex turbulent flows. These nonlinear equations have often been compared to experimental data through the outputs of numerical models. Here we perform a priori analysis of nonlinear eddy-viscosity models using direct numerical simulation (DNS) of simple shear flows. In this paper, the constitutive equation is directly checked using a tensor projection which involves several invariants of the flow. This provides a 3 terms development which is exact for 2D flows, and a best approximation for 3D flows. We provide the quadratic nonlinear constitutive equation for the near-wall region of simple shear flows using DNS data, and estimate their coefficients. We show that these coefficients have several common properties for the different simple shear flow databases considered. We also show that in the central region of pipe flows, where the shear rate is very small, the coefficients of the constitutive equation diverge, indicating the failure of this representation for vanishing shears.     

Large eddy simulation for turbulent magnetohydrodynamic flows

We investigate the mathematical properties of a model for the simulation of large eddies in turbulent, electrically conducting, viscous, incompressible flows. We prove existence and uniqueness of solutions for the simplest (zeroth) closed MHD model (1.7), we show that its solutions converge to the solution of the MHD equations as the averaging radii converge to zero, and derive a bound on the modeling error. Furthermore, we show that the model preserves the properties of the 3D MHD equations: the kinetic energy and the magnetic helicity are conserved, while the cross helicity is approximately conserved and converges to the cross helicity of the MHD equations, and the model is proven to preserve the Alfvén waves, with the velocity converging to that of the MHD, as δ₁,δ₂ tend to zero. We perform computational tests that verify the accuracy of the method and compare the conserved quantities of the model to those of the averaged MHD.     

Mathematical modelling of heat transfer in a catalytic reformer

In this work we analyse heat transfer in a system of channelsconnected by thin conducting walls. The channels are packedwith catalytic pellets that promote exothermic catalytic combustionreactions and endothermic reforming reactions in adjacent channels.A model is developed in which the thermal conductivity and thethickness of the interconnecting wall can be used as controlparameters characterizing the heat exchange between the neighbouringchannels. The model is to be used as a mathematical tool toanalyse design alternatives and develop accurate numerical techniques.Our objective is to study how the heat is transferred acrossthe conducting walls and how this influences the temperaturedistribution in the channels. We use an asymptotic techniqueto do this. The structure of the walls is then examined in detail,focusing on the case when we have layered walls.