Heat transfer and friction in turbulent vortex flow

Summary This paper presents experimentally measured heat transfer and friction coefficients for air and water flowing through a pipe with several types of inserts designed to induce a swirl in the flow. It was observed that inside-surface heat transfer coefficients in swirling flow can, under favourable conditions, be at least four times as large as heat transfer coefficients at the same mass flow rate in purely axial flow. At the same time the pumping power per unit rate of heat transfer can be reduced. The increase in heat transfer coefficients was found to depend on the degree of swirl and on the density or temperature gradient. However, at comparable Reynolds numbers and swirling motions the heat transfer coefficients for air were found to be smaller than the coefficients for water. The reason for this difference is not definitely known, but the phenomenon is qualitatively compatible with that causing the cooling effect in Ranque-Hilsch vortex tubes. The observed phenomena are analyzed qualitatively and it is shown that they are primarily the result of a centrifugal force which induces a radial inward motion of warmer fluid and a radial outward motion of cooler fluid. The application of vortex flow to boiling heat transfer and other high heat flux systems is discussed briefly.

Nomenclature

Symbols c _p Specific heat at constant pressure, BTU/(lb)(deg F) - D _H Hydraulic diameter, (ft) - D Tube diameter, (ft) - f ₀ Fanning friction factor for axial flow, - f Fanning friction factor for swirling flow, - g Acceleration due to gravity, ft/(sec)² - G Mass velocity, lb/(sec) (sq ft) - h _i Inside surface coefficient of heat transfer, BTU/(hr)(sq ft)(deg F) - k Thermal conductivity, BTU/(hr)(sq ft)(deg F/ft) - L Characteristic length used in Grashof numbers, ft - p Frictional pressure drop in a duct, lbs/sq ft - r Radius of tube, ft - t Temperature potential in Grashof number, deg F - U _i Over-all coefficient of heat transfer based on inside tube area, BTU/(hr)(sq ft)(deg F) - V Axial velocity, ft/sec - Coefficient of thermal expansion, (deg F)^–1 - Absolute viscosity, (lbs)/(ft)(hr) - Density, lbs/(ft)³ - Angular velocity of fluid, rad/sec Dimensionless Parameters Nu ₀ Nusselt Number in axial flow, h _i D _H /k - Nu Nusselt Number in swirling flow, h _i D _H /k - Re Reynolds Number, VD _Hp / - Pr Prandtl Number, c _p /k - j Colburn j-Factor, (Nu/RePr)Pr ^2/3 Member of Technical Staff, Bell Telephone Laboratories, Murray Hill, N. J. formerly Baldwin Research Fellow, Lehigh University.     

Heat transfer and friction factor of turbulent flow through a horizontal semi-circular duct

Forced convection heat transfer and pressure drop characteristics of air flow inside a horizontal semi-circular duct are investigated experimentally. The experiments are carried out on a semi-circular duct of 23 mm inner radius, 2 mm thickness, and 2,000 mm length within a range of Reynolds number (8,242 ≤ Re ≤ 57,794)., under uniform wall heat flux conditions. The friction factor is determined by measuring the axial static pressure at different selected axial stations along the semi-circular duct. The variations of surface and mean air temperatures, local heat transfer coefficient, local Nusselt number, and the friction factor with the axial dimensionless distance are presented. It is observed that, for a given value of Reynolds number, each of the local heat transfer coefficient and the friction factor has a relatively high value near the entrance of the semi-circular duct then it decreases with increasing the dimensionless axial distance until it approaches a nearly constant value at the fully developed region. Also, it is found that, with increasing the Reynolds number the average heat transfer coefficient is increased and the friction factor is decreased. Moreover, empirical correlations for the heat transfer coefficient and friction factor as a function of the Reynolds number are obtained.     

A manufactured solution for a two-dimensional steady wall-bounded incompressible turbulent flow

This paper presents a manufactured solution (MS), resembling a two-dimensional, steady, wall-bounded, incompressible, turbulent flow for RANS codes verification. The specified flow field satisfies mass conservation, but requires additional source terms in the momentum equations. To also allow verification of the correct implementation of the turbulence models transport equations, the proposed MS exhibits most features of a true near-wall turbulent flow. The model is suited for testing six eddy-viscosity turbulence models: the one-equation models of Spalart and Allmaras and Menter; the standard two-equation k–ε model and the low-Reynolds version proposed by Chien; the TNT and BSL versions of the k–ω model.     

Multigrid solution of compressible turbulent flow on unstructured meshes using a two-equation model

The steady state solution of the system of equations consisting of the full Navier-Stokes equations and two turbulence equations has been obtained using a multigrid strategy on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multistage Runge-Kutta time-stepping scheme with a stability-bound local time step, while the turbulence equations are advanced in a point-implicit scheme with a time step which guarantees stability and positivity. Low-Reynolds-number modifications to the original two-equation model are incorporated in a manner which results in well-behaved equations for arbitrarily small wall distances. A variety of aerodynamic flows are solved, initializing all quantities with uniform freestream values. Rapid and uniform convergence rates for the flow and turbulence equations are observed.     

Calculation of turbulent friction on a plate with distributed injection of polymer solution into the boundary layer

An expression is proposed for the mixing length on a permeable surface with polymer injection. Relations between mixing length and injection, useful for calculating boundary layers into which dilute polymer solutions are injected, are proposed. The numerical solution is compared with the available experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 169–171, March–April, 1991.Tha author wishes to thank V. A. Ioselevich for his interest in the work and V. A. Aleksin for useful comments on the numerical method.     

Particle collisions in a turbulent flow

An equation for the two-point probability density function of the two-particle the coordinate and velocity distribution is obtained. A closed system of equations for the first and second two-point moments of the velocity fluctuations of a pair of particles with allowance for the turbulent flow inhomogeneity is given. Boundary conditions for the equations of the particle concentration and the intensity of the relative random velocity during particle collision are obtained. A unified formula describing the interparticle collision process as a result of turbulent motion and the average relative particle velocity slip is obtained for the kernel of the coagulation equation. The effect of the average velocity slip of the particles and the carrier phase on the parameters of motion of the dispersed admixture and its coagulation is investigated on the basis of a two-point two-time velocity fluctuation autocorrelation function with two time and space scales representing the energy-bearing and small-scale motion of the fluid phase.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–116, March–April, 1996.     

Analysis of an organized turbulent structure using a pattern recognition technique in a drag-reducing surfactant solution flow

This work presents the investigation for an organized turbulent structure in a drag-reducing flow of dilute surfactant solution by utilizing a particle image velocimetry system to perform the pattern recognition technique on a trajectory in four quadrants of streamwise and wall-normal velocity fluctuations. The pattern recognition is added to a new algorithm in order to directly capture the spatial rotation motion. The Reynolds number based on the channel height and bulk mean velocity was set to 1.5 × 10⁴. Surfactant solution with a weight concentration of 150 ppm was employed and the drag reduction rate was 65%. In the drag-reducing flow, we observe increased frequencies of occurrence of the flow events that correspond to a meandering motion in the wall-normal direction of the high-and low-speed regions. Three findings from investigation of the ensemble-averaged Reynolds shear stress and vortex structure are as follows: (i) the Reynolds shear stress in the large fluctuation range occurs in the narrow region; (ii) Size, strength, arrangement and inclination in the spatial vortex structure in the drag-reducing flow differ from those of the water; and (iii) all trajectory contributions for the wall friction coefficient decrease. Finally, we interpreted that the viscoelasticity characterizing the viscoelastic stress and relaxation time in rheological properties of the flow changes specific elementary vortex for the drag-reducing flow, and the trajectories of each flow pattern change drastically.     

An improved EMMS model for turbulent flow in pipe and its solution

Based on the existing energy-minimization multi-scale (EMMS) model for turbulent flow in pipe, an improved version is proposed, in which not only a new radial velocity distribution is introduced but also the quantification of total dissipation over the cross-section of pipe is improved for the dominant mechanism of fully turbulent flow in pipe. Then four dynamic equality constraints and some other constraints are constructed but there are five parameters involved, leading to one free variable left. Through the compromise in competition between dominant mechanisms for laminar and fully turbulent flow in pipe respectively, the above four constructed dynamic equality constraints can be closed. Finally, the cases for turbulent flow in pipe with low, moderate and high Reynolds number are simulated by the improved EMMS model. The numerical results show that the model can obtain reasonable results which agree well with the data computed by the direct numerical simulation and those obtained by experiment. This illustrates that the improved EMMS model for turbulent flow in pipe is reasonable and the compromise in competition between dominant mechanisms is indeed a universal governing principle hidden in complex systems. Especially, one more EMMS model for a complex system is offered, promoting the further development of mesoscience.     

Hierarchical structures in a turbulent pipe flow

A hierarchical structure (HS) analysis (β-test and γ-test) is applied to a fully developed turbulent pipe flow. Velocity signals are measured at two cross sections in the pipe and at a series of radial locations from the pipe wall. Particular attention is paid to the variation of turbulent statistics at wall units 10<y⁺<3000. It is shown that at all locations the velocity fluctuations satisfy the She–Leveque hierarchical symmetry (Phys. Rev. Lett. 72 (1994) 336). The measured HS parameters, β and γ, are interpreted in terms of the variation of fluid structures. Intense anisotropic fluid structures generated near the wall appear to be more singular than the most intermittent structures in isotropic turbulence and appear to be more outstanding compared to the background fluctuations; this yields a more intermittent velocity signal with smaller γ and β. As turbulence migrates into the logarithmic region, small-scale motions are generated by an energy cascade and large-scale organized structures emerge which are also less singular than the most intermittent structures of isotropic turbulence. At the center, turbulence is nearly isotropic, and β and γ are close to the 1994 She–Leveque predictions. A transition is observed from the logarithmic region to the center in which γ drops and the large-scale organized structures break down. We speculate that it is due to the growing eddy viscosity effects of widely spread turbulent fluctuations in a similar way as in the breakdown of the Taylor vortices in a turbulent Couette–Taylor flow at high Reynolds numbers.     

Relative particle velocity in a turbulent flow

On the basis of a statistical approach using a probability density function for the coordinates of two particles in a turbulent flow, the parameters of the relative particle motion are investigated. For the functions describing particle entrainment in the turbulence, rigorous results are obtained using a 3D turbulence spectrum. A method of calculating the particle relative-velocity rate with account for particle trajectory correlation is presented. The effects of particle inertia and velocity slip on the parameters of the relative particle motion are studied. Simple approximating formulas for calculating the relative particle motion in a turbulent flow are proposed. The calculation results are compared with the data of direct numerical simulation of stochastic particle trajectories in an isotropic turbulent field.     

Velocity gradients in a turbulent jet flow

We present a selection of results from experiments on an air turbulent jet flow, which included measurements of all the three velocity components and their nine gradients with the emphasis on the properties of invariant quantities related to velocity gradients (enstrophy, dissipation, enstrophy generation, etc.). This has been achieved by a 21 hot wire probe (5 arrays x 4 wires and a cold wire), appropriate calibration unit and a 3-D calibration procedure [1]. A more detailed account on the results will be published elsewhere.     

Surface friction in a turbulent boundary layer with a positive pressure gradient

A dependence of the relative friction coefficient on the form parameter of the aerodynamic curvature is proposed on the basis of measurements of the tangential stress on the wall during the flow of a liquid in a diffusor using the electrodiffusion method. The intensity of fluctuations of the tangential stress at the wall and the instant of appearance of reverse flows are investigated. The results of measurement of the friction coefficient by the electrodiffusion method and Clauser method are compared.     

Numerical solution of 2D and 3D turbulent internal flow problems

The paper describes a method for solving numerically two-dimensional or axisymmetric, and three-dimensional turbulent internal flow problems. The method is based on an implicit upwinding relaxation scheme with an arbitrarily shaped conservative control volume. The compressible Reynolds-averaged Navier-Stokes equations are solved with a two-equation turbulence model. All these equations are expressed by using a non-orthogonal curvilinear co-ordinate system. The method is applied to study the compressible internal flow in modern power installations. It has been observed that predictions for two-dimensional and three-dimensional channels show very good agreement with experimental results.     

Modelling a recirculating density-driven turbulent flow

A mathematical model of turbulent density-driven flows is presented and is solved numerically. A form of the k–? turbulence model is used to characterize the turbulent transport, and both this non-linear model and a sediment transport equation are coupled with the mean-flow fluid motion equations. A partitioned, Newton–Raphson-based solution scheme is used to effect a solution. The model is applied to the study of flow through a circular secondary sedimentation basin.     

Particle deposition from a turbulent flow

The diffusion equations and boundary condition for particle deposition from a turbulent flow are obtained on the basis of the kinetic equation for the probability density of the particle distribution. This approach makes it possible to calculate the deposition fairly simply without introducing additional empirical information relating to the particles (empirical constants are needed only for calculating the characteristics of the turbulent carrier flow). Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 96–104, September–October, 1988.