Direct simulation of turbulent flow and heat transfer in a channel. Part I: Smooth walls

Interest in the use of supercomputers for the direct numerical calculation of turbulence prompts the development of efficient numerical techniques so that calculation at higher Reynolds numbers might be made. This paper presents an efficient pseudo-spectral technique, similar to but different from others that have recently appeared, for the calculation of momentum and heat transfer to a constant-property, turbulent fluid in a two-dimensional channel with walls at different, uniform temperature. The code uses no empiricism, although periodic boundary conditions are used for fluctuating quantities in the streamwise and spanwise directions. Calculations were made for a Prandtl number of 0·72 and Reynolds number based on friction velocity and channel half-height of 180 or 2800 based on channel half-height and average velocity. Calculations of mean velocity profile, turbulence intensities, skewness, flatness, Reynolds stress and eddy diffusivity of heat near a wall compare favourably with experimental results. Representative contour plots of the temperature field near the wall and of the spanwise and streamwise two-point velocity correlations are given. Deficiencies are that the calculation requires many hours on a fast computer with a large high-speed memory and that the grid size in each direction for appropriate resolution is approximately proportional to the square of the Reynolds number and to the Prandtl number raised to some power greater than one.     

Viscous incompressible flow in a narrow channel with one free wall and fluid supply through a porous insert

The problem investigated relates the plane unsteady flow of a viscous incompressible fluid in a narrow channel one of whose walls is free and acted upon by a given load, while the other is rigidly fixed. The fluid enters the channel through a porous insert in the stationary wall. A model of the flow of a thin film of viscous incompressible fluid and Darcy's law for flow in a porous medium are used to find the distribution of fluid pressure and velocity in the channel and the porous insert in the two-dimensional formulation for fairly general boundary conditions in the case where the length of the porous insert exceeds the length of the free wall. In the particular case where the length of the porous insert is equal to the length of the free wall an exact stationary solution of the problem is obtained for a given value of the channel height. The stability of the equilibrium position of the free wall supported on a hydrodynamic fluid film is examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–24, January–February, 1986.     

Effect of thermal dispersion on free convection in a fluid saturated porous medium

The present article considers a numerical study of thermal dispersion effect on the non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium. Forchheimer extension is considered in the flow equations. The coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. The non-dimensional governing equations are solved by the finite element method (FEM) with a Newton–Raphson solver. Numerical results for the details of the stream function, velocity and temperature contours and profiles as well as heat transfer rates in terms of Nusselt number are obtained. The study shows that the increase in thermal dispersion coefficient of the porous medium results in more heat energy to disperse away in the normal direction to the wall. This induces more fluid to flow along the wall, enhancing the heat transfer coefficient particularly near the wall.     

Combined influence of wall conductance,pressure work,viscous dissipation and joule heating on MHD channel flow heat transfer

To investigate the combined influence of viscous dissipation, pressure work, Joule heating, arbitrary voltage ratio, unequal wall conductances and wall heat fluxes on the fully developed laminar MHD channel flow heat transfer, the exact solution of the energy equations for fluid and channel walls are derived assuming the Hartmann velocity profile. It is concluded that there can be a substantial difference, depending upon Hartmann number, electric field and Brinkman number, between the Nusselt number considering the wall conductance and that neglecting it. Representative results are presented in diagrams.     

Heat transfer effects on the stability of low speed plane Couette-Poiseuille flow

The stability problem of low-speed plane Couette-Poiseuille flow of air under heat transfer effects is solved numerically using the linear stability theory. Stability equations obtained from two-dimensional equations of motion and their boundary conditions result in an eigenvalue problem that is solved using an efficient shoot-search technique. Variable fluid properties are accounted for both in the basic flow and the perturbation (stability) equations. A parametric study is performed in order to assess the roles of moving wall velocity and heat transfer. It is found that the moving wall velocity and the location of the critical layers play decisive roles in the instability mechanism. The flow becomes unconditionally stable whenever the moving wall velocity exceeds half of the maximum velocity in the channel. With wall heating and Mach number effects included, the flow is stabilized.     

Direct numerical simulation of particle behaviour in the wall region of turbulent flows in horizontal channels

The motion of small particles in the wall region of turbulent channel flows has been investigated using direct numerical simulation. It is assumed that the particle concentration is low enough to allow the use of one-way coupling in the calculations, i.e. the fluid moves the particles but there is no feedback from the particles on the fluid motion. The velocity of the fluid is calculated by using a pseudospectral, direct solution of the Navier-Stokes equations. The calculations indicate that particles tend to segregate into the low-speed regions of the fluid motion near the wall. The segregation tendency depends on the time constant of the particle made non-dimensional with the wall shear velocity and kinematic viscosity. For very small and very large time constants, the particles are distributed more uniformly. For intermediate time constants (of the order 3), the segregation into the low-speed fluid regions is the highest. The finding that segregation occurs for a range of particle time constants is supported by experimental results. The findings regarding the more uniform distributions, however, still remain to be verified against experimental data which is not yet available. For horizontal channel flows, it is also found that particles are resuspended by ejections (of portions of the low-speed streaks) from the wall and are, therefore, primarily associated with low-speed fluid. The smaller particles are flung further upwards and, as they fall back towards the wall, they tend to be accelerated close to the fluid velocity. The larger particles have greater inertia and, consequently, accelerate to lower velocities giving higher relative velocities. This velocity difference, as a function of wall-normal distance, follows the same trend as in experiments but is always somewhat smaller in the calculations. This appears to be due to the Reynolds number for the numerical simulation being smaller than that in the experiment. It is concluded that the average particle velocity depends not only on the wall variables for scaling, but also on outer variables associated with the mean fluid velocity and fluid depth in the channel. This is because fluid depth in combination with the wall shear velocity determines how much time a particle, of a given size and density, spends in the outer flow and, hence, how close it gets to the local fluid velocity.     

Application of the Galerkin method to the solution of the problem of combined (forced and free) turbulent convection in a vertical circular channel in a uniform solid mass

We consider steady-state combined (forced and free) turbulent convection in a vertical circular channel in a uniform solid medium for the case in which a constant vertical temperature gradientis maintained in the solid mass, far from the channel. The velocity and temperature distributions are found, and the critical values of the Rayleigh number for axisymmetric and antisymmetric fluid motions are calculated. The problem is solved by the Galerkin method.Notation v⁽⁰⁾ forced convection velocity - v⁽¹⁾ free convection velocity - v velocity with combination of forced and free convection - v average velocity across channel section - T temperature with combined forced and free convection - T_w channel wall temperature - y distance from channel wall - y_* dimensionless distance from wall - r₀ channel radius - r distance from centerline - v_t turbulent viscosity - _t turbulent thermal diffusivity - P₀ averaged pressure corresponding to constant fluid ternperature - z coordinate along channel axis, directed upward - Q quantity of heat released by internal sources per unit fluid volume per unit time - fluid thermal conductivity (_e for the surrounding mass) - R Reynolds number - R^* Rayleigh number - P Prandtl number - G Grashof number - V_* dynamic viscosity     

Turbulent film condensation on an isothermal cone surface

It is an investigation of turbulent film condensation on an isothermal cone. The present paper describes the eddy diffusivity of two turbulent models. And then it discusses the film thickness and heat transfer characteristics under the different turbulent models. The results show the mean heat transfer coefficient on two forms of eddy diffusivity, and there is a variation on the two models. Furthermore, the current results are compared with those generated by previous theoretical investigations. It is found that in high vapor velocity, the mean heat transfer was greater than that of the laminar flow theory. Under the high vapor velocity region, the eddy effect will be an important factor for the heat transfer of turbulent condensate film. Besides, in the low vapor velocity region, the eddy diffusivity seldom influences the heat transfer of condensate film.     

Turbulent Channel Flow with an Artificial Two-Dimensional Wall Layer

Turbulent flow of an incompressible fluid in a plane channel with parallel walls is considered. The three-dimensional time-dependent Navier-Stokes equations are solved numerically using the spectral finite-difference method. An artificial force which completely suppresses lateral oscillations of the velocity is introduced in the near-wall zone (10 % of the channel half-width in the neighborhood of each wall). Thus, the three-dimensional flow zone, in which turbulent oscillations can develop, is separated from the wall by a fluid layer. It is found that the elimination of three-dimensionality in the neighborhood of the walls leads to a significant reduction in the drag. However, complete laminarization does not occur. The flow in the stream core remains turbulent and can be interpreted as a turbulent flow in a channel with walls located on the boundary of the two-dimensional layer and traveling at the local mean-flow velocity. The oscillations developing inside the two-dimensional layer, which have significant amplitude, distort the flow only in the adjacent zone. Beyond this zone the distributions of the mean characteristics and the structure of instantaneous fields completely correspond to ordinary turbulent flow in a channel with rigid walls. The results obtained confirm the hypothesis of the unimportance of the no-slip boundary conditions for the fluctuating velocity component in the mechanism of onset and self-maintenance of turbulence in wall flows.     

Direct numerical simulation of particle-laden plane turbulent wall jet and the influence of Stokes number

The distribution and motion of inertial particles in plane turbulent wall jet are investigated using direct numerical simulation, under the assumption of one-way coupling. To our knowledge, this appears to be the first direct numerical simulation of a particle-laden plane turbulent wall jet. It is shown that, in outer part of the wall jet, the behaviour of particles closely resembles that of a free plane jet. Due to the streamwise decay of particle Stokes number, the particle streaks formed in the near wall region of the wall jet are characterized by their intensity variation, which differs significantly from those in the channel flow. The streamwise growth of the particle velocity half-width is approximately equal to that of the fluid velocity half-width and the maximum velocity of particles decays slower than that of fluid due to inertia. The outer scaling can collapse the mean particle velocity in both the inner and outer region for heavier particles. In the buffer region, the particle–fluid velocity difference can be negative or positive depending on the Stokes number since there are two competing effects, namely the memory effect and turbophoresis. In the viscous region, the larger particles are on average faster than fluid and the velocity difference is found to be self-similar depending on outer Stokes number. The near-wall distribution of velocity difference is significantly correlated with the presence of high-momentum particles which are entrained by vortical structures generated in the outer region of the wall jet. These results are useful for environmental and engineering applications.     

Effect of variable porosity on laminar convection in a uniformly heated vertical porous channel

The present investigation is devoted to the study of fully developed mixed convective flow through a vertical porous channel. The lateral variations of porosity and thermal diffusivity in the bed near the wall, are approximated by exponential functions. The correlation between permeability and porosity is brought through Kozney-Carman approximation. The volume averaged one dimensional low speed momentum equation proposed by Vafai is employed for the analysis of the problem. Results are obtained for steady heating of ascending cold fluid and steady cooling of ascending hot fluid. For the above physical situations it is observed that the heat transfer rate, and ratio of friction factor increases with increase in porous parameter, whereas the ratio of mass flow rate decreases with increase in porous parameter. The velocity profiles exhibit hydrodynamic channelling and peak velocity shifts towards the wall for higher values of the porous parameter. For steady heating of ascending could fluid increase in Rayleigh number enhances the heat transfer rate, and mass flow rate, while it reduces the ratio of friction factor. An opposite trend is observed for the case of steady cooling of ascending hot fluid.     

Heat dispersion in stationary mixed convection flow about horizontal surfaces in porous media

An analysis has been performed to study the influence of velocity dependent dispersion on transverse heat transfer in mixed convection flow above a horizontal wall of prescribed temperature in a saturated porous medium. The Boussinesq approximation and boundary layer analysis were used to numerically obtain gravity affected temperature and velocity distributions within the frames of Darcy's law and a total thermal diffusivity tensor comprising both of constant coefficient heat conduction and velocity proportional mechanical heat dispersion. Dependending on Pe_∞, the molecular Peclét number basing on the effective thermal diffusivity and the velocity of the oncoming flow, density coupling has distinct influences on heat transfer rates between the wall surface and the porous medium flow region. For small Peclét numbers, when heat conduction is the prevailing mechanism, wall heat fluxes are the higher the larger the density difference between the oncoming and the near wall fluid is. The opposite is true for larger Peclét numbers, when mechanical heat dispersion is the main cause of heat spreading. For Pe_∞ tending to infinity these wall heat fluxes approach finite maximum values in the total heat diffusivity model, they grow beyond any limit if only constant coefficient heat conduction is considered. Thus, the inclusion of mechanical heat dispersion effects yields physically more realistic predictions. Received on 18 September 1996     

Conjugated turbulent heat transfer with axial conduction in wall and convection boundary conditions in a parallel-plate channel

The steady-state conjugated turbulent heat transfer with axial conduction in the wall and convection boundary conditions is solved with the generalized integral transform technique for the flow of Newtonian fluid in parallel-plate duct. A lumped wall model that neglects transverse temperature gradients in the solid but that takes into account the axial heat conduction along the wall is adopted. Highly accurate results are presented for the fluid bulk and wall temperatures and Nusselt number. The effects of the conjugation parameter, Biot number, and the dimensionless channel length on Nusselt number and fluid bulk and wall temperatures are systematically investigated.     

The turbulent boundary layer of dissociated gas in the initial section of a tube

Many theoretical and experimental papers [1–4] have been devoted to investigating the turbulent boundary layer in the initial section of a channel. For the most part, however, the flow of an incompressible fluid with constant parameters is considered. There are many practical cases in which it is of interest to treat the development of the turbulent boundary layer of gas in the initial section of a pipe when conditions are strongly nonisothermal. A solution of a problem of this type, based on the theory of limit laws, is given in paper [1]. The present article extends this solution to the case of the flow of a high-enthalpy gas when the effect of gas dissociation on the turbulent boundary layer characteristics must be taken into account. We shall consider the flow of a mixture of i gases which is in a frozen state inside the boundary layer, and in an equilibrium state on its boundaries. Formulas are derived for the laws of friction and heat exchange, and a solution is given for the turbulent boundary layer equations in the initial section of the pipe when the wall temperature is constant and the gas flows at a subsonic velocity.Finally the authors are grateful to S. S. Kutateladze for discussing the paper.     

Development of a turbulent near-wall temperature model and its application to channel flow

A numerical study of fluid flow and heat transfer in a two-dimensional channel under fully developed turbulent conditions is reported. A computer program which is capable of treating both forced and natural convection problems under turbulent conditions has been developed. The code uses the high-Reynolds-number form of the two equation turbulent model(k-?) in which a turbulent kinetic energy near-wall model is incorporated in order to accurately represent the behavior of the flow near the wall, particularly in the viscous sublayer where the turbulent Reynolds number is small. A near-wall temperature model has been developed and incorporated into the energy equation to allow accurate prediction of the temperature distribution near the wall and, therefore, accurate calculation of heat transfer coefficients. The sensitivity of the prediction of flow and heat transfer to variations in the coefficients used in the turbulence model is investigated. The predictions of the model are compared to available experimental and theoretical results; good agreement is obtained. The inclusion of the near-wall temperature model has further improved the predictions of the temperature profile and heat transfer coefficient. The results indicate that the turbulent kinetic energy Prandtl number should be a function of Reynolds number.     

Performance of RNG Turbulence Modelling for Forced Convective Heat Transfer in Ducts

This investigation concerns numerical calculation of turbulent forced convective heat transfer and fluid flow in straight ducts using the RNG (Re-Normalized Group) turbulence method.

A computational method has been developed to predict the turbulent Reynolds stresses and turbulent heat fluxes in ducts with different turbulence models. The turbulent Reynolds stresses and other turbulent flow quantities are predicted with the RNG κ?ε model and the RNG non-linear κ-ε model of Speziale. The turbulent heat fluxes are modeled by the simple eddy diffusivity (SED) concept, GGDH and WET methods. Two wall functions are used, one for the velocity field and one for the temperature field. All the models arc implemented for an arbitrary three dimensional duct.

Fully developed condition is achieved by imposing cyclic boundary conditions in the main flow direction. The numerical approach is based on the finite volume technique with a non-staggered grid arrangement. The pressure-velocity coupling is handled by using the SIMPLEC-algorithm. The convective terms are treated by the QUICK, scheme while the diffusive terms are handled by the central-difference scheme. The hybrid scheme is used for solving the κ and ε equations.

The overall comparison between the models is presented in terms of friction factor and Nusselt number. The secondary flow generation is also of major concern.     

Momentum and heat transfer in two-phase bubble flow—I. Theory

A theory is proposed which describes the transfer process of momentum and heat in a two-phase bubble flow in channels. The eddy diffusivity to express the turbulent structure of the liquid phase is subdivided into the two components, one for the inherent wall turbulence independent of bubble agitation and the other for the additional turbulence caused by bubbles. On the basis of the theory, the velocity profile and the frictional pressure gradient for a given flow can be predicted when its void fraction profile is known. Furthermore, when a uniform heat flux is added to the system, its temperature distribution and heat transfer coefficient can be determined. A method for the numerical calculation of these parameters is also presented.     

Modeling the turbulent heat and momentum transfer in flows under different thermal conditions

Two-equation turbulence models for velocity and temperature (scalar) fields are developed to calculate wall shear flows under various flow conditions and related turbulent heat transfer under various wall thermal conditions. In the present models, we make the modified dissipation rates of both turbulent energy and temperature variance zero at a wall, though the wall limiting behavior of velocity and temperature fluctuations is reproduced exactly. Thus, the models assure computational expediency and convergence. Also, the present k- model is construted using a new type of expression for the Reynolds stress proposed by Abe et al. [Trans. JSME B 61 (1995) 1714–1721], whose essential feature lies in introducing the explicit algebraic stress model concept into the nonlinear k- formulation, and the present two-equation heat transfer model is constructed to properly take into account the effects of wall thermal conditions on the eddy diffusivity for heat. The models are tested with five typical velocity fields and four typical thermal fields. Agreement with experiment and direct simulation data is quite satisfactory.     

Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe

A numerical solution, for incompressible, steady-state, laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux and constant wall temperature. The development of velocity profile is obtained from Sparrow's entrance region solution. This velocity distribution is used in solving the energy equation numerically to obtain temperature profiles. Variation of the heat transfer coefficient for these two different boundary conditions for the early stages of boundary layer formation on the pipe wall is obtained. Local Nusselt numbers are calculated and the results are compared with those given byUlrichson andSchmitz. The effect of the thermal boundary conditions is studied by comparing the uniform wall heat flux results with uniform wall temperature.