Analytical solutions for laminar fully developed flows in double-sine shaped ducts

Fully developed, constant property, laminar flows in double-sine shaped ducts are considered. This cross section represents a limiting inter-plate channel geometry in plate heat exchangers. Accurate analytical solutions based on the Galerkin integral method are presented. Heat transfer with both T and H1 thermal boundary conditions is analyzed; they simulate the most fundamental practical heating/cooling applications. Velocity and temperature distributions, along withfRe, Nu _T , andNu _H1 results for flows in double-sine ducts of different aspect ratios (1/8 ≤γ ≤ 8) are presented. Effects of the relative cross-sectional geometry and thermal boundary conditions are delineated. A comparison of the thermal-hydraulic performance with that of other compact channel geometries is made. The results suggest an optimum (Nu/fRe) performance in a double-sine duct of aspect ratio near unity.     

Mixed convection adjacent to inclined flat surfaces embedded in a porous medium

An analysis is performed to study the flow and heat transfer characteristics of laminar mixed convection boundary layer flows from inclined (including horizontal and vertical) surfaces embedded in a saturated porous medium with constant aiding external flows and uniform surface temperature. Both the streamwise and normal components of the buoyancy forces are retained in the momentum equations. Nondimensionalization of the boundary layer equations results in the following three governing parameter: (1)Gr/Re, the ratio of the Grashof number to the Reynolds number; (2)Pe _x =Re _x Pr, the Peclet number; (3) φ, the angle of inclination from the horizontal. The resulting nonsimilar equations are solved by an efficient implicit finite-difference scheme. Numerical results are presented for flows with different values ofGr/Re in the range of 0 to 50, over a wide range of the Peclet numbersPe _x, and various values of φ ranging from 0 to 90 degrees. It is found that the local surface heat transfer rate increases with increasing the local Peclet number. In addition, as the plate is tilted from the horizontal to the vertical orientation, the local Nusselt number increases for a given Peclet number and the effect of the buoyancy force on the surface heat transfer rate increases.     

Effects of surface radiation on natural convection in a rayleigh-benard square enclosure: steady and unsteady conditions

Coupled laminar natural convection with radiation in air-filled square enclosure heated from below and cooled from above is studied numerically for a wide variety of radiative boundary conditions at the sidewalls. A numerical model based on the finite difference method was used for the solution of mass, momentum and energy equations. The surface-to-surface method was used to calculate the radiative heat transfer. Simulations were performed for two values of the emissivities of the active and insulated walls (ɛ₁=0.05 or 0.85, ɛ₂=0.05 or 0.85) and Rayleigh numbers ranging from 10³ to 2.3×10⁶ . The influence of those parameters on the flow and temperature patterns and heat transfer rates are analyzed and discussed for different steady-state solutions. The existing ranges of these solutions are reported for the four different cases considered. It is founded that, for a fixed Ra, the global heat transfer across the enclosure depends only on the magnitude of the emissivity of the active walls. The oscillatory behavior, characterizing the unsteady-state solutions during the transitions from bicellular flows to the unicellular flow are observed and discussed.     

Delay in response of turbulent heat transfer against acceleration or deceleration of flow in a pipe

To predict the heat transfer enhancements that result from the application of a pulsating flow in a pipe, we experimentally investigated the turbulent heat transfer variations produced in response to sudden accelerations or decelerations to flows within a pipe. To accomplish this, the Reynolds numbers with the valve open (Re₁) and close (Re₀) were systematically varied in the range of 8,000 ≤ Re₁ ≤ 34,000 and 700 ≤ Re₀ ≤ 23,000, respectively, and in-pipe spatiotemporal heat transfer variations were measured using infrared thermography simultaneously with temporal variations to the in-pipe flow properties. Based on the experimental results, it was found that the heat transfer delays that occur in response to accelerations or decelerations can be characterized using the corresponding time lag Δt and first-order time constant τ. The values of Δt and τ can be expressed as non-dimensional forms of Δt/(ν/u_τ²) and τ/(R/u_τ), respectively, where u_τ is the pipe wall friction velocity, ν is the kinematic viscosity of the fluid, and R is the pipe radius.     

The effect of variable properties on laminar boundary layer flow

The influence of temperature dependent fluid properties on laminar boundary layer flows is examined for wedge flows (Falkner-Skan). An asymptotic expansion for small heat transfer rates is applied under the boundary conditionsT _w= const andq _w=const. Linear deviations from the zero order solution (constant properties) can be given in a form known as the property ratio method. As a consequence this method is no longer an empirical one.     

Wall modeling via function enrichment within a high‐order DG method for RANS simulations of incompressible flow

We present a novel approach to wall modeling for the Reynolds‐averaged Navier‐Stokes equations within the discontinuous Galerkin method. Wall functions are not used to prescribe boundary conditions as usual, but they are built into the function space of the numerical method as a local enrichment, in addition to the standard polynomial component. The Galerkin method then automatically finds the optimal solution among all shape functions available. This idea is fully consistent and gives the wall model vast flexibility in separated boundary layers or high adverse pressure gradients. The wall model is implemented in a high‐order discontinuous Galerkin solver for incompressible flow complemented by the Spalart‐Allmaras closure model. As benchmark examples, we present turbulent channel flow starting from Re_τ=180 and up to Re_τ=100000 as well as flow past periodic hills at Reynolds numbers based on the hill height of Re_H=10595 and Re_H=19000.     

Particle convective heat transfer near the wall in a supercritical water fluidized bed by single particle model coupled with CFD-DEM

Supercritical water fluidized bed (SCWFB) is a promising reactor to gasify biomass or coal. Its optimization design is closely related to wall-to-bed heat transfer, where particle convective heat transfer plays an important role. This paper evaluates the particle convective heat transfer coefficient (h_pc) at the wall in SCWFB using the single particle model. The critical parameters in the single particle model which is difficult to get experimentally are obtained by the computational fluid dynamics-discrete element method (CFD-DEM). The contact statistics related to particle-to-wall heat transfer, such as contact number and contact distance, are also presented. The results show that particle residence time (τ), as the key parameter to evaluate h_pc, is found to decrease with rising velocity, while increase with larger thermal boundary layer thickness. τ follows a gamma function initially adopted in the gas–solid fluidized bed, making it possible to evaluate h_pc in SCWFB by a simplified single particle model. The theoretical predicted h_pc tends to increase with rising thermal gradient thickness at a lower velocity (1.5 U_mf), while first decreases and then increases at higher velocity (1.75 and 2 U_mf). h_pc occupies 30%–57% of the overall wall-to-bed heat transfer coefficient for a particle diameter of 0.25 mm. The results are helpful to predict the overall wall-to-bed heat transfer coefficient in SCWFB combined with a reasonable fluid convective heat transfer model from a theoretical perspective.     

Combined radiative and convective heat transfer in a three-dimensional rectangular channel at different wall temperatures

 This paper deals with a numerical study of combined convective and radiative heat transfer in a three-dimensional rectangular duct with hydrodynamically and thermally developing laminar flow. The gas is assumed to be an incompressible, absorbing, emitting, isotropically scattering, gray medium. Isothermal, gray, diffuse boundary walls at different temperatures are assumed. The finite-volume method (FVM) is adopted to describe both convective and radiative heat transfer. The coupled continuity and momentum equations are solved by means of SIMPLER algorithm. Numerical results for the radiative flux show very good agreement with the available data. The effects of aspect ratio, optical thickness, scattering albedo and wall emissivity on the mean bulk temperature are also investigated. By splitting the heat flux into convective and radiative contributions, the relative importance of these components is assessed for a typical range of values of the parameters. Received on 9 February 1999     

Computational study of combined effects of conduction-radiation and hydromagnetics on natural convection flow past magnetized permeable plate

The computational study of the combined effects of radiation and hydromagnetics on the natural convection flow of a viscous,incompressible,and electrically conducting fluid past a magnetized permeable vertical plate is presented.The governing non-similar equations are numerically solved by using a finite difference method for all values of the suction parameter ξ and the asymptotic solution for small and large values of ξ.The effects of varying the Prandtl number P r,the magnetic Prandtl number P r m,the magnetic force parameter S,the radiation parameter R d,and the surface temperature θ w on the coefficients of the skin friction,the rate of heat transfer,and the current density are shown graphically and in tables.An attempt is made to examine the effects of the above mentioned physical parameters on the velocity profile,the temperature distribution,and the transverse component of the magnetic field.     

On the accurate prediction of the wall-normal velocity in compressible boundary-layer flow

We consider a problem which arises in the numerical solution of the compressible two-dimensional or axisymmetric boundary-layer equations. Numerical methods for the compressible boundary-layer equations are facilitated by transformation from the physical (x, y) plane to a computational (ξ, η) plane in which the evolution of the flow is ‘slow’ in the time-like ξ direction. The commonly used Levy-Lees transformation results in a computationally well-behaved problem, but it complicates interpretation of the solution in physical space. Specifically, the transformation is inherently non-linear, and the physical wall-normal velocity is transformed out of the problem and is not readily recovered. Conventional methods extract the wall-normal velocity in physical space from the continuity equation, using finite-difference techniques and interpolation procedures. The present spectrally accurate method extracts the wall-normal velocity directly from the transformation itself, without interpolation, leaving the continuity equation free as a check on the quality of the solution. The present method for recovering wall-normal velocity, when used in conjunction with a highly accurate spectral collocation method for solving the compressible boundary-layer equations, results in a discrete solution which satisfies the continuity equation nearly to machine precision. As demonstration of the utility of the method, the boundary layers of three prototypical high-speed flows are investigated and compared: the flat plate, the hollow cylinder, and the cone. An important implication for classical linear stability theory is also briefly discussed.     

COUPLING BETWEEN LAMINAR FILM CONDENSATION AND NATURAL CONVECTION ON OPPOSITE SIDES OF A VERTICAL PLATE

Theoretical analyses which incorporate one-dimensional heat conduction along a plate and transverse heat conduction approximations are presented to predict the net heat transfer between laminar film condensation of a saturated vapour on one side of a vertical plate and boundary layer natural convection on the other side. It is assumed that countercurrent boundary layer flows are formed on the two sides. The governing boundary layer equations of this problem and their corresponding boundary conditions are all cast into dimensionless forms by using a non-similarity transformation. Thus the resulting system of equations can be solved by using the local non-similarity method for the boundary layer equations and a finite difference method for the heat conduction equation of the plate. The plate temperature and the heat flux through the plate are repetitively determined until the solutions for each side of the plate match. The predicted results show that the effect of Pr_c is not negligible for larger values of A* (thermal resistance ratio between natural convecti on side and condensing film side) and the approximation of transverse heat conduction overpredicts the plate temperature for lower values of R_t (thermal resistance ratio between plate and condensing film). However, no significant differences are observed between the two different approximations for higher values of R_t. © by 1997 John Wiley & Sons, Ltd.     

Transient two-dimensional radiative and conductive heat transfer in an axisymmetric medium

This work considers transient conductive and radiative heat transfer in a two-dimensional, cylindrical, scattering medium heated or cooled by internal heat source or boundary surface. A finite difference scheme is employed for handling the energy storage and the heat diffusion by conduction, while a discrete-ordinate method is used to analyze the radiative heat transfer. The effects of various parameters, including the conduction-radiation parameter, the scattering albedo and the emissivity of the boundary surfaces, are investigated. Received on 30 April 1997     

Thermal interaction between free convection and forced convection along a vertical conducting wall

A theoretical study is presented in this paper to investigate the conjugate heat transfer across a vertical finite wall separating two forced and free convection flows at different temperatures. It is assumed that the heat conduction in the wall is only in the transversal direction. We also assume that countercurrent boundary layers are formed on the both sides of the wall. The governing equations of this problem and their corresponding boundary conditions are all cast into a dimensionless form by using a non-similarity transformation. These resulting equations, which are singular at the points ξ_c=0 and 1, are solved numerically using a very efficient singular perturbation method. The effects of the resistance parameters and of the Prandtl numbers on heat transfer characteristics are investigated and presented in a table and ten figures. Received on 8 April 1998     

Radiative and Thermal Dispersion Effects on Non-Darcy Natural Convection with Lateral Mass Flux for Non-Newtonian Fluid from a Vertical Flat Plate in a Saturated Porous Medium

The problem of natural convective heat transfer for a non-Newtonian fluid from an impermeable vertical plate embedded in a fluid-saturated porous medium has been analyzed. Non-Darcian, radiative and thermal dispersion effects have been considered in the present analysis. The governing boundary layer equations and boundary conditions are cast into a dimensionless form and simplified by using a similarity transformation. The resulting system of equations is solved by using a double shooting Runge–Kutta method. The effect of viscosity index n, the conduction–radiation parameter R, the non-Darcy parameter Gr*, the thermal dispersion parameter Ds and the suction/injection parameter f_w on the fluid velocities, temperatures and the local Nusselt number are discussed.     

Radiative heat transfer calculations in real gases

A general formulation for radiative heat transfer calculations is presented, based on integrated quantities such as total emissivities and absorptivities. The procedure is intended particularly for combustion chamber applications of varying degree of complexity, the radiative active medium consisting of gases such as H₂O and CO₂ and of soot. First, some preliminary calculations are given for the often treated radiative equilibrium cases of plane parallel plates and infinite concentric cylinders. Then an example of a combustion chamber calculation is studied where the radiative heat transfer calculation is included in a system of partial differential equations describing momentum, heat and mass transfer with combustion.     

Turbulent transport of particles in a straight square duct

Turbulent flow through a duct of square cross-section gives rise to off-axis secondary flows, which are known to transfer momentum between fluid layers thereby flattening the velocity profile. The aim of this study is to investigate the role of the secondary flows in the transport and dispersion of particles suspended in a turbulent square duct flow. We have numerically simulated a flow through a square duct having a Reynolds number of Re_τ = 300 through discretization of the Navier–Stokes equations, and followed the trajectories of a large number of passive tracers and finite-inertia particles under a one-way coupling assumption. Snapshots of particle locations and statistics of single-particle and particle pair dispersion were analyzed. It was found that lateral mixing is enhanced for passive tracers and low-inertia particles due to the lateral advective transport that is absent in straight pipe and channels flows. Higher inertia particles accumulate close to the wall, and thus tend to mix more efficiently in the streamwise direction since a large number of the particles spend more time in a region where the mean fluid velocity is small compared to the bulk. Passive tracers tend to remain within the secondary swirling flows, circulating between the core and boundary of the duct.     

Wall modeling for implicit large-eddy simulation and immersed-interface methods

We propose and analyze a wall model based on the turbulent boundary layer equations (TBLE) for implicit large-eddy simulation (LES) of high Reynolds number wall-bounded flows in conjunction with a conservative immersed-interface method for mapping complex boundaries onto Cartesian meshes. Both implicit subgrid-scale model and immersed-interface treatment of boundaries offer high computational efficiency for complex flow configurations. The wall model operates directly on the Cartesian computational mesh without the need for a dual boundary-conforming mesh. The combination of wall model and implicit LES is investigated in detail for turbulent channel flow at friction Reynolds numbers from Re _τ  = 395 up to Re _τ =100,000 on very coarse meshes. The TBLE wall model with implicit LES gives results of better quality than current explicit LES based on eddy viscosity subgrid-scale models with similar wall models. A straightforward formulation of the wall model performs well at moderately large Reynolds numbers. A logarithmic-layer mismatch, observed only at very large Reynolds numbers, is removed by introducing a new structure-based damping function. The performance of the overall approach is assessed for two generic configurations with flow separation: the backward-facing step at Re _h = 5,000 and the periodic hill at Re _H = 10,595 and Re _H = 37,000 on very coarse meshes. The results confirm the observations made for the channel flow with respect to the good prediction quality and indicate that the combination of implicit LES, immersed-interface method, and TBLE-based wall modeling is a viable approach for simulating complex aerodynamic flows at high Reynolds numbers. They also reflect the limitations of TBLE-based wall models.     

Buoyancy-affected laminar film condensation

The flow and heat transfer in a laminar condensate flim on an isothermal vertical plate is modelled mathematically. The strict Boussinesq approximation is adopted to account for buoyancy due to local temperature variations within the film. A similarity transformation reduces the governing boundary-layer type equations to a coupled set of ordinary differential equations and the resulting three-parameter twopoint boundary value problem is solved numerically for Prandtl numbers,Pr, ranging from 0.001 to 1000 and Jakob numbers,Ja, between 0.0001 and 1.5. The principal effects of the favourable buoyancy are to reduce the thickness of the condensate film and increase the film velocity at the smooth liquid-vapour interface, whereas the friction and heat transfer at the plate are enhanced. In accordance with the classical Nusselt theory, it is found that the temperature varies nearly linearly across the film. The computed similarity profiles for velocity reveal, however, substantial departures from the parabolic distribution assumed in the simplified Nusselt analysis.     

Mixed convection boundary layer flow over a horizontal circular cylinder with Newtonian heating

The steady mixed convection boundary layer flow over a horizontal circular cylinder, generated by Newtonian heating in which the heat transfer from the surface is proportional to the local surface temperature, is considered in this study. The governing boundary layer equations are first transformed into a system of non-dimensional equations via the non-dimensional variables, and then into non-similar equations before they are solved numerically using a numerical scheme known as the Keller-box method. Numerical solutions are obtained for the skin friction coefficient Re ^1/2 C _f and the local wall temperature θ _w (x) as well as the velocity and temperature profiles with two parameters, namely the mixed convection parameter λ and the Prandtl number Pr.     

Buckling and its effect on the confined flow of a model capsule suspension

The rheology of confined flowing suspensions, such as blood, depends upon the dynamics of the components, which can be particularly rich when they are elastic capsules. Using boundary integral methods, we simulate a two-dimensional model channel through which flows a dense suspension of fluid-filled capsules. A parameter of principal interest is the equilibrium membrane perimeter, parameterized by ξ_o, which ranges from round capsules with ξ_o=1.0 to ξ_o=3.0 capsules with a dog-bone-like equilibrium shape. It is shown that the minimum effective viscosity occurs for ξ_o≈1.6, which forms a biconcave equilibrium shape, similar to a red blood cell. The rheological behavior changes significantly over this range; transitions are linked to specific changes in the capsule dynamics. Most noteworthy is an abrupt change in behavior for ξ_o≈2.0, which correlates with the onset of capsule buckling. The buckled capsules have a more varied orientation and make significant rotational (rotlet) contributions to the capsule–capsule interactions.