Prandtl Number Model Research Articles

Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization

We consider temporal approximation of stationary statistical properties of dissipative infinite-dimensional dynamical systems. We demonstrate that stationary statistical properties of the time discrete approximations, i.e., numerical scheme, converge to those of the underlying continuous dissipative infinite-dimensional dynamical system under three very natural assumptions as the time step approaches zero. The three conditions that are sufficient for the convergence of the stationary statistical properties are: (1) uniform dissipativity of the scheme in the sense that the union of the global attractors for the numerical approximations is pre-compact in the phase space; (2) convergence of the solutions of the numerical scheme to the solution of the continuous system on the unit time interval $[0,1]$ uniformly with respect to initial data from the union of the global attractors; (3) uniform continuity of the solutions to the continuous dynamical system on the unit time interval $[0,1]$ uniformly for initial data from the union of the global attractors. The convergence of the global attractors is established under weaker assumptions. An application to the infinite Prandtl number model for convection is discussed.

A Semi-Implicit Scheme for Stationary Statistical Properties of the Infinite Prandtl Number Model

We propose a semidiscrete in time semi-implicit numerical scheme for the infinite Prandtl model for convection. Besides the usual finite time convergence, this scheme enjoys the additional highly desirable feature that the stationary statistical properties of the scheme converge to those of the infinite Prandtl number model at vanishing time step. One of the key characteristics of the scheme is that it preserves the dissipativity of the infinite Prandtl number model uniformly in terms of the time step. So far as we know, this is the first rigorous result on convergence of stationary statistical properties of numerical schemes for infinite dimensional dissipative complex systems.

Effect of turbulent Prandtl number on the computation of film-cooling effectiveness

This paper presents a method to improve the accuracy of computed film cooling effectiveness spanwise distribution. The effect of turbulent Prandtl number in the flow field outside the near-wall region on the computation is studied. Realizable k – ε model with a one-equation model in near-wall region is employed. The results show that the variation of turbulent Prandtl number has great influence on the computation. Reducing turbulent Prandtl number increases film cooling effectiveness of the whole spanwise region remarkably under large blowing ratios. Under small blowing ratios, the reduction of turbulent Prandtl number decreases the cooling effectiveness of the center region, and increases the effectiveness of the lateral region off the centerline. Compared with the single value turbulent Prandtl number computation the agreement between computation and measured results is improved notably with varied one in different spanwise regions. A new laterally varying turbulent Prandtl number (LV- Pr t) model dependent on lateral location and blowing ratio has been suggested. Computation accuracy is improved greatly by LV- Pr t model. Compared with the TLVA- Pr model of Lakehal [D. Lakehal, Near-wall modeling of turbulent convective heat transport in film cooling of turbine blades with the aid of direct numerical simulation data, ASME J. Turbomach. 124 (2002) 485–498] which provides the best results in the calculated cases LV- Pr t model is an effective way to improve computation accuracy in the frame of the traditional isotropic turbulence models. More work on the information of the turbulent Prandtl number has to be done for the further development of the LV- Pr t model.

A uniformly dissipative scheme for stationary statistical properties of the infinite Prandtl number model

The purpose of this short communication is to announce that a class of numerical schemes, uniformly dissipative approximations, which uniformly preserve the dissipativity of the continuous infinite dimensional dissipative complex (chaotic) systems possess desirable properties in terms of approximating stationary statistics properties. In particular, the stationary statistical properties of these uniformly dissipative schemes converge to those of the continuous system at vanishing mesh size. The idea is illustrated on the infinite Prandtl number model for convection and semi-discretization in time, although the general strategy works for a broad class of dissipative complex systems and fully discretized approximations. As far as we know, this is the first result on rigorous validation of numerical schemes for approximating stationary statistical properties of general infinite dimensional dissipative complex systems.

Bound on vertical heat transport at large Prandtl number

We prove a new upper bound on the vertical heat transport in Rayleigh–Bénard convection of the form c Ra 1 3 ( ln Ra ) 2 3 under the assumption that the ratio of Prandtl number over Rayleigh number satisfies Pr Ra ≥ c 0 where the non-dimensional constant c 0 depends on the aspect ratio of the domain only. This new rigorous bound agrees with the (optimal) Ra 1 3 bound (modulo logarithmic correction) on vertical heat transport for the infinite Prandtl number model for convection due to Constantin and Doering [P. Constantin, C.R. Doering, Infinite Prandtl number convection, J. Stat. Phys. 94 (1) (1999) 159–172] and Doering, Otto and Reznikoff [C.R. Doering, F. Otto, M.G. Reznikoff, Bounds on vertical heat transport for infinite Prandtl number Rayleigh–Bénard convection, J. Fluid Mech. 560 (2006) 229–241]. It also improves a uniform (in Prandtl number) Ra 1 2 bound for the Nusselt number [P. Constantin, C.R. Doering, Heat transfer in convective turbulence, Nonlinearity 9 (1996) 1049–1060] in the case of large Prandtl number.

Stationary statistical properties of Rayleigh‐Bénard convection at large Prandtl number

AbstractThis is the third in a series of our study of Rayleigh‐Bénard convection at large Prandtl number. Here we investigate whether stationary statistical properties of the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number are related to those of the infinite Prandtl number model for convection that is formally derived from the Boussinesq system via setting the Prandtl number to infinity. We study asymptotic behavior of stationary statistical solutions, or invariant measures, to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number. In particular, we show that the invariant measures of the Boussinesq system for Rayleigh‐Bénard convection converge to those of the infinite Prandtl number model for convection as the Prandtl number approaches infinity. We also show that the Nusselt number for the Boussinesq system (a specific statistical property of the system) is asymptotically bounded by the Nusselt number of the infinite Prandtl number model for convection at large Prandtl number. We discover that the Nusselt numbers are saturated by ergodic invariant measures. Moreover, we derive a new upper bound on the Nusselt number for the Boussinesq system at large Prandtl number of the form which asymptotically agrees with the (optimal) upper bound on Nusselt number for the infinite Prandtl number model for convection. © 2007 Wiley Periodicals, Inc.

The Turbulent Prandtl Number for Temperature Analysis in Rod Bundle Subchannels

The turbulent Prandtl number (PrT) model for the temperature field analysis in rod bundle subchannels is proposed based on the experimental data. The proposed turbulent Prandtl number is a function of the position and has the characteristic of an anisotropy which may not be found in simple geometries such as circular tubes and flat plates. Also, the proposed turbulent Prandtl number is modeled by considering the effects of the pitch to diameter (P/D) and the molecular Prandtl number (Pr). The turbulent temperature field is analyzed by a detailed analysis code. The Nusselt numbers (Nu) obtained by the new model are compared with those obtained by the well-known experimental correlations. The new turbulent Prandtl number model can predict the experimental data more accurately than the previous turbulent Prandtl number model.

The Turbulent Prandtl Number for Temperature Analysis in Rod Bundle Subchannels

The turbulent Prandtl number (PrT) model for the temperature field analysis in rod bundle subchannels is proposed based on the experimental data. The proposed turbulent Prandtl number is a function of the position and has the characteristic of an anisotropy which may not be found in simple geometries such as circular tubes and flat plates. Also, the proposed turbulent Prandtl number is modeled by considering the effects of the pitch to diameter (P/D) and the molecular Prandtl number (Pr). The turbulent temperature field is analyzed by a detailed analysis code. The Nusselt numbers (Nu) obtained by the new model are compared with those obtained by the well-known experimental correlations. The new turbulent Prandtl number model can predict the experimental data more accurately than the previous turbulent Prandtl number model.

Large Prandtl number behavior of the Boussinesq system of Rayleigh-Bénard convection

We establish the validity of the inifinite Prandtl number model as an approximation of the Boussinesq system at large Prandtl number on finite and infinite time interval, as well as in some statistical sense.

Bounds on convection driven by internal heating

Bounds are derived for the space–time averaged temperature 〈T〉 of a fluid layer in the Boussinesq approximation between fixed-temperature horizontal boundaries subject to uniform heating H throughout the volume. The analysis is carried out for both finite and infinite Prandtl number fluids. While the average temperature 〈T〉∼H in the purely conductive state, convection enhances the heat transport beyond static conduction reducing the temperature. Lower bounds to the average temperature of the layer scale with the magnitude of the imposed heat flux, with one scaling exponent for the arbitrary Prandtl number case and another for the infinite Prandtl number model. Specifically, it is proven here that at large heating rates where convection is important, 〈T〉⩾c1H2/3 for finite Prandtl number fluids and 〈T〉⩾c2H5/7 for infinite Prandtl number fluids. Explicit prefactors c1 and c2 for the scaling bounds are computed as well.

Infinite Prandtl number limit of Rayleigh‐Bénard convection

AbstractWe rigorously justify the infinite Prandtl number model of convection as the limit of the Boussinesq approximation to the Rayleigh‐Bénard convection as the Prandtl number approaches infinity. This is a singular limit problem involving an initial layer. © 2003 Wiley Periodicals, Inc.

Scalar Variance Transport in the Turbulence Modeling of Propulsive Jets

A variable turbulent Prandtl number model is applied to the prediction of laboratory jets with signie cant temperature variations. The model involves solving supplemental scalar equations for the temperature variance anditsdissipationratealongwiththe k‐equations.Themodelisalsoappliedtojetswithvariablecompositionwith the aim of building a consolidated approach to model e uctuations of both temperature and species mass fraction. Comparisons are presented for subsonic jets for which scalar e uctuation measurements are available. The same coefe cientsareusedfortemperature/speciesvarianceanddissipationrateequations.Excellentagreementwithdata isobtained forround jets with respect to scalare uctuation intensitiesand prediction of turbulent Prandtl/Schmidt numbers.

Numerical study of three-dimensional mixed convection due to buoyancy and centrifugal force in an oxide melt for Czochralski growth

Mixed convection due to buoyant and centrifugal forces in the crucible of a Czochralski apparatus has a significant effect on the quality and stability of growing oxide single crystals. Thus, the present investigation is concerned with the time-dependent and three-dimensional simulation of the flow and heat transfer in an oxide melt with a rotating crystal for a fundamental understanding of mixed convection and its effect on the surface waves. Based on an efficient block-structured finite-volume Navier–Stokes solver, quasi-direct numerical simulations (quasi-DNS) were carried out. A cylindrical crucible filled with a high Prandtl number (Pr=10.45) model fluid was considered for different combinations of Rayleigh numbers (Ra=4.0×10 8 and 7.6×10 8) and Reynolds numbers (Re=235–461). It was found that the surface wave pattern is strongly dependent on the structure of the three-dimensional and time-dependent fluid flow in the crucible. For all cases predicted, the waves exhibit regularity and the flow structure is fully symmetric in the vertical midplane as long as the centrifugal force has a significant effect on the flow below the disc. Furthermore, a downward buoyant jet was found to form at the disc center in all cases when the upper flow column due to the centrifugal force becomes restricted to the top. The transient flow development can be characterized by Gr/Re 2. For Gr/Re 2=235 the flow field is always periodic whereas it attains a non-periodic state through a number of quasi-periodic modes for Gr/Re 2 larger than 334. Increasing values of Gr/Re 2 lead to an earlier onset of the non-periodicity. Except during the initial period, the hot fluid is pushed below the disc primarily by the radial buoyant convection. This hot fluid interacts with the cold column of the downward buoyant flow controlling the temperature at the crystal/melt interface. The shape, the rotation rate of the waves, and the transient nature of the flow field were found to be in agreement with experimental observations.

Test of an Eulerian–Lagrangian simulation of wall heat transfer in a gas-solid pipe flow

An Eulerian–Lagrangian model is proposed to numerically predict the heat transfer between a vertical pipe wall and a turbulent gas-solid suspension. The whole problem lies in the combined solution of mass, momentum, and energy equations written for each phase. Closure equations devoted to turbulence and heat transfer simulation are also needed for the continuous phase. This problem is treated using a k–ε model and a turbulent Prandtl number model for the gaseous phase. The coupling terms standing for the fluid–particle interactions are taken into account. The simulation of the dispersed phase dynamics is addressed using a Lagrangian model, taking the particle–wall and particle–particle interactions into account. Additionally, the temperature of each particle is tracked using a model for the convective heat exchange between the two phases. The accuracy of the thermal and dynamic solutions has been tested for particles of diameter 200 and 500 μm, and for a wide range of loading ratios (0–20). Corresponding velocity and turbulent kinetic energy profiles are presented for the dynamic modelling validation, while Nusselt numbers are calculated for the study of the thermal problem.

Numerical Prediction of Transitional Features of Turbulent Forced Gas Flows in Circular Tubes With Strong Heating

In order to treat strongly heated, forced gas flows at low Reynolds numbers in vertical circular tubes, the k-ε turbulence model of Abe, Kondoh, and Nagano (1994), developed for forced turbulent flow between parallel plates with the constant property idealization, has been successfully applied. For thermal energy transport, the turbulent Prandtl number model of Kays and Crawford (1993) was adopted. The capability to handle these flows was assessed via calculations at the conditions of experiments by Shehata (1984), ranging from essentially turbulent to laminarizing due to the heating. Predictions forecast the development of turbulent transport quantities, Reynolds stress, and turbulent heat flux, as well as turbulent viscosity and turbulent kinetic energy. Overall agreement between the calculations and the measured velocity and temperature distributions is good, establishing confidence in the values of the forecast turbulence quantities—and the model which produced them. Most importantly, the model yields predictions which compare well with the measured wall heat transfer parameters and the pressure drop.

Turbulent Heat Transport in a Perturbed Channel Flow

The recovering boundary layer downstream of a separation bubble is known to have a highly perturbed turbulence structure which creates difficulty for turbulence models. The present experiment addressed the effect of this perturbed structure on turbulent heat transport. The turbulent diffusion of heat downstream of a heated wire was measured in a perturbed channel flow and compared to that in a simple, fully developed channel flow. The turbulent diffusivity of heat was found to be more than 20 times larger in the perturbed flow. The turbulent Prandtl number increased to 1.7, showing that the turbulent eddy viscosity was affected even more strongly than the eddy thermal diffusivity. This result corroborates previous work showing that boundary layer disturbances generally have a stronger effect on the eddy viscosity, rendering prescribed turbulent Prandtl number models ineffective in perturbed flows.

An extended Kays and Crawford turbulent Prandtl number model

For theoretical predictions of turbulent heat transfer in boundary layers and in duct flows, the knowledge of the turbulent Prandtl number is crucial. This is especially true for fluids with low molecular Prandtl numbers (liquid metals). The present formulation, which is an extended Kays and Crawford ( Convective Heat and Mass Transfer, 3rd edn. McGraw-Hill, New York, 1993) turbulent Prandtl number model, can be used for accurately predicting the heat transfer for liquid metal flows. The Nusselt numbers calculated with the modified model for Pr t are found to be in good agreement with experimental data for fully-developed pipe flows as well as for thermally developing pipe flows for various wall boundary conditions. The present model reduces to the one given in the above reference for liquids and gases with higher molecular Prandtl numbers.

On the modeling of homogeneous turbulence in a stably stratified flow

The decay of incompressible, homogeneous turbulence in a stably stratified medium is investigated. A conventional high-Reynolds-number, two-equation model with suitable improvements for buoyancy effects is used to calculate the velocity field, while an explicit algebraic heat-flux model that depends on the velocity field and the transport of the temperature variance and its dissipation rate is invoked for the temperature field. The model thus derived could account for buoyancy flux and streamwise heat transport correctly and is used to investigate the decay of grid generated turbulence in a stably stratified medium. This class of flows displays countergradient transport of the vertical turbulent heat flux. Flows with buoyancy frequencies ranging from 0 to 2.42 are calculated. The results are compared with experimental data as well as with the calculations of a constant turbulent Prandtl number model, an algebraic-flux model and a second-order closure. The present model is found to be capable of correctly reproducing the countergradient heat transport phenomenon and the turbulence decay characteristics over the range of buoyancy frequency studied. However, the countergradient heat transport region is not captured by the algebraic flux and the constant turbulent Prandtl number model. Furthermore, the present results are in good agreement with data and with the predictions of a second-order closure.

Prediction of heat transfer in turbulent falling liquid films with or without interfacial shear

AbstractA unified approach is proposed for the prediction of heat transfer coefficients in turbulent falling films undergoing heating, evaporation or condensation for both of the cases with or without interfacial shear. A modified van Driest eddy viscosity model, which incorporated a damping factor f and takes into account the effect of variable shear stress, is used to predict the hydrodynamics of turbulent falling films. The calculated film thicknesses are in good agreement with the Nusselt‐Brauer correlations for the non‐sheared film and the Dukler prediction for highly sheared film. Also, by including a van Driest type turbulent Prandtl number model, the asymptotic heat transfer coefficients are accurately predicted and show better agreement with the extensive literature data and correlations than do most of the existing turbulence models proposed to date.

Numerical Solution of Two-Dimensional Turbulent Blunt Body Flows with an Impinging Shock

AN implicit finite-difference method has been developed to compute two-dimensional, turbulent, blunt-body flows with an impinging shock wave. The complete timeaveraged Navier-Stokes equations are solved with algebraic eddy viscosity and turbulent Prandtl number models employed for shear stress and heat flux. The irregular-shaped bow shock is treated as a discontinuity across which the Rankine-Hugoniot equations are applied. A Type III turbulent shock interference flowfield has been computed and the numerical results compare favorably with existing experimental data. Contents The problem of shock interference heating arising from an extraneous shock wave impinging on a blunt body has been studied extensively during the past several years. In the Type III interference pattern, as described by Edney,l a shear layer originates at the intersection of the impinging shock and bow shock and interacts with the wall boundary layer. It has been shown1;2 that the heat transfer is strongly dependent on whether the shear layer is laminar, transitional, or turbulent. A method for computing laminar shock interference flowfields has been developed previously and applied to both two-dimensional3'4 and three-dimensional5 configurations. This method numerically computes the entire shock layer flowfield using the standard, unsplit, MacCormack finitedifference scheme6 to solve the complete set of compressible Navier-Stokes equations. In order to calculate turbulent shock interference flowfields, a turbulence model was added to the method. Unfortunately, this explicit method suffers from the fact that the allowable time step is proportional to the square of the grid spacing in viscous regions. In order to overcome this step size limitation, an implicit, noniterative, approximate-factorization, finite-difference scheme developed by Lindemuth and Killeen,7 McDonald and Briley,8 and Beam and Warming9 is used in the present study to solve the complete time-averaged Navier-Stokes equations. Although this scheme increases the computation time per step over that of the previous explicit scheme by a factor of 1.8 for turbulent calculations, it permits a time step that is many times greater than the explicit time step when fine meshes are