Reynolds Analogy Research Articles

Relationship Between the Momentum and Scalar Fluxes Close to the Ground During the Indian Post-monsoon Period

Results of a comprehensive study of the characteristics of turbulent transport very close to the ground using micrometeorological eddy-covariance measurements from the Cloud Aerosol Interaction and Precipitation Enhancement Experiment (CAIPEEX) Integrated Ground Observational Campaign (IGOC) conducted in 2011 at a rural site over the Indian peninsular region are presented. The dataset used in the present study pertains to the morning hours when the convective boundary layer is at its growing phase. A new method, very similar to the traditional quadrant analysis, is introduced to investigate the relationship between the scalar and momentum flux transports by projecting the scalar and momentum flux contours onto the $$u-w$$ plane. Cospectral analysis is employed to study the scales of the transporting eddies; it was found that, during very unstable conditions, the scalar and momentum transports are poorly correlated with each other. However, for the near-neutral case, the correlation between the scalar and momentum transports becomes stronger, thus supporting the Reynolds analogy.

Near Wall Modeling for Trailing Edge Slot Film Cooling

Trailing edge slot film cooling is a widely used active cooling scheme for turbine blade trailing edges. Current Reynolds-Averaged Navier–Stokes (RANS) models are known to significantly overpredict the adiabatic effectiveness of these configurations. It is shown that this overprediction is due in part to the breakdown of the Reynolds analogy between turbulent shear stress and scalar transport in the near wall region. By examining previously reported direct numerical simulation (DNS) results for a wall-mounted cube in cross flow, it is seen that in a flow with a significantly perturbed outer boundary layer, the turbulent diffusivity is not as strongly damped as the turbulent viscosity in the viscous sublayer and buffer layer of the boundary layer. By removing the Van Driest damping function from the length scale model for the turbulent diffusivity, more accurate turbulent diffusivity predictions are possible. This near wall correction is applied to trailing edge slot film cooling flows and it is demonstrated that the predictive accuracy of the RANS models is significantly enhanced. Detailed comparisons between RANS results and experimental datasets for 15 different cases demonstrate that this correction gives significant improvement to the accuracy of the RANS predictions across a broad range of trailing edge slot film cooling configurations.

Nonlinear Shear and Heat Transfer in Hypersonic Rarefied Flows Past Flat Plates

This paper is a theoretical modeling study of the nonlinear transport of momentum and energy in a hypersonic rarefied flow past a sharp leading-edge flat plate at zero angle of attack. A characteristic parameter is introduced to describe the non-Newtonian shear and non-Fourier heat transfer features in the near-continuum flow regime via approximate analyses on the corresponding terms from the Burnett equation. Modified formulas based on are proposed to quantitatively predict the skin friction and heat flux along the plate, and the Reynolds analogy is proved to still be valid in the near-continuum flow regime. The physical meaning of and the flow mechanism are discussed. The direct simulation Monte Carlo method is also employed to validate the analytical results and to show the existence of a general analogy between the skin friction and heat flux in the whole flow regime from the continuum through the transition to the free molecular flow limit.

Generalized mathematical model for determination of intensity of disperse multiphase flows cleaning process in power plants systems

The three-dimensional mathematical model of dispersed multiphase flows cleaning process is the equations system, based on the equations of the Reynolds stress models, Reynolds analogy, heat and mass transfer to the momentum transfer, continuity, equation of dynamics of strengths balance when a particle is moving and unambiguous terms – geometric, physical, initial and boundary ones. Determination of the intensity of the cleaning process according to the solution results is performed according to the comprehensive measure of the intensity of the cleaning process which takes into account the mass transfer, hydrodynamic and energy coefficients of intensification by means of their values comparison during the movement process along the devices canals. The resulting value of the intensity coefficients of disperse multiphase flows cleaning processes is used to justify their usage in the environmental and energy resource saving technologies of separation, recovery, utilization, mist extraction in the power plants systems.

Series solutions for turbulent plumes evolving in a natural environment

Closed form solutions to the boundary layer equations for turbulent point and line thermal plumes evolving in natural convection have been obtained in the form of power series for the case when the turbulent viscosity is assumed to be proportional to the vertical height above the source. The initial values needed to generate the coefficients of the power series for different turbulent Prandtl number have been obtained numerically. To compliment these values the constants of proportionality between the turbulent viscosity and the height were determined using different methodologies including analysis of experimental data, computational fluid dynamics and numerical considerations. Evaluation of the results is primarily carried out by comparison to experimentally determined profiles of the temperature and velocity that are found in the literature. The best agreement was obtained when the Reynolds analogy—giving the turbulent Prandtl number as unity—was adopted. While the range of validity of the power series is limited, its radius of convergence can be extended using a suitable transformation.

Extended Reynolds analogy for slip and transition flow heat transfer in microchannels and nanochannels

It is significant to develop an expression for the rarefaction correction to the continuum solution for heat transfer. An extended Reynolds analogy for slip and transition flow heat transfer in microchannels and nanochannels is proposed. A relationship between the rate of the dimensionless mean wall shear stress reduction and the rate of the dimensionless mean wall heat flux reduction may exist. The extended Reynolds analogy can be utilized to obtain an initial engineering approximation of the Nusselt number. As slip flow heat transfer is inherently complex, the proposed simple model may provide an insight into the mechanism of slip flow heat transfer. This paper cultivates the facility to perform the kind of engineering analysis which, although not exact, still provides useful information concerning real engineering needs.

Turbulent thermal boundary layers with temperature-dependent viscosity

Direct numerical simulations (DNS) of turbulent boundary layers (TBLs) over isothermally heated walls were performed, and the influence of the wall-heating on the thermal boundary layers was investigated. The DNS adopt an empirical relation for the temperature-dependent viscosity of water. The Prandtl number therefore changes with temperature, while the Péclet number is constant. Two wall temperatures (Tw=70°C and 99°C) were considered relative to T∞=30°C, and a reference simulation of TBL with constant viscosity was also performed for comparison. In the variable viscosity flow, the mean and variance of the scalar, when normalized by the friction temperature deficit, decrease relative to the constant viscosity flow. A relation for the mean scalar which takes into account the variable viscosity is proposed. Appropriate scalings for the scalar fluctuations and the scalar flux are also introduced, and are shown to be applicable for both variable and constant viscosity flows. Due to the modification of the near-wall turbulence, the Stanton number and the Reynolds analogy factor are augmented by 10% and 44%, respectively, in the variable viscosity flow. An identity for the Stanton number is derived and shows that the mean wall-normal velocity and wall-normal scalar flux cause the increase in the heat transfer coefficient. Finally, the augmented near-wall velocity fluctuations lead to an increase of the wall-normal scalar flux, which contributes favorably to the enhanced heat transfer at the wall.

Mean structure of one-dimensional unstable detonations with friction

AbstractThis investigation deals with the study of the mean structure of a mildly unstable non-ideal detonation wave. The analysis is based on the integration of one-dimensional reactive Euler equations with friction forces using a third-order Runge–Kutta scheme and a fifth-order weighted essentially non-oscillatory (WENO5) spatial discretization. A one-step Arrhenius reaction mechanism is used for modelling the chemical reaction. When the frictional forces are active, the limit cycle based on the post-shock pressure reveals an enhanced pulsating behaviour of the downstream subsonic reaction zone compared to the ideal case. The results show that the detonation-velocity deficit increases as the mean reaction zone becomes thicker compared to the generalized ZND model. A new master equation, based on the Favre-averaged quantities, is derived and analysed along with new sonicity and thermicity conditions. The analysis of the species, momentum and energy balances reveals that the presence of mechanical fluctuations within the reaction zone constitutes another source of energy withdrawal, meaning that the detonation deviates from its laminar structure. Furthermore, the compressibility of the flow is analysed and the relationships between the fluctuations of temperature, velocity and reactive scalar are discussed in terms of strong Reynolds analogies.

Transient thermal boundary layers over rough surfaces

The transient heat transfer convection in a turbulent boundary layer is theoretically and experimentally studied when a time-periodical step-wise heat flux is imposed upon a rough surface. In particular, the behavior of the displacement height for the thermal boundary layer and the validity of the Reynolds analogy are analyzed. The local boundary layer characteristics, friction velocity, friction temperature and the displacement heights for the velocity and the temperature profiles, are evaluated through the graphical method of Perry and Joubert (1963) [3]. The results indicate that the displacement heights assume very different values and that the Reynolds analogy hypothesis is satisfied.

Experimental studies of aero-optical properties of subsonic turbulent boundary layers

AbstractThis paper gives the most-complete characterization to date of the optical aberrations imposed on a laser beam propagated through a subsonic, compressible, turbulent boundary layer in a zero-pressure gradient environment, over a range of boundary-layer thicknesses, oblique propagation angles and Mach numbers. This characterization is based on optical measurements using optical-wavefront-sensing instruments that have only become available in the last decade. The optical characterization includes and discusses in detail: optical-wavefront spectra, convective velocities of optically active large-scale structures and correlation functions in both streamwise and cross-stream directions, as well as root-mean-square optical path difference levels for different apertures. The scaling law based on the extended strong Reynolds analogy is derived and is shown to successfully collapse optical data collected in a number of facilities. Anisotropy of aero-optical distortions for different oblique viewing angles was experimentally quantified and is discussed.

Development and verification of a turbulent heat transport model for sodium-based liquid metal coolants

Numerical simulation of heat-and-mass transfer in fast neutron reactors with sodium coolant performed based on commercial Computational Fluid Dynamics (CFD) software products is considered. It is shown that the Reynolds analogy used in most turbulence models cannot be applied to study the peculiarities of heat transport in liquid metal coolants. We present here results for a turbulent LMS (Liquid Metal Sodium) heat transport model capable of taking into account the unique characteristics of sodium coolants. Within the model, the turbulent Prandtl number expression is implemented, a correction that takes into account gravitational anisotropy of the turbulent heat flow is made, and the thermal wall function is introduced. The model is implemented in the bundled CDF FlowVision code, is compatible with theturbulence models and can be used in both the high (with wall functions) and low (without wall functions) Reynolds number calculations of sodium flow. The proposed LMS model has been verified on the basis of experimental data obtained in the TEFLU test facility (Karlsruhe, Germany). The test facility is intended to simulate mixing processes of sodium coolant flows of various temperatures. In the experiment, three flow regimes have been studied: “free convection” regime, “transient” regime, and “forced convection” regime. For these regimes, simulation results gained from the application of such commercial CFD bundled software products as ANSYS CFX, Star-CD, Fluent, and FlowVision with and without the LMS model are presented. It is shown that the results obtained using the bundled FlowVision software with the LMS model demonstrate better agreement with the experimental data compared to the data obtained using other bundled software products.

A generalized Reynolds analogy for compressible wall-bounded turbulent flows

AbstractA generalized Reynolds analogy (GRA) is proposed for compressible wall-bounded turbulent flows (CWTFs) and validated by direct numerical simulations. By introducing a general recovery factor, a similarity between the Reynolds-averaged momentum and energy equations is established for the canonical CWTFs (i.e. pipes, channels, and flat-plate boundary layers that meet the quasi-one-dimensional flow approximation), independent of Prandtl number, wall temperature, Mach number, Reynolds number, and pressure gradient. This similarity and the relationships between temperature and velocity fields constitute the GRA. The GRA relationship between the mean temperature and the mean velocity takes the same quadratic form as Walz’s equation, with the adiabatic recovery factor replaced by the general recovery factor, and extends the validity of the latter to diabatic compressible turbulent boundary layers and channel/pipe flows. It also derives Duan & Martín’s (J. Fluid Mech., vol. 684, 2011, pp. 25–59) empirical relation for flows at different physical conditions (wall temperature, Mach number, enthalpy condition, surface catalysis, etc.). Several key parameters besides the general recovery factor emerge in the GRA. An effective turbulent Prandtl number is shown to be the reason for the parabolic profile of mean temperature versus mean velocity, and it approximates unity in the fully turbulent region. A dimensionless wall temperature, that we call the diabatic parameter, characterizes the wall-temperature effects in diabatic flows. The GRA also extends the analysis to the fluctuation fields. It recovers the modified strong Reynolds analogy proposed by Huang, Coleman & Bradshaw (J. Fluid Mech., vol. 305, 1995, pp. 185–218) and explains the variation of the temperature–velocity correlation coefficient with wall temperature. Thus, the GRA unveils a generalized similarity principle behind the complex nonlinear coupling between the thermal and velocity fields of CWTFs.

Analysis of heat transfer and stability of magnetohydrodynamic natural convection above a horizontal plate with heat flux boundary condition

A boundary layer analysis is performed for the steady laminar natural convection of an electrically conducting viscous incompressible fluid above a horizontal plate in the presence of a transverse magnetic field. The governing boundary layer equations are transformed to a set of non-linear ordinary differential equations which are then solved numerically using the shooting method. The effects of Prandtl number Pr and magnetic field parameter ζ on the velocity profiles, temperature profiles, skin friction coefficients and heat transfer coefficients are presented and discussed. A linear stability analysis based on the parallel flow model is also carried out to study the vortex instability of such flow. The present results indicate that the skin friction coefficient cfx decreases whereas the local Nusselt number Nux increases with increase in Pr if ζ is held constant. On the other hand, both the skin friction coefficient and Nusselt number decrease with increase in ζ for a fluid with constant Pr. Generic correlations for Nux and cfx have been developed in terms of Pr and ζ. A generic relation for magnetohydrodynamic free convection flow, equivalent to the Reynolds analogy for forced convection, has also been formulated.

Effect of wall temperature on hypersonic turbulent boundary layer

Turbulent boundary layers at Mach 4.9 with the ratio of wall temperature to recovery temperature from 0.5 to 1.5 are investigated by means of direct numerical simulation. Various fundamental properties relevant to the influence of wall temperature on Morkovin’s scaling, standard and modified strong Reynolds analogies, and coherent vortical structures have been studied. It is identified that the scaling relations proposed for cool and adiabatic wall conditions, such as Morkovin’s scaling and the modified strong Reynolds analogy, are also applicable for hot wall condition. Moreover, the relation between the density and temperature fluctuations under the second-order approximation is derived and verified to provide a reliable prediction. Based on the analysis of coherent vortical structures, it is found that the orientation of vortex core can be quantitatively determined by means of the vector with its direction and modulus using the local strain direction and the imaginary part of the eigenvalue of velocity gradient tensor, respectively. As the increase of wall temperature, the spanwise distance between the two legs of hairpin vortex increases, and the mean swirling strength and the angle of vortical structure with respect to the wall plane also increase in the inner layer. The statistical properties relevant to vortical structures are nearly insensitive to the wall temperature in the outer layer.

Analysis of available data from liquefied natural gas rollover incidents to determine critical stability ratios

Liquefied natural gas (LNG) rollover refers to the sudden mixing of stratified LNG layers, which can cause the generation of significant amounts of boil‐off gas. Such events are a significant safety concern in LNG storage but there are no reliable models for its description at industrial scales available in the open literature. In this article, the data and models for LNG rollover existing in the open literature are reviewed and a new framework for quantitatively analyzing the limited available data is presented. We extended the definition of the hydrostatic stability ratio for binary mixtures to allow its estimation for multicomponent mixtures, either from the reported LNG layer compositions or measurements of the LNG layer densities. By analyzing the graphical data of Bates and Morrison (Int J. Heat Mass Transfer. 1997;40:8) the critical value of the stability ratio, Rc, separating the diffusive phase of LNG rollover from the penetrative convection phase was estimated to be 3.8 ± 0.5. This is significantly larger than the critical ratio of 2 reported for saline solutions and is also larger than the initial stability ratio of 1.7 estimated from the best documented LNG rollover incident at La Spezia in 1971. Lumped‐parameter models for LNG rollover reported in the literature have successfully described the La Spezia incident by using the Reynolds analogy to estimate mass transfer rates from heat transfer correlations. However, these same models are unsuccessful when applied to other reported LNG rollover incidents, with the predicted rollover time being too short because the mass‐transfer coefficient is overestimated. The results presented here suggest that these limitations could be overcome by using a smaller mass‐transfer coefficient (estimated, e.g., from the Chilton‐Colburn analogy) and by tracking the multicomponent system's stability ratio until the critical value is reached whereupon the mass transfer regime changes. © 2013 American Institute of Chemical Engineers AIChE J, 60: 362–374, 2014

Nusselt number and friction factor in thermally stratified turbulent channel flow under Non-Oberbeck–Boussinesq conditions

In stably stratified turbulence, computations under Oberbeck–Boussinesq (OB) hypothesis of temperature-independent fluid properties may lead to inaccurate representation of the flow field and to wrong estimates of momentum/heat transfer coefficients. This is clearly assessed here comparing direct numerical simulations of stratified turbulence under OB conditions to simulations under NOB (Non-Oberbeck–Boussinesq) conditions of temperature-dependent fluid viscosity and thermal expansion coefficient. Compared to the OB case, NOB conditions may induce local flow relaminarization with significant variations (up to 30%) of heat and momentum transfer coefficients. Together with DNS results, we propose a phenomenological model (based on turbulent bursts) for heat transfer prediction in stratified turbulence under OB and NOB conditions. Implications of NOB assumptions on mixing efficiency (i.e. flux Richardson number Rif) and turbulent Prandtl number (Prt) are also discussed. These results are of specific importance in RANS modelling, where the condition Prt=1 is usually assumed (Reynolds analogy). Although this assumption is valid in some situations (i.e. boundary layer, pipe flow) there is uncertainty about its validity for stably-stratified turbulence. We demonstrate that this assumption is inaccurate when NOB effects become significant.

Creation and tidal advection of a cold salinity front in Storfjorden: 2. Supercooling induced by turbulent mixing of cold water

[1] Measurements near the edge of fast ice in Freemansundet, Svalbard, reveal mixing processes associated with tidal advection of a sharp front in salinity, including possible supercooling induced by double diffusion in a fully turbulent water column. The front translated back and forth with the semidiurnal tide between an area of mobile (drifting) ice in Storfjorden proper, and the narrow sound covered by fast ice. Water on each side of the front was near its salinity-determined freezing temperature. Instruments deployed about 400 m into the sound from the fast ice edge measured current, temperature, conductivity, and turbulence quantities through several tidal cycles. Turbulence data illustrate that as the steep horizontal salinity (density) gradient advected past the measurement site, vertical shear near the fast-ice base induced marked flood/ebb asymmetry in turbulent mixing. As fresher water entered the sound on the flood phase, inward transport of denser water near the upper boundary was retarded, leading to statically unstable conditions and enhanced turbulence. The opposite occurred during ebb tide, as denser water underran lighter. Transient episodes of supercooling accompanied frontal passage on both flood and ebb phases. The most likely explanation for a zone of supercooled water within the strongly mixed frontal region is that during mixing of fresher, slightly warmer (but still at freezing) water from outside with saltier, colder water in the sound, the former constituent lost heat faster than gaining salt. This interpretation (differing turbulent diffusivities for heat and salt) challenges strict application of Reynolds analogy for highly turbulent shear flow.

Direct Numerical Simulation and Large-Eddy Simulation of Supersonic Channel Flow

Compressible isothermal-wall channel flows are studied with direct numerical simulation and large-eddy simulation tools. Computations are carried out using a high-order, low-dissipation, bandwidth-optimized weighted essentially nonoscillatory numerical scheme to describe the hyperbolic terms of the Navier–Stokes equations. Periodic supersonic channel flow direct numerical simulation (, , , and ) is used to validate the procedure and the numerical scheme; a new subgrid term contribution based on pressure drop is proposed for the driving term required in momentum and energy equations for large-eddy simulation. Coherent structures of the flowfield are analyzed with scatter plots, criterion, and vorticity fields. As expected, the strong Reynolds analogy is not valid for this nonadiabatic flow. Streaks and horseshoe-like structures are highlighted and detailed. The authors propose a scenario for the formation of horseshoe-like structures. With large-eddy simulation tools, a dynamic procedure to evaluate the turbulent Prandtl number is required because results are found more accurate and computations more stable. Wall temperature and pressure impact are also emphasized on the normalized van Driest velocity profile in the logarithmic region. The classical log law is recovered: with , and a constant depending on and . An analog law is also recovered for the normalized temperature with the maximum of the Prandtl number . An a priori study on mesh requirements determination for a large range of pressure levels is realized through highly near-wall resolved large-eddy simulation.

DNS of a spatially evolving hypersonic turbulent boundary layer at Mach 8

This paper reports the direct numerical simulation (DNS) for hypersonic turbulent boundary layer over a flat-plate at Ma ∞=8 with the ratio of wall-to-freestream temperature equal to 1.9, which indicates an extremely cold wall condition. It is primarily used to assess the wall temperature effects on the mean velocity profile, Walz equation, turbulent intensity, strong Reynolds analogy (SRA), and compressibility. The present high Mach number with cold wall condition induces strong compressibility effects. As a result, the Morkovin’s hypothesis is not fully valid and so the classical SRA is also not fully consistent. However, some modified SRA is still valid at the far-wall region. It is also verified that the semi-local wall coordinate y* is better than conventional y + in analysis of statistics features in turbulent boundary layer (TBL) in hypersonic flow.

Magnetohydrodynamic free convection flow above an isothermal horizontal plate

In the present study a new similarity theory is developed to study the fluid flow and heat transfer characteristics for the steady laminar natural convection boundary layer flow of an incompressible and electrically conducting fluid past a semi-infinite horizontal plate subjected to a constant wall temperature under the action of transverse magnetic field. The governing parabolic boundary layer PDEs are transformed to ordinary differential equations using similarity transformation. This results in a set of three coupled, non-linear ordinary differential equations with variable coefficients (representing the interaction of the temperature and velocity fields) which are then solved by the shooting method. Asymptotic analyses and series solutions are also constructed to explore the mathematical behaviour of the solutions. The numerical results are obtained for various values of Prandtl number and magnetic field parameter ζ. The effects of various values of Prandtl number and magnetic field parameter ζ on the velocity profiles, temperature profiles, wall shear stress and heat transfer coefficients are presented. The results indicate that the wall shear stress τw decreases whereas the local Nusselt number Nux increases with increase in Prandtl number if the magnetic field parameter ζ is held constant at a particular value. On the other hand, both the wall shear stress and Nusselt number decrease with increase in the magnetic field parameter ζ for a fluid with constant Prandtl number. Generic correlations for Nux and τw have been developed in terms of the fluid Prandtl number Pr and magnetic field parameter ζ. A generic relation for free convection flow with magnetohydrodynamic effects, that is equivalent to the well-known Reynolds analogy for forced convection flow, has also been formulated in the present work.