Exit Reynolds Number Research Articles

Ignition and flame characteristics of cryogenic hydrogen releases

In this work, under-expanded cryogenic hydrogen jets were investigated experimentally for their ignition and flame characteristics. The test facility described herein, was designed and constructed to release hydrogen at a constant temperature and pressure, to study the dispersion and thermo-physical properties of cryogenic hydrogen releases and flames. In this study, a non-intrusive laser spark focused on the jet axis was used to measure the maximum ignition distance. The radiative power emitted by the corresponding jet flames was also measured for a range of release scenarios from 37 K to 295 K, 2–6 barabs through nozzles with diameters from 0.75 to 1.25 mm. The maximum ignition distance scales linearly with the effective jet diameter (which scales as the square root of the stagnant fluid density). A 1-dimensional (stream-wise) cryogenic hydrogen release model developed previously at Sandia National Laboratories (although this model is not yet validated for cryogenic hydrogen) was exercised to predict that the mean mole fraction at the maximum ignition distance is approximately 0.14, and is not dependent on the release conditions. The flame length and width were extracted from visible and infra-red flame images for several test cases. The flame length and width both scale as the square root of jet exit Reynolds number, as reported in the literature for flames from atmospheric temperature hydrogen. As shown in previous studies for ignited atmospheric temperature hydrogen, the radiative power from the jet flames of cold hydrogen scales as a logarithmic function of the global flame residence time. The radiative heat flux from jet flames of cold hydrogen is higher than the jet flames of atmospheric temperature hydrogen, for a given mass flow rate, due to the lower choked flow velocity of low-temperature hydrogen. This study provides critical information with regard to the development of models to inform the safety codes and standards of hydrogen infrastructure.

Reynolds stress distribution and turbulence generated secondary flow in the turbulent three-dimensional wall jet

The lateral half-width of the turbulent three-dimensional wall jet is typically five to eight times larger than the vertical half-width normal to the wall. Although the reason for this behaviour is not fully understood, it is caused by mean secondary flows that develop in the jet due to the presence of the wall. The origin of the secondary flow has been associated previously with both vorticity reorientation and also gradients in the Reynolds stresses, although this has not been directly quantified as yet. The present investigation focuses on a wall jet formed using a circular contoured nozzle with exit Reynolds number of 250 000. Stereoscopic particle image velocimetry measurements are used herein to measure the three-component velocity, thereby allowing access to the full Reynolds stress tensor that contributes to the secondary flow in a turbulent three-dimensional wall jet. Throughout the jet, the Reynolds normal stress ($\overline{u^{2}}$) makes the largest contribution to the Reynolds stress field whereas Reynolds shear stress ($\overline{vw}$) is found to be negligible when compared with other stresses. In particular, the differences in the Reynolds normal stresses ($\overline{v^{2}}-\overline{w^{2}}$) are found to be significantly larger than $\overline{vw}$; these terms are important for the generation of turbulence secondary flow in the wall jet. Above all, the differences in the Reynolds normal stresses are oriented to reinforce the near-wall streamwise vorticity, and thus contribute to the large lateral growth of this flow. The contours of the turbulent kinetic budget indicate that the turbulent energy budget obtained on the jet centreline is different from that obtained off of the jet centreline.

A New Test Facility to Investigate Film Cooling on a Nonaxisymmetric Contoured Turbine Endwall—Part II: Heat Transfer and Film Cooling Measurements

The present work is part of a comprehensive heat transfer and film-cooling study on a locally cooled nonaxisymmetric contoured turbine endwall. A new test rig consisting of a linear cascade of three prismatic vanes at unity scale and exchangeable endwall has been established. The rig is operated in an open-loop configuration at a reduced main gas temperature of 425 K, an exit Mach number of 0.5, and an exit Reynolds number of 1.6 × 106. Air is used both as main gas and coolant; a realistic density ratio is achieved by cooling the coolant below freezing. In the first part of the study, aerodynamic measurements are presented. This paper concentrates on film cooling of the contoured endwall with special emphasis on data acquisition and reduction for the application of the superposition principle of film cooling. The first experimental results from thermographic measurements are discussed.

Factors Influencing Computational Predictability of Aerodynamic Losses in a Turbine Nozzle Guide Vane Flow

This paper deals with the computational predictability of aerodynamic losses in a turbine nozzle guide vane (NGV) flow. The paper shows that three-dimensional (3D) computations of Reynolds-Averaged Navier Stokes (RANS) equations have the ability to adequately represent viscous losses in the presence of laminar flows, transitional regions, and fully turbulent flow areas in the NGV of an high pressure (HP) turbine stage. The Axial Flow Turbine Research Facility (AFTRF) used for the present experimental results has an annular NGV assembly and a 29-bladed HP turbine rotor spinning at 1330 rpm. The NGV inlet and exit Reynolds numbers based on midspan axial chord are around 300,000 and 900,000, respectively. A general purpose finite-volume 3D flow solver with a shear stress transport (SST) k–ω turbulence model is employed. The current computational study benefits from these carefully executed aerodynamic experiments in the NGV of the AFTRF. The grid independence study is performed with static pressure coefficient distribution at the midspan of the vane and the total pressure coefficient at the NGV exit. The effect of grid structure on aerodynamic loss generation is emphasized. The flow transition effect and the influence of corner fillets at the vane–endwall junction are also studied. The velocity distributions and the total pressure coefficient at the NGV exit plane are in very good agreement with the experimental data. This validation study shows that the effect of future geometrical modifications on the turbine endwall surfaces will be predicted reasonably accurately. The current study also indicates that an accurately defined turbine stage geometry, a properly prepared block-structured/body-fitted grid, a state-of-the-art transitional flow implementation, inclusion of fillets, and realistic boundary conditions coming from high-resolution turbine experiments are all essential ingredients of a successful turbine NGV aerodynamic loss quantification via computations. This validation study forms the basis for the successful future generation of nonaxisymmetric endwall surface modifications in AFTRF research efforts.

プラズマアクチュエータを用いた二次元オフセット噴流の 圧力特性と熱伝達特性

This study is concerned with the active flow control of two–dimension offset jet using a dielectric barrier discharge (DBD) plasma actuator (PA). The offset jet is produced by the flow of air which issues from the end of a long parallel channel, and the offset ratio H/h (H : step height, h : channel height) is 1.0. PA is installed on the lower wall of jet exit. The exit Reynolds number Re is changed in 1.5×103 to 6.0×103 . The wall static pressure and the heat transfer coefficient are measured. Flow visualization is carried out using a CCD camera and laser light. The results show that the pressure loss decreases and the heat transfer coefficient in the recirculating region increases with the induced flow due to the PA. These effects are remarkable at the low Reynolds number flow. In the downstream region, the heat transfer coefficient is smaller than without PA because the turbulent diffusion is promoted.

Steady interaction of a turbulent plane jet with a rectangular heated cavity

Turbulent heat transfer between a confined jet flowing in a hot rectangular cavity is studied numerically by finite volume method using the k-w SST one point closure turbulence model. The location of the jet inside the cavity is chosen so that the flow is in the non-oscillation regime. The flow structure is described for different jet-to-bottom-wall distances. A parametrical study was conducted to identify the influence of the jet exit location and the Reynolds number on the heat transfer coefficient. The parameters of this study are: the jet exit Reynolds number (Re, 1560< Re <33333), the temperature difference between the cavity heated wall and the jet exit (DT=60?C) and the jet location inside the cavity (Lf, 2? Lf? 10 and Lh 2.5<Lh?10). The Nusselt number increased and attained its maximum value at the stagnation points and then decreased. The flow structure is found in good agreement with the available experimental data. The maximum local heat transfer between the cavity walls and the flow occurs at the potential core end. The ratio between the stagnation point Nusselt numbers of the cavity bottom (NuB0) to the maximum Nusselt number on the lateral cavity wall (NuLmax) decreased with the Reynolds number for all considered impinging distances. For a given lateral confinement, the stagnation Nusselt number of the asymmetrical interaction Lh?10 is almost equal to that of the symmetrical interaction Lh=10.

Steady regime of a turbulent plane jet flowing into a rectangular hot cavity

Turbulent heat transfer between a confined jet and the walls of a hot rectangular cavity is studied numerically by finite volume method using the k-ω SST one point closure turbulence model. The parameters considered in this study were the jet exit Reynolds number, the temperature difference between the cavity heated wall and the jet exit and the jet location inside the cavity. The flow structure predictions were in good agreement with the available experimental data. The maximum local heat transfer between the cavity walls and the flow occurred at the potential core end. The ratio between the stagnation point Nusselt numbers of the cavity bottom to the maximum Nusselt number on the lateral cavity wall decreased with the Reynolds number for all considered impinging distances. For a given lateral confinement, the stagnation Nusselt number of the asymmetrical interaction is almost equal to that of the symmetrical interaction.

Dynamics and geometry of developing planar jets based on the invariants of the velocity gradient tensor

AbstractBased on the direct numerical simulation (DNS), the developing planar jets under different initial conditions, e.g., the conditions of the exit Reynolds number and the exit mean velocity profile, are investigated. We mainly focus on the characteristics of the invariants of the velocity gradient tensor, which provides insights into the evolution of the dynamics and the geometry of the planar jets along with the flow transition. The results show that the initial flow near the jet exit is strongly predominated by the dissipation over the enstrophy, the flow transition is accompanied by a severe rotation and straining of the flow elements, where the vortex structure evolves faster than the fluid element deformation, in the fully-developed state, the irrotational dissipation is dominant and the most probable geometry of the fluid elements should remain between the biaxial stretching and the axisymmetric stretching. In addition, with a small exit Re and a parabolic profile for the exit mean streamwise velocity, the decay of the mean flow field and the magnitude of the turbulent variables will be strengthened in the process of the flow transition, however, a large exit Re will promote the flow transition to the fully-developed state. The cross-impact between the exit Re and the exit mean velocity profile is also observed in the present study.

Flame volume and radiant fraction of jet diffusion methane flame at sub-atmospheric pressures

This paper reports an interesting experimental phenomenon regarding flame radiant fraction and flame volume of laminar diffusion methane flames under sub-atmospheric conditions. The quantitative formula of flame volume was derived by assuming that the shape of the flame was cylindrical, with which the relation between flame volume and ambient pressure was obtained. Then the dependence of flame radiant fraction on ambient pressure was analyzed, and the flame radiant fraction was seen to have a quarter-power law dependence on ambient pressure. Experiments were conducted using a circular nozzle (diameter of 0.012m) at 45–100kPa in a confined cabin with internal dimensions of 3m×2m×2m. Fuel mass flow rate was set to 5.98–20.91mg/s with jet exit Reynolds numbers of 57.66–201.8 and Froude numbers of 0.046–2.796. Experimental results obtained show an inverse relationship between flame volume and ambient pressure; the flame radiant fraction was at 0.194 power of ambient pressure; and the total absorption coefficient was at 1.78 power of ambient pressure. With increasing air pressure, the flames contracted progressively near the nozzle exit surface, which caused the location of the generated toroidal vortices to move downward, thus the oscillation amplitude was increased as IF=0.14p∞0.99.

Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach

The fluid flow and heat transfer characteristics of a free-surface liquid jet impingement cooling have been investigated numerically. The slot jet with water impinging normally on a flat plate is employed. To describe the turbulent structure, the turbulent governing equations are solved by a control-volume-based finite-difference method with a power-law scheme and the well-known turbulence model, which are associated with wall function. Numerical computations have been conducted with variations of jet exit Reynolds number (11,000≤Red≤17,000), dimensionless jet-to-surface distance (3≤H/d0≤12), dimensionless jet width (1≤B/d0≤2), and the heat flux (140kW/m2≤q″≤280kW/m2). The theoretical model developed is validated by comparing the numerical predictions with available experimental data in the literature. Under the studied ranges, the variations of local Nusselt numbers by hydraulic diameter Nud of the dimensionless jet-to-surface distance 3≤H/d0≤12 along the flat plate decrease monotonically from its maximum value at the stagnation point. In addition, the shape of the inlet area and jet-to-surface distance are optimized by using the response surface methodology (RSM) and the genetic algorithm (GA) method after solutions are carefully validated with available experimental results in the literature. Based on the optimal results, the optimum condition is in H/d0=7.86 and B/d0=2 for this physical model.

Heat Transfer From an Isothermally Heated Flat Surface Due to Confined Laminar Twin Oblique Slot-Jet Impingement

Convective heat transfer from a heated flat surface due to twin oblique laminar slot-jet impingement is investigated numerically. The flow domain is confined by an adiabatic surface parallel to the heated impingement surface. The twin slot jets are located on the confining surface. The flow and geometric parameters are the jet exit Reynolds number, distance between the two jets, distance between the jet exit and the impingement surface, and the inclination angle of the jet to the impingement surface. Numerical computations are done for various combinations of these parameters, and the results are presented in terms of the streamlines and isotherms in the flow domain, the distribution of the local Nusselt number along the heated surface, and the average Nusselt number at the heated surface. It is found that the peak and the average Nusselt number on the hot surface mildly decreases and the location of the stagnation point and the peak Nusselt number gradually moves downstream as the impingement angle is decreased from 90 deg. The heat transfer distribution from the impingement surface gets more uniform as the impingement angle is reduced to 45 deg and 30 deg at lager jet-to-plate distance (4–8) with a corresponding overall heat transfer reduction of about 40% compared to the normal impinging jet case. The specified jet exit velocity profile boundary condition has considerable effect on the predicted Nusselt number around the impingement location. Fully developed jet exit velocity profile correctly predicts the Nusselt number when compared to the experimental data.

Application of various RANS based models towards predicting turbulent slot jet impingement

Numerical simulation near the stagnation zone for an impinging jet is challenging because of the presence of sharp gradient due to the change of velocity from zero to the system scale velocity. A comparative study of four Reynolds averaged Navier–Stokes (RANS) based turbulence models has been done by numerically simulating a two-dimensional turbulent slot jet impinging air normally on a flat plate. Nozzle-to-plate spacings equal to 4 and 9.2 have been taken as two test cases with nozzle exit Reynolds number of 2×104. The turbulence intensity at the exit of the nozzle is kept constant at 1%. The capability to predict the fluid flow of the standard k−ε model, the low Reynolds number (LRN) k−ε models proposed by Launder and Sharma (LS) and Yang and Shih (YS), and the standard k−ω model have been evaluated by comparing the computational results with the experimental data available in the literature. The numerically obtained results have been compared with the experimental results on impingement surface pressure, free jet mean velocity profiles, spanwise velocity profiles, spanwise distribution of root mean square velocity fluctuation, skin friction coefficient distribution, non-dimensional velocity profile in wall coordinates. The standard k−ε model predicted the largest recirculation loop among the turbulence models used. The skin friction coefficient distribution is reproduced by YS and k−ω model for both the spacings whereas the other LRN model LS and the standard k−ε model are unable to predict the results. The linear law of the wall is found to be valid up to Y+≃4.0 in the recirculation region. The velocity profiles in wall coordinates in the recirculation region are behaving similar to those in the wall jet region of an impinging jet and the developed region of a wall jet.

Modeling mixing convection analysis of discrete partially filled porous channel for optimum design

Mixing convection flow inside a convergent horizontal channel partially filled with porous material and a clear channel are investigated numerically in the present study. Four discrete heat sources with uniform heat flux have been applied on the bottom surface of the channel. Three different channel exit heights are studied (He=1, 0.5 and 0.25). The thermal and flow-field analysis inside the channel is investigated for different wide range of Reynolds number (50⩽Re⩽300), Darcy number (10−2⩽Da⩽10−6), Richardson number (0⩽Ri⩽100) and Prandtl number (0.7⩽Pr⩽10). The present study carried out the effect of the channel exit height, Richardson number, Reynolds number, Darcy number and Prandtl number on the flow-field, the Nusselt number and the overall heat transfer performance. The Brinkman–Forchheimer–extended Darcy model is used to solve the governing equations of the fluid in the porous medium. The results reveal that the boundary layer thickness and flow velocity increase at high Richardson number for both porous and clear channels. The overall Nusselt number increases significantly for further increase in Darcy number, particularly for Ri>10. The smallest channel exit height (He=0.25) provides a high Nusselt number and low overall heat transfer performances. Furthermore, Richardson number has a small significant effect on overall Nusselt number and heat transfer performance at low Prandtl number.

Experimental Investigation of Louver Cooling Scheme on Gas Turbine Stator

An experimental investigation has been conducted to investigate the film cooling performance of the louver scheme over the surface of a gas turbine stator using a transient thermochromic liquid crystal technique. A two-dimensional airfoil cascade has been employed during the study. The exit Reynolds number based on the true chord length is 1.5E5 and the exit Mach number is 0.23. Two rows of an axially oriented louver scheme are distributed on both suction and pressure sides in a staggered arrangement. The effect of hole location on the cooling performance is investigated for each row individually; then row interaction is investigated at four different blowing ratios ranging from 1 to 2 and a 0.9 density ratio. The detailed local performance distribution and the lateral-averaged normalized performance are presented over both sides of the vane in terms of heat transfer coefficient and cooling effectiveness. The louver scheme provides a better cooling performance compared with the similar cylindrical scheme of the same base diameter at the same cooling amount. The blowing ratio does not influence significantly the performance for the louver scheme due to the considerable decrease in the jet momentum that impedes the jet lift-off at exit. The location of the scheme exit has a high impact on the cooling performance as it affects the development of the boundary layer. The double injection on the pressure side provides a superior effectiveness due to the blockage of the mainstream by the coolant injected from the first row. The louver scheme provides higher net heat flux reduction, which suits the cooling capacity needed for the next generation of gas turbines.

Large eddy simulation of rough and smooth liquid plunging jet processes

Large eddy simulation based on the Smagorinsky dynamic sub-grid scale model in combination with the multiphase volume of fluid (VOF) model, was used to simulate the flow of two turbulent plunging water jets. The jets were intended to simulate a weakly disturbed jet with low turbulence content and a highly disturbed jet with a profile almost fully-developed at the exit Reynolds number of 9000. The simulation captured successfully the previous experimentally-observed topological phenomena taking place during the transient impact and continuous-entrainment regime. It showed that surface instabilities in the free jet, due to its turbulence content have remarkable effect on submerged interfacial area, air volume, and air-entrainment rate. The simulation predicted an air entrainment rate within the range of semi-empirical correlations. The calculated mean velocity field exhibited almost identical trends for both. The combination of the LES-VOF models achieved a reasonably good level of agreement with experimental and empirical results.

Passive control of wall shear stress and mass transfer generated by submerged lobed impinging jet

Particle image velocimetry was used to investigate the flow field in an impinging lobed daisy hemispherical nozzle jet in comparison to its counterpart round jet, at a Reynolds number of 5620 based on the exit velocity and the equivalent diameter D e of the nozzle. The limitations of the PIV technique in the vicinity of the target wall due to the laser scattering were addressed by using the electrodiffusion (ED) technique to determine the wall shear rate distribution. The distribution of the mass transfer coefficient is also obtained using the ED technique. The target wall is placed at a distance H = 2D e from the plane tangent to the nozzle, at the center of the orifice. The entrainment of ambient fluid in the free jet region, which is larger in the lobed jet compared to the round jet, feeds in turn the wall jet region. The maximum wall shear rate was found significantly higher in the daisy jet, with an excess of 93 % compared to the reference round jet. The maximum mass transfer is 35 % higher in the former compared to the latter. Therefore, the hemispherical daisy nozzle is an excellent candidate in passive strategies to enhance local skin-friction and the subsequent local mass transfer at a constant exit Reynolds number

Planar laser-induced incandescence of turbulent sooting flames: the influence of beam steering and signal trapping

The influence of beam steering and signal trapping on the accuracy of soot volume fractions measured using planar laser-induced incandescence (LII) has been investigated in turbulent non-premixed sooting flames at atmospheric pressure. In turbulent non-premixed C2H4/air flames, the influence of local de-focusing/focusing of the laser sheet from beam steering can result in the underestimate of the averaged LII signal by 30 %, even when operating within the so-called plateau regime of laser fluence. Beam steering was also found to be significant in both the upstream region of C2H4/air flames and non-reacting C2H4 flows, because the fuel has a relatively high refractive index compared with ambient air. The extent of beam steering at different heights of reacting and isothermal flows as well as its dependence on exit Reynolds number (Re) has been measured. The measurements reveal that even at low turbulence levels (2000 < Re < 3000), beam steering effects can be significant. Also found is that the LII signal at a 450 nm wavelength can be attenuated by a few per cent at high soot loading regions in turbulent flames due to signal trapping. Finally, the feasibility of directly evaluating the signal attenuation via planar LII results was assessed by comparing the virtual soot attenuation calculated based on the planar LII result with that measured using light-of-sight extinction.

Role of Fluid-Dynamics in Soot Formation and Microstructure in Acetylene-Air Laminar Diffusion Flames

Residence time and thermo-chemical environment are important factors in the soot-formation process in flames. Studies have revealed that flow-dynamics plays a dominant role in soot formation process. For understanding the effect of flow dynamics on soot formation and physical structure of the soot formed in different combustion environments two types of laminar diffusion flames of Acetylene and air, a normal diffusion flame (NDF) and an inverse diffusion flame (IDF) have been investigated. The fuel and air supply in the reaction zone in two flame types were kept constant but the interchange of relative position of fuel and air altered the burner exit Reynolds and Froude numbers of gases, fuel/air velocity ratio and flame shape. Soot samples were collected using thermophoretic sampling on transmission electron microscope (TEM) grids at different flame heights and were analyzed off-line in a Transmission Electron Microscope. Soot primary particle size, soot aggregate size and soot volume fraction were measured using an image analysis software. In NDF the maximum flame temperature was about 1525 K and 1230 K for IDF. The soot primary particles are distinctly smaller in size in IDF (between 19 – 26 nm) compared to NDF (between 29–34 nm). Both NDF and IDF show chainlike branched structure of soot agglomerate with soot particles of a nearly spherical shape. The average number of soot primary particles per aggregate in NDF was in the range of 24 to 40 and in IDF it varied between 16 to 24. Soot volume fraction was between 0.6 to 1.5 ppm in NDF where as it was less than 0.2 ppm in IDF. The change in sooting characteristics of the two flame types is attributed to changed fuel/air velocity ratio, entrainment of gas molecules and thermophoresis on soot particles.

二次元オフセット噴流の流動特性と熱伝達

The flow field of an offset jet is complex and is frequently found in many engineering devices. In this paper, the fluid flow and heat transfer characteristics of two-dimensional offset jet are investigated experimentally. The offset jet is produced by the flow of air which issue from the end of a long parallel channel. The exit Reynolds number Re is changed in 1.5×103 to 7.0×103, and the offset ratio H/h (H : step height, h : channel height) is changed in 0.5, 1.0 and 1.5. The wall shear stress is measured using a micro flow sensor. Flow visualization is carried out using a CCD camera and laser light. The velocity profiles are measured by a PIV system. When the flow at the channel exit is laminar, fluid in the outer region has a counter-clockwise vorticity and fluid in the inner region has a clockwise vorticity, and a lifting-off of the wall jet flow into the ambient fluid is occurred. The Strouhal number of the vortex frequency is almost constant against the offset ratio. The formation of those vortices is not seen at the turbulent flow. Two peaks exist in the wall static pressure, Nusselt number and skin friction coefficient distributions of the laminar flow with H/h = 0.5 and 1.0. The 1st peak appears near the reattachment point, and the 2nd peak appears in the neighborhood of the lifting-off of the wall jet flow. The two peaks coalesce into one at H/h = 1.5. As the offset ratio increases, the reattachment point moves to downstream direction and Nu becomes large both the laminar and turbulent flows. The reattachment point of the turbulent flow moves to upstream direction compared with the laminar flow because Coanda-effect is strengthen.

Turbulent heat transfer for impinging jet flowing inside a cylindrical hot cavity

Convective heat transfer from an isothermal hot cylindrical cavity due to a turbulent round jet impingement is investigated numerically. Three-dimensional turbulent flow is considered in this work. The Reynolds stress second order turbulence model with wall standard treatment is used for the turbulence predictions the problem parameters are the jet exit Reynolds number, ranging from 2x104 to 105and the normalized impinging distance to the cavity bottom and the jet exit Lf, ranging from 4 to 35. The computed flow patterns and isotherms for various combinations of these parameters are analyzed in order to understand the effect of the cavity confinement on the heat transfer phenomena. The flow in the cavity is divided into three parts, the area of free jet, and the area of the jet interaction with the reverse flow and the semi-quiescent flow in the region of the cavity bottom. The distribution of the local and mean Nusselt numbers along the cavity walls for above combinations of the flow parameters are detailed. Results are compared against to corresponding cases for impinging jet on a plate for the case of the bottom wall. The analysis reveals that the average Nusselt number increases considerably with the jet exit Reynolds number. Finally, it was found that the average Nusselt number at the stagnation point could be correlated by a relationship in the form Nu=f(Lf,Re).