Fluid Pathlines Research Articles

A CFD Energetic study of the influence of the panel orientation in Open Joint Ventilated Façades

The arrangement of the panels in Open Joint Ventilated Façades (OJVF) is a potential factor in improving the energy efficiency of this building system. The distribution of joints in the façade influences the behaviour of the air flow in the channel which in turn could affect the overall heat exchanges with the envelope and thus the internal conditions of the building. Tiling panels can be installed on ventilated façades with different arrangement patterns according to the layout of the joints: lined up, staggered, stepped, diagonal or random, although manufacturers recommend a façade layout with in-line gaps to avoid costly façade maintenance. Thus, landscape and portrait layout with continuous joints are the most frequent arrangement in ventilated façades. This research assesses the benefit of the installation of OJVF panels in both layouts in order to reduce the cooling loads. Two real OJVFs with different panel arrangements, landscape and portrait, are analysed. Also, they are compared with a conventional façade with a sealed air cavity. All solutions are modelled and simulated using the commercial computational fluid dynamics software ANSYS FLUENT to evaluate the fluid-dynamic and thermal behaviour of the façades in summer and winter conditions. The energy performance of these solutions is evaluated, analysing different parameters such as panel’s temperature, mean air velocity inside the cavity, fluid pathlines through the open joints and thermal flux in the air cavity and to the room. The airflow inside the cavity is mainly driven by thermal buoyancy in all façades but differs from bi-dimensional convective loops in conventional façades to three-dimensional complex and asymmetrical airflows in OJVFs. The results obtained show that both OJVF configurations perform much better than the conventional sealed façade, reducing the heat transfer into the room by 30% in the summer period. In any case, the landscape OJVF façade reduces the transfer in the same period to a minimum value of 7.3 W/m2, which is 3% less than the flux transferred through the vertical one. This small difference in the energy performance of OJVFs makes the choice of panel orientation more based on other criteria such as aesthetics.

An anisotropic particle in a simple shear flow: an instance of chaotic scattering

Abstract

Deviation-angle and trajectory statistics for inertial particles in turbulence.

Small particles in suspension in a turbulent fluid have trajectories that do not follow the pathlines of the flow exactly. We investigate the statistics of the angle of deviation ϕ between the particle and fluid velocities. We show that, when the effects of particle inertia are small, the probability distribution function (PDF) P_{ϕ} of this deviation angle shows a power-law region in which P_{ϕ}∼ϕ^{-4}. We also find that the PDFs of the trajectory curvature κ and modulus θ of the torsion ϑ have power-law tails that scale, respectively, as P_{κ}∼κ^{-5/2}, as κ→∞, and P_{θ}∼θ^{-3}, as θ→∞: These exponents are in agreement with those previously observed for fluid pathlines. We propose a way to measure the complexity of heavy-particle trajectories by the number N_{I}(t,St) of points (up until time t) at which the torsion changes sign. We present numerical evidence that n_{I}(St)≡lim_{t→∞}N_{I}(t,St)/t∼St^{-Δ} for large St, with Δ≃0.5.

Non-linear interaction of buoyancy with von Kármán’s swirling flow in mixed convection above a heated rotating disc

In this paper, a systematic computational and theoretical study of the thermo-fluid-dynamics governing the flow above a heated horizontal rotating disc is presented. The fluid flow field is much more complex here as compared to von Kármán’s original solution (which took into account only the effect of disc rotation), and the effects of non-linear interaction between buoyancy and rotation are critically analysed here by studying the separate and combined roles of disc rotation and buoyancy on the fluid dynamic and heat transfer characteristics. The self-similarity of von Kármán’s flow field is lost, and the present paper establishes, for the first time, that the flow field above a heated rotating disc is divided into three distinct fluid dynamic regions. The three regions are demarcated by the loci of V̂z=0 and V̂r=0. In region 1 (R-1), V̂r is positive and V̂z is negative (such directions of the velocity components are characteristic of von Kármán’s flow or pure forced convection). In region 2 (R-2), V̂r is negative and V̂z is positive (such directions of the velocity components are characteristic of pure natural convection near a static disc surface). In region 3 (R-3), both V̂r and V̂z are negative. The forced convection results are obtained asymptotically at a large non-dimensional radius R within the region R-1 showing the dominance of forced convection mechanism, however, the fluid retains the signature of natural convection even at large values of R in the region R-3 where there is an inward radial velocity. Similarly, although a plume forms in the central portion of the disc where the solution is dominated by the effects of buoyancy, the fluid retains a signature of the disc rotation in the helical pathlines of fluid particles rising in the plume (whereas there is no swirl velocity present in pure natural convection above a static disc). The non-linear interaction between buoyancy and rotation results in several peculiar, rather non-intuitive, flow phenomena. Examples of such peculiarity include (i) the presence of a very sensitive spot on the upper boundary such that for a small change in this initial position the fluid pathlines may face drastically different final outcomes, (ii) the presence of a small portion near the centre of the disc where the fluid supplies energy to the disc, (iii) the effect of rotation on the rate of convective heat transfer being diminished by buoyancy over certain part of the disc while being enhanced over another part. This non-linear effect on Nusselt number is quantified here in terms of a Grashof number defined for mixed convection (Grmc).

Experimental Result Comparisons of Curve Fitting Algorithms on Fluid Path Lines Modeling in Strengthen Grinding Flow Field

Stirred turbulence with abrasive particles were widely employed for strengthen grinding in mechanical industries, therefore, the dynamic characteristic of turbulent flow comes up as a key element influencing grinding mechanism and operation performance in a long term. In this experimental research, we use flow path lines (turbulent path lines) to demonstrate the specific moving process of turbulent flow during strengthen grinding operation, which can be clearly identified by the real-time spatial movements of those abrasive particles, therefore a visual experimental approach can be provided to analyze the three-dimensionalmotion characteristics of strengthen grinding turbulence. The free movements of abrasive particles in three-dimensional grinding flow field exert an important influence on the geometrical properties of fluid path lines to be fitted, and the influence mechanism caused by those fluid path modeling algorithms should be studied quantitatively in detail. In this article, after extracting the image coordinates of those selected abrasive particles denoted as physical control points, and determining the three-dimensional moving path of turbulent flow by using imaging head array mounted on turbulence monitoring platform, several typical algorithms of curve fitting were sequentially employed for constructing the spatial path lines of turbulent flow, using coordinate transformation and flow motion vector computations. On the basis of computing the newly proposed curve mathematical properties, we made an experimental result comparison of those curve fitting algorithms on the established fluid path models. With several proposed experimental suggestions concerning turbulence inspection, flow path modeling processes and their correlated experimental mechanisms, the performance inspection of strengthen grinding turbulence experiments will be highly facilitated and remarkably optimized. Thereafter, new ideas for studying strengthen grinding process in future were also provided.

Autonomous obstacle avoidance for fixed-wing unmanned aerial vehicles

AbstractThis paper investigates a method for autonomous obstacle avoidance for fixed-wing unmanned aerial vehicles (UAVs), utilising potential fluid flow theory. The obstacle avoidance algorithm needs only compute the instantaneous local potential velocity vector, which is passed to the flight control laws as a direction command. The approach is reactive, and can readily accommodate real-time changes in obstacle information. UAV manoeuvring constraints on turning or pull-up radii, are accounted for by approximating obstacles by bounding rectangles, with wedges added to their front and back to shape the resulting fluid pathlines. It is shown that the resulting potential flow velocity field is completely determined by the obstacle field geometry, allowing one to determine a non-dimensional relationship between obstacle added wedge-length and the corresponding minimum pathline radius of curvature, which can then be readily scaled in on-board implementation. The efficacy of the proposed approach has been demonstrated numerically with an Aerosonde UAV model.

Flow of a nanofluid in the microspacing within co-rotating discs of a Tesla turbine

This article presents, for the first time, the fluid dynamics of the rotating flow of a nanofluid through the narrow spacings within co-rotating discs of a Tesla turbine. The inter-disc-spacing of multiple concentric discs of a Tesla disc turbine is usually of the order of 100μm. The study is conducted with the help of both mathematical analysis and computational fluid dynamic simulations. Numerical values are reported for a specific nanofluid which is a dilute solution of ferro-particles in water (maximum volume fraction considered is 0.05). The velocity field, pressure field and fluid pathlines are calculated in the three-dimensional, axi-symmetric flow domain for prescribed boundary values for the velocity components at inlet. Detailed comparisons between the analytical and computational solutions are provided. It is explained how the fluid dynamics of rotating flow within a Tesla disc turbine is influenced by the change in volume fraction of nanoparticles. The present study reveals that, with an increase in the volume fraction of nanoparticles, the pressure-drop in the radial direction increases; the tangential velocity at any point inside the computational domain tends to increase (even though its boundary value at inlet is kept fixed for each set of computation); however, the radial velocity field remains almost invariant. The present analysis shows that, with a suitable selection of the combination of geometric and flow parameters, the use of nanofluid leads to a significant improvement in the power output (the magnitude of increase would depend on the choice of nanofluid; the sample calculations show more than 30% increase in power output when the volume fraction of nanoparticles increases from 0 to 0.05). Moreover, the gain in power output is achieved without appreciably affecting the efficiency of the turbine. Indeed the present study shows that it is possible to achieve a high efficiency (a figure of 56% is included in the paper as a sample case), revealing the potential of the Tesla disc turbine to emerge as an attractive engineering product in the field of micro-turbines.

Visualization of Aspirated Flow Pathlines in a Thrombectomy Catheter with Three Extra Holes: A CFD Study

The aspiration catheter design requires the good understanding of blood flow, based on a complex analysis of its hemodynamic characteristics. Computational fluid dynamics (CFD) is an important approach in studying the blood flow. In this paper the effect of extra upper holes - designed for the aspiration catheter - on the aspirated fluid pathlines are studied and discussed.

Experimental Result Comparisons of Curve Fitting Algorithms on Fluid Path Lines Modeling in Strengthen Grinding Flow Field

Experimental Result Comparisons of Curve Fitting Algorithms on Fluid Path Lines Modeling in Strengthen Grinding Flow Field

Use of cosmogenic 129I to constrain numerical models of fluid flow in marine sediments: Application to the Blake Ridge Hydrate Province

AbstractConcentrations of cosmogenic iodine, 129I, in the pore fluid of marine sediments often indicate that the pore fluid is much older than the host sediment, even when vertical flow due to sediment compaction is taken into account. Old pore fluid has been used in previous studies to argue for pervasive upward fluid flow and a deep methane source for hydrate deposits. Alternatively, old pore fluid age may reflect more complex flow patterns. We use a two‐dimensional numerical transport model to account for the effects of topography and fractures on pore fluid pathlines when sediment permeability is anisotropic. We find that fluid focusing can cause significant lateral migration as well as regions where downward flow reverses direction and returns toward the seafloor. Longer pathlines can produce pore fluid ages much older than that expected with a one‐dimensional compaction model. For steady‐state models with geometry representative of Blake Ridge (USA), a well‐studied hydrate province, we find pore fluid ages beneath regions of topography and within fractured zones that are up to 70 Ma old. Our results suggest that the measurements of 129I/127I reflect a mixture of new and old pore fluid. However, old pore fluid need not originate at great depths. Methane within pore fluids can travel laterally several kilometers, implying an extensive source region around the deposit. This type of focusing should aid hydrate formation beneath topographic highs.

Trajectories and Deposition Sites of Spherical Particles Moving Inside Rhythmically Expanding Alveoli under Gravity-Free Conditions

Do fine particles (0.5-2 μm in diameter) deposit inside lung alveoli? This question is of particular interest in space flights where almost gravity-free conditions exist. Under such conditions, inhaled particles smaller than 0.5 μm in diameter or larger than 2 μm may deposit inside the alveoli due to Brownian motion or particle inertia, respectively. However, fine particles hardly affected by Brownian motion and of small mass can (wrongly) be perceived harmless, following closely fluid pathlines. The interplay between alveoli rhythmical expansion and the largely, previously disregarded geometrical interception mechanism was explored vis-à-vis predictions based on nonexpanding alveoli models. To this end, we employed a three-dimensional flow model that accounts for the rhythmical expansion of alveoli, and the trajectories of fine particles embedded in this flow were numerically calculated. Stochastic trajectories and deposition sites that are substantially different than those obtained for reversible Poiseuille-like flow models were widely used in the past. Indeed, small, inertialess, non-Brownian particles can hardly enter rigid alveoli in microgravity circumstances because the flow field consists of isolated closed streamlines that separate the cavities from the airways. However, for expanding alveoli, the streamline map is significantly altered, allowing diversion of particles from the airways toward the alveoli walls. As a result, collision with the alveoli wall due to geometrical interception may occur, revealing an additional mechanism that may control particle deposition inside alveoli. Fine particles 0.5-2 μm in diameter under zero gravity conditions may enter expanding alveoli and deposit due to the stochastic nature of the flow and the mechanism of geometrical interception. Their fate is very sensitive to their initial position. The majority of the particles tend to deposit inside alveoli located up the acinar tree, at the distal area of the alveoli and near its rim.

Numerical study on the performances of steam-jet vacuum pump at different operating conditions

A mathematic model based on realizable k- ɛ turbulence model for transonic flow was proposed to investigate the mixing flow behaviours of primary and secondary fluids in steam-jet vacuum pump and the affections to the pumping performances. The simulation was carried out to predict the static pressure distribution among mixing chamber wall by a commercial computational fluid dynamics (CFD) code FLUENT 6.2. Close agreements between the predicted results and experimental data validated the present theoretical model. Using the present approach, the velocity vectors and Mach number profiles in mixing chamber at different back pressures and the secondary fluid pathlines and mass flux profiles at different suction pressures were predicted. It is found that there are swirls separated from secondary fluid near the wall and the velocity of secondary fluid was fallen down obviously when the back pressure was bigger than critical back pressure. The above two factors lead to the entrainment ratio ( E m ) reduced rapidly. It is also found that the mass flux increased with the increasing of suction pressures and made the entrainment ratios increased. The prediction results show that the pressure ratio ( K) is a dominant position in affecting the pump's performances.

Flow Field Analysis in Expanding Healthy and Emphysematous Alveolar Models Using Particle Image Velocimetry

Particulates that deposit in the acinus region of the lung have the potential to migrate through the alveolar wall and into the blood stream. However, the fluid mechanics governing particle transport to the alveolar wall are not well understood. Many physiological conditions are suspected to influence particle deposition including morphometry of the acinus, expansion and contraction of the alveolar walls, lung heterogeneities, and breathing patterns. Some studies suggest that the recirculation zones trap aerosol particles and enhance particle deposition by increasing their residence time in the region. However, particle trapping could also hinder aerosol particle deposition by moving the aerosol particle further from the wall. Studies that suggest such flow behavior have not been completed on realistic, nonsymmetric, three-dimensional, expanding alveolated geometry using realistic breathing curves. Furthermore, little attention has been paid to emphysemic geometries and how pathophysiological alterations effect deposition. In this study, fluid flow was examined in three-dimensional, expanding, healthy, and emphysemic alveolar sac model geometries using particle image velocimetry under realistic breathing conditions. Penetration depth of the tidal air was determined from the experimental fluid pathlines. Aerosol particle deposition was estimated by simple superposition of Brownian diffusion and sedimentation on the convected particle displacement for particles diameters of 100-750 nm. This study (1) confirmed that recirculation does not exist in the most distal alveolar regions of the lung under normal breathing conditions, (2) concluded that air entering the alveolar sac is convected closer to the alveolar wall in healthy compared with emphysematous lungs, and (3) demonstrated that particle deposition is smaller in emphysematous compared with healthy lungs.

Large-Eddy Simulation of the Stable Atmospheric Boundary Layer using Dynamic Models with Different Averaging Schemes

Large-eddy simulation (LES) of a stable atmospheric boundary layer is performed using recently developed dynamic subgrid-scale (SGS) models. These models not only calculate the Smagorinsky coefficient and SGS Prandtl number dynamically based on the smallest resolved motions in the flow, they also allow for scale dependence of those coefficients. This dynamic calculation requires statistical averaging for numerical stability. Here, we evaluate three commonly used averaging schemes in stable atmospheric boundary-layer simulations: averaging over horizontal planes, over adjacent grid points, and following fluid particle trajectories. Particular attention is focused on assessing the effect of the different averaging methods on resolved flow statistics and SGS model coefficients. Our results indicate that averaging schemes that allow the coefficients to fluctuate locally give results that are in better agreement with boundary-layer similarity theory and previous LES studies. Even among models that are local, the averaging method is found to affect model coefficient probability density function distributions and turbulent spectra of the resolved velocity and temperature fields. Overall, averaging along fluid pathlines is found to produce the best combination of self consistent model coefficients, first- and second-order flow statistics and insensitivity to grid resolution.

Effects of implant design parameters on fluid convection, potentiating third-body debris ingress into the bearing surface during THA impingement/subluxation

Aseptic loosening from polyethylene wear debris is the leading cause of failure for metal-on-polyethylene total hip implants. Third-body debris ingress to the bearing space results in femoral head roughening and acceleration of polyethylene wear. How third-body particles manage to enter the bearing space between the closely conforming articulating surfaces of the joint is not well understood. We hypothesize that one such mechanism is from convective fluid transport during subluxation of the total hip joint. To test this hypothesis, a three-dimensional (3D) computational fluid dynamics (CFD) model was developed and validated, to quantify fluid ingress into the bearing space during a leg-cross subluxation event. The results indicated that extra-articular joint fluid could be drawn nearly to the pole of the cup with even very small separations of the femoral head (<0.60 mm). Debris suspended near the equator of the cup at the site of maximum fluid velocity just before the subluxation began could be transported to within 11° from the cup pole. Larger head diameters resulted in increased fluid velocity at all sites around the entrance to the gap compared to smaller head sizes, with fluid velocity being greatest along the anterosuperolateral cup edge, for all head sizes. Fluid pathlines indicated that suspended debris would reach similar angular positions in the bearing space regardless of head size. Increased inset of the femoral head into the acetabular cup resulted both in higher fluid velocity and in transport of third-body debris further into the bearing space.

Chaotic solute advection by unsteady groundwater flow

Solute mixing in fluids is enhanced significantly by chaotic advection, the phenomenon in which fluid pathlines completely fill the spatial domain explored by a laminar flow. Steady groundwater flows are, in general, not well conditioned for this phenomenon because Darcy's law confines them spatially to nonintersecting stream surfaces. Unsteady groundwater flows, however, may, in principle, induce chaotic solute advection if their time dependence is periodic and produces frequent reorientation of pathlines, an effect connected closely to the development of high solute mixing efficiencies. In this paper, a simple, two‐dimensional model groundwater system is studied theoretically to evaluate the possibility of inducing chaotic solute advection in the flow field near a recirculation well whose pumping behavior is time periodic. In the vernacular of dynamical systems theory the model studied is an example of a Hamiltonian system, thus allowing a mathematical formulation that facilitates consideration of whether chaotic advection can occur. When the model system is driven time periodically by alternate operation of the production and injection components of the recirculation well, numerical simulations of the fluid pathlines indicate that regions of chaotic solute advection will, indeed, develop in the flow field near the well, with expected major improvement in plume spreading.

Analysis of the flow field induced by the sessile peritrichous ciliate Opercularia asymmetrica

The feeding mechanism of the sessile protozoon Opercularia asymmetrica (Oligohymenophorea, Peritrichia) relies on the cilia beat generating a flow field that convectively transports suspended particles and dissolved substances to the oral cavity of the organism. By use of optical micro-flow measurement and theoretical methods the flow environment of two neighbouring peritrichous ciliate cells is studied. Both, yeast cells ( Saccharomyces cerevisiae) and artificial flow tracers are used for the visualisation of the flow field. Artificial tracers are rejected by the protozoa and deviate from the fluid path lines, while yeast cells follow the flow almost perfectly. This is shown through a dimensional analysis of the involved hydrodynamic forces on the tracers. The measured flow field exhibits maximum velocities of 25 μm/s at around 20 μm distance ahead of an individual ciliate. The flow field extends 200 μm from the location of the ciliate. A nicking motion of the protozoon is observed and found not to obey any periodic law. Multiples of protozoa exhibit most commonly an alternating cilia beat regime generating a non-stationary flow field. It can be shown through theoretical methods that fluid exchange is enhanced in this alternating regime compared to a flow field generated by a single ciliate. Fluid exchange depends on the distance of the ciliates from each other and on the alteration frequency of the cilia beat. The comparison of an analytical Stokes’ flow solution with the observed fluid flow serves to determine the force required to maintain the flow field against viscous dissipation. The force magnitude is in the order of magnitude of 10–100 pN.

The Effect of Work Processes on the Casing Heat Transfer of a Transonic Turbine

This paper considers the effect of the rotor tip on the casing heat load of a transonic axial flow turbine. The aim of the research is to understand the dominant causes of casing heat transfer. Experimental measurements were conducted at engine-representative Mach number, Reynolds number, and stage inlet to casing wall temperature ratio. Time-resolved heat-transfer coefficient and gas recovery temperature on the casing were measured using an array of heat-transfer gauges. Time-resolved static pressure on the casing wall was measured using Kulite pressure transducers. Time-resolved numerical simulations were undertaken to aid understanding of the mechanism responsible for casing heat load. The results show that between 35% and 60% axial chord the rotor tip-leakage flow is responsible for more than 50% of casing heat transfer. The effects of both gas recovery temperature and heat transfer coefficient were investigated separately and it is shown that an increased stagnation temperature in the rotor tip gap dominates casing heat transfer. In the tip gap the stagnation temperature is shown to rise above that found at stage inlet (combustor exit) by as much as 35% of stage total temperature drop. The rise in stagnation temperature is caused by an isentropic work input to the tip-leakage fluid by the rotor. The size of this mechanism is investigated by computationally tracking fluid path lines through the rotor tip gap to understand the unsteady work processes that occur.

Modeling CMP Transport and Kinetics at the Pad Groove Scale

AbstractA 3-D fluid flow and kinetics model of the full pad-wafer gap of a commercial CMP machine, including both grooves and land area flow properties, was developed to research the influence of these essential pad features at a scale not previously studied. Concentric circular grooves of two different dimensions were examined. Heat transfer and chemical reaction rates at the wafer surface were studied by including a combined frictional/chemical heat release model and firstorder kinetics for slurry activity. Results revealed sharp variations in velocity between the grooves and land areas, and a strong impact of groove depth and width on transient slurry mixing in the pad-wafer gap. These effects were more pronounced at higher polish pressures. Flow patterns in individual grooves beneath the rotating wafer were highly variable from one location to another such that fluid pathlines were dissimilar point to point even within the same groove. The effects of pad and wafer rotation, amplified by the unequal flow resistances of grooves and land areas, led to features in the steady-state temperature and concentration profiles at the same spacing as the groove pitch. This unusual finding identified a direct mechanism by which polish irregularities may be formed at the scale of the grooves.

Image quality or signal fidelity?: K.N. Bates, Advanced Technology Laboratories, Bellevue, WA 98007

In this paper, a systematic computational and theoretical study of the thermo-fluid-dynamics governing the flow above a heated horizontal rotating disc is presented. The fluid flow field is much more complex here as compared to von Kármán’s original solution (which took into account only the effect of disc rotation), and the effects of non-linear interaction between buoyancy and rotation are critically analysed here by studying the separate and combined roles of disc rotation and buoyancy on the fluid dynamic and heat transfer characteristics. The self-similarity of von Kármán’s flow field is lost, and the present paper establishes, for the first time, that the flow field above a heated rotating disc is divided into three distinct fluid dynamic regions. The three regions are demarcated by the loci of V̂z=0 and V̂r=0. In region 1 (R-1), V̂r is positive and V̂z is negative (such directions of the velocity components are characteristic of von Kármán’s flow or pure forced convection). In region 2 (R-2), V̂r is negative and V̂z is positive (such directions of the velocity components are characteristic of pure natural convection near a static disc surface). In region 3 (R-3), both V̂r and V̂z are negative. The forced convection results are obtained asymptotically at a large non-dimensional radius R within the region R-1 showing the dominance of forced convection mechanism, however, the fluid retains the signature of natural convection even at large values of R in the region R-3 where there is an inward radial velocity. Similarly, although a plume forms in the central portion of the disc where the solution is dominated by the effects of buoyancy, the fluid retains a signature of the disc rotation in the helical pathlines of fluid particles rising in the plume (whereas there is no swirl velocity present in pure natural convection above a static disc). The non-linear interaction between buoyancy and rotation results in several peculiar, rather non-intuitive, flow phenomena. Examples of such peculiarity include (i) the presence of a very sensitive spot on the upper boundary such that for a small change in this initial position the fluid pathlines may face drastically different final outcomes, (ii) the presence of a small portion near the centre of the disc where the fluid supplies energy to the disc, (iii) the effect of rotation on the rate of convective heat transfer being diminished by buoyancy over certain part of the disc while being enhanced over another part. This non-linear effect on Nusselt number is quantified here in terms of a Grashof number defined for mixed convection (Grmc).