Showing papers on "Oblique shock published in 1990"

Ion thermalization in quasi-perpendicular shocks involving reflected ions

Abstract: Ion thermalization mechanisms downstream of the quasi-perpendicular Earth's bow shock are examined by means of plasma and magnetic field data from the AMPTE/IRM spacecraft which include three-dimensional ion distributions, plasma fluid parameters derived every ∼4.3 s, and spectra of transverse and parallel magnetic fluctuations up to 16 Hz. The objects studied in detail are low-Mach number, low-β shocks in which reflected-gyrating ions are present and contribute to the downstream ion temperature but where processes beyond the ramp take place slowly, so that the basic phenomenology becomes apparent. In MHD terms, most of these shocks qualify as marginally critical. Downstream of the ramp, the initially separated core and ring ions slowly merge into a joint, less anisotropic distribution possessing a high-energy tail. The ion temperature ratio, T⊥/T∥, is high not only in the shock foot and ramp but also within some distance downstream; its speed of decline rises and the residual level lessens with increasing β. The ions diffuse about equally fast in energy and in pitch angle. An asymmetry of the distributions with respect to the field direction is present when the shock is slightly oblique. It decays only slowly, which might indicate that the pitch angle diffusion rate near zero pitch angle is reduced. Low-frequency electromagnetic waves are present below the proton gyrofrequency; they are characterized by strong left-hand-polarized emissions and a low level of parallel fluctuations except very close to the shock. The left-hand emissions are often concentrated into a narrow frequency band but sometimes they exhibit a double-humped structure. Waves and ion distributions approach a slowly varying equilibrium some distance downstream of the shock. After extending the analysis to one supercritical shock representing the majority of bow shock encounters, we conclude that our deductions are more generally valid, although the thermalization is faster, usually, and appears to involve nonlinear processes which tend to obscure most of the features noted.

Characteristics of multiple shock wave/turbulent boundary-layer interactions in rectangular ducts

Abstract: Multiple shock wave/turbulent boundary-layer interactions in a rectangular duct have been investigated using wall pressure measurements, surface oil flow visualization, spark schlieren photography, and laser Doppler velocimetry. Two undisturbed incoming Mach numbers were considered, Mach 2.45 and Mach 1.6. At Mach 2.45 the shock structure was a neutrally stable pattern of oblique shocks followed by repeated normal shocks with the level of flow confinement having only a small effect in the interaction. A large, three-dimensional separation region was observed. At Mach 1.6 the pattern consisted of a bifurcated normal shock followed by weaker, unbifurcated normal shocks. The boundary layer under the bifurcated shock was incipiently separated. In contrast to the Mach 2.45 case, the lower Mach number interaction was much steadier with the length of the interaction scaling directly with the level of flow confinement.

Re‐forming supercritical quasi‐parallel shocks: 2. Mechanism for wave generation and front re‐formation

Abstract: This paper continues the study of Thomas et al. (1990) in which hybrid simulations of quasi-parallel shocks were performed in one and two spatial dimensions. To identify the wave generation processes, the electromagnetic structure of the shock is examined by performing a number of one-dimensional hybrid simulations of quasi-parallel shocks for various upstream conditions. In addition, numerical experiments were carried out in which the backstreaming ions were removed from calculations to show their fundamental importance in reformation process. The calculations show that the waves are excited before ions can propagate far enough upstream to generate resonant modes. At some later times, the waves are regenerated at the leading edge of the interface, with properties like those of their initial interactions.

Shock enhancement and control of hypersonic mixing and combustion

Abstract: The possibility that shock enhanced mixing can substantially increase the rate of mixing between coflowing streams of hydrogen and air has been studied in experimental and computational investigations. Early numerical computations indicated that the steady interaction between a weak shock in air with a coflowing hydrogen jet can be well approximated by the two-dimensional time-dependent interaction between a weak shock and an initially circular region filled with hydrogen imbedded in air. An experimental investigation of the latter process has been carned out in the Caltech 17 Inch Shock Tube in experiments in which the laser induced fluorescence of byacetyl dye is used as a tracer for the motion of the helium gas after shock waves have passed across the helium cylinder. The flow field has also been studied using an Euler code computation of the flow field. Both investigations show that the shock impinging process causes the light gas cylinder to split into two parts. One of these mixes rapidly with air and the other forms a stably stratified vortex pair which mixes more slowly; about 60% of the light gas mixes rapidly with the ambient fluid. The geometry of the flow field and the mixing process and scaling parameters are discussed here. The success of this program encouraged the exploration of a low drag injection system in which the basic concept of shock generated streamwise vorticity could be incorporated in an injector for a Scramjet combustor at Mach numbers between 5 and 8. The results of a substantial computational program and a description of the wind tunnel model and preliminary experimental results obtained in the High Reynolds Number Mach 6 Tunnel at NASA Langley Research Center are given here.

Re‐forming supercritical quasi‐parallel shocks: 1. One‐ and two‐dimensional simulations

Abstract: The process of reforming supercritical quasi-parallel shocks is investigated using one-dimensional and two-dimensional hybrid (particle ion, massless fluid electron) simulations both of shocks and of simpler two-stream interactions. It is found that the supercritical quasi-parallel shock is not steady. Instread of a well-defined shock ramp between upstream and downstream states that remains at a fixed position in the flow, the ramp periodically steepens, broadens, and then reforms upstream of its former position. It is concluded that the wave generation process is localized at the shock ramp and that the reformation process proceeds in the absence of upstream perturbations intersecting the shock.

Ion reflection and dissipation at quasi‐parallel collisionless shocks

Abstract: Large scale one-dimensional hybrid simulations have been performed of a quasi-parallel (ΘBn = 20°) high Mach number collisionless shock. It is found that backstreaming reflected ions, i.e., upstream ions with velocities exceeding the shock ram velocity, originate from the outer part (v≳ 1.7vth) of the velocity space of the incident distribution. The backstreaming ions produce very low-frequency magnetosonic waves which propagate upstream with about 1.3VA (Alfven speed). As the wave crests convect toward the shock, they steepen up and the shock reforms itself. During shock reformation a large part of the incident ions are reflected. This, in turn, slows the incident ions down. The slowed down incident particle distribution and the reflected particle distribution merge and constitute the new thermalized downstream distribution. In the interval of a relatively stationary shock low-frequency whistler waves stand at the shock front. During these time intervals the whistler waves are probably responsible for dissipation by nonadiabatic compression of the incident ions. The whistler waves are destroyed by the incoming large amplitude wave crest and reemerge at the new shock front. The reapparance seems to be due to the nonlinear steepening of the incoming wave crest at the upstream side.

Anomalous reflection of a shock wave at a fluid interface

Abstract: Several wave patterns can be produced by the interaction of a shock wave with a fluid interface. We focus on the case when the shock passes from a medium of high to low acoustic impedance. Curvature of either the shock front or contact causes the flow to bifurcate from a locally self-similar quasi-stationary shock diffraction, to an unsteady anomalous reflection. This process is analogous to the transition from a regular to a Mach reflection when the reflected wave is a rarefaction instead of a shock. These bifurcations have been incorporated into a front tracking code that provides an accurate description of wave interactions. Numerical results for two illustrative cases are described; a planar shock passing over a bubble, and an expanding shock impacting a planar contact.

Diffuse ions at a quasi‐parallel collisionless shock: Simulations

Abstract: Large scale one-dimensional hybrid simulations of a quasi-parallel (ΘBn = 20°) high Mach number (MS ∼ 5) collisionless shock have been performed. The numerical system is 500 ion inertial lengths long and the shock has been followed up to 150 ion gyroperiods. The omnidirectional distribution function of the ions in the region upstream of the shock exhibits a high energy tail. About 5% of all ions in the upstream region have a velocity exceeding the shock ram velocity. Phase space contour plots show that these ions are diffuse. These are accelerated to high velocities during their first encounter with the shock. They stay very close to the shock for an extended period of time, about 50 gyroperiods, and experience the electric and magnetic field near the constantly reforming shock. Although the electric field at the particle position is highly fluctuating, the particle motion leads to a net energy gain. These particles constitute a seed particle population for a first order Fermi process. The initial state of such a Fermi process can indeed be seen: some particles escape upstream, are turned back by the upstream waves and recross the shock, thus increasing their energy even further.

Interaction of a normal shock wave with a compressible turbulent flow

Abstract: A speckle photographic method, which is sensitive to changes of gradients in fluid density, is applied for analyzing a compressible turbulent air flow with density fluctuations. Spatial correlation coefficients, turbulent length scales, and energy spectra are determined under the assumption of homogeneous isotropic turbulence. The experiments are performed in a shock tube where the flow is passed through a turbulence grid. Measurements are taken before and after the turbulent regime interacts with the normal shock wave reflected from the tube's end wall. Amplification of the turbulence intensity by the shock interaction process is verified quantitatively and is shown to be restricted to the lower wave numbers in the spectrum.

Turbulence phenomena in a multiple normal shock wave/turbulent boundary-layer interaction

Abstract: An experimental investigation of a Mach 1.61 multiple normal shock wave/turbulent boundary-layer interaction in a rectangular, nearly constant area duct is discussed with an emphasis on the turbulence phenomena. The two-component laser Doppler velocimeter measurements reveal a large amplification of the turbulence kinetic energy and Reynolds stress through the interaction. The leading shock in the multiple shock pattern causes a significant distortion of the turbulent stress tensor. Partial recovery occurs immediately downstream of the first shock. The trailing shocks in the system are much weaker than the first shock and tend to maintain the nonequilibrium turbulence structure, with complete recovery occurring well downstream of the interaction.

Flow downstream of the heliospheric terminal shock: 1. Irrotational flow

Abstract: Recent reports of remote detection of the heliospheric terminal shock place it near 50 AU. These conflict with standard models which, when combined with current data on the local interstellar medium, place the shock beyod 100 AU. Resolution of this discrepancy has led to hypotheses that invoke cosmic ray pressure, momentum exchange with interstellar neutrals, and magnetic field effects between the shock and the contact discontinuity dividing the solar wind from interstellar plasma. These hypotheses depend not only on properties of the interstellar medium, but also on the downstream three-dimensional flow between the shock and the contact discontinuity, in the region called the 'heliosheath'. The downstream flow field in the absence of magnetic fields is examined here under the assumptions that the flow everywhere outside the shock can be approximated as irrotational and incompressible. It is found, in particular, that the distance between the terminal shock and the contact discontinuity is less than the heliocentric distance to the terminal shock, effectively eliminating magnetic field effects in the heliosheath as being dynamically important.

Survey of coherent ion reflection at the quasi-parallel bow shock

Abstract: This paper presents the results of a survey of ion beam observations near the earth's quasi-parallel bow shock. Particular attention is given to three issues: (1) the spatial locations of the cold coherent ion beams relative to the shock, (2) the velocity space locations of the ion beams relative to that expected for specular reflection, and (3) the parameter regimes over which the beams are observed. Evidence is found that coherent ion reflection occurs commonly in the quasi-parallel regime, over essentially the full range of upstream plasma parameters normally found at the earth's bow shock. It was also observed that the cold coherent reflected beams spread rapidly in the velocity space upstream from the shock. The observations presented serve as an extension to those presented by Gosling et al. (1989) and further strengthen the conclusion that specularly reflected ions are contributing directly to the downstream thermalization at the earth's quasi-parallel bow shock.

MHD intermediate shocks in coronal mass ejections

Abstract: A simplified model of coronal mass ejections is considered in which at least a portion of the interaction with the background corona involves a shock wave, and the allowable shock solutions and their compressive signatures are examined. The MHD shock-jump equations have a maximum of three possible types of solutions with an entropy rise for fixed values of the physical variables (slow, intermediate, and fast shocks). However, one of the three solution classes (the intermediate shock) is widely believed to not occur in nature and is regarded as nonevolutionary or extraneous. Without the intermediate shock, there is no multiplicity of solutions in that only one shock (or none) can occur for given physical values. All three potential shock types are considered, and it is shown solely on the basis of the shock-jump equations, that intermediate shocks must exist along some segment of the shock front for certain parametric regimes and for conditions that probably occur in some coronal mass ejections.

The structure of resistive‐dispersive intermediate shocks

Abstract: The structure of intermediate shocks is studied on the basis of the resistive, nonviscous two-fluid equations. Electron inertia effects are neglected so that the generalized Ohm's law contains only the Hall current and the electron pressure terms in addition to the usual resistive term and the electric field. As for the case of purely resistive MHD, reported recently by Hau and Sonnerup (Journal of Geophysical Research, 94, 6539, 1989), fixed-point analysis is performed to examine the nature of the magnetic structure near the upstream and downstream states of the intermediate shock. The one-dimensional, steady state, resistive Hall MHD equations are then integrated numerically to generate complete shock structures which are presented in the form of magnetic hodograms. These hodograms describe fast and slow shocks in addition to intermediate shocks. As expected, the calculations show that the main effect of Hall currents is to remove the symmetry between left-hand and right-hand polarized shock structures found in the purely resistive case and sometimes to convert the smooth shock transitions obtained from the resistive MHD model into transitions that incorporate oscillatory standing wave train structures at their upstream and/or downstream edge. The magnetic structure in the plane of the shock near the possible upstream and downstream states of the intermediate shock, which in the case of purely resistive MHD is either a node or a saddle, can be either a node, a saddle or a spiral point, the latter corresponding to a standing wave train, when the Hall term is included. As a result, the number of possible types of magnetic hodogram topology increases from 3 in the resistive case, to a total of 20. However, it appears that the constraints provided by the shock jump conditions make certain of these topologies unattainable: only 13 of the 20 cases have been found and are reported in the paper. The relationship between small-amplitude dispersive waves in the flow upstream or downstream of a shock and the nature of the corresponding fixed point is also discussed.

Particle acceleration at shocks with surface ripples

Abstract: The present treatment of superthermal-ion acceleration on the surface of a fast-mode hydromagnetic shock gives attention to (1) small-amplitude surface ripples characterized by width L and amplitude A that are large relative to the energetic-ion gyroradius, and (2) shocks which are on average quasi-perpendicular. An investigation is made of the effects of the confinement, evolving geometry, and finite shock curvature associated with the ripple, by integrating along the orbits of the proton test particles. As an upstream magnetic field line convects through the surface ripple, it intersects the shock at two points, thereby forming a temporary magnetic trap. Flux-line profiles and angular distributions in a given ripple differ substantially, depending on the path it takes through the ripple and its distance from the shock.

Two dimensional hybrid simulation of a curved bow shock

Abstract: Results are presented from two-dimensional hybrid simulations of curved collisionless supercritical shocks, retaining both quasi-perpendicular and quasi-parallel sections of the shock in order to study the character and origin of the foreshock ion population. The simulations demonstrate that the foreshock ion population is dominated by ions impinging upon the quasi-parallel side of the shock, while nonlocal transport from the quasi-perpendicular side of the shock into the foreshock region is minimal. Further, it is shown that the ions gain energy by drifting significantly in the direction of the convection electric field through multiple shock encounters.

Shock/shock interference on a transpiration cooled hemispherical model

Abstract: Experimental results are presented which show the effectiveness of transpiration cooling in reducing the peak heat flux caused by an impinging shock on a bow shock of a hemispherical model. The 12-inch diameter hemispherical transpiration model with helium coolant was tested in the Calspan 48-inch Hypersonic Shock Tunnel at nominal Mach 12.1 and freestream unit Reynolds number of 0.33 x 10 to the 6th/ft. An incident shock wave, generated by a blunt flat-plate shock generator inclined at 10 deg to the freestream, intersected the bow shock of the model to produce shock/shock interference. The stagnation heat flux without coolant or shock/shock interference was about 1.6 times a smooth surface laminar prediction due to effective roughness of the coolant ejection slots. A coolant mass flux 31 percent of the freestream mass flux reduced the stagnation heat flux to zero without shock/shock interference. However, for the same coolant mass flux and with shock/shock interference the peak heat flux was only reduced 8.3 percent, even though the total integrated heat load was reduced.

Shock interference studies on a circular cylinder at Mach 16

Abstract: This paper presents the results of a computational study on the shock interference problems on a cylindrical body typical of an engine inlet cowl leading edge at a nominal Mach number of 16. The two-dimensional Navier-Stokes equations are solved assuming the flow to be in chemical and thermal equilibrium and using a finite element method. The algorithm employs a cell-centered fully implicit upwind scheme. Adaptively generated unstructured meshes of triangles and quadrilaterals are employed. Under certain conditions the finite element code resulted in oscillatory solutions for shock interference at Mach 16. Some of the causes of the unsteady behavior are identified, and to the extent possible, such situations are avoided in the present application. Two shock interference conditions involving a Type IV (supersonic jet) and a Type III (attaching shear layer) are solved. The results are compared with available experimental data and reasonable agreement is observed. A semi-empirical method is also used to estimate the maximum surface heat flux and static pressure.

Interaction of a plane shock wave with slitted wedges

Abstract: An experimental and numerical study was made of shock wave transition over slitted wedges. Experiments were conducted in a shock tube by using double exposure holographic interferometry. Shock Mach numbers ranged from 1.07 to 3.03 in air. Slitted wedge models having perforation ratios of 0.34 and 0.4 were installed in the test section. The critical transition angle was obtained analytically by the shock polar analysis where effects of boundary conditions, wall suction, and surface roughness were empirically taken into account. As the results, it was found that for stronger shock waves and a perforation ratio of 0.4, the critical transition angle was decreased by about 10° in comparison to the detachment criterion. A flow visualization study clarified various wave interaction mechanisms associated with the wall suction. The critical transition angle was successfully explained by the shock polar analysis. The PLM numerical simulation was also conducted. The numerical result agreed very well with the experimental findings.

Experimental and computational study of the effect of shocks on film cooling effectiveness in scramjet combustors

Abstract: This paper presents results from a study conducted to investigate the effect of incident oblique shocks on the effectiveness of a coolant film at Mach numbers, typical of those expected in a scramjet combustor at Mach 15 to 20 flight. Computations with a parabolic code are in good agreement with the measured pressures and heat fluxes, after accounting for the influence of the shock upstream of its point of impingement on the plate, and the expansion from the trailing edge of the shock generator. The test data shows that, for the blowing rates tested, the film is rendered largely ineffective by the shock. Computations show that coolant blowing rates five to ten times those tested are required to protect against shock-induced heating. The implications of the results to scramjet combustor design are discussed.

Investigation of Roe's 2D wave decomposition models for the Euler equations

Abstract: The subject of this work is the implementation of a new technique to solve the 2-D Euler equations by modeling the solution with a discrete number of simple waves. The theory of this decomposition was proposed by P. L. Roe in 1985 but was never put successfully into practice before. The global solution is obtained by superposition of elementary solutions consisting of plane waves. The traveling directions of these waves are not fixed in advance but depend on the local flow gradient at each time step. Three different wave decomposition models have been considered and implemented, based on four acoustic waves, one entropy wave and respectively vorticity (model A), or one shear wave propagating in the perpendicular direction to the streamlines (model B), or one shear wave propagating in the direct ion of the pressure gradient (model C). Different first order upwind fluctuation splitting schemes have been tested both for triangular and quadrilateral cells. Different test cases have been solved with the new method: shear flow, oblique shock reflection, ramp flow, nozzle flow; a comparison among the results obtained with the different decomposition models and numerical schemes has been made.

Coronal mass ejection shock fronts containing the two types of intermediate shocks

Abstract: Numerical solutions of the time-dependent, magnetohydrodynamic (MHD) equations in two dimensions are used to demonstrate the formation of both types of intermediate shocks in a single shock front for physical conditions that are an idealization of those expected to occur in some observed coronal mass ejections. The key to producing such a shock configuration in the simulations is the use of an initial atmosphere containing a magnetic field representative of that in a coronal streamer with open field lines overlying a region of closed field lines. Previous attempts using just open field lines (perpendicular to the surface) produced shock configurations containing just one of the two intermediate shock types. A schematic of such a shock front containing both intermediate shock types has been constructed previously based solely on the known properties of MHD shocks from the Rankine-Hugoniot equations and specific requirements placed on the shock solution at points along the front where the shock normal and upstream magnetic field are aligned. The shock front also contains, at various locations along the front, a hydrodynamic (nonmagnetic) shock, a switch-on shock, and a fast shock in addition to the intermediate shocks. This particular configuration occurs when the shock front speed exceeds themore » upstream (preshock) intermediate wave speed but is less than a critical speed defined in the paper (equation 1) along at least some portion of the shock front. A distinctive feature of the front is that it is concave upward (away from the surface) near the region where the field in the preshock plasma is normal to the front of near the central portion of the shock front.« less

Numerical Simulations of Collisionless Shocks

Abstract: This tutorial-style review presents recent developments of the numerical simulation activity devoted to the study of collisionless shock waves. First, background (theoretical and numerical) characteristics of non-linear waves and shock waves are reviewed. Second, a short description of the main features of shocks commonly observed in space plasmas is presented for both fields components and particles dynamics. Third, a review of the various appropriate simulation codes is presented with a particular emphasis of their respective advantages and disadvantages for the study of collisionless shock waves. Fourth, typical patterns of magnetosonic shock waves, such as the downstream (or upstream) “wavetrain”, the “ramp”, and the “foot” are shown to be well recovered in numerical results and are associated with particle acceleration mechanisms which are discussed in terms of dissipation sources. Fifth, a classification of the various plasma instabilities related to a magnetosonic shock is presented; these instabilities are strongly related to the particle dynamics presented in the previous section and are additional sources of energy dissipation; these are shown to affect strongly the overall structures of the shock and its nearest downstream and upstream neighborhood. Finally, some applications to astrophysical shocks will be presented. Numerical simulation appears to be a very powerful tool in order to interpret complex patterns of shock waves associated with some intricate wave-particle interaction mechanisms which are important sources of energy dissipation. In particular, such a tool is shown to represent, until now, the only way “to visualize” carefully the overall dynamics of particle species by the use of appropriate diagnostics.

Global mixing in SN 1987A by the asymmetric shock wave

Abstract: The possibility of global mixing of matter by the nonspherical shock wave in the SN explosion was investigated by the two-dimensional axial symmetric hydrodynamical simulation. It is found that (1) the asymmetric shock wave is generated by passing through the flattened core induced by stellar rotation, and (2) inner matter tends to move toward the symmetric axis, and global matter mixing is possible if the degree of the deformation of the core is large, since this asymmetry of shock is not dumped even if the outer envelope is almost spherically symmetric. If the initial shock is jetlike, this tendency is strengthened more. This asymmetric shock wave also may explain the nonspherically expanding envelope in SN 1987A (suggested by the speckle and polarization observations). 21 refs.

Experiments in a shock wave/homogeneous turbulence interaction

Abstract: Mast of the previous work on shock wave interaction with turbulent flows wes limited to shock wave/boundary layer interaction. The present study is f0cussir.g on the interactions of the normal shack wave with homogeneauslgrid-generated turbulence. A shack wave formed in a shack-tube is passing through a grid and the induced flow behind the shock has the features of a compressible flow with free stream turbulence. The decaying turbulence is subjected to an interaction with the reflected shock traveling in the opposite direction. Data were sampled simultaneously from four channels of high frequency response pressure transducers and hot-wires. Turbulence is considerably amplified during the interaction while spectral energy changed too. Results indicated that the amplification ratio is not the same for all investigated length scales and turbulence intensities during interactions with shock waves of the same strength.

Mach 6 testing of two generic three-dimensional sidewall compression scramjet inlets in tetrafluoromethane

Abstract: Three-dimensional sidewall compression scramjet inlets with leading edge sweeps of 30 and 70 degrees have been tested in the Langley Hypersonic CF4 Tunnel at Mach 6 and a ratio of specific heats of 1.2. The effects of cowl position, contraction ratio, and Reynolds number were investigated. The models were instrumented with 42 static pressure orifices distributed on the sidewalls, baseplate, and cowl. Schlieren movies were made of each test for flow visualization of the entrance plane and cowl region. In order to obtain an approximate characterization of the flow field, a modification to two-dimensional inviscid oblique shock theory was derived to accommodate the three-dimensional effects of leading edge sweep. This theory qualitatively predicted the reflected shock structure/sidewall impingement locations and the observed increase in spillage (flow upturning) with increasing leading edge sweep. The primary effect of moving the cowl forward is capturing the flow which would have otherwise spilled out ahead of the cowl. Increasing the contraction ratio (moving the sidewalls closer together) increases the number of internal shock reflections and hence incrementally increases the sidewall pressure distribution. Significant Reynolds number effects were noted over a small range of Reynolds number.

The Structural Stability of Oblique Detonation Waves

Abstract: We consider small two-dimensional spatial perturbations to a one-dimensional oblique detonation wave, and examine how they behave as the distance along the wave, in the downstream direction, is increased. With the kinetics modeled by one-step Arrhenius kinetics it is shown that, for large activation energy, the perturbations grow. This result is interpreted as a structural instability which will prevent the existence of one-dimensional oblique detonation waves. This result is compared with numerical results of Fujiwara,