Mit dieser Arbeit liegt eine ausführliche Studie zu den Integralverfahren der Grenzschichttheorie vor. Der Schwerpunkt liegt dabei auf den Dissipationsintegralverfahren. Im Vergleich zu anderen Grenzschichtverfahren bestehen die generellen Vorteile der Integralverfahren in ihrer Robustheit, ihrer hohen Praktikabilität sowie den im Sinne der aufzuwendenden Computerkapazität geringen Kosten. Ein spezieller Vorteil der Dissipationsintegralverfahren ist es zudem, das komplette Schubspannungsprofil der Grenzschicht zu berücksichtigen. Über eine entsprechende Ableitung der Dissipationsintegrale kann die Vorgeschichte der Grenzschicht erfassen werden. Ausgangspunkt der Arbeit ist eine Literaturanalyse, welche aufzeigt, dass Integralverfahren weit verbreitet sind und für vielfältige Typen von Grenzschichten Anwendung finden. Ausgehend von den Navier-Stokes-Gleichungen wird über die Grenzschichtgleichungen die allgemeine Form der Dissipationsintegralgleichungen hergeleitet. Auf der Basis dieser Gleichungen werden Berechnungsalgorithmen für zwei- und dreidimensionale Grenzschichten entworfen und ausführlich diskutiert. Die zur Komplettierung der Berechnungsalgorithmen benötigten Parameterzusammenhänge werden aus expliziten Geschwindigkeitsprofilen hergeleitet. Hierzu werden beruhend auf dem Zweischichtenmodell turbulenter Grenzschichten Geschwindigkeitsprofile turbulenter Grenzschichten diskutiert. Drei Kombinationen von Haupt- und Querströmungsprofil werden für den dreidimensionalen Fall ausgewählt und algorithmisch umgesetzt. Die Algorithmen für zweidimensionale Grenzschichten beruhen auf einem Geschwindigkeitsprofil. Dem Aufbau der Algorithmen schließen sich ausführliche Testrechnungen sowie eine Bewertung der Verfahren an. Es wird festgestellt, dass Dissipationsintegralverfahren für zwei- und dreidimensionale Grenzschichten mit gutem Erfolg angewandt werden können. Vergleiche der Rechenergebnisse für zweidimensionale Grenzschichten zeigen die zumindest Gleichwertigkeit mit Zweigleichungsmodellen an. / Calculation and prediction of turbulent boundary layers are among the most challenging tasks of present fluid mechanics. A strong demand exists for robust, easy-to-handle and in terms of computing effort cheap algorithms which can be used for technical applications. From an engineering point of view zonal methods and RANS are the most useful tools for solution of fluid mechanical problems. It is known that zonal methods which use integral approaches for the description of the boundary layer can be used successfully in manifold forms. One way to improve zonal methods further is the Basically three different types of integral algorithms - entrainment, momentum of momentum and dissipation integral method - can be derived from the three-dimensional boundary layer equations. If one compares the usual entrainment integral method with the dissipation integral method it turns out that the latter has the following physical advantages. While the entrainment method considers information about the shear stress distribution only at the outer edge of the boundary layer, the dissipation integral method uses the whole distribution. This work gives an overview over dissipation integral method and extends them to three-dimensional boundary layers. The general integral equations for the three-dimensional case are derived. Using two different sets of mean velocity profiles the hyperbolical character of a dissipation integral method is shown. Apart from the integral momentum balance, the dissipation integral method satisfies a second major balance with the integral balance of mechanical energy. It is found that for a practical calculation the integral momentum equation and the integral energy equation are most useful. Sixteen two-dimensional experimental test cases with none-zero pressure gradients were computed. It was found that the averaged relative deviation between measured and computed values for the skin friction coefficient is about 5 % and about 3 % for the shape parameter. Two three-dimensional fully turbulent boundary layers approaching an obstacle where computed. The agreement between experimental results and the calculation is reasonably good. The calculation allows the prediction of the velocity distributions. Flow angle and flow gradient angle distributions being additional results.

Couette, pipe, channel, and zero-pressure gradient (ZPG) turbulent boundary layer (TBL) flows have classically been considered as canonical wall-bounded turbulent flows since their near-wall behavior is generally considered to be universal, i.e. invariant of the flow case and the Reynolds number. Nevertheless, the idea that large-scale motions, being dominant in regions further away from the wall, might interact with and influence small-scale fluctuations close to the wall has not been disregarded. This view was mainly motivated due to the observed failure of collapse of the Reynolds normal stresses in viscous scaling. While this top-down influence has been studied extensively over the last decade, the idea of a bottom-up influence (backward energy transfer) is less examined. One exception was the recent experimental work on a Couette flow by Kawata, T. & Alfredsson, P. H. (Phys. Rev. Lett. 120, 244501, 2018). In the present work, a spectral representation of the Reynolds Stress transport equation is used to perform a scale-by-scale analysis of the terms in the equation. Two flow cases were studied: first, a Direct Numerical Simulation (DNS) of a Couette flow at a similar Reynolds number as Kawata and Alfredsson. The Reynolds number was ReT = 120, viscosity v. Second, a Large Eddy Simulation (LES) of a ZPG TBL at ReT = 730, 1270, and 2400. For both cases the classic interscale transport or turbulent kinetic energy was observed. However, also an inverse interscale transport of Reynolds shear stress was observed for both cases.

Modeling the effects of surface roughness is an area of concern in many practical engineering applications. Many current roughness models to this point have involved the use of empirical 'constants' and equivalent sand grain roughness. These underdeveloped concepts have little direct relationship to realistic roughness and cannot predict accurately and consistently the flow characteristics for different roughness shapes. In order to aid in the development of turbulence models, the present research is centered around the experimental investigation of seven various shaped single roughness elements and their effects on turbulence quantities in a two-dimensional turbulent boundary layer. The elements under scrutiny are as follows: cone, cone with spatial variations equal to the smallest sublayer structure length scale, cone with spatial variations equal to 2.5 times the smallest sublayer structure length scale, Gaussian-shaped element, hemisphere, cube aligned perpendicular to the flow (cube at 90°), and a cube rotated 45° relative to the flow. The roughness element heights, k+, non-dimensionalized by the friction velocity (U_tau) of the approaching turbulent boundary layer, are 145, 145, 145, 145, 80, 98, and 98 respectively. Analysis of a three-dimensional fetch of the same Gaussian-shaped elements described previously was also undertaken. In order to analyze the complex flow fields, detailed measurements were obtained using a fine-measurement-volume (50 micron diameter) three-velocity component laser-Doppler velocimetry (LDV) system. The data reveals the formation of a horseshoe vortex in front of the element, which induces the downwash of higher momentum fluid toward the wall. This 'sweep' motion not only creates high Reynolds stresses (v^2, w^2, -uv) downstream of the element, but also leads to higher skin-friction drag. Triple products were also found to be very significant near the height of the element. These parameters are important in regards to the contribution of the production and diffusion of the turbulent kinetic energy in the flow. The 'peakiness' of the roughness element was found to have a direct correlation to the production of circulation, whereas the spatial smoothing does not have an immense effect on this parameter. The peaked elements were found to have a similar trend in the decay of circulation in the streamwise direction. These elements tend to show a decay proportional to (x/d)^-1.12, whereas the cube elements and the hemisphere do not have a common trend. A model equation is proposed for a drag correlation common to all roughness elements. This equation takes into account the viscous drag and pressure drag terms in the calculation of the actual drag due to the roughness elements presence in the boundary layer. The size, shape, frontal and wetted surface areas of the roughness elements are related to one another via this model equation. Flow drawings related to each element are presented which gives rise to a deeper understanding of the physics of the flow associated with each roughness element. / Master of Science

The work presented in this thesis was on nominally two-dimensional turbulent boundary layers at zero pressure gradient subjected to high free-stream turbulent intensities of up to 7.9% in preparations for high free-stream turbulence studies on three-dimensional boundary layers, which will be done in the future in the Aerospace and Ocean Engineering Boundary Layer Wind Tunnel at Virginia Tech. The two-dimensional turbulent flow that will impinge three-dimensional bodies needed to be characterized, before the three-dimensional studies can be made. An active turbulence generator designed to create high free-stream turbulence intensities in the wind tunnel was tested and modified in order to obtain the lowest possible mean flow non-uniformities. A seven-hole pressure probe was used to obtain planes of mean velocity measurements. A three-component state of the art laser-Doppler velocimeter (LDV) was used to obtain mean and fluctuating velocities. Previous high free-stream turbulence studies have been reviewed and are discussed, and some of the previously published data of other authors have been corrected. Based on the measurements obtained with the LDV, it was also determined that the semi-log law of the wall is valid for high free-stream turbulence cases, but with different constants than the ones proposed by Coles, where the constants for the high free-stream cases may be dependent on the turbulence intensity. For the first time, the skin friction coefficient (Cf) was deduced from the viscous sublayer. The difference between the U_tau obtained in the viscous sublayer mean velocity profile and the U_tau obtained in the semi-log layer was 1.5%. The skin friction coefficient was determined to increase by 10.5% when the two-dimensional turbulent boundary layer was subjected to high free-stream turbulence effects. Spectral data obtained with the LDV, were compared to the von Kármán model spectrum and to the Pope's model spectrum, where the von Kármán spectrum was proven to fit the spectral data slightly better than the Pope's spectrum. Finally, the Hancock-Bradshaw-Blair parameter obtained for this experiment agreed very well with previously published data. / Master of Science

Computational Fluid Dynamics is a powerful and widely used tool for developing projectsthat concern flow motion, in very different fields. Industrial CFD solvers are continuouslydeveloped with the aim of improving accuracy and reducing the computational cost of thesimulations. Turbulent wall-flow cases are particular demanding as the presence of a solidsurfaceinterface generates steep gradients in the proximity of the wall. Resolving suchgradients can be crucial to obtain a consistent solution but also very expensive in terms ofgrid refinement, and hence computational time. Wall functions are widely used and offersignificant computational savings when it comes to near-wall flow resolution. Previous wallfunction implemented in the M-Edge solver suffered by poor performances in complex flowscharacterized by strong pressure-gradient phenomena, such as separation. A new formulationhas been developed and validated for k − omega and Spalart-Allmaras turbulence models. Testsimulations started from simple and near-ideal cases (2D zero pressure gradient flat plate)and advanced to always more complex flow cases and geometries (full 3D general fighter).Every case has been run coupling the wall-function boundary condition with three differentturbulence models: the Menter SST, the Menter BSL with an EARSM and the Spalart-Allmaras one-equation model. Overall results showed the upgraded performance of new wallfunction in flow resolution together with more agile grid requirements, faster and deeperconvergence of the residuals and a general reduction in computational time. / Berör strömmande fluider inom mycket olika områden. Industriella CFD-lösare utvecklaskontinuerligt i syfte att förbättra noggrannheten och minska beräkningskostnaderna försimuleringarna. Turbulent strömning nära väggar är särskilt krävande eftersom närvaron avett fast ytgränssnitt genererar stora gradienter i närheten av väggen. Att lösa upp sådanagradienter kan vara avgörande för att få en konsistent lösning men också mycket beräkningskrävandepå grund av nödvändig nätförfining.Väggfunktioner används ofta och ger betydandereduktioner i beräkningstid när det gäller att lösa upp strömningen nära vägg. En tidigareväggfunktion implementerad i M-Edge-lösaren led av dåliga prestanda i komplexa flödenmed starka tryckgradienter, såsom separation. En ny formulering har utvecklats och valideratsför k − omega och Spalart-Allmaras turbulensmodeller. Den har testats för enkla generiska fall(2D-plan platta utan tryckgradient) och för mer avancerade och komplexa strömningsfall ochgeometrier (komplett 3D-stridsflygplan).Varje fall har körts med väggfunktionens randvillkorkopplat med tre olika turbulensmodeller: Menter SST, Menter BSL med EARSM och Spalart-Allmaras enekvationsmodell. De övergripande resultaten visar att nya väggfunktionen gerbetydande förbättringar i att beskriva strömningen tillsammans med reducerade krav pånätupplösning, snabbare och djupare konvergens av lösningen och en allmän minskning avberäkningstiden.

Les écoulements décollés sont fortement instationnaires, l'objectif de cette thèse a été de localiser et d'identifier les phénomènes à la source de ces instationnarités et de comprendre les processus physiques permettant le transfert de l'information de ces zones sources au reste de l'écoulement. Pour ce faire une analyse des résultats issus de simulations numériques a été réalisée. En étudiant la corrélation et la cohérence entre les positions de choc et les fluctuations de pression, l'interaction a pu être séparée en plusieurs parties distinctes. A l'aide de la théorie des caractéristiques définissant les directions et les cinématiques de propagation de l'information, les liens spatio-temporels entre ces différentes régions ont pu être déterminés. Les résultats de ces études couplés avec ceux issus des expériences ont montré clairement que les phénomènes se produisant à l'intérieur de la zone de recirculation existant en aval du choc de décollement gouvernent la dynamique de la totalité de l'interaction, aussi bien à basse fréquence qu'à moyenne fréquence. Ainsi les mouvements de choc apparaissent comme le miroir des phénomènes se produisant à l'intérieur de la zone décollée. Une représentation équivalente en fluide non visqueux permettant une description du comportement instationnaire de l'interaction a aussi été proposée. / Separated flow are often strongly unsteady; the aim of this thesis is to localize and identify the sources of the unsteadiness and to understand the physical phenomena governing the information transfer from these source zones to the rest of the flow. To do this, data used for this analysis have been obtained from numerical simulations (LES). Both cross-correlation and coherence between shock motion and pressure fluctuations have shown that the interaction can be split in several distinct zones. The theory of characteristics is used to define the information paths and the propagation velocities, so that the space-time links between these regions have been determined. Both numerical and experimental studies have clearly shown that phenomena present within the recirculation buble govern the whole of the interaction, at low and intermediate frequencies. Indeed the shock motions appears as the mirror of phenomena present in the separated zone. An inviscid equivalent scenario has been proposed to represent the interaction.

This thesis deals with the description and development of two classical turbulent shear flows, namely free jet and flat plate turbulent boundary layer flows. In both cases new experimental data has been obtained and in the latter case comparisons are also made with data obtained from data bases, both of experimental and numerical origin. The jet flow studies comprise three parts, made in three different experimental facilities, each dealing with a specific aspect of jet flows. The first part is devoted to the effect of swirl on the mixing characteristics of a passive scalar in the near-field region of a moderately swirling jet. Instantaneous streamwise and azimuthal velocity components as well as the temperature were simultaneously accessed by means of combined X-wire and cold-wire anemometry. The results indicate a modification of the turbulence structures to that effect that the swirling jet spreads, mixes and evolves faster compared to its non-swirling counterpart. The high correlation between streamwise velocity and temperature fluctuations as well as the streamwise passive scalar flux are even more enhanced due to the addition of swirl, which in turn shortens the distance and hence time needed to mix the jet with the ambient air. The second jet flow part was set out to test the hypothesis put forward by Talamelli & Gavarini (Flow, Turbul. & Combust. 76), who proposed that the wake behind a separation wall between two streams of a coaxial jet creates the condition for an absolute instability. The experiments confirm the hypothesis and show that the instability, by means of the induced vortex shedding, provides a continuous forcing mechanism for the control of the flow field. The potential of this passive mechanism as an easy, effective and practical way to control the near-field of interacting shear layers as well as its effect towards increased turbulence activity has been shown. The third part of the jet flow studies deals with the hypothesis that so called oblique transition may play a role in the breakdown to turbulence for an axisymmetric jet.For wall bounded flows oblique transition gives rise to steady streamwise streaks that break down to turbulence, as for instance documented by Elofsson & Alfredsson (J. Fluid Mech. 358). The scenario of oblique transition has so far not been considered for jet flows and the aim was to study the effect of two oblique modes on the transition scenario as well as on the flow dynamics. For certain frequencies the turbulence intensity was surprisingly found to be reduced, however it was not possible to detect the presence of streamwise streaks. This aspect must be furher investigated in the future in order to understand the connection between the turbulence reduction and the azimuthal forcing. The boundary layer part of the thesis is also threefold, and uses both new data as well as data from various data bases to investigate the effect of certain limitations of hot-wire measurements near the wall on the mean velocity but also on the fluctuating streamwise velocity component. In the first part a new set of experimental data from a zero pressure-gradient turbulent boundary layer, supplemented by direct and independent skin friction measurements, are presented. The Reynolds number range of the data is between 2300 and 18700 when based on the free stream velocity and the momentum loss thickness. Data both for the mean and fluctuating streamwise velocity component are presented. The data are validated against the composite profile by Chauhan et al. (Fluid Dyn. Res. 41) and are found to fulfil recently established equilibrium criteria. The problem of accurately locating the wall position of a hot-wire probe and the errors this can result in is thoroughly discussed in part 2 of the boundary layer study. It is shown that the expanded law of the wall to forth and fifth order with calibration constants determined from recent high Reynolds number DNS can be used to fix the wall position to an accuracy of 0.1 and 0.25 l_ * (l_* is the viscous length scale) when accurately determined measurements reaching y+=5 and 10, respectively, are available. In the absence of data below the above given limits, commonly employed analytical functions and their log law constants, have been found to affect the the determination of wall position to a high degree. It has been shown, that near-wall measurements below y+=10 or preferable 5 are essential in order to ensure a correctly measured or deduced absolute wall position. A  number of peculiarities in concurrent wall-bounded turbulent flow studies, was found to be associated with a erroneously deduced wall position. The effect of poor spatial resolution using hot-wire anemometry on the measurements of the streamwise velocity is dealt with in the last part. The viscous scaled hot-wire length, L+, has been found to exert a strong impact on the probability density distribution (pdf) of the streamwise velocity, and hence its higher order moments, over the entire buffer region and also the lower region of the log region. For varying Reynolds numbers spatial resolution effects act against the trend imposed by the Reynolds number. A systematic reduction of the mean velocity with increasing L+ over the entire classical buffer region and beyond has been found. A reduction of around 0.3 uƬ, where uƬ is the friction velocity, has been deduced for L+=60 compared to L+=15. Neglecting this effect can lead to a seemingly Reynolds number dependent  buffer or log region. This should be taken into consideration, for instance, in the debate, regarding the prevailing influence of viscosity above the buffer region at high Reynolds numbers. We also conclude that the debate concerning the universality of the pdf within the overlap region has been artificially complicated due to the ignorance of spatial resolution effects beyond the classical buffer region on the velocity fluctuations. / QC 20100820

Near wake is the region of wake flow just behind the trailing edge of the body where the flow is strongly influenced by the upstream flow conditions and also perhaps by the charac­teristics of the body. The present work is concerned with the study of the development of turbulent near wake behind an infinitely yawed flat plate. The turbulent near wake behind an infinitely yawed flat plate is the simplest of the three-dimensional turbulent near wake flows. The present study aims at providing a set of data on the turbulent near wake behind an infinitely yawed flat plate and also at understanding the development and structure of the near wake. Detailed measurements of mean and turbulent quantities have been made using 3-hole probe, X-wire and 3-wire hotwire probes. Further an asymptotic analysis of the two-dimensional turbulent near wake flow has been formulated for the near wake behind an infinitely yawed flat plate. The feature that the near wake which is dominated by mixing of the oncoming turbulent boundary layer retains, to a large extent, the memory of the turbulent structure of the boundary layer, has been exploited to develop this analysis. The analysis leads to three regions of the wake flow (the inner near wake, the outer near wake and the far wake) for which the governing equations are derived. The matching conditions among these regions lead to logarithmic variations in both normal and longitudinal directions in the overlapping regions surrounding the inner wake. These features are validated by the present results. A computational study involving seven well known turbulence models was also under­taken in order to assess the performance of the existing turbulence models in the prediction of the turbulent near wake behind an infinitely yawed flat plate. In this study all the seven models are implemented into a common flow solver code, thus eliminating the influence of grid size, initial conditions and different numerical schemes while making the comparison. This study shows that the K - e model performs better than other models in predicting the near wake behind an infinitely yawed flat plate.

The laminar-turbulent transition in viscous fluid flows is one of the most intriguing problems in fluid dynamics today. In view of the enormous applications it has in a variety of fields such as aircraft design, turbomachinery, etc., scientists have now realized the importance of tackling this problem effectively. Three-dimensional flows are usually associated with pressure gradient, streamline curvature, streamline convergence / divergence etc., all acting simultaneously. Towards a better understanding of the transition process and modeling the transition zone, it is important to study the effect of each of these parameters on the transitional flow. The present work aims at studying experimentally the effect of lateral streamline convergence alone on the laminar-turbulent transition zone under constant stream-wise pressure. The experimental setup consists of a low turbulence wind tunnel with its test section modified to cause lateral streamline convergence under constant pressure. This is achieved by converging the side-walls and appropriately diverging the roof, thus maintaining a constant stream-wise pressure. The half angle of convergence is chosen as 100 , which is approximately the same as the half of the turbulent spot envelope in constant pressure two-dimensional flows. Experiments are carried out to analyze the development of the laminar and transitional boundary layers, intermittency distribution in the transition zone and the overall characteristics of an artificially induced turbulent spot. The laminar velocity profiles are found to be of the Blasius type for two-dimensional constant pressure flows. However, the converging streamlines are found to contribute to an increased thickness of the boundary layer as compared to the corresponding two-dimensional flow. The intermittency distribution in the transition zone is found to follow the universal intermittency distribution for two-dimensional constant pressure flow. A simple linear-combination model for two-dimensional flows is found to perform very well in predicting the measured velocity profiles in the transition zone. An artificially introduced turbulent spot is found to propagate along a conical envelope with an apex cone angle of 220 which is very nearly the value for a corresponding constant pressure two-dimensional flow. The spot shapes and celerities are also comparable to those in two-dimensional flow. In summary, the present study brings out many similarities between a constant pressure laterally converging flow and a constant pressure two-dimensional flow.

Lors du développement d’un nouvel avion, l’estimation des échanges d’énergie entre l’air ambiant et les parois est une donnée cruciale pour la conception aérothermique. Cette conception repose de plus en plus sur des simulations numériques mais certains phénomènes d’aérothermique externe, comme le jet débouchant du système de dégivrage des nacelles moteur, montrent les limites des modèles RANS classiques. La simulation des grandes échelles (LES) se révèle bien adaptée à ce type de phénomène mais se heurte à un coût de calcul extrêmement élevé pour ces écoulements pariétaux à très grand nombre de Reynolds. Pour lever cette limitation, cette thèse propose l’étude de deux briques fondamentales : la LES avec loi de paroi (WMLES) conjuguée à l’injection d’une couche limite turbulente à l’entrée du domaine. Pour une meilleure compréhension et une utilisation fiable de l’approche loi de paroi, on se concentre tout d’abord sur les sources d’erreur qui lui sont associées. Après les avoir identifiées, on propose une correction de l’erreur de sous-maille ainsi qu’une loi de paroi adaptée aux écoulements compressibles. Grâce à ces deux éléments, on obtient une estimation correcte du flux de chaleur pariétal sur des simulations WMLES de canal plan supersonique sur parois froides. Puis, pour préparer la transition vers des applications plus industrielles, on introduit un schéma numérique plus dissipatif ce qui nous permet d’étudier l’influence de la méthode numérique sur l’approche loi de paroi. Dans une seconde partie dédiée à l’injection de couche limite pour la WMLES, on sélectionne une méthode basée sur l’injection de perturbations combinée à un terme de contrôle volumique. On montre que des simulations WMLES utilisant cette méthode d’injection permettent d’établir une couche limite turbulente réaliste à une courte distance en aval du plan d’entrée, à la fois sur une plaque plane mais également sur un écoulement de jet débouchant à la géométrie plus complexe, représentative d’un cas avion. / During the design of a new aircraft, the prediction of energy exchanged between the ambient air and the aircraft walls is crucial regarding aerothermal design. Numerical simulations plays a role of increasing importance in this design. However classical RANS models reach their limits on some external aerothermal flows, like the jet-in-cross-flow from the anti-icing system oh the engine nacelles. The large eddy simulation (LES) is well suited to this kind of flow but faces an extremely large computational cost for such high Reynolds number wall-bounded flows. To remove this limitation, we propose two building blocks: the Wall Modeled LES (WMLES) combined with a turbulent inflow generation. For a better understanding and a reliable use of the WMLES, we first focus on the sources of error related to this approach. We propose a correction to the subgrid-scale error as well as a wall model suitable for compressible and anisothermal flows. Thanks to these two elements, we correctly predict the wall heat flux in WMLES computations of a supersonic isothermal-wall channel flow. Then, to allow the computation of more industrial flows, we introduce some numerical dissipation and study its effect on the wall modeling approach. The last part is dedicated to turbulent inflow generation for WMLES. We select a method based on synthetic perturbation combined with a dynamic control term. We validate this method on WMLES computations of a flat plate turbulent boundary layer and a hot jet-in-cross-flow representative of an industrial configuration. In both cases, we show that a realistic turbulent boundary layer is generated at a small distance downstream from the inlet plane.