Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “diffusion maps”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 397 records · Page 22
All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

Star formation in the Magellanic irregular galaxy NGC 4449

New NIR and FIR maps and J = 1-0 CO spectroscopy of the Magellanic irregular galaxy NGC 4449 are presented. The brighter 150-micron emission is concentrated along the central visual ridge of the galaxy, although there is lower intensity extended emission throughout the visible extent of the object. The maximum FIR emission is coincident, within the uncertainties, with the visual and NIR maxima, identified as the galactic nucleus. It is estimated that the IR luminosity of the 1-kpc-diameter central region in NGC 4449 is comparable to that for a similar-sized region at the center of the Galaxy. A large fraction of the 150-micron emission may arise from warm dust distributed throughout the galaxy and heated by the diffuse radiation field. Active star formation follows the NIR emission in part of the galaxy, but no coincidence is found in another region.

Thronson, Harley A., Jr.↗
NASA NTRS1989

Results of rocket measurements of D-region ionization over Thumba in MAP

Under MAP, two rockets were launched from Thumba (8.5 N, 76.8 E) around 1030 hrs Lt with identical payloads on 7 and 10 March 1986 for D region studies. Positive ion densities were measured by spherical probe and Gerdien condenser and electron densities were measured by Langmuir probe and propagation experiments. In both flights a valley in ionization height profile was noticed around 83 km. The density of ionization at this altitude was about 4 x 10(2) cu cm. A detailed positive ion-chemical scheme was used to reproduce the measured ionization height profiles. The density of NO needed to reproduce the valley in ionization at 83 km came around 5 x 10(5) cu cm. A photochemical treatment without diffusion process was found inadequate to explain this value of NO. Calculations showed that the value of vertical eddy diffusion needed to reproduce the value of NO was around 10(6)sq cm/s. Interestingly, the same value of eddy diffusion coefficient was obtained when derived in the manner described by Thrane and his coworkers using only the positive ion current data of spherical probes.

Chakrabarty, D. K.↗
NASA NTRS1991

An experimental trace gas investigation of fluid transport and mixing in a circular-to-rectangular transition duct

An ethylene trace gas technique was used to map out fluid transport and mixing within a circular to rectangular transition duct. Ethylene gas was injected at several points in a cross stream plane upstream of the transition duct. Ethylene concentration contours were determined at several cross stream measurement planes spaced axially within the duct. The flow involved a uniform inlet flow at a Mach number level of 0.5. Statistical analyses were used to quantitatively interpret the trace gas results. Also, trace gas data were considered along with aerodynamic and surface flow visualization results to ascertain transition duct flow phenomena. Convection of wall boundary layer fluid by vortices produced regions of high total pressure loss in the duct. The physical extent of these high loss regions is governed by turbulent diffusion.

Reichert, B. A.↗
NASA NTRS1991

An experimental trace gas investigation of fluid transport and mixing in a circular-to-rectangular transition duct

An ethylene trace gas technique was used to map out fluid transport and mixing within a circular-to-rectangular transition duct. Ethylene gas was injected at several points in a cross stream plane upstream of the transition duct. Ethylene concentration contours were determined at several cross stream measurement planes spaced axially within the duct. The flow involved a uniform inlet flow at a Mach number level of 0.5. Statistical analyses were used to quantitatively interpret the trace gas results. Also, trace gas data were considered along with aerodynamic and surface flow visualization results to ascertain transition duct flow phenomena. Convection of wall boundary layer fluid by vortices produced regions of high total pressure loss in the duct. The physical extent of these high loss regions is governed by turbulent diffusion.

Reichert, B. A.↗
NASA NTRS2020

Inter-species molecular attraction effect in the development of a two-species mixing layer

Simulations of fluid mixing occurring at high-pressures mandate the use of real-fluid equations of state to capture the effect of boh intermolecular repulsive forces and attractive forces which are neglected in perfect gases. Dedicated mixing rules account for these forces by providing a generic mapping of the mixture state-space onto a pure fluid domain. However, the specific inter-species molecular forces depend on the particular characteristics of the molecular pairs under consideration, and typically require the knowledge of binary interaction coefficients k_{αβ}. The impact of k_{αβ} on a flow field is as of now unclear; further, k_{αβ} is unknown for many molecular pairs and when unknown, set to be null. This study addresses the impact of k_{αβ} on the temporal evolution of a real-fluid mixing layer and uses Direct Numerical Simulation (DNS) to explore this impact. The results show differences between augmented attraction (k_{αβ}<0) and diminished attraction (k_{αβ}>0), affecting both fluid dynamics and diffusion processes. Specifically, in a binary mixing layer, it is observed that augmented attraction leads to delayed transition and mixing layer growth, manifested in the momentum layer thickness. The fractal dimension, which is a manifestation of interface corrugation, is calculated. These results provide an hitherto undocumented mechanism affecting the development of mixing layers.

Banuti, Daniel↗
NASA NTRS1987

A geostationary imaging spectrometer TOMS instrument

One design for a geostationary Total Ozone Mapping Spectrometer (TOMS) with many desirable features is an imaging spectrometer. A preliminary study makes use of a 0.25 m Czerny-Turner spectrometer with which the Earth is imaged on a charge-coupled device (CCD) in dispersed light. The wavelength is determined by a movable grating which can be set arbitrarily by ground control. The signal integration time depends on wavelength but this system allows arbitrary timing by command. Special circumstances such as a requirement to track a low-lying sulfur dioxide cloud or a need to discriminate high level ozone from total ozone at midlatitudes could be obtained by adding a particular wavelength to the normally pre-programmed time sequence. The incident solar irradiance is measured by deploying a diffuser plate in the field of view. Individual detector elements correspond to scene elements in which the several wavelengths are serially sampled and the Earth radiance is compared to the incident sunlight. Thus the problem of uncorrelated drift of multiple detectors is removed.

Krueger, Arlin J.↗
NASA NTRS1991

Long-term changes in the total ozone mapping spectrometer relative to world primary standard Dobson spectrometer 83

The stability of the calibration of the Nimbus 7 solar backscatter UV (SBUV) and total ozone mapping spectrometer (TOMS) instruments by comparing their ozone measurements with those made by a single, very stable Dobson instrument: the world primary standard Dobson spectrometer number 83. Measurements of ozone made with instrument 83 at Mauna Loa observatory in eight summers between 1979 and 1989 were compared with coincident TOMS ozone measurements. The comparison shows that relative to instrument 83, ozone measured by TOMS (and SBUV) was stable between 1979 and approximately 1983, had decreased by 3 percent by 1986, and had decreased by almost 7 percent by 1989. A similar time dependence is seen when data from an ensemble of 39 Dobson stations throughout the world is compared with TOMS over the period 1979-1987. The most likely reason for the relative drift is that the diffuser plate used by both SBUV and TOMS to measure solar flux has suffered an uncorrected wavelength-dependent degradation, with most of the degradation occurring after 1983. The recently released version 6 TOMS data, corrected using the internal 'pair justification' technique, show almost no drift relative to Dobson instrument 83. Accurate measurements of long-term global ozone change will require a coherent system incorporating both ground-based and satellite-based ozone measurements.

Mcpeters, Richard D.↗
NASA NTRS1990

Starburst outflows from nearby galaxies

Starburst outflows from NGC 5461, 1569 and M82 are discussed. The Sc I galaxy, M101, is reknowned for the kpc-size superassociations of star clusters and HII regions that dominate its spiral arms. NGC 5461 is one of the brightest of these superassociations, rivaling the Large Magellanic Cloud in H alpha luminosity. The NGC 5461 superassociation is dominated by a single unresolved HII region of outstanding luminosity (approx. 1000 Orion nebulae). Detailed examination of corresponding continuum images indicates that only the southern plume has any sort of stellar counterpart. The other plumes are clearly diffuse with no underlying hot stars. An image of NGC 1569 is discussed. Besides showing the peculiar arm noted by Zwicky (1971) and the filamentary extensions to the North and South (as noted by Hodge 1974), this image also reveals two arc-like features of diffuse ionized gas to the South. Both arcs are concentric with the bright center of the galaxy - where the super star clusters, A and B are located. The inner arc (Arc 1) appears to follow the same curve as the SW arm thus suggesting that the two features represent limb-brightened fragments of vast superbubble that was blown out by a central starburst sometime in the past. As the classic starburst galaxy, M82 displays all the luminous hallmarks of intense high-mass star formation and outflow activity. The diffuse H alpha and x ray emitting gas along the minor axis provides especially good evidence for a bipolar outflow of hot gas which is shock heating the swept-up interstellar medium (ISM) to temperatures of approx. 10(exp 4) K. An image shows the H alpha emission within the disk and along the minor axis. Another image shows the same field in the light of near-infrared. Both figures are based on charge coupled device images taken with the McGraw-Hill 1.3 m telescope (Waller 1989). The longer wavelength emission clearly shows a more extended morphology along the major axis. The morphological discrepancy is most likely due to the greater obscuration by dust suffered by the H alpha photons. A third image shows the map of visual extinction that researchers derive from a pixel-by-pixel comparison of the (SIII) and H alpha fluxes. The greatest extinctions are evident along an arc that includes two especially obscured regions on opposite sides of the bursting nucleus.

Waller, William H.↗
NASA NTRS2002

Dynamics of Quantum Adiabatic Evolution Algorithm for Number Partitioning

We have developed a general technique to study the dynamics of the quantum adiabatic evolution algorithm applied to random combinatorial optimization problems in the asymptotic limit of large problem size n. We use as an example the NP-complete Number Partitioning problem and map the algorithm dynamics to that of an auxiliary quantum spin glass system with the slowly varying Hamiltonian. We use a Green function method to obtain the adiabatic eigenstates and the minimum exitation gap, gmin = O(n2(sup -n/2)), corresponding to the exponential complexity of the algorithm for Number Partitioning. The key element of the analysis is the conditional energy distribution computed for the set of all spin configurations generated from a given (ancestor) configuration by simultaneous flipping of a fixed number of spins. For the problem in question this distribution is shown to depend on the ancestor spin configuration only via a certain parameter related to the energy of the configuration. As the result, the algorithm dynamics can be described in terms of one-dimensional quantum diffusion in the energy space. This effect provides a general limitation of a quantum adiabatic computation in random optimization problems. Analytical results are in agreement with the numerical simulation of the algorithm.

Smelyanskiy, Vadius↗
NASA NTRS2002

Dynamics of Quantum Adiabatic Evolution Algorithm for Number Partitioning

We have developed a general technique to study the dynamics of the quantum adiabatic evolution algorithm applied to random combinatorial optimization problems in the asymptotic limit of large problem size n. We use as an example the NP-complete Number Partitioning problem and map the algorithm dynamics to that of an auxiliary quantum spin glass system with the slowly varying Hamiltonian. We use a Green function method to obtain the adiabatic eigenstates and the minimum excitation gap. g min, = O(n 2(exp -n/2), corresponding to the exponential complexity of the algorithm for Number Partitioning. The key element of the analysis is the conditional energy distribution computed for the set of all spin configurations generated from a given (ancestor) configuration by simultaneous flipping of a fixed number of spins. For the problem in question this distribution is shown to depend on the ancestor spin configuration only via a certain parameter related to 'the energy of the configuration. As the result, the algorithm dynamics can be described in terms of one-dimensional quantum diffusion in the energy space. This effect provides a general limitation of a quantum adiabatic computation in random optimization problems. Analytical results are in agreement with the numerical simulation of the algorithm.

Smelyanskiy, V. N.↗
NASA NTRS2013

Mineralogical Composition of the Different Types of Bright Deposits on Vesta

VIR-MS, Dawn's Visible and Infrared Mapping Spectrometer, obtained hyperspectral images of a wide part of Vesta's surface at a variety of spatial resolutions [1]. Vesta spectra are similar to those of the howardite-eucrite-diogenite (HED) meteorites. Moreover, they are characterized by the two iron-bearing pyroxene bands at 0.9 (band I) and 1.9 microns (band II). Vesta surface's is dominated by eucrite/howardite with some diogenitic regions situated in the southern hemisphere near the Rheasilvia basin [2]. The surface is heavily craterized and impacts can expose fresh material, thus generating the Bright Material Deposits (BMD) observed within and surrounding certain craters. BMD can be classified into six different types based on their morphological characteristics: Crater Wall/Scarp Material (CWM), Radial Material (RM), Slope Material (SM), Patchy Material (PM), Spot Material (SpM) and Diffuse Plains Material (DPM) [3]. The most widespread BMD are CWM, SM and RM. CWM, SM, RM originate from impacts. CWM is situated on the edge of the craters. Mass wasting from the crater walls and generates the SM, while RM is associated with the ejecta of the craters [4]. BMD are characterized by albedo greater than that of the vestan average, 0.38 [5]. Therefore the different types of deposits present distinct levels of reflectance respect to the Surrounding Regions (SR), in particular: the CWM and SM is approx.40% brighter, the RM is approx.30- 40% brighter; the SpM is about 20-25% brighter and the PM is about 20% brighter. Near the edge of the Rheasilvia basin it is possible to find some extremely bright areas ~80% brighter than the vestan average [6].

Zambon, F.↗
NASA NTRS1995

Rapid Numerical Simulation of Viscous Axisymmetric Flow Fields

A two-dimensional Navier-Stokes code has been developed for rapid numerical simulation of axisymmetric flow fields, including flow fields with an azimuthal velocity component. The azimuthal-invariant Navier-Stokes equations in a cylindrical coordinate system are mapped to a general body-fitted coordinate system, with the streamwise viscous terms then neglected by applying the thin-layer approximation. Turbulence effects are modeled using an algebraic model, typically the Baldwin-Lomax turbulence model, although a modified Cebeci-Smith model can also be used. The equations are discretized using central finite differences and solved using a multistage Runge-Kutta algorithm with a spatially varying time step and implicit residual smoothing. Results are presented for calculations of supersonic flow over a waisted body-of-revolution, transonic flow through a normal shock wave in a straight circular duct of constant cross sectional area, swirling supersonic (inviscid) flow through a strong shock in a straight radial duct, and swirling subsonic flow in an annular-to-circular diffuser duct. Comparisons between computed and experimental results are in fair to good agreement, demonstrating that the viscous code can be a useful tool for practical engineering design and analysis work.

Tweedt, Daniel L.↗
NASA NTRS2007

Nanorod-Based Fast-Response Pressure-Sensitive Paints

A proposed program of research and development would be devoted to exploitation of nanomaterials in pressuresensitive paints (PSPs), which are used on wind-tunnel models for mapping surface pressures associated with flow fields. Heretofore, some success has been achieved in measuring steady-state pressures by use of PSPs, but success in measuring temporally varying pressures has been elusive because of the inherent slowness of the optical responses of these materials. A PSP contains a dye that luminesces in a suitable wavelength range in response to photoexcitation in a shorter wavelength range. The luminescence is quenched by oxygen at a rate proportional to the partial pressure of oxygen and thus proportional to the pressure of air. As a result, the intensity of luminescence varies inversely with the pressure of air. The major problem in developing a PSP that could be easily applied to a wind-tunnel model and could be useful for measuring rapidly varying pressure is to provide very high gas diffusivity for rapid, easy transport of oxygen to and from active dye molecules. Most PSPs include polymer-base binders, which limit the penetration of oxygen to dye molecules, thereby reducing responses to pressure fluctuations. The proposed incorporation of nanomaterials (somewhat more specifically, nanorods) would result in paints having nanostructured surfaces that, relative to conventional PSP surfaces, would afford easier and more nearly complete access of oxygen molecules to dye molecules. One measure of greater access is effective surface area: For a typical PSP as proposed applied to a given solid surface, the nanometer-scale structural features would result in an exposed surface area more than 100 times that of a conventional PSP, and the mass of proposed PSP needed to cover the surface would be less than tenth of the mass of the conventional PSP. One aspect of the proposed development would be to synthesize nanorods of Si/SiO2, in both tangle-mat and regular- array forms, by use of chemical vapor deposition (CVD) and wet chemical processes, respectively. The rods would be coated with a PSP dye, and the resulting PSP signals would be compared with those obtained from PSP dye coats on conventional support materials. Another aspect of the proposed development would be to seek to exploit the quantum properties of nanorods of a suitable semiconductor (possibly GaN), which would be synthesized by CVD. These quantum properties of semiconductor nanorods include narrow-wavelength-band optical absorption and emission characteristics that vary with temperature. The temperature sensitivity might enable simultaneous measurement of fluctuating temperature and pressure and to provide a temperature correction for the PSP response.

Bencic, Timothy↗
NASA NTRS1994

Hubble Space Telescope Planetary Camera observations of Arp 220

Planetary Camera images of peculiar galaxy Arp 220 taken with V, R, and I band filters reveal a very luminous object near the position of the western radio continuum source, assumed to be the major nucleus, ans seven lesser objects within 2 sec of this position. The most luminous object is formally coincident with the radio source to within the errors of Hubble Space Telescope (HST) pointing accuracy, but we have found an alternate, more compelling alignment of maps in which the eastern radio source coincides with one of the lesser objects and the OH radio sources reside near the surfaces of other optical objects. The proposed centering places the most luminous object 150 pc (0.4 sec) away from the western radio source. We explore the possibilities that the objects are either holes in the dense dust distribution, dusty clouds reflecting a hidden bright nucleus, or associations of bright young stars. We favor the interpretation that at least the brightest two objects are massive young star associations with luminosities 10(exp 9) to 10(exp 11) solar luminosity, but highly extinguished by intervening dust. These massive associations should fall into the nucleus on a time scale of 10(exp 8) yr. About 10% of the enigmatic far-IR flux arises from the observed objects. In addition, if the diffuse starlight out to a radius of 8 sec is dominated by stars with typical ages of order 10(exp 8) yr (the time since the alleged merger of two galaxies), as indicated by the blue colors at larger radius, then the lower limit to the reradiation of diffuse starlight contributes 3 x 10(exp 11) solar luminosity to the far-infrared flux, or greater than or equal to 25% of the total far-IR flux. Three additional bright objects (M(sub V) approximately equals -13) located about 6 sec from the core are likely young globular clusters, but any of these could be recently exploded supernovae instead. The expected supernovae rate, if the dominant energy source is young stars, is about one per month for the region where the intense far-infrared flux originates. Also, individual giant dust clouds are visible in these images. Their typical size is 300 pc (1 sec).

Shaya, Edward J.↗
NASA NTRS2008

Fast Plasma Instrument for MMS: Simulation Results

Magnetospheric Multiscale (MMS) mission will study small-scale reconnection structures and their rapid motions from closely spaced platforms using instruments capable of high angular, energy, and time resolution measurements. The Dual Electron Spectrometer (DES) of the Fast Plasma Instrument (FPI) for MMS meets these demanding requirements by acquiring the electron velocity distribution functions (VDFs) for the full sky with high-resolution angular measurements every 30 ms. This will provide unprecedented access to electron scale dynamics within the reconnection diffusion region. The DES consists of eight half-top-hat energy analyzers. Each analyzer has a 6 deg. x 11.25 deg. Full-sky coverage is achieved by electrostatically stepping the FOV of each of the eight sensors through four discrete deflection look directions. Data compression and burst memory management will provide approximately 30 minutes of high time resolution data during each orbit of the four MMS spacecraft. Each spacecraft will intelligently downlink the data sequences that contain the greatest amount of temporal structure. Here we present the results of a simulation of the DES analyzer measurements, data compression and decompression, as well as ground-based analysis using as a seed re-processed Cluster/PEACE electron measurements. The Cluster/PEACE electron measurements have been reprocessed through virtual DES analyzers with their proper geometrical, energy, and timing scale factors and re-mapped via interpolation to the DES angular and energy phase-space sampling measurements. The results of the simulated DES measurements are analyzed and the full moments of the simulated VDFs are compared with those obtained from the Cluster/PEACE spectrometer using a standard quadrature moment, a newly implemented spectral spherical harmonic method, and a singular value decomposition method. Our preliminary moment calculations show a remarkable agreement within the uncertainties of the measurements, with the results obtained by the Cluster/PEACE electron spectrometers. The data analyzed was selected because it represented a potential reconnection event as currently published.

Figueroa-Vinas, Adolfo↗
NASA NTRS2000

Radiation Heat Transfer Between Diffuse-Gray Surfaces Using Higher Order Finite Elements

This paper presents recent work on developing methods for analyzing radiation heat transfer between diffuse-gray surfaces using p-version finite elements. The work was motivated by a thermal analysis of a High Speed Civil Transport (HSCT) wing structure which showed the importance of radiation heat transfer throughout the structure. The analysis also showed that refining the finite element mesh to accurately capture the temperature distribution on the internal structure led to very large meshes with unacceptably long execution times. Traditional methods for calculating surface-to-surface radiation are based on assumptions that are not appropriate for p-version finite elements. Two methods for determining internal radiation heat transfer are developed for one and two-dimensional p-version finite elements. In the first method, higher-order elements are divided into a number of sub-elements. Traditional methods are used to determine radiation heat flux along each sub-element and then mapped back to the parent element. In the second method, the radiation heat transfer equations are numerically integrated over the higher-order element. Comparisons with analytical solutions show that the integration scheme is generally more accurate than the sub-element method. Comparison to results from traditional finite elements shows that significant reduction in the number of elements in the mesh is possible using higher-order (p-version) finite elements.

Gould, Dana C.↗
NASA NTRS2004

Flow field topology of transient mixing driven by buoyancy

Transient mixing driven by buoyancy occurs through the birth of a symmetric Rayleigh-Taylor morphology (RTM) structure for large length scales. Beyond its critical bifurcation the RTM structure exhibits self-similarity and occurs on smaller and smaller length scales. The dynamics of the RTM structure, its nonlinear growth and internal collision, show that its genesis occurs from an explosive bifurcation which leads to the overlap of resonance regions in phase space. This event shows the coexistence of regular and chaotic regions in phase space which is corroborated with the existence of horseshoe maps. A measure of local chaos given by the topological entropy indicates that as the system evolves there is growth of uncertainty. Breakdown of the dissipative RTM structure occurs during the transition from explosive to catastrophic bifurcation; this event gives rise to annihilation of the separatrices which drives overlap of resonance regions. The global bifurcation of explosive and catastrophic events in phase space for the large length scale of the RTM structure serves as a template for which mixing occurs on smaller and smaller length scales. Copyright 2004 American Institute of Physics.

Nonlinear Dynamics↗
NASA NTRSDate not supplied

Influence of Thermophysical Property Variability on Thermal Predictions of Ti-6Al-4V

The additive manufacturing (AM) industry has experienced rapid growth in recent decades as industrial interest in the process has grown. Because of this, there are many groups interested in simulations of the AM process. However, the influence of thermophysical property variability on the melt pool geometry during processing of various AM alloys is unclear. The goal of my work at NASA Langley Research Center (LaRC) was to characterize this influence on Ti-6Al-4V during AM processing alongside developing a tool to enable equivalent studies on other AM materials. To facilitate this process, a database tool was developed to contain and organize thermophysical data previously reported by primary sources. The specific thermophysical properties of interest for this study were density, specific heat capacity, and conductivity. Thermal diffusivity was calculated using the other three properties. This data was then fit with a Gaussian distribution and then randomly sampled from using Monte Carlo random value sampling. Using the Rosenthal equation, this sample data was used to simulate the temperature field during additive manufacturing and extract the melt pool geometry. Specifically, the melt pool’s length, width, and depth. This process was repeated an arbitrary number of times, with the default being 1000. Once this information was obtained, histograms were made showing the distributions of the sizes of the melt pool’s length, width, and depth. Representative statistical metrics of the distributions were calculated (mean, standard deviation, and coefficient of variance). This work found that at the simulated processing values, the melt pool’s width and depth had a 5.7% variation while the length had a 3.2% variation. Additionally, a tool was constructed to graph a thermal color map representation of the melt pool for specific, arbitrary values of density, specific heat, and conductivity. This tool allowed for more efficient plotting of individual queries of the thermophysical properties. The results that were observed in the research were that values reported in literature vary and this variance can have a significant impact on the simulation. Predictions of the melt pool’s dimensions show that length has a standard deviation of approximately 6 μm, width has a standard deviation of approximately 5 μm, and depth can vary by approximately 3 μm. Considering the mean sizes of length, width, and depth are 185 μm, 88 μm, and 44 μm respectively, such a deviation is significant. This shows that depth, for example, could be more than 10% larger or smaller than expected. This demonstrates that variability in the reported thermophysical properties are not negligible and should be expected to have an influence on the results of the laser powder bed fusion additive manufacturing process. When simulating this process in the future, measures should be taken to account for this discrepancy and the uncertainty involved

Justin Martin↗