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At least 19 records

Low-Cost Flow Visualization for a Supersonic Ejector

Shadowgraph techniques were applied to the cold flow ejector facility at the Propulsion Research Center at the University of Alabama in Huntsville. The setup for the experiments was relatively simple and was accomplished at very little cost. Series of shadowgraph images were taken of both dual nozzle and single nozzle strut based ejectors operating over a range of chamber pressures. The density gradient patterns in the shadowgraphs were compared to pressure data measured along the top and side walls of the mixing duct. The shadowgraph images showed the presence of barrel shocks emanating from the nozzles which at low pressures terminated in Mach disks and at higher pressures extended beyond the barrel shape and reflected off the walls of the duct. Based on pressure data from previous testing, reflected shocks were expected on the walls of the duct. The shadowgraph images confirmed the locations of these reflected shocks on the top wall of the duct. The shadowgraph images also showed the structure change which correlated to a change in pitch of the ejector noise, and corresponded to a change in trend of the duct wall pressure ratio distributions. The images produced from the setup provided insight into the complex flow behavior inside the ejector duct. In addition, the techniques were a valuable tool as an educational device for students.

Olden, George W.↗

Validation of Shadowgraph Spectral Analysis using an SLS Block 2 Wind-Tunnel Model

A flow quantification method has previously been developed that uses high-speed shadowgraph images to extract frequency content of a transonic flow field around a wind-tunnel model. In this method, a classical spectral analysis is performed on a set of shadowgraph image pixels to identify their intensity fluctuation frequencies. The spatial change of the image intensity is related to the second derivative of the density gradient. Examining the change of the density gradient as a function of time can be used to understand the flow field pressure fluctuations. Pressure measurements collected during a Booster Obsolescence Life Extension Forward Attachment Structure Trade Study Test are used to calibrate the shadowgraph-based measurements. The test was performed at the NASA Ames Research Center Unitary Plan Wind Tunnel 11- by 11-foot test section and investigated aerodynamic environment downstream of various booster forward attachment hardware geometries of the Space Launch System Block 2 Cargo configuration. Time-correlated shadowgraph video and model surface pressure measurements were acquired during the test and served as an ideal opportunity to calibrate and validate the shadowgraph image intensity method.

Shadowgraph↗

Stability and Heat Transfer Characteristics of Condensing Films

The overall objective of this research is to investigate the fundamental physics of film condensation in reduced gravity. The condensation of vapor on a cool surface is important in many engineering problems,including spacecraft thermal control and also the behavior of condensate films that may form on the interior surfaces of spacecraft. To examine the effects of body force on condensing films, two different geometries have been tested in the laboratory: (1) a stabilizing gravitational body force (+1g, or condensing surface facing 'upwards') and (2) de-stabilizing gravitational body force (-1g, or 'downwards'). For each geometry, different fluid configurations are employed to help isolate the fluid mechanical and thermal mechanisms operative in condensing films. The fluid configurations are (a) a condensing film, and (b) a non-condensing film with film growth by mass addition by through the plate surface. Condensation experiments are conducted in a test cell containing a cooled copper or brass plate with an exposed diameter of 12.7 cm. The metal surface is polished to allow for double-pass shadowgraph imaging, and the test surface is instrumented with imbedded heat transfer gauges and thermocouples. Representative shadowgraph images of a condensing, unstable (-1g) n-pentane film are shown. The interfacial disturbances associated with the de-stabilizing body force leading to droplet formation and break-off can be clearly seen. The heat transfer coefficient associated with the condensing film is shown. The heat transfer coefficient is seen to initially decrease, consistent with the increased thermal resistance due to layer growth. For sufficiently long time, a steady value of heat transfer is observed, accompanied by continuous droplet formation and break-off. The non-condensing cell consists of a stack of thin stainless steel disks 10 cm in diameter mounted in a brass enclosure. The disks are perforated with a regular pattern of 361 holes each 0.25 mm in diameter. Non-condensing experiments in -1g have employed 50 cSt and 125 cSt silicone oil pumped through the perforated disks at a specified rate by a syringe micropump. The time to droplet break-off and the disturbance wavelengths appear to decrease with increasing pumping rate. The ability to reliably perform multi-point, ultrasonic measurements of the film thickness has been demonstrated. A linear array of eight transducers of 6 mm diameter (with a beam footprint of comparable size) are pulsed with a square-wave signal at a frequency of 5 MHz and a pulse duration of approximately 0.3 s. For thin films (60 m to 2-3 mm in thickness) the layer thickness is determined by frequency analysis, where the received ultrasound pulse is Fourier transformed and the spacing between the peaks in the frequency spectrum is analyzed. For thicker layers (up to at least 1 cm in thickness), time-domain analysis is performed of the received ultrasound pulses to generate directly the layer thickness. A time-trace of the film thickness at a point using a single transducer in the linear array is shown for the case of an unstable (-1g) n-pentane film. The oscillations in film thickness are evidently due to the passage and/or shedding of droplets from the cooled plate surface. The entire transducer array was used to measure the changes in film thickness resulting from the passage of gravity waves generated either by an oscillating wall or the impact of a single droplet on the free surface of a film. The enclosure in both cases was 14 cm square and the transducer spacing was 12 mm. Best results were obtained using as test fluid a mixture of 50% glycerol and 50% water with a fluid layer thickness of 3-5 mm. In both cases the measured wavelengths and wave propagation speeds using the ultrasound technique compared reasonably well with those observed by optical imaging. Additional information can be found in the original extended abstract.

Hermanson, J. C.↗

Experimental Visualizations of a Generic Launch Vehicle Flow Field: Time-Resolved Shadowgraph and Infrared Imaging

Time-Resolved shadowgraph and infrared (IR) imaging were performed to investigate off-body and on-body flow features of a generic, 'hammer-head' launch vehicle geometry previously tested by Coe and Nute (1962). The measurements discussed here were one part of a large range of wind tunnel test techniques that included steady-state pressure sensitive paint (PSP), dynamic PSP, unsteady surface pressures, and unsteady force measurements. Image data was captured over a Mach number range of 0.6 less than or equal to M less than or equal to 1.2 at a Reynolds number of 3 million per foot. Both shadowgraph and IR imagery were captured in conjunction with unsteady pressures and forces and correlated with IRIG-B timing. High-speed shadowgraph imagery was used to identify wake structure and reattachment behind the payload fairing of the vehicle. Various data processing strategies were employed and ultimately these results correlated well with the location and magnitude of unsteady surface pressure measurements. Two research grade IR cameras were positioned to image boundary layer transition at the vehicle nose and flow reattachment behind the payload fairing. The poor emissivity of the model surface treatment (fast PSP) proved to be challenging for the infrared measurement. Reference image subtraction and contrast limited adaptive histogram equalization (CLAHE) were used to analyze this dataset. Ultimately turbulent boundary layer transition was observed and located forward of the trip dot line at the model sphere-cone junction. Flow reattachment location was identified behind the payload fairing in both steady and unsteady thermal data. As demonstrated in this effort, recent advances in high-speed and thermal imaging technology have modernized classical techniques providing a new viewpoint for the modern researcher

Shadowgraph↗

Experimental Study of a Pulse Detonation Engine Driven Ejector

Results of an experimental effort on pulse detonation driven ejectors are presented and discussed. The experiments were conducted using a pulse detonation engine (PDE)/ejector setup that was specifically designed for the study. The results of various experiments designed to probe different aspects of the PDE/ejector setup are reported. The baseline PDE was operated using ethylene (C2H4) as the fuel and an oxygen/nitrogen (O2 + N2) mixture at an equivalence ratio of one. The PDE only experiments included propellant mixture characterization using a laser absorption technique, high fidelity thrust measurements using an integrated spring-damper system, and shadowgraph imaging of the detonation/shock wave structure emanating from the tube. The baseline PDE thrust measurement results are in excellent agreement with experimental and modeling results reported in the literature. These PDE setup results were then used as a basis for quantifying thrust augmentation for various PDE/ejector setups with constant diameter ejector tubes and various detonation tube/ejector tube overlap distances. The results show that for the geometries studied here, a maximum thrust augmentation of 24% is achieved. Further increases are possible by tailoring the ejector geometry based on CFD predictions conducted elsewhere. The thrust augmentation results are complemented by shadowgraph imaging of the flowfield in the ejector tube inlet area and high frequency pressure transducer measurements along the length of the ejector tube.

Santoro, Robert J.↗

Laser Light Scattering by Shock Waves

Scattering of coherent light as it propagates parallel to a shock wave, formed in front of a bluff cylindrical body placed in a supersonic stream, is studied experimentally and numerically. Two incident optical fields are considered. First, a large diameter collimated beam is allowed to pass through the shock containing flow. The light intensity distribution in the resultant shadowgraph image, measured by a low light CCD camera, shows well-defined fringes upstream and downstream of the shadow cast by the shock. In the second situation, a narrow laser beam is brought to a grazing incidence on the shock and the scattered light, which appears as a diverging sheet from the point of interaction, is visualized and measured on a screen placed normal to the laser path. Experiments are conducted on shocks formed at various free-stream Mach numbers, M, and total pressures, P(sub 0). It is found that the widths of the shock shadows in a shadowgraph image become independent of M and P(sub 0) when plotted against the jump in the refractive index, (Delta)n, created across the shock. The total scattered light measured from the narrow laser beam and shock interaction also follows the same trend. In the numerical part of the study, the shock is assumed to be a 'phase object', which introduces phase difference between the upstream and downstream propagating parts of the light disturbances. For a given shape and (Delta)n of the bow shock the phase and amplitude modulations are first calculated by ray tracing. The wave front is then propagated to the screen using the Fresnet diffraction equation. The calculated intensity distribution, for both of the incident optical fields, shows good agreement with the experimental data.

Panda, J.↗

Thrust Augmentation Measurements Using a Pulse Detonation Engine Ejector

Results of an experimental effort on pulse detonation driven ejectors are presented and discussed. The experiments were conducted using a pulse detonation engine (PDE)/ejector setup that was specifically designed for the study and operated at frequencies up to 50 Hz. The results of various experiments designed to probe different aspects of the PDE/ejector setup are reported. The baseline PDE was operated using ethylene (C2H4) as the fuel and an oxygen/nitrogen O2 + N2) mixture at an equivalence ratio of one. The PDE only experiments included propellant mixture characterization using a laser absorption technique, high fidelity thrust measurements using an integrated spring-damper system, and shadowgraph imaging of the detonation/shock wave structure emanating from the tube. The baseline PDE thrust measurement results at each desired frequency agree with experimental and modeling results reported in the literature. These PDE setup results were then used as a basis for quantifying thrust augmentation for various PDE/ejector setups with constant diameter ejector tubes and various ejector lengths, the radius of curvature for the ejector inlets and various detonation tube/ejector tube overlap distances. For the studied experimental matrix, the results showed a maximum thrust augmentation of 106% at an operational frequency of 30 Hz. The thrust augmentation results are complemented by shadowgraph imaging of the flowfield in the ejector tube inlet area and high frequency pressure transducer measurements along the length of the ejector tube.

Santoro, Robert J.↗

Validation of Shadowgraph Spectral Analysis using an SLS Block 2 Wind-Tunnel Model

A flow quantification method has previously been developed that uses high-speed shadowgraph images to extract frequency content of a transonic flow field around a wind-tunnel model. In this method, a classical spectral analysis is performed on shadowgraph video pixels to identify their intensity fluctuation frequencies. The spatial change of the image intensity is related to the second derivative of the density gradient. Examining the change of the density gradient as a function of time can be used to understand the flow-field pressure fluctuations. Time-varying pressure measurements collected during a Space Launch System (SLS) unsteady aerodynamic test of an advanced Block 2 booster design are used to calibrate the shadowgraph-based measurements. The test was performed at the NASA Ames Research Center Unitary Plan Wind Tunnel 11- by 11-foot test section and investigated the aerodynamic environment downstream of candidate booster forward attachment hardware geometries of the SLS Block 2 Cargo configuration. Time-correlated shadowgraph video and model surface pressure measurements were acquired during the test and served as an ideal opportunity to calibrate and validate the shadowgraph image intensity method.

Shadowgraph↗

Validation of Shadowgraph Spectral Analysis using an SLS Block 2 Wind-Tunnel Model

A flow quantification method has previously been developed that uses high-speed shadowgraph images to extract frequency content of a transonic flow field around a wind-tunnel model. In this method, a classical spectral analysis is performed on shadowgraph video pixels to identify their intensity fluctuation frequencies. The spatial change of the image intensity is related to the second derivative of the density gradient. Examining the change of the density gradient as a function of time can be used to understand the flow-field pressure fluctuations. Time-varying pressure measurements collected during a Space Launch System (SLS) unsteady aerodynamic test of an advanced Block 2 booster design are used to calibrate the shadowgraph-based measurements. The test was performed at the NASA Ames Research Center Unitary Plan Wind Tunnel 11- by 11-foot test section and investigated the aerodynamic environment downstream of candidate booster forward attachment hardware geometries of the SLS Block 2 Cargo configuration. Time-correlated shadowgraph video and model surface pressure measurements were acquired during the test and served as an ideal opportunity to calibrate and validate the shadowgraph image intensity method.

Shadowgraph↗

Flame-Vortex Interactions Imaged in Microgravity - To Assess the Theory Flame Stretch

The goals of this research are to: 1) Assess the Theory of Flame Stretch by operating a unique flame-vortex experiment under microgravity conditions in the NASA Glenn 2.2 Second Drop Tower (drops to identify operating conditions have been completed); 2) Obtain high speed shadowgraph images (500-1000 frames/s) using the drop rig (images were obtained at one-g, and the NASA Kodak RO camera is being mounted on the drop rig); 3) Obtain shadowgraph and PIV images at 1-g while varying the effects of buoyancy by controlling the Froude number (completed); 4) Numerically model the inwardly-propagating spherical flame that is observed in the experiment using full chemistry and the RUN 1DL code (completed); 5) Send images of the flame shape to Dr. G. Patniak at NRL who is numerically simulating the entire flame-vortex interaction of the present experiment (data transfer completed); and 6) Assess the feasibility of obtaining PIV velocity field images in the drop rig, which would be useful (but not required) for our assessment of the Theory of Flame Stretch (PIV images were obtained at one-g using same low laser power that is available from fiber optic cable in drop tower). The motivation for the work is to obtain novel measurement needed to develop a physically accurate model of turbulent combustion that can help in the control of engine pollutants. The unique experiment allows, for the first time, the detailed study of a negatively-curved (negatively stretched) flame, which is one of the five fundamental types of premixed flames. While there have been studies of flat flames, positively-curved (outwardly-propagating) cases and positively-strained (counterflow) cases, this is the first detailed study of a negatively-curved (inwardly-propagating) flame. The first set of drops in the 2.2 Second Drop Tower showed that microgravity provides more favorable conditions for achieving inwardly-propagating flames (IPFs) than 1-g. A vortex interacts with a flame and creates a spherical pocket, which burns inwardly. Shadowgraphs at 1000 frames/sec quantify the Markstein number and flame speed. A Low-Laser Power PIV System was developed and is being added to the drop package. Numerical computations were required to explain why the Markstein numbers measured for the inwardly-propagating flames differ from those of outward propagating flames; this is an important research issue in the assessment of the Theory of Flame Stretch. The RUN-1DL code (developed by Prof. B. Rogg) was run for IPF and OPFs with complex methane and propane chemistry. Results confirmed that Ma for the IPFs are larger than for OPFs as was observed experimentally. Physical reasons for these new findings about the Theory of Flame Stretch are being determined from the experiments and the computations. Several journal papers have been published; the drop package is described in the AIAA Journal, while the one-g results appear in three other journal papers.

Driscoll, James F.↗

Background Oriented Schlieren Applied to Study Shock Spacing in a Screeching Circular Jet

Background oriented schlieren (BOS) is a recent development of the schlieren and shadowgraph methods. The BOS technique has the ability to provide visualizations of the density gradient in both the axial and radial directions. The resultant magnitude of the density gradients allows for comparison with shadowgraph images. This paper first compares data obtained by the BOS and shadowgraph techniques at identical conditions in a free jet. The patterns and spacing of the shock trains obtained by the two techniques are found to be consistent with one another. This provides confidence in the shock spacing measurement by the BOS technique. Due to its simpler setup, BOS is then applied to investigate the shock spacing associated with the screech phenomenon, especially during stage jumps. Screech frequencies from a 37.6 mm convergent nozzle, as a function of jet Mach number (M(sub j)), are shown to exhibit various stages. As many as eight stages are identified with the present nozzle over the range 1.0 < M(sub j) <1.7. BOS images are acquired at various screech conditions and the shock spacing is examined as a function of M(sub j).

Clem, Michelle M.↗

A New Approach to Measure Contact Angle and Evaporation Rate with Flow Visualization in a Sessile Drop

The contact angle and the spreading process of sessile droplet are very crucial in many technological processes, such as painting and coating, material processing, film-cooling applications, lubrication, and boiling. Additionally, as it is well known that the surface free energy of polymers cannot be directly, measured for their elastic and viscous restraints. The measurements of liquid contact angle on the polymer surfaces become extremely important to evaluate the surface free energy of polymers through indirect methods linked with the contact angle data. Due to the occurrence of liquid evaporation is inevitable, the effects of evaporation on the contact angle and the spreading become very important for more complete understanding of these processes. It is of interest to note that evaporation can induce Marangoni-Benard convection in sessile drops. However, the impacts of the inside convection on the wetting and spreading processes are not clear. The experimental methods used by previous investigators cannot simultaneously measure the spreading process and visualize the convection inside. Based on the laser shadowgraphic system used by the present author, a very simple optical procedure has been developed to measure the contact angle, the spreading speed, the evaporation rate, and to visualize inside convection of a sessile drop simultaneously. Two CCD cameras were used to synchronously record the real-time diameter of the sessile drop, which is essential for determination of both spreading speed and evaporation rate, and the shadowgraphic image magnified by the sessile drop acting as a thin plano-convex lens. From the shadowgraph, the inside convection of the drop can be observed if any and the image outer diameter, which linked to the drop profile, can be measured. Simple equations have been derived to calculate the drop profile, including the instantaneous contact angle, height, and volume of the sessile drop, as well as the evaporation rate. The influence of the inside convection on the wetting and spreading processes can be figured out through comparison of the drop profiles with and without inside convection when the sessile drop is placed at different evaporation conditions.

Zhang, Nengli↗

Control of Wind Tunnel Operations Using Neural Net Interpretation of Flow Visualization Records

Neural net control of operations in a small subsonic/transonic/supersonic wind tunnel at Lewis Research Center is discussed. The tunnel and the layout for neural net control or control by other parallel processing techniques are described. The tunnel is an affordable, multiuser platform for testing instrumentation and components, as well as parallel processing and control strategies. Neural nets have already been tested on archival schlieren and holographic visualizations from this tunnel as well as recent supersonic and transonic shadowgraph. This paper discusses the performance of neural nets for interpreting shadowgraph images in connection with a recent exercise for tuning the tunnel in a subsonic/transonic cascade mode of operation. That mode was operated for performing wake surveys in connection with NASA's Advanced Subsonic Technology (AST) noise reduction program. The shadowgraph was presented to the neural nets as 60 by 60 pixel arrays. The outputs were tunnel parameters such as valve settings or tunnel state identifiers for selected tunnel operating points, conditions, or states. The neural nets were very sensitive, perhaps too sensitive, to shadowgraph pattern detail. However, the nets exhibited good immunity to variations in brightness, to noise, and to changes in contrast. The nets are fast enough so that ten or more can be combined per control operation to interpret flow visualization data, point sensor data, and model calculations. The pattern sensitivity of the nets will be utilized and tested to control wind tunnel operations at Mach 2.0 based on shock wave patterns.

Buggele, Alvin E.↗

Reference-Free, Projection Background-Oriented Schlieren

A projection background-oriented schlieren (P-BOS) system is developed and demonstrated. Instead of a background that has a speckle pattern printed, painted, or otherwise deposited onto its surface, and can thus not be altered, the pattern here is projected onto the background. This allows for changes to the speckle pattern without replacement of the background material, which can be time-consuming and expensive. Reference images are acquired simultaneously with flow images. A pre-test transformation between the reference and flow images allows for on-the-fly changes to the speckle pattern during a test without requiring a stoppage of the flow, allowing for optimization of the BOS signal. Using a programmable LCD screen as the speckled optic allows for remote control of the speckle pattern. Because the background is not speckled, the system can be easily transformed to acquired shadowgraph images by removing the speckled optic and closing the aperture of the light source, which is useful for achieving measurements with higher spatial resolution. The system can be nearly as compact as a conventional BOS system, and can be assembled with polarized optics to reduce or eliminate window reflections and glare, making this system particularly well-suited for wind tunnel testing.

Joshua M Weisberger↗

Seeding Method for Velocimetry and Visualization of Supersonic Retropropulsion Nozzle Plumes

In the current work, the “Venturi seeder” method for nozzle plumes is improved and studied to determine suitability for particle image velocimetry. This seeding method involves independently pressurizing the primary flow path and a separate seed liquid reservoir, connecting them at the throat of a Venturi contraction upstream of the nozzle. The pressure differential at the throat causes the liquid to enter the nozzle channel where it is atomized into particles by strong shearing forces. This new seeding system was characterized by conducting a series of diagnostics using different exit nozzle pressure ratios and seed reservoir pressures. Shadowgraph imaging confirmed the nozzle plumes to be underexpanded jets. A particle sizing device determined that generated DEHS oil particles had median aerodynamic diameters of 0.67, 0.69, and 0.73 μm for seed liquid reservoir pressure supplies of 0.67, 0.60, and 0.93 MPa and primary flow path pressures of 0.67, 0.76, and 0.93 MPa measured downstream of the Venturi throat, respectively. These diameters were within the threshold for acceptable response time in typical gas flows (≤ 1 μm). PIV experiments were conducted on the nozzle plume for all presented cases; mean axial and transverse velocities appeared as expected for the underexpanded jet structure, including the Mach disk and re-acceleration regions. Additional sizing analysis based on the particle response times across the normal shock again showed particle diameters to be well less than 1 μm for all cases. The results suggest that this method is an inexpensive and relatively simple solution to the problem of seeding nozzle plumes. The method will be used in testing of supersonic retropropulsion models in the Unitary Plan Wind Tunnel at The NASA Langley Research Center in 2022.

particle image velocimetry↗

Supersonic jet shock noise reduction

Shock-cell noise is identified to be a potentially significant problem for advanced supersonic aircraft at takeoff. Therefore NASA conducted fundamental studies of the phenomena involved and model-scale experiments aimed at developing means of noise reduction. The results of a series of studies conducted to determine means by which supersonic jet shock noise can be reduced to acceptable levels for advanced supersonic cruise aircraft are reviewed. Theoretical studies were conducted on the shock associated noise of supersonic jets from convergent-divergent (C-D) nozzles. Laboratory studies were conducted on the influence of narrowband shock screech on broadband noise and on means of screech reduction. The usefulness of C-D nozzle passages was investigated at model scale for single-stream and dual-stream nozzles. The effect of off-design pressure ratio was determined under static and simulated flight conditions for jet temperatures up to 960 K. Annular and coannular flow passages with center plugs and multi-element suppressor nozzles were evaluated, and the effect of plug tip geometry was established. In addition to the far-field acoustic data, mean and turbulent velocity distributions were measured with a laser velocimeter, and shadowgraph images of the flow field were obtained.

Stone, J. R.↗

Supersonic jet shock noise reduction

Shock-cell noise is identified to be a potentially significant problem for advanced supersonic aircraft at takeoff. Therefore NASA conducted fundamental studies of the phenomena involved and model-scale experiments aimed at developing means of noise reduction. The results of a series of studies conducted to determine means by which supersonic jet shock noise can be reduced to acceptable levels for advanced supersonic cruise aircraft are reviewed. Theoretical studies were conducted on the shock associated noise of supersonic jets from convergent-divergent (C-D) nozzles. Laboratory studies were conducted on the influence of narrowband shock screech on broadband noise and on means of screech reduction. The usefulness of C-D nozzle passages was investigated at model scale for single-stream and dual-stream nozzles. The effect of off-design pressure ratio was determined under static and simulated flight conditions for jet temperatures up to 960 K. Annular and coannular flow passages with center plugs and multi-element suppressor nozzles were evaluated, and the effect of plug tip geometry was established. In addition to the far-field acoustic data, mean and turbulent velocity distributions were measured with a laser velocimeter, and shadowgraph images of the flow field were obtained.

Stone, J. R.↗

Visualization, Extraction and Quantification of Discontinuities in Compressible Flows

Scientific visualizations of two-dimensional compressible flow of a gas with discontinuities are presented. The numerical analogue to experimental techniques such as schlieren imaging, shadowgraphs, and interferograms are discussed. Edge detection techniques are utilized to identify the discontinuities. In particular, the zero crossing of the Laplacian of a field (usually density) is recommended for extracting the discontinuities. An algorithm to extract and quantify the discontinuities is presented. To illustrate the methods developed in the report, the example chosen is that of an unsteady interaction of a shock wave with a contact discontinuity.

Samtaney, Ravi↗