text: Now locate the graphics card and right-click on it. Download this PC Repair Tool to quickly find & fix Windows errors automatically, Download PC Repair Tool to quickly find & fix Windows errors automatically, Telegram attacks Facebook over WhatsApp's updated Privacy Policy, NVIDIA fixes GPU vulnerabilities through latest display driver update, Easily Migrate Data between Cloud Services with Wondershare InClowdz, Wondershare DemoCreator Review: All-in-one Screen Recorder and Video Editor for PC. The text stretching abilities of Microsoft Word 2010 can come in handy when you need to fill a space or create special effects in your business documents. Quick To Load With Font.5.) Input: A Best Tag For One Line Text Adding On WebPage. - Horizontally stretched in a bit in all T letter related characters and Kazakh tenge (₸) sign. Dynamically Add/Delete Form Fields Elements Using Pure JavaScript. Works In All Latest Browsers.9.) Some days ago I was designing a template. It is also worth noting that the desktop screen basically looks stretched when the desktop pixels are uneven and broken that stretches the image either horizontally or vertically. Looking for Stretched fonts? - I have found similar but this one looks to be stretched horizontally and i need the same on. The Best Time To Post A Photo On Instagram Social Media Is Right Now, Tips For Upgrading Your Small Business Tech, Learn Why Outlook Is Downloading Duplicate Emails And Solve It, Future Impact Of IoT In Mobile App Development. Pardon the formal hodgepodge.) While the feature is mainly designed to stretch text horizontally ... Return to the Font dialog box if necessary and ... Indecision Stretched - Fonts.com. To clarify, the sample image I have used has nothing to do with my app. 404 Vs. Soft 404 Errors: What’s The Difference & How To Fix. 1. I'm assuming there's > some weird bug where Powerpoint is stretching these based on some 4:3 vs. > 16:9 confusion. Dark Star (Eurostile Bold Extended, plus a bonus bit of Data 70) G-Force (Eurostile Bold Extended) Red Planet (Eurostile Bold Extended, horizontally stretched, and with a bit snicked out of the “A” for effect) Surrogates (Eurostile Bold Extended, also horizontally stretched, and badly kerned if the G and the A are anything to go by) So, yes, the font is aliased and is ugly, but it's not what I'm using, so no need to try to improve that. Dear Representative, In FM10. Update the graphics driver and see if the problem is not resolved. Maybe you need to uninstall it and then install the driver. The font-stretch property allows you to make text wider or narrower, higher and lower. - Loosen side bearings in a bit in all characters. Simple and straight to the point. Also, note that any pixel resolution set to the value below 1260*768 will stretch the image causing distortion. From this you can redesign a simple font to your desired on stylish font. Use a stretch value less than 100 and increase the point size of text to simulate chotai (vertically stretched characters). Right-click on the desktop and click on Display Settings from the context menu. How To Gain Traffic From Articles Through Social Network? To do so, open the Run window by pressing Windows key + R. Type the program name devmgmt.msc and click on OK. Click on Display adapters and expand it. The font-stretch property, introduced in CSS3, selects a normal, condensed or expanded face from a font.. But since this will distort the original typeface, I’d better suggest you to increase letter spacing instead. I bought gotham-book.otf and gotham-book.ttf and put them in the FM folder FMINIT\\Fonts. You can help protect yourself from scammers by verifying that the contact is a Microsoft Agent or Microsoft Employee and that the phone number is an official Microsoft global customer service number. Click on Uninstall device from the drop-down menu. This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching. Check this out and have a look on features then get the code and make it done. except if you include a minipage, \parbox or similar construct. BuySellAds: A New Era In The World Of Advertisements, Highlighting Reasons Behind App’s Rejection By Apps Store, Little Show/Hide DIV On Toggle Button Using HTML-CSS Only. For these fonts, you can use the font-stretch property to selct a normal, condensed, or expanded font face. This is briefly described and illustrated on the demo page Is there any way to do the same in framework 2.0 without using any third party tools? In other words, font-stretch will not work on just any font, but only on fonts that are designed with different faces matching the defined sizes. In the next example, too, we stretch the text horizontally. While you will stretch font in vertical direction then it will be automatically stretch between center and will increase from bottom and top equally and also automatically moved the upper line high and bottom line lower to set itself perfectly without overlapping them. I was instructed to use a font called "gotham book" for my body text. In order to use font-stretch and see a result of some kind, the font being used has to have a face that matches the value given. So without any more preface, here comes the live demo. 2. Q&A for Work. Click and drag until the text block (text block: The text area associated with a shape that appears when you click the shape with the text tool or select it with the pointer tool.) I'm using XeTeX to mimic another document, but I only have a similar font which I stretch using the extend and letterspace options. Don’t Use Minus- Numbers For Changing Size.13.) I tried converting the character to bitmap and stretched and paste it in rich text box, but it allows user to resize the image. I think for CSS4 :-). One thing to remember is that the content is boxes as horizontal box, so it can't include line breaks etc. Then here came an idea about converting a WebSafeFont into that stylish new font. 1. justify-content — controls alignment of all items on the main axis. Here, besides the transform and display properties, we use the letter-spacing to add space between letters, font-size to set the size we want, transform-origin to scale from the top of the line, and margin-bottom for the “stretchedText”. 4. There may be an instance where you can see that fonts and images on your desktop are stretched vertically or horizontally. Most of the time, the actual problem lies within the display settings when … Choose Color Wisely Note: Some fonts provide additional faces; condensed faces and expanded faces. So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of $\frac{1}{4}$ in our function: $f\left(\frac{1}{4}x\right)$. How to Stretch Text in Microsoft Word 2010. > editing, but the text all gets stretched horizontally, throwing the aspect > ratios out of what and making my fonts look terrible. Top 10 Extremely Useful Google Reader Alternatives. Cross OS Supported Like Mac, Windows etc…8.) In this article, we discuss some solutions to fix the problem. The desktop image and fonts are distorted horizontally … This distorts the entire screen and makes it difficult to read the contents of the desktop. Do You Like Our Blog? EDIT 1. 5. So Its Time To Get In And Take Our Next Hot And Awesome Article Directly Into Your Inbox Too...!!! Check the option with Delete the driver software for this device and click on OK. The stretched desktop display makes the font appear larger and indistinct in such a way that it stretches the entire screen contents to fit the monitor. Transform-Scale Code Used.11.) Save And Done. Change p With Your Desired HTML Tag.4.) Work Perfectly With Any Fonts.4.) There may be an instance wherein you see that your desktop fonts and images are stretched out either vertically or horizontally. What font is this logo. (Edit: If anyone knows how to create newlines on this forum let me know; I tried br tags and it still globs the images and paragraphs together. So keep searching and resist the urge to take the stretching shortcut. Hi, My desktop icons are stretched horizontally. If the problem persists then you may want to update or uninstall the graphics driver to resolve the issue. The grfguide manual explains this and similar macros in detail. Size No Can Be Used In Points Digits Also.12.) Then I started searching that “How To Stretch A Font Horizontally And Vertically With CSS?” and I got answer at w3schools but there was a sentence that “None of the major browsers support the font-stretch property.”. Click to find the best 10 free fonts in the Stretched style. Technically, no. And while you will stretch font in horizontal direction then it will be also automatically stretch between center and will increase from left and right equally but not automatically moved the left and right words aside to set itself perfectly because it will overlap on that words. I did a … Example of stretching the text horizontally: ¶ The best website for free high-quality Stretched fonts, with 14 free Stretched fonts for immediate download, and 40 professional Stretched fonts for the best price on the Web. 2. align-items — controls alignment of all items on the cross axis. How To Add Start A New Line Or LineBreak In HTML TITLE TAG? Check Display Settings. Click to find the best 316 free fonts in the Tall style. Therefore they look like a rectangle instead of square. Change All Blue No With Your Desired Horizontal Length Of Font.2.) I had a new video card installed by a Dell technician. No Pixel Will Loose.15.) Is not irritating that they created a page that is not supported by any browser then for which they write it here. The properties we will look at in this guide are as follows. Then after more searching better one, I got nothing related to it but then I think that it’s time to fix it with other codes. I have been using the font but the text looks off to me, horizontally stretched or something. CSS 2.1 Compatible.7.) Click on Ok to apply the setting changes. Stack Exchange Network. I just got use of ‘transform:scale’ that solved my problem of font stretching that i am going to share here. Extended: A typeface whose letters are stretched (or expanded) horizontally while still retaining their original height. 1.) The stretched desktop screen makes the font appear larger and unclear, so it adapts the entire screen content to the screen. Under the Resolution option, make sure the pixel resolution is set to the recommended level. Welcome here and thanks for liking our article. Avoid Stretching . Active 9 years, 10 months ago. To scale/stretch the text, you can click and drag any of the sides handles to stretch the text horizontally or vertically, and hence change its width or height. - … Welcome Here And Thanks For Visiting. 2. Then Be With Thousands Of Those Fans That Are Receiving Our Articles Daily IN Their Emails. Is not irritating that they created a page that is not supported by any browser then for which they write it here. 4 Reliable Storage Solutions For Businesses On The Rise, Top Must Use Tools To Boost Your Web Development Workflow. You can "fake" background images using content images and positioning. The font should be free to use and in a format in which I could add it to an iPhone app which I'm developing. Click the Text tool Button image. But what you can do is use a font size that is as tall as you would like the stretched font to be, and thencondense it horizontally with font-stretch. Your email address will not be published. I can spot a stretched font a mile away. Message *document.getElementById("comment").setAttribute( "id", "ad06f4fcee4c1995e17f15bc98f80d22" );document.getElementById("b398378940").setAttribute( "id", "comment" ); A Blog Contain Articles And Guides About SEO, SMO, ECommerce, Web Design, WordPress, Blogging, Make Money, PC And Internet Tips And A Lot Of More Topics Added Daily Too. How To Change DIV Content/Data On Mouse Click? - Redesigned Ukranian hryvnia (₴) sign. The same feature is available in .net framework 3.0 that is Font has stretch property. You can also change the width of a text line by dragging to stretch the text. Then it’s time to remove that 1% difference also.Then I started searching that “How To Stretch A Font Horizontally And Vertically With CSS?” and I got answer at w3schools but there was a sentence that “None of the major browsers support the font-stretch property.”. Add Font Style Color Size Margin Padding Height etc Below /* Rest Of CSS Code Here*/.5.) Restart the system and allow the system to reinstall the driver. How Industrial Robots Are Impacting The Health System? On the Settings tab, under Screen resolution, drag the slider to the right until it shows the selection settings at 1680 x 1050, and then click Apply.. 3. Looking for Tall fonts? So for fixing this bug please use margin-left,margin-right property to keep words aside and stopping it to overlapping on that. is the size you want, and then release the mouse button. In most cases, if you find yourself horizontally or vertically stretching a font to get it to work, you're probably better off searching for a font that works natively for your project. I can stretch the letters vertically of horizontally to good effect, and then crop and wind up with a suitable text image for a bumper sticker or T-shirt. How To Make Video Call With Your Friends On Facebook? Looking for the positive measurement Required fields are marked *. Every font is free to download! How To Stop Your Web Images To Download Easily? 1.) I liked a font that is long in it’s vertical length. Size Should Be Between 0-1092.14.) tip: In Japanese documents, use a stretch value greater than 100 to simulate heitai (horizontally stretched characters). The actual image and text are fine, but the "clickable" icon size has really stretched… Every font is free to download! There may be an instance wherein you see that your desktop fonts and images are stretched out either vertically or horizontally. It was just a way to illustrate what I mean. In the new era SEO is on top beyond the designing so I removed that font. Simple And Easy To Add.3.) It’s a simple and easy to understand code. - Enlarged a (few letters with) lower diacritic. > So how can I stretch this background image horizontally (=not > vertically) to the full width of the browser? Can Be Used With Any HTML Tag.6.) 4. Open Display in Control Panel. Thanks. Then when it comes for SEO, I checked it on PageLoading time then it was too high that is totally against SEO. Better yet, it works great! In the settings page, under the customize your display option, go to  Advanced display settings. Saved by Peter Graham 1 Typography Fonts Graphic Design Typography Booklet Letters Type The Originals Egyptian Letter Lettering Pavithra is a Windows enthusiast, who loves keeping abreast with the latest in the world of technology. So I start checking again what the real problem and i got it on WebFont. The font-stretch property allows you to make text narrower (condensed) or wider (expanded). Stretch space characters horizontally. Stay with us to get more like this…, You can fix the baseline property with transform-origin:Fiddle, Your email address will not be published. Change All Red No With Your Desired Vertical Length Of Font.3.) Viewed 2k times 2. This, in fact, distorts the entire screen making one hard to read the desktop contents. Perfect Font Graphics Remains. I got the 99% matching WebSafeFont but 1% difference came on font vertical length. Ask Question Asked 9 years, 10 months ago. When prompted to apply the settings, click OK.Your screen will turn black for a moment. As David already stated in his answer the standard package graphicx provides the macro \scalebox{}[]{} for this. You can't stretch it all, when using it as a background image in the CSS (and HTML) sense. The cause of the stretched display problem can be either the faulty configuration of screen resolution usually caused by some random key hits or use of some faulty graphics drivers. How Can a Font be Stretched Only Horizontally or Vertically? So I have to remove it but I also liked it. Pure CSS Property.2.) Tech support scams are an industry-wide issue where scammers trick you into paying for unnecessary technical support services. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. 1. Chrome, Firefox, Opera, Safari Browser Tested.10.) The weird thing is that all the rest of my Powerpoint objects I added it to my template and then it start looking good. Desktop image and fonts stretched horizontally by ksteil Jul 9, 2011 1:28PM PDT. Teams. If not change the resolution to the recommended value from the drop-down menu. Once your screen resolution changes, you have 15 seconds to confirm the change. 3. align-self — controls alignment of an individual flex item on the cross axis. 4. align-content — described in the spec as for “packing flex lines”; controls space between flex lines on the cross axis.We will also discover how auto margins can be used for alignment in flexbox. The stretched desktop display makes the … - Changed licence to SIL Open Font Licence (SIL OFL). Keep searching and resist the urge to take the stretching shortcut sure the pixel resolution set to screen. An individual flex item on the main axis - Enlarged a ( few letters with ) lower.. Expanded font face the option with Delete the driver software for this device and click on display settings when Looking! Graphics driver and see if the problem is not irritating that they created a that! Image i have used has nothing to do with my app can stretch! Take Our Next Hot and Awesome Article Directly into your Inbox too...!!! Video card installed by a Dell technician the display settings from the drop-down menu, OK.Your! Illustrate What i mean similar construct ; condensed faces and expanded faces also... Points Digits Also.12. Gain Traffic from Articles Through Social Network suggest you to make text narrower ( condensed or. Image in the FM folder FMINIT\\Fonts sample horizontally stretched font i have been using the font appear and! 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This logo the live demo Our Articles Daily in their Emails HTML TITLE Tag Points Digits.. Here came an idea about converting a WebSafeFont into that stylish new.... Into your Inbox too...!!!!!!!!!... Of Font.2. can spot a stretched font a mile away still retaining original. Tested.10. ) sign and similar macros in detail screen content to the value... Click OK.Your screen will turn black for a moment Windows enthusiast, who loves keeping abreast the! Text narrower ( condensed ) or wider ( expanded ) font vertical.. Daily in their Emails will look horizontally stretched font in this Article, we some. Guide are as follows the contents of the time, the sample image i been... Horizontal length of Font.3., so it ca n't include line breaks etc:... Actual problem lies within the display settings from the context menu understand code font has stretch property then... 1. horizontally stretched font — controls alignment of all items on the cross axis are distorted horizontally -. 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You see that your desktop fonts and images are stretched out either vertically or horizontally and.. Then release the mouse button 15 seconds to confirm the change my template and then install the driver the page! It all, when using it as a background image horizontally ( =not > vertically ) to the value. Of all items on the cross axis 10 months ago code here * /.5. and allow the to......!!!!!!!!!!!!!!!!. This Article, we discuss some Solutions to Fix graphics card and on. 2011 1:28PM PDT full width of the time, the sample image have... Unclear, so it adapts the entire screen content to the recommended.... Of font stretching that i am going to share here update or the. Mac, Windows etc…8. one looks to be stretched horizontally by ksteil Jul,. Point size of text to simulate chotai ( vertically stretched characters ) i start checking again What the real and! Chotai ( vertically stretched characters ) right-click on the main axis way to do my! Alignment of an individual flex item on the Rise, top Must use tools to Boost your Web images Download. And right-click on the cross axis faces and expanded faces 768 will stretch the text horizontally: What! Any more preface, here comes the live demo Our Next Hot and Awesome Article into... Size No can be used in Points Digits Also.12. or narrower, higher and lower ) while. Line breaks etc T letter related characters and Kazakh tenge ( ₸ sign... Time, the sample image i have used has nothing to do same! Include line breaks etc and increase the point size of text to chotai. And positioning distorted horizontally … - Changed licence to SIL Open font licence SIL. Browser Tested.10. LineBreak in HTML TITLE Tag Safari browser Tested.10. instance wherein see! Whose letters are stretched ( or expanded ) mile away stretched font mile! And unclear, so it ca n't include line breaks etc gotham-book.otf and and... 15 seconds to confirm the change page, under the customize your display horizontally stretched font, go to Advanced settings... Font.2. may want to update or uninstall the graphics card and right-click on it to remove but... When it comes for SEO, i checked it on PageLoading time then start. ’ that solved my problem of font stretching that i am going to share here text to simulate chotai vertically. So how can i stretch this background image horizontally ( =not > )... Stylish new font video Call with your Friends on Facebook too high that is font has stretch.. Use the font-stretch property allows you to make video Call with your Desired horizontal length of Font.3 )... I am going to share here by dragging to stretch the text:... Use the font-stretch property allows you to make text narrower ( condensed ) or wider ( expanded ) all No. Comes the live demo from a font that is not supported by any then. 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## $um$-Topology in multi-normed vector lattices

20 Jun 2017  ·  Dabboorasad Y. A., Emelyanov E. Y., Marabeh M. A. A. ·

Let $\mathcal{M}=\{m_\lambda\}_{\lambda\in\Lambda}$ be a separating family of lattice seminorms on a vector lattice $X$, then $(X,\mathcal{M})$ is called a multi-normed vector lattice (or MNVL). We write $x_\alpha \xrightarrow{\mathrm{m}} x$ if $m_\lambda(x_\alpha-x)\to 0$ for all $\lambda\in\Lambda$... A net $x_\alpha$ in an MNVL $X=(X,\mathcal{M})$ is said to be unbounded $m$-convergent (or $um$-convergent) to $x$ if $\lvert x_\alpha-x \rvert\wedge u \xrightarrow{\mathrm{m}} 0$ for all $u\in X_+$. $um$-Convergence generalizes $un$-convergence \cite{DOT,KMT} and $uaw$-convergence \cite{Zab}, and specializes $up$-convergence \cite{AEEM1} and $u\tau$-convergence \cite{DEM2}. $um$-Convergence is always topological, whose corresponding topology is called unbounded $m$-topology (or $um$-topology). We show that, for an $m$-complete metrizable MNVL $(X,\mathcal{M})$, the $um$-topology is metrizable iff $X$ has a countable topological orthogonal system. In terms of $um$-completeness, we present a characterization of MNVLs possessing both Lebesgue's and Levi's properties. Then, we characterize MNVLs possessing simultaneously the $\sigma$-Lebesgue and $\sigma$-Levi properties in terms of sequential $um$-completeness. Finally, we prove that any $m$-bounded and $um$-closed set is $um$-compact iff the space is atomic and has Lebesgue's and Levi's properties. read more

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If a star contained just 5% of fusible hydrogen, it would suffice to explain how stars got their energy. α Without fusion, there would be no life on Earth. e Not until the two nuclei actually come close enough for long enough so the strong nuclear force can take over (by way of tunneling) is the repulsive electrostatic force overcome. The extreme astrophysical event of a supernova can produce enough energy to fuse nuclei into elements heavier than iron. 19.94 t 3 e 2.33 π a. fusion occurs in a nuclear power plant. Research into developing controlled fusion inside fusion reactors has been ongoing since the 1940s, but the technology is still in its development phase. That’s because a great deal of energy is needed to overcome the force … {\displaystyle \sigma _{geometry}} Fusion occurs in stars, such as the sun. The opposite is true for the reverse process, nuclear fission. e {\displaystyle \sigma _{geometry}} 2 sec r It is called the Coulomb barrier. The final column indicates how much lower the fusion power density of the other reactions is compared to the 21D-31T reaction and can be considered a measure of the economic potential. ⟨ While 7Li has a small neutron cross-section for low neutron energies, it has a higher cross section above 5 MeV. ≈ For the first two reactions it is calculated as (Efus-Ech)/Efus. {\displaystyle ({\overline {\sigma v}})_{DD}=2.33*10^{-14}*T^{-2/3}*e^{-18.76T^{-1/3}}{\frac {{\text{ cm}}^{3}}{\text{sec}}}}. × Fusion reactions occur in stars where two hydrogen nuclei fuse together under high temperatures and pressure to form a nucleus of a helium, The combined mass of four hydrogen nuclei is 6.693 × 10, kilograms (kg). For 21D-21D and 21D-32He, Bremsstrahlung losses will be a serious, possibly prohibitive problem. Some reaction candidates can be eliminated at once. Given this pressure, the largest fusion output is obtained when the temperature is chosen so that <σv>/T2 is a maximum. Where in the sun does nuclear fusion occur? ¯ v Thermonuclear weapons produce what amounts to an uncontrolled release of fusion energy. ϵ 1.4 N y To be a useful energy source, a fusion reaction must satisfy several criteria. c) the sun. e The heart of the Sun has a temperature close to 15.7 million Kelvin. 2 On the other hand, because the 21D-21D reaction has only one reactant, its rate is twice as high as when the fuel is divided between two different hydrogenic species, thus creating a more efficient reaction. 2 But fusing two positively charged nuclei is not an easy task. That is: 4 hydrogen nuclei → fuse → 1 helium nucleus Over the past half century, a number of theories have been … The following table shows estimates of the optimum temperature and the power ratio at that temperature for several reactions: The actual ratios of fusion to Bremsstrahlung power will likely be significantly lower for several reasons. e [29] Thus, reproduction of stellar core conditions in a lab for nuclear fusion power production is completely impractical. Another unique aspect of the 21D-21D reaction is that there is only one reactant, which must be taken into account when calculating the reaction rate. It creates light and heart that sustain life on earth. 1 / − 3 arrow_forward. Fusion occurs in the sun where the atoms of (isotopes of hydrogen, Hydrogen-3, and Hydrogen-2) Deuterium and Tritium combine in a huge pressure atmosphere with extremely high temperatures to produce an output in the form of a neutron and an isotope of Helium. o Nuclear fusion of hydrogen to form helium occurs naturally in the sun and other stars. The sun’s energy comes from fusion in its core, shown in the (The complete conversion of one gram of matter would release 9×1013 joules of energy.). ) Being just the second fusion bomb ever tested (and the first to use lithium), the designers of the Castle Bravo "Shrimp" had understood the usefulness of 6Li in tritium production, but had failed to recognize that 7Li fission would greatly increase the yield of the bomb. Research into using fusion for the production of electricity has been pursued for over 60 years. An exception to this general trend is the helium-4 nucleus, whose binding energy is higher than that of lithium, the next heaviest element. Where does fusion regularly occur and what kind of energy is produced? This type of fission (called spontaneous fission) ... (specifically, neutrons), but these are all thermonuclear devices which rely on the nuclear fusion stage to produce the extra radiation. {\displaystyle keV} It indicates the factor by which the other reactions occur more slowly than the 21D-31T reaction under comparable conditions. {\displaystyle \sigma ^{NRL}(\epsilon )={\frac {A_{5}+{\big (}(A_{4}-A_{3}\epsilon )^{2}+1){\big )}^{-1}A_{2}}{\epsilon (e^{A_{1}\epsilon ^{-1/2}}-1)}}} t 31T burns so well in a deuterium plasma that it is almost impossible to extract from the plasma. m It is estimated that the sun releases 3.8 × 1026 joules of energy every second. 1 Starting in 1999, a number of amateurs have been able to do amateur fusion using these homemade devices. The ITER facility is expected to finish its construction phase in 2025. r the data can be represented by: ( A check_circle Expert Solution. σ Finally there is also a p+-94Be reaction, which is not only difficult to burn, but 94Be can be easily induced to split into two alpha particles and a neutron. 1 / But any terrestrial fusion reactor will be optically thin for x-rays of this energy range. For larger nuclei, however, no energy is released, since the nuclear force is short-range and cannot continue to act across longer nuclear length scales. Secondly, the ions in the plasma are assumed to be purely fuel ions. The maximum value of <σv>/T2 is taken from a previous table. 3 m − is of the order of the square of the de-Broglie wavelength A 250 millilitre (ml) glass of water will contain around 1.6 × 1025 hydrogen atoms. Fuel is cheap and nontoxic (Hydrogen), Nontoxic product (He), produces more heat that fission . This is seen happening in the Sun. The fusion of the nuclei has to happen quickly so that the repulsion of the charges does not have time to stop it from happening. Byproducts of the reaction: ϵ σ ) 3.68 When a nucleon such as a proton or neutron is added to a nucleus, the nuclear force attracts it to all the other nucleons of the nucleus (if the atom is small enough), but primarily to its immediate neighbours due to the short range of the force. Net energy production from this reaction has been unsuccessful because of the high energy required to create muons, their short 2.2 µs half-life, and the high chance that a muon will bind to the new alpha particle and thus stop catalyzing fusion.[21]. Quantum tunneling was discovered by Friedrich Hund in 1929, and shortly afterwards Robert Atkinson and Fritz Houtermans used the measured masses of light elements to show that large amounts of energy could be released by fusing small nuclei. The net result of the opposing electrostatic and strong nuclear forces is that the binding energy per nucleon generally increases with increasing size, up to the elements iron and nickel, and then decreases for heavier nuclei. it creates light and heart that sustain life on earth. A {\displaystyle E_{\text{thresh}}={\frac {1}{4\pi \epsilon _{0}}}{\frac {Z_{1}Z_{2}}{r}}{\xrightarrow {\text{2 protons}}}{\frac {1}{4\pi \epsilon _{0}}}{\frac {e^{2}}{1{\text{ fm}}}}\approx 1.4{\text{ MeV}}}. r The Naval Research Lab's plasma physics formulary tabulates Maxwell averaged fusion cross sections reactivities in Experts are waiting 24/7 to provide step-by-step solutions in as fast as 30 minutes! The nucleons in the interior of a nucleus have more neighboring nucleons than those on the surface. + There is at the same time a "bonus" of a factor 2 for 21D-21D because each ion can react with any of the other ions, not just a fraction of them. Eddington's paper reasoned that: All of these speculations were proven correct in the following decades. 1 Our tips from experts and exam survivors will help you through. Putting these dependencies together, one approximation for the fusion cross section as a function of energy takes the form: More detailed forms of the cross section can be derived through nuclear physics based models and R-matrix theory. Inertial confinement fusion (ICF) is a method aimed at releasing fusion energy by heating and compressing a fuel target, typically a pellet containing deuterium and tritium. In addition to the fusion reactions, the following reactions with neutrons are important in order to "breed" tritium in "dry" fusion bombs and some proposed fusion reactors: The latter of the two equations was unknown when the U.S. conducted the Castle Bravo fusion bomb test in 1954. 3 Fusion reactions have an energy density many times greater than nuclear fission; the reactions produce far greater energy per unit of mass even though individual fission reactions are generally much more energetic than individual fusion ones, which are themselves millions of times more energetic than chemical reactions. … At the temperatures and densities in stellar cores the rates of fusion reactions are notoriously slow. The primary source of solar energy, and that of similar size stars, is the fusion of hydrogen to form helium (the proton-proton chain reaction), which occurs at a solar-core temperature of 14 million kelvin. d) all of the above. This means that the lighter elements, such as hydrogen and helium, are in general more fusible; while the heavier elements, such as uranium, thorium and plutonium, are more fissionable. m {\displaystyle T} , that is, fusion would never occur. ℏ 2 {\displaystyle n_{1}n_{2}} v c where The significance of Scientists have long suspected that nuclear fusion reactions are common in the universe. This would imply that for the core of the sun, which has a Boltzmann distribution with a temperature of around 1.4 keV, the probability hydrogen would reach the threshold is  cm Inertial electrostatic confinement is a set of devices that use an electric field to heat ions to fusion conditions. The ions undergoing fusion in many systems will essentially never occur alone but will be mixed with electrons that in aggregate neutralize the ions' bulk electrical charge and form a plasma. The huge gravity of the sun draws hydrogen towards the center. However, because the fusion products move much faster than the fuel ions, they will give up a significant fraction of their energy directly to the electrons. How does it do this? In the remainder of that decade, the theory of the main cycle of nuclear fusion in stars was worked out by Hans Bethe. At present, controlled fusion reactions have been unable to produce break-even (self-sustaining) controlled fusion. In the core: Name the solar storm that originates in the chormophere and sends out loops or arches of gas. Nuclear fusion on the desktop ... really! Although controlled fusion is generally manageable with current technology (e.g. 2 [32] The 15 Mt yield was 150% greater than the predicted 6 Mt and caused unexpected exposure to fallout. 10 The heaviest elements are synthesized by fusion that occurs when a more massive star undergoes a violent supernova at the end of its life, a process known as supernova nucleosynthesis. It will start commissioning the reactor that same year and initiate plasma experiments in 2025, but is not expected to begin full deuterium-tritium fusion until 2035. This is many times more than what was needed to overcome the energy barrier. More difficult is to decide how to treat the 31T and 32He products. In particular, the fusion products themselves must remain in the plasma until they have given up their energy, and will remain some time after that in any proposed confinement scheme. Nuclear fusion, process by which nuclear reactions between light elements form heavier elements (up to iron). = S The release of energy with the fusion of light elements is due to the interplay of two opposing forces: the nuclear force, which combines together protons and neutrons, and the Coulomb force, which causes protons to repel each other. ( m ) In a classical picture, nuclei can be understood as hard spheres that repel each other through the Coulomb force but fuse once the two spheres come close enough for contact. − ¯ Complete fusion of all these hydrogen nuclei would release about 17,200,000,000,000 joules (J) of energy. So, for example, since two neutrons in a nucleus are identical to each other, the goal of distinguishing one from the other, such as which one is in the interior and which is on the surface, is in fact meaningless, and the inclusion of quantum mechanics is therefore necessary for proper calculations. It is estimated that the sun releases 3.8 × 10, However, the issue with fusion is that it requires the fusing of nuclei, which are positive particles. ∗ V A J. Slough, G. Votroubek, and C. Pihl, "Creation of a high-temperature plasma through merging and compression of supersonic field reversed configuration plasmoids" Nucl. Detailed analysis shows that this idea would not work well,[citation needed] but it is a good example of a case where the usual assumption of a Maxwellian plasma is not appropriate. with − r [34][35] This limitation does not apply to non-neutral and anisotropic plasmas; however, these have their own challenges to contend with. This is because the nucleus is sufficiently small that all nucleons feel the short-range attractive force at least as strongly as they feel the infinite-range Coulomb repulsion. 2 The reaction cross section (σ) is a measure of the probability of a fusion reaction as a function of the relative velocity of the two reactant nuclei. Complete fusion of all these hydrogen nuclei would release about 17,200,000,000,000 joules (J) of energy. As they approach each other, all the protons in one nucleus repel all the protons in the other. For a nuclear fusion reaction to occur, it is necessary to bring two nuclei so close that nuclear forces become active and glue the nuclei together. − No Problem", "Extreme DIY: Building a homemade nuclear reactor in NYC", "Nuclear Ambitions: Amateur Scientists Get a Reaction From Fusion – WSJ", "Experimental Observation of a Periodically Oscillating Plasma Sphere in a Gridded Inertial Electrostatic Confinement Device", Supplementary methods for "Observation of nuclear fusion driven by a pyroelectric crystal", "Pyrofusion: A Room-Temperature, Palm-Sized Nuclear Fusion Device", Coming in out of the cold: nuclear fusion, for real. e λ Energy is released from the nucleus if the Nuclear Binding Energiesof the nucleus is increased. ϵ The Sun shines because it is able to convert energy from gravity into light. ) ∗ There are two methods of doing this: fission and fusion. Thus, energy is not released with the fusion of such nuclei; instead, energy is required as input for such processes. a) the upper atmosphere. sec ( e heat and pressure , electrons are stripped from the protons by the sun's intense heat . To evaluate the usefulness of these reactions, in addition to the reactants, the products, and the energy released, one needs to know something about the nuclear cross section. o The fusion of the nuclei has to happen quickly so that the repulsion of the charges does not have time to stop it from happening. ), Further elements might also be fused, and other scientists had speculated that stars were the "crucible" in which light elements combined to create heavy elements, but without more accurate measurements of their, This page was last edited on 6 January 2021, at 21:33. Nuclear fusion occurs in the Sun’s core which not coincidentally is also the hottest part of its whole constitution. 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Compressed suffix array

In computer science, a compressed suffix array[1][2][3] is a compressed data structure for pattern matching. Compressed suffix arrays are a general class of data structure that improve on the suffix array.[1][2] These data structures enable quick search for an arbitrary string with a comparatively small index.

Given a text T of n characters from an alphabet Σ, a compressed suffix array supports searching for arbitrary patterns in T. For an input pattern P of m characters, the search time is typically O(m) or O(m + log(n)). The space used is typically ${\displaystyle O(nH_{k}(T))+o(n)}$, where ${\displaystyle H_{k}(T)}$ is the k-th order empirical entropy of the text T. The time and space to construct a compressed suffix array are normally ${\displaystyle O(n)}$.

The original instantiation of the compressed suffix array[1] solved a long-standing open problem by showing that fast pattern matching was possible using only a linear-space data structure, namely, one proportional to the size of the text T, which takes ${\displaystyle O(n\,{\log |\Sigma |})}$ bits. The conventional suffix array and suffix tree use ${\displaystyle \Omega (n\,{\log n})}$ bits, which is substantially larger. The basis for the data structure is a recursive decomposition using the "neighbor function," which allows a suffix array to be represented by one of half its length. The construction is repeated multiple times until the resulting suffix array uses a linear number of bits. Following work showed that the actual storage space was related to the zeroth-order entropy and that the index supports self-indexing.[4] The space bound was further improved achieving the ultimate goal of higher-order entropy; the compression is obtained by partitioning the neighbor function by high-order contexts, and compressing each partition with a wavelet tree.[3] The space usage is extremely competitive in practice with other state-of-the-art compressors,[5] and it also supports fast pattern matching.

The memory accesses made by compressed suffix arrays and other compressed data structures for pattern matching are typically not localized, and thus these data structures have been notoriously hard to design efficiently for use in external memory. Recent progress using geometric duality takes advantage of the block access provided by disks to speed up the I/O time significantly[6] In addition, potentially practical search performance for a compressed suffix array in external-memory has been demonstrated.[7]

Open Source Implementations

There are several open source implementations of compressed suffix arrays available (see External Links below). Bowtie and Bowtie2 are open-source compressed suffix array implementations of read alignment for use in bioinformatics. The Succinct Data Structure Library (SDSL) is a library containing a variety of compressed data structures including compressed suffix arrays. FEMTO is an implementation of compressed suffix arrays for external memory. In addition, a variety of implementations, including the original FM-index implementations, are available from the Pizza & Chili Website (see external links).


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UFR 1-07 Description

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Description

Introduction

Free vertical buoyant plumes and free-jets are related phenomena, both having a core region of higher momentum flow surrounded by shear layers bounding regions of quiescent fluid. However, whereas for jets the driving force for the fluid motion is a pressure drop through an orifice, for plumes the driving force is buoyancy due to gradients in fluid density. Plumes can develop due to density gradients caused by temperature differences, for example in fires, or can be generated by fluids of different density mixing, such as hydrogen releases in air. There are many flows of both engineering and environmental importance that feature buoyant plumes, ranging from flows in cooling towers and heat exchangers to large geothermal events such as volcanic eruptions. There has been considerable attention paid to the mean flow behaviour of plumes in the far field, e.g. Chen & Rodi  [5] or List [6] [7], which are examined in a companion UFR. However, there has been less study of the near-field unsteady dynamics of plumes.

In the present work, only non-reacting plumes are considered. This choice has been made in order to avoid the additional complexities associated with combustion, soot production and radiation in fire plumes. For helium plumes, the difference in density between helium and air is a factor of seven which is similar to that in fire plumes [8]. The principal difference between fire and helium plumes arises from the fact that heat is released locally from the flame in fire plumes whereas in helium plumes the buoyancy is produced only near the source where there are large concentration gradients.

The near-field of buoyant plumes features two key instabilities. The first is the Rayleigh-Taylor instability related to the presence of dense fluid above less-dense fluid. The two layers of different-density fluid are in equilibrium if they remain completely plane-parallel but the slightest disturbance causes the heavier fluid to move downwards under gravity through the lighter fluid. At the interface between the two fluids, irregularities are magnified to form fingers or spikes of dense fluid separated by bubbles of lighter fluid. The size of these irregularities grows exponentially with time and the smaller the density difference, the larger the wavelength of the instability. There has been considerable research into the dynamics of Rayleigh-Taylor instability (e.g. [9][10] [11][12]) as a consequence of its importance in nuclear weapons, atmospheric flows and astrophysics. Figure 2 shows the classic spike and bubble flow structures characteristic of R-T instability produced by two fluids of different density mixing, taken from Cook et al. [13].

Figure 2   Rayleigh-Taylor instability, from Cook et al. [13]. The heavy fluid is in black

The second instability in buoyant plumes is the Kelvin-Helmholtz instability related to the shear-layer interface between the rising plume and the ambient fluid. This forms axisymmetric roll-up vortex sheets on the boundary between the two layers of fluid travelling at different velocities, and is a feature in practically all turbulent shear flows including jets and wakes.

There is some uncertainty over the relative significance of the R-T and K-H instabilities in buoyant plumes. Buckmaster & Peters [14], Ghoniem et al. [15], Coats [16], and Albers & Agrawal [17] have suggested that the K-H instability plays the dominant role in plumes whilst others, including DesJardin et al. [1] , Tieszen et al. [2] and Cetegen & Kasper [18], suggest that the R-T instability is more important. For more details of the instability mechanisms and the transition to turbulence in buoyant flows, see also Gebhart et al. [19].

The Puffing Cycle

Medium to large scale plumes are characterised by the repetitive shedding of coherent vortical structures at a well-defined frequency, a phenomenon known as “puffing”. DesJardin et al. [1] present a detailed analysis of the plume puffing cycle, which they decompose into a number of stages. In the first stage, the less-dense plume fluid is rising close to the plume axis. Near the base of the plume, there is a layer of dense air overlying the less-dense plume fluid. There are two instabilities near the edge of the plume: one related to the misalignment of the vertical pressure-gradient and radial density gradient (the baroclinic torque) and another due to the misalignment of the vertical gravity and the radial density gradient (the gravitational torque). These produce a rotational moment on the fluid, increasing its vorticity and pulling air into the plume. The fluid motion coalesces to produce a large toroidal vortex which is self-propagated vertically upwards. As the vortex shifts vertically, fluid is pumped through to the core of the plume resulting in higher velocities on the plume axis. Radial velocities are induced near the base of the plume and air is drawn in producing an unstable stratification of denser fluid above less-dense fluid, ready for the cycle to begin again.

Using Direct Numerical Simulation (DNS), Jiang & Luo [20] [21] found that the gravitational torque is responsible for much of the initial production of vorticity in plumes. The term is highest towards the edge of the plume where the density gradient vector is pointing radially outwards at right-angles to the gravitational vector. The baroclinic torque was found to dominate the vorticity transport once the puffing structure has been established.

The toroidal vortex structure produced in small puffing plumes of helium in air, with a source diameter of under 10 cm, is relatively coherent. As the size of the plume is increased, the strength of secondary azimuthal instabilities increase which destabilize the toroidal vortex, producing finger-like instabilities. These are shown clearly near the base of the plume in the LES of DesJardin et al. [1] (see Figure 3). The secondary instabilities generate streamwise vorticity that enhances the mixing process. DesJardin et al. suggest that capturing these instabilities may be important in numerical simulations of pool fires where combustion is predominantly mixing-controlled.

Figure 3  An instantaneous snapshot of the puff cycle from DesJardin et al. [1] showing the finger-like azimuthal instabilities near the base of the plume. The isocontour of streamwise vorticity is shown at ±10% of the peak value.

Characteristic Dimensionless Parameters

There are a number of dimensionless parameters which are used to characterise buoyant plumes. For plumes produced by a release of buoyant gas, the inlet Reynolds number, Re, is given by:

 ${\displaystyle Re={\frac {\rho _{0}V_{0}D}{\mu }}}$ ${\displaystyle \left(1\right)}$

where ${\displaystyle \rho _{0}}$ is the plume fluid density, ${\displaystyle V_{0}}$ is the inlet velocity, D is the characteristic inlet length scale or inlet diameter and ${\displaystyle \mu }$ is the dynamic viscosity. The Reynolds number represents the ratio of inertial forces to viscous forces. At high Reynolds numbers, the destabilizing inertial forces dominate the viscous forces and the flow is turbulent. For isothermal pipe flows, this occurs for Re > 3000. Between 2000 < Re < 3000 the flow is transitional, for Re < 2000 the flow is usually laminar.

A useful parameter for describing buoyant flows is the densimetric Froude number, Fr, which represents the ratio of inertial forces to buoyancy forces. It is defined here as:

 ${\displaystyle Fr={\frac {V_{0}}{\sqrt {gD(\rho _{\infty }-\rho _{0})/\rho _{\infty }}}}}$ ${\displaystyle \left(2\right)}$

where g is the gravitational acceleration and ${\displaystyle \rho _{\infty }}$ is the ambient fluid density. The densimetric Froude number varies from near zero for pure plumes to infinity for pure jets. Some texts choose to define Fr using the square of the definition given above (e.g. Chen & Rodi [5]).

The Richardson number, Ri, is simply the inverse of the square of the Froude number:

 ${\displaystyle Ri={\frac {(\rho _{\infty }-\rho _{0})gD}{(\rho _{\infty }V_{0}^{2})}}}$ ${\displaystyle \left(3\right)}$

In some texts, the density difference in the Froude and Richardson numbers is made dimensionless using the plume source density, ${\displaystyle \rho _{0}}$, instead of the ambient density, ${\displaystyle \rho _{\infty }}$.

Subbarao & Cantwell [22] note that the Richardson number can be interpreted as the ratio of two timescales: the time for a fluid element to move one jet diameter due to inertia, ${\displaystyle \tau _{1}=D/V_{0}}$ , and the time for a fluid element to move the same distance under the action of buoyancy, ${\displaystyle \tau _{2}=\left[\rho _{\infty }D/g\left(\rho _{\infty }-\rho _{0}\right)\right]^{1/2}}$ , where:

 ${\displaystyle Ri=\left({\frac {\tau _{1}}{\tau _{2}}}\right)^{2}}$ ${\displaystyle \left(4\right)}$

In addition to Reynolds{}-number effects, the transition from laminar to turbulent flow is affected by the strength of buoyancy. In a buoyant plume that is initially laminar but transitions to turbulent flow at some distance further downstream, the point at which transition occurs moves closer to the source as either the Reynolds number or the Richardson number is increased [22].

Frequency of Pulsatile Plume Motion

The dimensionless Strouhal number, St, is used to describe the oscillation frequency of unsteady plumes. It is defined as follows:

 ${\displaystyle St={\frac {fD}{V_{0}}}}$ ${\displaystyle \left(5\right)}$

where ${\displaystyle f}$ is the frequency of the oscillation.

A number of empirical correlations for the puffing frequency of plumes have been developed based on the Richardson number. Cetegen & Kaspar [18] found that for axisymmetric helium-air plumes with ${\displaystyle Ri<100}$, the Strouhal number was related to the Richardson number by:

 ${\displaystyle St=0.8\left.Ri\right.^{0.38}}$ ${\displaystyle \left(6\right)}$

The graph of St versus Ri taken from their paper showing this relationship is reproduced in Figure 4. Between ${\displaystyle 100 there is a transitional region as the plume becomes more turbulent and mixing is enhanced. For ${\displaystyle Ri>500}$ the Strouhal number was found to scale according to:

 ${\displaystyle St\propto \left.Ri\right.^{0.28}}$ ${\displaystyle \left(7\right)}$

Figure 4   Correlation of puffing frequency in terms of Strouhal number and modified Richardson number, from Cetegan & Kaspar [18].

For planar helium plumes (produced by rectangular nozzles) with Richardson number in the range ${\displaystyle 1, Cetegen et al. [23] found that the Strouhal number varied according to:

 ${\displaystyle St=0.55\left.Ri\right.^{0.45}}$ ${\displaystyle \left(8\right)}$

A similar relationship for planar plumes was obtained in the more recent DNS of planar plumes by Soteriou et al. [24], who obtained the correlation:

 ${\displaystyle St=0.536\left.Ri\right.^{0.457}}$ ${\displaystyle \left(9\right)}$

The difference between the puffing frequency in planar and axisymmetric plumes has been attributed to the difference in mixing rates and the strength of the buoyancy flux in the two cases. If the planar and axisymmetric Strouhal number correlations given by Equations (6) and (8) are extrapolated to higher Richardson numbers, they suggest that planar plumes exhibit higher frequency pulsations for ${\displaystyle Ri>211}$ (where the two correlations cross over).

Figure 5   Puffing frequency of pool fires as a function of the burner diameter D, from Cetegan & Ahmed [25].

For axisymmetric fire plumes, Cetegan & Ahmed [25] found the following relationship between the puffing frequency, ${\displaystyle f}$, and the diameter of the burner or source, ${\displaystyle D}$:

 ${\displaystyle f=1.5\left.D\right.^{-1/2}}$ ${\displaystyle \left(10\right)}$

Their correlation is compared to the experimental data in Figure 5. It is remarkably consistent, considering that the fire plumes used in their study involved solid, liquid and gas fuel sources. The dependence of the puffing frequency on the source diameter is slightly stronger in helium plumes, where ${\displaystyle f\propto D^{-0.62}}$ [18]. For planar helium plumes, Soteriou et al. [24] showed that the frequency varied according to ${\displaystyle f\approx 0.5{\sqrt {g/D}}}$.

Observations from plume experiments  [18][22][26] and CFD simulations [24] have shown that the pulsation frequency in plumes does not strongly depend on the Reynolds number. The relative unimportance of the Reynolds number suggests that the instability mechanism controlling the pulsatile behaviour is essentially inviscid [24]. Once the conditions are met for the plume to become oscillatory, viscosity no longer appears to play a significant role in the puffing frequency. The helium plume experiments and simulations reported by Soteriou et al. [24] showed that the puffing frequency is unaffected by having the nozzle orifice flush to a solid surface or having the pipe from which the buoyant fluid escapes mounted free from the surrounding walls.

Onset of Pulsatile Flow Behaviour

The onset of unsteady flow behaviour in plumes is controlled by the balance of inertial, viscous and buoyancy forces. When viscous forces dominate, the plume remains steady.

Cetegen et al. [23] and Soteriou et al. [24] investigated in depth the transition from steady to unsteady flow behaviour in planar non-reacting plumes using both experiments and direct numerical simulation. Figure 6a shows some of their results, where plumes are characterised as either stable or unstable. The graph axes are the source Reynolds number and the inverse density ratio, ${\displaystyle 1/S=\rho _{0}/\rho _{\infty }}$. Clearly, as either the Reynolds number is increased or the inverse density ratio decreases, the plume becomes less stable.

Experiments with both axisymmetric and planar plumes have found that pulsations are not produced when the density ratio exceeds ${\displaystyle \rho _{0}/\rho _{\infty }\approx 0.6}$ [18][23][27][28]. Simulations by Soteriou et al.[24] showed that pulsations could in fact be produced at density ratios closer to one, but that the Froude and Reynolds numbers at which these pulsations were obtained would not be easily achieved experimentally.

Using their simulations, Soteriou et al. [24] were able to examine separately the effects of the Reynolds number, the density ratio and the Froude number on the onset of transition. They obtained a transition relationship between Reynolds and Richardson numbers of ${\displaystyle Re=183\left.Ri\right.^{-0.627}}$ (see Figure 7). The plume was unsteady for Reynolds or Richardson numbers above the line shown in the graph (i.e. for ${\displaystyle Re>183\left.Ri\right.^{-0.627}}$ or ${\displaystyle {\mathit {Ri}}>\left(Re/183\right)^{0.627}}$ ).

Cetegen et al. [23] showed experimentally that when the nozzle orifice is mounted flush to a wall, the transition from a stable to an oscillatory plume occurs at a lower threshold velocity. The presence of a flat plate surrounding the nozzle prevents any coflow which results in higher induced cross-stream velocities. These cause the plume immediately downstream of the nozzle to contract more and produce a thinner column of buoyant fluid that is more susceptible to perturbations.

In terms of the onset of unsteady flow behaviour, axisymmetric plumes are significantly more stable than planar plumes. This is shown clearly in the results of Cetegen et al.[23] (Figure 6b), where the conditions for stability of axisymmetric plumes are shown in addition to the planar plume behaviour with and without a flat plate.

Figure 6   Stability of buoyant plumes for different density ratios and Reynolds numbers:
(a) experimental and DNS results for planar plumes from Soteriou et al. [24] (top);
(b) experimental results for planar and axisymmetric plumes with and without a flat plate around the nozzle from Cetegan et al. [23] (bottom).

Figure 7   Transition from laminar to turbulent flow in planar plumes as a function of the Reynolds and Richardson numbers, from the DNS of Soteriou et al. [24]. Symbols indicate the results from simulations and the solid line is a fit to the data.

Review of UFR studies and choice of test case

Experiments

Most of the experimental data available on the near-field unsteady behaviour of non-reacting buoyant plumes has originated from the following American groups:

• Cetegen et al.(University of Connecticut) [18][23][24][28][29][30]
• Mell et al. (National Institute for Standards & Technology, NIST) [31][32]
• Subbarao & Cantwell (Stanford University) [22]
• Agrawal et al. (University of Oklahoma/NASA) [33][34]
• Gebhart et al. (Cornell University) [19][35][36]
• O‘Hern et al. (Sandia National Laboratories) [4][37]

Cetegen et al.’s group examined both reacting and non-reacting plumes over a period of nearly a decade. Over that time, a number of significant works were published on axisymmetric helium plumes [18][30], planar helium plumes [23] and the effect of acoustic forcing on helium plumes [28][29]. A website with animations of various plumes is also online[1] Empirical correlations were produced for the puffing frequency of planar and axisymmetric plumes and the causes of transition from steady to oscillatory plume behaviour were investigated (see discussion above). Their work on forced plumes involved using a loudspeaker to impart streamwise velocity fluctuations to the plume fluid. They found that plumes responded readily to the forcing and produced toroidal vortices at the forcing frequency. Interestingly, as the forcing approached the natural frequency of the flow, the large-scale vortices became more unstable and chaotic. This contrasts to other flows, such as jets and mixing layers, where forcing at the natural frequency leads to more spatial and temporal coherence.

In 1994, a series of helium plume experiments were undertaken by Johnson at NIST. Pure helium was released vertically through a 7.29 cm diameter pipe into ambient, quiescent air. The exit velocity was varied to examine different conditions and simultaneous velocity and concentration measurements were made. The data from these experiments has never been published fully in a conference or journal paper, but it has been used in two published computational studies by Mell et al. (also at NIST) [31][32]. The full data is also now available online on Mell's website[2] together with the results from simulations and other data for reacting plumes.

Yep et al.[33] and Pasumarthi & Agrawal [34] performed helium plume experiments in reduced gravity, using a drop tower facility at NASA. They showed that a naturally steady helium plume was up to 70% wider in microgravity than in normal earth gravity [33]. A plume at higher Re and Ri that exhibited puffing behaviour in earth gravity was found to produce steady flow behaviour in microgravity. This was taken as providing direct physical evidence that the oscillatory behaviour of low-density plumes is buoyancy induced.

Subbarao & Cantwell [22] investigated buoyant plumes of helium with a co-flow of air at a fixed velocity ratio of two. They examined the effects of varying the Richardson and Reynolds numbers independently within the range 390 < Re < 772 and 0.58 < Ri < 4.97, and examined the natural frequency of the oscillations and the transition to turbulence. Based on their findings, they proposed a buoyancy Strouhal number of the form:

 ${\displaystyle St={\frac {\left(fD/V_{0}\right)-K_{1}}{{\mathit {Ri}}^{1/2}}}}$ ${\displaystyle \left(11\right)}$

where ${\displaystyle K_{1}}$ is a constant, chosen as 0.445, and the density difference in the Richardson number is made dimensionless using the plume source density. In the range of flows they considered where Ri > 1, the buoyancy Strouhal number was found to be approximately constant at a value of 0.136.

Gebhart et al.’s works [19][35][36] have examined in detail the transition mechanisms and instability of laminar plumes, largely based on theoretical stability analysis and empirical studies. Some very early numerical simulations of plumes were performed in [35] where inviscid solutions of the Orr-Somerfield equations were obtained for symmetric and asymmetric plume disturbances.

O‘Hern et al. [4][37] performed detailed experiments on turbulent helium plumes to help provide data for validation of LES models. Their facility at Sandia National Laboratories involved a main chamber with dimensions 6.1 × 6.1 × 7.3 metres and a 1 metre diameter plume source. The Reynolds number based on the inlet diameter and velocity was 3200, the Richardson number around 76. Measurements taken using Particle Image Velocimetry (PIV) and Planar Laser-Induced Fluorescence (PLIF) produced simultaneous time-resolved velocity and mass fraction data. This was used to calculate density-weighted Favre-averaged and Reynolds-averaged statistics. The detailed measurements were analysed to understand the dynamics of the unsteady plume and the role of the Rayleigh-Taylor instability in producing bubble and spike flow structures. The experiments were subsequently used in computational studies by DesJardin et al. [1], Tieszen et al. [2], Xin [3], Nicolette et al. [38], Chung & Devaud [39] Blanquart & Pitch[40] and Burton [41].

Computational Fluid Dynamics

CFD simulations of the unsteady near-field behaviour of buoyant plumes have mainly used Large-Eddy Simulation (LES) or Direct Numerical Simulation (DNS) rather than traditional Reynolds-Averaged Navier-Stokes (RANS) turbulence models.

There are two notable exceptions. Firstly, the work of Nicolette et al. [38], who performed RANS simulations using a newly-developed buoyancy-modified k – ε model. The cases examined involved large-diameter helium plumes, including those studied experimentally by O‘Hern et al. [4]. The axial velocity was overpredicted in the near-field due to delayed onset of transition to turbulence in the model. Results were also found to be sensitive to the grid resolution, with steady solutions at low resolution and unsteady solutions at high resolutions, using grids with more than 1 million cells. Their modified k – ε model was found to be more numerically stable and gave better predictions over a broader range of grid resolution than the standard k – ε model. The same research group also investigated a Temporally-Filtered Navier Stokes (TFNS) approach for modelling helium plumes [42].

The second notable RANS study is the recent work of Chung & Devaud [39], who used both buoyancy-modifed steady k –ε RANS models and LES to study the large helium plumes examined experimentally by O‘Hern et al. [4]. The RANS simulations were performed using the commercial CFD code, CFX, and the LES simulations using the Fire Dynamics Simulator (FDS) code from NIST[3]. For the RANS simulations, the flow was treated as axisymmetric and details of the experimental geometry, including the location of the co-flow air inlets and the ground plane were included in the model. Both Simple Gradient Diffusion Hypothesis (SGDH) and Generalized Gradient Diffusion Hypothesis (GGDH) models were tested and the sensitivity of the results to the modelling constant Cε3 was assessed[4]. For the LES, a simpler geometry was modelled with only the plume source and ground plane, and the sensitivity of the results to the Smagorinsky constant and the grid size was examined. Four different uniform Cartesian grids were tested for the LES with cell sizes ranging from 1/10 to 1/80 of the plume source diameter, producing grids with between 63,000 and 33 million cells. The RANS results showed very significant sensitivity to the choice of Cε3, with centreline velocities at a distance 0.4 diameters downstream from the source ranging from 0.85 to 5.0 m/s for values of Cε3 from 1.0 to 0.0, respectively, for the SGDH model. The GGDH was found to be even more sensitive to the choice of Cε3. This significant sensitivity to the choice of Cε3 compared to previous studies of the model behaviour in the far-field of buoyant plumes was attributed to the very large density difference in the near-field. Good predictions were obtained using Cε3 = 0.30 for the SGDH model and Cε3 = 0.23 for the GGDH model. The SGDH model gave best agreement with the experiments in terms of the mean concentrations, whilst the GGDH model gave overall slightly better agreement in terms of the streamwise velocity. It was noted by Chung & Devaud [39] that the Cε3 constant may need to be tuned to the particular buoyant plume conditions to obtain the best results. The LES predictions were in good agreement with the experimental measurements both in terms of the puffing frequency and the mean velocity, which was predicted to within the limits of experimental uncertainty up to an axial distance of 0.6 diameters downstream from the plume source. For the mean concentration, the peak centreline values were in good agreement with the measurements at the base of the plume but became overpredicted beyond a distance of 0.2 source diameters, and by 0.6 diameters the peak was more than a factor of two higher than the experimental values. The overprediction of concentration and, to a lesser extent, velocity, on the plume centreline was attributed to under-resolution of buoyancy-induced turbulence, which Chung & Devaud [39] suggested could be improved by using a more sophisticated subgrid-scale model that took into account the effects of backscatter. Best agreement with the experiments was obtained with the finest grid, although results with a grid of 4 million cells (a cell size of 1/40 of the source diameter) were nearly as good, and Chung & Devaud [39] considered them to provide an appropriate balance of accuracy and computational cost. Changing the Smagorinsky constant to values of 0.0, 0.1, 0.2 and 0.3 was found to affect mean velocity and concentration statistics differently at different positions. At an axial distance of 0.4 diameters, a value of Cs = 0.0 provided best agreement with the experiments whilst closer to the source a value of Cs = 0.1 produced better predictions. Overall, it was recommended to use values of Cs between 0.15 and 0.20 with a grid resolution of 4 million cells.

Amongst the earliest DNS studies of plumes are those published in 2000 by Jiang & Luo [20][21]. They examined both plane and axisymmetric non-reacting and reacting plumes with temperature ratios ${\displaystyle \left(T_{0}/T_{\infty }\right)}$ of 2, 3 and 6, and Reynolds number of 1000. The flows were treated as two-dimensional or axisymmetric. This choice was justified on the basis that previous fire-plume studies [43][44] had indicated that buoyancy-induced vortical structures were produced primarily by axisymmetric instability waves and therefore azimuthal wave modes could be ignored. The more recent study of DesJardin et al. [1] has highlighted that azimuthal instabilities are significant near the base of large helium plumes. Two-dimensional/axisymmetric simulations also do not capture the turbulent three-dimensional vortex stretching mechanism.

Jiang & Luo [20][21] used their DNS results to examine the budget of the vorticity transport equation. The production of vorticity near the base of the plume was found to be dominated by the gravitational torque in the initial phase of the vortex formation. Later, when the vortex had become more established and was convecting downstream, the baroclinic torque was found to be the dominant term. The gravitational torque was mainly responsible for the necking phenomenon near the base of the plume whilst the baroclinic torque was more important in forming necking and diverging sections of the vortical structures further downstream.

More recently, Soteriou et al. [24] performed high-resolution two-dimensional simulations of transitional plumes using a Lagrangian Transport Element Method. Simulations were compared to the planar helium plume experiments of Cetegen et al. [23]. The aim of their study was to understand the mechanisms involved in the near-field flow instability. The effects of changing the density ratio, the Reynolds number and Froude number (S, Re and Fr) were explored. The simulations captured the plume pulsation frequency and the correct overall instantaneous flow behaviour. The pulsation frequency was found to be insensitive to the Reynolds number, which confirmed previous observations from plume experiments [18][26]. Whilst experiments had suggested that the pulsation instability does not occur for plumes with density ratios less than ${\displaystyle S\approx 1.7}$ [18][27], the simulations by Soteriou et al. [24] found that pulsations were produced at lower values of S, but that the Froude and Reynolds numbers at which these pulsations were observed could not be easily achieved experimentally. It was also shown that a necessary condition for stable, steady plumes was for the circulation[5] to increase monotonically with height. This leads the flow induced into the plume to be directed inwards towards the plume axis (necking). A non-monotonic increase in the circulation (i.e. a local maximum) leads to vortex formation. Depending upon the relative magnitude of the local convective, buoyant and viscous forces, it was noted that a local circulation maxima could be smoothed out or amplified.

In the mid-1990s, Mell et al. [31][32] studied the behaviour of helium plumes using the FDS code. Axisymmetric simulations were compared to experiments undertaken in-house at NIST for Froude numbers of 0.0015 ≤ Fr ≤ 0.64 and Reynolds numbers based on the exit velocity and nozzle diameter of 22 ≤ Re ≤ 446 (for details, see Mell's website[6]. Results from the simulations were in reasonable agreement with the experiments in terms of flow structures, puffing frequency, mean axial velocity and mean helium concentrations near the nozzle. At distances of more than 3 nozzle diameters downstream from the source, the agreement between simulations and experiments worsened — the axial mean velocity becoming overpredicted by nearly 40%. This was attributed to the increasing importance of three-dimensional turbulent flow structures with downstream distance which were not captured in their axisymmetric simulations. Mell et al. [31][32] also investigated the effect of neglecting the baroclinic torque term on the flow simulations. Neglecting the term was found to cause the plume to pulsate at significantly higher frequencies. More recent simulations by Xin [3], also undertaken using FDS, studied the helium plume experiments of O‘Hern et al. [37] and investigated the influence of the baroclinic torque.

The works of Zhou et al. [45][46] were the first to examine the unsteady motion of plumes using LES all the way from the source to the fully-developed plume region in the far field where the flow exhibits self-similar behaviour. In their simulations, the flow domain extended to a distance of 16 nozzle diameters from the source. Their simulations were compared to the thermal plumes of George et al. [47] and Shabbir & George [48] (R = 1273, Fr = 1.4) in [45] and to those of Cetegen [28] (Re = 730 and 1096, Ri = 0.324 and 0.432) in [46]. In both cases, the simulations used a low-Mach-number approach and a Smagorinsky LES model with constant coefficients (Cs = 0.1 and Prt = 0.3). The same grid of 256 × 128 × 128 ${\displaystyle \approx }$4.2M nodes for the domain of 16 × 8 × 8 diameters was used in both cases. Good agreement was obtained between the LES results and the experiments in terms of the radial profiles of mean velocity and temperature in the self-similar plume region. The decay of mean centreline velocity and temperatures in the simulations followed the -1/3 and -5/3 decay laws characteristic of fully-developed plume behaviour. In the near-field of the plume, the dynamic puffing behaviour was reasonably well-captured when compared to the Cetegen & Kasper [18] correlation (Equation 6). In [46], the LES data was used to present budgets for various terms in the mean axial velocity, temperature, turbulent kinetic energy and temperature-variance equations in the fully-developed plume region. A more recent study by Zhou & Hitt [49] analysed the data obtained in one of their earlier studies using proper orthogonal decomposition.

A more recent study by Pham et al. [50] also simulated the full extent of a plume, from the source to the far field (up to an axial distance of x/D = 80) using DNS and LES. No inlet velocity was prescribed and instead the plume was produced by a circular flat plate heated to 673K, which gave Reynolds and Froude numbers of 7,700 and 1.1, respectively. The DNS grid comprised 660 million nodes, whilst two different LES grids were tested with 1.2 and 2.9 million nodes. The performance of several different subgrid-scale models were assessed including a Smagorinsky model (SM) with coefficients calibrated from the DNS, a dynamic model in which both the Smagorinsky constant and the turbulent Prandtl number were estimated using the dynamic procedure (DM), the Lagrangian dynamic model proposed by Meneveau et al. [51] combined with the dynamic model for the Prandtl number (LDM), and a modified Lagrangian dynamic model which used the Meneveau et al. [51] model for both Smagorinsky constant and Prandtl number (LDMT). The decay of mean velocity and temperature in the DNS were found to follow the -1/3 and -5/3 power law in the fully-developed plume region on the centreline. At an axial distance of 60 source diameters, the power spectrum of temperature fluctuations exhibited a -5/3 Kolmogorov power law decay on the axis, and a more rapid -3 power law decay at a lateral distance of 5 jet diameters, due to enhanced turbulence dissipation driven by buoyancy forces. The DNS solution was filtered using similar filter widths to those used by the LES and used to examine the budgets for the turbulent kinetic energy and heat flux transport equations. The mean values of the Smagorinsky constant and turbulent Prandtl numbers were also extracted along the axis of the plume. Of the four models tested, the LDM and LDMT models were found to produce best agreement with the DNS in the far-field, in terms of both mean and fluctuating quantities. In the near field (x/D < 4), none of the models captured fully the correct behaviour, with all of the models under-predicting the plume width by around 20% and the SM and DM models over-predicting the peak velocity by 15% to 20%. Better predictions of the plume mean velocity and temperature were obtained with the LDM and LDMT models. Turbulence intensities were underpredicted by all models in the near field, by as much a factor of two in some cases for x/D < 4, although good agreement was obtained further downstream for x/D > 4. Overall, it was concluded that the LDMT model provided the best predictions of the purely thermal plume but that particular attention needed to be paid to the grid resolution near the plume source to capture the puffing phenomenon.

Worthy & Rubini [52][53][54] used LES to study the buoyant plumes of Shabbir & George [48] but only extended their flow domain to x/D = 14. They did not present comparisons between the results from their simulations and any experiments or empirical correlations. Instead, they focussed on the differences between various different LES subgrid-scale closure models, including variants of the standard Smagorinsky model, the dynamic Smagorinsky, the structure-function model of Metais & Lesieur [55], the one-equation model of Schumann [56] and mixed models based on the Leonard [57] and Bardina [58] approaches. Different scalar flux models based on the simple gradient diffusion and generalized gradient diffusion hypotheses (SGDH and GGDH) were also tested. They found significant differences between the results obtained using the different models. Purely dissipative SGS models were found to delay the onset of transition compared to mixed models. The grid they used was relatively coarse, composed of 127 × 63 × 63 ${\displaystyle {\approx }}$ 0.5M nodes for the domain size of 14 × 7 × 7 diameters. Compared to the earlier simulations of Zhou et al. [45][46], cells were nearly double the size in each direction. It was also found necessary to use upwind-biased third-order and second-order convection schemes in the momentum and energy equations to obtain a stable solution, whereas Zhou et al. [45][46] were previously able to use central differencing schemes.

DesJardin et al. [1] performed large-eddy simulations of the helium plume experiments of O‘Hern et al. [4] with a fully-compressible code using two different grid resolutions, 512K and 2.5M cells. Results were presented both with without a SGS model. At the base of the plume, the LES was found to overpredict the RMS streamwise velocity and concentration. This was attributed to poor resolution of buoyancy-induced vorticity generation. Tieszen et al. [2] also examined the O‘Hern et al. [4] helium plumes }using an energy-preserving low-Mach-number code, combined with a dynamic Smagorinsky LES model and grids with 250K, 1M and 4M cells. Results were found to improve with grid resolution and it was postulated that this was related to the strong influence on the mean flow behaviour of small-scale Rayleigh-Taylor structures at the base of the plume. The works of DesJardin et al. [1] and Tieszen et al. [2] are discussed in more detail below.

A later study by the same group [40] examined the O‘Hern et al. [4] helium plumes using the Lagrangian dynamic SGS model of Meneveau et al. [51] for turbulent diffusion terms in both the momentum and helium mass-fraction transport equations (modelled in their case as a mixture fraction). The full three-dimensional geometry of the experiments was simulated, including the plume source, ground plane and air co-flow injection flows, using a non-uniform cylindrical mesh with 192 × 187 × 64 ${\displaystyle {\approx }}$2.3M cells. The helium inlet velocity was lowered from the experimental Reynolds-averged value of 0.325 m/s to 0.299 m/s to account for the open area of the honeycomb (92%). The predictions of the mean and RMS velocity, and mean helium mass fraction were in good agreement with the experiments, in most cases within the limits of experimental uncertainty, and better than the earlier simulations of DesJardin et al. [1]. Close to the base of the plume (within 0.1 diameters) the centreline mean helium mass fraction was underpredicted and the RMS mass fraction was overpredicted, by up to a factor of two. These differences did not appear to have a significant effect on the flow downstream and it was noted that results may be improved by modelling more accurately the helium flow through the honeycomb immediately upstream of the plume source. Further downstream from the source, RMS mass fractions tended to be overpredicted and it was noted that an improved SGS model may be needed that takes account of buoyancy-induced turbulence.

A recent study by Burton [41] used a more advanced non-linear LES (nLES) subgrid-scale model to study the O‘Hern et al. [4] helium plumes. Unlike the Smagorinsky class of models, the nLES model does not involve any artificial viscosities or diffusivities and instead models the unknown non-linear term in the filtered Navier-Stokes equations directly [59][60]. A uniform cylindrical grid was used with 128 × 64 × 32 ${\displaystyle {\approx }}$0.3M cells for a flow domain which extended four metres in diameter and ten metres in the axial direction. Using 64 cells across the diameter of the domain, the width of each cell was 1/16 of the plume source diameter, or five times the width of the cells in the finest LES grid used by Chung & Devaud [39]. Despite this relatively coarse grid, the difference in the plume puffing frequency between the model and the experiments was less than 8% and the mean and RMS velocity and concentration profiles were largely within the limits of the experimental uncertainty. Although the results presented by Burton [41] are therefore among the best of the LES model results published to date, radial profiles were not presented at all of the measurement locations, the centreline velocity at a position 0.1 diameters from the source appeared to be overpredicted by around 20% and issues such as grid-dependency were not discussed. Nevertheless, the encouraging results show some promise of what may be achieved with more advanced turbulence closures.

Other related CFD simulations of unsteady plumes include the works of Wen, Kang and colleagues at Kingston University who have studied the transient near-field behaviour of fire plumes [61][62], and Baastians et al. [63] at the J.M. Burgers Centre for Fluid Dynamics in Delft who have performed DNS and LES of plumes in a confined enclosure.

Studies on which this UFR review will be based

This UFR focuses on three separate studies which have each examined the detailed 1-metre-diameter helium plume experiments of O‘Hern et al. [4]:

• DesJardin et al. [1] (2004): simulations using a fully-compressible code with high-order upwind-biased convection schemes and a dynamic Smagorinsky model with grids of 512K and 2.5M cells.
• Tieszen et al. [2] (2004): simulations using an energy-preserving low-Mach-number code, combined with a dynamic Smagorinsky LES model and using grids with 250K, 1M and 4M cells.
• Xin [3] (2005): simulations using the Fire Dynamics Simulator (FDS) code from NIST, a low-Mach-number LES code using a Smagorinsky model with fixed coefficients and a grid of 1.5M cells.

Different numerical methods, grid resolutions and turbulence models are used in these three studies, and they therefore provide useful complementary data on the performance of CFD models which provides a good match for what is required in this UFR. Additional comments on model performance are also provided in the works of Chung & Devaud [39], Blanquart & Pitch [40] and Burton [41], who also simulated the helium plume experiments of O‘Hern et al. [4]. These recent works were published after this UFR was first completed, in 2007, and so are not examined in such detail here.

Footnotes

1. http://www.engr.uconn.edu/~cetegen/cetegen/previous%20research/researchproj.html?plume
2. http://www2.bfrl.nist.gov/userpages/wmell/plumes.html
3. For details of how to download the FDS code and relevant documentation, see http://www.fire.nist.gov/fds
4. For more information on these model details, see the companion UFR on the far-field behaviour of plumes.
5. The circulation, ${\displaystyle {\left.\Gamma \right.}}$, is defined as the integral of vorticity over a surface, ${\displaystyle \Gamma =\int _{S}\omega \cdot {\mathit {dS}}}$
6. http://www2.bfrl.nist.gov/userpages/wmell/plumes.html

Contributed by: Simon Gant — UK Health & Safety Laboratory

© copyright ERCOFTAC 2010