31:Tb995,32:T838,gobstopper = { let tDomain = [d3.min(waterLevel.map(x => x.datetime)), d3.max(waterLevel.map(x => x.datetime))] let rDomain = [d3.min(waterLevel.map(x => x.waterLvlPct)), d3.max(waterLevel.map(x => x.waterLvlPct))] let timeScale = waterLevelTimeCurve(tDomain[0], tDomain[1]) let simTime = simPct / 100; let momentum = timeScale(simTime) * wallAreaForce let hydrostatic = timeScale(simTime) * hydroAreaForce let cumulative = momentum + hydrostatic; let max = timeScale(.18) * wallAreaForce + timeScale(.18) * hydroAreaForce; let circleShift = .7 let labelShift = .05 let rScale = d3.scaleLinear().domain([0, max]).range([0, 10]) let circleData = [ {v: max, label: "Maximum force"}, {v: cumulative, label: "Combined force"}, {v: momentum, label: "Momentum force"}, {v: hydrostatic, label: "Hydrostatic force"} ] let colors = d3.scaleOrdinal(d3.schemeCategory10) let circleOpts = { x: circleShift, y: .5, r: d => rScale(d.v), fill: d => d.label === "Maximum force" ? "#1f1f1f" : colors(d.label), stroke: "white" // stroke: d => d.label === "Maximum force" ? "white" : null } let textData = circleData.slice(1) let textYPos = d3.scaleLinear().domain([0, 3]).range([.8, 0]) let textOpts = {text: "label", x: labelShift, y: (d,i) => textYPos(i), textAnchor: "start", fill: "white", fontSize: 16, fontWeight: "bold", fill: d => colors(d.label)} // return textData let results = circleData.slice(1) let resultsOpts = Object.assign({}, textOpts, { text: d => `${fmtInt(d.v)} pound force`, dy: 30, dx: 20, fontSize: 20 }) return Plot.plot({ r: { domain: rDomain, range: [0, 30] }, x: { ticks: null, domain: [0, 1] }, y: { ticks: null, domain: [0, 1] }, marks: [ Plot.frame({fill: "#1f1f1f"}), Plot.circle(circleData, circleOpts), Plot.text(textData, textOpts), Plot.text(results, resultsOpts), Plot.text([{t: "Max force"}], {x: circleShift, y: .15, text: "t", fill: "white", fontSize: 14}) ] }) }33:[{"id":"17b340c89a03508cee09d3f9b9ef9a8161c3088500a15d836c19e274083906fa99194674a7e656eadd60992a987ef0f61c7ee3a1a2a0a6bb564f71c4abdf3dd1","href":"https://static.observableusercontent.com/files/17b340c89a03508cee09d3f9b9ef9a8161c3088500a15d836c19e274083906fa99194674a7e656eadd60992a987ef0f61c7ee3a1a2a0a6bb564f71c4abdf3dd1","name":"Storm speeds - Sheet1.csv","type":"text/csv","lastModified":1670460606157.211,"size":545},{"id":"5011920f5214b1844511f424e410db40af056c9c138e6efee2935b3d59c46c2f46b339b5d676db8ed9c74c22c1debd6b08975da23aff5e8041f5b8782c433088","href":"https://static.observableusercontent.com/files/5011920f5214b1844511f424e410db40af056c9c138e6efee2935b3d59c46c2f46b339b5d676db8ed9c74c22c1debd6b08975da23aff5e8041f5b8782c433088","name":"Storm speeds - Sheet1 (1).csv","type":"text/csv","lastModified":1670460774903.056,"size":529},{"id":"772e91190ddca379b002ca63d6762c3048095957a37b0c65eec8a0df616a51f2a40d81c81e13e5a500fde533c97217242b555c09581f58487b2b2285d9682288","href":"https://static.observableusercontent.com/files/772e91190ddca379b002ca63d6762c3048095957a37b0c65eec8a0df616a51f2a40d81c81e13e5a500fde533c97217242b555c09581f58487b2b2285d9682288","name":"CO-OPS_8725520_wl (2).csv","type":"text/csv","lastModified":1670600913507.725,"size":38227},{"id":"c90fcb605ea81aed266e38bf1e49d3b6ee2aee99e8b9ec5dd5eb2c80070fa16221244096e4c2113f694c263f838261ee483893b45133cc03853c16f29f56f10a","href":"https://static.observableusercontent.com/files/c90fcb605ea81aed266e38bf1e49d3b6ee2aee99e8b9ec5dd5eb2c80070fa16221244096e4c2113f694c263f838261ee483893b45133cc03853c16f29f56f10a","name":"report-averaged-1-second-force-rfile.csv","type":"text/csv","lastModified":1671053612003.373,"size":283334},{"id":"7e10abf8236e347c5bb8e5ae9733daf79ff967e4d0776aa952e2eb201a589170b7b57b380d232d310171073f9089e83bf5ef9d668e874184be52802a1076ffa0","href":"https://static.observableusercontent.com/files/7e10abf8236e347c5bb8e5ae9733daf79ff967e4d0776aa952e2eb201a589170b7b57b380d232d310171073f9089e83bf5ef9d668e874184be52802a1076ffa0","name":"storm surge (1).svg","type":"image/svg+xml","lastModified":1669751818676.925,"size":1728}] 2f:{"title":"The force of storm surge methodology","cells":[{"id":0,"value":"# The force of storm surge methodology","pinned":false,"hidden":false,"mode":"md"},{"id":1189,"value":"## [Read the article here!](https://projects.tampabay.com/projects/2022/hurricane/rising-threat/ian-storm-surge-model)","pinned":false,"hidden":false,"mode":"md"},{"id":5,"value":"In this document we are going to explain 3 different methods for calculating the force of storm surge as well as a few comparisons we can make. The code on this page powers the interactives on the article page.","pinned":false,"hidden":false,"mode":"md"},{"id":1155,"value":"toc()","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"## Model 1","pinned":false,"hidden":false,"mode":"md"},{"id":9,"value":"This calculation attempts to estimate the force of storm surge as the storm crashed into Florida. This calculator estimates **only the force of the wave** atop [storm tide](https://www.wunderground.com/prepare/storm-surge).","pinned":false,"hidden":false,"mode":"md"},{"id":11,"value":"In order to simplify, this calculation assumes the item being struck by the surge is exactly level with the tide. That means the building would be level with the sea **at the time the storm arrives**. We also assume the surge is arriving perpendicular to the object being struck and its velocity is equal to the velocity of the storm itself.","pinned":false,"hidden":false,"mode":"md"},{"id":13,"value":"First we need to calculate the volume of water. Imagine the storm surge as a cross-section. A right angle triangle where the bottom edge falls along sea-level the length of the distance inland. The triangle extends up from the leading edge to the height of the storm surge.","pinned":false,"hidden":false,"mode":"md"},{"id":552,"value":"$31","pinned":false,"hidden":false,"mode":"html"},{"id":21,"value":"viewof waveHeightFeet = Inputs.range([1, 20], {label: \"Wave height (feet):\", value: 8, step: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":24,"value":"viewof waveLengthFeet = Inputs.range([10, 100], {label: \"Distance inland (feet):\", value: 20, step: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":27,"value":"If we define a length for the wall, we can calculate the volume of the wave in cubic feet.","pinned":false,"hidden":false,"mode":"md"},{"id":29,"value":"viewof wallWidthFeet = Inputs.range([1, 50], {label: \"Wall width (feet):\", value: 20, step: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":32,"value":"waveVolume = (waveHeightFeet * waveLengthFeet / 2) * wallWidthFeet","pinned":true,"hidden":false,"mode":"ojs"},{"id":40,"value":"Next, we multiply the volume by the mass of one cubic foot of seawater. We will use a density of 1025 kilograms per cubic meter of water and convert that to pounds per cubic foot.","pinned":false,"hidden":false,"mode":"md"},{"id":43,"value":"seawaterMassPounds = kgM3TolbsF3(1025)","pinned":true,"hidden":false,"mode":"ojs"},{"id":46,"value":"waveMassPounds = waveVolume * seawaterMassPounds","pinned":true,"hidden":false,"mode":"ojs"},{"id":56,"value":"To calculate the momentum of the wave, we also need to know how fast it's moving. To approximate, we will use the forward speed of the storm itself as it crossed land.","pinned":false,"hidden":false,"mode":"md"},{"id":1110,"value":"viewof stormVelMPH = Inputs.range([1,15], {label: \"Storm forward velocity mph:\", value: 8, step: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":62,"value":"We'll convert that to feet per second.","pinned":false,"hidden":false,"mode":"md"},{"id":64,"value":"waveSpeedFPS = mphToFPS(stormVelMPH)","pinned":true,"hidden":false,"mode":"ojs"},{"id":72,"value":"The formula we will use to calculate the force of the wave is the Impulse-Momentum force equation.","pinned":false,"hidden":false,"mode":"md"},{"id":83,"value":"`Force = velocity * mass / time`","pinned":false,"hidden":false,"mode":"md"},{"id":78,"value":"We can plug in velocity and mass for our wave but we still need to solve for time. This time is the time that it takes for the entire wave to crash into the wall. We can approximate the time the collision took using the initial speed of the wave and the length of the wave.","pinned":false,"hidden":false,"mode":"md"},{"id":87,"value":"`time = (distance / average speed of wave) = (width of wave) / ((initial speed - final speed) / 2)`","pinned":false,"hidden":false,"mode":"md"},{"id":96,"value":"Since we want the final wave speed to be 0, we can further simplify and calculate the average speed as half of the initial speed.","pinned":false,"hidden":false,"mode":"md"},{"id":85,"value":"crashTime = waveLengthFeet / (waveSpeedFPS / 2)","pinned":true,"hidden":false,"mode":"ojs"},{"id":90,"value":"Now we can plug all of these into our Impulse-Momentum force equation.","pinned":false,"hidden":false,"mode":"md"},{"id":92,"value":"waveLBSForce = waveSpeedFPS * waveMassPounds / crashTime","pinned":true,"hidden":false,"mode":"ojs"},{"id":100,"value":"That means it would take **${fmtInt(roundDown(3)(waveLBSForce))} pounds of force** to stop this wave. Newton's third law tells us that every force has an equal and opposite reaction - so if that's the force that it would take to stop the wave, that's the force that a structure would need to withstand.","pinned":false,"hidden":false,"mode":"md"},{"id":177,"value":"## Model 2","pinned":false,"hidden":false,"mode":"md"},{"id":179,"value":"The first model is a decent back-of-the-envelope calculation. The problem with it is that it doesn't take into consideration the surface area of the wall, it instead calculates force at a single, averaged point along the wall.","pinned":false,"hidden":false,"mode":"md"},{"id":181,"value":"Our second model will take the entire surface area of the wall into consideration.","pinned":false,"hidden":false,"mode":"md"},{"id":183,"value":"The formula to calculate the drag force across an area is as follows:\n\n`Force = (material density) * (velocity^2) * (Surface area)`","pinned":false,"hidden":false,"mode":"md"},{"id":188,"value":"For material density and velocity, we'll use the same inputs as before:","pinned":false,"hidden":false,"mode":"md"},{"id":190,"value":"`Force = (${seawaterMassPounds}) * (${Math.pow(waveSpeedFPS, 2)}) * (Surface area)`","pinned":false,"hidden":false,"mode":"md"},{"id":193,"value":"To calculate the surface area, let's define a length and width for our wall.","pinned":false,"hidden":false,"mode":"md"},{"id":195,"value":"viewof wallWidthFeet2 = Inputs.range([1, 50], {label: \"Wall width (feet):\", value: 20, step: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":198,"value":"viewof wallHeightFeet = Inputs.range([1, 20], {label: \"Wall height (feet):\", value: 10, step: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":204,"value":"And when we plug that into our formula:","pinned":false,"hidden":false,"mode":"md"},{"id":206,"value":"`Force = (${seawaterMassPounds}) * (${Math.pow(waveSpeedFPS, 2)}) * (${wallWidthFeet2 * wallHeightFeet})`","pinned":false,"hidden":false,"mode":"md"},{"id":214,"value":"wallAreaForce = (seawaterMassPounds) * (Math.pow(waveSpeedFPS, 2)) * (wallWidthFeet2 * wallHeightFeet)","pinned":false,"hidden":false,"mode":"ojs"},{"id":218,"value":"That means that, at the moment the wall is completely consumed by our wave, the total force against the wall is **${fmtFloat(wallAreaForce)} lbs**.","pinned":false,"hidden":false,"mode":"md"},{"id":221,"value":"That's an instantaneous force. Measuring the total force over the course of the storm will require some calculus.","pinned":false,"hidden":false,"mode":"md"},{"id":223,"value":"First, we have to know how much of the wall is in contact with our wave and for how long. In order to estimate this, we'll use the water level data from the [Caloosahatchee River gauge in Ft. Myers](https://waterdata.usgs.gov/monitoring-location/02293202/#parameterCode=62620&period=P7D) during Hurricane Ian. We will assume that when the water gauge recorded its max reading, our wall was completely consumed by the wave.","pinned":false,"hidden":false,"mode":"md"},{"id":228,"value":"waterLevel = (await FileAttachment(\"CO-OPS_8725520_wl (2).csv\").csv({typed: true})).map(x => {\n let [y, m, d] = x.Date.split(\"/\");\n let [h, mins] = x[\"Time (LST/LDT)\"].split(\":\");\n return {\n datetime: d3.isoParse(`${y}-${m}-${d}T${h}:${mins}:00`),\n waterLevel: x[\"Preliminary (ft)\"] + x[\"Predicted (ft)\"]\n }\n}).map((x, _, c) => {\n let scale = d3.scaleLinear().domain([\n d3.min(c.filter(x => x.datetime > d3.isoParse(\"2022-09-28T11:59:00\") && x.datetime < d3.isoParse(\"2022-09-30T00:00:00\")).map(x => x.waterLevel)),\n d3.max(c.filter(x => x.datetime > d3.isoParse(\"2022-09-28T11:59:00\") && x.datetime < d3.isoParse(\"2022-09-30T00:00:00\")).map(x => x.waterLevel))\n ]).range([0, 1]);\n return {\n ...x,\n waterLvlPct: scale(x.waterLevel)\n }\n}).filter(x => x.datetime > d3.isoParse(\"2022-09-28T11:59:00\") && x.datetime < d3.isoParse(\"2022-09-30T00:00:00\"))","pinned":false,"hidden":false,"mode":"ojs"},{"id":232,"value":"{\n let tDomain = [d3.min(waterLevel.map(x => x.datetime)), d3.max(waterLevel.map(x => x.datetime))];\n let ts = d3.scaleTime().domain(tDomain).range([0, 1]);\n const loc = [{datetime: ts.invert(simTime), waterLvlPct: waterLevelTimeCurve(tDomain[0], tDomain[1])(simTime)}];\n return Plot.plot({\n y: {\n tickFormat: \"%\",\n label: \"Water level ↑\"\n },\n grid: true,\n marks: [\n Plot.dot(loc, {x: \"datetime\", y: \"waterLvlPct\", stroke: \"red\", r: 5}),\n Plot.line(waterLevel, {x: \"datetime\", y: \"waterLvlPct\"}),\n Plot.text([{datetime: ts.invert(.8), height: .8}], {x: \"datetime\", y: \"height\", text: d => d3.format(\".2%\")(simTime) + \" storm complete\", fontSize: 16, fontWeight: \"bold\", fill: \"red\", textAnchor: \"left\"}),\n Plot.text([{datetime: ts.invert(.8), height: .7}], {x: \"datetime\", y: \"height\", text: d => d3.format(\".2%\")(waterLevelTimeCurve(tDomain[0], tDomain[1])(simTime)) + \" of wall in contact with wave\", fontSize: 16, fontWeight: \"bold\", fill: \"red\"}),\n ]\n })\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":234,"value":"viewof simTime = Inputs.range([0, 1], {label: \"Simulation time (normalized)\", step: .01, value: 0})","pinned":false,"hidden":false,"mode":"ojs"},{"id":255,"value":"function waterLevelTimeCurve(isoStart, isoEnd) {\n let scale = d3.scaleTime().domain([isoStart, isoEnd]).range([0, 1])\n return function curve(t) {\n \n let d = scale.invert(t); // get date from time value\n let results = waterLevel.filter(x => fmtDt(x.datetime) === fmtDt(d)) // filter for exact match\n \n // If we get a single result we have an exact match\n if (results.length === 1) {\n return results[0].waterLvlPct\n }\n\n // Otherwise grab all results insert our time\n results = waterLevel.concat([{datetime: d}])\n // Sort them and then find the index of the one we inserted\n results = results.sort((a, b) => d3.ascending(a.datetime, b.datetime))\n let idx = results.findIndex(x => fmtDt(x.datetime) === fmtDt(d))\n \n // Return the average of the element before and after\n return (results[idx - 1].waterLvlPct + results[idx + 1].waterLvlPct) / 2\n }\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":384,"value":"Then, we'll accumulate this force over time.","pinned":false,"hidden":false,"mode":"md"},{"id":678,"value":"We'll break our simulation into 100 periods of time.","pinned":false,"hidden":false,"mode":"md"},{"id":672,"value":"periods = d3.range(.01, 1, .01).concat([1])","pinned":true,"hidden":false,"mode":"ojs"},{"id":682,"value":"And then aggregate the force against the wall for each of those periods.","pinned":false,"hidden":false,"mode":"md"},{"id":386,"value":"cumulativeForceLBS = {\n let isoStart = d3.min(waterLevel.map(x => x.datetime));\n let isoEnd = d3.max(waterLevel.map(x => x.datetime));\n let curve = waterLevelTimeCurve(isoStart, isoEnd);\n \n let momentumOfT = (t) => curve(t) * wallAreaForce;\n \n let milliElapsed = (isoEnd.getTime() - isoStart.getTime())\n let seconds = milliElapsed / 1000;\n let stepTime = seconds / 100;\n\n // Integrate!\n return periods.reduce((a, t) => {\n return momentumOfT(t) * (stepTime) + a\n }, 0)\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":392,"value":"That gives us a total force of **${fmtFloat(cumulativeForceLBS)} lbs** of force against our wall during the entire storm.","pinned":false,"hidden":false,"mode":"md"},{"id":382,"value":"### Adding hydrostatic pressure","pinned":false,"hidden":false,"mode":"md"},{"id":319,"value":"We can also add to this force the hydrostatic pressure of water. This is a constant force that standing water exerts on nearby objects. We can get the hydrostatic pressure of water across an area with the formula:\n\n`Force = (material density) * g * (water depth / 2) * (Surface area)`","pinned":false,"hidden":false,"mode":"md"},{"id":567,"value":"Our water depth will be a function of the wall height. The water depth in this case refers to how deep the storm tide is. We'll default to say the storm tide comes halfway up our wall, but you can tweak the slider and see how it affects the force.","pinned":false,"hidden":false,"mode":"md"},{"id":569,"value":"viewof waterDepthFactor = Inputs.range([0,1], {label: \"Water depth factor:\", step: .01, value: .5})","pinned":false,"hidden":false,"mode":"ojs"},{"id":335,"value":"hydroAreaForce = seawaterMassPounds * 9.8 * ((wallHeightFeet * waterDepthFactor) / 2) * ((wallHeightFeet * waterDepthFactor) * wallWidthFeet2)","pinned":true,"hidden":false,"mode":"ojs"},{"id":822,"value":"viewof simPct = Inputs.range([0, 100], {value: 0, label: \"Storm complete %:\", step: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":306,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":694,"value":"As we did above, we can sum these forces together and accumulate it over time.","pinned":false,"hidden":false,"mode":"md"},{"id":657,"value":"totalCumulativeForce = {\n let isoStart = d3.min(waterLevel.map(x => x.datetime));\n let isoEnd = d3.max(waterLevel.map(x => x.datetime));\n let curve = waterLevelTimeCurve(isoStart, isoEnd);\n \n let momentumOfT = (t) => curve(t) * wallAreaForce;\n let hydroOfT = (t) => curve(t) * hydroAreaForce;\n \n let milliElapsed = (isoEnd.getTime() - isoStart.getTime())\n let seconds = milliElapsed / 1000;\n let stepTime = seconds / 100;\n \n return periods.reduce((a, t) => {\n return momentumOfT(t) * (stepTime) + hydroOfT(t) * (stepTime) + a\n }, 0)\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":655,"value":"If we include this hydrostatic force in our calculation, we get **${fmtFloat(totalCumulativeForce)} total pound•seconds** of force.","pinned":false,"hidden":false,"mode":"md"},{"id":832,"value":"## Model 3","pinned":false,"hidden":false,"mode":"md"},{"id":906,"value":"### Numerical solution","pinned":false,"hidden":false,"mode":"md"},{"id":834,"value":"We can improve our estimate by using finite volume analysis and a different governing equation that takes into consideration more than hydrostatic and drag forces of water on a plane.","pinned":false,"hidden":false,"mode":"md"},{"id":837,"value":"We teamed up with water researchers from the University of South Florida — Juan Penaloza Gutierrez, Thathsarani Dilini Herath Herath Mudiyanselage, and Dr. Andres Tejada-Martinez — to make this happen.","pinned":false,"hidden":false,"mode":"md"},{"id":839,"value":"They built a virtual wave tank in Ansys Fluent. This virtual model is very similar to a [physical model by JBA Group](https://www.youtube.com/watch?v=3yNoy4H2Z-o).","pinned":false,"hidden":false,"mode":"md"},{"id":842,"value":"Let's load the output force data from this simulation and graph it to see average force along the boundary over time.","pinned":false,"hidden":false,"mode":"md"},{"id":844,"value":"numericalModelData = (await FileAttachment(\"report-averaged-1-second-force-rfile.csv\").csv({typed: true})).map(x => ({step: x[\"Time,Step\"], avgForce: x[\"report-averaged-1-second-force\"], instantaneousForce: x[\"report-averaged-1-second-force(instantaneous,values)\"], flowTime: x[\"flow-time\"], pct: x[\"pct-max\"]}))","pinned":false,"hidden":false,"mode":"ojs"},{"id":848,"value":"Plot.plot({\n y: {\n domain: [0, 7]\n },\n grid: true,\n marks: [\n Plot.line(numericalModelData, {x: \"flowTime\", y: \"avgForce\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":854,"value":"This force seems pretty low at first glance, but that's because our simulated wave was only 0.066 feet tall and 1.312 feet long. Our simulation is uniform along the y axis so we will scale only x and z. ","pinned":false,"hidden":false,"mode":"md"},{"id":857,"value":"viewof scaleFactor = Inputs.range([1, 200], {value: 50, label: \"Scale factor:\", step: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":861,"value":"Scaling our data by ${scaleFactor} gives us a wave that's **${fmtFloat(scaleFactor * 0.066)} feet tall** and **${fmtFloat(scaleFactor * 1.312)} feet long**.","pinned":false,"hidden":false,"mode":"md"},{"id":869,"value":"If we scale our time-averaged, instantaneous force we get:","pinned":false,"hidden":false,"mode":"md"},{"id":871,"value":"scaled = numericalModelData.map(x => ({...x, \"scaledAvgForce\": x[\"avgForce\"] * Math.pow(scaleFactor, 2)}))","pinned":true,"hidden":false,"mode":"ojs"},{"id":874,"value":"Plot.plot({\n marginLeft: 70,\n y: {\n domain: [0, d3.max(scaled.map(x => x.scaledAvgForce)) + d3.scaleLinear().domain([1,200]).range([0, 10000])(scaleFactor)]\n },\n grid: true,\n marks: [\n Plot.line(scaled, {x: \"flowTime\", y: \"scaledAvgForce\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":879,"value":"Then, to get the cumulative force, we will take the integral of this function.","pinned":false,"hidden":false,"mode":"md"},{"id":881,"value":"Our simulation ran for 60 seconds, and our force is the 1-second average, so to keep things clean we'll integrate from [0, 60] with a 1-second timestep.","pinned":false,"hidden":false,"mode":"md"},{"id":885,"value":"timeStep = 1","pinned":true,"hidden":false,"mode":"ojs"},{"id":883,"value":"steps = d3.range(0, 61, timeStep)","pinned":true,"hidden":false,"mode":"ojs"},{"id":888,"value":"timeStepData = scaled.filter(x => steps.includes(x.flowTime))","pinned":true,"hidden":false,"mode":"ojs"},{"id":890,"value":"scaledForceOfT = (t) => timeStepData.filter(x => x.flowTime === t)[0].scaledAvgForce","pinned":true,"hidden":false,"mode":"ojs"},{"id":892,"value":"minuteForce = steps.reduce((a, t) => scaledForceOfT(t) * timeStep + a)","pinned":true,"hidden":false,"mode":"ojs"},{"id":894,"value":"hourForce = minuteForce * 60","pinned":true,"hidden":false,"mode":"ojs"},{"id":899,"value":"That gives us a total of **${fmtFloat(hourForce)} pound•seconds** of force over the course of an hour.","pinned":false,"hidden":false,"mode":"md"},{"id":902,"value":"That unit will make little sense to someone without a physics background, but we will explain it with a comparison in a moment.","pinned":false,"hidden":false,"mode":"md"},{"id":904,"value":"First, we need to check our numerical solution with an analytical solution. We don't expect them to match, but they should be in the same ball park.","pinned":false,"hidden":false,"mode":"md"},{"id":909,"value":"### Analytical solution","pinned":false,"hidden":false,"mode":"md"},{"id":911,"value":"We are going to solve this analytically using the drag force and hydrostatic force equations from Model 2 with one substitution. Instead of calculating the wave speed from the storm forward velocity, we are going to use a formula that gives us wave speed in shallow water as a function of depth.","pinned":false,"hidden":false,"mode":"md"},{"id":913,"value":"Our equations:\n* `drag force = (material density) * (wave speed) * (surface area)`\n* `wave speed = sqrt(g * waterdepth)`\n* `hydrostatic force = (material density) * g * (water depth / 2) * (surface area)`","pinned":false,"hidden":false,"mode":"md"},{"id":919,"value":"We need a **wall surface area**, and a **water depth** to solve these. To get wall surface area, we will use the boundary wall of our model's domain.\n\nOur model was .328 feet wide and tall.\n\nSince we scaled our wave on x and z, we need to scale our plane on z (it has no x).","pinned":false,"hidden":false,"mode":"md"},{"id":924,"value":"checkWallWidth = .328","pinned":true,"hidden":false,"mode":"ojs"},{"id":926,"value":"checkWallHeight = scaleFactor * checkWallWidth","pinned":true,"hidden":false,"mode":"ojs"},{"id":930,"value":"We filled the tank with .196 feet of water (our water depth) and generated waves that were 0.066 feet tall. \n\nWe will scale those and use them to get water depth and surface area in our equations:","pinned":false,"hidden":false,"mode":"md"},{"id":937,"value":"checkWaterDepth = scaleFactor * .196","pinned":true,"hidden":false,"mode":"ojs"},{"id":941,"value":"checkHighWaterMark = checkWaterDepth + scaleFactor * 0.066","pinned":true,"hidden":false,"mode":"ojs"},{"id":946,"value":"checkSurfaceArea = checkHighWaterMark * checkWallWidth","pinned":true,"hidden":false,"mode":"ojs"},{"id":948,"value":"Now we can plug surface area and water depth into our equations:","pinned":false,"hidden":false,"mode":"md"},{"id":950,"value":"`drag force = (material density) * (wave speed) * (surface area)`\n\n`drag force = (63.99) * (sqrt(32.17 * checkWaterDepth)) * checkSurfaceArea`","pinned":false,"hidden":false,"mode":"md"},{"id":954,"value":"checkDragForce = 63.99 * Math.sqrt(32.17 * checkWaterDepth) * checkSurfaceArea","pinned":true,"hidden":false,"mode":"ojs"},{"id":958,"value":"`hydro force = (63.99) * g * ( checkWaterDepth / 2) * checkSurfaceArea`","pinned":false,"hidden":false,"mode":"md"},{"id":962,"value":"checkHydroForce = 63.99 * 32.17 * (checkWaterDepth/2) * (checkWaterDepth * checkWallWidth)","pinned":true,"hidden":false,"mode":"ojs"},{"id":964,"value":"Now we can turn this into a differential equation by applying this force over time using the normalized force curve from the fluent model.","pinned":false,"hidden":false,"mode":"md"},{"id":966,"value":"We will assume that when force reaches 100%, our wall will be 100% in contact with the wave.","pinned":false,"hidden":false,"mode":"md"},{"id":968,"value":"Plot.plot({\n y: {\n domain: [0, 1],\n tickFormat: \"%\",\n tickSize: 0,\n },\n grid: true,\n marks: [\n Plot.line(scaled, {x: \"flowTime\", y: \"pct\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":971,"value":"pctAtTime = (t) => scaled.filter(x => x.flowTime === t)[0].pct","pinned":true,"hidden":false,"mode":"ojs"},{"id":973,"value":"This will give us two differential equations, one for hydro force and one for drag force. Now we just integrate their sum to get the cumulative force of our analytical model.","pinned":false,"hidden":false,"mode":"md"},{"id":976,"value":"dragForceAtT = (t) => checkDragForce * pctAtTime(t)","pinned":true,"hidden":false,"mode":"ojs"},{"id":979,"value":"hydroForceAtT = (t) => checkHydroForce * pctAtTime(t)","pinned":true,"hidden":false,"mode":"ojs"},{"id":981,"value":"checkMinForce = steps.reduce((a, t) => dragForceAtT(t) * timeStep + hydroForceAtT(t) * timeStep + a)","pinned":true,"hidden":false,"mode":"ojs"},{"id":983,"value":"checkHourForce = checkMinForce * 60","pinned":true,"hidden":false,"mode":"ojs"},{"id":987,"value":"That gives our analytical solution, calculated with a simpler governing equation, an hourly force of **${d3.format(\".4s\")(checkHourForce)} pound•seconds**","pinned":false,"hidden":false,"mode":"md"},{"id":991,"value":"Our numerical solution had an hourly force of **${d3.format(\".4s\")(hourForce)} pound•seconds**","pinned":false,"hidden":false,"mode":"md"},{"id":396,"value":"## Comparisons","pinned":false,"hidden":false,"mode":"md"},{"id":398,"value":"### Model 1 comparison","pinned":false,"hidden":false,"mode":"md"},{"id":400,"value":"Let's compare our model 1 force, **${fmtFloat(waveLBSForce)} lbf**, to the collision force of Tampa Bay Buccaneers defensive tackle, Vita Vea.","pinned":false,"hidden":false,"mode":"md"},{"id":405,"value":"veaStats = ({weightLBS: 347, speedFPS: 40 * 3 / 5.1, armLengthFeet: 32/12})","pinned":false,"hidden":false,"mode":"ojs"},{"id":410,"value":"We can calculate Vea's momentum force by simulating him running top speed at a wall with his arms outstretched. We can measure the force from the moment his hands hit the wall until his chest collides and he comes to a complete stop.","pinned":false,"hidden":false,"mode":"md"},{"id":412,"value":"`Force = velocity * mass / time`\n\n`time = distance / ((initial speed - final speed) / 2)`\n\n`Force = velocity * mass / (distance / (initial speed / 2))`\n\n`Force = ${veaStats.speedFPS} * ${veaStats.weightLBS} / (${veaStats.armLengthFeet} / (${veaStats.speedFPS} / 2))`","pinned":false,"hidden":false,"mode":"md"},{"id":418,"value":"veaForceLBS = veaStats.speedFPS * veaStats.weightLBS / (veaStats.armLengthFeet / (veaStats.speedFPS / 2))","pinned":true,"hidden":false,"mode":"ojs"},{"id":449,"value":"numVeas = waveLBSForce / veaForceLBS","pinned":true,"hidden":false,"mode":"ojs"},{"id":445,"value":"That means it would take **${fmtFloat(numVeas)} Veas** to match the strength of the wave in our first model.","pinned":false,"hidden":false,"mode":"md"},{"id":455,"value":"### Model 3 comparison","pinned":false,"hidden":false,"mode":"md"},{"id":1007,"value":"Model 2 and 3 give us forces over an area. We can compare this to the amount of force required to stop a train car.","pinned":false,"hidden":false,"mode":"md"},{"id":1014,"value":"We'll focus on Model 3 because it's our better model, but the technique is the same to solve for Model 2.","pinned":false,"hidden":false,"mode":"md"},{"id":543,"value":"We got **${fmtInt(hourForce)} pound•seconds** for our cumulative hourly force on our numerical model.","pinned":false,"hidden":false,"mode":"md"},{"id":546,"value":"In order to make a comparison, we are going to figure out what speed an ${html`Amtrak Superliner`} would need to be traveling to impart the same amount of force on a wall.","pinned":false,"hidden":false,"mode":"md"},{"id":1090,"value":"We'll measure the force across the front plane of the train car as it smashes into an immovable wall.","pinned":false,"hidden":false,"mode":"md"},{"id":459,"value":"train = {\n let t = {widthFeet: 16, heightFeet: 10, lengthFeet: 85, massLBS: 150000}\n return {\n ...t,\n density: t.massLBS / (t.widthFeet * t.heightFeet * t.lengthFeet)\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":476,"value":"`Force = density * velocity^2 * surface area`\n\n`velocity = sqrt(Force / (density * surface area))`","pinned":false,"hidden":false,"mode":"md"},{"id":468,"value":"trainHourVelocityFPS = Math.sqrt(hourForce / (train.density * (train.widthFeet * train.heightFeet)))","pinned":true,"hidden":false,"mode":"ojs"},{"id":1028,"value":"trainMinuteVelocityFPS = Math.sqrt(minuteForce / (train.density * (train.widthFeet * train.heightFeet)))","pinned":true,"hidden":false,"mode":"ojs"},{"id":1021,"value":"That means our accumulated force has the same force as a train car crashing into a wall at **${fmtInt(fpsToMPH(trainHourVelocityFPS))} miles per hour!**","pinned":false,"hidden":false,"mode":"md"},{"id":1025,"value":"That's the equivalent to a train car hitting a wall at **${fmtInt(fpsToMPH(trainMinuteVelocityFPS))} miles per hour**, every minute for an hour!","pinned":false,"hidden":false,"mode":"md"},{"id":102,"value":"## Appendix","pinned":false,"hidden":false,"mode":"md"},{"id":175,"value":"This section includes code to do simple conversions and to drive the interactives on the article page.","pinned":false,"hidden":false,"mode":"md"},{"id":530,"value":"function kgM3TolbsF3(v) {\n return v / 16.018;\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":525,"value":"function mphToFPS(v) {\n return v * 1.466;\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":502,"value":"function fpsToMPH(v) {\n return v / 1.466;\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":497,"value":"function hoursBetween(d1, d2) {\n // https://stackoverflow.com/questions/19225414/how-to-get-the-hours-difference-between-two-date-objects\n return Math.abs(d1 - d2) / 36e5;\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":104,"value":"fmtFloat = d3.format(\",.2f\")","pinned":true,"hidden":false,"mode":"ojs"},{"id":700,"value":"function roundDown(places) {\n return function(v) {\n return Math.floor(v / Math.pow(10, places)) * Math.pow(10, places)\n }\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":130,"value":"waveMassPoundsRounded = d3.format(\",\")(roundDown(2)(waveMassPounds))","pinned":true,"hidden":false,"mode":"ojs"},{"id":132,"value":"wavePoundForceRounded = d3.format(\",\")(roundDown(2)(waveLBSForce))","pinned":true,"hidden":false,"mode":"ojs"},{"id":138,"value":"numVeasRounded = roundDown(0)(numVeas)","pinned":true,"hidden":false,"mode":"ojs"},{"id":153,"value":"viewWaveHeightFeet = waveHeightFeet","pinned":true,"hidden":false,"mode":"ojs"},{"id":155,"value":"viewWaveLengthFeet = waveLengthFeet","pinned":true,"hidden":false,"mode":"ojs"},{"id":157,"value":"viewWallWidthFeet = wallWidthFeet","pinned":true,"hidden":false,"mode":"ojs"},{"id":161,"value":"storms = [{n: \"Ian\", velocity: 8},{n: \"Michael\", velocity: 14},{n: \"Irma\", velocity: 12},{n: \"Katrina\", velocity: 15},]","pinned":false,"hidden":false,"mode":"ojs"},{"id":257,"value":"fmtDt = d3.timeFormat(\"%Y-%m-%dT%H:%M:%s\")","pinned":true,"hidden":false,"mode":"ojs"},{"id":815,"value":"fmtInt = d3.format(\",d\")","pinned":true,"hidden":false,"mode":"ojs"},{"id":1049,"value":"viewHourlyTrainSpeedMPH = fmtInt(fpsToMPH(trainHourVelocityFPS))","pinned":true,"hidden":false,"mode":"ojs"},{"id":1053,"value":"viewMinuteTrainSpeedMPH = fmtInt(fpsToMPH(trainMinuteVelocityFPS))","pinned":true,"hidden":false,"mode":"ojs"},{"id":1059,"value":"viewFEAwaveHeight = fmtInt(scaleFactor * 0.066)","pinned":true,"hidden":false,"mode":"ojs"},{"id":1065,"value":"viewFEAwaveLength = fmtInt(scaleFactor * 1.312)","pinned":true,"hidden":false,"mode":"ojs"},{"id":1082,"value":"viewHourForce = fmtInt(hourForce)","pinned":true,"hidden":false,"mode":"ojs"},{"id":1084,"value":"viewMinuteForce = fmtInt(minuteForce)","pinned":true,"hidden":false,"mode":"ojs"},{"id":1146,"value":"import {toc} from \"@mbostock/toc\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}