U.S. Standard Atmosphere<\/a>). The ratios of properties are convenient:<\/p>\n$$\\delta=\\frac{p}{p_{ssl}}$$
\n$$ \\theta=\\frac{T}{T_{ssl}}$$
\n$$\\sigma=\\frac{\\rho}{\\rho_{ssl}} = \\frac{\\delta}{\\theta}$$<\/p>\n
From compressible flow theory, an isentropic process gives a stagnation pressure (aka. total pressure) of $$p_{t}=p\\left(1+\\frac{\\gamma-1}{2}M^{2}\\right)^{\\frac{\\gamma}{\\gamma-1}}$$<\/p>\n
Rearranging with $$V=aM$$
\nand $$\\Delta p=p_{t}-p$$
\nresults in $$V^{2}=\\frac{2a^{2}}{\\gamma-1}\\left[\\left(\\frac{p_{t}}{p}\\right)^{\\frac{\\gamma-1}{\\gamma}}-1\\right]$$
\nwhich is identical to $$V^{2}=\\frac{2a^{2}}{\\gamma-1}\\left[\\left(\\frac{\\Delta p}{p}+1\\right)^{\\frac{\\gamma-1}{\\gamma}}-1\\right]$$<\/p>\n
Rearranging gives $$\\frac{p_{t}}{p}\\frac{p}{p_{ssl}}=\\left(\\frac{V^{2}}{a_{ssl}^{2}}\\frac{\\gamma-1}{2}+1\\right)^{\\frac{\\gamma}{\\gamma-1}}$$
\nSubstitution gives $$\\frac{p_{t}}{p}=\\frac{1}{\\delta}\\left(\\frac{V^{2}}{a_{ssl}^{2}}\\frac{\\gamma-1}{2}+1\\right)^{\\frac{\\gamma}{\\gamma-1}}$$<\/p>\n
Mach Meter<\/strong><\/p>\nA Mach meter determines the freestream mach number from a pressure ratio. $$M=\\sqrt{\\frac{2}{\\gamma-1}\\left[\\left(\\frac{p_{t}}{p}\\right)^{\\frac{\\gamma-1}{\\gamma}}-1\\right]}$$
\nor the inverse $$\\frac{p_{t}}{p}=\\left(M^{2}\\frac{\\gamma-1}{2}+1\\right)^{\\frac{\\gamma}{\\gamma-1}}$$
\nHowever, when the upstream Mach number exceeds 1, a shock will be generated forward of the pitot tube. Given the freestream is point 1 and the pitot is point 3, the pressure ratio upstream in terms of the shock’s M
\nare $$\\frac{p_{t_{1}}}{p_{1}}=\\left(\\frac{p_{t_{3}}}{p_{1}}\\right)\\left(\\frac{p_{t_{1}}}{p_{t_{3}}}\\right)$$
\nWhere from compressible flow of a shock, the stagnation pressure ratios are $$\\left(\\frac{p_{t_{3}}}{p_{t_{1}}}\\right)=\\left(\\frac{\\frac{\\gamma+1}{2}M_{1}^{2}}{1+\\frac{\\gamma-1}{2}M_{1}^{2}}\\right)^{\\frac{\\gamma}{\\gamma-1}}\\left(\\frac{2\\gamma}{\\gamma+1}M_{1}^{2}-\\frac{\\gamma-1}{\\gamma+1}\\right)^{\\frac{-1}{\\gamma-1}}$$
\nSubstitute into $$\\frac{p_{t_{1}}}{p_{1}}=\\left(M^{2}\\frac{\\gamma-1}{2}+1\\right)^{\\frac{\\gamma}{\\gamma-1}}$$
\ngiving $$\\left(\\frac{p_{t_{3}}}{p_{3}}\\right)=\\left(M^{2}\\frac{\\gamma-1}{2}+1\\right)^{\\frac{\\gamma}{\\gamma-1}}\\left(\\frac{p_{t_{3}}}{p_{t_{1}}}\\right)$$
\nwhich is $$\\left(\\frac{p_{t_{3}}}{p_{3}}\\right)=\\left(M_{1}^{2}\\frac{\\gamma-1}{2}+1\\right)^{\\frac{\\gamma}{\\gamma-1}}\\left(\\frac{\\frac{\\gamma+1}{2}M_{1}^{2}}{1+\\frac{\\gamma-1}{2}M_{1}^{2}}\\right)^{\\frac{\\gamma}{\\gamma-1}}\\left(\\frac{2\\gamma}{\\gamma+1}M_{1}^{2}-\\frac{\\gamma-1}{\\gamma+1}\\right)^{\\frac{-1}{\\gamma-1}}$$
\nThis reduces to the Reighleigh Pitot Equation $$\\left(\\frac{p_{t_{3}}}{p_{3}}\\right)=\\left(\\frac{\\gamma+1}{2}M_{1}^{2}\\right)^{\\frac{\\gamma}{\\gamma-1}}\\left(\\frac{2\\gamma}{\\gamma+1}M_{1}^{2}-\\frac{\\gamma-1}{\\gamma+1}\\right)^{\\frac{-1}{\\gamma-1}}$$
\nGiven a particular pressure ratio, only one Mach number satisfies this equation. Even better, the subsonic value is a good approximation for this supersonic value.<\/p>\n
Calibrated Airspeed<\/strong><\/p>\nThe airspeed indicator operates by receiving static pressure from a static port and stagnation pressure from a pitot system. The airspeed indicator measures the pressure difference: $$\\Delta p=p_{t}-p_{static}$$<\/p>\n
The reference pressure and speed of sound are the sea level values. $$V_{cal}=\\sqrt{\\frac{2a_{ssl}^{2}}{\\gamma-1}\\left[\\left(\\frac{\\Delta p_{cal}}{p_{ssl}}+1\\right)^{\\frac{\\gamma-1}{\\gamma}}-1\\right]}$$
\nThe inverse mapping indicates the pressure differential generated for a given calibrated airspeed. $$\\Delta p_{cal}=p_{ssl}\\left[\\left(\\frac{\\gamma-1}{2}\\left(\\frac{V_{cal}}{a_{ssl}}\\right)^{2}+1\\right)^{\\frac{\\gamma}{\\gamma-1}}-1\\right]$$
\nSubsonic Case (ONLY):<\/strong><\/p>\nFor the subsonic case only, computing the true airspeed simplifies to an explicit equation. From the calibrated airspeed, we found a pressure differential. This pressure differential is applied to the same stagnation pressure relation for the local properties. $$V_{true}=\\sqrt{\\frac{2a^{2}}{\\gamma-1}\\left[\\left(\\frac{\\Delta p_{cal}}{p}+1\\right)^{\\frac{\\gamma-1}{\\gamma}}-1\\right]}$$
\nThus the ratio of true to calibrated airspeed is $$\\frac{V_{true}}{V_{cal}}=\\sqrt{\\frac{\\frac{2a^{2}}{\\gamma-1}\\left[\\left(\\frac{\\Delta p_{cal}}{p}+1\\right)^{\\frac{\\gamma-1}{\\gamma}}-1\\right]}{\\frac{2a_{sl}^{2}}{\\gamma-1}\\left[\\left(\\frac{\\Delta p_{cal}}{p_{ssl}}+1\\right)^{\\frac{\\gamma-1}{\\gamma}}-1\\right]}}$$
\nCancelling terms gives with $$a^{2}=\\gamma RT$$
\ngives $$\\frac{V_{true}}{V_{cal}}=\\sqrt{\\theta\\frac{\\left[\\left(\\frac{\\Delta p_{cal}}{p_{ssl}}\\frac{1}{\\delta}+1\\right)^{\\frac{\\gamma-1}{\\gamma}}-1\\right]}{\\left[\\left(\\frac{\\Delta p_{cal}}{p_{ssl}}+1\\right)^{\\frac{\\gamma-1}{\\gamma}}-1\\right]}}$$<\/p>\n","protected":false},"excerpt":{"rendered":"
This post provides a visual characterization of a generic flight envelope with a standard atmosphere. The following figure shows a generic flight envelope map. A pdf version is available at airspeed-2014c.pdf. This file plots altitude (0 to 50 thousand feet), calibrated airspeed (0 to 1000 KCAS), true airspeed, Mach number, dynamic pressure, static pressure, and […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/posts\/152"}],"collection":[{"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/comments?post=152"}],"version-history":[{"count":6,"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/posts\/152\/revisions"}],"predecessor-version":[{"id":160,"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/posts\/152\/revisions\/160"}],"wp:attachment":[{"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/media?parent=152"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/categories?post=152"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/tags?post=152"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
![\"KCAS,](\"http:\/\/charles-oneill.com\/blog\/wp-content\/uploads\/2014\/10\/figure_1.png\")
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