Flow Equations

The governing flow equations for the variable-area formulation of the compressible Euler equations are as follows:

var element = document.getElementById("moose-equation-a8bbfa18-5485-42d0-9239-80caa2e88884");katex.render("\\pd{\\rho A}{t} + \\pd{\\rho u A}{x} = 0 \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

var element = document.getElementById("moose-equation-a693c641-a990-496f-9773-81d24ae203d3");katex.render("\\pd{\\rho u A}{t} + \\pd{(\\rho u^2 + p) A}{x} = p \\pd{A}{x} - F A + \\rho g_x A \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

var element = document.getElementById("moose-equation-7913b13a-bb3d-4dce-88a6-f310398653a5");katex.render("\\pd{\\rho E A}{t} + \\pd{u (\\rho E + p) A}{x} = \\rho u g_x A + q''' A \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where

$var element = document.getElementById("moose-equation-3facb0b7-b441-4729-81fe-59bb39c47b2f");katex.render("t", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is time,
$var element = document.getElementById("moose-equation-cc7ecb3c-1d86-4643-b69f-4f856c3a38b3");katex.render("x", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the spatial position along the chosen axis,
$var element = document.getElementById("moose-equation-a7776a46-d373-47b1-8f53-b591d7884ee6");katex.render("A", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the cross-sectional area,
$var element = document.getElementById("moose-equation-e8a4926a-4160-4858-b88a-5f7a819f8627");katex.render("\\rho", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the density,
$var element = document.getElementById("moose-equation-e119bba4-2fa1-4ab5-8f44-50de0b424e9e");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the axial velocity,
$var element = document.getElementById("moose-equation-5b120aee-d4e5-45cc-96e9-87d03f3ff6a9");katex.render("E", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the specific total energy,
$var element = document.getElementById("moose-equation-ad1d420f-6e89-46b9-b404-b6ce982a2a5e");katex.render("p", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the pressure,
$var element = document.getElementById("moose-equation-8c154e84-40bb-4f63-ac45-eb7419eee2bf");katex.render("F", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the viscous drag force density,
$var element = document.getElementById("moose-equation-48160a18-c0b8-426c-a920-daeb47e606c6");katex.render("g_x", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the component of acceleration due to gravity in the axial direction, and
$var element = document.getElementById("moose-equation-eb36110a-760b-47cf-bee3-e9689435596b");katex.render("q'''", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the heat source rate density.

The viscous drag force density is computed as

var element = document.getElementById("moose-equation-0cffe7fd-bb1d-45a5-9c2d-5e33733055ab");katex.render("F = \\frac{f \\rho u |u| A}{2 D_h} \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where $var element = document.getElementById("moose-equation-2c5a3ec1-6997-4052-8330-02ada3159288");katex.render("f", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the Darcy friction factor and $var element = document.getElementById("moose-equation-1cc2f8d9-928d-4233-a314-d41555313d6b");katex.render("D_h", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the hydraulic diameter.

Flow Equations