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-4d53cc27-3827-4b08-a496-db39ff4bc036");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-01a1ac2c-a569-4822-b169-52262380060f");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-762a1463-4537-4bd1-a261-5cea64455be8");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

$var element = document.getElementById("moose-equation-4ede6a50-4c47-43ff-bcd6-3144c4b2a8c4");katex.render("t", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is time,
$var element = document.getElementById("moose-equation-79edc553-7c8d-45fc-b262-f70b4633a3fb");katex.render("x", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the spatial position along the chosen axis,
$var element = document.getElementById("moose-equation-ab286150-7a52-4ae1-9d24-3542860e0f0d");katex.render("A", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the cross-sectional area,
$var element = document.getElementById("moose-equation-04fb3243-9535-4dda-a8ed-2af910649320");katex.render("\\rho", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the density,
$var element = document.getElementById("moose-equation-9a1c55e3-8da1-41da-957a-4d102b76fb6f");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the axial velocity,
$var element = document.getElementById("moose-equation-42a0cb20-7764-4475-a3b6-f1f7dcf597f4");katex.render("E", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the specific total energy,
$var element = document.getElementById("moose-equation-3376bb2c-c18e-4a13-a587-3ab202990bf2");katex.render("p", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the pressure,
$var element = document.getElementById("moose-equation-a7222818-c8c5-418b-a8bd-d93cd9242e8d");katex.render("F", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the viscous drag force density,
$var element = document.getElementById("moose-equation-f05ef044-3a7a-49c3-a26e-65b9219da221");katex.render("g_x", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the component of acceleration due to gravity in the axial direction, and
$var element = document.getElementById("moose-equation-03c096d2-0c42-4c3d-9b70-0a7b544ab1eb");katex.render("q'''", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the heat source rate density.

The viscous drag force density is computed as

var element = document.getElementById("moose-equation-3f3c3a4f-1af2-4e64-9c8c-e44d2f97a5a8");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 $var element = document.getElementById("moose-equation-4738ad3f-c58e-4e51-aaf5-480d5533d6ea");katex.render("f", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the Darcy friction factor and $var element = document.getElementById("moose-equation-4c9c349e-7f3d-4673-b47a-e8ffbccef82a");katex.render("D_h", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the hydraulic diameter.

Flow Equations