User contributions for Nian

16 March 2026

• 22:4122:41, 16 March 2026 diff hist −125 The Q1D equations No edit summary
• 22:1222:12, 16 March 2026 diff hist +37 N Thermodynamics Nian moved page Thermodynamics to Thermodynamic processes Tag: New redirect
• 22:1222:12, 16 March 2026 diff hist 0 m Thermodynamic processes Nian moved page Thermodynamics to Thermodynamic processes
• 22:1222:12, 16 March 2026 diff hist −3,769 Thermodynamic processes No edit summary
• 22:1022:10, 16 March 2026 diff hist +2,650 N Isentropic relations Created page with "Category:Compressible flow Category:Thermodynamics __TOC__ === First law of thermodynamics === First law of thermodynamics: <math display="block"> de=\delta q - \delta w </math> For a reversible process: <math>\delta w=pd(1/\rho)</math> and <math>\delta q=Tds</math> <math display="block"> de=Tds-pd\left(\frac{1}{\rho}\right) </math> Enthalpy is defined as: <math>h=e+p/\rho</math> and thus <math display="block"> dh=de+pd\left(\frac{1}{\rho}\right)+\left(\f..."
• 22:0722:07, 16 March 2026 diff hist +1,203 N Specific heat Created page with "Category:Compressible flow Category:Thermodynamics __TOC__ For thermally perfect and calorically perfect gases <math display="block"> \begin{aligned} &C_p=\frac{dh}{dT}\\ &C_v=\frac{de}{dT} \end{aligned} </math> From the definition of enthalpy and the equation of state <math>p=\rho RT</math> <math display="block"> h=e+\frac{p}{\rho}=e+RT </math> Differentiate Eqn. \ref{eq:enthalpy} with respect to temperature gives <math display="block"> \frac{dh}{dT}=\fra..."
• 17:1217:12, 16 March 2026 diff hist −303 Thermodynamic processes No edit summary
• 16:5916:59, 16 March 2026 diff hist +302 Compressible flow No edit summary
• 16:5616:56, 16 March 2026 diff hist +3 Incompressible flow No edit summary current
• 16:5516:55, 16 March 2026 diff hist +48 Compressible flow No edit summary
• 16:5416:54, 16 March 2026 diff hist −36 Flowpedia No edit summary
• 16:5116:51, 16 March 2026 diff hist +12 Compressible flow No edit summary
• 16:5016:50, 16 March 2026 diff hist +59 Flowpedia No edit summary
• 16:3816:38, 16 March 2026 diff hist −193 Shock-tube relations No edit summary
• 15:3615:36, 16 March 2026 diff hist +136 Compressible flow No edit summary
• 15:3515:35, 16 March 2026 diff hist +1,840 N Shock-tube relations Created page with "Category:Compressible flow Category:Unsteady waves Category:Inviscid flow __TOC__ \subsection{The Shock Tube Relations} \begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter07/pdf/shock-tube.pdf} \caption{traveling waves in a shock tube} \label{fig:shocktube} \end{center} \end{figure} \noindent From the analysis of the incident shock, we have a relation for the induced flow behind the shock\\ \begin{equation} u_2=u_p=\f..."
• 15:3415:34, 16 March 2026 diff hist +8,013 Moving shock waves No edit summary
• 15:3315:33, 16 March 2026 diff hist −8,035 Moving expansion waves No edit summary
• 15:3215:32, 16 March 2026 diff hist +10,729 N Moving shock waves Created page with "Category:Compressible flow Category:Unsteady waves Category:Inviscid flow __TOC__ \section{Moving Normal Shock Waves} \noindent The starting point is the governing equations for stationary normal shocks (repeated here for convenience). \begin{equation} \rho_1 u_1 = \rho_2 u_2 \label{eq:stationary:cont} \end{equation} \begin{equation} \rho_1 u_1^2+p_1 = \rho_2 u_2^2 + p_2 \label{eq:stationary:mom} \end{equation} \begin{equation} h_1 + \frac{1}{2}u_1^2 =..."
• 15:3115:31, 16 March 2026 diff hist +10,729 N Moving expansion waves Created page with "Category:Compressible flow Category:Unsteady waves Category:Inviscid flow __TOC__ \section{Moving Normal Shock Waves} \noindent The starting point is the governing equations for stationary normal shocks (repeated here for convenience). \begin{equation} \rho_1 u_1 = \rho_2 u_2 \label{eq:stationary:cont} \end{equation} \begin{equation} \rho_1 u_1^2+p_1 = \rho_2 u_2^2 + p_2 \label{eq:stationary:mom} \end{equation} \begin{equation} h_1 + \frac{1}{2}u_1^2 =..."
• 15:3015:30, 16 March 2026 diff hist +8,659 N Finite non-linear waves Created page with "Category:Compressible flow Category:Unsteady waves Category:Inviscid flow __TOC__ \section{Finite Nonlinear Waves} \noindent Starting point: the governing flow equations on partial differential form\\ \noindent Continuity equation: \begin{equation} \frac{\partial \rho}{\partial t}+u\frac{\partial \rho}{\partial x}+\rho\frac{\partial u}{\partial x}=0 \label{eq:pde:cont} \end{equation}\\ \noindent Momentum equation: \begin{equation} \frac{\partial u}{\pa..."
• 15:2915:29, 16 March 2026 diff hist +11,388 N Acoustic theory Created page with "Category:Compressible flow Category:Unsteady waves Category:Inviscid flow __TOC__ \section{Acoustic Theory} \noindent In the following we are going to derive the linear acoustic wave equation starting from the continuity and momentum equations on non-conservation differential form. The equations are repeated here for convenience.\\ \[\dfrac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0\] \[\rho\dfrac{D\mathbf{v}}{Dt}+\nabla p=0\]\\ \noindent Remember that $..."
• 15:1015:10, 16 March 2026 diff hist +166 N Nozzle flow Created page with "Category:Compressible flow Category:Quasi-one-dimensional flow Category:Inviscid flow __TOC__ == Nozzle flow == add description of nozzle flows here..."
• 15:0715:07, 16 March 2026 diff hist +1,886 N Choked flow Created page with "Category:Compressible flow Category:Quasi-one-dimensional flow Category:Inviscid flow __TOC__ \section{Geometric Choking} \noindent For steady-state nozzle flow, the massflow is obtained as \\ \begin{equation} \dot{m}=\rho uA=const \label{eq:massflow:a} \end{equation}\\ \noindent Eqn. \ref{eq:massflow:a} can be evaluated at any location inside the nozzle and if evaluated at sonic conditions we get\\ \begin{equation} \dot{m}=\rho^* u^* A^* \label{eq:mass..."
• 15:0515:05, 16 March 2026 diff hist +160 Compressible flow No edit summary
• 15:0315:03, 16 March 2026 diff hist +5,141 N The Q1D equations Created page with "Category:Compressible flow Category:Quasi-one-dimensional flow Category:Governing equations Category:Inviscid flow __TOC__ \section{Governing Equations} \begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter06/pdf/control-volume.pdf} \caption{Quasi-one-dimensional flow - control volume} \label{fig:cv} \end{center} \end{figure} \noindent In the following quasi-one-dimensional flow will be assumed. That means that the c..."
• 15:0115:01, 16 March 2026 diff hist +161 N Diffusers Created page with "Category:Compressible flow Category:Quasi-one-dimensional flow Category:Inviscid flow __TOC__ == Diffusers == Add description and examples here..."
• 15:0015:00, 16 March 2026 diff hist +2,653 N Area-velocity relation Created page with "Category:Compressible flow Category:Quasi-one-dimensional flow Category:Inviscid flow __TOC__ \section{The Area-Velocity Relation} \noindent Starting point - the continuity equation (Eqn. \ref{eq:governing:cont}):\\ \[d(\rho uA)=0 \Rightarrow \rho u dA+\rho Adu +uAd\rho=0\]\\ \noindent divide by $\rho uA$ gives\\ \begin{equation} \frac{d\rho}{\rho}+\frac{du}{u}+\frac{dA}{A}=0 \label{eq:governing:cont:b} \end{equation}\\ \noindent As the name suggests,..."
• 14:5814:58, 16 March 2026 diff hist +5,296 N Area-Mach relation Created page with "Category:Compressible flow Category:Quasi-one-dimensional flow Category:Inviscid flow __TOC__ \section{The Area-Mach-Number Relation} \noindent Starting point - the continuity equation (Eqn. \ref{eq:governing:cont}):\\ \[d(\rho uA)=0 \Rightarrow \rho u A=const\]\\ \noindent This applies everywhere in the nozzle and therefore the sonic conditions can be used as a reference\\ \[\rho uA=\rho^*u^*A^*=\left\{u^*=a^*\right\}=\rho^*a^*A^*\]\\ \noindent divide..."
• 14:5414:54, 16 March 2026 diff hist +162 Compressible flow No edit summary
• 14:5314:53, 16 March 2026 diff hist +655 N Shock-expansion theory Created page with "Category:Compressible flow Category:Two-dimensional flow Category:Inviscid flow __TOC__ \section{Shock-Expansion Theory} \begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter05/pdf/shock-expansion-diamond-airfoil.pdf} \caption{Supersonic flow over a symmetric diamond wedge airfoil at zero angle of attack} \label{fig:shock:expansion:wedge} \end{center} \end{figure} \begin{figure}[ht!] \begin{center} \includegraphics[]{fig..."
• 14:5114:51, 16 March 2026 diff hist +6,396 N Expansion waves Created page with "Category:Compressible flow Category:Two-dimensional flow Category:Inviscid flow __TOC__ \section{Prandtl-Meyer Expansion Waves} \begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter05/pdf/Mach-wave.pdf} \caption{Mach wave flow turning} \label{fig:machwave} \end{center} \end{figure} \noindent A single Mach wave has a insignificant effect on the flow passing it but an expansion region constitutes an infinite number of Mach..."
• 14:5014:50, 16 March 2026 diff hist +10,498 N Oblique shocks Created page with "Category:Compressible flow Category:Two-dimensional flow Category:Inviscid flow Category:Wave solution __TOC__ \section{Oblique Shock Relations} \begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter05/pdf/oblique-shock-0.pdf} \caption{Definition of oblique shock angles} \label{fig:oblique:shock:angles} \end{center} \end{figure} %\begin{figure}[ht!] %\begin{center} %\includegraphics[]{figures/standalone-figures/Chapt..."
• 14:4514:45, 16 March 2026 diff hist +24,219 N One-dimensional flow with heat addition Created page with "Category:Compressible flow Category:One-dimensional flow Category:Inviscid flow Category:Continuous solution __TOC__ \section{One-Dimensional Flow with Heat Addition} \noindent The aim is to derive relations for pressure ratio and temperature ratio as a function of Mach numbers. We will do that starting from the momentum equation.\\ \begin{equation} p_2-p_1=\rho_1 u_1^2 - \rho_2 u_2^2 \label{eq:governing:mom} \end{equation}\\ \noindent Assuming calo..."
• 14:4314:43, 16 March 2026 diff hist +19,536 N One-dimensional flow with friction Created page with "Category:Compressible flow Category:One-dimensional flow Category:Inviscid flow Category:Continuous solution __TOC__ \section{One-Dimensional Flow with Friction} \noindent The starting point is the governing equations for one-dimensional steady-state flow\\ \subsection{Continuity} \begin{equation} \rho_1 u_1=\rho_2 u_2 \label{eq:governing:cont} \end{equation}\\ \subsection{Momentum} \begin{equation} \rho_1 u_1^2+p_1-\bar{\tau}_w\frac{bL}{A}=\rho_2..."
• 13:4213:42, 16 March 2026 diff hist +1,483 N The entropy equation Created page with "Category:Compressible flow Category:Governing equations __TOC__ \section{The Entropy Equation} \noindent From the second law of thermodynamics\\ \begin{equation} \frac{De}{Dt}=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right) \label{eq:second:law} \end{equation}\\ \noindent From the energy equation on differential non-conservation form internal energy formulation\\ \[\frac{De}{Dt} = \dot{q} - \frac{p}{\rho}(\nabla\cdot\mathbf{v})\]\\ \noindent The co..."
• 13:4113:41, 16 March 2026 diff hist +1,481 N Crocco's equation Created page with "Category:Compressible flow Category:Governing equations __TOC__ \section{Crocco's Equation} \noindent The momentum equation without body forces\\ \[\rho\frac{D\mathbf{v}}{Dt}=-\nabla p\]\\ \noindent Expanding the substantial derivative\\ \[\rho\frac{\partial \mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\nabla\mathbf{v}=-\nabla p\]\\ \noindent The first and second law of thermodynamics gives\\ \[T\nabla s =\nabla h-\frac{\nabla p}{\rho}\]\\ \noindent Insert..."
• 13:3813:38, 16 March 2026 diff hist +54 N Governing equations on differential form Nian moved page Governing equations on differential form to Governing equations on differential form current Tag: New redirect
• 13:3813:38, 16 March 2026 diff hist 0 m Governing equations on differential form Nian moved page Governing equations on differential form to Governing equations on differential form
• 13:3713:37, 16 March 2026 diff hist +39 Compressible flow No edit summary
• 13:3613:36, 16 March 2026 diff hist +251 Compressible flow No edit summary
• 13:2413:24, 16 March 2026 diff hist +11,896 N Governing equations on differential form Created page with "Category:Compressible flow Category:Governing equations __TOC__ \section{Governing Equations on Differential Form} \subsection{Conservation of Mass} \noindent Apply Gauss's divergence theorem on the surface integral in Eqn. \ref{eq:governing:cont:int} gives\\ \[\oiint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=\iiint_{\Omega}\nabla\cdot(\rho\mathbf{v})d\mathscr{V}\]\\ \noindent Also, if $\Omega$ is a fixed control volume\\ \[\frac{d}{dt}\iiint_{\O..."
• 13:2213:22, 16 March 2026 diff hist +8,432 N Governing equations on integral form Created page with "Category:Compressible flow Category:Governing equations __TOC__ \section{Governing Equations on Integral Form} \begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter02/pdf/control-volume.pdf} \caption{Generic control volume} \label{fig:generic:cv} \end{center} \end{figure} \noindent The governing equations stems from mass conservation, conservation of momentum and conservation of energy \subsection{The Continuity Equation}..."
• 12:2112:21, 16 March 2026 diff hist +368 N Reference flow states Created page with "Category:Compressible flow Category:One-dimensional flow Category:Reference flow states __TOC__ \section{Reference Flow States} \subsection{Stagnation Flow Properties} \[h_o=h+\dfrac{1}{2}u^2\Rightarrow Cp T_o=C_p T +\dfrac{1}{2}u^2\Rightarrow \dfrac{T_o}{T}=1+\dfrac{1}{2C_p}M^2\gamma RT=1+\dfrac{\gamma-1}{2}M^2\] \subsection{Sonic Flow Properties}"
• 12:1912:19, 16 March 2026 diff hist +6,468 N Thermodynamic processes Created page with "Category:Compressible flow Category:Thermodynamics __TOC__ \section{Thermodynamics} \subsection{Specific Heat Relations} \noindent For thermally perfect and calorically perfect gases\\ \begin{equation} \begin{aligned} &C_p=\frac{dh}{dT}\\ &C_v=\frac{de}{dT} \end{aligned} \label{eq:specificheat} \end{equation}\\ \noindent From the definition of enthalpy and the equation of state $p=\rho RT$\\ \begin{equation} h=e+\frac{p}{\rho}=e+RT \label{eq:enthalpy} \end..."
• 10:1510:15, 16 March 2026 diff hist +2 Normal-shock relations →The Hugoniot Equation
• 10:1410:14, 16 March 2026 diff hist −21,439 One-dimensional inviscid flow Replaced content with "Category:Compressible flow Category:One-dimensional flow Category:inviscid flow __TOC__ {{:Acoustic waves}} {{:Shock waves}} {{:Normal-shock relations}}" Tag: Replaced
• 10:1110:11, 16 March 2026 diff hist +128 Normal-shock relations No edit summary
• 10:1010:10, 16 March 2026 diff hist +8,654 N Normal-shock relations Created page with " ==Normal Shock Relations== Rewriting the continuity equation (Eqn. \ref{eq:governing:cont}) <math display="block"> \frac{\rho_2}{\rho_1}=\frac{u_1}{u_2}=\frac{u_1^2}{u_1 u_2}=\left\{{a^*}^2=u_1u_2\right\}=\frac{u_1^2}{{a^*}^2}={M^*_1}^2 </math> Eqn. \ref{eq:MachStar} in Eqn. \ref{eq:Normal:density:a} gives <math display="block"> \frac{\rho_2}{\rho_1}=\frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2} </math> To get a corresponding relation for the pressure ratio over the s..."
• 10:0810:08, 16 March 2026 diff hist +28 N Acoustic waves Nian moved page Acoustic waves to Acoustic waves current Tag: New redirect