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21:5921:59, 20 March 2026 Unsteady waves (hist | edit) [648 bytes] Nian (talk | contribs) (Created page with "Category:Collection Category:Compressible flow Category:Unsteady waves Category:inviscid flow <noinclude> Category:Compressible flow:Section </noinclude> __TOC__ == Moving shock waves == {{:Moving shock waves}} == Acoustic theory == {{:Acoustic theory}} == Finite non-linear waves == {{:Finite non-linear waves}} == Moving expansion waves == {{:Moving expansion waves}} == Shock-tube relations == {{:Shock-tube relations}}")
21:5321:53, 20 March 2026 Quasi-one-dimensional flow (hist | edit) [654 bytes] Nian (talk | contribs) (Created page with "Category:Collection Category:Compressible flow Category:Quasi-one-dimensional flow Category:inviscid flow <noinclude> Category:Compressible flow:Section </noinclude> __TOC__ == The Q1D equations == {{:The Q1D equations}} == Area-velocity relation == {{:Area-velocity relation}} == Area-Mach relation == {{:Area-Mach relation}} == Choked flow == {{:Choked flow}} == Nozzle flow == {{:Nozzle flow}} == Diffusers == {{:Diffusers}}")

18:4518:45, 17 March 2026 Governing equations (hist | edit) [632 bytes] Nian (talk | contribs) (Created page with "Category:Compressible flow Category:Governing equations Category:inviscid flow Category:Collection __TOC__ == Governing equations on integral form == {{:Governing equations on integral form}} == Governing equations on differential form == {{:Governing equations on differential form}} == The entropy equation == {{:The entropy equation}} == Crocco's equation == {{:Crocco's equation}}")
13:4413:44, 17 March 2026 Oblique shocks and expansion waves (hist | edit) [532 bytes] Nian (talk | contribs) (Created page with "Category:Compressible flow Category:Two-dimensional flow Category:inviscid flow Category:Collection __TOC__ == Oblique shocks == {{:Oblique shocks}} == Expansion waves == {{:Expansion waves}} == Shock-expansion theory == {{:Shock-expansion theory}}")

22:1022:10, 16 March 2026 Isentropic relations (hist | edit) [3,085 bytes] Nian (talk | contribs) (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 Specific heat (hist | edit) [1,540 bytes] Nian (talk | contribs) (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...")
15:3515:35, 16 March 2026 Shock-tube relations (hist | edit) [1,879 bytes] Nian (talk | contribs) (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...")
15:3215:32, 16 March 2026 Moving shock waves (hist | edit) [16,896 bytes] Nian (talk | contribs) (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 Moving expansion waves (hist | edit) [2,807 bytes] Nian (talk | contribs) (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 Finite non-linear waves (hist | edit) [8,143 bytes] Nian (talk | contribs) (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 Acoustic theory (hist | edit) [11,412 bytes] Nian (talk | contribs) (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 Nozzle flow (hist | edit) [422 bytes] Nian (talk | contribs) (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 Choked flow (hist | edit) [1,856 bytes] Nian (talk | contribs) (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...")
15:0315:03, 16 March 2026 The Q1D equations (hist | edit) [5,155 bytes] Nian (talk | contribs) (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 Diffusers (hist | edit) [417 bytes] Nian (talk | contribs) (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 Area-velocity relation (hist | edit) [2,859 bytes] Nian (talk | contribs) (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 Area-Mach relation (hist | edit) [5,372 bytes] Nian (talk | contribs) (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...")
14:5314:53, 16 March 2026 Shock-expansion theory (hist | edit) [925 bytes] Nian (talk | contribs) (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 Expansion waves (hist | edit) [6,109 bytes] Nian (talk | contribs) (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 Oblique shocks (hist | edit) [11,027 bytes] Nian (talk | contribs) (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 One-dimensional flow with heat addition (hist | edit) [23,561 bytes] Nian (talk | contribs) (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 One-dimensional flow with friction (hist | edit) [18,814 bytes] Nian (talk | contribs) (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 The entropy equation (hist | edit) [1,678 bytes] Nian (talk | contribs) (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 Crocco's equation (hist | edit) [1,807 bytes] Nian (talk | contribs) (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...")
13:2413:24, 16 March 2026 Governing equations on differential form (hist | edit) [12,939 bytes] Nian (talk | contribs) (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...") originally created as "Governing equations in differential form"
13:2213:22, 16 March 2026 Governing equations on integral form (hist | edit) [8,479 bytes] Nian (talk | contribs) (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 Reference flow states (hist | edit) [606 bytes] Nian (talk | contribs) (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 Thermodynamic processes (hist | edit) [2,695 bytes] Nian (talk | contribs) (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...") originally created as "Thermodynamics"
10:1010:10, 16 March 2026 Normal-shock relations (hist | edit) [8,831 bytes] Nian (talk | contribs) (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...")
10:0610:06, 16 March 2026 Shock waves (hist | edit) [7,949 bytes] Nian (talk | contribs) (Created page with "==Shock Waves==  The starting point is to set up the governing equations for one-dimensional steady compressible flow over a control volume enclosing the normal shock (Fig. \ref{fig:shock:cv}). <!-- \begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-fig...") originally created as "Shock Waves"
10:0410:04, 16 March 2026 Acoustic waves (hist | edit) [5,611 bytes] Nian (talk | contribs) (Created page with "Category:Compressible flow Category:One-dimensional flow Category:inviscid flow __TOC__ ==Acoustic Waves==  In Fig. \ref{fig:soundwave}, station 1 represents the flow state ahead of the sound wave and station 2 the flow state behind the sound wave. Set up the continui...") originally created as "Acoustic Waves"

14:0314:03, 13 March 2026 One-dimensional inviscid flow (hist | edit) [755 bytes] Nian (talk | contribs) (Created page with " == Acoustic Wave ==  In Fig. \ref{fig:soundwave}, station 1 represents the flow state ahead of the sound wave and station 2 the flow state behind the sound wave. Set up the continuity equation for one-dimensional flows between 1 and 2. If we could change frame of reference and fol...")
13:4813:48, 13 March 2026 Flowpedia (hist | edit) [249 bytes] Nian (talk | contribs) (Created page with "* Incompressible Flow * Compressible Flow")
08:4208:42, 13 March 2026 Incompressible flow (hist | edit) [58 bytes] Nian (talk | contribs) (init) Tag: Visual edit
08:4108:41, 13 March 2026 Compressible flow (hist | edit) [3,475 bytes] Nian (talk | contribs) (init) Tag: Visual edit originally created as "Compressible flow"

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