One-dimensional flow with friction: Difference between revisions
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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..." |
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__TOC__ | __TOC__ | ||
==== Flow-station data ==== | |||
\noindent The starting point is the governing equations for one-dimensional steady-state flow\\ | \noindent The starting point is the governing equations for one-dimensional steady-state flow\\ | ||
===== Continuity ===== | |||
\begin{equation} | \begin{equation} | ||
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\end{equation}\\ | \end{equation}\\ | ||
===== Momentum ===== | |||
\begin{equation} | \begin{equation} | ||
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\end{equation}\\ | \end{equation}\\ | ||
===== Energy ===== | |||
\begin{equation} | \begin{equation} | ||
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\end{equation}\\ | \end{equation}\\ | ||
==== Differential Form ==== | |||
In order to remove the integral term in the momentum equation, the governing equations are written in differential form\\ | In order to remove the integral term in the momentum equation, the governing equations are written in differential form\\ | ||
===== Continuity ===== | |||
\[\rho_1 u_1=\rho_2 u_2=const\Rightarrow\]\\ | \[\rho_1 u_1=\rho_2 u_2=const\Rightarrow\]\\ | ||
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\end{equation}\\ | \end{equation}\\ | ||
===== Momentum ===== | |||
\[(\rho_2 u_2^2+p_2-\rho_1 u_1^2+p_1)=-\frac{4}{D}\int_0^L\tau_w dx\Rightarrow\]\\ | \[(\rho_2 u_2^2+p_2-\rho_1 u_1^2+p_1)=-\frac{4}{D}\int_0^L\tau_w dx\Rightarrow\]\\ | ||
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\end{equation}\\ | \end{equation}\\ | ||
===== Energy ===== | |||
\[h_1 + \frac{1}{2}u_1^2=h_2 + \frac{1}{2}u_2^2=const\]\\ | \[h_1 + \frac{1}{2}u_1^2=h_2 + \frac{1}{2}u_2^2=const\]\\ | ||
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\end{equation}\\ | \end{equation}\\ | ||
==== Summary ==== | |||
\noindent continuity:\\ | \noindent continuity:\\ | ||
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\end{itemize} | \end{itemize} | ||
==== Continuity equation ==== | |||
\noindent We start with the continuity equation which for one-dimensional steady flows reads\\ | \noindent We start with the continuity equation which for one-dimensional steady flows reads\\ | ||
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\end{equation}\\ | \end{equation}\\ | ||
==== Energy equation ==== | |||
\noindent For an adiabatic one-dimensional flow we have that \\ | \noindent For an adiabatic one-dimensional flow we have that \\ | ||
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\end{equation}\\ | \end{equation}\\ | ||
==== Momentum equation ==== | |||
By combining the above derived relations and the momentum equation on the form given by (3.95), we can get an expression where the friction force is a function of Mach number only\\ | By combining the above derived relations and the momentum equation on the form given by (3.95), we can get an expression where the friction force is a function of Mach number only\\ | ||
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\end{equation*} | \end{equation*} | ||
==== Differential Relations ==== | |||
\noindent In analogy with the heat addition process discussed in the previous section, one-dimensional flow with heat addition is a continuous process. We will derive the differential relations for one-dimensional flow with friction, which will lead to trends for supersonic and supersonic flow with friction. | \noindent In analogy with the heat addition process discussed in the previous section, one-dimensional flow with heat addition is a continuous process. We will derive the differential relations for one-dimensional flow with friction, which will lead to trends for supersonic and supersonic flow with friction. | ||
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\end{figure} | \end{figure} | ||
==== Friction Choking ==== | |||
\begin{figure}[ht!] | \begin{figure}[ht!] | ||
