The Q1D equations: Difference between revisions

Latest revision as of 18:53, 1 April 2026

Governing Equations

In the following quasi-one-dimensional flow will be assumed. That means that the cross-section is allowed to vary smoothly but flow quantities varies in one direction only. The equations that are derived will thus describe one-dimensional flow in axisymmetric tubes. Let's assume flow in the $x$ -direction, which means that all flow quantities and the cross-section area will vary with the axial coordinate $x$ .

A = A (x), ρ = ρ (x), u = u (x), p = p (x), ...

(Eq. 5.1)

We will further assume steady-state flow, which means that unsteady terms will be zero.

The equations are derived with the starting point in the governing flow equations on integral form

Continuity Equation

Applying the integral form of the continuity equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives

\underset{= 0}{\underset{⏟}{\frac{d}{d t} \underset{Ω}{∭} ρ d V}} + \iint_{\partial Ω} ρ 𝐯 \cdot 𝐧 d S = 0

(Eq. 5.2)

\iint_{\partial Ω} ρ 𝐯 \cdot 𝐧 d S = - ρ_{1} u_{1} A_{1} + ρ_{2} u_{2} A_{2}

(Eq. 5.3)

ρ_{1} u_{1} A_{1} = ρ_{2} u_{2} A_{2}

(Eq. 5.4)

Momentum Equation

Applying the integral form of the momentum equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives

\underset{= 0}{\underset{⏟}{\frac{d}{d t} \underset{Ω}{∭} ρ 𝐯 d V}} + \iint_{\partial Ω} [ρ (𝐯 \cdot 𝐧) 𝐯 + p 𝐧] d S = 0

(Eq. 5.5)

\iint_{\partial Ω} ρ (𝐯 \cdot 𝐧) 𝐯 d S = - ρ_{1} u_{1}^{2} A_{1} + ρ_{2} u_{2}^{2} A_{2}

(Eq. 5.6)

\iint_{\partial Ω} p 𝐧 d S = - p_{1} A_{1} + p_{2} A_{2} - \int_{A_{1}}^{A_{2}} p d A

(Eq. 5.7)

collecting terms

(ρ_{1} u_{1}^{2} + p_{1}) A_{1} + \int_{A_{1}}^{A_{2}} p d A = (ρ_{2} u_{2}^{2} + p_{2}) A_{2}

(Eq. 5.8)

Energy Equation

Applying the integral form of the energy equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives

\underset{= 0}{\underset{⏟}{\frac{d}{d t} \underset{Ω}{∭} ρ e_{o} d V}} + \iint_{\partial Ω} [ρ h_{o} (𝐯 \cdot 𝐧)] d S = 0

(Eq. 5.9)

\iint_{\partial Ω} [ρ h_{o} (𝐯 \cdot 𝐧)] d S = - ρ_{1} u_{1} h_{o_{1}} A_{1} + ρ_{2} u_{2} h_{o_{2}} A_{2}

(Eq. 5.10)

ρ_{1} u_{1} h_{o_{1}} A_{1} = ρ_{2} u_{2} h_{o_{2}} A_{2}

(Eq. 5.11)

Now, using the continuity equation $ρ_{1} u_{1} A_{1} = ρ_{2} u_{2} A_{2}$ gives

h_{o_{1}} = h_{o_{2}}

(Eq. 5.12)

Differential Form

The integral term appearing the momentum equation is undesired and therefore the governing equations are converted to differential form.

The continuity equation (Eqn. \ref{eq:governing:cont:b}) is rewritten in differential form as

ρ_{1} u_{1} A_{1} = ρ_{2} u_{2} A_{2} = c o n s t

(Eq. 5.13)

d (ρ u A) = 0

(Eq. 5.14)

The momentum equation (Eqn. \ref{eq:governing:mom:b}) is rewritten in differential form as

(ρ_{1} u_{1}^{2} + p_{1}) A_{1} + \int_{A_{1}}^{A_{2}} p d A = (ρ_{2} u_{2}^{2} + p_{2}) A_{2} \Rightarrow d [(ρ u^{2} + p) A] = p d A

(Eq. 5.15)

d (ρ u^{2} A) + d (p A) = p d A

(Eq. 5.16)

u d (ρ u A) + ρ u A d u + A d p + p d A = p d A

(Eq. 5.17)

From the continuity equation we have $d (ρ u A)$ and thus

ρ u A d u + A d p = 0 \Rightarrow

(Eq. 5.18)

d p = - ρ u d u

(Eq. 5.19)

which is the momentum equation on differential form. Also referred to as Euler's equation. Finally, the energy equation (Eqn. \ref{eq:governing:cont:b}) is rewritten in differential form as

h_{o_{1}} = h_{o_{2}} = c o n s t \Rightarrow d h_{o} = 0

(Eq. 5.20)

h_{o} = h + \frac{1}{2} u^{2} \Rightarrow d h + \frac{1}{2} d (u^{2}) = 0

(Eq. 5.21)

d h + u d u = 0

(Eq. 5.22)

Summary

Continuity:

d (ρ u A) = 0

Momentum:

d p = - ρ u d u

Energy:

d h + u d u = 0

The equations are valid for:

quasi-one-dimensional flow
steady state
all gas models (no gas model assumptions made)
inviscid flow

It should be noted that equations are exact but they are applied to a physical model that is approximate, i.e., the approximation that flow quantities varies in one dimension with a varying cross-section area. In reality, a variation of cross-section area would imply flow in three dimensions.

@@ Line 1: / Line 1: @@
-[[Category:Compressible flow]]
+<!--
-[[Category:Quasi-one-dimensional flow]]
+-->[[Category:Compressible flow]]<!--
-[[Category:Governing equations]]
+-->[[Category:Quasi-one-dimensional flow]]<!--
-[[Category:Inviscid flow]]
+-->[[Category:Inviscid flow]]<!--
+--><noinclude><!--
+-->[[Category:Compressible flow:Topic]]<!--
+--></noinclude><!--
-__TOC__
+--><nomobile><!--
+-->__TOC__<!--
+--></nomobile><!--
-\section{Governing Equations}
+--><noinclude><!--
+-->{{#vardefine:secno|5}}<!--
+-->{{#vardefine:eqno|0}}<!--
+--></noinclude><!--
+-->
+=== Governing Equations ===
+<!--
 \begin{figure}[ht!]
 \begin{center}
@@ Line 15: / Line 26: @@
 \end{center}
 \end{figure}
+-->
-\noindent In the following quasi-one-dimensional flow will be assumed. That means that the cross-section is allowed to vary smoothly but flow quantities varies in one direction only. The equations that are derived will thus describe one-dimensional flow in axisymmetric tubes. Let's assume flow in the $x$-direction, which means that all flow quantities and the cross-section area will vary with the axial coordinate $x$.
+In the following quasi-one-dimensional flow will be assumed. That means that the cross-section is allowed to vary smoothly but flow quantities varies in one direction only. The equations that are derived will thus describe one-dimensional flow in axisymmetric tubes. Let's assume flow in the <math>x</math>-direction, which means that all flow quantities and the cross-section area will vary with the axial coordinate <math>x</math>.
-\[A=A(x),\ \rho=\rho(x),\ u=u(x),\ p=p(x),\ ...\]\\
+{{NumEqn|<math>
+A=A(x),\ \rho=\rho(x),\ u=u(x),\ p=p(x),\ ...
+</math>}}
-\noindent We will further assume steady-state flow, which means that unsteady terms will be zero.\\
+We will further assume steady-state flow, which means that unsteady terms will be zero.
-\noindent The equations are derived with the starting point in the governing flow equations on integral form\\
+The equations are derived with the starting point in the governing flow equations on integral form
-%\newpage
+==== Continuity Equation ====
-\subsection{Continuity Equation}
+Applying the integral form of the continuity equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives
-\noindent Applying the integral form of the continuity equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives\\
+{{NumEqn|<math>
+\underbrace{\frac{d}{dt}\iiint_{\Omega}\rho d{V}}_{=0}+\iint_{\partial \Omega}\rho {\mathbf{v}}\cdot {\mathbf{n}}dS=0
+</math>}}
-\begin{equation}
+{{NumEqn|<math>
-\underbrace{\frac{d}{dt}\iiint_{\Omega}\rho d{\mathscr{V}}}_{=0}+\varoiint_{\partial \Omega}\rho {\mathbf{v}}\cdot {\mathbf{n}}dS=0
+\iint_{\partial \Omega}\rho {\mathbf{v}}\cdot {\mathbf{n}}dS=-\rho_1 u_1 A_1+\rho_2 u_2 A_2
-\label{eq:governing:cont:a}
+</math>}}
-\end{equation}\\
-\[\varoiint_{\partial \Omega}\rho {\mathbf{v}}\cdot {\mathbf{n}}dS=-\rho_1 u_1 A_1+\rho_2 u_2 A_2\]\\
+{{NumEqn|<math>
-\begin{equation}
 \rho_1 u_1 A_1=\rho_2 u_2 A_2
-\label{eq:governing:cont:b}
+</math>}}
-\end{equation}\\
-\subsection{Momentum Equation}
+==== Momentum Equation ====
-\noindent Applying the integral form of the momentum equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives\\
+Applying the integral form of the momentum equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives
-\begin{equation}
+{{NumEqn|<math>
-\underbrace{\frac{d}{dt}\iiint_{\Omega}\rho{\mathbf{v}} d{\mathscr{V}}}_{=0}+\varoiint_{\partial \Omega}\left[\rho ({\mathbf{v}}\cdot {\mathbf{n}}){\mathbf{v}}+p{\mathbf{n}}\right]dS=0
+\underbrace{\frac{d}{dt}\iiint_{\Omega}\rho{\mathbf{v}} d{V}}_{=0}+\iint_{\partial \Omega}\left[\rho ({\mathbf{v}}\cdot {\mathbf{n}}){\mathbf{v}}+p{\mathbf{n}}\right]dS=0
-\label{eq:governing:mom:a}
+</math>}}
-\end{equation}\\
-\[\varoiint_{\partial \Omega} \rho ({\mathbf{v}}\cdot {\mathbf{n}}){\mathbf{v}}dS=-\rho_1u_1^2A_1+\rho_2u_2^2A_2\]\\
+{{NumEqn|<math>
+\iint_{\partial \Omega} \rho ({\mathbf{v}}\cdot {\mathbf{n}}){\mathbf{v}}dS=-\rho_1u_1^2A_1+\rho_2u_2^2A_2
+</math>}}
-\[\varoiint_{\partial \Omega} p{\mathbf{n}}dS=-p_1A_1+p_2A_2-\int_{A_1}^{A_2}pdA\]\\
+{{NumEqn|<math>
+\iint_{\partial \Omega} p{\mathbf{n}}dS=-p_1A_1+p_2A_2-\int_{A_1}^{A_2}pdA
+</math>}}
-\noindent collecting terms\\
+collecting terms
-\begin{equation}
+{{NumEqn|<math>
 \left(\rho_1u_1^2+p_1\right)A_1+\int_{A_1}^{A_2}pdA=\left(\rho_2u_2^2+p_2\right)A_2
-\label{eq:governing:mom:b}
+</math>}}
-\end{equation}\\
-\subsection{Energy Equation}
+==== Energy Equation ====
-\noindent Applying the integral form of the energy equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives\\
+Applying the integral form of the energy equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives
-\begin{equation}
+{{NumEqn|<math>
-\underbrace{\frac{d}{dt}\iiint_{\Omega}\rho e_o d{\mathscr{V}}}_{=0}+\varoiint_{\partial \Omega}\left[\rho h_o ({\mathbf{v}}\cdot{\mathbf{n}})\right]dS=0
+\underbrace{\frac{d}{dt}\iiint_{\Omega}\rho e_o d{V}}_{=0}+\iint_{\partial \Omega}\left[\rho h_o ({\mathbf{v}}\cdot{\mathbf{n}})\right]dS=0
-\label{eq:governing:energy:a}
+</math>}}
-\end{equation}\\
-\[\varoiint_{\partial \Omega}\left[\rho h_o ({\mathbf{v}}\cdot{\mathbf{n}})\right]dS=-\rho_1u_1h_{o_1}A_1+\rho_2u_2h_{o_2}A_2\]\\
+{{NumEqn|<math>
+\iint_{\partial \Omega}\left[\rho h_o ({\mathbf{v}}\cdot{\mathbf{n}})\right]dS=-\rho_1u_1h_{o_1}A_1+\rho_2u_2h_{o_2}A_2
+</math>}}
-\[\rho_1u_1h_{o_1}A_1=\rho_2u_2h_{o_2}A_2\]\\
+{{NumEqn|<math>
+\rho_1u_1h_{o_1}A_1=\rho_2u_2h_{o_2}A_2
+</math>}}
-\noindent Now, using the continuity equation $\rho_1u_1A_1=\rho_2u_2A_2$ gives\\
+Now, using the continuity equation <math>\rho_1u_1A_1=\rho_2u_2A_2</math> gives
-\begin{equation}
+{{NumEqn|<math>
 h_{o_1}=h_{o_2}
-\label{eq:governing:energy:b}
+</math>}}
-\end{equation}\\
-\subsection{Differential Form}
+==== Differential Form ====
-\noindent The integral term appearing the momentum equation is undesired and therefore the governing equations are converted to differential form.\\
+The integral term appearing the momentum equation is undesired and therefore the governing equations are converted to differential form.
-\noindent The continuity equation (Eqn. \ref{eq:governing:cont:b}) is rewritten in differential form as\\
+The continuity equation (Eqn. \ref{eq:governing:cont:b}) is rewritten in differential form as
-\[\rho_1u_1A_1=\rho_2u_2A_2=const\]\\
+{{NumEqn|<math>
+\rho_1u_1A_1=\rho_2u_2A_2=const
+</math>}}
-\begin{equation}
+{{NumEqn|<math>
 d(\rho uA)=0
-\label{eq:governing:cont:c}
+</math>}}
-\end{equation}\\
-\noindent The momentum equation (Eqn. \ref{eq:governing:mom:b}) is rewritten in differential form as\\
+The momentum equation (Eqn. \ref{eq:governing:mom:b}) is rewritten in differential form as
-\[\left(\rho_1u_1^2+p_1\right)A_1+\int_{A_1}^{A_2}pdA=\left(\rho_2u_2^2+p_2\right)A_2\Rightarrow d\left[(\rho u^2+p)A\right]=pdA\]\\
+{{NumEqn|<math>
+\left(\rho_1u_1^2+p_1\right)A_1+\int_{A_1}^{A_2}pdA=\left(\rho_2u_2^2+p_2\right)A_2\Rightarrow d\left[(\rho u^2+p)A\right]=pdA
+</math>}}
-\[d(\rho u^2A)+d(pA)=pdA\]\\
+{{NumEqn|<math>
+d(\rho u^2A)+d(pA)=pdA
+</math>}}
-\[ud(\rho uA)+\rho uAdu+Adp+\cancel{pdA}=\cancel{pdA}\]\\
+{{NumEqn|<math>
+ud(\rho uA)+\rho uAdu+Adp+\cancel{pdA}=\cancel{pdA}
+</math>}}
-\noindent From the continuity equation we have $d(\rho uA)$ and thus\\
+From the continuity equation we have <math>d(\rho uA)</math> and thus
-\[\rho u\cancel{A}du+\cancel{A}dp=0\Rightarrow\]\\
+{{NumEqn|<math>
+\rho u\cancel{A}du+\cancel{A}dp=0\Rightarrow
+</math>}}
-\begin{equation}
+{{NumEqn|<math>
 dp=-\rho udu
-\label{eq:governing:mom:c}
+</math>}}
-\end{equation}\\
-\noindent which is the momentum equation on differential form. Also referred to as Euler's equation. Finally, the energy equation (Eqn. \ref{eq:governing:cont:b}) is rewritten in differential form as\\
+which is the momentum equation on differential form. Also referred to as Euler's equation. Finally, the energy equation (Eqn. \ref{eq:governing:cont:b}) is rewritten in differential form as
-\[h_{o_1}=h_{o_2}=const\Rightarrow dh_o=0\]\\
+{{NumEqn|<math>
+h_{o_1}=h_{o_2}=const\Rightarrow dh_o=0
+</math>}}
-\[h_o=h+\frac{1}{2}u^2\Rightarrow dh+\frac{1}{2}d(u^2)=0\]\\
+{{NumEqn|<math>
+h_o=h+\frac{1}{2}u^2\Rightarrow dh+\frac{1}{2}d(u^2)=0
+</math>}}
-\begin{equation}
+{{NumEqn|<math>
 dh+udu=0
-\label{eq:governing:energy:c}
+</math>}}
-\end{equation}\\
-%\newpage
+==== Summary ====
-\subsection{Summary}
+<div style="border: 1px solid;">
+{{OpenInfoBox|<math>
+d(\rho uA)=0
+</math>|description=Continuity:}}
-\noindent Continuity:
+{{OpenInfoBox|<math>
+dp=-\rho udu
+</math>|description=Momentum:}}
-\[d(\rho uA)=0\]
+{{OpenInfoBox|<math>
+dh+udu=0
-\noindent Momentum:
+</math>|description=Energy:}}
+</div>
-\[dp=-\rho udu\]
-\noindent Energy:
-\[dh+udu=0\]
+The equations are valid for:
-\noindent The equations are valid for:\\
+* quasi-one-dimensional flow
+* steady state
+* all gas models (no gas model assumptions made)
+* inviscid flow
-\begin{itemize}
-\item quasi-one-dimensional flow
-\item steady state
-\item all gas models (no gas model assumptions made)
-\item inviscid flow
-\end{itemize}
-\noindent It should be noted that equations are exact but they are applied to a physical model that is approximate, i.e., the approximation that flow quantities varies in one dimension with a varying cross-section area. In reality, a variation of cross-section area would imply flow in three dimensions.
+It should be noted that equations are exact but they are applied to a physical model that is approximate, i.e., the approximation that flow quantities varies in one dimension with a varying cross-section area. In reality, a variation of cross-section area would imply flow in three dimensions.

The Q1D equations: Difference between revisions

Latest revision as of 18:53, 1 April 2026

Contents

Governing Equations

Continuity Equation

Momentum Equation

Energy Equation

Differential Form

Summary

Navigation menu

The Q1D equations: Difference between revisions

Latest revision as of 18:53, 1 April 2026

Governing Equations

Continuity Equation

Momentum Equation

Energy Equation

Differential Form

Summary

Navigation menu

Search