Area-velocity relation: Difference between revisions
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=== The Area-Velocity Relation === | === The Area-Velocity Relation === | ||
Starting point - the continuity equation (Eqn. \ref{eq:governing:cont}): | Starting point - the continuity equation (Eqn. \ref{eq:governing:cont}): | ||
<math | {{NumEqn|<math> | ||
d(\rho uA)=0 \Rightarrow \rho u dA+\rho Adu +uAd\rho=0 | d(\rho uA)=0 \Rightarrow \rho u dA+\rho Adu +uAd\rho=0 | ||
</math> | </math>}} | ||
divide by <math>\rho uA</math> gives | divide by <math>\rho uA</math> gives | ||
<math | {{NumEqn|<math> | ||
\frac{d\rho}{\rho}+\frac{du}{u}+\frac{dA}{A}=0 | \frac{d\rho}{\rho}+\frac{du}{u}+\frac{dA}{A}=0 | ||
</math> | </math>}} | ||
As the name suggests, the area-velocity relation is a relation including the area and the flow velocity. Therefore, the next step is to replace the density terms. | As the name suggests, the area-velocity relation is a relation including the area and the flow velocity. Therefore, the next step is to replace the density terms. | ||
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This can be achieved using the momentum equation (Eqn. \ref{eq:governing:mom}) | This can be achieved using the momentum equation (Eqn. \ref{eq:governing:mom}) | ||
<math | {{NumEqn|<math> | ||
dp=-\rho udu\Leftrightarrow \frac{dp}{\rho}=-udu | dp=-\rho udu\Leftrightarrow \frac{dp}{\rho}=-udu | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
\frac{dp}{\rho}=\frac{dp}{d\rho}\frac{d\rho}{\rho}=-udu | \frac{dp}{\rho}=\frac{dp}{d\rho}\frac{d\rho}{\rho}=-udu | ||
</math> | </math>}} | ||
If we assume adiabatic and reversible flow processes, i.e., isentropic flow | If we assume adiabatic and reversible flow processes, i.e., isentropic flow | ||
<math | {{NumEqn|<math> | ||
\frac{dp}{d\rho}=\left(\frac{dp}{d\rho}\right)_s=a^2\Rightarrow a^2\frac{d\rho}{\rho}=-udu | \frac{dp}{d\rho}=\left(\frac{dp}{d\rho}\right)_s=a^2\Rightarrow a^2\frac{d\rho}{\rho}=-udu | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
a^2\frac{d\rho}{\rho}=-udu=-u^2\frac{du}{u} | a^2\frac{d\rho}{\rho}=-udu=-u^2\frac{du}{u} | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
\frac{d\rho}{\rho}=-M^2\frac{du}{u} | \frac{d\rho}{\rho}=-M^2\frac{du}{u} | ||
</math> | </math>}} | ||
Eqn. \ref{eq:governing:mom:b} inserted in Eqn. \ref{eq:governing:cont:b} gives | Eqn. \ref{eq:governing:mom:b} inserted in Eqn. \ref{eq:governing:cont:b} gives | ||
<math | {{NumEqn|<math> | ||
-M^2\frac{du}{u}+\frac{du}{u}+\frac{dA}{A}=0 | -M^2\frac{du}{u}+\frac{du}{u}+\frac{dA}{A}=0 | ||
</math> | </math>}} | ||
or | or | ||
<math | {{NumEqn|<math> | ||
\frac{dA}{A}=(M^2-1)\frac{du}{u} | \frac{dA}{A}=(M^2-1)\frac{du}{u} | ||
</math> | </math>}} | ||
which is the area-velocity relation. | which is the area-velocity relation. | ||
