Euler Equations (fluid Dynamics) - Steady Flow in Streamline Coordinates

Steady Flow in Streamline Coordinates

In the case of steady flow, it is convenient to choose the Frenet–Serret frame along a streamline as the coordinate system for describing the momentum part of the Euler equations:

${\mathrm{D} \boldsymbol{v} \over \mathrm{D}t} = -{1\over\rho}\nabla p,$

where, and denote the velocity, the pressure and the density, respectively.

Let be a Frenet–Serret orthonormal basis which consists of a tangential unit vector, a normal unit vector, and a binormal unit vector to the streamline, respectively. Since a streamline is a curve that is tangent to the velocity vector of the flow, the left-handed side of the above equation, the substantial derivative of velocity, can be described as follows:

$\begin{align} {\mathrm{D} \boldsymbol{v} \over \mathrm{D}t} &= \boldsymbol{v}\cdot\nabla \boldsymbol{v} \\ &= v{\partial \over \partial s}(v\boldsymbol{e}_s) &(\boldsymbol{v} = v \boldsymbol{e}_s ,~ {\partial / \partial s} \equiv \boldsymbol{e}_s\cdot\nabla)\\ &= v{\partial v \over \partial s}\boldsymbol{e}_s + {v^2 \over R} \boldsymbol{e}_n &(\because~ {\partial \boldsymbol{e}_s \over \partial s}={1\over R}\boldsymbol{e}_n), \end{align}$

where is the radius of curvature of the streamline.

Therefore, the momentum part of the Euler equations for a steady flow is found to have a simple form:

$\begin{cases} \displaystyle v{\partial v \over \partial s} = -{1 \over \rho}{\partial p \over \partial s},\\ \displaystyle {v^2 \over R} = -{1 \over \rho}{\partial p \over \partial n} &({\partial / \partial n}\equiv\boldsymbol{e}_n\cdot\nabla),\\ \displaystyle 0 = -{1 \over \rho}{\partial p \over \partial b} &({\partial / \partial b}\equiv\boldsymbol{e}_b\cdot\nabla). \end{cases}$

For barotropic flow ( ), Bernoulli's equation is derived from the first equation:

${\partial \over \partial s}\left({v^2\over 2} + \int {\mathrm{d}p \over \rho}\right) =0.$

The second equation expresses that, in the case the streamline is curved, there should exist a pressure gradient normal to the streamline because the centripetal acceleration of the fluid parcel is only generated by the normal pressure gradient.

The third equation expresses that pressure is constant along the binormal axis.

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