Question

In a steady two-dimensional flow, particle trajectories are given byx(t) =x0cosγt;y(t) =y0sinγta. From these trajectories determine the Lagrangian particle velocity componentsu(t) =dx/dtandv(t) =dy/dt. Convert these to Eulerian velocity componentsu(x,y) andv(x,y). Note that the Eulerian velocity does not depend on time.

Answer 1

Answer with Explanation:

Given that

$x(t)=x_ocos(\gamma t)............(i)$

$y(t)=y_osin(\gamma t)..............(ii)$

Thus by definition x component of velocity in Lagrangian system is given by

$u=\frac{dx}{dt}\\\\u=\frac{d(x_ocos(\gamma t)}{dt}=-\gamma x_osin(\gamma t)$

Thus by definition y component of velocity in Lagrangian system is given by

$v=\frac{dy}{dt}\\\\v=\frac{d(y_osin(\gamma t)}{dt}=\gamma y_ocos(\gamma t)$

Since in eulerian system we need to eliminate time from the equations

From euations 'i' and 'ii' we can write

$cos(\gamma t)=\frac{x}{x_o}\\\\sin(\gamma t)=\frac{y}{y_o}$

Applying these values in the velocity components as obtained in Lagrangian system we get

$u=\frac{-\gamma x_o}{y_o}\cdot y\\\\v=\frac{\gamma y_o}{x_o}\cdot x$