Suppose we have an aircraft flying in steady, level flight at zero angle of attack. In this configuration, the velocity vector $\vec{V}$ is perfectly aligned with the $x$ axis. At some time $t=t_0$, the pilot applies a finite positive rudder input $\delta_{r}$ which produces a yawing moment on the aircraft. My question is what will happen to the velocity vector immediately after the rudder input gets in.
From my understanding, as soon as the the pilot inputs the rudder, the plane produces a positive side force (in the $y$ direction), which causes a negative moment about the $z$ direction. We can write this as something like $N = -l_vY_v$, where $l_v$ is the distance from the vertical tail to the CG, and $Y_v = C_{L_v}Q_vS_v$ is the side force as mentioned above. We can write this in non-dimensional form as the yaw moment coefficient, i.e. \begin{equation} C_n \equiv \frac{N}{Q_wSb} = -\eta_vV_v\frac{\partial C_L}{\partial\delta_r}\delta_r, \end{equation} where $\eta_v = Q_v/Q_w$ and $V_v = l_vS_v/(Sb)$. So, a positive rudder deflection will give a negative yaw moment, as seen in that equation, proportional to $C_{L_{\delta_r}}$. However, this doesn't tell me anything about the velocity vector. Intuitively, I would think that after a long time when the aircraft has time to stabilize itself, the velocity vector would point along some angle $\beta$ from the $x$-axis relative to where to was before. However, I am not sure what would happen immediately. Maybe there is an immediate sideslip angle, and over time it decreases (assuming the aircraft is stable). Thanks for the help in advance!
