Is this how one calculates convective derivative of velocity?
Supposed I 've a body which plunges up/down, given by the equation:
y = y0 * sin(2*Pi*f*t)
where y is the displacement, y0 is the amplitude, f is the frequency, and t is the time
Differentiating, I will get:
dy/dt = v = 2*Pi*f*y0 * cos(2*Pi*f*t)
dv/dt = -(2*Pi*f*y0)^2 * sin(2*Pi*f*t)
How can I calculate convective derivative of velocity for a particular point on the body?
Dv/dt = dv/dt + u*dv/dx + v*dv/dy
I can get dv/dt, but what about dv/dx and dv/dy?
Using chain rule, for e.g. dv/dy :
dv/dy = dv/dt * dt/dy
dv/dt is known from above.
t = acos (y/y0) / (2*Pi*f)
dt/dy = -1 / ((2*Pi*f*y0) * sqrt(1-(y/y0)^2))
Is this how it is done?
If you are looking at a fixed mass point and following its motion, the convective derivative has no meaning. The convective derivative arises mathematically as a result of watching a little window in space and having different mass points move through the window.
Ok I understand now. But how can I compute this numerically, at a certain point. What info do I need? Thanks
The velocity and acceleration of the airfoil points are just the time derivatives of the displacements of the points. So if you have X(t), where X is the displacement of a point, then differentiate with respect to time. The only reason to compute a convective derivative is if you have the displacement of a point given as a field quantity, i.e. the displacement is given by X(x(t),y(t),z(t),t). Then the time derivative is the sum of the local piece and the piece that comes from the implicit dependence on time. The second piece is the convective derivative. But unless the airfoil is deforming there is no reason to compute a convective derivative because the airfoil's motion can be expressed explicity as only a function of time, for example by giving the translational displacement of the center of mass and the rotation of airfoil around the center of mass. If you want to describe the motion as a field quantity then you would evaluate the convective derivative just like you do for any other field quantity - numerically compute the gradient of the velocity and form the scalar product with the velocity, as u.del(u). If you are using a structured grid then the simplest approach is to use finite difference forms for components of del(u). On an unstructured grid you could use a least squares approach to compute del(u).
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