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2.2  Problem definition in a fixed frame

Fully axisymmetric rotors can be modeled in a fixed frame using an Eulerian representation, where particles are moving under a deformed mesh. Particles located at point {p(p₀,t)}={p₀}+{u(p₀,t)} in the deformed Eulerian frame have a velocity given by xxx

(2.6)

The body fixed frame verifies θ=φ+Ω t, which can be written as p_G={r,θ,z}=R_{Ω t}{p}. The matching of displacements in both frames is given by

(2.7)

The velocity of a particle in the disk is given by

(2.8)

Validation example : One first considers a disk that has a steady state deformation in the global frame. That is the Eulerian frame, one has ∂ u/∂ t=0 and {u(r,θ,z)}={u(p)}=(2−cos²θ){e_θ}.

One now considers a disk that has a steady state deformation in the rotating frame: ∂ u/∂ t=0 and {u(r,φ,z)}=(1−cos²φ){e_φ}. The displacement of a particle located at at time t is given by

its velocity is