Hi Bahadir,
You’re not alone—this type of animation can definitely be tricky to recreate without third-party plugins, but it’s absolutely possible to achieve a similar look in After Effects using built-in tools (just takes some layering and patience).
The key to that “wave-like particle motion” is combining Fractal Noise or Turbulent Displace with Particle World or CC Particle Systems II — both are native to AE. You’ll need to animate the emitter and play with velocity and gravity to get that flowing trail effect. Adding a subtle Turbulent Displace to the layer can give you the organic swirl you’re after.
Since you already tried with Fractal Noise, you’re halfway there. Try precomposing your fractal shape and using it as a displacement map or mask for a particle layer. Also experiment with time remapping and motion blur to smooth out the motion.
If you’re familiar with Blender, that’s a solid plan B. Blender’s particle system with force fields could give you very fluid results — plus it’s free and much more flexible for 3D motion.
Keep pushing, you’re close! These kinds of effects often come from trial and error. And don’t worry, your post is totally fine — searching “particles” always turns into a black hole! 😊
Good luck!