Curl Noise by Robert Bridson is a widely known method that can generate divergence-free noise. This divergence-free property makes it extremely suitable for driving particles to move like real fluid motion.

Here is another divergence-free noise generator that is computationally cheaper than curl noise. I thought it was new and named it Bitangent Noise, but later I found it was already invented by Ivan DeWolf in 2005. (I'm wondering why it is so less popular comparing to curl noise.)

The implementation is carefully optimized, and both HLSL and GLSL codes are provided for your convenience. More details can be found here. Following image shows a particle system that is updated using bitangent noise.

Here is a shadertoy example shows how to use bitangent noise to make a smoke ball.

These performance data are measured on a Nvidia GTX 1060 card, where each noise function is executed 1280 * 720 * 10 times.