  Turbulent flows have historically presented formidable challenges to
predictive computational modeling. Traditional numerical simulations often
require vast computational resources, making them infeasible for numerous
engineering applications. As an alternative, deep learning-based surrogate
models have emerged, offering data-drive solutions. However, these are
typically constructed within deterministic settings, leading to shortfall in
capturing the innate chaotic and stochastic behaviors of turbulent dynamics. We
introduce a novel generative framework grounded in probabilistic diffusion
models for versatile generation of spatiotemporal turbulence. Our method
unifies both unconditional and conditional sampling strategies within a
Bayesian framework, which can accommodate diverse conditioning scenarios,
including those with a direct differentiable link between specified conditions
and generated unsteady flow outcomes, and scenarios lacking such explicit
correlations. A notable feature of our approach is the method proposed for
long-span flow sequence generation, which is based on autoregressive
gradient-based conditional sampling, eliminating the need for cumbersome
retraining processes. We showcase the versatile turbulence generation
capability of our framework through a suite of numerical experiments,
including: 1) the synthesis of LES simulated instantaneous flow sequences from
URANS inputs; 2) holistic generation of inhomogeneous, anisotropic wall-bounded
turbulence, whether from given initial conditions, prescribed turbulence
statistics, or entirely from scratch; 3) super-resolved generation of
high-speed turbulent boundary layer flows from low-resolution data across a
range of input resolutions. Collectively, our numerical experiments highlight
the merit and transformative potential of the proposed methods, making a
significant advance in the field of turbulence generation.
