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Turbulent flows, characterized by their chaotic and stochastic nature, have historically presented formidable challenges to predictive computational modeling. Traditional eddy-resolved numerical simulations often require vast computational resources, making them impractical or 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. In this study, we introduce a novel generative framework grounded in probabilistic diffusion models for versatile generation of spatiotemporal turbulence under various conditions. 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, as well as 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 evaluate and showcase the versatile turbulence generation capability of our framework through a suite of numerical experiments, including: (1) the synthesis of Large Eddy Simulations (LES) simulated instantaneous flow sequences from unsteady Reynolds-Averaged Navier–Stokes (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. 

Direct numerical simulations (DNS) are performed to investigate the spatial evolution of flat-plate zero-pressure-gradient turbulent boundary layers over long streamwise domains ( $${>}300\delta _i$$ , with $$\delta _i$$ the inflow boundary-layer thickness) at three different Mach numbers, $2.5$ , $4.9$ and $10.9$ , with the surface temperatures ranging from quasiadiabatic to highly cooled conditions. The settlement of turbulence statistics into a fully developed equilibrium state of the turbulent boundary layer has been carefully monitored, either based on the satisfaction of the von Kármán integral equation or by comparing runs with different inflow turbulence generation techniques. The generated DNS database is used to characterize the streamwise evolution of multiple important variables in the high-Mach-number, cold-wall regime, including the skin friction, the Reynolds analogy factor, the shape factor, the Reynolds stresses, and the fluctuating wall quantities. The data confirm the validity of many classic and newer compressibility transformations at moderately high Reynolds numbers (up to friction Reynolds number $$Re_\tau \approx 1200$$ ) and show that, with proper scaling, the sizes of the near-wall streaks and superstructures are insensitive to the Mach number and wall cooling conditions. The strong wall cooling in the hypersonic cold-wall case is found to cause a significant increase in the size of the near-wall turbulence eddies (relative to the boundary-layer thickness), which leads to a reduced-scale separation between the large and small turbulence scales, and in turn to a lack of an outer peak in the spanwise spectra of the streamwise velocity in the logarithmic region.