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A novel method for assimilating and extending measured turbulent Rayleigh–Bénard convection data is presented, which relies on the fractional step method also used to solve the incompressible Navier–Stokes equation in direct numerical simulations. Our approach is used to make measured tomographic particle image velocimetry (tomo PIV) fields divergence-free and to extract temperature fields. Comparing the time average of the extracted temperature fields with the temporally averaged temperature field, measured using particle image thermometry in a subdomain of the flow geometry, shows that extracted fields correlate well with measured fields with a correlation coefficient of    $$C_{T\tilde{T}}=0.84$$  C T T ~ = 0.84 . Additionally, extracted temperature fields as well as divergence-free velocity fields serve as initial fields for subsequent direct numerical simulations with and without feedback which generate small-scale turbulence initially absent in the experimental data. Although the tomo PIV data set was spatially under-resolved and did not include any information on the boundary layers, the here-proposed method successfully generates velocity and temperature fields featuring small-scale turbulence and thermal as well as kinetic boundary layers, without disturbing the large-scale circulation contained in the original experimental data significantly. The latter is underpinned by high vertical and horizontal velocity correlation coefficients—computed from velocity fields averaged in time and horizontal  x -direction obtained from the measurement and from the simulation without feedback—of    $$C_{v\tilde{v}}=0.92$$  C v v ~ = 0.92 and    $$C_{w\tilde{w}}=0.91$$  C w w ~ = 0.91 representing the large-scale structure. For simulations with feedback, the generated velocity fields resemble the experimental data increasingly well for higher feedback gain values, whereas the temperature fluctuation intensity deviates noticeably from the values obtained from a direct numerical simulation without feedback for gain values    $$\alpha \ge 1$$  α ≥ 1 . Thus, a feedback gain of    $$\alpha =0.1$$  α = 0.1 was found optimal with correlation coefficients of    $$C_{v\tilde{v}}=0.96$$  C v v ~ = 0.96 and    $$C_{w\tilde{w}}=0.95$$  C w w ~ = 0.95 as well as a realistic temperature fluctuation intensity profile. The  xt -averaged temperature fields obtained from the direct numerical simulations with and without feedback correlate somewhat less with the extracted temperature field (   $$C_{T\tilde{T}}\approx 0.6$$  C T T ~ ≈ 0.6 ), which is presumably caused by spatially under-resolved and temporally oscillating initial tomo PIV fields reflected by the extracted temperature field. Graphical abstract

Assimilation and extension of particle image velocimetry data of turbulent Rayleigh–Bénard convection using direct numerical simulations