A strategy to construct identities between components of the Reynolds stress tensor: Physics-to-geometry transformation

Numerical solutions of the Reynolds-averaged Navier--Stokes (RANS) equations have been popularly employed to understand turbulent flows in engineering applications owing to computational cost-efficiency and tractability, despite that small-scale flow structures are not resolved. In order to obtain the numerical solutions, it is necessary to construct a closure for the Reynolds stress tensor, which has been resolved to date by modeling. Although Reynolds-stress modeling is able to implement the effect of Reynolds stresses with physically plausible accuracy in RANS simulations, it has suffered from uncertainties in predictive ability. The reason can be attributed to the empiricism or physical intuition applied in formulating the Reynolds stress tensor, but inherently to the absence of mathematical exactness. We present identities between Reynolds stresses, constructed on the basis of the shift of perspective on physical scalar quantities from physics to differential geometry. Further, we show that the second-order statistics of velocity fluctuations in turbulent pipe flow at pipe-diameter-based Reynolds number $Re_{D}=5300$ satisfy the identities.