Multi-scale turbulence modeling and maximum information principle. Part 3: Homogeneous shear turbulence as a SOCP

摘要

To resolve the issues of non-realizability and restriction to homogeneityfaced by the analytical theories of turbulence and to make the frameworkmanageable, we explore the possibility to treat three-dimensional homogeneousshear turbulence of an incompressible Newtonian fluid as a second-order coneprogram, composed of multi-point spatial correlations of velocity and pressurefluctuations up to the degenerated fourth order. Two models are formulated indetail or outlined: The second-order model takes the second order correlationsas the state variables and the contracted and degenerated third ordercorrelations as the control variables; The third-order model contains thecorrelations up to the fourth order with the second and third ordercorrelations as the state variables and the fourth order correlations as thecontrol variables. The sources of the constraints are discussed like thecorrelation definitions, the divergence-free condition, the Cauchy-Schwarzinequality and the non-negative variance of products; One significance of thenon-negative variance comes from its imposition of $\max\sigma=0$ on theexponential growth rate $\sigma$ of the correlations. The candidates for theobjective function are discussed and tested. The asymptotic state solution ofthe second-order model under $\sigma=0$ is obtained numerically with paralleland distributed computing; The predicted anisotropic tensor values areconsistent qualitatively with the experimental data, albeit with a significantquantitative difference which is attributed to the non-enforceability of thenon-negative variance within the model. The third-order model accommodatescertain constraints of the non-negative variance and is expected to improve theprediction, but there is the difficulty to solve this huge-scale problem whichneeds to be tackled. The relevance of the present work to inhomogeneousturbulence is commented.