SENSITIVITY OF A NEW VELOCITY/PRESSURE-GRADIENT MODEL TO THE REYNOLDS NUMBER

Abstract

The paper presents results of validation of a new model for velocity/pressure-gradient correlations in planar wall-bounded incompressible turbulent flows. The model belongs to a class of data-driven models and was obtained from the analysis of direct numerical simulation (DNS) data collected in a fully developed channel flow at Reτ = 392. It includes only the terms originally present in the transport equations for the Reynolds stresses. In the paper, sensitivity of the model coefficients to the flow Reynolds number and geometry is tested using three other DNS databases, with the Reynolds numbers, Reτ and Reθ , varying from 392 to 5200 in a fully developed channel flow and from 300 to 6500 in a zeropressure-gradient boundary-layer over a flat plate, respectively. In addition, the model is validated using DNS data in a strained channel flow at eight strain times including those that cause the flow to separate. It is demonstrated that no tuning of the model coefficients is required to accurately reproduce the behaviour of velocity/pressure-gradient correlations right to the wall in all considered flows. INTRODUCTION When simulating turbulent flows with the Reynolds-averaged Navier-Stokes (RANS) equations, models for unknown terms in the equations are required. Among such terms are velocity/pressure-gradient (VPG) correlations. Traditional approach to modelling the VPG correlations can be traced back to Chou (1945) and is based on the analysis of the exact integro-differential expressions for the VPG correlations. In the approach, the goal is to preserve tensor properties of the twopoint turbulence statistics that are integrated over the entire flow domain in models for one-point VPG correlations. Various linear and non-linear models were developed in such a manner over the years (see, e.g., Hanjalić and Launder, 2011). However, as demonstarted by many studies, values of model coefficients have to vary in any of these models. Ad-hoc or empirical corrections have to be implemented into a model to describe the turbulent flow features in different flow geometries and at different Reynolds numbers, also near a wall, under rotation etc. A presence of the flow separation reduces predictive capability of such models as well. In Poroseva and Murman (2014a), we analysed the major assumptions used in the traditional approach to modelling onepoint VPG correlations, proposed new models for the VPG correlations up to the fourth order, which are free of the homogeneous flow assumption, and analysed predictive capabilities of the derived models in a priori testing with DNS data in planar wall-bounded flows (Spalart, 1988; Sillero et al., 2013; Jeyapaul et al., 2015). The study led us to conclude that the traditional approach is unlikely to result in models for VPG correlations that are accurate near a wall. A more promising approach to modelling VPG correlations in wall-bounded flows was proposed in Poroseva and Murman (2014a,b). In the approach, model expressions for VPG correlations up to the fourth order were obtained from the direct analysis of DNS data in a fully developed channel flow at Reτ = 392 based on the friction velocity and the channel half-width (Jeyapaul et al., 2015). When these new model expressions for the second-order VPG correlations were tested using DNS data in a zero-pressure gradient boundary layer (ZPGBL) over a flat plate at different Reynolds numbers (Spalart, 1988; Sillero et al., 2013), the results of the same level of accuracy as in the channel flow were obtained without changing the model coefficients (Poroseva and Murman, 2014a). Due to the lack of DNS data for the budget terms in transport equations for higher-order velocity moments, the model expressions for higher-order VPG correlations could not be tested when the study was conducted. Motivated by success of the initial models, a new set of datadriven models for VPG correlations in the transport equations for the Reynolds stresses 2 u < > , 2 v < > , and uv < > were proposed in Poroseva et al. (2015), with the models performance being improved in the flow area at 10 y+ < , that is, in immediate vicinity to a wall. Here, / y yuτ ν + = is the normal-to-wall coordinate y in the wall units; uτ and ν are the friction velocity and the kinematic viscosity, respectively; u and v are velocity fluctuations in streamwise and normal-to-wall directions. The new models were also obtained from the analysis of DNS data in a fully developed channel flow at Reτ = 392 (Jeyapaul et al., 2015). The current paper presents results of validation of these models using different DNS datasets in a fully-developed channel flow and in a ZPGBL at various Reynolds numbers. Also, predictive capability of the models in a separated flow is evaluated. VELOCITY/PRESSURE-GRADIENT CORRELATION MODELS The models for VPG correlations proposed in Poroseva et al. (2015) and tested in the current study are: 10 International Symposium on Turbulence and Shear Flow Phenomena (TSFP10), Chicago, USA, July, 2017 2 6A-3 0.92 0.92 0.3 T M xy xy xy xy D P D Π = − − − 0.78 0.7 0.25 0.01 T M xx xy yy xy xx D D Π = − Π − Π − + (1) 0.45 0.031 1.35 1.15 0.47 0.2 T T yy xy xx yy zz