NASA TM Elliptic Relaxation of a Tensor Representation for the Redistribution Terms in a Reynolds Stress Turbulence Model

A formulation to include the e ects of wall proximity in a second moment closure model that utilizes a tensor representation for the redistribution terms in the Reynolds stress equations is presented The wall proximity e ects are modeled through an elliptic relaxation process of the tensor expansion coef cients that properly accounts for both correlation length and time scales as the wall is approached Direct numerical simulation data and Reynolds stress solutions using a full di erential approach are compared to the tensor repre sentation approach for the case of fully developed channel ow INTRODUCTION The theoretical development of higher order closure models such as Reynolds stress models have primarily been formulated based on high Reynolds number as sumptions The in uence of solid boundaries on these closure models has usually been accounted for through either a wall function approach or a modi cation to the high Reynolds number form of the pressure related correlations and tensor dissipation rate and predicated on the near wall asymptotic behavior of the various velocity second moments So et al Hanjali c A broader based attempt to account for the proximity of a solid boundary is the elliptic relaxation approach introduced over a decade ago Durbin and further developed for second moment closures Durbin a Wizman et al Manceau and Hanjali c Manceau Carlson and Gatski In its two equa tion form the v f model has been applied to a variety of ows e g Durbin b Pettersson Reif et al The new approach outlined here introduces a ten sor representation for the combined e ects of a near wall velocity pressure gradient correlation and anisotropic dissipation rate that asymptotes to a high Reynolds num ber form away from solid boundaries through an elliptic equation for the polynomial expansion coe cients The development of a generalized methodology for determin ing the polynomial expansion coe cients of representations for the turbulent stress anisotropies by Gatski and Jongen is extended to an elliptic relaxation proce dure for these expansion coe cients Although the material presented here introduces tensor representations and a ten sor projection methodology into the elliptic relaxation formulation this work can also be viewed as an intermediate step between a fully explicit elliptic relaxation algebraic Reynolds stress formulation and the full di erential elliptic relaxation Reynolds stress formulation The predictive capabilities of the new model are assessed through comparisons with direct numerical simulation channel ow data Moser et al These com parisons include both mean and turbulent ow quantities Theoretical Background and Development In this section a mathematical framework is developed for the Reynolds stress transport equations and the corresponding elliptic relaxation equation when a tensor representation of the redistribution terms is used in the formulation The method ology introduces a set of elliptic relaxation equations for the polynomial expansion coe cients of the chosen representation The f model uses the redistributive terms in the elliptic equations while the n model uses the expansion coe cients in the elliptic equations Both models use the Reynolds stress transport equations Transport Equations The transport of the Reynolds stresses ij uiuj is governed by the equation D ij Dt ik Uj xk jk Ui xk ij ij DT ij D ij where Ui is the mean velocity ij is the pressure redistribution term ij is the tensor dissipation rate and DT ij and D ij are the turbulent transport and viscous di usion respectively In the development outlined here it is best to have ij given by ij ui p xj uj p xi puk xk ij so that the trace of the pressure redistribution term is zero In the application of the elliptic relaxation method it is also necessary to account for the e ect of the dissi pation rate anisotropy as the wall is approached This accounting for the dissipation rate anisotropy is accomplished e g Manceau by a relaxation of the dissipa tion rate anisotropy to its wall value which is assumed to be equal to the Reynolds stress anisotropy This assumption allows the Reynolds stress transport equation in to be written as D ij Dt ik Uj xk jk U i xk Kfij ij K DT ij D ij where Kfij ij dij bij with the Reynolds stress anisotropy bij and dissipation rate anisotropy dij de ned as bij ij K ij dij ij ij The original scaling of the relaxation function fij was solely through the turbulent kinetic energy K however Manceau Carlson and Gatski have recently shown that an added dissipation rate factor to the scaling Kfij eliminates an unwanted ampli cation e ect inherent in the original scaling Equation is closed when the model for the turbulent transport DT ij is used In previous elliptic relaxation studies that used the Reynolds stress transport equations the viscous di usion and turbulent transport terms were modeled as D ij r ij DT ij xk uiujuk puk xl C lk K c ij xk with K and C The composite time scale