DLR.de Chart 1 Perspectives on Reynolds Stress Modeling Part I: General Approach Umich/NASA Symposium on Advances in Turbulence Modeling Bernhard Eisfeld, Tobias Knopp 11 July 2017
DLR.de Chart 2 Overview Introduction Reynolds Stress Modeling Flow Physics Modeling Perspectives
DLR.de Chart 3 Overview Introduction Reynolds Stress Modeling Flow Physics Modeling Perspectives
DLR.de Chart 4 Introduction Why RANS? Aerodynamic design/optimization è short response times required RANS (EVM) based CFD very successful Why RSM? Virtual product è CFD at off-design conditions è Lack of accuracy Representation of more complex flow physics required è RSM naturally provides opportunities Virtual product
DLR.de Chart 5 Introduction Strategy Physics-based modeling High quality data base (experiments, DNS/LES) derive predict (Empirical) laws of turbulence derive predict Physics-based RANS improvement General approach for shear flows è now APG boundary layers è presentation by Tobias Knopp
DLR.de Chart 6 Overview Introduction Reynolds Stress Modeling Flow Physics Modeling Perspectives
DLR.de Chart 7 Reynolds Stress Modeling Transport equation Incompressible formulation R t + U k R x k = P + P -e + D Production P U x j i = -Rik - R jk Exact, no modeling involved k U x k Dissipation e High Re è isotropic e from length-scale equation Anisotropy effects near walls è e.g. Jakirlic Diffusion D Gradient driven modeling Minor influence on overall performance
DLR.de Chart 8 Reynolds Stress Modeling Pressure-strain correlation P Traceless (incompressible) è no contribution to k-budget è re-distribution of Reynolds stresses Rotta s analysis (1951) P = P + P + P ( s) ( r) ( b) Slow term (s) P Rapid term r P ) Return to isotropy (independent of mean flow) è (non-linear) function of all Reynolds stress anisotropies b kl U ( k = M kl x Influence of mean flow l Influence of boundaries Wall-reflexion terms (slow + rapid) (b) P
DLR.de Chart 9 Reynolds Stress Modeling Rapid-term modeling M kl = function of all Reynolds stress anisotropies b mn Constraints, e.g. symmetry (Rotta) è reduction of terms/coefficients Approaches Standard M kl linear in b mn (e.g. LRR) Non-linear extension M kl = power series in b mn è More degrees of freedom Opportunities Additional physics (realizability, two-component limit) Concerns Rapid Distortion Theory Numerics
DLR.de Chart 10 Reynolds Stress Modeling Calibration of RSM Boundary layer theory è Turbulent equilibrium (Rotta/Hinze) P -e + P = 0 è trace (k ) P = e (two-equation models) RSM: 3 equations for 3 coefficients = f(b mn ) Independent of velocity profile Shear stress anisotropy by Bradshaw hypothesis è in boundary layers b xy = 0.15 Normal stress ratios by rule of thumb, e.g. Wilcox 4:2:3 Why is the Bradshaw hypothesis valid?
DLR.de Chart 11 Overview Introduction Reynolds Stress Modeling Flow Physics Modeling Perspectives
DLR.de Chart 12 Flow Physics Theoretical considerations Turbulent equilibrium P -e + P = 0 Self-similarity of U and R xy + isotropic dissipation (high Re) è (k ) P, e, P, P,e Self-similar with identical profile function Scaling arguments for P Slow term ( ) ( b ) const. s s P ( ) µ e F ( ) bmn è F è b mn = const. ( s) mn = Rapid term è consistent with b mn = const. è Flow physics principle Boundary layer equations (turb. equilibrium) + self-similarity/self-preservation è constant Reynolds stress anisotropy Applies to various shear flows Bradshaw hypothesis is special case è Is the theory correct?
DLR.de Chart 13 Reynolds Stress Anisotropy Experimental confirmation: Plane jet Indicator function: b = f xy ( x ) [ ( ) ] 2 R DU max f xy ( ) x exp Theoretical profile of R xy /(DU max ) 2 All b = const. è identical profiles è constant anisotropy Exp. data confirm theory B. Eisfeld: Reynolds Stress Anisotropy in Self-Preserving Turbulent Shear Flows, DLR-IB-AS-BS-2017-106
DLR.de Chart 14 Reynolds Stress Anisotropy Experimental confirmation: Axisymmetric jet Region of constant indicator function è Exp. data confirm theory B. Eisfeld: Reynolds Stress Anisotropy in Self-Preserving Turbulent Shear Flows, DLR-IB-AS-BS-2017-106
DLR.de Chart 15 Reynolds Stress Anisotropy Experimental confirmation: Plane mixing layer Region of constant indicator function è Exp. data confirm theory B. Eisfeld: Reynolds Stress Anisotropy in Self-Preserving Turbulent Shear Flows, DLR-IB-AS-BS-2017-106
DLR.de Chart 16 Flow Physics Reynolds stress anisotropy Provided by indicator function in constant region b = b b kk 1 - d 3 Shear stress anisotropy (estimates) Flow b xy Boundary layer 0.150 Plane jet 0.147 Axisym. jet 0.131 Mixing layer 0.164±0.012 Calibration value Similar è spreading o.k. Smaller è R xy overestimated (spreading) Larger è R xy underestimated Plane jet axisymmetric jet è round jet/plane jet anomaly Boundary layer mixing layer è Re-attachment delayed
DLR.de Chart 17 Overview Introduction Reynolds Stress Modeling Flow Physics Modeling Perspectives
DLR.de Chart 18 Modeling Perspectives Mismatch of R xy Scaling approach Modify k and keep anisotropy (boundary layer calibration) è Modify length-scale determining equation Scaling Alternative approach (experimental observation) Modify anisotropy and keep k (length-scale) = change orientation of principle axes è Consistent with RSM technology è Adapt model calibration Rotation+ deformation Combination required? Note: Self-adaptation of model è zonal approach
DLR.de Chart 19 Modeling Perspectives Example Baseline = SSG/LRR-w Rough recalibration of SSG-part for mixing layer (Delville et al. data) è get R right at most downstream position Note: for demonstration only
DLR.de Chart 20 Modeling Perspectives Example Application to separated flows è Separation length reduced (for demonstration only!) Half-jet mixing layer backward facing step è model of reattachment
DLR.de Chart 21 Modeling Perspectives Requirements for future improvement Reliable anisotropy data for free shear flows and boundary layers Highly accurate experiments DNS Requirement for self-preservation High enough Re Downstream development documented Sensors for self-adaptation e.g. SAS-related? Application of machine-learning methods Requirements Reliability Suitability for RANS-based CFD Model analysis Calibration Interaction of Reynolds stress anisotropy and length-scale equation Improvement for APG boundary layers è presentation by Tobias Knopp Joint effort required