Multi-layer analytic solution for k-ωmodel equations via a symmetry approach

Despite being one of the oldest and most widely-used turbulence models in engineering computational fluid dynamics(CFD),the k-ωmodel has not been fully understood theoretically because of its high nonlinearity and complex model parameter setting.Here,a multi-layer analytic expression is postulated for two lengths(stress and kinetic energy lengths),yielding an analytic solution for the k-ωmodel equations in pipe flow.Approximate local balance equations are analyzed to determine the key parameters in the solution,which are shown to be rather close to the empirically-measured values from the numerical solution of the Wilcox k-ωmodel,and hence the analytic construction is fully validated.The results provide clear evidence that the k-ωmodel sets in it a multilayer structure,which is similar to but different,in some insignificant details,from the Navier-Stokes(N-S)turbulence.This finding explains why the k-ωmodel is so popular,especially in computing the near-wall flow.Finally,the analysis is extended to a newlyrefined k-ωmodel called the structural ensemble dynamics(SED)k-ωmodel,showing that the SED k-ωmodel has improved the multi-layer structure in the outer flow but preserved the setting of the k-ωmodel in the inner region.

The nature of near-wall convection velocity in turbulent channel flow

A novel notion of turbulent structure the local cascade structure-is introduced to study the convection phenomenon in a turbulent channel flow. A space-time cross-correlation method is used to calculate the convection velocity. It is found that there are two characteristic convection speeds near the wall, one associated with small-scale streaks of a lower speed and another with streamwise vortices and hairpin vortices of a higher speed. The new concept of turbulent structure is powerful to illustrate the dominant role of coherent structures in the near-wall convection, and to reveal also the nature of the convection-the propagation of patterns of velocity fluctuations-which is scale-dependent.

Accurately predicting hypersonic transitional flow on cone via a symmetry approach

A new algebraic transition model is proposed based on a Structural Ensemble Dynamics(SED)theory of wall turbulence,for accurately predicting the hypersonic flow heat transfer on cone.The model defines the eddy viscosity in terms of a two-dimensional multi-regime distribution of a Stress Length(SL)function,and hence is named as SED-SL.This paper presents clear evidence of precise predictions of transition onset location and peak heat flux of a wide range of hypersonic Transitional Boundary Layers(TrBL)around straight cone at zero incidence,to an unprecedented accuracy as validated by over 70 measurements for varying five crucial influential factors(Mach number,temperature ratio,cone half angle,nose Reynolds number and noise level).The results demonstrate the universality of the postulated multi-regime similarity structure,in characterizing not only the spatial non-uniform distribution of the eddy viscosity in hypersonic TrBL on cone,but also the dependence of the transition onset location on the five influential factors.The latter yields a novel correlation formula for transition center Reynolds number which takes similar functional form as the SL function within the symmetry approach.It is concluded that the SED-SL model simulates TrBL around cone with uniformly high accuracy,and then points out to an optimistic alternative way to construct hypersonic transition model.

β-distribution for Reynolds stress and turbulent heat flux in relaxation turbulent boundary layer of compression ramp

A challenge in the study of turbulent boundary layers(TBLs) is to understand the non-equilibrium relaxation process after separation and reattachment due to shock-wave/boundary-layer interaction. The classical boundary layer theory cannot deal with the strong adverse pressure gradient, and hence, the computational modeling of this process remains inaccurate. Here, we report the direct numerical simulation results of the relaxation TBL behind a compression ramp, which reveal the presence of intense large-scale eddies, with significantly enhanced Reynolds stress and turbulent heat flux. A crucial finding is that the wall-normal profiles of the excess Reynolds stress and turbulent heat flux obey a β-distribution, which is a product of two power laws with respect to the wall-normal distances from the wall and from the boundary layer edge. In addition, the streamwise decays of the excess Reynolds stress and turbulent heat flux also exhibit power laws with respect to the streamwise distance from the corner of the compression ramp. These results suggest that the relaxation TBL obeys the dilation symmetry, which is a specific form of self-organization in this complex non-equilibrium flow. The β-distribution yields important hints for the development of a turbulence model.