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Register a new userThe colour of forcing statistics in resolvent analyses of turbulent channel flows
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The cross-spectral density (CSD) of the non-linear forcing in resolvent analyses is here quantified for the first time for turbulent channel flows. Direct numerical simulations (DNS) at $Re_{tau} =179$ and $Re_{tau} =543$ are performed. The CSDs are computed for highly energetic structures typical of buffer-layer and large-scale motions, for different temporal frequencies. The CSD of the non-linear forcing is shown not to be uncorrelated (white) in space, which implies the forcing is structured. Since the non-linear forcing is non-solenoidal by construction and the velocity of an incompressible flow is affected only by the solenoidal part of the forcing, this solenoidal part is evaluated. It is shown that the solenoidal part of the non-linear forcing is the combination of oblique streamwise vortices and a streamwise component which counteract each other, as in a destructive interference. It is shown that a rank-2 approximation of the forcing, with only the most energetic SPOD (spectral proper orthogonal decomposition) modes, leads to the bulk of the response. The projections of the non-linear forcing onto the right-singular vectors of the resolvent are evaluated. The left-singular vectors of the resolvent associated with very low-magnitude singular values are non-negligible since the non-linear forcing term has a non-negligible projection onto the linear sub-optimals of resolvent analysis. The same projections are computed when the forcing is modelled with an eddy-viscosity approach. It is clarified that this modelling improves the accuracy of the prediction since the projections are closer to those associated with the non-linear forcing from DNS data.
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An analysis of the statistics of the non-linear terms in resolvent analysis is performed in this work for turbulent Couette flow at low Reynolds number. Data from a direct numerical simulation of a minimal flow unit, at Reynolds number 400, is post-processed using Fourier analysis in both time and space, leading to the covariance matrix of the velocity. From the same data, we computed the non-linear terms of the Navier-Stokes equations (treated as forcing in the present formulation), which allowed us to compute the covariance matrix of the forcing for this case. The two covariances are related exactly by the resolvent operator; based on this, we explore the recovery of the velocity statistics from the statistics of the forcing as a function of the components of the forcing term. This is carried out for the dominant structures in this flow, which participate in the self-sustaining cycle of turbulence: (i) streamwise vortices and streaks, and (ii) spanwise coherent fluctuations of spanwise velocity. The present results show a dominance by four of the non-linear terms for the prediction of the full statistics of streamwise vortices and streaks; a single term is seen to be dominant for spanwise motions. A relevant feature observed in these cases is that forcing terms have significant coherence in space; moreover, different forcing components are also coherent between them. This leads to constructive and destructive interferences that greatly modify the flow response, and should thus be accounted for in modelling work.
The ability of linear stochastic response analysis to estimate coherent motions is investigated in turbulent channel flow at friction Reynolds number Re$_tau$ = 1007. The analysis is performed for spatial scales characteristic of buffer-layer and large-scale motions by separating the contributions of different temporal frequencies. Good agreement between the measured spatio-temporal power spectral densities and those estimated by means of the resolvent is found when the effect of turbulent Reynolds stresses, modelled with an eddy-viscosity associated to the turbulent mean flow, is included in the resolvent operator. The agreement is further improved when the flat forcing power spectrum (white noise) is replaced with a power spectrum matching the measures. Such a good agreement is not observed when the eddy-viscosity terms are not included in the resolvent operator. In this case, the estimation based on the resolvent is unable to select the right peak frequency and wall-normal location of buffer-layer motions. Similar results are found when comparing truncated expansions of measured streamwise velocity power spectral densities based on a spectral proper orthogonal decomposition to those obtained with optimal resolvent modes.
We seek possible statistical consequences of the way a forcing term is added to the Navier--Stokes equations in the Direct Numerical Simulation (DNS) of incompressible channel flow. Simulations driven by constant flow rate, constant pressure gradient and constant power input are used to build large databases, and in particular to store the complete temporal trace of the wall-shear stress for later analysis. As these approaches correspond to different dynamical systems, it can in principle be envisaged that these differences are reflect by certain statistics of the turbulent flow field. The instantaneous realizations of the flow in the various simulations are obviously different, but, as expected, the usual one-point, one-time statistics do not show any appreciable difference. However, the PDF for the fluctuations of the streamwise component of wall friction reveals that the simulation with constant flow rate presents lower probabilities for extreme events of large positive friction. The low probability value of such events explains their negligible contribution to the commonly computed statistics; however, the very existence of a difference in the PDF demonstrates that the forcing term is not entirely uninfluential. Other statistics for wall-based quantities (the two components of friction and pressure) are examined; in particular spatio-temporal autocorrelations show small differences at large temporal separations, where unfortunately the residual statistical uncertainty is still of the same order of the observed difference. Hence we suggest that the specific choice of the forcing term does not produce important statistical consequences, unless one is interested in the strongest events of high wall friction, that are underestimated by a simulation run at constant flow rate.
We present direct numerical simulations of turbulent channel flow with passive Lagrangian polymers. To understand the polymer behavior we investigate the behavior of infinitesimal line elements and calculate the probability distribution function (PDF) of finite-time Lyapunov exponents and from them the corresponding Cramers function for the channel flow. We study the statistics of polymer elongation for both the Oldroyd-B model (for Weissenberg number $Wi <1$) and the FENE model. We use the location of the minima of the Cramers function to define the Weissenberg number precisely such that we observe coil-stretch transition at $Wiapprox1$. We find agreement with earlier analytical predictions for PDF of polymer extensions made by Balkovsky, Fouxon and Lebedev [Phys. Rev. Lett., 84, 4765 (2000).] for linear polymers (Oldroyd-B model) with $Wi<1$ and by Chertkov [Phys. Rev. Lett., 84, 4761 (2000).] for nonlinear FENE-P model of polymers. For $Wi>1$ (FENE model) the polymer are significantly more stretched near the wall than at the center of the channel where the flow is closer to homogenous isotropic turbulence. Furthermore near the wall the polymers show a strong tendency to orient along the stream-wise direction of the flow but near the centerline the statistics of orientation of the polymers is consistent with analogous results obtained recently in homogeneous and isotropic flows.
We use direct numerical simulations to calculate the joint probability density function of the relative distance $R$ and relative radial velocity component $V_R$ for a pair of heavy inertial particles suspended in homogeneous and isotropic turbulent flows. At small scales the distribution is scale invariant, with a scaling exponent that is related to the particle-particle correlation dimension in phase space, $D_2$. It was argued [1, 2] that the scale invariant part of the distribution has two asymptotic regimes: (1) $|V_R| ll R$ where the distribution depends solely on $R$; and (2) $|V_R| gg R$ where the distribution is a function of $|V_R|$ alone. The probability distributions in these two regimes are matched along a straight line $|V_R| = z^ast R$. Our simulations confirm that this is indeed correct. We further obtain $D_2$ and $z^ast$ as a function of the Stokes number, ${rm St}$. The former depends non-monotonically on ${rm St}$ with a minimum at about ${rm St} approx 0.7$ and the latter has only a weak dependence on ${rm St}$.
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