Synthetic Velocity Fields

Synthetic velocity fields that approximate the DNS propagating waves but have a known frequency content were generated using the framework of McKeon & Sharma (2010). Generation of these fields required low computational and storage resources compared to the DNS flow fields, thus they were used to identify the influence of the sampling parameters on the frequencies extracted via compressive sampling, for a known set of input frequencies.

McKeon & Sharma (2010) considered a gain analysis of the Navier-Stokes (NS) equations to obtain a basis for the wall-normal coherence of propagating waves in turbulent pipe flow. In this approach, the full synthetic velocity field ˆu(x, r, θ, t) was decomposed into a series of propagating waves defined by their streamwise and azimuthal wavenumbers (k, n) and angular frequencyω= 2πf,

ˆ

u(x, r, θ, t) = X

k,n,ω

ak,n,ωck,n,ω(r)ei(kx+nθ−ωt), (3.2)

where theak,n,ω are complex-valued coefficients denoting the relative magnitude and phase of the

waves. The wavenumbers and frequencies are non-dimensionalized based on the pipe radius and bulk velocity. Berkooz et al. (1993) showed that the Fourier series decomposition in the homogeneous (spatial and temporal) directions (equation (3.2)) is optimal in the sense that it maximizes the turbulent kinetic energy captured for a given number of basis functions and the Fourier modes are known to be the eigenmodes of the linear NS equations. Under this formulation, the nonlinear terms in the NS equations for the velocity fluctuations act as an unknown forcing on the linear part of

the equations. A divergence-free basis was used in the radial direction to project the NS equations, resulting in the elimination of the pressure term. The radial distribution of momentumck,n,ω(r) for

the propagating waves is described by the forced NS equations

iωck,n,ω(r) =Lk,n,ω(r)ck,n,ω(r) +fk,n,ω(r), (3.3)

wherefk,n,ω(r) andLk,n,ω(r) are the projection of the nonlinear terms and of the linear part of the

NS equations onto the Fourier basis, respectively. The propagating waveck,n,ω(r)ei(kx+nθ−ωt) can

be seen as a response of the mean turbulent flow to the forcing fk,n,ω(r) via the transfer function

(iωI− Lk,n,ω(r))−1, i.e.,

ck,n,ω(r) = (iωI− Lk,n,ω(r))−1fk,n,ω(r). (3.4)

The analysis identifies the forcing and response modes that are most amplified based on a turbulent kinetic energy norm, at each given wavenumber pair and frequency, via singular value decomposition (SVD) of the transfer function.

The framework of McKeon & Sharma (2010) was used to generate response modes to represent wall-bounded turbulence. The synthetic velocity fields were constructed by superposition of response modes computed with three different (k, n) wavenumber pairs chosen to represent the large scales (1,10) modes, the near-wall type modes (4.3,43) corresponding to a streamwise and azimuthal extent of 1,000 and 100 viscous wall-units at Re = 24,580, and the small dissipative scales (16.75,60), respectively. For each wavenumber pair, several convection velocities (frequencies) were chosen arbitrarily in a range varying from a few viscous units to the centerline velocity. The propagating waves generated with the three wavenumber pairs and the various frequencies span the broad range of scales present in wall-bounded turbulence, and their superposition results in realistic velocity fields due to the coherence in the wall-normal direction. The relative phase of the waves was randomly chosen and the magnitude was set to one. The propagating waves were computed using the spectral code of Meseguer & Trefethen (2003) modified by McKeon & Sharma (2010) to allow for the input of any velocity profile and the decomposition into singular modes. The velocity profile from the DNS of Wu & Moin (2008) atRe= 24,580 was used as an input to the model. The synthetic fields can be sampled at any time instant and require low computational and storage resources compared to the DNS flow fields, allowing for the test of numerous sampling parameters to use compressive sampling in wall-bounded turbulence.

Compressive sampling was applied to the streamwise velocity component of the synthetic velocity fields sampled randomly in time. The velocity fields were decomposed as a Fourier series in the streamwise and azimuthal directions, and each 2D Fourier modeck,n(r, ts) was written as a Fourier

series in time with unknown coefficients, ck,n(r, ts) = Nopti X j=1 ck,n,2πjdf(r)ei2πjdf ts, (3.5)

whereNopti anddf are optimization variables (to be determined) corresponding respectively to the

number of frequencies and the frequency increment for the optimization. The coefficients of the temporal Fourier series ck,n,2πjdf(r) are the solution of an optimization problem that consists of

minimizing the sum of the absolute value of the Fourier coefficientsck,n,2πjdf(r), i.e., minimizing

Nopti

X

j=1

|ck,n,2πjdf(r)|, (3.6)

at each wall-normal location separately, under the constraint that the reconstructed signal

Nopti

X

j=1

ck,n,2πjdf(r)ei2πjdf ts

matches the input signalck,n(r, ts) at every sampling time instantts. The dominant Fourier coeffi-

cientsck,n,2πjdf(r) identified via compressive sampling correspond to the streamwise component of

the propagating wavesck,n,ω(r) constituting the synthetic flow field.

Contrarily to periodic sampling, the frequency range for the optimization [df, Nopti×df] is not

directly related to the sampling parameters, and was chosen to maximize the frequency resolution, while still satisfying the sparsity relationship (equation (3.1)). For simplicity, the frequencies were chosen to be equi-spaced within the optimization frequency range (but this is not required in order to apply compressive sampling). The maximum and minimum frequencies can be respectively higher than the mean sampling rate and lower than the inverse of the sampling duration. The lower the number of sparse frequencies present in the input signal the broader the frequency range that can be resolved with compressive sampling.

The minimization problem was solved with Matlab using the CVX toolbox for convex opti- mization, and an example of the code used is shown in appendix A. The frequency spectrum was computed at each wall-normal location, and integrated in the radial direction to identify the ener- getically dominant frequencies, and to check whether the dominant frequencies matched the input frequencies. The optimization frequency range and sampling parameters were adjusted to find the optimal number of samples and sampling duration leading to the recovery of the signal.