Predicting structural and statistical features of wall turbulence

The majority of practical flows, particularly those flows in applications of importance to transport, distribution and climate, are turbulent and as a result experience complex three-dimensional motion with increased drag comparedwith the smoother, laminar condition. In this study, we describe the development of a simple model that predicts important structural and scaling features of wall turbulence. We show that a simple linear superposition of modes derived from a forcing-response analysis of the Navier-Stokes equations can be used to reconcile certain key statistical and structural descriptions of wall turbulence. The computationally cheap approach explains and predicts vortical structures and velocity statistics of turbulent flows that have previously been identified only in experiments or by direct numerical simulation. In particular, we propose an economical explanation for the meandering appearance of very large scale motions observed in turbulent pipe flow, and likewise demonstrate that hairpin vortices are predicted by the model. This new capability has clear implications for modeling, simulation and control of a ubiquitous class of wall flows

Scaling and interaction of self-similar modes in models of high-Reynolds number wall turbulence

Previous work has established the usefulness of the resolvent operator that maps the terms nonlinear in the turbulent fluctuations to the fluctuations themselves. Further work has described the selfsimilarity of the resolvent arising from that of the mean velocity profile. The orthogonal modes provided by the resolvent analysis describe the wall-normal coherence of the motions and inherit that self-similarity. In this contribution, we present the implications of this similarity for the nonlinear interaction between modes with different scales and wall-normal locations. By considering the nonlinear interactions between modes, it is shown that much of the turbulence scaling behaviour in the logarithmic region can be determined from a single arbitrarily chosen reference plane. Thus, the geometric scaling of the modes is impressed upon the nonlinear interaction between modes. Implications of these observations on the self-sustaining mechanisms of wall turbulence,modelling and simulation are outlined

Determination of flow resistance coefficient for vegetation in open channel: laboratory study

This study focused on determination of flow resistances coefficient for grass in an open channel. Laboratory works were conducted to examine the effects of varying of roughness elements on the flume to determine flow resistance coefficient and also to determine the optimum flow resistance with five different flow rate, Q. Laboratory study with two type of vegetation which are Cow Grass and Pearl Grass were implementing to the bed of a flume. The roughness coefficient, n value is determine using Manning’s equation while Soil Conservation Services (SCS) method was used to determine the surface resistance. From the experiment, the flow resistance coefficient for Cow Grass in range 0.0008 - 0.0039 while Pearl Grass value for the flow resistance coefficient are in between 0.0013 - 0.0054. As a conclusion the vegetation roughness value in open channel are depends on density, distribution type of vegetation used and physical characteristic of the vegetation itsel

Low-dimensional representations of exact coherent states of the Navier-Stokes equations from the resolvent model of wall turbulence

We report that many exact invariant solutions of the Navier-Stokes equations for both pipe and channel flows are well represented by just few modes of the model of McKeon & Sharma J. Fl. Mech. 658, 356 (2010). This model provides modes that act as a basis to decompose the velocity field, ordered by their amplitude of response to forcing arising from the interaction between scales. The model was originally derived from the Navier-Stokes equations to represent turbulent flows and has been used to explain coherent structure and to predict turbulent statistics. This establishes a surprising new link between the two distinct approaches to understanding turbulence

On the structure and origin of pressure fluctuations in wall turbulence: predictions based on the resolvent analysis

We generate predictions for the fluctuating pressure field in turbulent pipe flow by re-formulating the resolvent analysis of McKeon & Sharma (2010) in terms of the so-called primitive variables. Under this analysis, the nonlinear convective terms in the Fourier-transformed Navier-Stokes equations are treated as a forcing that is mapped to a velocity and pressure response by the resolvent of the linearized Navier-Stokes operator. At each wavenumber-frequency combination, the turbulent velocity and pressure field are represented by the most-amplified (rank-1) response modes, identified via a singular value decomposition of the resolvent. We show that these rank-1 response modes reconcile many of the key relationships between the velocity field, coherent structure (i.e., hairpin vortices), and the high-amplitude wall-pressure events observed in previous experiment and DNS. A Green’s function representation shows that the pressure fields obtained under this analysis correspond primarily to the fast pressure contribution arising from the linear interaction between the mean shear and the turbulent wall-normal velocity. Recovering the slow pressure requires an explicit treatment of the nonlinear interactions between the Fourier response modes. By considering the velocity and pressure fields associated with the triadically-consistent mode combination studied by Sharma & McKeon (2013), we identify the possibility of an apparent amplitude modulation effect in the pressure field, similar to that observed for the streamwise velocity field. However, unlike the streamwise velocity, for which the large scales of the flow are in phase with the envelope of the small-scale activity close to the wall, we expect there to be a ?/2 phase difference between the large scale wall-pressure and the envelope of the small-scale activity. Finally, we generate spectral predictions based on a rank-1 model assuming broadband forcing across all wavenumber-frequency combinations. Despite the significant simplifying assumptions, this approach reproduces trends observed in previous DNS for the wavenumber spectra of velocity and pressure, and for the scale-dependence of wall-pressure propagation speed

A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization

We combine resolvent-mode decomposition with techniques from convex optimization to optimally approximate velocity spectra in a turbulent channel. The velocity is expressed as a weighted sum of resolvent modes that are dynamically significant, non-empirical, and scalable with Reynolds number. To optimally represent DNS data at friction Reynolds number 2003, we determine the weights of resolvent modes as the solution of a convex optimization problem. Using only 12 modes per wall-parallel wavenumber pair and temporal frequency, we obtain close agreement with DNS-spectra, reducing the wall-normal and temporal resolutions used in the simulation by three orders of magnitude

On the origin of frequency sparsity in direct numerical simulations of turbulent pipe flow

The possibility of creating reduced-order models for canonical wall-bounded turbulent flows based on exploiting energy sparsity in frequency domain, as proposed by Bourguignon et al. [Phys. Fluids26, 015109 (2014)], is examined. The present letter explains the origins of energetically sparse dominant frequencies and provides fundamental information for the design of such reduced-order models. The resolvent decomposition of a pipe flow is employed to consider the influence of finite domain length on the flow dynamics, which acts as a restriction on the possible wavespeeds in the flow. A forcing-to-fluctuation gain analysis in the frequency domain reveals that large sparse peaks in amplification occur when one of the possible wavespeeds matches the local wavespeed via the critical layer mechanism. A link between amplification and energy is provided through the similar characteristics exhibited by the most energetically relevant flow structures, arising from a dynamic mode decomposition of direct numerical simulation data, and the resolvent modes associated with the most amplified sparse frequencies. These results support the feasibility of reduced-order models based on the selection of the most amplified modes emerging from the resolvent model, leading to a novel computationally efficient method of representing turbulent flows

On the design of optimal compliant walls for turbulence control

This paper employs the resolvent framework to consider the design of compliant walls for turbulent skin friction reduction. Specifically, the effects of simple spring–damper walls are contrasted with the effects of more complex walls incorporating tension, stiffness and anisotropy. In addition, varying mass ratios are tested to provide insight into differences between aerodynamic and hydrodynamic applications. Despite the differing physical responses, all the walls tested exhibit some important common features. First, the effect of the walls (positive or negative) is the greatest at conditions close to resonance, with sharp transitions in performance across the resonant frequency or phase speed. Second, compliant walls are predicted to have a more pronounced effect on slower moving structures because such structures generally have larger wall-pressure signatures. Third, two-dimensional (spanwise constant) structures are particularly susceptible to further amplification. These features are consistent with many previous experiments and simulations, suggesting that mitigating the rise of such two-dimensional structures is essential to designing performance-improving walls. For instance, it is shown that further amplification of such large-scale two-dimensional structures explains why the optimal anisotropic walls identified in previous direct numerical simulations only led to drag reduction in very small domains. The above observations are used to develop design and methodology guidelines for future research on compliant walls

Compact representation of wall-bounded turbulence using compressive sampling

Compressive sampling is well-known to be a useful tool used to resolve the energetic content of signals that admit a sparse representation. The broadband temporal spectrum acquired from point measurements in wall-bounded turbulence has precluded the prior use of compressive sampling in this kind of flow, however it is shown here that the frequency content of flow fields that have been Fourier transformed in the homogeneous spatial (wall-parallel) directions is approximately sparse, giving rise to a compact representation of the velocity field. As such, compressive sampling is an ideal tool for reducing the amount of information required to approximate the velocity field. Further, success of the compressive sampling approach provides strong evidence that this representation is both physically meaningful and indicative of special properties of wall turbulence. Another advantage of compressive sampling over periodic sampling becomes evident at high Reynolds numbers, since the number of samples required to resolve a given bandwidth with compressive sampling scales as the logarithm of the dynamically significant bandwidth instead of linearly for periodic sampling. The combination of the Fourier decomposition in the wall-parallel directions, the approximate sparsity in frequency, and empirical bounds on the convection velocity leads to a compact representation of an otherwise broadband distribution of energy in the space defined by streamwise and spanwise wavenumber, frequency, and wall-normal location. The data storage requirements for reconstruction of the full field using compressive sampling are shown to be significantly less than for periodic sampling, in which the Nyquist criterion limits the maximum frequency that can be resolved. Conversely, compressive sampling maximizes the frequency range that can be recovered if the number of samples is limited, resolving frequencies up to several times higher than the mean sampling rate. It is proposed that the approximate sparsity in frequency and the corresponding structure in the spatial domain can be exploited to design simulation schemes for canonical wall turbulence with significantly reduced computational expense compared with current technique