Zero-pressure Gradient Turbulent Boundary Research Articles

Energy transfer in the formation and evolution of energy-containing motions in turbulent boundary layers

This paper examines using direct numerical simulation (DNS) the recovery of the energy-containing scales of motion (energy-containing motions) when these are removed at the inflow boundary condition of a flat plate zero-pressure gradient turbulent boundary layer (ZPG-TBL). The energy-containing motions are identified from the spanwise spectral energy density of the interscale turbulent transport. Analysis of the resulting spanwise spectral energy densitiy of velocity and vorticity suggests that near-wall streaks are the initial energy-containing motions formed with quasi-streamwise vortices only developing further downstream. This result is in agreement with the previous study on turbulent channel flows by Kannadasan et al. [1].

Response of a Turbulent Boundary Layer to an Imposed Synthetic Large-Scale Structure

The dynamic response of a zero-pressure gradient turbulent boundary layer (TBL) to an active flow control actuator was experimentally investigated. The experimental TBL had a sufficiently low momentum thickness Reynolds number, such that there were no energetic organized large-scale structures in the outer region, as evidenced from measurements of the premultiplied wavenumber–frequency spectra. The periodically pulsed plasma actuator, placed inside the outer region of TBL, introduced a synthetic large-scale structure, and the boundary-layer response to this synthetic structure downstream of the actuator at select wall-normal and streamwise locations was investigated. The turbulence amplitude modulating effect of the synthetic large-scale structure, within the inner and log-linear regions of the boundary layer, was isolated and analyzed using a phase-locked analysis. The dynamic interaction of the synthetic large-scale structure and smaller-scale turbulent motions was quantified using a modulation coefficient, and a strong positive correlation within the inner and log regions of the boundary layer was measured. The streamwise development of the synthetic large-scale structure and its modulating effect on the near-wall turbulence across several streamwise locations are described. Profiles of the phase speed at these streamwise locations were extracted and were found to be constant within the log region, further confirming a strong coupling between the near-wall fluctuations in turbulence intensity and the synthetic large-scale motions introduced in the outer region. Overall, the turbulence amplitude modulation effect induced by the synthetic large-scale structure was found to be dynamically similar to the large-scale modulation measured in canonical TBLs at higher Reynolds numbers.

Turbulent superstructure statistics in a turbulent boundary layer with pressure gradients

In turbulent boundary layers, streamwise elongated regions of high- and low-momentum in the log-law layer that can extent up to several boundary layer thicknesses are often referred to as turbulent superstructures. These structures contain a relatively large portion of the layer’s turbulent kinetic energy and have been shown to interact with the near-wall flow structures. In the last few decades extensive research on zero-pressure gradient (ZPG) turbulent boundary layers has been done, however by comparison, the structural characteristics for adverse pressure gradient turbulent (APG) boundary layer flows are much less studied despite their strong significance aero-hydrodynamic vehicle design. Therefore, the three-dimensional dynamics of turbulent superstructures in a turbulent boundary layer flow are investigated in the Atmospheric Wind Tunnel Munich (AWM) using a multi-camera 3D time-resolved Lagrangian particle tracking approach. In this study, Lagrangian and Eulerian statistics will be used to characterize the dynamics and interaction of turbulent superstructures within a zero pressure gradient (ZPG) turbulent boundary layer at Reτ = 5000 or Reθ = 14 000 that then flows over a curved plate subjected to a favorable (FPG) and strong adverse (APG) pressure gradient, which eventually separates. An Eulerian analysis, using multi-point correlations of 3D velocity fields, found that the average superstructure topology is modulated by decelerating flow in the APG region when compared to the ZPG region, however the basic shape and spanwise pattern is preserved. Looking into the behavior of individual trajectories, it was found that the dispersion of single particles along trajectories in the log-law layer are capable of moving more than the average Eulerian superstructure spacing in the spanwise direction. Furthermore, the mean square of the single particle dispersion indicates that the maximum dispersion in the spanwise direction comes from particles released at the wall-normal location corresponding to the so-called “second-peak/plateau” region in the streamwise normal Reynolds stress.

Sensitivity of TBL Wall-Pressure over the Flat Plate on Numerical Turbulence Model Parameter Variations

A two-fold sensitivity of the zero-pressure gradient (ZPG) turbulent boundary layer (TBL) wall-pressure spectrum to different RANS model parameters is investigated for a flat plate case, which is a close approximation to the aircraft fuselage or wing. The alteration in the mean square pressure fluctuations due choice of semi-empirical pressure model and the choice of computational model parameters like solver, near wall grid clustering, measuring location, and flow velocity are separately studied. The underlying effect of different TBL parameters in the said sensitivity has been studied while numerically replicating wind tunnel experiments and in-flight tests considering different RANS configurations. Initially, the best-predicting pressure spectrum models are selected by comparing them with available in-flight and wind tunnel test data. Subsequently, the accuracy of all the individual model parameters in predicting mean TBL flow quantities like wall shear stress, boundary layer thickness, displacement thickness, momentum thickness, etc., and eventually mean square pressure (MSP) is estimated. The sensitivity of the mean square pressure fluctuations value to the TBL flow quantities and the near-wall grid clustering is observed to be significant. In general, family of models is found to be best in terms of numerical convergence and closeness when compared to the experimental MSP values. family of models is suggested to be avoided while estimating MSP in flat plate TBL case

A Modified Shielding and Rapid Transition DDES Model for Separated Flows

In this paper, the major problems associated with detached eddy simulation (DES) (namely, modeled stress depletion (MSD) and slowing of the RANS to LES transition (RLT)) are discussed and reviewed, and relevant improvements are developed. A modified version for the delayed DES (DDES) method with adaptive modified adequate shielding and rapid transition is proposed; this is called MSRT DDES. The modified shielding strategy can be adjusted adaptively according to the local flow conditions: keeping the RANS behavior in the whole boundary layer when there is no resolved turbulence, and weakening the shielding function when resolved turbulence exists in the mainstream over the boundary layer. This strategy can significantly ameliorate the MSD in the RANS boundary layer, regardless of the mesh refinement, and avoid excessive shielding in the fully developed resolved turbulence that may otherwise delay the development of the separated and reattached flow. Three cases are designed to test the modified DDES, namely, complete shielding in the RANS zone of a boundary layer (the zero-pressure gradient turbulent boundary layer with the refined mesh), modified adaptive improved shielding with a rapid transition (the flow over a hump), and the overall performance in a complex 3D separation (the corner separation in a compressor cascade). The results show that the modified shielding function is more physical than earlier proposals compared to shielding functions, and according to detailed comparisons of the wall skin friction coefficients, velocity profiles, total pressure-loss coefficients, entropy production analyses, and so on, the MSD and RLT problems are moderately alleviated by the MSRT DDES.

Investigation of Large Scale Motions in Zero and Adverse Pressure Gradient Turbulent Boundary Layers Using High-Spatial-Resolution PIV

Particle image velocimetry (PIV) has been used to capture the high-spatial-resolution (HSR) two-component, two-dimensional (2C-2D) velocity fields of a zero-pressure-gradient (ZPG) turbulent boundary layer (TBL) and of an adverse-pressure-gradient (APG) TBL. Proper Orthogonal Decomposition (POD) is performed on the measured velocity fields to characterize the velocity fields as large or small scale motions (LSMs or SSMs), with further characterisation of the LSMs into high and low momentum events. This paper reports the findings of the PIV experiment and the subsequent analysis of the high Reynolds number ZPG and APG TBLs

Improved wall model treatment for aerodynamic flows in LBM

The article deals with an improved treatment of wall models for the simulation of turbulent flows in the framework of Immersed Wall Boundaries on Cartesian grids. The emphasis is put on the implementation in a Lattice-Boltzmann Method solver without loss of generality, since the proposed approach can be used in Navier–Stokes-based solvers in a straightforward way. The proposed improved wall model implementation relies on the combination of several key elements, namely i) the removal of grid points too close to the solid surface and ii) an original computation of wall normal velocity gradient and iii) the interpolation scheme. The new method is successfully assessed considering URANS simulations focusing on steady solutions of the Zero Pressure Gradient turbulent flat plate boundary layer and the turbulent flow around a NACA0012 airfoil at several angles of attack.

STREAmS: A high-fidelity accelerated solver for direct numerical simulation of compressible turbulent flows

We present STREAmS, an in-house high-fidelity solver for direct numerical simulations (DNS) of canonical compressible wall-bounded flows, namely turbulent plane channel, zero-pressure gradient turbulent boundary layer and supersonic oblique shock-wave/boundary layer interaction. The solver incorporates state-of-the-art numerical algorithms, specifically designed to cope with the challenging problems associated with the solution of high-speed turbulent flows and can be used across a wide range of Mach numbers, extending from the low subsonic up to the hypersonic regime. From the computational viewpoint, STREAmS is oriented to modern HPC platforms thanks to MPI parallelization and the ability to run on multi-GPU architectures. This paper discusses the main implementation strategies, with particular reference to the CUDA paradigm, the management of a single code for traditional and multi-GPU architectures, and the optimization process to take advantage of the latest generation of NVIDIA GPUs. Performance measurements show that single-GPU optimization more than halves the computing time as compared to the baseline version. At the same time, the asynchronous patterns implemented in STREAmS for MPI communications guarantee very good parallel performance especially in the weak scaling spirit, with efficiency exceeding 97% on 1024 GPUs. For overall evaluation of STREAmS with respect to other compressible solvers, comparison with a recent GPU-enabled community solver is presented. It turns out that, although STREAmS is much more limited in terms of flow configurations that can be addressed, the advantage in terms of accuracy, computing time and memory occupation is substantial, which makes it an ideal candidate for large-scale simulations of high-Reynolds number, compressible wall-bounded turbulent flows. The solver is released open source under GPLv3 license. Program summaryProgram Title: STREAmSCPC Library link to program files:https://doi.org/10.17632/hdcgjpzr3y.1Developer’s repository link:https://github.com/matteobernardini/STREAmSCode Ocean capsule:https://codeocean.com/capsule/8931507/tree/v2Licensing provisions: GPLv3Programming language: Fortran 90, CUDA Fortran, MPINature of problem: Solving the three-dimensional compressible Navier–Stokes equations for low and high Mach regimes in a Cartesian domain configured for channel, boundary layer or shock-boundary layer interaction flows.Solution method: The convective terms are discretized using a hybrid energy-conservative shock-capturing scheme in locally conservative form. Shock-capturing capabilities rely on the use of Lax–Friedrichs flux vector splitting and weighted essentially non-oscillatory (WENO) reconstruction. The system is advanced in time using a three-stage, third-order RK scheme. Two-dimensional pencil distributed MPI parallelization is implemented alongside different patterns of GPU (CUDA Fortran) accelerated routines.

High-performance simulations of turbulent boundary layer flow using Intel Xeon Phi many-core processors

Direct numerical simulations (DNS) of turbulent flows have increasing importance because they not only provide fundamental understanding of turbulent flows but also complement and extend experimental results. DNS of high Reynolds numbers, however, require huge computing cost so high-performance computing has been strongly pursued. In this study, we examine the feasibility of cost-efficient DNS on Intel Xeon Phi many-core processors that are currently adopted by 10% of the 100 largest supercomputers in the world as listed in the Top500 site. For this purpose, we port and optimize our in-house turbulent flow solver named as DNS-TBL (direct numerical simulation-turbulent boundary layer) on Xeon Phi Knights Landing (KNL) many-core processors and conduct benchmark tests on KNL and conventional multicore processors. The key architectural features of KNL processors and strategies to exploit them for performance enhancement are discussed. The optimized code is validated by conducting numerical simulations of zero-pressure gradient turbulent boundary layers at high Reynolds numbers and by comparing simulated turbulent statistics to those reported in previous studies. With the details of optimization strategies and validation processes, this work can serve as a practical guideline for acceleration of large-scale and precise DNS with many-core computing.

Boundary layer turbulence statistics over transitionally rough marine fouling control coatings

The results of a dedicated boundary layer measurement campaign, which was conducted by using a 2D LDV system over transitionally rough and irregular marine surfaces in a large flow-channel, are presented and discussed in this study. The experimental campaign aimed to investigate the turbulence statistics for zero-pressure gradient turbulent boundary layers developed over marine fouling control coatings, which represent transitionally rough and irregular surfaces, along with hydraulically smooth and fully-rough reference surfaces. A particular emphasis was placed on the wall outer layer similarity. At the same time, the main characteristics of the boundary layer flow such as the velocity defect profiles, Reynolds stresses, triple correlations of turbulence velocities, skewness distributions and flatness factors were comparatively discussed for six different surfaces. The surface roughness properties, boundary layer characteristic parameters, wake parameters and frictional drag properties were also presented for the tested surfaces along with their roughness functions. The outer layer similarity was observed for the surfaces coated with the fouling control coatings in general. The current study presented, perhaps for the first time in such details, the higher-order statistics of turbulence for the turbulent boundary layer flow developed over the state-of-the-art marine fouling control coatings.

Drag Reduction and Energy Saving by Spanwise Traveling Transversal Surface Waves for Flat Plate Flow

Wall-resolved large-eddy simulations are performed to study the impact of spanwise traveling transversal surface waves in zero-pressure gradient turbulent boundary layer flow.Eighty variations of wavelength, period, and amplitude of the space- and time-dependent sinusoidal wall motion are considered for a boundary layer at a momentum thickness based Reynolds number of {rm Re}_theta = 1000. The results show a strong decrease of friction drag of up to 26% and considerable net power saving of up to 10%. However, the highest net power saving does not occur at the maximum drag reduction. The drag reduction is modeled as a function of the actuation parameters by support vector regression using the LES data.A dependence of the spanwise pressure drag on the wavelength is found. A substantial attenuation of the near-wall turbulence intensity and especially a weakening of the near-wall velocity streaks are observed. Similarities between the current actuation technique and the method of a spanwise oscillating wall without any normal surface deflection are reported. In particular, the generation of a directional spanwise oscillating Stokes layer is found to be related to skin-friction reduction.

Reynolds Number Dependence of Zero Pressure Gradient Turbulent Boundary Layers Including Third-Order Moments and Spatial Correlations

Abstract Measurements were made in zero pressure gradient turbulent boundary layers on a smooth wall, at momentum thickness Reynolds numbers, ranging from 800 to 6340. The experiments were conducted in a recirculating water tunnel. Two-component velocity profiles were acquired using laser Doppler velocimetry at five streamwise stations and three different freestream velocities. Velocity field measurements were acquired using particle image velocimetry in streamwise-wall normal and streamwise–spanwise planes. Profiles of mean velocity and turbulence statistics including the Reynolds normal and shear stresses, and triple products of the velocity fluctuations are presented in both inner and outer coordinates. Variations in the profiles at representative distances from the wall are presented and quantified as functions of Reynolds number. The triple products are explained in terms of transport of Reynolds stresses though motions associated with quadrant analysis, and variation with Reynolds number is consistent with that of Reynolds stresses. The structure of turbulence was considered through two-point correlations of the fluctuations in velocity fields. In general, the shape and inclination angles of the structures did not change with Reynolds number, but some streamwise and spanwise growth was observed as Reynolds number increased.

On mean flow universality of turbulent wall flows. II. Asymptotic flow analysis

ABSTRACTUnderstanding of the structure of turbulent flows at extreme Reynolds numbers (Re) is relevant because of several reasons: almost all turbulence theories are only valid in the high Re limit, and most turbulent flows of practical relevance are characterized by very high Re. Specific questions about wall-bounded turbulent flows at extreme Re concern the asymptotic laws of the mean velocity and turbulence statistics, their universality, the convergence of statistics towards their asymptotic profiles, and the overall physical flow organization. In extension of recent studies focusing on the mean flow at moderate and relatively high Re, the latter questions are addressed with respect to three canonical wall-bounded flows (channel flow, pipe flow, and the zero-pressure gradient turbulent boundary layer). Main results reported here are the asymptotic logarithmic law for the mean velocity and corresponding scale-separation laws for bulk flow properties, the Reynolds shear stress, the turbulence production and turbulent viscosity. A scaling analysis indicates that the establishment of a self-similar turbulence state is the condition for the development of a strict logarithmic velocity profile. The resulting overall physical flow structure at extreme Re is discussed.

Influence of spanwise transversal surface wave on coherent structures in turbulent boundary layers

Coherent structures in an actively drag-reduced zero-pressure gradient turbulent boundary layer are investigated using particle-image velocimetry (PIV) and micro-particle-tracking velocimetry (μ-PTV). The friction drag reduction is achieved by spanwise traveling waves via wall-normal deflection of an aluminum surface generated by an electromagnetic actuator system. For a momentum thickness based Reynolds number Reθ=1200 flow, the measurements yield a maximum local drag reduction of 5.3%. The coherent structures in the streamwise vertical and streamwise horizontal planes are measured by phase-averaged PIV. The spanwise traveling transversal surface waves manipulate the coherent structures by interrupting the regeneration cycle of the wall-bounded turbulence. The Reynolds stresses are enhanced in the outer part of the boundary layer above the wave trough. The turbulence production is shifted off the wall during the upward motion of the wall due to the wall-normal momentum induced by the transversal surface waves. This finding is substantiated by the distributions of the wall-normal vorticity fluctuations and two-point correlations. A secondary spanwise flow induced by the waves generates a net spanwise fluid transport in the direction of the wave propagation. This secondary flow prevents the low-speed streaks to follow the downward motion. It rearranges them in the outer region of the boundary layer resulting in the shift of the near-wall turbulence production off the wall.

On mean flow universality of turbulent wall flows. I. High Reynolds number flow analysis

ABSTRACTThe universality and mathematical physical structure of wall-bounded turbulent flows is a topic of discussions over many decades. There is no agreement about questions like what is the physical mean flow structure, how universal is it, and how universal are theoretical concepts for local and global flow variations. These questions are addressed by using latest direct numerical simulation (DNS) data at moderate Reynolds numbers Re and experimental data up to extreme Re. The mean flow structure is explained by analytical models for three canonical wall-bounded turbulent flows (channel flow, pipe flow, and the zero-pressure gradient turbulent boundary layer). Thorough comparisons with DNS and experimental data provide support for the validity of models. Criteria for veritable physics derived from observations are suggested. It is shown that the models presented satisfy these criteria. A probabilistic interpretation of the mean flow structure shows that the physical constraints of equal entropies and equally likely mean velocity values in a region unaffected by boundary effects impose a universal log-law structure. The structure of wall-bounded turbulent flows is much more universal than previously expected. There is no discrepancy between local logarithmic velocity variations and global friction law and bulk velocity variations. Flow effects are limited to the minimum: the difference of having a bounded or unbounded domain, and the variation range of mean velocity values allowed by the geometry.

Parametric investigation of friction drag reduction in turbulent flow over a flexible wall undergoing spanwise transversal traveling waves

Active friction drag reduction by spanwise transversal traveling surface waves is investigated experimentally in a fully developed zero-pressure gradient (ZPG) turbulent boundary layer (TBL). The spanwise transversal traveling wave of an aluminum surface is generated by an electromagnetic actuator system. A parametric study focusing on the influence of the wave amplitude (A+) and wave period (T+) is performed to analyze the impact of the wave parameters on drag reduction. Within the range of the parameters investigated, the maximum local drag reduction of 4.5% is found at A+ = 11.8 and T+ = 110. Furthermore, the TBL flows above the wave crest and trough are investigated by phase-locked PIV and µ-PTV measurements. The results evidence that the drag reduction effect is not only enhanced by increasing the amplitude, but also by reducing the period in the range of the current parameters. The turbulence statistics show that the velocity fluctuations and the Reynolds shear stresses in the streamwise and in the wall-normal direction are damped by the traveling surface wave motion in the near-wall region. The outer velocity distribution deviates from the inner scaling based on the actuated friction velocity, i.e., it possesses a slight tendency of a varying slope in the log region. The phase-locked measurements of the velocity profiles above the crest and the trough show that only above the crest the inner scaling property is valid. Above the moving surface a non-zero spanwise secondary flow is induced. The quadrant decomposition of the turbulent productions shows that the sweep and ejection events are weakened.

Experimental measurement of wall shear stress in strongly disrupted flows

ABSTRACTMean and fluctuating wall shear stress is measured in strongly disrupted cases generated by various low-porosity wall-mounted single- and multi-scale fences. These grids generate a highly turbulent wake which interacts with the wall-bounded flow modifying the wall shear stress properties. Measurement methods are validated first against a naturally growing zero pressure gradient turbulent boundary layer showing accuracies of 1% and 4% for extrapolation and direct measurement of the mean shear stress respectively. Uncertainty associated with the root mean square level of the fluctuations is better than 2% making it possible to measure small variations originating from the different fences. Additionally, probability density functions and spectra are also measured providing further insight into the flow physics. Measurement of shear stress in the disrupted cases (grid+TBL) suggest that the flow characteristics and turbulence mechanisms remain unaltered far from the grid even in the most disrupted cases. However, a different root mean square level of the fluctuations is found for different grids. Study of the probability density functions seem to imply that there are different degrees of interaction between the inner and outer regions of the flow.

Turbulent boundary layer separation control using plasma actuator at Reynolds number 2000000

An experimental investigation was conducted to evaluate the effect of symmetrical plasma actuators on turbulent boundary layer separation control at high Reynolds number. Compared with the traditional control method of plasma actuator, the whole test model was made of aluminum and acted as a covered electrode of the symmetrical plasma actuator. The experimental study of plasma actuators’ effect on surrounding air, a canonical zero-pressure gradient turbulent boundary, was carried out using particle image velocimetry (PIV) and laser Doppler velocimetry (LDV) in the 0.75m×0.75m low speed wind tunnel to reveal the symmetrical plasma actuator characterization in an external flow. A half model of wing-body configuration was experimentally investigated in the ∅ 3.2m low speed wind tunnel with a six-component strain gauge balance and PIV. The results show that the turbulent boundary layer separation of wing can be obviously suppressed and the maximum lift coefficient is improved at high Reynolds number with the symmetrical plasma actuator. It turns out that the maximum lift coefficient increased by approximately 8.98% and the stall angle of attack was delayed by approximately 2° at Reynolds number 2×106. The effective mechanism for the turbulent separation control by the symmetrical plasma actuators is to induce the vortex near the wing surface which could create the relatively large-scale disturbance and promote momentum mixing between low speed flow and main flow regions.

Turbulent drag reduction by spanwise traveling ribbed surface waves

The combination of passive and active drag reduction techniques is investigated by particle-image velocimetry (PIV) and micro-particle tracking velocimetry (μ-PTV) measurements. To be more precise, the impact of a riblet-structured surface, which represents the passive control technique, undergoing spanwise transversal wave motion, which defines the active flow control method, on the wall-shear stress distribution of a zero-pressure gradient turbulent boundary layer is analyzed. The experimental setup consists of a flat plate equipped with an insert to generate a spanwise traveling wave on a riblet-structured aluminum sheet. The PIV and μ-PTV measurements are conducted downstream of the actuated riblet surface at two momentum thickness based Reynolds numbers Reθ=1200 and 2080. The surface deformation is generated by an electromagnetic actuator system at three amplitudes A= 0.25, 0.30, and 0.375 mm with a wavelength of λ=160 mm and a frequency of f=81 Hz. The nondimensional quantities in inner wall units defined by the friction velocity of the non-actuated flat plate flow at Reθ=1200 are the amplitudes A+=6, 7, and 9, the wavelength λ+=3862, and the frequency T+=110 and the corresponding quantities for Reθ=2800 are A+=11, 14 and 17, λ+=7170, and T+=380. The results are twofold. On the one hand, the combination of the riblet surface with the transversal wave motion increases the local drag reduction (DR). That is, compared to a boundary layer flow over a non-actuated smooth surface a local drag reduction of up to DR =9.4 % is determined at the chosen measurement position whereas the drag reduction due to the riblet surface is DR =4.7%. On the other hand, the dependence of the local drag reducing efficiency of the riblets on their geometric dimensions with respect to the flow field could be decreased. That is, the transversal wave motion extends the positive drag reducing efficiency range of the riblets.

Subgrid-scale turbulence in shock–boundary layer flows

Data generated by direct numerical simulation (DNS) for a Mach 2.75 zero-pressure gradient turbulent boundary layer interacting with shocks of different intensities are used for a priori analysis of subgrid-scale (SGS) turbulence and various terms in the compressible filtered Navier–Stokes equations. The numerical method used for DNS is based on a hybrid scheme that uses a non-dissipative central scheme in the shock-free turbulent regions and a robust monotonicity-preserving scheme in the shock regions. The behavior of SGS stresses and their components, namely Leonard, Cross and Reynolds components, is examined in various regions of the flow for different shock intensities and filter widths. The backscatter in various regions of the flow is found to be significant only instantaneously, while the ensemble-averaged statistics indicate no significant backscatter. The budgets for the SGS kinetic energy equation are examined for a better understanding of shock–tubulence interactions at the subgrid level and also with the aim of providing useful information for one-equation LES models. A term-by-term analysis of SGS terms in the filtered total energy equation indicate that while each term in this equation is significant by itself, the net contribution by all of them is relatively small. This observation is consistent with our a posteriori analysis.