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  • 1.
    Saemi, Simindokht
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran.
    Sundström, Joel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Cervantes, Michel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Raisee, Mehrdad
    Hydraulic Machinery Research Institute, School of Mechanical Engineering, College of Engineering, University of Tehran, Iran.
    Evaluation of transient effects in the pressure-time method2019In: Flow Measurement and Instrumentation, ISSN 0955-5986, E-ISSN 1873-6998, Vol. 68, article id 101581Article in journal (Refereed)
    Abstract [en]

    The pressure-time is a method for measuring the flow rate in closed conduits and is typically used in hydropower applications. The scope of the present paper is to examine the flow physics in the pressure-time method using experimental measurements and two-dimensional numerical simulations. The Unsteady Reynolds-averaged Navier–Stokes (URANS) equations and the low-Re k-ω SST turbulence model are employed for the simulations. The contributions of inertia, pressure gradient, viscous and turbulent shear stresses are investigated in the flow during a pressure-time measurement. It is shown that away from the wall and at the first times, the turbulent shear stress balances with the pressure gradient. By increasing the time, the inertia effect becomes dominant and balances with the pressure gradient and turbulent shear stress. Close to the wall, both viscous and turbulent shear stresses are the dominant terms which are decreasing by increasing the time. It is also shown that the prediction of the friction losses can be improved by modeling the dependence of the friction factor on the dimensionless parameter instead of the Reynolds number.

  • 2.
    Sundström, Joel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Studies of Transient and Pulsating flows with application to Hydropower2018Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The rotational motion of a hydraulic turbine runner makes pulsating flows ubiquitous in different locations of the machine. The cyclic loading thus induced may generate large pressure forces acting periodically on both stationary and rotating parts. In addition to the presence of pulsating flows in a turbine runner, transient flows are encountered at an increasingly higher rate due to the continual installation of intermittent sources of renewable energy, such as wind and solar power. To mitigate the imbalance that these unpredictable sources induce on the frequency of the electrical grid, hydropower turbines are enforced to regulate their power production, and consequently flow rate, thus leaving them to operate under transient conditions. In terms of wear and fatigue, a startup or shutdown of a hydraulic turbine corresponds to 10-20 hours of steady state operation at the design point. Transient operation of a hydraulic machine can, however, also be used in favour for measuring the discharge through the turbine using the pressure-time method. A better understanding of pulsating and transient flows thus has the potential both to mitigate problems associated with them, and to increase the accuracy with which the turbine flow rate can be measured; two great merits for the hydropower community. In light of this observation, the following work constitutes a fundamental investigation of transient and pulsating flows performed in a straight pipe.Studies have been performed experimentally using particle image velocimetry, hot-film anemometry, laser Doppler velocimetry and pressure sensors.

    A chief finding is that the time-development of the wall shear stress and near-wall turbulence fields exhibit significant similarity between transient and pulsating flows, despite the different conditions of the mean flow. Whereas the former is initiated from a statistically steady state, the latter is constantly subjected to a time-varying forcing. Both types of unsteady flows have previously been investigated in detail; however, any potential similarity between them has, largely, been unexplored. An important implication of this finding, then, is that knowledge acquired in one type of unsteady flow can be used, if not interchangeably, at least as a guidance for the expected behaviour in the other type of flow. An example is the development of unsteady turbulence models. Another important finding is that the frictional losses arising during the late stage of a pressure-time flow rate measurement can be accurately modelled using an analytical laminar formulation of the wall shear stress, despite the bulk of the flow being turbulent. The formulation of the wall shear stress has potential to be further improved by incorporating a damping-function.

  • 3.
    Sundström, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Amiri, Kaveh
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Bergan, Carl
    Norwegian University of Science and Technology (NTNU), Trondheim.
    Cervantes, Michel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Dahlhaug, Ole Gunnar
    Department of Energy and Process Engineering, Water Power Laboratory, Norwegian University of Science and Technology, Trondheim, Norwegian University of Science and Technology (NTNU), Trondheim.
    LDA measurements in the Francis-99 draft tube cone2014In: 27th IAHR Symposium on Hydraulic Machinery and Systems, IAHR:: Montreal, Canada, 22-26 September 2014, IOP Publishing Ltd , 2014, article id 22012Conference paper (Refereed)
    Abstract [en]

    Velocity measurements were performed in the draft tube cone of a 1:5.1 scaled model of the Tokke hydropower plant, Norway; also known as the Francis-99 model. Results from the laser Doppler anemometry measurements undertaken at three operating points will be used as validation data for an upcoming workshop on the state of the art of Francis turbine numerical simulation. With the turbine operating at the best efficiency point, a sensitivity analysis of the flow parameters head, flow rate and runner rotational speed shows that the effects on the dimensionless velocity profiles are small as long as nED and QED are held constant. The results indicate a well-functioning turbine at the best efficiency point and high load. At the part load operating point, a vortex breakdown occurs which distorts the velocity profiles and significantly lowers the turbine’s hydraulic efficiency. Frequency spectrums of each LDA signal at part load reveals a peak which is asynchronous to that of the runner angular speed. The peaks might be related to the precession of a rotating vortex rope but the characteristics of the LDA signals are different compared to previous studies involving rotating vortex ropes.

  • 4.
    Sundström, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Cervantes, Michel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Evaluation of the volumetric methods with application to low head hydraulic machines2013In: 16th International Flow Measurement Conference 2013: Flomeko 2013 ; Paris, France, September 24 - 26, Red Hook, NY: Curran Associates, Inc., 2013, p. 278-281Conference paper (Refereed)
    Abstract [en]

    The introduction of electricity certificates has increased the interest in development of flow measuring techniques to verify efficiency improvements after refurbishments in Swedish low head hydraulic machines. At Luleå Univeristy of Technology Jonsson [1] and Lövgren [2] have been working with development of the pressure-time method. The present work presents an evaluation of a flow measuring method known as the volumetric method. The method consists in running a turbine at a constant load during an extended period of time which depends upon the reservoir geometry. The flow rate through the turbine is determined from the volume change in the reservoir(s) by measuring the water level change. Through a pilot study in a full scale machine it is shown that the method reproduces reasonable results. The measured flow rate deviates by less than 3% from the value reported in the hydropower plant. The accuracy of the reported flow rate and the one measured in here is however difficult to determine and left unconsidered in this work. For future studies of the method it is recommended to more thoroughly investigate how accurately the rate of change of water level along with the reservoir area can be determined.

  • 5.
    Sundström, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Cervantes, Michel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Laminar similarities between accelerating and decelerating turbulent flows2018In: International Journal of Heat and Fluid Flow, ISSN 0142-727X, E-ISSN 1879-2278, Vol. 71, p. 13-26Article in journal (Refereed)
    Abstract [en]

    An experimental study of a pipe flow ramping monotonically between two turbulent states has been undertaken. Ensemble-averaged mean and turbulent flow quantities obtained from two-component particle image velocimetry and hot-film anemometry measurements have been presented. It is shown that the initial developments of the mean and turbulence quantities in linearly as well as impulsively accelerating and decelerating flows are similar. Specifically, the mean perturbation velocity (defined as the surplus/deficit from the initial value) can be described using self-similar expressions. The duration of this initial stage, when normalized appropriately, is shown to be approximately invariant of the type of transient imposed on the bulk flow. Data from studies of linearly accelerating and decelerating flows as well as impulsively accelerating and decelerating flows have been used to validate the results, covering four orders of magnitude of the dimensionless parameter . The highest initial Reynolds number (31,000) is, however, relatively low thus requiring further studies at high Reynolds numbers to assure the universality of the results. We have also shown that the time-development of the mean and turbulent quantities between an accelerating and a decelerating flow looses their similarity as the transient proceeds beyond the initial stage. The departure was explained by the time-evolvement of the production of turbulence kinetic energy, which exhibit differences between the two types of transients.

  • 6.
    Sundström, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Cervantes, Michel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    The response of the wall shear stress in uniformly and nonuniformly accelerating pipe flows2017Conference paper (Refereed)
    Abstract [en]

    Wall shear stress measurements in an accelerating turbulent pipe flow have been performed. Four different imposed accelerations have been studied including one uniform and three nonuniform ones. The initial-to-final Reynolds numbers as well as the acceleration time were approximately the same for each case, thereby isolating the effect of the type of acceleration. Data have been taken using hot-film anemometry, and it has been established that the time-development for each case is qualitatively similar, although there are significant quantitative differences between each case. The previously established view that the time-development of an accelerating flow resemble a laminar-to-turbulent bypass transition is confirmed. An explanation for the transitional behavior is sought through the Poisson equation describing the pressure fluctuations. It is postulated that the fast pressure induces asymmetries in the time-development of the wall-normal velocity fluctuations thereby leading to a route to transition. Due to lack of data, however, the proposed explanation cannot be confirmed or rejected, for that, further experimental as well as numerical studies have to be performed

  • 7.
    Sundström, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Cervantes, Michel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    The self-similarity of wall-bounded temporally accelerating turbulent flows2018In: Journal of turbulence, ISSN 1468-5248, E-ISSN 1468-5248, Vol. 19, no 1, p. 49-60Article in journal (Refereed)
    Abstract [en]

    From the study of viscous flow it is known that certain time-dependent laminar problems, such as the impulsively started flat plate and the diffusion of a vortex sheet, possess self-similar solutions. Previous studies of turbulent channel and pipe flows accelerating between two steady states have shown that the flow field evolves in three distinct stages. Furthermore, recent direct numerical simulations have shown that the perturbation velocity, i.e. the surplus velocity from the initial value, in an impulsively accelerating turbulent channel and pipe flow also possesses a self-similar distribution during the initial stage. In here, these results are developed analytically and it is shown that accelerating flows in which the centreline velocity develops as Uc(t) = U0(t/t0)m will possess a self-similar velocity distribution during the initial stage. The displacement thickness of the perturbation velocity is shown to be dependent only on the type of acceleration, and not on the initial Reynolds number, the acceleration rate or the change in Reynolds number. The derived formulas are verified with good agreement against measurements performed in a linearly accelerating turbulent pipe flow and with data from channel flow simulations.

  • 8.
    Sundström, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Cervantes, Michel
    Department of Energy and Process Engineering, Norwegian University of Science and Technology.
    Transient wall shear stress measurements and estimates at high Reynolds numbers2017In: Flow Measurement and Instrumentation, ISSN 0955-5986, E-ISSN 1873-6998, Vol. 58, p. 112-119Article in journal (Refereed)
    Abstract [en]

    Transient wall shear stress measurements using hot-film anemometry have been performed in a large-scale laboratory setup at high Reynolds numbers. Starting from Reynolds numbers 1.7×1061.7×106 and 0.7×1060.7×106, the flow was brought to a complete rest by closing a knife gate thus replicating a pressure-time (also know as Gibson) flow rate measurement in a hydropower plant. Ensemble-averaged mean wall shear stresses obtained from 22 repeated runs have been compared with estimates obtained using the pressure-time method. The objective of the work has been to assess the accuracy of the frictional formulation entering the pressure-time integral. It is shown that both the standard method, a quasi-steady approach as well as the recently introduced unsteady method all reproduce the measured wall shear stresses quantitatively during most of the transient. The last phase, following the complete closure of the gate, which is characterized by a slow decay towards zero shear stress at the wall is, however, not captured by the available methods. In general, the unsteady formulation produces the smallest flow rate estimation error, which in turn, implies the best modeling of the frictional losses.

  • 9.
    Sundström, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Cervantes, Michel J.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Department of Energy and Process Engineering, Norwegian University of Science and Technology, Trondheim, Norway.
    On the Similarity of Pulsating and Accelerating Turbulent Pipe Flows2018In: Flow Turbulence and Combustion, ISSN 1386-6184, E-ISSN 1573-1987, Vol. 100, no 2, p. 417-436Article in journal (Refereed)
    Abstract [en]

    The near-wall region of an unsteady turbulent pipe flow has been investigated experimentally using hot-film anemometry and two-component particle image velocimetry. The imposed unsteadiness has been pulsating, i.e., when a non-zero mean turbulent flow is perturbed by sinusoidal oscillations, and near-uniformly accelerating in which the mean flow ramped monotonically between two turbulent states. Previous studies of accelerating flows have shown that the time evolution between the two turbulent states occurs in three stages. The first stage is associated with a minimal response of the Reynolds shear stress and the ensemble-averaged mean flow evolves essentially akin to a laminar flow undergoing the same change in flow rate. During the second stage, the turbulence responds rapidly to the new flow conditions set by the acceleration and the laminar-like behavior rapidly disappears. During the final stage, the flow adapts to the conditions set by the final Reynolds number. In here, it is shown that the time-development of the ensemble-averaged wall shear stress and turbulence during the accelerating phase of a pulsating flow bears marked similarity to the first two stages of time-development exhibited by a near-uniformly accelerating flow. The stage-like time-development is observed even for a very low forcing frequency; ω+=ων/u¯τ2=0.00073" role="presentation">ω+=ων/u¯¯¯2τ=0.00073 (or equivalently, ls+=2/ω+=52" role="presentation">l+s=2/ω+−−−−√=52), at an amplitude of pulsation of 0.5. Some previous studies have considered the flow to be quasi-steady at ls+=52" role="presentation">l+s=52; however, the forcing amplitude has been smaller in those studies. The importance of the forcing amplitude is reinforced by the time-development of the ensemble-averaged turbulence field. For, the near-wall response of the Reynolds stresses showed a dependence on the amplitude of pulsation. Thus, it appears to exist a need to seek alternative similarity parameters, taking the amplitude of pulsation into account, if the response of different flow quantities in a pulsating flow are to be classified correctly.

  • 10.
    Sundström, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Mulu, Berhanu
    Vattenfall Research and Development, Älvkarleby.
    Cervantes, Michel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Wall friction and velocity measurements in a double-frequency pulsating turbulent flow2016In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 788, p. 521-548Article in journal (Refereed)
    Abstract [en]

    Wall shear stress measurements employing a hot-film sensor along with laser Doppler velocimetry measurements of the axial and tangential velocity and turbulence profiles in a pulsating turbulent pipe flow are presented. Time-mean and phase-averaged results are derived from measurements performed at pulsation frequencies ω+ = ων/u¯τ 2 over the range of 0.003-0.03, covering the low-frequency, intermediate and quasi-laminar regimes. In addition to the base case of a single pulsation imposed on the mean flow, the study also investigates the flow response when two pulsations are superimposed simultaneously. The measurements from the base case show that, when the pulsation belongs to the quasi-laminar regime, the oscillating flow tends towards a laminar state in which the velocity approaches the purely viscous Stokes solution with a low level of turbulence. For ω+ < 0.006, the oscillating flow is turbulent and exhibits a region with a logarithmic velocity distribution and a collapse of the turbulence intensities, similar to the time-averaged counterparts. In the low-frequency regime, the oscillating wall shear stress is shown to be directly proportional to the Stokes length normalized in wall units ls + (=√2/ω+), as predicted by quasi-steady theory. The base case measurements are used as a reference when evaluating the data from the double-frequency case and the oscillating quantities are shown to be close to superpositions from the base case. The previously established view that the time-averaged quantities are unaffected by the imposition of small-amplitude pulsed unsteadiness is shown to hold also when two pulsations are superposed on the mean flow

  • 11.
    Sundström, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Mulu, Berhanu
    Cervantes, Michel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Wall friction and velocity measurements in multiple frequencies pulsating flow2014Conference paper (Refereed)
    Abstract [en]

    Turbulent pulsating flow in a 100 mm diameter pipe has been studied experimentally at a Reynolds number of 14,500. The work covers four different flow conditions; steady, pulsating flow at oscillation frequencies 0.08 and 0.4 Hz, and pulsating flow with simultaneous imposition of both frequencies. The amplitudes of the pulsations were 10 and 7.5% of the bulk flow at 0.08 and 0.4 Hz, respectively. Laser Doppler anemometry, hot-film and pressure measurements show that the mean values of velocity, wall shear stress and pressure gradient are unaffected by the imposed pulsations. In agreement with previous studies of pulsating flow, the phase averaged pressure gradient leads both the velocity and wall shear stress when a single pulsation is imposed on the mean flow. These phase leads remain virtually unchanged when the two frequencies are imposed simultaneously. The amplitude responses of the velocity, wall shear stress and pressure gradient in the combined pulsating flow is shown to be superpositions of the amplitudes from the cases of separate pulsations. There are no signs of non-linear interactions between the harmonics.

  • 12.
    Sundström, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Saemi, Simindokht
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran.
    Raisee, M.
    Hydraulic Machinery Research Institute, School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran.
    Cervantes, Michel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Improved frictional modeling for the pressure-time method2019In: Flow Measurement and Instrumentation, ISSN 0955-5986, E-ISSN 1873-6998, Vol. 69, article id 101604Article in journal (Refereed)
    Abstract [en]

    The pressure-time method is classified as a primary method for measuring discharge in hydraulic machinery. The uncertainty in the discharge determined using the pressure-time method is typically around ±1.5 %; however, despite dating back almost one hundred years in time, there still exists potential to reduce this uncertainty. In this paper, an improvement of the pressure-time method is suggested by implementing a novel formulation to model the frictional losses arising in the evaluation procedure. By analyzing previously obtained data from CFD, laboratory and full-scale pressure-time measurements it is shown that the new friction model improves the accuracy of the flow rate calculation by approximately 0.1–0.2% points, compared to currently utilized friction models. Despite being a small absolute improvement, the new friction model presents an important development of the pressure-time method because the relative improvement is significant.

  • 13.
    Sundström, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Saemi, Simindokht
    School of Mechanical Engineering, College of Engineering, University of Tehran.
    Raisee, Mehrdad
    Hydraulic Machinery Research Institute, School of Mechanical Engineering, College of Engineering, University of Tehran.
    Cervantes, Michel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Improved frictional modelling for the pressure-time methoManuscript (preprint) (Other academic)
    Abstract [en]

    The pressure-time method is classified as a primary method for measuring discharge in hydraulic machinery. The uncertainty in the discharge determined using the pressure-time method is typically around %; however, despite dating back almost one hundred years in time, there still exists potential to reduce this uncertainty. In this paper, an improvement of the pressure-time method is suggested by implementing a novel formulation to model the frictional losses arising in the evaluation procedure. By analyzing previously obtained data from CFD, laboratory and full-scale pressure-time measurements it is shown that the new friction model improves the accuracy of the flow rate calculation by approximately 0.1-0.2 percentage points, compared to currently utilized friction models. Despite being a small absolute improvement, the new friction model presents an important development of the pressure-time method because the relative improvement is significant.

  • 14.
    Sundström, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Saemi, Simindokht
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Raisee, Mehrdad
    Cervantes, Michel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Improved frictional modelling for the pressure-time method2019In: Flow measurement and instrumentationArticle in journal (Refereed)
    Abstract [en]

    The pressure-time method is classified as a primary method for measuring discharge in hydraulic machinery. The uncertainty in the discharge determined using the pressuretime method is typically around±1.5%; however, despite dating back almost one hundred years in time, there still exists potential to reduce this uncertainty. In this paper, an improvement of the pressure-time method is suggested by implementing a novel formulation to model the frictional losses arising in the evaluation procedure. By analyzing previously obtained data from CFD, laboratory and full-scale pressure-time measurements it is shown that the new friction model improves the accuracy of the flow rate calculation by approximately 0.1-0.2 percentage points, compared to currently utilized friction models. Despite being a small absolute improvement, the new friction model presents an important development of the pressure-time method because the relative improvement is significant.

  • 15.
    Sundström, L.R. Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Cervantes, Michel
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Department of Energy and Process Engineering, Norwegian University of Science and Technology.
    Characteristics of the wall shear stress in pulsating wall-bounded turbulent flows2018In: Experimental Thermal and Fluid Science, ISSN 0894-1777, E-ISSN 1879-2286, Vol. 96, p. 257-265Article in journal (Refereed)
    Abstract [en]

    A pulsating turbulent pipe flow has been investigated experimentally using hot-film anemometry and particle image velocimetry. Particularly, a `paradoxical phenomenon' that is known to occur for a range of forcing frequencies and time-averaged Reynolds numbers has been investigated in detail. The paradoxical phenomenon is that the oscillating component of the wall shear stress exhibits a smaller amplitude in a turbulent flow compared to in a laminar flow exposed to the same oscillation in the pressure gradient. In here, the phenomenon is explained by splitting the response of the wall shear stress into one contribution resulting from the imposed pressure gradient , and a second contribution resulting from the oscillating Reynolds shear stress, At the conditions of maximum reduction of the wall shear stress amplitude,  and  are nearly 136 degrees out of phase. The contributions are thus interfering destructively, this being the ultimate reason for the reduced amplitude. It is also shown that the level of reduction is dependent on the imposed forcing amplitude, this in turn residing from a dependence of the time-development of the oscillating Reynolds shear stress on the forcing amplitude.

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