Effects of Rotation on Turbulence Production

Direct numerical simulations (DNS) of non-rotating and rotating turbulent channel flow were conducted. The data base obtained from these DNS simulations was used to investigate the prominent coherent structures involved in the turbulence generation cycle. Predictions from three theoretical models concerning the formation and evolution of sublayer streaks, three-dimensional hairpin vortices and propagating plane waves were validated using visualizations from the present DNS data. Quadrant analysis was used to determine a phase shift between the fluctuating streamwise and wall-normal velocities as a characteristic of turbulence production in the suction region at a low rotation number.


Introduction
The scientific field of turbulence has posed long-standing challenges to researchers due to the inherent chaotic and irregular motions which define turbulent flows.Since turbulent flows, not laminar flows, are predominantly found in nature and with the prevalence of rotation-dependent machinery in engineering, a physical understanding of turbulent flow in these systems is a necessity for engineering and scientific analyses.The fundamental structures involved in the turbulence sustenance cycle for simple turbulent channel flow, defined as a turbulent flow field possessing only one mean flow gradient, have been well-documented in experimental [1] and DNS studies [2].Near the channel walls, interactions between streamwise vortices and streamwise elongated sublayer streaks result in ejections of the latter structures from the near-wall region.
The streaks break down into smaller instabilities which are swept back towards the wall and reform into vortices and streaks.This cyclical process is referred to as the bursting cycle and the mechanism through which turbulence is sustained [3].Despite the lack of a constitutive theoretical model for governing the overall structure of the turbulence sustenance cycle, theoretical models have been proposed for selected components of the cycle.Using direct numerical simulation (DNS), it is instructive to assess the validity of the theoretical model predictions.
The present work examines the fundamental structures of turbulence with comparison to the theoretical models proposed by [4], [5] and [6] for the formation and evolution of such structures with the DNS data base.
When turbulent channel flow is subject to rotation in the spanwise direction, rotation-induced body forces (Coriolis, centrifugal) generate secondary flows.
Regions known as the suction and pressure sides emerge, distinguished by reduced and elevated turbulence levels, respectively [7].In the present work, a comprehensive investigation of the effects of rotation on channel turbulence including general turbulence statistics, higher-order statistics, energy budgets and coherent structures, was conducted.In the pressure region, the streaky and vortical structures associated with the turbulence sustenance cycle persist and are the primary structures which contribute towards the energetic ejection and sweep events near the channel walls [2] [4].In the suction region, rotational forces induce flow re-laminarization and hence the turbulence sustenance cycle disappears along with the coherent structures which define it.With the absence of the turbulence generation cycle in the suction region, a sudden shift in phase relationship between streamwise and wall-normal fluctuating velocity was identified and examined using quadrant analyses.Additional aspects of the laminarized flow regime were also investigated including characteristical structures of laminar-to-turbulence transition and new observations into the mechanisms of turbulence production are provided.
The present work also provides an examination of higher-Reynolds number effects on rotational turbulence through direct comparison of lower and higher-Reynolds number simulations subject to spanwise rotation, specifically in regards to the characteristics of the coherent structures which contribute to turbulence production.Consideration of higher-Reynolds number effects is necessary due to the prevalence of higher-Reynolds number flows in engineering applications such as gas turbine blade and rotating turbomachinery design [8].This research has been conducted by members of Prof. Sedat Biringen's computational fluid dynamics group within the Ann and H.J. Smead Department of Aerospace Engineering Sciences at the University of Colorado-Boulder.

Numerical Method and Case Descriptions
The time-dependent, three-dimensional incompressible Navier-Stokes and energy equations were numerically integrated in a doubly periodic (in x and z-directions) channel flow using a fractional step method [9] [10].With all spatial coordinates non-dimensionalized by the channel half-height δ and veloci- ( ) where , ν is the kinematic viscosity, κ is the ther- mal diffusivity and the vector , , u u v w = is composed of three velocity components in the x (streamwise), y (wall-normal) and z (spanwise) directions, respectively.The rotation number (or Rossby number) is defined as where Ω is the spanwise angular rotation vector.p is the non-dimensional effective pressure ( ( )  A. S.More information on the code scheme and verification may be found in [9] and [10].
The full listing of simulations and their corresponding domain lengths and grid resolutions are found in the case descriptions (Table 1).Four simulation cases A-D were conducted for Rossby numbers 0 b Ro = , 0.2, 0.5 and 0.9.An additional higher-Reynolds number simulation case E was conducted for  The data base used in the present work is the same one used in [12] and [13] and more information regarding the verification of simulation cases A-E via comparison to other experimental and computational studies may be found there.
In this section, visualizations from turbulent channel flow simulation case A are used to substantiate the theoretical model predictions proposed by [4] [5] [14].The Landahl model proposes the formation of sublayers streaks, or streamwise elongated u′ structures, is a consequence of algebraic instabilities com- monly found in turbulent flows [15].The streaks are generated by the continuous linear growth of these algebraic instabilities in the streamwise (x) direction, necessitated by a linear temporal growth of total streamwise momentum which continues indefinitely until viscous forces impede growth.[4] used the conditional sampling technique of variable interval time averaging (VITA) to obtain flow visualizations demonstrating the temporal structural evolution suggested by the theoretical model.The model proposed two structure classes which formed from the original algebraic instabilities: symmetrical and asymmetrical structures which correspond with oblique defomation angles in the spanwise direction of 0 θ =  and 5 θ =  , respectively.As the symmetrical structures did not demonstrate streamwise elongation over time, [4] related the asymmetrical structures to sublayer streaks.The asymmetrical structures demonstrated a consistent pattern dominated by a high-speed structure side-by-side in the spanwise direction with a low-speed structure.Once these structures elongated, an irregular wavy appearance was observed consistent with the oscillatory motion of streaks [1].
The present DNS results are also used to evaluate the coherent structure formation scheme proposed in the experimental study by [5] which used Particle Image Velocimetry (PIV) to examine a low-pressure-turbine blade flow regime.In contrast to the study by [4] which studied the initial development and evolution of sublayer streaks, [5] examined the formation of three-dimensional coherent structures which accompanied streak breakdown into turbulence.It was proposed that the breakdown of elongated sublayer streaks in the near-wall region induced three-dimensional vortical structures which manifested on the streak flanks.These large-scale structures were related to hairpin vortices [16], characterized by spanwise vorticity on the top and wall-normal vorticity on the bottom legs.Smaller observed vortical structures, such as vorticity tubes, were proposed to be residuals of the hairpin vortices which contributed to the sinuous motion of the streaks.The breakdown of streaks and these vortical structures generate high velocity fluctuations and lead to the formation of other large-scale turbulence structures.
The study of [14] postulated the existence of secondary instabilities, mainly propagating plane waves, which serve as a trigger for the interactions between these primary turbulence structures.[17] was one of the first to qualitatively analyze the role of these secondary instabilities to the transition process and later studies of [18] [19] supported this mechanism by discovering these secondary instabilities travel obliquely to the streamwise direction and contributed to flow de-stabilization.The flow control studies by [20] and [21] additionally affirmed the proposal by demonstrating that the phase randomization of a small subset of propagating modes, primarily within the energy-containing scales, reduced turbulent kinetic energy and drag by significant amounts.The present work aims to substantiate the low-Reynolds number results ( 120 Re τ = ) of [6] for a turbulent flow field at 200 Re τ = and demonstrate the existence of these propagating waves.

Landahl Model
The Landahl model qualitatively examined the formation and evolution of sublayers streaks in the turbulence system cycle and used the variable interval time averaging (VITA) method to predict structural characteristics suggested by the theoretical model.From the present DNS results in case A, maps of fluctuating streamwise velocity u′ are obtained in the x-z plane of peak turbulent kinetic energy production ( 15 y + = ).To obtain Figure 2(a), the variable interval spatial averaging (VISA) method [22] was applied with an averaging length of 200 wall units in the streamwise and spanwise directions.In accordance with the VISA method, a detection criterion was used to isolate islands of high local u′ va- riance and the space-time position of these islands was tracked to visualize the temporal evolution of sublayer streaks.These VISA-educed structures are compared to the asymmetrical structures obtained from the modeled VITA results of [4] and numerical VISA results of [23] in Figure 2  Similarities with the theoretical model of [4] are also observed in the temporal evolution of VISA-educed structures from the present DNS in Figure 3.The familiar spanwise array of aligned high-speed and low-speed streaks is observed and this pattern demonstrates the expected streamwise advection and elongation for increasing t + .Similar to the modeled asymmetrical structures, the present DNS results show a consistent oscillatory shape with an increasing amount of inflection points over time.

Lengani-Simoni Model
The [5] model proposed three-dimensional coherent structures accompanied the sublayer streaks of turbulence sustenance cycle.These three-dimensional vortical structures manifested as vorticity tubes on the streak flanks or hairpin vortices which envelop the streak.In Figure 4, a three-dimensional contour representation of the near-wall coherent structures is shown for a single low-speed sublayer streak near the bottom channel wall ( 0 y = ) for the periodic channel.The blue streaky structure denotes high levels of negative streamwise fluctuating

Sirovich Model
To visualize the spanwise-propagating plane waves proposed in [6], the method of principal orthogonal decomposition was applied to a periodic channel.Principal orthogonal decomposition (POD), also known as the Karhunen-Loeve decomposition, is a procedure for extracting the coherent motions from two-point velocity correlations which contain the most energy [24].Further detailed in [25], this method is based on the decomposition of the velocity field where q a and ( ) y z φ are the basis-function coefficients and basis functions (eigenfunctions), respectively.q is the quantum number which refers to two-point separations in the inhomogeneous direction y.In addition, the basis-function coefficients correspond to their respective eigenvectors via with satisfaction of the orthonormality condition for the eigenfunctions.The average mean energy of the velocity field is defined by and through substitution of i u in Equation ( 5), the contribution of energy from various N modes is shown through the partial sum Using the above relations, an eigenvalue problem is created using the two-point autocorrelation tensor ij R and if a structure contributes energy to the Reynolds stress tensor, it will domi- For multi-dimensional POD analysis and application to three-dimensional turbulent channel flow, it is fitting to convert the two-point correlation tensor ij R into the spectral density correlation tensor ij Φ [26]   ( ) ( ) (11) such that the flow field may be expressed as a function of streamwise ( x k ) and spanwise wavenumbers ( z k ).For all wavenumber combinations, a Φ matrix of dimensions Hence, a new eigenvalue problem ( Aφ λφ = ) is generated where a, φ and λ represent the integrated spectral density correlation tensor, corresponding eigenfunctions and eigenvalues, respectively.To approximate the integral of Φ in the wall-normal y direction, the weighting function matrix D is calculated using the following trapezoidal numerical approximation [27] ( ) ( ) ( ) where i Φ is the value of Φ at a discrete grid point and i ω is the corres- ponding weight function.To apply standard numerical eigenproblem solution techniques, it is required that the matrix-valued function a, or D Φ in the eigenproblem, be symmetric.This is accomplished through the following convolution Once the D and Φ matrices are created, the resultant eigenproblem is solved to return a system of eigenvalues and eigenfunctions for various modal combinations: ( ) , , k k q .
For analysis, instantaneous fluctuating velocity fields were collected from simulation case A for a large time window.For an accurate and relevant comparison to the results of [6] regarding the wavenumber ranges, these fields were interpolated onto an approximately equivalent domain: In Table 2, the top fifteen energetic modes and their corresponding energy fractions for simulation case A are compared to the results of [6] for a low Reynolds number 125 Re τ = . m and n refer to indexes for the streamwise ) and spanwise wavenumbers ( 3 2π z k n L = ), respectively.It is demonstrated that nonpropagating modes, despite being a small fraction of the total number of modes, are the most energetic.Table 2 also demonstrates identical propagating modes which possess the highest energetical content such as the (1,3,1) and (1,2,1) modes, and that a small range of spanwise wavenumbers (n = 2 -4) captures most of the highly energetic propagating modes.This finding supports the flow control design of [21], which found that randomizing a small range of inertial scales reduced turbulent drag by significant amounts.
To visualize the presence and interactions of these propagating plane waves, [14] derived a frequency and corresponding wave speed for these structures.
Using the methodology of [14] to calculate the wave speed for the most energetic propagating modes from Table 2, we plot a normal speed locus for a discrete number of points for the periodic channel in Figure 5.It is demonstrated that most waves do propagate at an oblique angle to the streamwise direction in accordance with the expectation of [6].(1,3) mode.
[6] additionally proposed a relationship between these plane wave envelopes and other energetic turbulence structures which advected with the mean flow velocity in the near-wall region, mainly the streaky structures and three-dimensional vortices discussed in the previous theoretical models, which form the bursting process.As the energy contained within the propagating modes is relatively small compared to the non-propagating modes (Table 2), these plane wave modes were proposed by [14] to be a triggering mechanism for bursting events with the non-propagating modes providing the energy cascade necessary for the bursts to occur.Through a decomposition of the Reynolds shear stress into separate contributions from the non-propagating and propagating modes, [6] demonstrated the presence of both modes were necessary for strong turbulent activity.These discoveries bode well for future flow control efforts as the present work and [14] have collectively shown that most of the turbulent kinetic energy is contained to a small range of modes.) are examined for effects of rotational forces on turbulence over a wide range of rotation rates.In spanwise-rotating turbulent channel flow, the Coriolis force acts in the wall-normal direction, resulting in asymmetry across the channel and the creation of two distinct flow regimes: the pressure and suction regions.In the pressure region of the channel, secondary flow circulation and high levels of turbulence are present and in the suction region, re-laminarization of the regime results in low levels of turbulence.

General Turbulence and Thermal Statistics
Rotational effects on the mean velocity and temperature profiles are shown for simulation cases A-D in Figure 6 and Figure 7, respectively.In Figure 6, the mean velocity profile is symmetric about the channel centerline ( 1 y = ) for case A ( 0 b Ro = ).With system rotation, the mean velocity distributions become asymmetric as the flow regime is separated into the pressure and suction regions.For rotational cases B-D, a laminar-like (parabolic) profile is observed near the suction wall ( 0 y = ) which is characteristic of the suppressed turbu- lence in the suction region.As the flow progressively relaminarizes with increasing rotation number, the suction region expands.Near the pressure wall − Ω is shown in the mean velocity profiles at all rotation rates which is consistent with previous DNS results in [7] and [11] In  To illustrate the effects of rotation on wall shear stress and heat transfer, the dimensionless friction Reynolds ( Re τ ) and Nusselt ( Nu ) numbers for both channel walls are provided for the present simulation cases A-D in Table 3.The introduction of rotation is shown to initially decrease Re τ on the suction wall while increasing Re τ on the pressure wall.At higher rotation numbers, these Re τ trends are shown to significantly weaken or even reverse in case D ( 0.9 b Ro = ) for Re τ on the pressure wall.These results correspond well with   [7] which demonstrated that Re τ on both walls trended towards convergence at high rotation numbers until the eventual full re-laminarization of the flow regime.For the Nusselt number, a dimensionless number generally used to represent surface heat transfer, increasing system rotation is shown to continually decrease Nu on both channel walls.This trend was similarly observed in [28].
A strong correlation between streamwise velocity and temperature fluctuations is also observed.In Figure 9

Higher-Order Statistics
The third-moment of a fluctuating velocity component normalized by the cube of the root-mean-square (r.m.s.) velocity component is known as the skewness: ( ) ( ) The skewness quantifies the asymmetry of a variable's probability density function (PDF) distribution about its mean and measures extreme events occurring in a velocity field.For example, positive skewness indicates large amplitude positive fluctuations have a greater likelihood for occurrence than negative regardless of rotation number.In the kurtosis distributions, rotation is not shown to significantly alter the profile values in the pressure region even at high rotation numbers.In the suction region, there is a significant increase of kurtosis

Energy Budgets
It is instructive to examine the various energy budget components in order to discover which force dynamics are primarily affected by rotational forces.The Reynolds stress equation displays the intercomponent energy transfer [31] with the terms on the right-hand side of Equation ( 17) representing, respectively, the production In the present work, the production, Coriolis and pressure-strain budgets are investigated due to their high contribution level compared to the other budget terms and correspondence with the turbulence generation cycle [1].These quantities are expressed in tensor form in Equations ( 18), ( 19) and (20).
( ) and the total (summation of all tensor components) distributions are shown in ), the observed peak in Figure 12(a) is demonstrated to be primarily composed of contributions from the other budget terms with a small supplement from P, demonstrating a fundamental alteration to the dominant processes which contribute towards turbulence production which manifests at low rotation numbers.At higher rotation numbers, re-laminarization mechanisms suppress turbulence production in the near-wall region of the suction side to negligible amounts.Journal of Applied Mathematics and Physics   ).The roll cells appear as streamwise-elongated cylindrical structures which persist throughout the pressure region of the channel.The study by [11] showed the number of roll cell pairs increased with increasing rotation number although the wall-normal length of the circulation region is reduced from progressive re-laminarization.The study by [11] showed the number of roll cell pairs increased with increasing rotation number although the wall-normal length of the circulation region is reduced from progressive re-laminarization.Journal of Applied Mathematics and Physics

Quadrant Analysis
Quadrant analysis of the Reynolds shear stress, which divides u v ′ ′ into four quadrants according to the signs of fluctuating streamwise ( u′ ) and wall-normal ( v′ ) velocity where 0 v′ > signifies motion away from the wall, provides im- portant information on contributions to TKE production [36].In Equation ( 18), the Reynolds shear stress is shown to have a significant contribution towards the production (P) term.For non-rotating flow, the second and fourth quadrant events dominate the near-wall region of peak TKE production and at the location of peak production ( 12 y + ≈ ), the contributions from both events are approximately equal [1].The second quadrant event, ( ) For simulation case A ( 0 b Ro = ), Figure 18 shows maps of significant qua- drant events in an x-z cross-section at 0.06 y = , the location of peak production in Figure 12(b).The bursting event criterion recommended by [37] was used to define significant quadrant events, where H is the threshold level.This H value was set to 1 in the present work and chosen to be similar to the selected threshold values in the bursting event studies by [3] and [38].A significantly larger number of powerful second (Q2) and fourth (Q4) quadrant events are observed in comparison to the small number of powerful first (Q1) and third (Q3) quadrant events; the number of significant Q2 and Q4 events is also approximately equal.In Figure 18(b), the streamwise-elongated structures correspond with the sublayer streaks of the turbulence generation cycle.Journal of Applied Mathematics and Physics ).This alignment is also corroborated by the quasi-periodic motions in Figure 20(b) which occur concurrently for all four quadrants.

Rotational Turbulence: Higher-Reynolds Number Effects
In engineering applications, turbulent flows often encounter higher-Reynolds number effects.Hence it is prudent to investigate how turbulence production and the corresponding turbulence structures are different in higher-Reynolds number flows compared to lower-Reynolds number flows.The higher-Reynolds number simulation case E ( 406 Re τ = ) is examined and compared to the lower-Reynolds number simulation case B ( 197 Re τ = ).In Figure 22( ) and E ( 406 Re τ = ), respectively.In Figure 22(a), a higher-Reynolds number is shown to significantly decrease the amplitudes of the mean velocity distribution although other distribution characteristics such as the shape and slope in the pressure region are preserved.The decreased amplitudes were expected due to the significant increase of Re τ (and subsequently u τ ) in simulation case E ( 406 Re τ = ), which affected the scaling of the distribution.In Figure 22(b), the mean temperature distribution for case E ( 406 Re τ = ) is shown to be significantly less asymmetric than the distribution for case B ( 197 Re τ = ) and resembles the distribution for the no-rotation case A shown in Figure 7. ) profile.
In Figure 23(c), the amplitudes of the pressure-strain distribution for case E are shown to be very similar to that of case B with exception of the region near the pressure wall.Similar to the effects of high-rotation numbers, a higher-Reynolds number increases the amplitude of the pressure-strain budget near the pressure wall significantly.In Figure 23(d), the near-wall peaks of the Coriolis energy budget distribution are also shown to shift towards the channel walls with a higher-Reynolds number, similar to the production budget, although there is less amplitude increase compared to the other energy budget distributions.
It is also imperative to look at how turbulence structures in both the pressure and suction regions are altered by higher-Reynolds number effects.In Figure 24, x-z planar contours of fluctuating streamwise velocity are shown for simulation cases B ( 197 Re τ = ) and E ( 406 Re τ = ) in the region near the suction wall ( 0.05 y = ).In Figure 24 ) and E ( 406 Re τ = ) in the region near the pressure ) and E ( 406 Re τ = ).In the regions near both channel walls, the overall turbulence structure is preserved: although the number of structures has increased, elongated slanted structures and clusters of high vorticity concentrations are seen near the suction and pressure walls, respectively.In the center of the channel however, vorticity has clearly increased in the case of simulation case E ( 406 Re τ = ) perhaps resulting from the alterations to the flow dynamics near both channel walls.

Conclusions
In summary, the theoretical model predictions proposed by [4] and [5] for the coherent structures of sublayer streaks and accompanying vortical structures were validated through comparison with the present DNS results.For the [4] model, the appearance and evolution of the sublayer streaks obtained using the VISA method from the DNS data corresponded very well with the modeled VITA results.For the [5] model, snapshots of a sublayer streak and its surrounding vortical structures were obtained from both the periodic and spatial DNS simulations which matched the pictorial representation of the theoretical model.In agreement with the results of [6] for a lower-Reynolds number, non-propagating modes were found to possess the highest energetical content and a normal speed locus was generated in order to visualize the interaction and movement of propagating plane waves.With most of the turbulent kinetic energy contained to a small range of modes, this validation study bodes well for future flow control work.
Since propagating waves have been established in both periodic and spatial models despite the lack of a streamwise wavenumber in the spatial channel, flow control designs in future work should aim to inhibit these particular modes to reduce turbulent kinetic energy and drag.
An examination of the coherent structures which contribute to turbulence production for various rotation rates is of great importance to understand the effects of rotational forces on turbulence.In the present work, a comprehensive investigation involving low and high-order statistics, coherent structures, budget analysis and quadrant analysis was conducted.Quadrant analysis was used in the suction region to elucidate a phase shift in the relationship between the streamwise and wall-normal fluctuating velocities, a significant finding as these components form the principal contribution to turbulence production and are involved in the ejection and sweep events which dominate the near-wall turbulence generation cycle.This investigation of the contributions of coherent structures to turbulence production was extended for a higher-Reynolds number ro-

Figure 1 .
Figure 1.Geometry of DNS computational domain with rotation in the spanwise direction.
examine higher-Reynolds number effects on turbulence production with rotation.For simulation cases A-D, the Reynolds number based on the laminar centerline velocity, 8000 c Re = , was kept constant which resulted in a friction Reynolds number 200 Re τ = for the no-rotation case ( 0 b Ro = ).The low Reynolds number was chosen such that reasonable comparisons could be made between the present DNS and other DNS studies of rotating turbulent channel flow with similar Reynolds numbers [7] [11].The asymmetric velocity distributions due to the rotational effects decreased the value of Re τ which was calculated as an average between the two walls.Simulations were performed at constant mass flux which resulted in a bulk Reynolds number u denotes a plane-averaged quantity.The domain lengths for the simuspatial autocorrelations in the streamwise and spanwise directions converged to zero at the largest separations.
(b) and Figure 2(c), respectively.All three patterns demonstrate a structural inclination of approximately 5 θ =  in accordance with the theory of [4].The spanwise lengths of the streak patterns are also shown to be very similar at approximately 100 z + = , corresponding to the accepted mean spacing between sublayer streaks [23].The numerical and modeled results display a comparable streamwise streak length of approximately 500 x + = , demonstrating the three streak patterns are in a similar stage of development.The present DNS results in Figure 2(a) show more similarity with the theoretical model of [4], demonstrating an abrupt onset of the

Figure 4 .
Figure 4.An isolated three-dimensional field of coherent structures in the near-wall region for simulation case A. Blue contours: sublayer streak; yellow contours: vorticity field.(a) Front view; (b) Top view.
the previously outlined procedure for periodic POD analysis, eigenvalues and eigenfunctions were obtained for a large number of modes.A. S. Hsieh, S. Biringen DOI: 10.4236/jamp.2019.72024308 Journal of Applied Mathematics and Physics

3. 1 .
IntroductionIn this section, the DNS data base from simulation cases A (

Figure 7 ,Figure 6 .
Figure 7, the thickness of the thermal diffusive sublayer, characterized by large

Figure 10 (
Figure 10(a) and Figure 10(b), respectively.Re-laminarization of the suction region is demonstrated to suppress significant fluctuations of both quantities near the suction wall.Near the pressure wall, arrays of coherent u′ and θ ′ structures are observed.In Figure 10(a), the sublayer streaks are elongated in the wall-normal direction, corresponding to increased v′ for case C compared to case A (Figure 8(b)) in the pressure region.The thermal structures exhibit similar characteristics and continue to correlate with the sublayer streaks in the presence of rotational effects.

Figure 9 .
Figure 9. Contours in the y-z plane for simulation case A ( 0 b Ro = ).(a) Streamwise

Figure 10 .
Figure 10.Contours in the y-z plane for simulation case C ( 0.5 b Ro = ).(a) Streamwise

Figures 12 (
Figures 12(b)-(d), respectively.In Figure 12(a), the turbulent kinetic energy (k) distributions for simulation cases A-D are shown.The expected suppression of k-amplitudes in the suction region is demonstrated for case B ( 0.2 b Ro = ), but a significant peak near the suction wall continues to persist despite the elimination of the turbulence sustenance cycle; no near-wall peak is observed for cases C ( 0.5 b Ro = ) and D

Figure 15 .
Figure 15.Instantaneous z ω′ map for a x-y section at π z = for case B ( 0.2 b Ro =).

Figure 16 .
Figure 16.(a) Time-averaged v and w velocity vectors for a y-z section at 2π x = for

Figure 17 .
Figure 17.Three-dimensional contours of time-averaged v and w velocity for case C ( 0.5 b Ro = ).Blue and yellow contours denote clockwise and counter-clockwise motion, respectively.
0 v′ > ), contains the motion attributed with ejections of low-speed fluid away from the wall.The fourth quadrant event, ( ) 0 v′ < ), contains the motion attributed to an inrush of high-speed fluid into the wall region.In the present work, quadrant analysis is used to compare cases A ( to the relationship between u′ and v′ whose outer product forms the dominant contribution towards the near-wall P peaks observed in Figure12(b).
(b), the number of powerful fluctuations comprising the turbulent spots have increased significantly for simulation case E ( 406 Re τ = ) and the shape of the overall structure is much more defined.In Figure 25, contours of fluctuating streamwise velocity are shown for simulation cases B ( 197 Re τ =

Table 1 .
Case descriptions and initial conditions.
[26]point correlation statistics and manifest in the POD[25].For the periodic channel with periodicity in the streamwise (x) and spanwise (z) directions, the three-dimensional two-point correlation tensor is[26] Hsieh, S. Biringen DOI: 10.4236/jamp.2019.72024307 Journal of Applied Mathematics and Physics nate the

Table 2 .
Energy content of the first 15 eigenfunctions obtained from POD analysis of si- 0090 Journal of Applied Mathematics and Physics

Table 3 .
Friction Reynolds ( Re τ ) and Nusselt ( Nu ) numbers for present DNS cases A-D.