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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.

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 [

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 [

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 [

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 [

∂ u i ∂ x i = 0 (1)

∂ u i ∂ t + ∂ u i u j ∂ x j = − ∂ p ∂ x i + 1 R e c ∂ 2 u i ∂ x j ∂ x j − R o c ε i j k Ω j Ω u k (2)

∂ θ ∂ t + ∂ ( θ u j ) ∂ x j = 1 R e c P r ∂ 2 θ ∂ x i ∂ x j (3)

where R e c = u c δ / ν , P r = ν / κ , ν is the kinematic viscosity, κ is the thermal 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

R o c = 2 Ω δ / u c

where Ω is the spanwise angular rotation vector. p is the non-dimensional effective pressure ( p = p 0 − ( 1 / 8 ) R o c 2 r 2 ) which combines the static pressure ( p 0 ) and centrifugal force, r represents the nondimensional distance away from the axis of rotation. Also, t is non-dimensional time and θ is non-dimensional temperature ( θ = [ T − T L ] / [ T U − T L ] ) with T U and T L representing dimensional temperatures on the upper and lower walls, respectively. Equation (2) is uncoupled from Equation (3) and buoyancy effects are neglected. The Prandtl number ( P r ) was kept constant at 0.71.

The flow geometry of the present DNS is shown in

scheme is a modified semi-implicit Adams-Bashforth/Crank-Nicolson method, which is second order accurate in time and fourth order accurate in space. More information on the code scheme and verification may be found in [

The full listing of simulations and their corresponding domain lengths and grid resolutions are found in the case descriptions (

u b = 1 2 δ ∫ 0 2 δ u ¯ d y (4)

where u ¯ denotes a plane-averaged quantity. The domain lengths for the simulations were selected as L x = 4 π δ , L y = 2 δ and L z = 2 π δ such that two-point spatial autocorrelations in the streamwise and spanwise directions converged to zero at the largest separations.

Mesh independence was established by designating two high-resolution cases with grid numbers n x × 2 n y × n z for case A ( R o b = 0 ) and 2 n x × n y × 2 n z for case D ( R o b = 0.9 ). For both cases, the distributions of mean velocity, Reynolds stresses and turbulent kinetic energy budgets compared very favorably to those of the original simulations, demonstrating the selected meshes of the present DNS cases were mesh invariant. The grid spacing is also comparable to other DNS studies of spanwise-rotating turbulent channel flow [

Case | R e τ | R o b | n x × n y × n z |
---|---|---|---|

A | 200 | 0 | 256 × 129 × 256 |

B | 197 | 0.2 | 256 × 129 × 256 |

C | 192 | 0.5 | 256 × 129 × 256 |

D | 183 | 0.9 | 256 × 383 × 256 |

E | 406 | 0.2 | 512 × 513 × 512 |

Superscript + refers to nondimensionalization by the friction velocity, u τ = ν ∂ u ¯ / ∂ y | wall or friction temperature, T τ = ( κ / u τ ) ∂ θ / ∂ y | wall . The global friction velocity for the rotational cases is denoted as u τ = u τ s 2 / 2 + u τ p 2 / 2 where u τ s and u τ p are the local friction velocities at the suction and pressure walls, respectively; an equivalent calculation of the global friction temperature was used for the thermal statistics. Unless otherwise specified, all coordinate directions are non-dimensionalized by δ . For all cases, the governing equations were integrated until both the friction Reynolds number R e τ and friction temperature converged availing a sufficiently long time window ( t + = t u τ 2 / ν ≥ 1000 ) to calculate statistics.

The data base used in the present work is the same one used in [

In this section, visualizations from turbulent channel flow simulation case A are used to substantiate the theoretical model predictions proposed by [

The Landahl model proposes the formation of sublayers streaks, or streamwise elongated u ′ structures, is a consequence of algebraic instabilities commonly found in turbulent flows [

The present DNS results are also used to evaluate the coherent structure formation scheme proposed in the experimental study by [

The study of [

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 ( y + = 15 ). To obtain

All three patterns demonstrate a structural inclination of approximately θ = 5 ∘ in accordance with the theory of [

disturbed region and more discernable wavy structure in correspondence with the modeled results in

Similarities with the theoretical model of [

The [

velocity and the yellow hairpin vortex is composed from high levels of combined spanwise ( ω ′ z ) and wall-normal vorticity ( ω ′ y ). The smaller yellow vortical structures aligned with the streak in the spanwise direction denote high levels of streamwise vorticity ( ω ′ x ) and the vorticity field is filtered such that the coherent structures are isolated from one another.

Both views of the streak in

To visualize the spanwise-propagating plane waves proposed in [

u i = ∑ q = 1 n y a q ϕ i q (5)

where a q and ϕ i q ( y , z ) 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

a n = 1 2 δ ∫ 0 2 δ u i ϕ i q d y (6)

with satisfaction of the orthonormality condition for the eigenfunctions. The average mean energy of the velocity field is defined by

E i i = 1 2 δ ∫ 0 2 δ 1 2 u i 2 d y (7)

and through substitution of u i in Equation (5), the contribution of energy from various N modes is shown through the partial sum

E N = ∑ q = 1 N 1 2 a q 2 (8)

Using the above relations, an eigenvalue problem is created using the two-point autocorrelation tensor R i j

∫ 0 2 δ R i j ϕ j q d y = λ ϕ i q (9)

and if a structure contributes energy to the Reynolds stress tensor, it will dominate the two-point correlation statistics and manifest in the POD [

R i j ( r x , y , y ′ , r z , t ) = 〈 u i ( x , y , z , t ) u j ( x + r x , y ′ , z + r z , t ) 〉 (10)

where r x and r z represent the two-point separations x − x ′ and z − z ′ in the streamwise and spanwise directions, respectively. The brackets 〈 〉 denote ensemble averaging in time and the homogeneous x and z directions.

For multi-dimensional POD analysis and application to three-dimensional turbulent channel flow, it is fitting to convert the two-point correlation tensor R i j into the spectral density correlation tensor Φ i j [

Φ i j ( k x , y , y ′ , k z ) = 1 4 π 2 ∫ ∫ e − i k x r x − i k z r z R i j ( r x , y , y ′ , r z ) d r x d r z (11)

such that the flow field may be expressed as a function of streamwise ( k x ) and spanwise wavenumbers ( k z ). For all wavenumber combinations, a Φ matrix of dimensions 3 n y × 3 n y is assembled

Φ = [ Φ 11 Φ 12 Φ 13 Φ 21 Φ 22 Φ 23 Φ 31 Φ 32 Φ 33 ] (12)

Hence, a new eigenvalue problem ( A ϕ = λ ϕ ) 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 [

∫ 0 2 δ Φ d y = ∑ i = 1 n y ω i Φ i = 1 2 ∑ i = 2 n y ( y i − y i − 1 ) [ Φ ( y i − 1 ) + Φ ( y i ) ] (13)

where Φ i is the value of Φ at a discrete grid point and ω i is the corresponding weight function. To apply standard numerical eigenproblem solution techniques, it is required that the matrix-valued function a, or Φ D in the eigenproblem, be symmetric. This is accomplished through the following convolution

Φ D ϕ = D Φ D ϕ = λ ϕ (14)

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 1 , k 3 , 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 [

In

To visualize the presence and interactions of these propagating plane waves, [

Index | Sirovich | Case A | ||
---|---|---|---|---|

(m,n,q) | Energy Frac. | (m,n,q) | Energy Frac. | |

1 | (0,3,1) | 0.0428 | (0,2,1) | 0.0484 |

2 | (0,1,1) | 0.0399 | (0,1,1) | 0.0299 |

3 | (0,4,1) | 0.0327 | (0,4,1) | 0.0286 |

4 | (0,5,1) | 0.0287 | (0,3,1) | 0.0246 |

5 | (0,4,2) | 0.0229 | (0,5,1) | 0.0221 |

6 | (0,1,2) | 0.0210 | (0,6,1) | 0.0208 |

7 | (0,3,2) | 0.0206 | (0,4,2) | 0.0144 |

8 | (0,2,1) | 0.0197 | (0,7,1) | 0.0136 |

9 | (0,2,2) | 0.0188 | (1,7,1) | 0.0116 |

10 | (0,6,1) | 0.0138 | (1,4,1) | 0.0105 |

11 | (0,5,2) | 0.0131 | (1,3,1) | 0.0100 |

12 | (1,3,1) | 0.0125 | (1,3,2) | 0.0092 |

13 | (1,2,1) | 0.0095 | (1,1,1) | 0.0092 |

14 | (1,4,1) | 0.0084 | (1,2,1) | 0.0092 |

15 | (1,5,1) | 0.0083 | (0,8,1) | 0.0090 |

[

In this section, the DNS data base from simulation cases A ( R o b = 0 ), B ( R o b = 0.2 ), C ( R o b = 0.5 ) and D ( R o b = 0.9 ) 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.

Rotational effects on the mean velocity and temperature profiles are shown for simulation cases A-D in

near-wall temperature gradients, is broader near the suction wall than the pressure wall for rotating simulation cases B-D. As the rotation number increases, the size of the diffusive layer in the suction region increases and the mean temperature profile shifts towards the pressure wall [

In

To illustrate the effects of rotation on wall shear stress and heat transfer, the dimensionless friction Reynolds ( R e τ ) and Nusselt ( N u ) numbers for both channel walls are provided for the present simulation cases A-D in

Case | ( R e τ ) s | ( R e τ ) p | N u s | N u p |
---|---|---|---|---|

A | 200 | 200 | 6.8 | 6.8 |

B | 150 | 235 | 5.7 | 5.8 |

C | 138 | 235 | 4.3 | 4.6 |

D | 137 | 219 | 3.2 | 3.4 |

[

A strong correlation between streamwise velocity and temperature fluctuations is also observed. In

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:

S ( u ′ i ) = u ′ i u ′ i u ′ i ¯ ( u ′ i u ′ i ¯ ) 3 / 2 (15)

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

fluctuations of similar strength. The flatness (F), also known as the kurtosis, represents the fourth-order moment of a fluctuating velocity component normalized by the square of its corresponding Reynolds stress component:

F ( u ′ i ) = u ′ i u ′ i u ′ u ′ i i ¯ ( u ′ i u ′ i ¯ ) 2 (16)

The flatness is the measure of a variable’s peakedness and represents the frequency at which extreme events occur as a deviation from the Gaussian distribution ( F = 3 ). For example, a high value of flatness ( F > 3 ) indicates relatively large values at the edges of the PDF distribution and a higher concentration directly around the mean. In [

In

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 values for cases B ( R o b = 0.2 ) and C ( R o b = 0.2 ) from the non-rotational case but a decrease for case D ( R o b = 0.9 ).

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 [

D u ′ i u ′ j ¯ D t = P i j + Π i j − ε i j + C i j + D i j T (17)

with the terms on the right-hand side of Equation (17) representing, respectively, the production ( P i j ), pressure-strain ( Π i j ), dissipation ( ε i j ), Coriolis ( C i j ) and diffusion terms ( D i j T ).

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 [

P i j = − u ′ i u ′ k ¯ ∂ U ¯ j ∂ x k − u ′ j u ′ k ¯ ∂ U ¯ i ∂ x k (18)

C i j = 2 Ω k ( u ′ i u ′ m ¯ ϵ m j k − ϵ i m k u ′ m u ′ j ¯ ) (19)

Π i j = − ( u ′ i ∂ p ′ ∂ x j + u ′ j ∂ p ′ ∂ x i ¯ ) (20)

and the total (summation of all tensor components) distributions are shown in Figures 12(b)-(d), respectively.

In

In

In this section, visualizations of various coherent structures are extracted from DNS cases A-C to ascertain the role of these structures to turbulence production amidst contributions from rotational forces. Although the bursting cycle is maintained in the pressure region [

streaks as these structures are composed of significant concentrations, and therefore variations, of streamwise fluctuating velocity u ′ [

With the introduction of spanwise rotation in case B ( R o b = 0.2 ), the wall-normal vorticity field in

In

System rotation also generates rotation-induced structures in the pressure region in the form of a spanwise array of longitudinal roll cells, known as Taylor-Gortler vortices [

The three-dimensional structure of the Taylor-Gortler vortices is shown across the entire channel in

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 v ′ > 0 signifies motion away from the wall, provides important information on contributions to TKE production [

For simulation case A ( R o b = 0 ),

| u ′ v ′ | u r m s v r m s ≥ H (21)

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 [

The spatially-averaged (in x and z) quadrant contributions are also shown in

To analyze the phase differences between the streamwise ( u ′ ) and wall-normal ( v ′ ) fluctuating velocities for cases A ( R o b = 0 ) and B ( R o b = 0.2 ), the temporal distributions of u ′ v ′ ¯ are decomposed using Fourier transforms. The separate u ′ and v ′ signals in

and wall-normal fluctuating velocity, which manifests as the two quantities being almost completely in-phase or out-of-phase for all four quadrants, is demonstrated in the suction region for case B ( R o b = 0.2 ). This alignment is also corroborated by the quasi-periodic motions in

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 ( R e τ = 406 ) is examined and compared to the lower-Reynolds number simulation case B ( R e τ = 197 ).

In

In

In

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

In

wall ( y = 1.95 ). Similar to

In

In summary, the theoretical model predictions proposed by [

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 rotational simulation in which notable characteristical changes to various turbulence structures were identified.

This work was supported in part by the Air Force Office of Scientific Research under grant number No. FA9550-15-0495.

The authors declare no conflicts of interest regarding the publication of this paper.

Hsieh, A.S. and Biringen, S. (2019) Effects of Rotation on Turbulence Production. Journal of Applied Mathematics and Physics, 7, 298-330. https://doi.org/10.4236/jamp.2019.72024