# Statistical Structure and Deviations from Equilibrium in Wavy Channel Turbulence

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Simulation Design

- The major focus of the current work is on slope-dependent dynamics without presence of significant separation. This is part of a broader analysis theme where we expect to characterize how separation related dynamics impact turbulence structure differently from when significant separation exists.
- At the same time, we wanted to realize high enough Reynolds numbers to characterize well known turbulent behavior, and yet,
- minimize computational/storage cost.

#### 2.2. Convergence of Turbulence Statistics

#### 2.3. Assessment of Simulation Accuracy

## 3. Results

#### 3.1. Streamwise Averaging of Turbulence Statistics

#### 3.2. Outer Layer Similarity and Mean Velocity Profiles

#### 3.3. Quantification of Mean Velocity Gradients and Inertial Sublayer

#### 3.4. Characterization of the Roughness Function and Roughness Scales

#### 3.5. Characterization of Horizontal Flow Stress and Implications to Drag

#### 3.6. Characterization of Reynolds Stress Tensor and its Production

#### 3.6.1. Streamwise Variance

#### 3.6.2. Vertical Variance

#### 3.6.3. Spanwise Variance

#### 3.6.4. Mean Turbulent Kinetic Energy

#### 3.6.5. Characterization of Reynolds Stress Anisotropy

#### 3.6.6. Vertical Turbulent Momentum Flux

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DNS | Direct numerical simulation |

IBM | Immersed boundary method |

TKE | Turbulent kinetic energy |

TBL | Turbulent boundary layer |

PPE | Pressure Poisson equation |

6OCCS | 6th-order central compact scheme |

KMM87 | Kim, Moin, and Moser (1987) [39] |

## Appendix A

#### Appendix A.1. Nikuradse’s Correlations

#### Appendix A.2. Colebrook’s Correlation

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**Figure 1.**Illustration of 1D polynomial reconstruction based on Lagrangian polynomial. In (

**a**), we show the 2D grid distribution with the immersed boundary while (

**b**) illustrates the 1D velocity reconstruction along the vertical line enclosed by the dashed rectangle in (a).

**Figure 2.**Schematic illustration of the Cartesian grid with the immersed boundaries of different shapes (

**a**) and a close-up of the buffer region (

**b**). The solid thick curve represents the wave for $\lambda =4\pi $ and the dashed line for $\lambda =\frac{8\pi}{3}$. A similar setup is used for other surface shapes as well.

**Figure 3.**Quantification of statistical stationarity for the different DNS datasets using the residual of mean horizontal stress from 2500 samples collected over ~$12\frac{\delta}{{u}_{\tau}}$.

**Figure 4.**Comparison of mean velocity and RMS velocity fluctuation between DNS of flat channel turbulent flow with IBM and the Kim et al. [39] DNS without IBM.

**Figure 5.**Comparison of instantaneous flow separation for the different wave steepness, $\zeta $. The wavy surface is denoted in cyan and the separation is denoted in red. The coordinate system (

**a**) is kept consistent (

**b**,

**c**).

**Figure 6.**Inner scaled mean (

**a**) streamwise velocity, (

**b**) vertical velocity, and (

**c**) defect velocity computed using local coordinate-based average; zoomed version near the surface (

**d**). The thick lines represent averaging at constant ${y}_{local,1}$ and the thin lines with markers represent averaging at scaled ${y}_{local,2}$. Three vertical straight lines correspond to the different ${a}^{+}$ for $\zeta >0$ (see Table 1).

**Figure 7.**Spanwise and temporally averaged (

**a**) streamwise velocity, (

**b**) vertical velocity, and (

**c**) spanwise vorticity over wavy surfaces in turbulent channel flow.

**Figure 8.**Variation of nondimensional mean streamwise velocity gradients (

**a**) $\gamma ={y}^{+}\frac{d{\langle u\rangle}_{x,z,t}^{+}}{d{y}^{+}}$ and (

**b**) $\mathsf{\Phi}=\frac{\kappa y}{{u}_{\tau}}\frac{d{\langle u\rangle}_{x,z,t}}{dy}$. The thin dashed black line (

**a**) corresponds to the mean $\gamma $ valued 2.5604 computed based on ${y}^{+}=60-110$.

**Figure 9.**Schematic illustration of the wall-normal variation of streamwise averaged production of TKE in (

**a**) inner variable nondimensionalized and (

**b**) dimensional (${m}^{2}/{s}^{3}$) forms. Corresponding nondimensional and dimensional forms of TKE dissipation are shown (

**c**,

**d**).

**Figure 10.**Variation of the different roughness quantifications with $\zeta $ (

**a**–

**c**) and wall normal variation of mean roughness function (

**d**).

**Figure 11.**Variation of mean roughness function (

**a**) with roughness Reynolds number and (

**b**) with effective slope in comparison with reported data from known literature. In (

**a**) we compare the roughness function from current DNS with the correlations of Nikuradse [3],Colebrook [4] and the fully rough asymptote (see Appendix A for the correlations). In (

**b**) we show comparison with data from Napoli et al. [17] and Schultz et al. [12].

**Figure 12.**The schematic shows the inner scaled mean (

**a**) horizontal stress, (

**b**) viscous stress, and (

**c**) Reynolds stress in the top row and the dimensional mean (

**d**) horizontal stress, (

**e**) viscous stress, and (

**f**) Reynolds stress in the bottom row. The vertical lines correspond to the different ${a}^{+}$ values.

**Figure 13.**Inner scaled mean (

**a**) streamwise variance and (

**b**) production of streamwise variance, ${\langle {P}_{11}\rangle}_{x}^{+}$. We further split the streamwise production term into its dominant components, (

**c**) ${\langle {P}_{11}^{{u}^{\prime}{u}^{\prime}}\rangle}_{x}^{+}$ and (

**d**) ${\langle {P}_{11}^{{u}^{\prime}{v}^{\prime}}\rangle}_{x}^{+}$. The horizontal lines correspond to height with maximum value of the statistics along the profile.

**Figure 14.**Inner scaled mean (

**a**) covariance ${\langle {u}^{\prime}{v}^{\prime}\rangle}_{x,z,t}^{+}$, (

**b**) covariance ${\langle {u}^{\prime}{v}^{\prime}\rangle}_{x,z,t}^{+}$ (zoomed near the surface), and (

**c**) vertical gradient of streamwise velocity, $d{\langle {u}^{+}\rangle}_{x,z,t}/d{y}^{+}$. The black horizontal line corresponds to the average of the maximum magnitude of ${\langle {u}^{\prime}{v}^{\prime}\rangle}_{x,z,t}^{+}$ for the different $\zeta $. Note that the individual peak values were too close to each other to be shown separately.

**Figure 15.**Inner-scaled mean (

**a**) vertical variance and (

**b**) production of vertical variance, ${\langle {P}_{22}\rangle}_{x}^{+}$. We further split the vertical production term into its cominant components, (

**c**) ${\langle {P}_{22}^{{u}^{\prime}{u}^{\prime}}\rangle}_{x}^{+}$ and (

**d**) ${\langle {P}_{22}^{{u}^{\prime}{v}^{\prime}}\rangle}_{x}^{+}$. The horizontal lines correspond to height with maximum value of the statistics along the profile.

**Figure 16.**Inner scaled mean (

**a**) spanwise variance and (

**b**) turbulent kinetic energy (TKE). The horizontal lines correspond to height with maximum value of the statistics along the profile.

**Figure 17.**(

**a**) Representation of the Reynolds stress structure in a Lumley triangle on the plane of invariants $\xi $ and $\eta $ of the Reynolds stress anisotropy tensor. Lines and vertices correspond to different states of turbulent stresses (i.e., iso, axi, 1C, and 2C represent isotropic, axisymmetric, one-component, and two-component states of turbulence, respectively). (

**b**) Zoomed version (

**a**) near the 1C corner; (

**c**) even more zoomed to clearly illustrate the trend.

**Figure 18.**Schematic illustration of the wall-normal variation of the inner–scaled, streamwise-averaged diagonal elements of pressure–rate-of-strain tensor, ${\langle {\mathcal{R}}_{11}\rangle}^{+}$ in $\left(\mathbf{a}\right)$, ${\langle {\mathcal{R}}_{22}\rangle}^{+}$ in $\left(\mathbf{b}\right)$ and ${\langle {\mathcal{R}}_{33}\rangle}^{+}$ in $\left(\mathbf{c}\right)$.

**Table 1.**Tabulation of different design parameters for the simulations such as: wavelength ($\lambda $), amplitude (a), and steepness ($\zeta =\frac{2a}{\lambda}$) of the wavy surface, friction velocity (${u}_{\tau}$), Reynolds numbers ($Re$) based on boundary layer height ($\delta $), and different velocities expressed as the subscripts (‘cl’ = centerline velocity; ‘b’ = bulk velocity; ‘$\tau $’ = friction velocity) and the grid spacing in different directions (’$\Delta x$’ = streamwise; ‘$\Delta z$’ = spanwise; ‘$\Delta {y}_{w}$’ = wall normal near the wall; ‘$\Delta {y}_{cl}$’ = wall normal near the flow centerline). Superscript ‘+’ refers to inner scaled quantity (scaled with respect to dynamic viscosity ($\nu $) and friction velocity (${u}_{\tau}$)).

Case | $\mathit{\lambda}$ | ${\mathit{\lambda}}^{+}$ | ${\mathbf{a}}^{+}$ | $\mathit{\zeta}$ | $\mathbf{\Delta}{\mathbf{x}}^{+}$ | $\mathbf{\Delta}{\mathbf{y}}_{\mathbf{w}}^{+}$ | $\mathbf{\Delta}{\mathbf{y}}_{\mathbf{cl}}^{+}$ | $\mathbf{\Delta}{\mathit{z}}^{+}$ | ${\mathit{Re}}_{\mathit{cl}}$ | ${\mathit{Re}}_{\mathit{b}}$ | ${\mathit{Re}}_{\mathit{\tau}}$ | ${\mathit{u}}_{\mathit{\tau}}\times {10}^{3}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

A | ∞ | ∞ | 0 | 0 | 8.94 | 1.05 | 2.00 | 4.55 | 3263 | 2800 | 180.9 | 43.07 |

B | $4\pi $ | 2354 | 13.07 | 0.011 | 9.23 | 1.15 | 2.25 | 4.70 | 3277 | 2800 | 186.8 | 44.48 |

C | $\frac{8}{3}\pi $ | 1618 | 13.48 | 0.017 | 6.34 | 1.19 | 2.32 | 4.84 | 3285 | 2800 | 192.6 | 45.85 |

D | $2\pi $ | 1252 | 13.92 | 0.022 | 9.82 | 1.23 | 2.39 | 5.00 | 3298 | 2800 | 198.7 | 47.32 |

**Table 2.**Tabulation of estimated turbulence parameters, namely, von Kármán constants for the different cases and commonly used roughness parameters.

$\mathit{\zeta}$ | $\mathit{\kappa}$ | $\langle \mathbf{\Delta}\langle \mathit{u}\rangle $${}_{\mathit{x},\mathit{z},\mathit{t}}^{+}$〉${}_{\mathit{y}}$ | ${\mathit{k}}_{\mathit{s}}^{+}$ | ${\mathit{k}}_{0}^{+}$ | $\mathit{a}+$ | ${\mathit{\lambda}}^{+}$ |
---|---|---|---|---|---|---|

0.000 | 0.3954 | 0.0000 | 3.2838 | 0.1139 | 0.0000 | ∞ |

0.011 | 0.3805 | 0.8150 | 4.5867 | 0.1592 | 13.070 | 2354 |

0.017 | 0.3839 | 1.3242 | 5.6514 | 0.1961 | 13.480 | 1618 |

0.022 | 0.4033 | 1.7655 | 6.7725 | 0.2350 | 13.920 | 1252 |

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**MDPI and ACS Style**

Khan, S.; Jayaraman, B.
Statistical Structure and Deviations from Equilibrium in Wavy Channel Turbulence. *Fluids* **2019**, *4*, 161.
https://doi.org/10.3390/fluids4030161

**AMA Style**

Khan S, Jayaraman B.
Statistical Structure and Deviations from Equilibrium in Wavy Channel Turbulence. *Fluids*. 2019; 4(3):161.
https://doi.org/10.3390/fluids4030161

**Chicago/Turabian Style**

Khan, Saadbin, and Balaji Jayaraman.
2019. "Statistical Structure and Deviations from Equilibrium in Wavy Channel Turbulence" *Fluids* 4, no. 3: 161.
https://doi.org/10.3390/fluids4030161