^{1}and Hyung Jin Sung

^{1,a)}

### Abstract

A direct numerical simulation of a fully developed turbulent pipe flow was performed to investigate the similarities and differences of very-large-scale motions (VLSMs) to those of turbulent boundary layer (TBL) flows. The Reynolds number was set to Re D = 35 000, and the computational domain was 30 pipe radii in length. Inspection of instantaneous fields, streamwise two-point correlations, and population trends of the momentum regions showed that the streamwise length of the structures in the pipe flow grew continuously beyond the log layer (y/δ < 0.3–0.4) with a large population of long structures (>3δ), and the maximum length of the VLSMs increased up to ∼30δ. Such differences between the TBL and pipe flows arose due to the entrainment of large plumes of the intermittent potential flow in the TBL, creating break-down of the streamwise coherence of the structures above the log layer with the strong swirling strength and Reynolds shear stress. The average streamwise length scale of the pipe flow was approximately 1.5–3.0 times larger than that of the TBL through the log and wake regions. The maximum contribution of the structures to the Reynolds shear stress was observed at approximately 6δ in length, whereas that of the TBL was at 1δ–2δ, indicating a higher contribution of the VLSMs to the Reynolds shear stress in the pipe flow than in the TBL flow.

This work was supported by the Creative Research Initiatives (No. 2012-0000246) program of the National Research Foundation and was partially supported by KISTI under the Strategic Supercomputing Support Program.

I. INTRODUCTION

II. NUMERICAL METHOD

III. VALIDATION OF DNS RESULTS

IV. INSTANTANEOUS ANALYSIS

V. TWO-POINT CORRELATIONS OF THE VELOCITY FLUCTUATIONS

VI. POPULATION TRENDS IN THE MOMENTUM REGIONS

VII. SUMMARY AND CONCLUSIONS

### Key Topics

- Rotating flows
- 32.0
- Channel flows
- 21.0
- Reynolds stress modeling
- 21.0
- Turbulent pipe flows
- 21.0
- Potential flows
- 17.0

## Figures

Mean velocity profiles U + in (a) inner and (b) outer coordinates. Solid line: present DNS; dashed line: Wu et al. 19 at Re D = 24 580; dotted line: Wu and Moin 26 at Re D = 44 000; circles: TBL data, Schlatter and Örlü 27 at Re θ = 3030.

(a)–(d) Turbulence intensities and Reynolds shear stress along the outer coordinates. The legend is the same as in Fig. 1 .

(a)–(d) Turbulence intensities and Reynolds shear stress along the outer coordinates. The legend is the same as in Fig. 1 .

Very-large-scale low-momentum regions with 2D signed swirling strengths (Λ ci = λ ci ω y /|ω y |) and second-quadrant Reynolds shear stress (Q2 event). (a) y/δ = 0.15, (b) y/δ = 0.3, and (c) y/δ = 0.5. Blue and red contours indicate the positive and negative values of Λ ci with magnitudes of 30% of the maximum and minimum values. The luminous contours indicate a Q2 event with magnitude − . The flow is from left to right.

Very-large-scale low-momentum regions with 2D signed swirling strengths (Λ ci = λ ci ω y /|ω y |) and second-quadrant Reynolds shear stress (Q2 event). (a) y/δ = 0.15, (b) y/δ = 0.3, and (c) y/δ = 0.5. Blue and red contours indicate the positive and negative values of Λ ci with magnitudes of 30% of the maximum and minimum values. The luminous contours indicate a Q2 event with magnitude − . The flow is from left to right.

Very-large-scale high-momentum regions with 2D signed swirling strengths (Λ ci = λ ci ω y /|ω y |) and fourth-quadrant Reynolds shear stress (Q4 event). (a) y/δ = 0.15, (b) y/δ = 0.3, and (c) y/δ = 0.5. Blue and red contours indicate the positive and negative values of Λ ci with a magnitude of 30% of the maximum and minimum values. The luminous contours indicate a Q4 event with magnitude − .

Very-large-scale high-momentum regions with 2D signed swirling strengths (Λ ci = λ ci ω y /|ω y |) and fourth-quadrant Reynolds shear stress (Q4 event). (a) y/δ = 0.15, (b) y/δ = 0.3, and (c) y/δ = 0.5. Blue and red contours indicate the positive and negative values of Λ ci with a magnitude of 30% of the maximum and minimum values. The luminous contours indicate a Q4 event with magnitude − .

Intermittency coefficient γ in the boundary layer flow.

Intermittency coefficient γ in the boundary layer flow.

Contour of the vorticity magnitude |ω| in the (a) boundary layer and (b) pipe flows. The contour level employed here is |ω|+ = |ω|ν/u τ 2 = 0.0165.

Contour of the vorticity magnitude |ω| in the (a) boundary layer and (b) pipe flows. The contour level employed here is |ω|+ = |ω|ν/u τ 2 = 0.0165.

Probability density functions of the wall-normal vorticity fluctuations ω y ν/u τ 2. The black solid, dashed, and dotted lines depict the present pipe data at y/δ = 0.15, 0.3, and 0.5, and the circles, squares, and triangles indicate the TBL data of Lee and Sung 5 at the same wall-normal heights, respectively.

Probability density functions of the wall-normal vorticity fluctuations ω y ν/u τ 2. The black solid, dashed, and dotted lines depict the present pipe data at y/δ = 0.15, 0.3, and 0.5, and the circles, squares, and triangles indicate the TBL data of Lee and Sung 5 at the same wall-normal heights, respectively.

The probability density functions for the (a) Q2 and (b) Q4 Reynolds shear stresses normalized by the friction velocity. The legend is the same as in Fig. 7 .

The probability density functions for the (a) Q2 and (b) Q4 Reynolds shear stresses normalized by the friction velocity. The legend is the same as in Fig. 7 .

Spatial correlations of the streamwise velocity fluctuations in the streamwise/spanwise planes at several wall-normal heights: (a) y/δ = 0.1, (b) y/δ = 0.2, (c) y/δ = 0.3, (d) y/δ = 0.4, (e) y/δ = 0.5, (f) y/δ = 0.6, (g) y/δ = 0.7, and (h) y/δ = 0.8. The contour levels were varied from 0.1 to 1.0 with a negative value of −0.05.

Spatial correlations of the streamwise velocity fluctuations in the streamwise/spanwise planes at several wall-normal heights: (a) y/δ = 0.1, (b) y/δ = 0.2, (c) y/δ = 0.3, (d) y/δ = 0.4, (e) y/δ = 0.5, (f) y/δ = 0.6, (g) y/δ = 0.7, and (h) y/δ = 0.8. The contour levels were varied from 0.1 to 1.0 with a negative value of −0.05.

Spatial correlations among the streamwise velocity fluctuations in the spanwise/wall-normal plane in the turbulent pipe flow: (a) y/δ = 0.1, (b) y/δ = 0.2, (c) y/δ = 0.3, (d) y/δ = 0.4, (e) y/δ = 0.5, (f) y/δ = 0.6, (g) y/δ = 0.7, and (h) y/δ = 0.8. The contour levels are the same as those described in Fig. 9 .

Spatial correlations among the streamwise velocity fluctuations in the spanwise/wall-normal plane in the turbulent pipe flow: (a) y/δ = 0.1, (b) y/δ = 0.2, (c) y/δ = 0.3, (d) y/δ = 0.4, (e) y/δ = 0.5, (f) y/δ = 0.6, (g) y/δ = 0.7, and (h) y/δ = 0.8. The contour levels are the same as those described in Fig. 9 .

(a) Streamwise line plot of R uu at y/δ = 0.15 with r z /δ = 0 and (b) variation of the characteristic streamwise length scale based on the range of streamwise displacement where R uu > 0.05.

(a) Streamwise line plot of R uu at y/δ = 0.15 with r z /δ = 0 and (b) variation of the characteristic streamwise length scale based on the range of streamwise displacement where R uu > 0.05.

(a) Spanwise line plot of R uu at y/δ = 0.15 with r x /δ = 0 and (b) variation of the characteristic spanwise length scale based on the range of spanwise displacement where R uu > 0.05.

(a) Spanwise line plot of R uu at y/δ = 0.15 with r x /δ = 0 and (b) variation of the characteristic spanwise length scale based on the range of spanwise displacement where R uu > 0.05.

Conditional two-point correlations of the streamwise velocity fluctuations at y/δ = 0.1, 0.3, 0.5, and 0.7 (top to bottom) for the pipe flow. (a) u > 0 and (b) u < 0. The contour levels are varied from 0.3 to 1.0 with interval of 0.1.

Conditional two-point correlations of the streamwise velocity fluctuations at y/δ = 0.1, 0.3, 0.5, and 0.7 (top to bottom) for the pipe flow. (a) u > 0 and (b) u < 0. The contour levels are varied from 0.3 to 1.0 with interval of 0.1.

Variations in the characteristic (a) streamwise and (b) spanwise length scales based on the range of streamwise and spanwise displacements where R uu > 0.3. Open and closed circles indicate the data of conditioned events u > 0 and u < 0, respectively. For reference, the data from the two-point correlation by the definition in (5.1) is indicated by solid line.

Variations in the characteristic (a) streamwise and (b) spanwise length scales based on the range of streamwise and spanwise displacements where R uu > 0.3. Open and closed circles indicate the data of conditioned events u > 0 and u < 0, respectively. For reference, the data from the two-point correlation by the definition in (5.1) is indicated by solid line.

Comparison of the population trends for (a) the streamwise length and (b) the spanwise width in the pipe and boundary layer flows in the logarithmic layer. The conditional events were the negative streamwise fluctuations (u < 0).

Comparison of the population trends for (a) the streamwise length and (b) the spanwise width in the pipe and boundary layer flows in the logarithmic layer. The conditional events were the negative streamwise fluctuations (u < 0).

Contribution to the total second-quadrant Reynolds shear stress identified by the feature extraction algorithm as a function of the length for both the pipe and boundary layer flows in the logarithmic layer. The length is normalized by the outer unit.

Contribution to the total second-quadrant Reynolds shear stress identified by the feature extraction algorithm as a function of the length for both the pipe and boundary layer flows in the logarithmic layer. The length is normalized by the outer unit.

## Tables

Domain size and mesh resolution.

Domain size and mesh resolution.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content