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Fig. 16 High- and low-speed streaks and the second invariant of the deformation tensor for Re,=180 (u’+<—3.0; light-gray, u’*>3.0; dark-gray, //'t<—0.03; white)

Figure 16 High- and low-speed streaks and the second invariant of the deformation tensor for Re,=180 (u’+<—3.0; light-gray, u’*>3.0; dark-gray, //'t<—0.03; white)

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with them. However, there exists a small but discernible differ- ence between the present and the KMM87 [2] and KMM90 [5] results. This is a reflection of the slight difference in the mean velocity distribution in the channel center region.
with them. However, there exists a small but discernible differ- ence between the present and the KMM87 [2] and KMM90 [5] results. This is a reflection of the slight difference in the mean velocity distribution in the channel center region.
The von Karman constant « can be obtained from Eq. (10) as
The von Karman constant « can be obtained from Eq. (10) as
Table 2 Mean flow variables
Table 2 Mean flow variables
and is plotted in Fig. 3. It is well-known that « obtained from the experiments ranges from 0.40—0.42. The present calculation indi- cates that « is not completely constant but it stays at a roughly constant value of 0.40 around y * =50— 100. Moreover, the region of the approximate constant tends to expand with the increase of the Reynolds number.
and is plotted in Fig. 3. It is well-known that « obtained from the experiments ranges from 0.40—0.42. The present calculation indi- cates that « is not completely constant but it stays at a roughly constant value of 0.40 around y * =50— 100. Moreover, the region of the approximate constant tends to expand with the increase of the Reynolds number.
Table 3 Near-wall expansion coefficient
Table 3 Near-wall expansion coefficient
Fig. 6 Reynolds shear stress and total shear stress distributions
Fig. 6 Reynolds shear stress and total shear stress distributions
Fig. 7 Two-point correlation coefficients of velocity fluctua- tions for Re,=640: (a), (b) streamwise, (c), (d) spanwise corre- lation coefficients
Fig. 7 Two-point correlation coefficients of velocity fluctua- tions for Re,=640: (a), (b) streamwise, (c), (d) spanwise corre- lation coefficients
Fig. 8 Spanwise two-point correlation coefficient R,, at y* =11
Fig. 8 Spanwise two-point correlation coefficient R,, at y* =11
Figure 12 shows the budget terms of the Reynolds stresses for Re,=640 compared with those of Re,=180 and 395. For u’*u'* component, as the Reynolds number becomes higher, the peak value of the production and the wall values of the molecular dif- fusion and dissipation increase. The production almost balances with the some of the dissipation and the velocity pressure-gradient correlation (v.p.g, hereafter) terms. On the other hand, for v'*v'* and w’*w’* components, the v.p.g and dissipation terms are dominant and increase significantly with the increase of the Rey- nolds number. These indicate that the Reynolds-number effect on vty’? and w’*w’* t+ components is more enhanced than that of Pete u'*u ee RI I ! Energy k. Budget terms of Reynolds stress uj u j v/ us are expressed as follows: normalized by
Figure 12 shows the budget terms of the Reynolds stresses for Re,=640 compared with those of Re,=180 and 395. For u’*u'* component, as the Reynolds number becomes higher, the peak value of the production and the wall values of the molecular dif- fusion and dissipation increase. The production almost balances with the some of the dissipation and the velocity pressure-gradient correlation (v.p.g, hereafter) terms. On the other hand, for v'*v'* and w’*w’* components, the v.p.g and dissipation terms are dominant and increase significantly with the increase of the Rey- nolds number. These indicate that the Reynolds-number effect on vty’? and w’*w’* t+ components is more enhanced than that of Pete u'*u ee RI I ! Energy k. Budget terms of Reynolds stress uj u j v/ us are expressed as follows: normalized by
Fig. 11 Premultiplied energy spectra for Re,=640 (4 k,Eyy/u'u’, (b) kyEyy/u'u’ Fig. 10 Streamwise one-dimensional energy dissipation spec- tra normalized by Kolmogorov scale It is well-known that the pressure strain term plays a dominant role on the energy redistribution. Figure 12 indicates that all the components for the pressure strain term exhibit a prominent in- crease with increasing Reynolds number. It is interesting to note that the increase in the pressure strain of u’*u’* from Re, Fig. 9 One-dimensional energy spectra of velocity fluctuations for Re,=640 in comparison with Re,=180: (a) streamwise, (b) spanwise
Fig. 11 Premultiplied energy spectra for Re,=640 (4 k,Eyy/u'u’, (b) kyEyy/u'u’ Fig. 10 Streamwise one-dimensional energy dissipation spec- tra normalized by Kolmogorov scale It is well-known that the pressure strain term plays a dominant role on the energy redistribution. Figure 12 indicates that all the components for the pressure strain term exhibit a prominent in- crease with increasing Reynolds number. It is interesting to note that the increase in the pressure strain of u’*u’* from Re, Fig. 9 One-dimensional energy spectra of velocity fluctuations for Re,=640 in comparison with Re,=180: (a) streamwise, (b) spanwise
term is reduced to the pressure diffusion one due to the continuity condition. Figure 13 indicates that all terms gradually increase with the increase of Re,. Especially, the wall values of the dissi- pation and molecular diffusion increase appreciably with the in- creasing Reynolds number. The peak value of the production term Pe max becomes 0.25 as the Reynolds number goes to the infinity. In the present results, Py max 18 0.218 for Re,=180; while, it reaches 0.239 for Re,=640. The increase in P; wx from Re, =180 to 640 is small but significant. This point will be discussed later together with the pressure strain term. The v.p.g term can be split into the pressure strain and the pressure diffusion terms as follows:
term is reduced to the pressure diffusion one due to the continuity condition. Figure 13 indicates that all terms gradually increase with the increase of Re,. Especially, the wall values of the dissi- pation and molecular diffusion increase appreciably with the in- creasing Reynolds number. The peak value of the production term Pe max becomes 0.25 as the Reynolds number goes to the infinity. In the present results, Py max 18 0.218 for Re,=180; while, it reaches 0.239 for Re,=640. The increase in P; wx from Re, =180 to 640 is small but significant. This point will be discussed later together with the pressure strain term. The v.p.g term can be split into the pressure strain and the pressure diffusion terms as follows:
= 180-640 is about 0.02 around its peak. This value is roughly equal to the increase in the peak of P,. Since P, is a half of the P,,, it means that about half of the increase in the production rate of u’*u'* is consumed by u’*u'* itself and the rest half is transferred to the other components. This is the reason why v/,” and w/.” increase significantly with the increase of the Reynolds number.
= 180-640 is about 0.02 around its peak. This value is roughly equal to the increase in the peak of P,. Since P, is a half of the P,,, it means that about half of the increase in the production rate of u’*u'* is consumed by u’*u'* itself and the rest half is transferred to the other components. This is the reason why v/,” and w/.” increase significantly with the increase of the Reynolds number.
The present results are compared with the above approximation in Fig. 14. The agreement is generally good for all the normal com- ponents. If examined more precisely, however, the agreement is less satisfactory in the central region of i=1 and 2 and also in the near-wall region of i=2. In the near-wall region, referring to Launder and Reynolds [38], the relation of Eq. (23) is exactly valid for i=1 and 3; while, for i=2, the wall asymptotic value of E) becomes
The present results are compared with the above approximation in Fig. 14. The agreement is generally good for all the normal com- ponents. If examined more precisely, however, the agreement is less satisfactory in the central region of i=1 and 2 and also in the near-wall region of i=2. In the near-wall region, referring to Launder and Reynolds [38], the relation of Eq. (23) is exactly valid for i=1 and 3; while, for i=2, the wall asymptotic value of E) becomes
Fig. 14 Dissipation rate of the normal Reynolds stresses for Re,=640 symbol, ¢; by Eq. (23); ——, ujjui; (elk); , (2/3)E; , 4c2y*? To examine the above approximation further, the anisotropy tensors are defined for the Reynolds stress and its dissipation rate as follows:
Fig. 14 Dissipation rate of the normal Reynolds stresses for Re,=640 symbol, ¢; by Eq. (23); ——, ujjui; (elk); , (2/3)E; , 4c2y*? To examine the above approximation further, the anisotropy tensors are defined for the Reynolds stress and its dissipation rate as follows:
Fig. 15 Relation between anisotropy tensors b;, and dj,
Fig. 15 Relation between anisotropy tensors b;, and dj,
Fig. 17 High- and low-speed streaks and the second invariant of the deformation tensor for Re,=640 (u’+<—3.0; light-gray, u’*>3.0; dark-gray, //'t<—0.03; white) As for the streaks, high- and low-speed streaks are obtained for both Re,’s. The low-speed streaks are more elongated than the high-speed ones for both of the Reynolds numbers observed. The spanwise two-point correlation Rj, is generally used to estimate the spacing of the streaks. It is known that the position of the negative peak of R,, provides an estimation of the mean separa- tion between the high- and low-speed streaks; that is, the streak spacing becomes twice of the distance to the negative peak. KMM87 [2] obtained the minimum value of R,, at z* ~50 anc
Fig. 17 High- and low-speed streaks and the second invariant of the deformation tensor for Re,=640 (u’+<—3.0; light-gray, u’*>3.0; dark-gray, //'t<—0.03; white) As for the streaks, high- and low-speed streaks are obtained for both Re,’s. The low-speed streaks are more elongated than the high-speed ones for both of the Reynolds numbers observed. The spanwise two-point correlation Rj, is generally used to estimate the spacing of the streaks. It is known that the position of the negative peak of R,, provides an estimation of the mean separa- tion between the high- and low-speed streaks; that is, the streak spacing becomes twice of the distance to the negative peak. KMM87 [2] obtained the minimum value of R,, at z* ~50 anc
Fig. 18 Mean velocity distribution by fourth-order calculation
Fig. 18 Mean velocity distribution by fourth-order calculation
Fig. 20 One-dimensional energy spectra of velocity fluctuations by fourth-order calculation: (a) streamwise, (b) spanwise Fig. 19 Rms _ of velocity fluctuations by fourth-order calculation
Fig. 20 One-dimensional energy spectra of velocity fluctuations by fourth-order calculation: (a) streamwise, (b) spanwise Fig. 19 Rms _ of velocity fluctuations by fourth-order calculation
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EngineeringPhysicsMechanicsDirect Numerical SimulationChannel Flow
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