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Turbulent secondary flows in wall turbulence: vortex forcing, scaling arguments, and similarity solution

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Abstract

Spanwise surface heterogeneity beneath high-Reynolds number, fully-rough wall turbulence is known to induce a mean secondary flow in the form of counter-rotating streamwise vortices—this arrangement is prevalent, for example, in open-channel flows relevant to hydraulic engineering. These counter-rotating vortices flank regions of predominant excess(deficit) in mean streamwise velocity and downwelling(upwelling) in mean vertical velocity. The secondary flows have been definitively attributed to the lower surface conditions, and are now known to be a manifestation of Prandtl’s secondary flow of the second kind—driven and sustained by spatial heterogeneity of components of the turbulent (Reynolds averaged) stress tensor (Anderson et al. J Fluid Mech 768:316–347, 2015). The spacing between adjacent surface heterogeneities serves as a control on the spatial extent of the counter-rotating cells, while their intensity is controlled by the spanwise gradient in imposed drag (where larger gradients associated with more dramatic transitions in roughness induce stronger cells). In this work, we have performed an order of magnitude analysis of the mean (Reynolds averaged) transport equation for streamwise vorticity, which has revealed the scaling dependence of streamwise circulation intensity upon characteristics of the problem. The scaling arguments are supported by a recent numerical parametric study on the effect of spacing. Then, we demonstrate that mean streamwise velocity can be predicted a priori via a similarity solution to the mean streamwise vorticity transport equation. A vortex forcing term has been used to represent the effects of spanwise topographic heterogeneity within the flow. Efficacy of the vortex forcing term was established with a series of large-eddy simulation cases wherein vortex forcing model parameters were altered to capture different values of spanwise spacing, all of which demonstrate that the model can impose the effects of spanwise topographic heterogeneity (absent the need to actually model roughness elements); these results also justify use of the vortex forcing model in the similarity solution.

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Acknowledgements

This work was supported by the U.S. Air Force Office of Scientific Research, Grant # FA9550-14-1-0394 (WA, JY) and Grant # FA9550-14-1-0101 (WA, AA), and by the Texas General Land Office, Contract # 16-019-0009283 (WA, KS).

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Correspondence to William Anderson.

Appendices

Appendix A: Datapoints from Yang and Anderson (2017)

Figure 1 showed datapoints of compensated circulation from Yang and Anderson [32]. For that study, LES with an IBM was used to model flow over a series of topographies composed of streamwise-aligned, vertically-truncated pyramid obstacles. For the simulations, we used the LES code already described in Sect. 6 (while the IBM has been outlined in many previous articles [68]). Figure 9a shows a sample arrangement, while Fig. 9b is a close-up sketch of the elements. Two sets of simulations were considered: (a) Set 1 featured elements with \(H/h = 15\); (b) Set 2 featured elements with \(H/h = 20\).

For Set 1, we considered \(s_2/H = \{ 0.1, 0.2, 0.32, 0.46, 0.53, 0.64, 0.8, 1.0, \frac{1}{2} \pi , 2 \pi \}\), while for Set 2 we considered \(s_2/H = \{ 0.32, 0.46, 0.64, 1.0, \pi , 2 \pi \}\). Figure 9b shows detailed attributes of the elements; for all cases, \(w_b/H = 0.0756\), \(l_b/H = 0.0756\), \(w_t/H = 0.025\), \(l_t/H = 0.025\), and \(s_x/H = 0.0756\). For Set 1 and Set 2, \(\theta = 69.30\) and 58.52, respectively, though we stress that \(\theta \) is not expected to have a material affect on conclusions of this study, given the fully rough flow conditions [2, 69].

Fig. 9
figure 9

Illustration of spanwise heterogeneous topography composed of streamwise-aligned rows of pyramidal obstacles: a perspective image showing spacing between adjacent rows, \(s_2/H\), and obstacle height, h / H and b close up of obstacles showing streamwise spacing, \(s_1\), inclination, \(\theta \), width and length at base, \(w_b\) and \(w_t\), respectively, and width and length and top, \(w_t\) and \(l_t\), respectively. Pressure-gradient forcing aligned with \(x_1\) direction, as shown with \(U_0\). In figure, all lengths normalized by channel half height, H

Appendix B: Faá di Bruno Formula

The univariate Faá di Bruno formula:

$$ \dfrac{\partial ^n g(f(x))}{\partial x} = \displaystyle \sum \dfrac{n !}{b_1 ! b_1 ! \ldots b_n !} g^{\left( k \right) } \left( f(x) \right) \displaystyle \prod _{j=1}^n \left( \dfrac{f^{\left( k \right) } (x)}{j!} \right) ^{b_j}, $$
(49)

where the summation is over all possible solutions with non-negative integer input arguments to the relations, \(b_1 + 2 b_2 + \ldots + n b_n = n\) and \(k = \displaystyle \sum\nolimits _{i=1}^{n} b_i\). With this, the first-order derivatives in \(x_2\) and \(x_3\) are:

$$ \partial _{2} \mathrm {\Psi } = \mathrm {\Psi }^{\prime } \partial _2 \eta , $$
(50)

and

$$ \partial _{3} \mathrm {\Psi } = \mathrm {\Psi }^{\prime } \partial _3 \eta , $$
(51)

where the summation is made over a single partition: (1) \(b_1 = 1\), and \(k = 1\). The second-order derivatives in \(x_2\) and \(x_3\) are:

$$ \partial _{22} \mathrm {\Psi } = \mathrm {\Psi }^{\prime \prime } \left[ \partial _2 \eta \right] ^2 + \mathrm {\Psi }^\prime \partial _{22} \eta , $$
(52)

and

$$ \partial _{33} \mathrm {\Psi } = \mathrm {\Psi }^{\prime \prime } \left[ \partial _3 \eta \right] ^2 + \mathrm {\Psi }^\prime \partial _{33} \eta , $$
(53)

where the summation is made over two partitions: (1) \(b_1 = 2\), \(b_2 = 0\), and \(k = 2\); and (2) \(b_1 = 0\), \(b_2 = 1\), and \(k = 1\). The third-order derivatives in \(x_2\) and \(x_3\) are:

$$ \partial _{222} \mathrm {\Psi } = \mathrm {\Psi }^{\prime \prime \prime } \left[ \partial _2 \eta \right] ^3 + 3 \mathrm {\Psi }^{\prime \prime } \partial _2 \eta \partial _{22} \eta + \mathrm {\Psi }^\prime \partial _{222} \eta , $$
(54)

and

$$ \partial _{333} \mathrm {\Psi } = \mathrm {\Psi }^{\prime \prime \prime } \left[ \partial _3 \eta \right] ^3 + 3 \mathrm {\Psi }^{\prime \prime } \partial _3 \eta \partial _{33} \eta + \mathrm {\Psi }^\prime \partial _{333} \eta , $$
(55)

where the summation is made over three partitions: (1) \(b_1 = 3\), \(b_2 = 0\), \(b_3 = 0\), and \(k = 3\); (2) \(b_1 = 0\), \(b_2 = 0\), \(b_3 = 1\), and \(k = 1\); and (3) \(b_1 = 1\), \(b_2 = 1\), \(b_3 = 0\), and \(k = 2\). The fourth-order derivatives in \(x_2\) and \(x_3\) are:

$$ \partial _{2222} \mathrm {\Psi } = \mathrm {\Psi }^{\prime \prime \prime \prime } \left[ \partial _2 \eta \right] ^4 + 6 \mathrm {\Psi }^{\prime \prime \prime } \left[ \partial _2 \eta \right] ^2 \partial _{22} \eta + 4 \mathrm {\Psi }^{\prime \prime } \partial _{2} \eta \partial _{222} \eta + \mathrm {\Psi }^{\prime } \partial _{2222} \eta , $$
(56)

and

$$ \partial _{3333} \mathrm {\Psi } = \mathrm {\Psi }^{\prime \prime \prime \prime } \left[ \partial _3 \eta \right] ^4 + 6 \mathrm {\Psi }^{\prime \prime \prime } \left[ \partial _3 \eta \right] ^2 \partial _{33} \eta + 4 \mathrm {\Psi }^{\prime \prime } \partial _{3} \eta \partial _{333} \eta + \mathrm {\Psi }^{\prime } \partial _{3333} \eta , $$
(57)

where the summation is made over four partitions: (1) \(b_1 = 4\), \(b_2 = 0\), \(b_3 = 0\), \(b_4 = 0\), and \(k = 4\); (2) \(b_1 = 0\), \(b_2 = 0\), \(b_3 = 0\), \(b_4 = 1\), and \(k = 1\); (3) \(b_1 = 1\), \(b_2 = 0\), \(b_3 = 1\), \(b_4 = 0\), and \(k = 2\); and (4) \(b_1 = 2\), \(b_2 = 1\), \(b_3 = 0\), \(b_4 = 0\), and \(k = 3\). Finally, a multivariate version of Eq. 49 is needed for terms in Eq. 25. The multivariate form of Eq. 49 yields,

$$ \partial _{223} \mathrm {\Psi } = \mathrm {\Psi }^{\prime \prime \prime } \left[ \partial _2 \eta \right] ^2 \partial _3 \eta + \mathrm {\Psi }^{\prime \prime } \left[ \partial _3 \left( \partial _{2} \eta \right) ^2 + \partial _{22} \eta \partial _3 \eta \right] + \mathrm {\Psi }^{\prime } \partial _{223} \eta , $$
(58)
$$ \partial _{332} \mathrm {\Psi } = \mathrm {\Psi }^{\prime \prime \prime } \left[ \partial _3 \eta \right] ^2 \partial _2 \eta + \mathrm {\Psi }^{\prime \prime } \left[ \partial _2 \left( \partial _{3} \eta \right) ^2 + \partial _{33} \eta \partial _2 \eta \right] + \mathrm {\Psi }^{\prime } \partial _{332} \eta , $$
(59)

and

$$ \begin{aligned} \partial _{3322} \mathrm {\Psi }& = \mathrm {\Psi }^{\prime \prime \prime \prime } \left[ \partial _2 \eta \right] ^2 \left[ \partial _3 \eta \right] ^2 + \mathrm {\Psi }^{\prime \prime \prime } \left( 2 \partial _2 \eta \partial _2 \left[ \partial _3 \eta \right] ^2 + \partial _{22} \eta \left[ \partial _3 \eta \right] ^2 + \partial _{33} \eta \left[ \partial _2 \eta \right] ^2 \right) \nonumber \\&\quad + \mathrm {\Psi }^{\prime \prime } \left( 2 \left[ \{ \partial _{23} \eta \}^2 + \partial _3 \eta \partial _{223}\eta + \partial _2 \eta \partial _{332} \eta \right] + \partial _{22} \eta \partial _{33} \eta \right) + \partial _{2233} \eta \mathrm {\Psi }^\prime ,\end{aligned} $$
(60)

where it has been presumed throughout that the partial differential operators commute. Note that the efficacy of Eqs. 58 and 59 can be quickly established by setting index 3 to 2 or setting index 2 to 3 in the former and latter, respectively, and comparing against Eqs. 54 and 55. In addition, the denominators of prefactors A to F (Eqs. 30 to 35) make use of the spatial derivatives of the Eq. 3 Stokes drift function, which are defined here for completeness:

$$ \partial _{3} u_1^s = - U^s \dfrac{2 x_3}{\alpha _2 H^2} \left( \exp \left[ - \dfrac{x_2^2}{\alpha _1 H^2}\right] - \exp \left[ - \dfrac{\{x_2 - \lambda \}^2}{\alpha _1 H^2}\right] \right) \exp \left( - \dfrac{x_3^2}{\alpha _2 H^2} \right) , $$
(61)

and,

$$ \partial _{2} u_1^s = U^s \dfrac{2 }{\alpha _1 H^2} \left( \{x_2 - \lambda \} \exp \left[ - \dfrac{\{x_2 - \lambda \}^2}{\alpha _1 H^2}\right] - x_2 \exp \left[ - \dfrac{x_2^2}{\alpha _1 H^2}\right] \right) \exp \left( - \dfrac{x_3^2}{\alpha _2 H^2} \right) . $$
(62)

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Anderson, W., Yang, J., Shrestha, K. et al. Turbulent secondary flows in wall turbulence: vortex forcing, scaling arguments, and similarity solution. Environ Fluid Mech 18, 1351–1378 (2018). https://doi.org/10.1007/s10652-018-9596-6

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