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ZDES Simulation and Spectral Analysis of a High-Reynolds-Number Out-of-Equilibrium Turbulent Boundary Layer

Abstract

A test-case for the assessment of Zonal Detached Eddy Simulation (ZDES) mode 3 [which corresponds to a Wall-Modelled Large Eddy Simulation approach (WMLES)] for turbulent boundary layers in pressure gradient conditions is presented. The demanding test-case corresponds to an experiment at high Reynolds number, reaching up to \(Re_\theta \approx 13000\), probably too expensive for Direct Numerical Simulation or Wall-Resolved Large Eddy Simulation, but still affordable using ZDES mode 3 (WMLES). At the considered station, the boundary layer is in out-of-equilibrium conditions. The presented results prove the advantage of the scale-resolving approach, the ZDES mode 3, with respect to the RANS approach, as evidenced by the better representation observed for the mean velocity and Reynolds stress profiles, in particular in the outer layer where non-canonical effects are more evident. Thanks to the resolved turbulence, a more physically realistic flow is predicted by ZDES mode 3 and more in depth analysis of turbulence is accessible. In particular, spectral analysis of turbulence is performed in this study, and a scale-dependent convection velocity is also assessed for the first time with a hybrid RANS/LES approach in out-of-equilibrium conditions. Such analysis allow to identify some features of the turbulent scales distribution within the boundary layer, which seem responsible for some uncommon features observed in the present mean flow.

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Acknowledgements

The authors are greatly thankful to all the people involved in the evolution of the FLU3M and elsA solvers. The authors would also like to express their gratitude to the committee of the 13th International ERCOFTAC symposium on engineering, turbulence, modelling and measurements, ETMM13 for encouraging this publication.

Funding

The thesis of J. Vaquero is partly funded by DGA (French defence procurement agency). Support from the framework of the ONERA research project FROTTEMENT is also acknowledged.

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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by JV under the supervision of NR and SD. The first draft of the manuscript was written by JV and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Jaime Vaquero.

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Appendix: Reminder About the Scale-Dependent Convection Velocity (Renard and Deck 2015b)

Appendix: Reminder About the Scale-Dependent Convection Velocity (Renard and Deck 2015b)

This appendix presents a reminder of the definition and evaluation of the convection velocity \(U_c(f)\) of turbulent structures dependent on their streamwise length scale and, hence, on their frequency f, following the study of Renard and Deck (2015b). This convection velocity is studied in their work for a non-homogeneous mean flow in the streamwise direction, x, which is the case for instance in flat plate boundary layer flows. Their study is based on that of del Álamo and Jiménez (2009) made for channel flows, for which, on the contrary, the mean flow is homogeneous in the streamwise direction. The idea is to find a convection velocity \(\mathcal {C}\) such that the residual of the advection equation is minimised:

$$\begin{aligned} \frac{1}{\mathcal {C}} \frac{\partial u^\prime }{\partial t} + \frac{\partial u^\prime }{\partial x} = 0. \end{aligned}$$
(11)

Thus, the global convection velocity, which takes into account the whole range of scales (or frequencies), \(\mathcal {C}_u\), corresponds to that giving the minimum of the quantity \(\mathcal {D}(\mathcal {C})\), defined as:

$$\begin{aligned} \mathcal {D}(\mathcal {C}) = \frac{E \left[ \left( \frac{1}{\mathcal {C}} \frac{\partial u^\prime }{\partial t} + \frac{\partial u^\prime }{\partial x} \right) ^2 \right] }{E \left[ \left( \frac{\partial u^\prime }{\partial x} \right) ^2 \right] } \end{aligned}$$
(12)

where \(E\left[ \bullet \right]\) denotes the mathematical expectation operator. Since the denominator of \(\mathcal {D}(\mathcal {C})\) does not depend on \(\mathcal {C}\), minimising either \(\mathcal {D}(\mathcal {C})\) or \(E \left[ \left( \frac{1}{\mathcal {C}} \frac{\partial u^\prime }{\partial t} + \frac{\partial u^\prime }{\partial x} \right) ^2 \right]\) gives the same condition, which is:

$$\begin{aligned} \frac{\partial \mathcal {D}(\mathcal {C}_u)}{\partial \mathcal {C}} = 0 \Rightarrow \mathcal {C}_u = - \frac{E \left[ \left( \frac{\partial u^\prime }{\partial t} \right) ^2 \right] }{E \left[ \frac{\partial u^\prime }{\partial t} \frac{\partial u^\prime }{\partial x} \right] } . \end{aligned}$$
(13)

Moreover, for \(\mathcal {C}_u\) it is possible to rewrite (12) such that:

$$\begin{aligned} 1-\gamma _{cu} ^2 = \mathcal {D}(\mathcal {C}_u), \quad \gamma _{cu} \ge 0, \quad \Rightarrow \gamma _{cu} = \frac{ |E\left[ \frac{\partial u^\prime }{\partial t} \frac{\partial u^\prime }{\partial x} \right] |}{\sqrt{E \left[ \left( \frac{\partial u^\prime }{\partial t} \right) ^2 \right] E \left[ \left( \frac{\partial u^\prime }{\partial x} \right) ^2 \right] }} \end{aligned}$$
(14)

where \(\gamma _{cu}\) is a correlation coefficient ranging from 0 to 1 and corresponds to a validity indicator of Taylor’s hypothesis of frozen turbulence (\(\gamma _{cu}=1\) in the case of perfect convection).

By considering the estimation of the PSD through Welch’s method (Welch 1967), as previously stated, together with some properties of the correlation functions (Renard and Deck 2015b; Bendat and Piersol 2010), one may write \(\mathcal {C}_u\) as a function of the two-sided cross-PSD:

$$\begin{aligned} \mathcal {C}_u = \frac{- \int _{-\infty } ^{+\infty } S_{\partial _t u \partial _t u;f}(f) \mathrm {d} f}{\int _{-\infty } ^{+\infty } S_{\partial _t u \partial _x u;f} (f) \mathrm {d} f} = \frac{- \int _{-\infty } ^{+\infty } (2 \pi f) ^2 S_{uu;f} (f) \mathrm {d} f}{\int _{-\infty } ^{+\infty } -2 i \pi f S_{ u \partial _x u;f} (f) \mathrm {d} f} \end{aligned}$$
(15)

where \(S_{\partial _t u \partial _t u;f} (f)\) represents the two-sided PSD of the temporal derivative of \(u^\prime\) and \(i^2=-1\). It is possible to show that \(S_{uu;f}(f)\) is a real function (Bendat and Piersol 2010) and also that, for statistically stationary signals, \(S_{uu;f}(f)\) is a symmetric function and therefore \(S_{uu;f}(-f) = S_{uu;f}(f)\). It may also be written \(S_{ u \partial _x u;f} (-f) = S_{ u \partial _x u;f}^* (f)\) and thus:

$$\begin{aligned} \mathcal {C}_u = - \frac{\int _{0}^{+\infty } (2 \pi f) ^2 S_{uu;f}(f) \mathrm {d} f}{\int _{0}^{+\infty } 2 \pi f ~\text {Im} (S_{ u \partial _x u;f} (f)) \mathrm {d} f} \end{aligned}$$
(16)

where \(\text {Im}(\bullet )\) corresponds to the imaginary part. The expression for \(\gamma _{cu}\) may be then expressed as:

$$\begin{aligned} \gamma _{cu} = \frac{|\int _0^{+\infty } 2 \pi f ~ \text {Im} (S_{ u \partial _x u;f} (f)) \mathrm {d} f |}{\sqrt{\int _0 ^{+\infty } (2 \pi f) ^2 S_{uu;f} (f) \mathrm {d} f} \sqrt{\int _0 ^{+\infty } S_{\partial _x u \partial _x u;f}(f) \mathrm {d} f}}. \end{aligned}$$
(17)

The final expressions for the convection velocity and the correlation coefficient as a function of the frequency (scale) \(U_c(f)\) and \(\gamma _{cu} (f)\) are obtained from Eqs. (16) and (17) respectively by integrating in an infinitely small interval around a given frequency \(f_0\) (Renard and Deck 2015b):

$$\begin{aligned} U_c(f) = - \frac{2 \pi f S_{uu;f}(f)}{\text {Im}(S_{u \partial _x u;f} (f))}, \quad \gamma _{cu} (f) = \frac{|\text {Im}(S_{u \partial _x u;f}(f)) |}{\sqrt{S_{uu;f}(f)} \sqrt{S_{\partial _x u \partial _x u;f}(f)}}. \end{aligned}$$
(18)

In all the expressions here shown, the dependency on the wall distance has been omitted for clarity purposes. It is however important to remind that these expressions are given at a specified wall distance, so that for a given frequency there is a dependency of both \(U_c(f)\) and \(\gamma _{cu}(f)\) on the wall distance.

In Renard and Deck (2015b), the convection velocity \(U_c(f)\) and the correlation coefficient \(\gamma _{cu} (f)\) are presented for a zero-pressure-gradient turbulent boundary layer at \(Re_\theta = 13000\). The convection velocity is compared to the velocity based on the two-point two-time correlation \(U_{corr}\), which is the one used in the present paper for the spectral analysis of turbulence. Quite close values between \(U_c(f)\) and \(U_{corr}\) are obtained by Renard and Deck (2015b) thus suggesting that \(U_{corr}\) is a good alternative for the convection velocity, since its computation is easier than that of \(U_c(f)\). Indeed, \(U_{corr}\) may be seen as a weighted harmonic average of \(U_c(f)\) (Renard and Deck 2015b), and is coincident with \(\mathcal {C}_u\) (16).

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Vaquero, J., Renard, N. & Deck, S. ZDES Simulation and Spectral Analysis of a High-Reynolds-Number Out-of-Equilibrium Turbulent Boundary Layer. Flow Turbulence Combust (2022). https://doi.org/10.1007/s10494-022-00361-7

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Keywords

  • Wall turbulence
  • ZDES simulation
  • Hybrid RANS/LES
  • Out-of-equilibrium flows
  • Spectral analysis