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Register a new userVelocity Derivatives in Turbulent Boundary Layers. Part II: Statistical Properties
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Michel Stanislas Pr.
Publication date
2020
fields
Physics
and research's language is
English
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An experiment was performed using Dual-plane-SPIV in the LMFL boundary layer facility to determine all of the derivative moments needed to estimate the average dissipation rate of the turbulent kinetic energy, $varepsilon$, and its Reynolds stress counterpart the dissipation tensor, $varepsilon_{ij}$. For this experiment, the Reynolds number was $Re_theta = 7500$ or $Re_tau = 2300$. Part I of this contribution cite{stanislas20} presented in short the experiment and discussed in detail the dissipation profile and all twelve derivative moments required to compute it. The data were compared to a channel flow DNS at approximately the same Reynolds number and to previous results. They were also used to evaluate recent theoretical results for the overlap region. In this Part II the experimental and DNS results are used to evaluate the assumptions of `local isotropy, `local axisymmetry, and `local homogeneity. They are extended to include the full dissipation tensor, $varepsilon_{ij}$ and the `pseudo-dissipation tensor, $mathcal{D}_{ij}$ and explain the strong anisotropy of the dissipation tensors observed. Two important results of the present study are that {it local isotropy} is never valid inside the outer limit of the overlap region, $y/delta_{99} approx 0.1$; and that the assumptions of {it local axisymmetry} and {it local homogeneity} fail inside of $y^+ =100$. The implications of {it homogeneity in planes parallel to the wall} is introduced to partially explain observations throughout the wall layer. The dissipation characteristics in this very near wall region show that $varepsilon_{ij}$ is close to but different from $mathcal{D}_{ij}$ .
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An experiment was performed using SPIV in the LMFL boundary layer facility to determine all the derivative moments needed to estimate the average dissipation rate of the turbulence kinetic energy, $varepsilon = 2 u langle s_{ij}s_{ij} rangle$ where $s_{ij}$ is the fluctuating strain-rate and $langle~rangle$ denotes ensemble averages. Also measured were all the moments of the full average deformation rate tensor, as well as all of the first, second and third fluctuating velocity moments except those involving pressure. The Reynolds number was $Re_theta = 7500$ or $Re_tau = 2300$. The results are presented in three separate papers. This first paper (Part I) presents the measured average dissipation, $varepsilon$ and the derivative moments comprising it. It compares the results to the earlier measurements of cite{balint91,honkan97} at lower Reynolds numbers and a new results from a plane channel flow DNS at comparable Reynolds number. It then uses the results to extend and evaluate the theoretical predictions of cite{george97b,wosnik00} for all quantities in the overlap region. Of special interest is the prediction that $varepsilon^+ propto {y^+}^{-1}$ for streamwise homogeneous flows and a nearly indistinguishable power law, $varepsilon propto {y^+}^{gamma-1}$, for boundary layers. In spite of the modest Reynolds number, the predictions seem to be correct. It also predicts and confirms that the transport moment contribution to the energy balance in the overlap region, $partial langle - pv /rho - q^2 v/2 rangle/ partial y$ behaves similarly. An immediate consequence is that the usual eddy viscosity model for these terms cannot be correct. The second paper, Part II, examines in detail the statistical character of the velocity derivatives. The details of the SPIV methodology is in Part III, since it will primarily be of interest to experimentalists.
An SPIV experiment using two orthogonal planes simultaneously was performed in the LML boundary layer facility to specifically measure all of the derivative moments needed to estimate the dissipation rate of the Turbulence Kinetic Energy. The Reynolds number was $Re_theta = 7500$ or $Re_tau = 2300$. A detailed analysis of the errors in derivative measurements was carried out, as well as applying and using consistency checks derived from the continuity equation. The random noise error was quantified, and used to ``de-noise the derivative moments. A comparison with a DNS channel flow at comparable Reynolds number demonstrated the capability of the technique. The results were further validated using the recent theory developed by George and Stanislas 2020. The resulting data have been extensively used in parts I and II of the present contribution to study near wall dissipation. An important result of the present work is the provision of reliable rules for an accurate assessment of the dissipation in future PIV experiments.
The effect of rotation on the boundary layers (BLs) in a Rayleigh-Benard (RB) system at a relatively low Rayleigh number, i.e. $Ra = 4times10^7$, is studied for different Pr by direct numerical simulations and the results are compared with laminar BL theory. In this regime we find a smooth onset of the heat transfer enhancement as function of increasing rotation rate. We study this regime in detail and introduce a model based on the Grossmann-Lohse theory to describe the heat transfer enhancement as function of the rotation rate for this relatively low Ra number regime and weak background rotation $Rogtrsim 1$. The smooth onset of heat transfer enhancement observed here is in contrast to the sharp onset observed at larger $Ra gtrsim 10^8$ by Stevens {it{et al.}} [Phys. Rev. Lett. {bf{103}}, 024503, 2009], although only a small shift in the Ra-Ro-Pr phase space is involved.
We investigate the structures of the near-plate velocity and temperature profiles at different horizontal positions along the conducting bottom (and top) plate of a Rayleigh-B{e}nard convection cell, using two-dimensional (2D) numerical data obtained at the Rayleigh number Ra=10^8 and the Prandtl number Pr=4.4 of an Oberbeck-Boussinesq flow with constant material parameters. The results show that most of the time, and for both velocity and temperature, the instantaneous profiles scaled by the dynamical frame method [Q. Zhou and K.-Q. Xia, Phys. Rev. Lett. 104, 104301 (2010) agree well with the classical Prandtl-Blasius laminar boundary layer (BL) profiles. Therefore, when averaging in the dynamical reference frames, which fluctuate with the respective instantaneous kinematic and thermal BL thicknesses, the obtained mean velocity and temperature profiles are also of Prandtl-Blasius type for nearly all horizontal positions. We further show that in certain situations the traditional definitions based on the time-averaged profiles can lead to unphysical BL thicknesses, while the dynamical method also in such cases can provide a well-defined BL thickness for both the kinematic and the thermal BLs.
Four well-resolved LESs of the turbulent boundary layers around a NACA4412 wing section, with Rec ranging from 100,000 to 1,000,000, were performed at 5 degree angle of attack. By comparing the turbulence statistics with those in ZPG TBLs at approximately matching Re_tau, we find that the effect of the adverse pressure gradient (APG) is more intense at lower Reynolds numbers. This indicates that at low Re the outer region of the TBL becomes more energized through the wall-normal convection associated to the APG. This is also reflected in the fact that the inner-scaled wall-normal velocity is larger on the suction side at lower Reynolds numbers. In particular, the wing cases at Rec = 200,000 and 400,000 exhibit wall-normal velocities 40% and 20% larger, respectively, than the case with Rec = 1,000,000. Consequently, the outer-region energizing mechanism associated to the APG is complementary to that present in high-Re TBLs.
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