Bulletin of the American Physical Society
65th Annual Meeting of the APS Division of Fluid Dynamics
Volume 57, Number 17
Sunday–Tuesday, November 18–20, 2012; San Diego, California
Session G20: Turbulent Boundary Layers IV: Scaling, logarithmic layer |
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Chair: Ron Panton, University of Texas Room: 30A |
Monday, November 19, 2012 8:00AM - 8:13AM |
G20.00001: Fluctuating Vorticity in Turbulent Boundary Layers Ron Panton Profiles of fluctuating vorticity from the channel flow DNS (Del Alamo, et al. (P of F \textbf{15}, L-41; JFM, \textbf{500}, p135, P of F, \textbf{18}) are correlated in Panton (Phys. Fluids, \textbf{21}, 2009). In the inner region, a two-term expansion represents the vorticity profiles; $\langle \omega \omega \rangle$ $^{\#}$ = $\langle \omega \omega \rangle$ $_{0}^{\# }$+ $\langle \omega \omega \rangle$ $_{1}^{+}u_{\tau }$/$U_{0}$. The scaling $\langle \omega \omega \rangle$ $_{0}^{\# }$ = $\langle \omega \omega \rangle$ $_{0}$/($u_{\tau}^{3}U_{0}$/$\nu $$^{2})$ for inactive motions applies only to the streamwise and spanwise components. This term is zero for the normal vorticity component. The scrubbing of the inactive motions over the wall generates vorticity, which is a maximum at the wall, and diffuses to about $y^{+}$ = 50 before it decays. The fluctuating wall shear stress is due entirely to this motion, and the stress ratio (rms/mean) depends on \textit{Re}. The second scaling $\langle \omega \omega \rangle$ $_{1}^{+}$ = $\langle \omega \omega \rangle$ $_{1}$/($u_{\tau }^{4}$/$\nu^{2})$, the same scaling as the Reynolds shear stress, is active motions. These motions are zero at the wall, peak about $y^{+}$ = 13--20, and fall to zero about $y^{+}$ = 400. The outer region is correlated by a third scaling using the Kolmogorov time scale; $\langle \omega \omega \rangle$ /($u_{\tau}^{3}$/$\delta$ $\nu$). Matching between the inner and outer regions has an overlap law (common part) of $\sim C$/$y^{+}$ or $\sim $ $C$/$Y$ for all components. In this paper DNS boundary layer data of Schlatter et al. (Phys. Fluids, \textbf{21,} 2009) is correlated in the manner previously used for channel flows. [Preview Abstract] |
Monday, November 19, 2012 8:13AM - 8:26AM |
G20.00002: Von Karman re-visited Donald M. McEligot, Kevin P. Nolan, Edmond J. Walsh A number of authors have presented extended versions of the integral momentum equation, allowing for perturbations or fluctuations in the boundary layer. ``Conventional wisdom'' is that these added terms can be neglected and one can apply the von Karman version directly. For two-dimensional turbulent boundary layers at high Reynolds numbers, experience shows this assumption to be reasonable. However, recent examination of entropy generation in bypass transition with zero pressure gradient shows a term for turbulence energy convection can be important in determining the energy dissipation coefficient [Walsh et al., JFE 2011]. The present study employs the direct numerical simulations of Zaki and Durbin [JFM 2006] for bypass transition with streamwise pressure gradients to quantify the additional normal stress term when estimating the skin friction coefficient via a momentum balance. It is found that this term becomes noticeable in the pre-transitional laminar boundary layer and can exceed forty per cent of C$_{f}$ in the transition region. Thus, it should be included in such calculations. [Preview Abstract] |
Monday, November 19, 2012 8:26AM - 8:39AM |
G20.00003: The quest for the von K\'arm\'an constant P.H. Alfredsson, R. \"Orl\"u, A. Segalini Already in 1930 von K\'{a}rm\'{a}n presented an expression for the mean velocity distribution in channel and pipe flows, that can be transformed in the today well known logarithmic velocity distribution. He was also able to obtain a value of 0.38 for the inverse of its slope, what we now know as the von K\'{a}rm\'{a}n constant ($\kappa$). In his case the value was obtained from pressure drop measurements and the, at the same time, formulated logarithmic skin friction law. Since then different values of $\kappa$ have been suggested ranging from 0.37 to 0.44. Different approaches to determine $\kappa$ have been suggested over the years, and also the range of the wall normal coordinate of the boundary layer over which the logarithmic law is valid have been debated. Not until independent measurements of the wall shear stress were available has there been a possibility to actually determine $\kappa$ accurately from the measured mean velocity distribution. We discuss various pitfalls and error sources and based on a new straightforward method to determine $\kappa$, we use data from the literature to show that von K\'{a}rm\'{a}n's original suggestion of the value of $\kappa$ seems to be valid also today. [Preview Abstract] |
Monday, November 19, 2012 8:39AM - 8:52AM |
G20.00004: A universal logarithmic region in wall turbulence Ivan Marusic, Jason Monty, Marcus Hultmark, Alexander Smits Considerable discussion over the past few years has been devoted to the question of whether the logarithmic region in wall turbulence is indeed universal. Here, we analyse recent experimental data in the Reynolds number range of nominally $2\times10^3 < Re_\tau < 6\times10^5$ for boundary layers, pipe flow and the atmospheric surface layer, and show that within the experimental uncertainty, the data support the existence of a universal logarithmic region. The results support the theory of Townsend (1976) and Perry \& Chong (1982) that an inertial region requires both a logarithmic profile for the mean flow and the streamwise turbulence intensities. The experimental data are unique given the high Reynolds numbers presented and the fidelity of the measurement techniques where both the mean velocity and streamwise turbulence intensities are measured with the same instrument. [Preview Abstract] |
Monday, November 19, 2012 8:52AM - 9:05AM |
G20.00005: Unified description of logarithmic profiles in a turbulent channel and pipe Zhen-Su She, Xi Chen, Fazle Hussain A similarity is discovered between the transports of the mean momentum and turbulent kinetic-energy, based on empirical analysis of the two balance equations in DNS data. It yields a new invariant distribution characterizing universal bulk flow dynamics in a channel or a pipe. The theory derives a logarithmic law for the mean kinetic-energy profile at high enough Reynolds numbers (Re). In particular, a Karman-like constant (0.8) for energy is obtained, which yields a quantitative explanation for a recent discovery of Hulkmark et al. (PRL, 2012) with right empirical constants. Together with the momentum Karman constant (0.45), we offer a unified description of the logarithmic distribution for both momentum and kinetic energy. Finally, the newly-found similarity governs also the temperature variations in Rayleigh-Benard convection, and the common log law originates from a sub-leading-order effect of turbulent transport in balancing the difference between turbulence production and dissipation. [Preview Abstract] |
Monday, November 19, 2012 9:05AM - 9:18AM |
G20.00006: Spectral analogue of the law of the wall Gustavo Gioia, Carlo Zuniga Zamalloa, Pinaki Chakraborty We use a recently-proposed spectral model (Gioia et al., PRL, 2010) of the Reynolds shear stress in smooth wall-bounded, uniform turbulent flows to derive a scaling relation for the turbulent energy spectra. This scaling relation is the spectral analogue of Prandtl's scaling relation for the mean velocity profiles (the ``law of the wall''). To test the scaling relation for the turbulent energy spectra, we use data from direct numerical simulations of channel flow. [Preview Abstract] |
Monday, November 19, 2012 9:18AM - 9:31AM |
G20.00007: Mesolayer analysis in a turbulent boundary layer and DNS data Noor Afzal The intermediate layer (mesolayer) in turbulent boundary layer has been analysed by the matched asymptotic expansions where matching is implemented by Izakson-Millikan-Kolmogorov hypothesis. The large-scale motions and very large scale motion are modifying the influences of the outer geometries, and most significantly near the locus of the peak in shear stress in the mesolayer. The mesolayer is formed by the interaction of inner and outer layer scales, whose length (time) scale is the geometric mean of the inner and outer length (time) scales, and is also proportional to Taylor micro length (time) scale. The mesolayer variable is proportional to inverse square root of appropriate friction Reynolds number, provided Reynolds number is large. It is shown that the shape factor and Reynolds shear maxima scale with mesolayer scale equivalent to Taylor micro length scale. Further, the turbulent bursting time period scales is shown to mesolayer time scale which is equivalent to Taylor micro time scale. The implications of mesolayer on higher order effects on skin friction law for lower Reynolds number have also been analyzed. The implications of shift origin are proposed by the Prandtl's transposition theorem, and consequently without any closure model. [Preview Abstract] |
Monday, November 19, 2012 9:31AM - 9:44AM |
G20.00008: Model-based scaling and prediction of streamwise energy spectrum at high Reynolds numbers Rashad Moarref, Ati S. Sharma, Joel A. Tropp, Beverley J. McKeon To better understand the behavior of wall-bounded turbulent flows at high Reynolds numbers, we study the Reynolds number scaling of the low-rank approximation to turbulent channel flows. Following McKeon and Sharma (J. Fluid Mech. 2010), the velocity is decomposed into propagating waves (with single streamwise and spanwise wavelengths and wave speed) whose wall-normal shapes are determined from the principal singular function of the corresponding resolvent operator. We identify three regions of wave parameters that induce intrinsic Reynolds number scales on the low-rank model, reveal the universal shape of the streamwise energy spectrum for the model subject to broadband forcing, and show that this model captures the dominant near-wall turbulent structures. The model-based streamwise spectrum is then shaped by optimal weight functions to match direct numerical simulations throughout the channel at low Reynolds numbers. Representation of the resulting weight functions using similarity laws facilities predictions of the streamwise energy spectra at high Reynolds numbers ($R_\tau \approx 10^3 - 10^6$) which are shown to agree closely with experiments. [Preview Abstract] |
Monday, November 19, 2012 9:44AM - 9:57AM |
G20.00009: Multi-layer prediction of mean velocity profiles in turbulent boundary layers Xi Chen, Fazle Hussain, Zhen-Su She A multi-layer prediction of the mean velocity profile (MVP) is developed for the zero pressure gradient (ZPG) turbulent boundary layer (TBL), in good agreement with empirical data over a wide range of the Reynolds number (Re). The theory builds on our model of the mixing length for channel and pipe flows, in which all of the physical parameters characterizing the viscous sublayer, buffer layer and bulk layer are held universal, as well as the Karman constant 0.45. The theory predicts a logarithmic law constant B of 6.5. The identified differences between the channel/pipe and TBL are the absence of a wall-confined central core layer and a fractional scaling of the total stress for the latter. Then, the theory yields an analytic expression for the wake function and friction coefficient in excellent agreement with measurements. In conclusion, a unified theory is presented for the MVPs of all canonical wall-bounded turbulent flows. [Preview Abstract] |
Monday, November 19, 2012 9:57AM - 10:10AM |
G20.00010: A new scaling for the streamwise broadband turbulence intensity profiles of ZPG turbulent boundary layers Vigneshwaran Kulandaivelu, Nicholas Hutchins Turbulent boundary layers under zero pressure gradient are investigated experimentally with the aim of proposing a new scaling for the streamwise turbulence intensity. The streamwise intensity normalized by the inner and outer scales seems to collapse the profiles near the wall and in the wake region respectively. We here suggest a new scaling that aims to collapse these profiles across both the inner and outer regions. This is done by assuming a logarithmic variation between the viscous-scaling at the wall and outer scaling in the wake region. It is defined as $\hat{z} = \log_{10}\left(z^{+}/C\right)/\log_{10}\left(\delta^{+}/C\right)$, with $\hat{z}$ = 0, at $z^{+}$ = $C$ and $\hat{z}$ = 1, at $z^{+} = \delta^{+}$= $Re_{\tau}$. A very good collapse of the data is observed from $z^{+}\approx15$ to $z/\delta \approx 1$. The constant ``$C$'' is chosen to be 15 which signifies the inner normalised wall location $z^{+}$, where the peak in turbulence intensity is observed. [Preview Abstract] |
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