Asymptotic theory of turbulent shear flows

1970 ◽  
Vol 42 (2) ◽  
pp. 411-427 ◽  
Author(s):  
Kirit S. Yajnik
Keyword(s):  
Shear Flows ◽  
Functional Relationship ◽  
Turbulent Shear ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
Underdetermined System ◽  
Different Types ◽  
Turbulent Shear Flows ◽  
The Mean ◽  
The Law

A theory is proposed in this paper to describe the behaviour of a class of turbulent shear flows as the Reynolds number approaches infinity. A detailed analysis is given for simple representative members of this class, such as fully developed channel and pipe flows and two-dimensional turbulent boundary layers. The theory considers an underdetermined system of equations and depends critically on the idea that these flows consist of two rather different types of regions. The method of matched asymptotic expansions is employed together with asymptotic hypotheses describing the order of various terms in the equations of mean motion and turbulent kinetic energy. As these hypotheses are not closure hypotheses, they do not impose any functional relationship between quantities determined by the mean velocity field and those determined by the Reynolds stress field. The theory leads to asymptotic laws corresponding to the law of the wall, the logarithmic law, the velocity defect law, and the law of the wake.

A unified approach for symmetries in plane parallel turbulent shear flows

2001 ◽  
Vol 427 ◽  
pp. 299-328 ◽  
Author(s):  
MARTIN OBERLACK
Keyword(s):  
Shear Flows ◽  
Stokes Equations ◽  
Turbulent Shear ◽  
Large Set ◽  
Mean Velocity ◽  
Velocity Fluctuations ◽  
Law Of The Wall ◽  
Turbulent Shear Flows ◽  
The Mean ◽  
Logarithmic Law

A new theoretical approach for turbulent flows based on Lie-group analysis is presented. It unifies a large set of ‘solutions’ for the mean velocity of stationary parallel turbulent shear flows. These results are not solutions in the classical sense but instead are defined by the maximum number of possible symmetries, only restricted by the flow geometry and other external constraints. The approach is derived from the Reynolds-averaged Navier–Stokes equations, the fluctuation equations, and the velocity product equations, which are the dyad product of the velocity fluctuations with the equations for the velocity fluctuations. The results include the logarithmic law of the wall, an algebraic law, the viscous sublayer, the linear region in the centre of a Couette flow and in the centre of a rotating channel flow, and a new exponential mean velocity profile not previously reported that is found in the mid-wake region of high Reynolds number flat-plate boundary layers. The algebraic scaling law is confirmed in both the centre and the near-wall regions in both experimental and DNS data of turbulent channel flows. In the case of the logarithmic law of the wall, the scaling with the distance from the wall arises as a result of the analysis and has not been assumed in the derivation. All solutions are consistent with the similarity of the velocity product equations to arbitrary order. A method to derive the mean velocity profiles directly from the two-point correlation equations is shown.

Experimental Investigation of a Turbulent Boundary Layer Subjected to an Adverse Pressure Gradient

2011 ◽  
Author(s):  
Takanori Nakamura ◽  
Takatsugu Kameda ◽  
Shinsuke Mochizuki
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Friction Velocity ◽  
Adverse Pressure Gradient ◽  
Turbulent Intensity ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
Velocity Scale ◽  
The Mean ◽  
The Law

Experiments were performed to investigate the effect of an adverse pressure gradient on the mean velocity and turbulent intensity profiles for an equilibrium boundary layer. The equilibrium boundary layer, which makes self-similar profiles, was constructed using a power law distribution of free stream velocity. The exponent of the law was adjusted to −0.188. The wall shear stress was measured with a drag balance by a floating element. The investigation of the law of the wall and the similarity of the streamwise turbulent intensity profile was made using both a friction velocity and new proposed velocity scale. The velocity scale is derived from the boundary layer equation. The mean velocity gradient profile normalized with the height and the new velocity scale exists the region where the value is almost constant. The turbulent intensity profiles normalized with the friction velocity strongly depend on the nondimensional pressure gradient near the wall. However, by mean of the local velocity scale, the profiles might be achieved to be similar with that of a zero pressure gradient.

scholarly journals Wind Turbulence Statistics of the Atmospheric Inertial Sublayer under Near-Neutral Conditions

Atmosphere ◽  
2020 ◽  
Vol 11 (10) ◽  
pp. 1087
Author(s):  
Eslam Reda Lotfy ◽  
Zambri Harun
Keyword(s):  
Boundary Layer ◽  
Atmospheric Stability ◽  
Turbulent Velocity ◽  
Stability Conditions ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
Turbulent Statistics ◽  
The Mean ◽  
The Law ◽  
Velocity Variances

The inertial sublayer comprises a considerable and critical portion of the turbulent atmospheric boundary layer. The mean windward velocity profile is described comprehensively by the Monin–Obukhov similarity theory, which is equivalent to the logarithmic law of the wall in the wind tunnel boundary layer. Similar logarithmic relations have been recently proposed to correlate turbulent velocity variances with height based on Townsend’s attached-eddy theory. The theory is particularly valid for high Reynolds-number flows, for example, atmospheric flow. However, the correlations have not been thoroughly examined, and a well-established model cannot be reached for all turbulent variances similar to the law of the wall of the mean-velocity. Moreover, the effect of atmospheric thermal condition on Townsend’s model has not been determined. In this research, we examined a dataset of free wind flow under a near-neutral range of atmospheric stability conditions. The results of the mean velocity reproduce the law of the wall with a slope of 2.45 and intercept of −13.5. The turbulent velocity variances were fitted by logarithmic profiles consistent with those in the literature. The windward and crosswind velocity variances obtained the average slopes of −1.3 and −1.7, respectively. The slopes and intercepts generally increased away from the neutral state. Meanwhile, the vertical velocity and temperature variances reached the ground-level values of 1.6 and 7.8, respectively, under the neutral condition. The authors expect this article to be a groundwork for a general model on the vertical profiles of turbulent statistics under all atmospheric stability conditions.

Spectral analogues of the law of the wall, the defect law and the log law

2014 ◽  
Vol 757 ◽  
pp. 498-513 ◽  
Author(s):  
Carlo Zúñiga Zamalloa ◽  
Henry Chi-Hin Ng ◽  
Pinaki Chakraborty ◽  
Gustavo Gioia
Keyword(s):  
Turbulent Energy ◽  
Wall Turbulence ◽  
Spectral Domain ◽  
Energy Spectra ◽  
Scaling Relations ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
The Mean ◽  
Log Law ◽  
The Law

AbstractUnlike the classical scaling relations for the mean-velocity profiles of wall-bounded uniform turbulent flows (the law of the wall, the defect law and the log law), which are predicated solely on dimensional analysis and similarity assumptions, scaling relations for the turbulent-energy spectra have been informed by specific models of wall turbulence, notably the attached-eddy hypothesis. In this paper, we use dimensional analysis and similarity assumptions to derive three scaling relations for the turbulent-energy spectra, namely the spectral analogues of the law of the wall, the defect law and the log law. By design, each spectral analogue applies in the same spatial domain as the attendant scaling relation for the mean-velocity profiles: the spectral analogue of the law of the wall in the inner layer, the spectral analogue of the defect law in the outer layer and the spectral analogue of the log law in the overlap layer. In addition, as we are able to show without invoking any model of wall turbulence, each spectral analogue applies in a specific spectral domain (the spectral analogue of the law of the wall in the high-wavenumber spectral domain, where viscosity is active, the spectral analogue of the defect law in the low-wavenumber spectral domain, where viscosity is negligible, and the spectral analogue of the log law in a transitional intermediate-wavenumber spectral domain, which may become sizable only at ultra-high$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_{\tau }$), with the implication that there exist model-independent one-to-one links between the spatial domains and the spectral domains. We test the spectral analogues using experimental and computational data on pipe flow and channel flow.

The law of the wall in turbulent flow

1995 ◽  
Vol 451 (1941) ◽  
pp. 165-188 ◽  
Keyword(s):  
Turbulence Models ◽  
Turbulence Theory ◽  
Turbulent Shear ◽  
Wall Region ◽  
Reynolds Stresses ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
Logarithmic Law ◽  
The Law ◽  
Simple Flows

The ‘law of the wall’ for the inner part of a turbulent shear flow over a solid surface is one of the cornerstones of fluid dynamics, and one of the very few pieces of turbulence theory whose results include a simple analytic function for the mean velocity distribution, the logarithmic law. Various aspects of the law have recently been questioned, and this paper is a summary of the present position. Although the law of the wall for velocity has apparently been confirmed by experiment well outside its original range, the law of the wall for temperature seems to apply only to very simple flows. Since the two laws are derived by closely analogous arguments this throws suspicion on the law of the wall for velocity. Analysis of simulation data, for all the Reynolds stresses including the shear stress, shows that law-of-the-wall scaling fails spectacularly in the viscous wall region, even when the logarithmic law is relatively well behaved. Virtually all turbulence models are calibrated to reproduce the law of the wall in simple flows, and we discuss whether, in practice or in principle, their range of validity is larger than that of the law of the wall itself: the present answer is that it is not; so that when the law of the wall (or the mixing-length formula) fails, current Reynolds-averaged turbulence models are likely to fail too.

scholarly journals The law of the wake in the turbulent boundary layer

1956 ◽  
Vol 1 (2) ◽  
pp. 191-226 ◽  
Author(s):  
Donald Coles
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Velocity Profile ◽  
Three Dimensional ◽  
Shearing Stress ◽  
Universal Functions ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
The Mean ◽  
The Law

After an extensive survey of mean-velocity profile measurements in various two-dimensional incompressible turbulent boundary-layer flows, it is proposed to represent the profile by a linear combination of two universal functions. One is the well-known law of the wall. The other, called the law of the wake, is characterized by the profile at a point of separation or reattachment. These functions are considered to be established empirically, by a study of the mean-velocity profile, without reference to any hypothetical mechanism of turbulence. Using the resulting complete analytic representation for the mean-velocity field, the shearing-stress field for several flows is computed from the boundary-layer equations and compared with experimental data.The development of a turbulent boundary layer is ultimately interpreted in terms of an equivalent wake profile, which supposedly represents the large-eddy structure and is a consequence of the constraint provided by inertia. This equivalent wake profile is modified by the presence of a wall, at which a further constraint is provided by viscosity. The wall constraint, although it penetrates the entire boundary layer, is manifested chiefly in the sublayer flow and in the logarithmic profile near the wall.Finally, it is suggested that yawed or three-dimensional flows may be usefully represented by the same two universal functions, considered as vector rather than scalar quantities. If the wall component is defined to be in the direction of the surface shearing stress, then the wake component, at least in the few cases studied, is found to be very nearly parallel to the gradient of the pressure.

Weakly sheared turbulent flows generated by multiscale inhomogeneous grids

2018 ◽  
Vol 848 ◽  
pp. 788-820 ◽  
Author(s):  
Shaokai Zheng ◽  
P. J. K. Bruce ◽  
J. M. R. Graham ◽  
J. C. Vassilicos
Keyword(s):  
Velocity Profile ◽  
Turbulence Intensity ◽  
Shear Flows ◽  
Vertical Direction ◽  
Turbulent Shear ◽  
Incoming Flow ◽  
Mean Velocity ◽  
Free Stream Turbulence ◽  
The Mean ◽  
Vertical Bars

A group of three multiscale inhomogeneous grids have been tested to generate different types of turbulent shear flows with different mean shear rate and turbulence intensity profiles. Cross hot-wire measurements were taken in a wind tunnel with Reynolds number$Re_{D}$of 6000–20 000, based on the width of the vertical bars of the grid and the incoming flow velocity. The effect of local drag coefficient$C_{D}$on the mean velocity profile is discussed first, and then by modifying the vertical bars to obtain a uniform aspect ratio the mean velocity profile is shown to be predictable using the local blockage ratio profile. It is also shown that, at a streamwise location$x=x_{m}$, the turbulence intensity profile along the vertical direction$u^{\prime }(y)$scales with the wake interaction length$x_{\ast ,n}^{peak}=0.21g_{n}^{2}/(\unicode[STIX]{x1D6FC}C_{D}w_{n})$($\unicode[STIX]{x1D6FC}$is a constant characterizing the incoming flow condition, and$g_{n}$,$w_{n}$are the gap and width of the vertical bars, respectively, at layer$n$) such that$(u^{\prime }/U_{n})^{2}\unicode[STIX]{x1D6FD}^{2}(C_{D}w_{n}/x_{\ast ,n}^{peak})^{-1}\sim (x_{m}/x_{\ast ,n}^{peak})^{b}$, where$\unicode[STIX]{x1D6FD}$is a constant determined by the free-stream turbulence level,$U_{n}$is the local mean velocity and$b$is a dimensionless power law constant. A general framework of grid design method based on these scalings is proposed and discussed. From the evolution of the shear stress coefficient$\unicode[STIX]{x1D70C}(x)$, integral length scale$L(x)$and the dissipation coefficient$C_{\unicode[STIX]{x1D716}}(x)$, a simple turbulent kinetic energy model is proposed that describes the evolution of our grid generated turbulence field using one centreline measurement and one vertical profile of$u^{\prime }(y)$at the beginning of the evolution. The results calculated from our model agree well with our measurements in the streamwise extent up to$x/H\approx 2.5$, where$H$is the height of the grid, suggesting that it might be possible to design some shear flows with desired mean velocity and turbulence intensity profiles by designing the geometry of a passive grid.

Diffusion in free turbulent shear flows

1957 ◽  
Vol 3 (1) ◽  
pp. 67-80 ◽  
Author(s):  
G. K. Batchelor
Keyword(s):  
Time Scales ◽  
Particle Velocity ◽  
Random Function ◽  
Shear Flows ◽  
Fluid Particle ◽  
Turbulent Shear ◽  
Mean Flow ◽  
Turbulent Shear Flows ◽  
The Mean ◽  
Stationary Random Function

This paper is concerned with some statistical properties of the displacement of a marked fluid particle released from a given position in a turbulent shear flow, and in particular with the dispersion about the mean position after a long time. It is known that the dispersion takes a simple asymptotic form when the particle velocity is a stationary random function of time, and that analogous results are obtainable when the particle velocity can be transformed to a stationary random function by suitable stretching of the velocity and time scales. The basic hypothesis of the paper is that, in steady free turbulent shear flows which are generated at a point and have a similar structure at different stations downstream, the velocity of a fluid particle exhibits a corresponding Lagrangian similarity and can be so transformed to a stationary random function.The velocity and time scales characterizing the motion of a fluid particle at time t after release at the origin are determined in terms of the powers with which the Eulerian length and velocity scales of the turbulence vary with distance x from the origin. The time scale has the same dependence on t for all jets, wakes and mixing layers (and also for decaying homogeneous turbulence) possessing the usual kind of Eulerian similarity. The dispersion of a particle in the longitudinal or mean-flow direction (and likewise that in the lateral direction in cases of two-dimensional mean flow) is found to vary with t in such a way as to be proportional to the thickness of the shear layer at the mean position of the particle. The way in which the maximum value of the mean concentration of marked fluid falls off with t (for release of a single particle) or with x (for continuous release) is also found.

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