Turbulence may be defined as the most general case of unsteady fluid motion allowed by the Navier-Stokes equations. It is irregular and disorderly, it causes rapid mixing of heat and momentum, it is rotational and three-dimensional, it occurs over a range of scales, and, it is dissipative. The irregularity is illustrated in Figure 1 which shows the turbulent velocity signal from a hot-wire anemometer 18 m above the ground in an atmospheric surface layer.
Figure 1. Turbulent velocity signal plotted in arbitrary units of time and velocity. [From Sreenivasan (1991)]. Reproduced with permission, from the Annual Review of Fluid Mechanics, Vol. 23, © 1991, by Annual Reviews Inc.
Turbulence is not random in the Gaussian sense, and significant departures from normal probability distributions are observed. The diffusivity of turbulence at high Reynolds numbers is generally orders of magnitudes greater than molecular diffusivity and on scales comparable to the dimensions of the flow field. Turbulent momentum transfer is the source of increased drag on immersed bodies and resistance to flow in pipelines. A vortex-stretching mechanism maintains turbulent fluctuations in three dimensions. This mechanism is absent in two-dimensional flow.
Turbulence occurs at large Reynolds numbers, usually as a result of the instability of laminar flow to small disturbances. In nature and in engineering practice, the Reynolds number will usually be high enough for flows to be turbulent. The fluctuations occur over a range of scales, from those compatible with the flow domain to the smallest scale allowed by viscous dissipation. The vortex-stretching mechanism maintains an "energy cascade" from large-scale motion where most of the turbulent kinetic energy is generated and contained, to motion at smaller scales where it is dissipated by viscosity. An important result of this process is that the smaller eddies lose directional orientation. The Navier–Stokes equations cannot be solved in closed form save for a few linear problems. Numerical solutions are possible at present only for low-Reynolds-numbers flows in which the range of scales is narrow. Such solutions require many hours of supercomputer time.
For reasons of space, attention is restricted to selected topics relating to turbulence in incompressible, uniform-property, Newtonian fluids. These are:
stability and transition from laminar flow,
the equations of the mean motion,
length and time scales of turbulence,
the energy spectrum,
the statistical treatment of isotropic turbulence,
the development of free turbulent shear flow,
the characteristics of turbulent wall flow,
gravitational effects on turbulence in density-stratified shear flow,
coherent structures and the proper orthogonal decomposition.
Statistical methods are described in the standard texts:
Batchelor (1953), Monin and Yaglom (1971), Bradshaw (1971, 1976), Tennekes and Lumley (1972), Hinze (1975), Lesieur (1990).
Its early development may be traced in the classic papers collected by Freidlander and Topper (1961). For the identification of coherent structures see Lumley (1981), Fiedler (1987) and Gatski et al. (1992), who also deal with compressibility effects. Atmospheric turbulence affected by density stratification is described by Wyngaard (1992). Developments in the chaotic behavior of non-linear systems, with application to turbulence, are to be found in Comte-Bellot and Mathieu (1987), Lumley (1990), Ottino (1990) and Sreenivasan (1991).
The systematic study of turbulence dates from the 19th century investigations of Osborne Reynolds who found that the character of pipe flow changed completely when the Reynolds Number Re ≡ ρUD/μ, exceeded a critical value. In this definition r and m are the fluid density and viscosity, U is the mean velocity and D the pipe diameter. Re is therefore a measure of the relative importance of inertia and viscous forces. At low values of Re, Reynolds found the flow to be laminar, the pressure drop moderate, and the velocity profile conforming to the steady-flow Hagen-Poiseuille distribution. For Re > 2300 the flow became unsteady, or turbulent, the resistance substantially increased, and the velocity distribution became more uniform on account of increased mixing. In uniform-pressure external boundary layers of thickness δ, the onset of turbulence occurs at values of ρUδ/μ of about 5000; in free shear flows at very low Reynolds numbers.
Bradshaw (1976) summarizes the stages in the development of turbulence from an initial unstable shear layer as:
The growth of disturbances with periodic fluctuations of vorticity,
Their secondary instability to three-dimensional disturbance if the primary fluctuations are two-dimensional,
The growth of three-dimensionality and higher harmonics of the disturbance, with spectral broadening by vortex-line interaction and
The onset of fully-developed turbulence with energy transfer across the spectrum to smaller and smaller scales.
The initial step in this process, the instability to small disturbances, has been treated with the aid of linearized theory [Lin (1955)], Hinze (1975)]. This is only the first stage in a sequence of events. Stuart [in Comte-Beliot and Mathieu (1987)] summarizes some recent developments, and Landahl and Mollo-Christensen (1986) provide a useful short survey, including the possibility of universality in transition to chaos.
Turbulence in an incompressible, uniform-property, fluid is described by the continuity and Navier-Stokes equations:
In the traditional treatment due to Reynolds, time-dependent quantities are decomposed into their mean and fluctuating components. Thus = U_{i} + u_{i} and = P + p. Substitution in Eqs. (1) and (2), followed by time or ensemble averaging, produces for the mean flow field:
These equations are indeterminate, there being now ten unknowns: three mean velocity components U_{i}, the pressure P, and six independent components of the Reynolds-stress tensor, ρu_{i}u_{j}. The equations for contain further unknowns of higher order.
The conservation of kinetic energy, is described by
where the terms on the right are
production by interaction with mean velocity gradients (here after P),
viscous dissipation (hereafter ξ),
turbulent, viscous and pressure diffusion.
The practice of formulating additional equations to form a closed set is known as Turbulence Modeling
The largest length scales in a turbulent flow are set by the dimensions of the flow field or the size of the body generating the flow disturbance. If the characteristic dimension and velocity are L and U respectively, a mean flow advection time scale is L/U. The characteristic time for viscous diffusion across a length L is L^{2}/ν and the ratio of these times is the Reynolds number, Re_{L} = UL/ν. The smallest scales, η and η^{2}/ν, are set by the dissipation rate of turbulent energy.
Figure 2 shows how turbulent kinetic energy is distributed in wavenumber space. E(κ,t) is the three-dimensional energy-spectrum function, κ is the wavenumber 2π(frequency)/U and
The large eddies at low wave numbers retain at least some of the characteristics of their origins and generally will be highly anisotropic in shear flows. Near isotropy is achieved only in grid turbulence in the absence of mean velocity-gradients. These eddies have also a quasi-permanent character with relatively long turnover times and distances, particularly in decaying turbulence when the motion at smaller scales is dissipated first.
Figure 2. Distribution of turbulent energy in wave number space. From Landahl and Mollo-Christensen (1986) by permission of Cambridge University Press.
As wavenumber κ → 0, E(κ) α κ^{n}, with n ranging from 3 to 5 according to different theories. In the energy-containing range, where E(κ) reaches a peak, the characteristic length and time scales are formed from k and its dissipation rate ε, i.e., = k^{3/2}/ε and t = k/ε. In this case ε is to be interpreted as the rate at which turbulent energy leaves the eddies in this range to travel down the spectral cascade to be dissipated at higher wave numbers. According to Kolmogorov's hypothesis, there is, at high Reynolds numbers, a range of high k where the turbulence is statistically in a state of universal equilibrium uniquely determined by ν and ε, with length and velocity scales η = (ν^{3}/ε)^{1/4} and υ = (νε)^{1/4}. The viscous diffusion time is equal to the eddy advection time, giving Re_{η} = υη/ν = 1.0. The energy-containing and dissipation ranges are widely separated on the high-Reynolds-number spectrum and the small eddies in the latter tend to be locally isotropic because the directional orientation of the large eddies is not transmitted in the vortex-stretching process of the energy cascade. A theoretical result for isotropic turbulence relates the dissipation rate to the energy spectrum through
Typically the peak in the dissipation spectrum occurs at κη ≈ 0.2. Between the energy-containing and dissipation ranges there exists an inertial subrange in which, at large Reynolds numbers, E(κ,t) is independent of η and determined solely by ε. It follows then that in this range:
where A is a constant of order unity.
An intermediate scale between L and η is the Taylor microscale λ. This is defined with respect to the dissipation rate through the relation ε = 15ν(u'/λ)^{2} where u' is the rms velocity fluctuation in isotropic turbulence. The ratios of the scales may be expressed in terms of the microscale Reynolds number R_{λ} ≡ u'λ/ν. For isotropic turbulence
where, with = k^{3/2}/ξ as before, B = 1.5^{1.5}. Equation (10) shows why direct numerical simulations are limited to low-Reynolds-number turbulence where the scale range is narrow. R_{λ} values in grid-generated wind-tunnel turbulence usually range from about 25 to 50 with, typically, ≈ 25 mm, η ≈ 0.5 mm. In an experimental verification of Kolmogorov's hypothesis for high-Reynolds-number turbulence in a laboratory air jet, Gibson (1963) measured ≈ 650 mm, η = 0.14 mm for R_{λ} = 780. In environmental flows R_{λ} may be of order 10^{4} or more, and the range of turbulence scales is correspondingly greater.
In the absence of mean-velocity gradients, inhomogeneous turbulence tends to isotropy, interactions with the fluctuating part of the pressure serving to equalize the energy content in each of the three component directions. Nearly-isotropic turbulence generated by a wind-tunnel grid is homogeneous in directions at right angles to the mean flow. The energy, Equation (5), reduces to Udk/dx = –ε, and the data are fitted by the power law k ≈ ax^{−n}, with n ≈ –1.25 initially and n ≈ –2.5 in the final period. Statistical theory and experiment have focused on the behavior of velocity correlation tensors for two points separated by the space vector r, e.g.:
from which the energy spectrum tensor is obtained:
If a Fourier analysis is made of the variation of velocity along a line in the x_{1} direction, the resulting spectrum function is related to R_{ij} by
For i = j = 1, Θ_{ij} is a longitudinal one-dimensional spectrum; i = j = 2 or 3 gives a lateral spectrum. Batchelor (1953) further defines the tensors obtained by averaging R_{ij} and Ф_{ij} over spherical surfaces in physical and wavenumber space. In particular,
and
Dimensionless spatial correlation coefficients are defined with respect to velocity components u_{p} and u_{n} parallel to and normal to the vector separation r:
A measure of the largest eddies in the flow is then given by the integral scale:
As r → 0 g(r) = 1 – (r/λ)^{2}, where λ is the Taylor microscale. The variation of g with r, and the relationship to Λ and λ, are shown in Figure 3
Figure 3. Relationship of the correlation function g(r) to the integral and microscales Λ, λ. (From Landahl and Mollo-Christensen (1986) by permission of Cambridge University Press.)
The single point correlations are components of the Reynolds-stress tensor. The dynamical equations for these and other double velocity correlations necessarily contain triple products, as in the k, Equation (5), and the Karman–Howarth equation in, for example, Batchelor (1953) and Hinze (1975).
Free turbulent shear flows unidirectional in the x-direction are described by the reduced form of Equation (4):
For a two-dimensional shear layer bounded by an irrotational free stream of velocity U_{∞}
Many flows described by this equation are at least approximately self-preserving in the sense that their velocity and shear-stress distributions may be scaled with velocity U_{s}(x) and thickness δ(x) to the self-similar forms: U/U_{s} = F(η), = g(η), where F and g are universal functions of η ≡ y/δ. Experimental evidence suggests that if self-preserving solutions are possible then the flows in question will tend to develop in this way. The stream function Ψ ≡ δU_{s}f(η) is used to transform Equation (19) into an ordinary differential equation. With –dP/dx = pU_{∞}dU_{∞}/dx
and f'(η) ≡ df/dη = F(η) = U/U_{s}, g'(η) = dg/dη. For the special case U_{s} = constant, g = 0, δ = (νx/U_{∞})^{1/2}, the result is the Blasius equation for the flat-plate laminar boundary layer: 2f''' + ff'' = 0. For self-similar development the coefficients in Eq. (20) must be independent of x. This condition is met by
linear growth, δ α x,
power-law development of the scaling velocity, U_{s} α x^{m},
negligible viscous diffusion, Re → ∞.
When the pressure is uniform, dP/dx = 0:
The result must also satisfy the requirements of momentum conservation. In the case of a plane jet in a co-flowing stream of velocity U_{∞}, for example, the condition
shows that self-preserving development is only possible when U_{∞} is zero. Then
Since δ is required to vary linearly with x, the scaling, or maximum velocity, must therefore decay as x^{−1/2}. The turbulent jet in a co-flowing stream (U_{∞} ≠ 0) is only approximately self-preserving. For a round jet, (δU_{S})^{2} = constant and U_{max} x^{−1}. In order to solve Equation (21) an assumption must be made for the shear-stress distribution g(η). It turns out that the rate of growth and velocity distributions can be predicted quite accurately using the mixing-length assumption:
when the mixing length m is taken as a constant of the order of 0.1δ. Exactly self-similar wake flow is impossible but it is approached far downstream of the generating body where the velocity deficit is vanishingly small. For the plane wake δ x^{1/2}, U_{min} x^{–1/2} and for the round wake δ x^{1/3}, U_{min} x^{–2/3}.
The experimental data collected and analyzed by Rodi (1975) have been supplemented by a few later measurements, notably Hussein et al. (1994). Spreading rates dδ/dx are of order 0.1, the width δ being chosen as the distance from the center to the point y_{1/2} where the velocity U = 0.5 (U_{max} – U_{min}). Figures 4 and 5 show self-preserving profiles and the energy balance in a plane jet, for which
Energy production reaches a maximum at about y_{1/2} where and ∂U/∂y are both large.
Turbulence is significantly affected by the presence of a solid boundary, whether in external flow, where a "Boundary Layer" forms, or in the developed flow through pipes and channels. In each case the turbulence is generated in a high-shear region close to the surface whence it diffuses outwards. Between the turbulence and the wall lies a thin sublayer in which the fluctuations are heavily damped by viscosity. Further from the wall the direct effects of viscosity are negligible but the influence of the wall is propagated by the shear stress. In a boundary layer this wall layer extends to about one fifth of the boundary layer thickness, δ, which is defined as the distance from the wall to the point where the mean velocity reaches 99% of its free-stream value, U_{∞}. In pipe flow the wall layer extends almost to the centerline, the flow being turbulent throughout. Boundary-layer turbulence, by contrast, is highly intermittent (Figures 6 and 7). The interface between the turbulent outer region and the irrotational free stream consists of a viscous superlayer whose thickness is of the order of the Kolmogorov microscale. In the outer layer, 0.2δ < y < 0.8δ the turbulence structure resembles that of free shear flow.
Figure 6. Schematic diagram of a boundary layer by means of smoke. Flow from left to right. From Tennekeg and Lumley (1972): A First Course in Turbulence, by permission of M.I.T. Press.
Figure 7. Visualization of a boundary layer flow from right to left. From Fernholz in Bradshaw P (Ed.) (1976); Turbulence. Topics in Applied Physics, 12, by permission of Springer Verlag GMBH and Co. KG.
In the viscous and turbulent layers close to the wall, Equation (19) reduces to a balance between the gradients of shear stress and pressure: ∂τ/∂y ≈ dP/dx. In fully-developed duct flow τ = τ_{w} + ydP/dx. When stress and velocity changes in the x-direction are relatively small, inner-layer properties depend on y, τ_{w}, p and v. Then, from dimensional analysis:
where the velocity scale is u_{τ} ≡ , is commonly called the "Friction Velocity". The dimensionless quantities U/u_{τ} and yu_{τ}/v are denoted by u^{+} and . Very close to the wall, for y^{+} < 5, and u^{+} = y_{+}. For y^{+} > 30 approximately, the direct effects of viscosity are negligibly small and f_{2} → constant; ∂U/∂y = u_{τ}/κy and:
For uniform-pressure flow on a smooth plate, von Karman constant κ ≈ 0.41 and C ≈ 5.2. In pipe flow the logarithmic zone extends almost to the centerline. The logarithmic dependence of U on y apparently holds regardless of pressure gradient, wall roughness or Reynolds number [but see George and Castillo (1993) for an opposing view]. The logarithmic and viscous sublayers are separated by a buffer layer in which neither viscous nor turbulent stresses are negligible.
In the outer region of a boundary layer the velocity distribution obeys the defect law:
Since the defect and logarithmic laws must overlap, the functions f_{3} and f_{i}, must be logarithms. Experimental data are correlated by
with C_{2} ≈ 2.35 for boundary layers and C_{2} ≈ 0.65 for pipes and channels.
An energy balance, Figure 8, shows that in the constant-stress wall layer, , the dominant terms in the k-Equation (25) are P ≡ and ξ, thus ξ ≈ P ≈ . Advection and diffusion are relatively small, though the latter, in transporting turbulent energy away from the wall, is responsible for the growth of the shear layer. Energy production reaches a maximum at the edge of the viscous layer and approximately half of the total occurs in the inner 5% of the boundary layer. The production mechanism close to the wall is associated with the bursting of the low-velocity fluid streaks in a regular spanwise array of alternate high- and low-velocity streaks. Favorable pressure gradients (dP/dx < 0) tend to reduce the rate and intensity of bursting and vice versa.
Figure 8. Energy balance in a boundary layer [results of Townsend (1956) reproduced by Hinze J. (1975): Turbulence, by permission of McGraw Hill, Inc.].
The effect of a positive streamwise pressure gradient is to decelerate the flow, possibly to the point where it separates and τ_{w} falls to zero. The result is a loss of pressure in, for example, diffusers and pipe fittings, or a loss of lift on wings, where the separation on the suction surface is referred to as stall. Conversely, reverse turbulent-to-laminar transition may occur in highly accelerated flow.
Turbulence is significantly affected by gravity in the presence of density fluctuations. If density changes are important only in the inertia terms (the Boussinesq approximation), the k-Equation 25 for a horizontal shear layer becomes:
where the "flux Richardson number" R_{f} is the ratio of k-production by buoyancy to shear production of k, viz
T and are respectively the mean and fluctuating parts of the temperature and, following geophysical practice, z is chosen as the vertical coordinate instead of y. Equation (31) shows that turbulence is suppressed in conditions of stable stratification (light fluid above heavy) and augmented when buoyancy is destabilizing, as in a shear layer heated from below. The critical value of R_{f} above which turbulence cannot be maintained is as low as 0.2 approximately. This is becau.se in the horizontal shear layer, turbulence is generated by the mean velocity gradient only in the streamwise (u) component, and by buoyancy only in the vertical (w) component. It follows that in stably stratified flow there is a strong tendency to two-dimensionality as .
The frequency of gravity waves in a stable atmosphere is the Brunt–Väisälä frequency:
When ∂p/∂z > 0 gravity waves are unstable and break up into turbulence. The gradient Richardson number is defined as the ratio of N^{2} to a typical turbulence frequency, ∂U/∂z (= u_{τ}/κz) in the logarithmic layer when buoyancy effects are small. Then:
For ∂T/∂z = in the logarithmic layer:
For large buoyancy effects the gradients of U and T are expressed more generally as
where H is the temperature flux in the z direction and L is the Monin-Obukov length scale L ≡ ; (z/L → R_{f} when both are small). Also for small positive z/L, the truncated Taylor series expansion φ;_{m} = 1 + βz/L leads to the wind profile corresponding to Equation (28) :
where the wall-law constant C in Eq. (33) is absorbed in z_{0} and β is a constant ≈7.
Measurements, flow visualization and the results of direct simulations are used to support a view that there are organized motions in turbulence which are not random. Coherent structures vary in form and scale from flow to flow and there are significant differences in the structure of wall and free shear flows. In an influential book, Townsend (1956) postulated the existence of a double structure of large eddies and small-scale turbulence and emphasized the role of the former in controlling turbulent transport. Kline et al. (1967) studied the near-wall structure and the "bursting" mechanism responsible for much of the production of turbulent energy. Cantwell (1981, 1990) and Fiedler (1987) survey the field, Robinson (1992) deals with structures in the boundary layer, and the symposium volume edited by Gatski et al. (1992) contains several useful papers, including the application of the "proper orthogonal decomposition theorem" [Lumley (1981), Berkooz et al. (1993)].
Fiedler (1987) defines coherent structures as spontaneously formed, nonstationary motional systems of correlated vorticity which
are in most cases of large scale, comparable to the lateral flow dimension, and flow specific in shape and composition,
are recurrent, having a characteristic life-span, typically of the average passage time of a structure,
exhibit a high measure of organization in structure as well as in dynamics although their appearance is at best quasi-periodic,
are similar to the corresponding structures in the laminar- turbulent transition.
Kline et al. (1967) found that the region very close to the wall, 5 < y^{+} < 70, is one of high turbulence activity which is associated with the behavior of low-speed streaks spaced at intervals of about 100v/υ_{τ} in the spanwise (z) direction, interacting with the outer flow in a sequence of four events starting deep in the viscous sublayer: gradual outflow, lift up, oscillation, and breakup. The oscillation occurs at about 8–12 wall units from the wall, the breakup in the region 10 < y^{+} < 30. These events are described as "bursting". They contribute as much as 70% of the turbulence production. Later measurements have shown that stress could be produced by a combination of ejections involving rapid outflow of low-speed fluid from the wall region, and sweeps involving the inflow of high-speed fluid toward the wall. These motions are highly intermittent. Two views of the mechanics of streak breakup are shown in Figures 9 and 10. Cantwell (1981), after surveying the data to 1980, identified four main constituents of the organized structure including the outer layer (Figure 7) — nearest to the wall is a fluctuating array of streamwise counter-rotating vortices. Above these, but still close to the wall, is a layer subjected to bursts of high-intensity small-scale motions. Intense small-scale motion is found also in the outer layer, primarily on the upstream-facing portions of the turbulent–non-turbulent interface, the backs of the bulges in the outer part of the layer. These are part of an overall transverse rotation on a scale comparable with the layer thickness.
Figure 9. Mechanics of streak breakup as visualized by Kline et al. (1976) (From Landahl and Mollo-Christensen (1986) by permission of Cambridge University Press.)
Figure 10. Breakup of a low-speed streak generating hairpin vortices. Reproduced, with permission, from the Annual Review of Fluid Mechanics © 1991 by Annual Reviews Inc.
Aubry et al. (1988) model this wall region by expanding the instantaneous field in so-called empirical eigenfunctions as permitted by the proper orthogonal decomposition theorem. The method leads to low-dimensional sets of ordinary differential equations, from the Navier-Stokes equations, via Galerkin projection. The equations exhibit intermittency as well as a chaotic regime, capturing major aspects of the ejection and bursting events. Robinson (1991) proposes an idealized model for low-Reynolds-number boundary layers based on vortical structures: quasi-streamwise vortices dominate the buffer region while archlike vortices are the most common vortical structure in the wake.
For the relationship to chaos, reference may be made to recent papers by Sreenivasan (1991), on fractals, and Ottino (1990) on mixing and chaotic advection.
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