Technology & Engineering
Eddy Viscosity
Eddy viscosity is a measure of the turbulent diffusion of momentum in a fluid. It is used to model the effects of turbulence in fluid flow, particularly in engineering applications such as aerodynamics and hydrodynamics. Eddy viscosity is a key parameter in many computational fluid dynamics simulations.
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9 Key excerpts on "Eddy Viscosity"
eBook - ePubStatistical Turbulence Modelling for Fluid Dynamics — Demystified
An Introductory Text for Graduate Engineering Students
- Michael Leschziner(Author)
- 2015(Publication Date)
- ICP(Publisher)
Chapter Seven 7. The Eddy Viscosity 7.1 Conceptual foundation In this chapter, we turn to the key question of how to determine the unknown turbulence correlations — the Reynolds stresses — in the Reynolds-averaged Navier–Stokes (RANS) equations, i.e. Eq. (3.9). The discerning reader may well already recognise that the most rational and straightforward approach to this question would be to attempt to solve more general forms of equations of the type Eq. (4.6), which govern the evolution of the Reynolds stresses. Indeed, this is a possible route that will be discussed in detail in Chapter 12. However, it is also a highly challenging route, because many terms in these equations are unknown, and their modelling is very difficult. Even after closure is achieved, there remains the far from trivial task of solving, numerically, the highly coupled, non-linear set of equations for the Reynolds stresses — a task that is computationally often more challenging than solving the RANS equations themselves. This is certainly not the route that was adopted, therefore, in early efforts to model the correlations and solve the RANS equations numerically or analytically — the latter only possible for a few especially simple shear flows. An extremely influential idea, introduced by Joseph Valentin Boussinesq in Essai sur la théorie des eaux courantes (1877), is the eddy-viscosity concept. This is based on the notion that the action of turbulence is analogous to the Brownian motion that is responsible for fluid viscosity — an analogy that is, ultimately, flawed, for it overlooks the spatial coherence of turbulent structures (two-point correlations and vortical motions, in particular). Thus, the proposal simply states that, similar to fluid viscosity in laminar flows, a flow-properties-dependent turbulent viscosity may be added to the molecular agitation to represent turbulent mixing or diffusion
- Andrew L. Gerhart, John I. Hochstein, Philip M. Gerhart(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Unlike the absolute viscosity, μ, which is a known value for a given fluid, the Eddy Viscosity is a function of both the fluid and the flow conditions. That is, the Eddy Viscosity of water cannot be looked up in handbooks—its value changes from one turbulent flow condition to another and from one point in a turbulent flow to another. The inability to accurately determine the Reynolds stress, −ρ ‾ u ′ υ ′ , is equivalent to not knowing the Eddy Viscosity. Several semiempirical theories have been proposed (Ref. 3) to determine approximate values of η. L. Prandtl (1875–1953), a German physicist and aero- dynamicist, proposed that the turbulent process could be viewed as the random transport of bundles of fluid particles over a certain distance, ℓ m , the mixing length, from a region of one velocity to another region of a different velocity. By the use of some ad hoc assump- tions and physical reasoning, it was concluded that the Eddy Viscosity was given by η = ρℓ m 2 | d _ u ___ dy | Thus, the turbulent shear stress is τ turb = ρℓ m 2 ( d _ u ___ dy ) 2 (8.28) The problem is thus shifted to that of determining the mixing length, ℓ m . Further considerations indicate that ℓ m is not a constant throughout the flow field. Near a solid surface the turbulence is dependent on the distance from the surface. Thus, additional assumptions are made regarding how the mixing length varies throughout the flow. The net result is that as yet there is no general, all-encompassing, useful model that can accurately predict the shear stress throughout a general incompressible, viscous Various ad hoc assumptions have been used to approx- imate turbulent shear stresses. 0 R r 0 (b) u(r) V c Outer layer Overlap layer Viscous sublayer τ w lam turb τ τ τ τ Pipe wall Pipe centerline 0 R r 0 (a) (r) FIGURE 8.15 Structure of turbulent flow in a pipe. (a) Shear stress. (b) Average velocity. 8.3 Fully Developed Turbulent Flow 329 turbulent flow.
- Andrew L. Gerhart, John I. Hochstein, Philip M. Gerhart(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Unlike the absolute viscosity, μ, which is a known value for a given fluid, the Eddy Viscosity is a function of both the fluid and the flow conditions. That is, the Eddy Viscosity of water cannot be looked up in handbooks—its value changes from one turbulent flow condition to another and from one point in a turbulent flow to another. The inability to accurately determine the Reynolds stress, −ρ ‾ u ′ υ ′ , is equivalent to not knowing the Eddy Viscosity. Several semiempirical theories have been proposed (Ref. 3) to determine approximate values of η. L. Prandtl (1875–1953), a German physicist and aero- dynamicist, proposed that the turbulent process could be viewed as the random transport of bundles of fluid particles over a certain distance, ℓ m , the mixing length, from a region of one velocity to another region of a different velocity. By the use of some ad hoc assump- tions and physical reasoning, it was concluded that the Eddy Viscosity was given by η = ρℓ m 2 | d _ u ___ dy | Thus, the turbulent shear stress is τ turb = ρℓ m 2 ( d _ u ___ dy ) 2 (8.28) The problem is thus shifted to that of determining the mixing length, ℓ m . Further considerations indicate that ℓ m is not a constant throughout the flow field. Near a solid surface the turbulence is dependent on the distance from the surface. Thus, additional assumptions are made regarding how the mixing length varies throughout the flow. The net result is that as yet there is no general, all-encompassing, useful model that can accurately predict the shear stress throughout a general incompressible, viscous Various ad hoc assumptions have been used to approximate turbulent shear stresses. 0 R r 0 (b) u(r) V c Outer layer Overlap layer Viscous sublayer τ w lam turb τ τ τ τ Pipe wall Pipe centerline 0 R r 0 (a) (r) FIGURE 8.15 Structure of turbulent flow in a pipe. (a) Shear stress. (b) Average velocity. 404 CHAPTER 8 | Viscous Flow in Pipes turbulent flow.
- Steven N. Shore(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
C H A P T E R 2 Viscosity and Diffusion No one means all he says, and yet very few say all they mean, for words are slippery and thought is viscous. Henry Adams, The Education of Henry Adams 2.1 Introduction The action of internal friction is the most important dissipation mechanism in fluids. It acts on all scales, from molecular to megaparsec sizes, and powers most of the internal energy conversion processes. In the laboratory it is often an annoyance, making flows difficult to control and unstable. In the cosmos, it appears that viscous action, whether by molecular or turbu-lent processes, shapes most of the hydrodynamic structure we see. Here we will look at a general approach to what viscosity is, and how it acts, in a rather traditional fashion. The generalizations will have to wait for now. But once you have a feel for what viscous coupling is about, it should be possible to anticipate some applications even before we reach them. The particle jumps and momentum coupling within a continuous medium can have many causes. On the kinetic level, they result from collisions. On a macroscopic level, they may be due to turbulent eddies transporting material through the fluid or gas. Collisions of waves within a fluid can simulate a turbulence, indeed power it, and these will produce a macroscopic diffusion. Because such waves can transport momentum as well as mass, they can be thought of as a source of friction, just as the kinetic scale can produce a similar effect. This friction is called the viscosity and gives rise to most of the interesting effects in fluids. For instance, Feynman referred to two kinds of fluid-mechanical treatments, those con-cerned with dry water being ones which neglect viscosity (so-called ideal fluids) and those concerned with wet water ones which include such 3 4 2.2 The Navier-Stokes Equation 3 5 effects. In fact, it is viscosity that produces the phenomenon of wetting, the adhesion of a fluid to a surface, so the labels are most apt.
- Richard C. Farmer, Ralph W. Pike, Gary C. Cheng, Yen-Sen Chen(Authors)
- 2009(Publication Date)
- CRC Press(Publisher)
C H A P T E R 4 Turbulence Modeling Concepts 4.1 REYNOLDS AVERAGING AND Eddy Viscosity MODELS Physically, turbulent flow has some meaning to most of us as some sort of chaotic fluid motion. Tree concepts allow us to begin obtaining a quan-titative measure of turbulence. In 1883, O. Reynolds observed that a dye jet injected cocurrently into water flowing in a smooth tube retained is jet-like stream until some critical velocity afer which the dye stream rapidly broke-up and mixed with the water (McKusick and Wiskind 1959). A larger pressure drop in the pipe was also observed afer this critical velocity was reached. Pursuing his studies on turbulence, Reynolds (1895) averaged the Navier–Stokes equations in time to define mean and fluctuating properties, i.e., the “Reynolds stresses.” Prandtl in 1904 (Schlichting, 1979) explained the effects of friction as a modification of the mean velocity profile in the near vicinity to the wall, i.e., the boundary layer for laminar and turbulent flow. Tus, turbulent flow mixes more rapidly than laminar flow, has fluc-tuating fluid properties imposed on an otherwise laminar-like velocity field, and produces more friction when these ragged flows are slowed down by a wall to produce a no-slip condition. Turbulent flow has other characteristics, but these are the most important for influencing transport phenomena. Early investigators defined turbulence as: “an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surfaces or even when neighboring streams of the same fluid flow past or over one another.” Taylor and Von Karman initiated the 191 192 ■ Computational Transport Phenomena for Engineering Analyses application of statistical theories to describe these irregular motions. A collection of their works is compiled in Friedlander and Topper (1961).
eBook - PDFTurbulence Phenomena
An Introduction to the Eddy Transfer of Momentum, Mass, and Heat, Particularly at Interfaces
- J.T. Davies(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
They are often known as the Eddy Viscosity and eddy kinematic viscosity. In words, the drag μ Ε on the fluid due to momentum effects is proportional to the velocity fluctuations and to their distances (as with the product of the average molecular velocity and the mean free path in the kinetic theory of gases). In practice, the term v y r is difficult to measure, and it is therefore required to express i> E in a form not involving v v '. Prandtl achieved this by assuming v y ' to be simply related to the velocity fluctuation v x [= I dvjdy, cf. Eq. (1.15)]. Toward the wall of the pipe, dvjdy is large for the fluid in turbulent flow, and it is clear that a pulse of fluid moving away from the wall with 16 7. Velocities and Stresses in Turbulent Flows positive v y will be entering a faster-moving part of the stream, where it will decrease locally the velocity in the x direction. Thus a positive v y should correlate with a negative v X9 particularly near the walls of the tube. Toward the center, where dvjdy is small, the correlation will be much less pro-nounced. Accordingly, much of the stress τ (= — Qv x r v y ) will originate in the turbulent fluid fairly near the walls (but not immediately adjacent to the walls). Figures 1-7 and 3-3 show typical experimental results. By Eq. (1.15), v x = I dvjdy, and by analogy with this, Prandtl (1925, 1967) wrote that for fluctuations in the y direction, v y oc I dvjdy. He then >0.5 > 0.5 FIG. 1-7. Experimental distribution of —v x 'v y ' in turbulent pipe flow. This represents the turbulence shear stress, since τ = —QVx'v y '. Very close to the wall (a), the stress being generated falls rapidly to zero, but the turbulence stress is effectively constant (at a mean value denoted τ 0 ) from y + > 20 [the dimensionless distance y + being defined by Eq. (1.36)]. This constant stress persists out to about y = 0.1a, thereafter falling to zero at the center of the pipe (b).
- George E. Totten, Victor J. De Negri(Authors)
- 2011(Publication Date)
- CRC Press(Publisher)
The importance of viscosity is far more complex than described here, but these simple descriptions are a starting point for a discussion of viscosity and its relevance to hydraulic fluids. 4.1.1 A BSOLUTE V ISCOSITY AND N EWTON ’ S L AW Viscosity is a fluid’s resistance to flow. The viscosity of a fluid, either liquid or gas, describes the opposition to a change in shape or to movement. A mathematical description was first developed by Sir Isaac Newton as a special case of his second law of motion: Shear stress (Coefficient of viscosity) (Shear rate), = × (4.1) where the shear stress is the force per area and the shear rate is the velocity gradient. This leads to a description of viscosity as the force that must be overcome to cause a given fluid motion. Newton’s viscosity law applies when laminar, or non-turbulent, flow occurs. Laminar flow can be imagined as numerous, discrete layers (lamina) of the fluid moving in the same direction, but with the velocity of each layer varying, depending on the distance from a system boundary. The concept of velocity as a function of distance is known as the “velocity gradient.” Fluid flow through a tube caused by an applied pressure, known as “Poiseuelle flow,” is repre-sented in Figure 4.1. The arrows represent layers moving in streamlines between stationary, parallel walls. The layer closest to a wall is assumed to adhere to the wall and, in turn, will cause a frictional drag on the next closest layer, which is moving because of the applied force. Successive layers expe-rience gradually reduced frictional drag and thus flow at successively higher velocities. The central layer encounters the least friction from adjacent layers and, consequently, has the highest velocity. This change in velocity—from zero at the boundaries to the highest at the center—describes the term velocity gradient.
eBook - ePub- David Ting(Author)
- 2016(Publication Date)
- Academic Press(Publisher)
Chapter 8 .1.1.6. Three-Dimensional
Turbulence is intrinsically three-dimensional. The term “two-dimensional turbulence” is only used to describe the simplified case where flow is restricted to two dimensions. Based on this description, we can note that two-dimensional turbulence is not true turbulence. Vorticity fluctuations cannot be two-dimensional because vortex stretching, an important vorticity-maintenance mechanism, is not present in a two-dimensional flow.1.1.7. Highly Diffusive
Turbulent flows are also highly diffusive, and their diffusivity is much greater than that of a laminar flow (molecular diffusivity). The highly diffusive turbulence causes rapid mixing and increased rates of momentum, heat, and/or mass transfer. An easy example for students in a flow turbulence class to remember is also a smelly one. Suppose a student in the class ate too many beans or other healthy foods that promote flatulence and happened to release some of the by-product in the middle of the class. Most of the students, other than those seated next to the culprit, would probably not be aware of the release if the air in the classroom was largely stagnant. Under these conditions, the by-product could only spread via molecular diffusivity. In reality, thanks to turbulent air motion, everyone gets a dose of the gas within a minute or two. This dissipation is proof that the air in the room is turbulent. As such, even an apparently random flow pattern is not turbulent if it does not exhibit the spreading of velocity fluctuations throughout the surrounding fluid, as is the case with, for example, a constant diameter jet.1.1.8. Turbulent Flows are Flows
Turbulence is not a feature of fluids but of fluid flows. Turbulence is different for different flows, even though all turbulent flows have many common characteristics. Thus, the research approach of borrowing from molecular diffusivity and/or gas kinetic theories and applying them to flow turbulence is fundamentally unfounded. Notwithstanding that, flows can be compared, provided they are not too dissimilar.
eBook - PDF- Hillel Rubin(Author)
- 2001(Publication Date)
- CRC Press(Publisher)
(5 .5. 15) . The process of energy transfer from larger to successively smaller eddies continues until the eddies become so small that viscous effects become important and the energy is dissipated. This stage is characterized by an eddy Reynolds number approximately equal to one, where the eddy Reynolds number is defined using the characteristic length and velocity of the smallest eddies. This reflects the idea that at these smallest scales of motion, the inertial strength of the eddy is approximately equal to its viscous transport strength, or the Eddy Viscosity, is approximately equal to the kinematic viscosity. Thus (5.6.4) Introduction to Turbulence 245 where J) and 1') are the velocity and length scales, respectively, for the smallest eddies. We also know (see Eq. 5.5.24) £ ~ (~r (5.6. 5) J) By combining Eqs . (5 .6 .4) and (5 .6.5 ), we obtain estimates for the smallest eddies, in terms of the dissipation ra te. These are called the Kolmogorov microscales, J) = (J)£)1/4 (5 .6.6) A micro-time scale, t', also can be defined as the ratio of 1') to J) , t' = !!.. (5 .6.7) J) As will be seen in later chapters, the scales of turbulent motion have strong influences on transport and mixing properties for water quality modeling. They also control a number of processes of direct interest in environmental flow modeling, such as particle-particle interactions and contaminant desorption phenomena. Some of these applications are described further in Part 2 of this text. PROBLEMS Solved Problems Prob le m 5.1 Consider a turbulent flow of water with a measured power spectral density curve as shown and listed in Fig. 5.17. ( a) A common estimate for the turbulent velocity scale (denoted by u for this problem) is the root-mean-square value of the fluctuations, u = u~s = (u '2 )1/2, where the u' are the fluctuating velocities. Calculate u for this flow, using the fact that the average value for U '2 is the autocorrelation for a time lag of O.
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