Fluid Mechanics III

Reynolds experiments

First qualitative and quantitative results for turbulent flows were obtained by Osborn Reynolds in 1883, when he published results of his classical experiments on viscous flows in pipes (figure 1).1 In first set of experiments flow behaviour for different pipe diameters and flow rates was observed by using dye for flow visualisation. For small flow rates the coloured fluid did not mixed with the rest of the flow and moved within a distinct smooth streamline (fig. 1 A). For increased flow rate the fluid could exhibit sudden transition to the chaotic motion with intensive mixing of the coloured and not coloured fluids (fig. 1 B). When such flows where observed in a stro- boscopic light of an electric spark, complicated structure of the flow with multiple eddies of different scales have been discovered (fig. 1 C). Reynolds have therefore clearly distinguished between two different flow regimes: laminar and turbulent. It was established that transition to turbulent motion depends on a certain non-dimensional parameter known now as Reynolds number (Re = Ud/ν), and on the level of flow perturbations on the pipe inlet. For small perturbations laminar flow was obtained for Re 13000, for higher level of perturbations the transition occurred for smaller∼ values of , ℜ but turbulence never occurred for Re . 2000. Another set of Reynolds’ experiments was similar to your first year lab on pipe flows. He measured the head loss along pipes (h) for different flow rates (Q). He found that the relation between h and Q obtained in the experiment for gradually increasing flow rate is different from the results for decreasing flow rate (figure 2) When the flow rate was increased, the transition occurred at larger flow rates, and the critical flow rate in this case depends on initial flow perturbations. For decreasing flow rates the transition from turbulent to laminar motion always occurred at the flow rate corresponding to Re 2000.

1 O.Reynolds (1883) An experimental investigation of the circumstances which deter- mine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels, Phil. Trans. Roy. Soc. 174, 935–982 http://www.jstor.org/stable/114354

1 2

Fig. 1

The explanation of this behaviour is given by the modern theory of non- linear stability of fluid flows. The very simplistic version of this explanation is as follows. The starting point of developing of turbulence is fluid instability under certain flow conditions. Such conditions could be a Reynolds number (Re) and an amplitude of flow perturbations A caused by various external sources (noise, vibrations, etc.). A certain curve on the A-Re diagram subdivide regions of stable and unstable flows (figure 3) If we increase the Reynolds number for a given amplitude of external perturbations, the flow becomes unstable at a certain value of Re, and unstable perturbations grow fast to the maximal unstable amplitude causing the transition to turbulence. If we now start to decrease the Reynolds number, the high amplitude unstable oscillations will continue until the critical Reynolds number is reached, where instability disappears, and all perturbations decay except ones cause by external sources.

Mean ﬂow and turbulent pulsations

Turbulent flows are essentially unsteady and three-dimensional. However, a typical period of chaotic turbulent motion is much smaller then the time scale of the entire flow. This makes possible to represent turbulent flow as a superposition of chaotic, unsteady, three-dimensional turbulent pulsation and of a steady or slowly changing mean flow. The idea was proposed by Reynolds 3

Turbulent flow Decaying perturbations ( stable ) Reducing Re

Reducing Q Increasing Re

Growing perturbations Increasing Q Laminar flow ( unstable ) Amplitude of perturbations Pressure gradient ( dP / dx )

Initial perturbations Pipe flow ( Q or U ) Reynolds number ( Re ) Fig. 2 Fig. 3 in his 1895 paper.2 He derived equations describing the mean flow which are now known as Reynolds-averaged Navier-Stokes equations (RANS). See the paper and lecture slides for details. For a steady, parallel, two-dimensional mean flow we can write u = U(y)+ u′(x,y,z,t); v = v′(x,y,z,t); w = w′(x,y,z,t), where U = u is the mean velocity of the flow and u′, v′,w′ are three compo- nents of turbulent pulsations. Mean values are specified as ∆T 1 f = f(t)dt , ∆T Z 0 where the time used for averaging should be much larger then the typical period of turbulent pulsations, but much smaller than the time when the mean flow will change. The means of turbulent pulsations are zero u′ = v′ = w′ =0 , but the means of squares and products of turbulent pulsations are in general different from zero u′v′ = 0; u′2 = 0; v′2 =0 6 6 6 and are important characteristics of turbulent motion.

Turbulent stresses

Let us consider a small period of time ∆t, which is still much larger then the period of turbulent pulsations and can be used to specify mean quantities 2 O.Reynolds (1895) On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion, Phil. Trans. Roy. Soc. 186, 123–164 http://www.jstor.org/stable/90643 4

y x δx z v′(x,y,z,t) δz τt

U(y)+ u′(x,y,z,t)

Fig. 4 of turbulent motion. The momentum transferred through a small element δxδy of a plane parallel to the direction of the mean flow (figure 4) during this time is: ′ ′ ∆Ix = ρδxδz v (U + u ) ∆t . The momentum transfer will increase the momentum of the fluid above the plane. This is equivalent to the action of the mean force ∆I F = x = ρδxδz v′ (U + u′) ∆t on the lower surface of the plane. From the properties of mean values and the fact the U does not depends on time it follows that

v′ (U + u′)= v′U + v′u′ = v′ U + u′v′ = u′v′ and the force becomes F = ρδxδz u′v′ . The force is parallel to the flow and its direction is defined by the sign of u′v′. For positive u′v′ the force applied to the lower surface is positive and acts in the positive x-direction. The equal opposite force will act on the upper surface of the plane. Therefore, the mean flow behaves as if there is a shear force acting between the upper and lower layers of the fluid and the corresponding shear stress F τ = = ρ u′v′ . t −δxδy − is called a turbulent stress or Reynolds stress. The total effective shear stress acting on the mean flow of turbulent fluid is then the sum of the laminar stress dU τ = ρ ν l dy 5

y Turbulent U(y)

τ = τt Turbulent flow

Laminar

τ = τ0 Viscous sublayer x

Fig. 5 Fig. 6

and the turbulent stress τ = ρ u′v′ . t − Momentum interchange by turbulent velocity pulsations can be very intensive, and the resulting turbulent stresses are much larger then laminar stresses, which can be neglected in such cases. However, near a solid wall velocity pulsations become smaller and completely disappear on the wall due to the no-slip boundary condition, and in a thin layer near the wall viscous stresses become dominant (ﬁgure 5). This layer is called laminar sublayer or viscous sublayer. Because of the small thickness of the viscous sublayer the shear stress there can be assumed constant and equal to the wall shear stress τ0. Since turbulent stresses are very small inside the viscous sublayer and vanish on the wall, τ0 can be calculated by the usual formula for laminar stresses dU τ0 = µ . dy y=0

Turbulent velocity profiles are more uniform on the main part of the flow than laminar analogues. This can be explained by greater efficiency of turbulent stresses, which provide the same dynamical effect with smaller velocity gradients. This leads to higher velocity gradients near solid walls, where no-slip condition to be satisfied, increasing the wall shear stress and the total resistance of the flow. For example, a turbulent flow through a pipe will require higher pressure gradient to provide the same flow rate (figure 6). Prediction of turbulent stresses is an extremely complicated problem, and there is no a simple theory which could be used to calculate them. However, numerous approximate turbulent models are developed for solving problems of turbulent fluid motion. An important feature of these models is that they 6

U + ∆U τt v l

δx U U δz

Fig. 7 usually include parameters which can not be speciﬁed by theoretical consideration and should be found in the process model veriﬁcation by experiments.

Eddy viscosity

When size of turbulent vortexes is much smaller then the linear scale of an entire flow, it becomes possible to use the concept of turbulent viscosity or eddy viscosity, that is to represent turbulent stress in the form dU dU τ = µ = ρ ν , t t dy t dy where µt is the eddy viscosity, and νt is the kinematic eddy viscosity. The principal difference of the eddy viscosity from the fluid viscosity µ, is that it is not a property of a fluid, but depends on characteristics of the mean flow and is different for different points of the flow. Therefore, unlike the fluid viscosity, the eddy viscosity can not be measured in experiment and applied to other problems. This makes the direct application of eddy viscosity concept rather problematic.

Mixing length theory

The concepts of a mixing length incorporates the following model of momentum transfer between layers of the mean flow. A fluid particle from a layer with a mean velocity U drifts across the flow due to pulsations of vertical velocity v′. After travelling a certain distance l called the mixing length, the particle transmits its extra momentum to the surrounding fluid (figure 7). The mean velocity of the drift is proportional to the mean square velocity of vertical pulsations v = v′2 . p 7

Then the force caused by the momentum exchange is

F = ρ v δxδy ∆U , − where ∆U can be expressed as ∆U = ldU/dy. The corresponding stress is

dU τ = ρ v l = ρ u′v′ . t dy − The developed turbulence can often be considered as locally isotropic and homogeneous, that is in the vicinity of any point properties of turbulent motion do not depend on the position and direction. For such ﬂows

u′2 = v′2; u′v′ = v′2 = v2 . | | Combining this with the previous expression we obtain

dU v = l . dy

and the equation for the turbulent stress becomes

dU dU τ = ρ l2 , t dy dy

where the kinematic eddy viscosity is

dU ν = l2 . t dy

The mixing length l is not a universal constant of the flow and further consideration is required to reduce it to the form convenient to experimental verification. Let us consider the flow near the wall but outside of the viscous sublayer. The size of turbulent eddies in this region will be restricted by the distance to the wall. We can assume that the mixing length is proportional to the size of the turbulent eddies, that is

l = K y , where the vertical coordinate y is measured from the wall. The resulting equation for the turbulent stress near the wall is then

dU dU τ = ρK2 y2 . t dy dy

8 τ U/U

log10(Uτ y/ν) Fig. 8

Close to the wall the shear stress is almost constant and is approximately equal to the wall shear stress. For the positive velocity gradient we then have

dU τ0 1 = . dy r ρ Ky Integration of this equation gives the classical logarithmic proﬁle of mean velocity in a turbulent ﬂow close to the wall:

τ0 1 U = log y + C , r ρ Ky where the value τ0 Uτ = r ρ is called the friction velocity. It is conventional to write the expression for the turbulent logarithmic velocity proﬁle in the following canonical form:

U Uτ y = B log10 + A , Uτ ν and for a pipe flow the comparison with experiment gives (figure 8) A =5.5; B =5.75 . Note that the solution goes to minus infinity when approaching the wall (y 0). This means that the logarithmic is not valid in the viscous sublayer → 9 close to the wall. The logarithmic profile still can be used to derive the equation for friction coefficient f of a smooth pipe. The resulting equation was found to be 1 f = 2 , (4 log10(Re√f) 0.4) − which represents the lower bound of Moody diagram, and is in a good agreement with experiment in the range of Reynolds numbers 3 103 . Re . 6 × 3 10 . This equation does not give an explicit relation between Re and f ×and can be solved by iterations. An alternative experimental formula for friction coefficient f =0.0791 (Re)−1/4 gives good agreement with experiments for turbulent flows in smooth pipes with Reynolds numbers smaller then 105. 10

Reading:

Massey,B.S. Mechanics of Fluids, 8th edition

Chapter 1: Fundamental concepts 1.9 Classification and description of fluid flow 1.9.2 Laminar and turbulent flows

Chapter 7: Flow and losses in pipes and ﬁttings 7.2 Flow in pipes of circular cross section 7.2.1 Aspects of laminar and turbulent ﬂow in pipes

Chapter 8: Boundary layers, wakes and other shear layers 8.11 Eddy viscosity and the mixing length hypothesis 8.12 Velocity distribution in turbulent ﬂow 8.13 Free turbulence

Douglas,J.F., etal. Fluid Mechanics

Chapter 4: Motion of Fluid Particles and Streams 4.10 Laminar and Turbulent Flow Chapter 10: Laminar and Turbulent Flows in Bounded Systems 10.6 Velocity distribution in turbulent fully developed pipe flow. Chapter 11: Boundary Layer 11.3 Factors affecting transmission from laminar to turbulent flow regimes 11.4 Discussion of flow patterns and regions within the turbulent boundary layer 11.5 Prandtl mixing length theory