Chapter 12-2 The Effect of Turbulence on Momentum Transfer

12.8 DESCRIPTION OF TURBULENCE Turbulent flow is the most frequently encountered type of viscous flow (yet not nearly well developed). In a turbulent flow the fluid and flow variables vary with time, for example, small random fluctuations in velocity occur about the mean value. F 12.11

V = f ( x , y , z , t ) = V ( x , y , z , t ) ：time-averaged velocity (12-44) V = f ( x , y , z , t ) = V ( x , y , z , t ) x x ：time-averaged velocity (12-45) t1 >> duration of any fluctuation. The mean value of is zero, (12-46)

Fluctuations contribute to the mean value of certain flow quantities. For example, the mean kinetic energy per unit volume The average of sum is the sum of the averages

dimensionless group or parameter A fraction of the total kinetic energy：magnitude of the turbulent fluctuations. The rms (root mean square) values of the fluctuations, is a significant quantity. Intensity of turbulence is defined as (12-48) dimensionless group or parameter v∞：mean velocity of the flow. The intensity of turbulence is a measure of the kinetic energy of the turbulence. In model testing, simulation of turbulent flows requires not only duplication of Reynolds number but also duplication of the turbulent kinetic energy.

12.9 TURBULENT SHEARING STRESSES In Chapter 7, the random molecular motion of the molecules was shown to result in a net momentum transfer between two adjacent layers of fluid in laminar flow. Large-scale fluctuation will result in a net transfer of momentum in the turbulent flow. F 7.4 F 12.12

The control-volume expression for linear mometum (5-4) momen The flux of x-directional momentum across the top of the control surface is (12-49) In the case of steady mean flow, the time derivative in equation (5-4) is zero. (12-50)

The turbulent fluctuations is seen to contribute a mean x directional momentum flux of per unit area. Their analytical description has not been achieved, even for the simple case. Analogy between the molecular exchange of momentum in laminar flow, and the macroscopic exchange of momentum in turbulent flow, the total shear stress： (12-51) The turbulent contribution to the shear is called Reynolds stress. In turbulent flows it is found that the magnitude of the Reynolds stress is much greater than the molecular contribution except near the walls.

By analogy with the form of Newton’s viscosity relation 【for laminar flow, 】, Boussinesq introduces the concept At (turbulent) (laminar) At：eddy viscosity. With introduction of eddy diffusivity of momentum, ( turbulent ) (laminar) (12-52)

12.10 THE MIXING-LENGTH HYPOTHESIS Prandtl in 1925： Mean free path in molecular momentum exchange (laminar flow) mixing length (turbulent flow) f 7.4 f 12.12 The velocity fluctuation is hypothesized as being due to the y-directional motion of a “lump” of fluid through a distance L. F 12.13 The instantaneous value of is then , (12-52) (12-51)

Pandtl assumed that must be proportional to . (experimental data show that there is some degree of proportionally between and .) Using , Prandtl expressed the time average, , as (12-53) The constant in (12-53), which is unknown, may be incorporated into the mixing length, (12-54) (12-51)

Comparison with Boussinesq’s expression for the eddy diffusivity ( i Comparison with Boussinesq’s expression for the eddy diffusivity ( i.e,(12-52) ) (12-55) (12-54) (12-51) =ρ (12-52)

12.11 VELOCITY DISTRIBUTION FROM THE MIXING-LENGTH THEORY Assumptions： 𝜏 𝑦𝑥 =𝜇 𝑑 𝜈 𝑥 𝑑𝑦 − 𝜌 𝜈 𝑥 𝜈 𝑦 = 𝜇 𝑑 𝜈 𝑥 𝑑𝑦 +ρ 𝐿 2 𝑑 𝜈 𝑥 𝑑𝑦 𝑑 𝜈 𝑥 𝑑𝑦 In the neighborhood of the wall the mixing length is assumed to vary as L=Ky. The shear stress is assumed to be entirely due to turbulence and to remain constant over the region of interest. The velocity 𝜈 𝑥 is assumed to increase in the y direction, and thus 𝑑 𝜈 𝑥 𝑑𝑦= 𝑑 𝜈 𝑥 𝑑𝑦 =𝜌 𝑘 2 𝑦 2 𝑑 𝜈 𝑥 𝑑𝑦 𝑑 𝜈 𝑥 𝑑𝑦 =𝜌 𝑘 2 𝑦 2 𝑑 𝜈 𝑥 𝑑𝑦 2

(dimensionless group) f(y/h) = ρ (12-56) by setting at y=h, whereby (12-57) (dimensionless group) f(y/h)

The constant K was evaluated by Prandtl and Nikuradse to have a value of 0.4. The agreement of experimental data for turbulent flow in smooth tubes with equation (12-57) is quite good (Figure 12.14.). F 12.14

12.12 THE UNIVERSAL VELOCITY DISTRIBUTION By introducing a dimensionless velocity . (12-58) we may write equation (12-56) as (12-59) Defining a pseudo-Reynolds number： (12-60)

Equation (12-61) indicates that or equation (12-59) becomes (12-61) Equation (12-61) indicates that or (12-62) F 12.15 F12.5

For turbulent core, y+≧30： Universal velocity distribution. (12-63) For the buffer layer, 30 ≧y+ ≧5 (12-64) For the laminar sublayer, 5＞y+＞0 (12-65) In rough tubes, the scale of the roughness is found to affect the flow in the turbulent core, but not in the laminar sublayer. The constant β becomes In β= for rough tubes.

12.13 FURTHER EMPIRICAL RELATIONS FOR TURBULENT FLOW Two important experimental results for turbulent flows： the power-law relation for velocity profiles and a turbulent-flow shear-stress relation due to Blasius. (A) velocity profile For flow in smooth circular tubes, the velocity profile may be correlated by (12-66) R：radius of the tube n：vary from a value of 6 at Re=4000 to 10 at Re= 3,200,000.

For flow over a flat plate At Re=105, n is 7. one-seven-power law, For flow over a flat plate with boundary layer of thickness δ, (12-67) Two obvious difficulties： the velocity gradient at the wall is infinite, the velocity gradient at δ is nonzero.

(B) Blasius’s correlation for shear stress For pipe-flow Reynolds numbers up to 105 and flat-plate Reynolds numbers up to 107, the wall shear stress： (12-68) ymax = R in pipes；ymax = δ for flat surfaces.

12.14 THE TURBULENT BOUNDARY LAYER ON A FLATE PLATE Boundary-layer thickness for turbulent flow over a smooth flat plate may be obtained from the von Karman momentum integral (similar to that in a laminar flow). In a laminar flow, a simple polynomial was assumed to represent the velocity profile. In a turbulent flow, velocity profile depends upon the wall shear stress and no single function adequately represents the velocity profile. For a zero pressure gradient the von Karman integral relation is 34 (12-38)

Employing the one-seventh-power law for υx and the Blasius relation, equation (12-68) for τ0 , equation (12-38) becomes (12-69) Where is written in place of . Performing the integration and differentiation (12-70) upon integration (12-71)

The local skin-friction coefficient If the boundary layer is assumed to be turbulent from the leading edge, x=0 (a poor assumption), 35 (12-72) The local skin-friction coefficient 36 (12-73) Several things are to be noted： First, they are limited to values of Rex＜107, Second, they apply only to smooth flate plates. Last, a major assumption the boundary layer to be turbulent from the leading edge.

12.15 Comparison of Laminar and Turbulent Boundary Layer At the same Reynolds number, the turbulent boundary layer is observed to be thicker, and is associated with a larger skin friction coefficient. In most cases of engineering interest, a turbulent boundary layer is desired because it resists separation better than a laminar boundary layer. F 12.16 (thus being able to remain unseparated for a greater distance in the presence of an adverse pressure gradient) The turbulent boundary layer has a greater mean velocity, hence both greater momentum and energy than laminar boundary layer.

12.15 FACTORS AFFECTING THE TRASITION FROM LAMINAR TO TURBULENT FLOW So far the occurrence of transition has been expressed in terms of the Reynolds number alone, while a variety of factors other than Re actually influence transition. The Reynolds number remains the principal parameter for predicting transition. T 12.2

(12-44)

instant velocity =average velocity (no fluctuations) Vx(Y)=Vx(Y), Vy=Vy=0 , Vz =Vz=0 indirect and macroscopic description of momentum transfer +X momentum is transferred to the CV along －y direction direct and microscopic description of momentum transfer dy x d yx u m t =

Turbulent shear stress (τyx) turb direct and microscopic description of turbulent momentum transfer Instant velocity with fluctuations of Vx’ , Vy’ and Vz’

y+L ⊕ L y-L V’xly=Vxly±L-Vxly =

Vmax h y

From Eqs.(5-5a) and (4-1), we obtain (12-35) The boundary-layer concept assumes inviscid flow outside the boundary layer, for which we may write Bernoulli’s equation, which may be rearranged to the form (12-36) the left-hand sides of equations (12-35) and (12-36) are similar.

The significant results of Blasius’ work： (a) The boundary thickness, δ, (b) The velocity gradient at the surface (c) The drag force and drag coefficient F 12.6

The coefficient of skin (local coefficient) (12-31) The mean coefficient of skin friction： consider a plate of uniform width W, and length L, (12-32)

－ =

(Vx’,Vy’=? , =? )

How regularity How element Shear stress Steady VX=VX V=V Fluctuation V=V+V’ Vx=Vx+Vx’ Vy=Vy+Vy’ Vz=Vz+Vz’ (Vx’,Vy’,Vz’=?) Molecules Fluid packets Diffasion of molecules Diffusion of molcules mixing of fluid packets Laminar Turbulent In the neighborhood of the wall ρ L=ky = y=0 =

=f(y+) f(y/h) V +~y+

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