John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 45
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In the turbulent flow case, pictured in Fig. 6.18, we can think ofPrandtl’s parcels of fluid (rather than individual molecules) as carryingthe x-momentum. Let us rewrite eqn. (6.45) in the following way: ,• The shear stress τyx becomes a fluctuation in shear stress, τyxresulting from the turbulent movement of a parcel of fluid• changes from the mean free path to the mixing length• C is replaced by v = v + v , the instantaneous vertical speed of thefluid parcel• The velocity fluctuation, u , is for a fluid parcel that moves a distance through the mean velocity gradient, ∂u/∂y.
It is given by(∂u/∂y).Then"! = (constant) ρ v + v uτyx(6.84)Equation (6.84) can also be derived formally and precisely with thehelp of the Navier-Stokes equation. When this is done, the constantcomes out equal to −1. The average of the fluctuating shear stress isτyxρ=−TT0 vu + v u dt = −ρv u −ρv u=0(6.85)Turbulent boundary layers§6.7317Figure 6.18 The shear stress, τyx , in a laminar or turbulent flow.Notice that, while u = v = 0, averages of cross products of fluctuations(such as u v or u 2 ) do not generally vanish.
Thus, the time average ofthe fluctuating component of shear stress isτyx= −ρv u(6.86)In addition to the fluctuating shear stress, the flow will have a mean shearstress associated with the mean velocity gradient, ∂u/∂y. That stress isµ(∂u/∂y), just as in Newton’s law of viscous shear.It is not obvious how to calculate v u (although it can be measured),so we shall not make direct use of eqn.
(6.86). Instead, we can try to modelv u . From the preceding discussion, we see that v u should go to zerowhen the velocity gradient, (∂u/∂y), is zero, and that it should increasewhen the velocity gradient increases. We might therefore assume it to beproportional to (∂u/∂y). Then the total time-average shear stress, τyx ,can be expressed as a sum of the mean flow and turbulent contributionsthat are each proportional to the mean velocity gradient.
Specifically,∂u− ρv u∂ysome other factor, which ∂u∂u+=µreflects turbulent mixing ∂y∂yτyx = µ(6.87a)τyx(6.87b)≡ ρ · εmorτyx = ρ (ν + εm )∂u∂y(6.87c)318Laminar and turbulent boundary layers§6.7where εm is called the eddy diffusivity for momentum. We shall use thischaracterization in examining the flow field and the heat transfer.The eddy diffusivity itself may be expressed in terms of the mixinglength. Suppose that u increases in the y-direction (i.e., ∂u/∂y > 0).Then, when a fluid parcel moves downward into slower moving fluid,it has u (∂u/∂y). If that parcel moves upward into faster fluid,the sign changes.
The vertical velocity fluctation, v , is positive for anupward moving parcel and negative for a downward motion. On average,u and v for the eddies should be about the same size. Hence, we expectthat ∂u ∂u± ∓ ∂y ∂y ∂u ∂u= ρ(constant) 2 ∂y ∂y∂u= −ρv u = −ρ(constant)ρεm∂y(6.88a)(6.88b)where the absolute value is needed to get the right sign when ∂u/∂y < 0.Both ∂u/∂y and v u can be measured, so we may arbitrarily set theconstant in eqn.
(6.88) to unity to obtain a measurable definition of themixing length. We also obtain an expression for the eddy diffusivity:εm.= ∂y 2 ∂u (6.89)Turbulence near wallsThe most important convective heat transfer issue is how flowing fluidscool solid surfaces. Thus, we are principally interested in turbulence nearwalls. In a turbulent boundary layer, the gradients are very steep nearthe wall and weaker farther from the wall where the eddies are largerand turbulent mixing is more efficient.
This is in contrast to the gradualvariation of velocity and temperature in a laminar boundary layer, whereheat and momentum are transferred by molecular diffusion rather thanthe vertical motion of vortices. In fact,the most important processes inturbulent convection occur very close to walls, perhaps within only afraction of a millimeter. The outer part of the b.l. is less significant.Let us consider the turbulent flow close to a wall. When the boundarylayer momentum equation is time-averaged for turbulent flow, the resultTurbulent boundary layers§6.7319is∂u∂u∂u∂ρ u+vµ− ρv u=∂x∂y∂y∂y(6.90a)neglect very near wall∂τyx∂y∂∂u=ρ (ν + εm )∂y∂y=(6.90b)(6.90c)In the innermost region of a turbulent boundary layer — y/δ 0.2,where δ is the b.l.
thickness — the mean velocities are small enoughthat the convective terms in eqn. (6.90a) can be neglected. As a result,∂τyx /∂y 0. The total shear stress is thus essentially constant in y andmust equal the wall shear stress:τw τyx = ρ (ν + εm )∂u∂y(6.91)Equation (6.91) shows that the near-wall velocity profile does not depend directly upon x. In functional form u = fn τw , ρ, ν, y(6.92)(Note that εm does not appear because it is defined by the velocity field.)The effect of the streamwise position is carried in τw , which varies slowlywith x.
As a result, the flow field near the wall is not very sensitiveto upstream conditions, except through their effect on τw . When thevelocity profile is scaled in terms of the local value τw , essentially thesame velocity profile is obtained in every turbulent boundary layer.Equation (6.92) involves five variables in three dimensions (kg, m, s),so just two dimensionless groups are needed to describe the velocityprofile: ∗ uu y= fn∗uν(6.93)3where the velocity scale u∗ ≡ τw /ρ is called the friction velocity. Thefriction velocity is a speed characteristic of the turbulent fluctuations inthe boundary layer.320Laminar and turbulent boundary layers§6.7Equation (6.91) can be integrated to find the near wall velocity profile:uτwdu =ρ 0 y0dyν + εm(6.94)=u(y)To complete the integration, an equation for εm (y) is needed.
Measurements show that the mixing length varies linearly with the distance fromthe wall for small y = κyfory/δ 0.2(6.95)where κ = 0.41 is called the von Kármán constant. Physically, this saysthat the turbulent eddies at a location y must be no bigger that the distance to wall. That makes sense, since eddies cannot cross into the wall.The viscous sublayer. Very near the wall, the eddies must become tiny; and thus εm will tend to zero, so that ν εm . In other words, inthis region turbulent shear stress is negligible compared to viscous shearstress.
If we integrate eqn. (6.94) in that range, we findu(y) =τwρy0dyτw y=ρ νν(u∗ )2 y=ν(6.96)Experimentally, eqn. (6.96) is found to apply for (u∗ y/ν) 7, a thin region called the viscous sublayer. Depending upon the fluid and the shearstress, the sublayer is on the order of tens to hundreds of micrometersthick. Because turbulent mixing is ineffective in the sublayer, the sublayer is responsible for a major fraction of the thermal resistance of aturbulent boundary layer. Even a small wall roughness can disrupt thisthin sublayer, causing a large decrease in the thermal resistance (but alsoa large increase in the wall shear stress).The log layer.
Farther away from the wall, is larger and turbulentshear stress is dominant: εm ν. Then, from eqns. (6.91) and (6.89)∂u2 ∂u ∂u= ρ (6.97)τw ρεm ∂y ∂y∂yTurbulent boundary layers§6.7321Assuming the velocity gradient to be positive, we may take the squareroot of eqn. (6.97), rearrange, and integrate it:2du =τwρu(y) = u∗=dydy+ constantκyu∗ln y + constantκ(6.98a)(6.98b)(6.98c)Experimental data may be used to fix the constant, with the result that ∗ 1u(y)u y+Bln=∗uκν(6.99)for B 5.5. Equation (6.99) is sometimes called the log law. Experimentally, it is found to apply for (u∗ y/ν) 30 and y/δ 0.2.Other regions of the turbulent b.l.
For the range 7 < (u∗ y/ν) < 30,the so-called buffer layer, more complicated equations for , εm , or u areused to connect the viscous sublayer to the log layer [6.7, 6.8]. Here, actually decreases a little faster than shown by eqn. (6.95), as y 3/2 [6.9].In contrast, for the outer part of the turbulent boundary layer (y/δ 0.2), the mixing length is approximately constant: 0.09δ. Gradientsin this part of the boundary layer are weak and do not directly affecttransport at the wall. This part of the b.l. is nevertheless essential tothe streamwise momentum balance that determines how τw and δ varyalong the wall.
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