Algebraic turbulence models for non-isothermal boundary layers

Now we assume that ˙q^V vanishes in the boundary layer. The remaining difficulty arises from the pressure gradient. As an immediate consequence of Bernoulli’s equation (−dP/dx_i = P3

j=1Uj∂Ui∂xj), decelerating flow (dU_∞/dx <0) corresponds to a positive, so-called ad-versepressure gradient, which can lead to separation of the boundary layer from the surface.

But also in attached boundary-layer flows the effect of a pressure gradient on the mean flow and on the Reynolds stresses can be significant. This is described in more details e.g.

in [DPR01], pp.66. Moreover, pressure gradients might affect the value ofP r_T, see [KC93], although such effects are small. Nevertheless, in this thesis we restrict ourselves to zero pressure gradient boundary layers. For our purposes this approximation is reasonable since in indoor-air flow problems, buoyance is the driving force of the air movement and pressure gradients are hopefully small. (However, in natural convection in a closed cavity, adverse pressure gradients occur as the flow approaches the corners, see [TODB98], p.290.) Then (5.25)-(5.26) reduce to the following coupled system of ordinary differential equations:

− d momentum and heat transfer equation look analogously. As shown in Section C.2, when suitably normalized, both have the same profile, an observation calledReynolds analogy.

5.3. Algebraic turbulence models for non-isothermal boundary layers

5.3.1. The substructure of a forced convection turbulent boundary layer

The total stress is the sum of the viscous stressν∇U and the Reynolds stress −hu⁰⊗u⁰i. The fundamental observation is that in attached boundary layers the profiles of the viscous and Reynolds stresses are universal, i.e., for different Reynolds numbers their normalized profiles collapse when they are plotted againsty⁺. On the basis of the relative magnitude of the stresses and motivated by the profile for U several regions can be distinguished (cf.

[Pope00], p. 275), see Table 5.7.

In the literature, the near-wall edge of the log layer varies from y⁺ = 40 (Durbin) to y⁺ = 50 (Pope). An understanding of the names ”log-law” and ”law of the wake” will

5. Near-wall treatment in turbulence modelling

Region Location Characteristic property

Near-wall region y⁺<40 The viscous contribution to the shear stress

(viscous wall region) is significant

Viscous sublayer y⁺<5 The viscous stress highly dominates the Reynolds shear stress

Buffer layer 5< y⁺<40 large production, large Reynolds stress anisotropy

Outer layer y⁺>40 Direct effects of the viscous stresses onU are negligible

Log-law region y⁺>40, y/δ <0.2 The log-law holds

Defect layer 0.2< y/δ <1.0 The law of the wake holds

Figure 5.7.: Location and defining properties of the near-wall regions and layers.

be postponed to Subsection 5.3.3. A computational method should be able to identify the boundary layer and to distinguish between its subregions. In ParallelNS the thick-ness of the boundary layerδ_layer is estimated before the calculation when the mesh for the numerical computation is chosen. This estimate is checked during the calculation. The sublayers can be identified within the numerical solution process. For this purpose, recall that if using the eddy-viscosity hypothesis, the turbulent viscosityν_t should be a measure for the magnitude of the Reynolds stresses. Thus, if we have an appropriate model forν_tin the boundary layer, on the basis of νt we can distinguish between its subregions. For this purpose, we review algebraic turbulence models for the boundary layer. Algebraic models are the simplest turbulence models, calculating ν_t from analgebraic expression.

5.3.2. Algebraic models based on Prandtl’s mixing-length hypothesis

In 1925, Prandtl proposed his famous mixing-length hypothesis as a relation for νt for an isothermal turbulent boundary layer. It is a reasonable model in the region 40δν ≤ y ≤0.2δ_layer, later referred to as thelog-layer. A quite heuristic derivation of this model will be given in the sequel, cf. [Pope00], p. 289. Therefore we consider a two dimensional boundary-layer flow withU =U(y)e1. Then the turbulent-viscosity hypothesis reduces to

hu⁰v⁰iE = −ν_tdU (5.29) dy

(3.11) motivates that ν_t is the product of a velocity scale u^∗ and a lengthscale l_m, viz., ν_t = u^∗ l_m . Choosingu^∗ =|hu⁰v⁰iE|^1/2 and substituting this into (5.29) gives

u^∗ = lm |dU dy|. (5.30)

Measurements and DNS data show that in the log-law region (i) hu⁰v⁰iE is approximately constant, viz., |hu⁰v⁰iE|^1/2 ≈ u_τ and (ii) dU/dy = u_τ/(κy) . Inserting these two semi-empirical relations into (5.30) givesPrandtl’s mixing-length hypothesis :

ν_t = u^∗l_m , with l_m =κy , and u^∗ = l_m|dU

dy| = u_τ . (5.31)

5.3. Algebraic turbulence models for non-isothermal boundary layers

y+

100 200 300

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

DNSPrandtl Prandtl dir Albring

Figure 5.8.:ν_t for simple algebraic models.

Note that (5.31) can be written as

νt = uτκy . (5.32)

Figure 5.8 provides some illustration. However, this sketch needs some explanation. The solid line shows the profile of νt obtained from DNS data for a turbulent channel flow at Re_τ = 395: Given DNS data for U and hu⁰v⁰iE, ν_t can be computed numerically using formula (5.29). Secondly, given DNS data forU we can calculateνtfrom Prandtl’s relation νt= (κy)²dU/dy (dashed line). Next, we can plot νt from (5.32) (dot-dashed line). Note that the difference between the second and the third profile is that the latter explicitely uses assumption (ii) whereas the former does not. Finally, from the DNS data for U we can plot νt fromAlbring’s proposal (5.38) (dot-dot-dashed line), which is covered in the next section. Apparently,ν_tis modelled reasonably in the log-layer (40< y⁺ <80 = 0.2δ).

However, in applications it may appear that we need an extension of our model forν_t (i) to the near-wall region y⁺ < 40 or (ii) to the region 0.2 < y/δ < 1.0. An example is the wall function concept presented in the next chapter: An intrinsic objective therein is to estimate the velocity at a certain distance y_P from the wall in the boundary layer (to be more precise: at the first node above the wall). On the one hand, when the Reynolds number is large, we only can afford 0.2 < y_P/δ < 1.0. On the other hand, in simple test cases we are also interested in the model’s behaviour ify_Pν/u_τ <40, see also (5.17).

Concerning (i) van Driest proposed to multiply lm with a suitable damping function D(y⁺) = 1−exp(−y⁺/A⁺), known as the van Driest damping function, viz.,

l_m = δ_νκy⁺D(y⁺) , D(y⁺) = 1−exp(−y⁺/A⁺) , A⁺= 26. (5.33)

Then ν_t is effectively multiplied with (D(y⁺))². Some heuristic physical support for this modification stems from the fact that the no-penetration conditionv⁰= 0 and the continu-ity equation imply that asymptoticallyu⁰ ∼y,v⁰ ∼y²near the wall and hencehu⁰v⁰iE ∼y³. But (5.31) predictshu⁰v⁰iE ∼y² and thus has to be damped. (5.33) giveshu⁰v⁰iE ∼y⁴ (see [Pope00], Exercise 7.19), which is in much better agreement with DNS data.

5. Near-wall treatment in turbulence modelling

The deviation in the defect layer is much more severe. Figure 5.8 reveals that a con-stant eddy viscosity is a much better approximation. One modification to accomplish this is to choose νt = 0.2κuτd99 (y > 0.2δ) (cf. [DPR01], p.116) as originated by Clauser (see [Wilcox98], p.73). Alternatively, Escudier proposed to limit l_m by setting l_m = min{κy,0.09d99}. Note that Escudiers modification is quite simple and can be included in both Prandtl’s and Albring’s model immediately once d99 can be estimated. An addi-tional improvement for approaching the freestream from within the boundary layer can be achieved by multiplying ν_t by a so-called intermittency factor [1 + 5.5(0.3y/d₉₉)⁶]⁻¹, see [DPR01], p.117 or [Wilcox98], pp. 73.

Remark 5.3

Concerning the isothermal case, a survey of more advanced algebraic turbulence models

can be found in [Wilcox98], Chapters 3.4-3.8. ♦

Nevertheless, one should be aware that all these modifications have been conceived for isothermal boundary layers.

5.3.3. Forced convection solution in the viscous sublayer and in the log layer

In this subsection we elaborate on the mean velocity profiles in forced convection. ThenU is the solution of the boundary-layer equation (5.27) with zero right hand side. Substituting (5.31) into (5.27) simple integration yields the famous log law (see C.3)

u⁺(y⁺) =

y⁺, ify⁺≤11.06

1

κln(y⁺) +B, ify⁺>11.06 , with κ= 0.41, B= 5.2.

(5.34)

The name log-layer originates from the logarithmic profile of the velocity in that layer.

From the definition of y⁺ and (5.34), it can be checked during the calculation whether a point with distance y⁺ to the wall is located in the viscous sublayer or in the log layer.

Another well-established solution is given by Reichardt’s law u⁺ = 1

Since the simple algebraic models for ν_t fail in the defect layer, it is evident that the predicted profiles for U deviate from the true profiles. For the boundary layer, in good agreement with experimental data is the modified log-law, cf. [Dea76], viz.,

u⁺ = 1

with Π ≈ 0.4757 for a zero pressure gradient boundary layer. The term w(·) in (5.36) is called wake contribution.

Figure 5.9 visualises the profiles. Apparently the deviation of (5.34), (5.35) and (5.41) from the log-wake law in the defect layer is significant.