Fuentes

Dean and Bradshaw (1976) proposed an experimental correlation between Cf and

5.2. WALL SHEAR STRESS

Equation 5.3 can be rewritten in terms of Re:

Cf =aRe−0.25, (5.4)

where a= 0.073×2−0.25_{≈0.061. Then, the rate of change ofC}

f can be calculated using Equation 5.1: dCf dξ = dCf dRe dRe dξ , (5.5) =−0.25aRe−1.25dRe dξ , =−10.7Re−1.25. (5.6)

Similarly, Equation 5.4 can be rewritten for Reτ:

Reτ =cRe7/8, (5.7)

where c = pa/2 ≈ 0.175. The rate of change of Reτ can be calculated using

Equation 5.1: dReτ dξ = dReτ dRe dRe dξ , (5.8) = 7 8cRe −1/8dRe dξ , = 107.4Re−1/8. (5.9)

dReτ/dξ is about 38.7 at Re0 = 3500, and reduces to 32.3 at Ref = 15000. From

Equation 5.7, U+ m can be calculated: Um/uτ = 1 cRe 1/8_{. (5.10)}

a) b)

Figure 5.1: Variation of the wall shear stress during the acceleration. a) Reτ, and

b) duτ/dξ, the rate of change of Reτ. Steady DNS data and Dean and Bradshaw

(1976) correlation from Equations 5.7 and 5.9 are included for comparison.

of near-wall turbulence response can be observed here: the initial transient (IT) stage (stage I), the weak time-dependence (WT) stage (stage II), and the strong time-dependence (ST) stage (stage III). In chapter 7, it is found that the ST stage is followed by the pseudo-steady (PS) stage (stage IV). The similar response of the wall shear stress was reported in the previous LES study of a pipe flow subjected to temporal acceleration (Jung and Chung, 2007).

At the onset of acceleration, uτ increases rapidly in the stage I (3500< Re <4300,

or 0< ξ <1). Initially,duτ/dtis several times larger than the steady corresponding

value, and this is due to a uniform increase in the mean velocity across the channel. As a result, uτ is significantly larger than the steady corresponding values at the

same Renumbers. duτ/dt decreases sharply and the stage I range is defined where

the rate of change of uτ is larger than the steady value calculated in Equation 5.9.

In the stage II (4300 < Re < 12000, or 1 < ξ < 12), the rate of change of uτ is

5.2. WALL SHEAR STRESS

The largestduτ/dt is observed at ξ = 13.9 with its value approximately an order of

magnitude smaller than the imposed acceleration rate (f = 0.2). The rate of change of uτ starts decreasing towards the end of the stage III, and Reτ has almost the

steady value at the end of the stage III (Re= 15000). In the PS stage (Re >15000, or ξ > 17), the near-wall turbulence approaches the pseudo-steady state (this is shown in Chapter 7).

Figure 5.2 shows the variation of the ratio of the bulk-mean velocity to uτ i.e.,

(Um/uτ = Um+) and the skin friction coefficient, Cf = τw/(1₂ρUm2). In the steady

flow, the skin friction coefficient decreases with the Re number: Cf = 0.061Re−0.25

as in Equation 5.4. Cf decreases by 30% from Cf = 7.98×10−3 at Re = 3500 to

Cf = 5.54×10−3 atRe= 15000. As the bulk-mean velocity increases linearly during

the acceleration, U+

m decreases sharply at the start of acceleration and lowest value

is reached during the stage I at Re= 3950. It starts to increase afterwards due to the subsequent near-wall flow adjustment to the imposed acceleration until the end of the stage II (Re= 11800). Finally,U_m+decreases towards the steady value during the stage III (Equation 5.10). Cf exhibits exactly the opposite trends as observed

for U_m+, showing an initial increase during the stage I followed by a reduction in

the stage II at exactly the same time as that for U+

m during acceleration. This

phenomenon of an initial increase and a subsequent reduction in the skin friction coefficient is reported in previous studies of the boundary layer subjected to the acceleration by favourable pressure gradient (FPG) (Sreenivasan, 1982; Fernholz and Warnack, 1998).

Figure 5.3 shows the variation of boundary layer parameters during acceleration. The displacement thickness (δ∗) and the momentum thickness (θ ) decrease after the onset of acceleration in an identical manner until both reach a minimum level atRe= 11350 and Re= 11100 respectively. It is followed by a subsequent increase during the stage III. It is worth noting that the timing for the minimum values of

a) b)

Figure 5.2: Variation of a) Um/uτ =Um+, and b) Cf during acceleration. Dean and

Bradshaw (1976) correlation from Equation 5.10 is included for comparison.

Cf, δ∗ and θ occur approximately at the end of the stage II. On the other hand,

the shape factor exhibits a different trend; it decreases from the start and reaches

its minimum value at Re = 5500 and a subsequent gradual increase is observed

during the stage II with a local maximum atRe= 10850. This two-stage behaviour

of shape factor has been reported in several studies of boundary layer subjected to FPG (Blackwelder and Kovasznay, 1972; Fernholz and Warnack, 1998; Bourassa and Thomas, 2009). It is suggested that the flow depicting the two-stage shape factor variation accompanied with a significant reduction in Cf value is representative of