Basic flow dynamics and universal law of natural frequency

1.3.1 Basic flow visualization and velocity fields

The flow visualization of unforced mixing layer is plotted in Figure 1.2. It can be seen, the typical Kelvin-Helmholtz (K-H) instability induced vortex structures are easily found.

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So, intuitively, in this confined mixing layer, the instability mechanism has no apparently difference from conventional free mixing layer.

The mean velocity distributions (normalized by U) at Re=2939 and λ=1/3 are plotted

in Figure 1.3.

According to the development of wall boundary layer on splitter plate, the influence of its wake spread to almost x*=0.85 (in this paper, x*=x/D, y*=y/D, z*=z/D). After that,

as shown in Figure 1.3(a), the streamwise velocity U in the central plane exhibits a typical

“tanh-shaped” profile similar as in free shear layer. To compare with conventional free mixing layer, the momentum thickness of mixing layer is investigated, which is:

= ∫ 1 − (1.1)

where = [ = + ( − )], hence, = ( = ) and = ( = ). The variation of along x direction is plotted in Figure 1.3(c). In the field of interest (FOI, marked by the window with gray dashed line in Figure 1.6 and Figure 1.8), the local momentum thickness Reynolds number = ( − ) ⁄ varies with streamwise positions from 21 to 81. If in free mixing layer, this belongs to the laminar and transition region (Winant and Browand 1974; Ho and Huerre 1984). However, in confined mixing

(a) (b)

Figure 1.2 Visualization of unforced flow at Re=2939, =1/3. (a) The length of mixing chamber is L=0.25 m and (b) L=1.753 m.

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layer, the flow seems more “stable”. Through a power fitting, the scaling exponent is found to be about 0.5 (θm~x0.5), which is consistent with the researches by Winant and Browand

and Ho and Huerre in laminar (or pre-transition) region of unforced free mixing layer. Flow visualization in Figure 1.6(a) also verifies the laminar character in the FOI even at Reθ =

81.

Similar conclusions are also found when increasing Re to 8816 and keeping λ=1/3

Figure 1.3(c). In the FOI, Reθ varies from 97 to 251 which apparently set down into the

turbulent region. However, from Figure 1.3(c), the scaling behavior of momentum thickness (θm~x0.5) indicates the flow is still laminar, only except at the downstream end of

FOI where transition begins as shown in Figure 1.8(a). Thus, the transient is obviously postponed compared with free mixing layer.

(a) (b)

(c) (d)

Figure 1.3. (a) The distribution of U in unforced mixing layer, Re=2939, =1/3, (b) The distribution of V in unforced mixing layer, Re=2939, =1/3, (c) Momentum thickness of unforced mixing layer vs x, fitted by power-distribution curves, where =1/3. (d) The width of mixing layer vs x, where =1/3.

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In the FOI, the averaged dθm/dx in this region is 0.022 for Re=2939 and 0.023 for

Re=8816 respectively. Both of them are two times larger than that in free mixing layer investigated by Winant et al (1974) and even comparable with the value in forced free mixing layer (Ho and Huang 1982; Oster and Wygnanski 1982; Roberts 1985; Weisbrot and Wygnanski 1988). As the slope of spreading momentum thickness is proportional to velocity ratio, at the equivalent Re, a higher dθm/dx is achieved even under a smaller

velocity ratio compared to Winant et al’s work (1974). This is really surprising. The higher dθm/dx means a higher rate of momentum loss. It may be attributed to the relatively high

area-to-volume ratio of the channel. As known, in laminar pipe flow with same bulk velocity, due to the viscosity, the rate of momentum loss in unit length can be estimated as:

⁄ = −8 〈 〉 (1.2)

which is of the order O(D0). But the total momentum of flow is = 〈 〉 4⁄ which is at the order of O(D2_{). Hence the relative rate of momentum loss, П, defined as:}

= (1.3)

should be proportional to D-2. This indicates, under the same bulk flow velocity, the smaller the scale of cross section, the faster the momentum loss is. That's maybe why in confined mixing layer, the momentum thickness has larger value. The rapid loss of momentum is also the result of secondary flow and its vortex structures near side wall. This will be detailed discussed later. The width of mixing layer δm (=y0.95-y0.1, by Liepmann & Laufer

(Liepmánn, 1947 #168)) under both Reynolds numbers is plotted in Figure 1.3(d). As the

14 1.3.2 Natural frequency

As in free mixing layer, the natural frequency (estimated by vortex passage frequency) of K-H instability is proportional to <U>3/2_{. Here, the vortex passage frequency (λ=1/3) at}

Re~1200, 3000, 6000 and 9000 are investigated and compared with <U>3/2 law, as shown

in Figure 1.4. It can be seen, in confined mixing layer, the experimental natural frequency approximately fits the 3/2 law. This means in this confined shear layer with circular transverse mixing chamber, K-H instability is still the most important instability mechanism for mean flow. Besides, with different length of mixing chamber, the difference of frequency is neglectable, as shown in Figure 1.4(a) and (b). The shortened mixing chamber will not affect the character of basic flow by introducing more disturbances from downstream. This is consistent with K-H instability which is convective instability, not absolute instability.

All these founds mean that there is no other instability mechanism which may introduce un-expected self-sustain frequency in this flow field. The parametric investigations in section 1.5 on the effect of 1-D acoustic resonance have the same basic flow.

1.3.3 Short discussion

From above investigations, we can see, the instability of flow in free mixing layer and confined mixing layer has no intrinsic difference. The fast mixing in confined mixing layer

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is unique which indicates the specialty of velocity field in forced confined mixing layer. This will be investigated in details in the following sections.

Figure 1.4 The experimental natural frequency (λ=1/3) vs 3/2 law in free shear layer. (a) mixing chamber of 0.25 m long. (b) mixing chamber of 1.753 m long.