Flow Separation Using the Integral Approach

For a flow about separation, the left hand side of equation (4.2) at the wall must be zero, then: 1 ρ · ∂p ∂x w = ν∂ 2_u ∂y2 w

where the subscript w denotes values near wall.

[∂2u/∂y2]_w >0 is a necessary condition for a steady two-dimensional boundary layer to separate and the contrary is also a necessary condition for flow to stay attached, as discussed by Gad-el-Hak and Bushnell [49]. Then [∂2_u/∂y2_]

w should be kept as

Figure 4.3: Illustration of a short and long separation bubbles with inviscid flow -Cp distri- butions

The observation of [∂u/∂y]_w, as reported by Gad-el-Hak and Bushnell [49] can tell if flow is about to separate or not. If [∂u/∂y]_w = 0, it means that separation is about to start.

Analysing terms l and mfrom Thwaites method, in equations (4.9) and (4.10), the first one (l) is the slope of boundary layer profile near the wall. l must be zero at the separation point. The second term (m) is basically the pressure gradient and it is related to the curvature of velocity profile. Using the same correlation of equation (4.11) and comparing with the integral equation (4.8), the following can be concluded:

θ2 ue · ∂2_u ∂y2 w ≡ 1 ue ∂ue ∂sθ and ∂2_u ∂y2 w = 1 θ · ∂ue ∂s (4.40)

also, using the empirical equation (4.12) l value is obtained by equation (4.41).

l= 0.225−m·(H−1) (4.41)

Then, observing equation (4.41), flow will separate when m·(H−1) approaches 0.225 using Thwaites method.

4.3.3.1 H Shape Parameter Investigation

H, according to Gad-el-Hak [1] is an important parameter to identify laminar flow sep- aration. Drela [35] and Gad-el-Hak [1] presented different limits ofH where separation is considered to occur based in wind tunnel experiments. In Drela’s [35] work, separa- tion was considered to occur when H= 2.5, while in Gad-el-Hak [1] it was considered

H = 3.7.

Here the H laminar distribution downstream the separation point is considered to be valid for engineering purposes, although it must be clear thatHlaminar distribution cannot be used to compare with real flow boundary layer in separated flow regions.

Using a Thwaites method in a laminar separated region, as discussed by Duncan et al [42],H increases from 2.5 to values near 10 asymptotically, decreasing sharply to values near 2. When H decreases to a value near 2, although it is in a separated flow region, it approaches the initial turbulent boundary layer values, according to Cebeci and Bradshaw [63].

If the point on surface where H ≈ 2 after the peak region is close to the place on surface where transition is detected in experimental data then, the H distribution can be used as an estimate for the transition point in partially separated flows.

4.3.3.2 The Criteria to Identify Transition after Laminar Separation

Existent methods used to identify transition, such as the criteria of Cebeci and Smith [63] and Wazzan et al [66] do not consider separated flow. The objective of the inves- tigation of H after the separation point is the creation of an estimate for transition in partially separated flows.

A first step of H shape parameter investigation is the observation of its behaviour using an initially attached inviscid flow on a body that presents a separated flow in real conditions. The circular cylinder flow is a classical case of a bluff body separated flow. An inviscid attached flow will be used to illustrate H shape parameter development on this circular cylinder.

A two-dimensional potential flow is used to calculate the initial velocity distribution. A Thwaites method will be applied to this potential flow, considering Re∞ ≥ 5·104.

Inviscid velocity u/U∞is obtained by equation (4.42). Transforming s/Rinto degrees,

Figure 4.4 shows the inviscid pressure distribution on the cylinder surface.

u U∞ = 2·sins R (4.42) For boundary layer calculation, it was considered that, a priori, all cylinder flow was laminar and attached to body. The initial H is calculated using the Hiemenz solution H= 2.35.

In Figure 4.5, H was set to zero in the last 25% of circle length, near trailing edge region, in order to avoid numerical problems with very small inviscid velocities. It does not interfere in calculations as, according to Parkinson and Jandali [67], real flow

Figure 4.4: Inviscid Cp distribution on a circular cylinder

in this part of the circle would be completely separated and integral methods are not valid.

Observing figures 4.4 and 4.5, while −CP is growing, H hardly changes. When

−CP starts to decrease, H starts the asymptotical growth.

IfH = 2.5 is the value where separation occurs, according to Drela [62], then Table 4.2 shows the separation points for different Re∞.

Re∞ (×105) Separation angle

0.50 90.9o

5.00 87.3o

10.0 83.6o

15.0 78.2o

Table 4.2: Separation points on an infinite cylinder section varying Re

Further downstream, H will make a sharp peak with high values approaching 10 and decreasing to values near 2, as predicted. By increasing Re∞, it can be noticed

in Figure 4.5 that the sharp peaks appear more downstream, leading to the previous conclusion that it has a different trend than the rough rule for separation, discussed before, where transition would be expected to occur nearer the separation point.

However, the circle, as discussed by Parkinson and Jandali [67], is a case of a bluff body separation where flow will continue separated. As the method for transition detection has to be applied to regions where flow will reattach to surface at some length, according to Wilkinson [4], sail and mast configurations should be used as examples of

Figure 4.5: H variation withRe∞ on a circular cylinder

partially separated flow instead.

4.3.4 Numerical Identification of Separation and Reattachment on Sail and