Cálculo sub-pixélico del centro de las elipses

Three-dimensional separation is radically different from two-dimensional separation which was described earlier. In two-dimensional flows separation is identified by a region of backflow, and at the wall by a zero shear stress solution. This difference is illustrated in Figure 2.6: in two-dimensional separation the separation point and reattachment point are connected by a (u= 0) velocity line and a dividing streamline. The region is closed and the streamlines inside are closed. In a three-dimensional separation, the separation and reattachment points cannot be connected by a streamline to form a closed vortex [13]. The concepts of separation and reattachment points, recirculation regions or dividing streamlines as mentioned above need careful redefinition.

( b ) primary separation dividing streamline secondary separation F N S N2 S1 1 2 N S 1 2 N2 S1 u = 0 ( a )

Figure 2.6: (a) A flow which exhibits two-dimensional separation (b) A flow with three- dimensional separation [45]

To identify three-dimensional separated regions, one focuses on the shear stress at the wall, which is a vector field in three-dimensional flows. The trajectories of this field are the skin friction lines, sometimes called limiting streamlines because they are the limit of a streamline when the distance to the wall becomes zero. The set of the skin friction lines covering a body constitutes a skin friction line pattern, and separation is defined from

2.3 Shock/Boundary Layer Interaction S.J.VITHANA examination of this pattern. However, the sole inspection of the skin friction line pattern is far from being sufficient to define and describe three-dimensional separation. Considering orthogonal coordinates in the two-dimensional space (x, z) of the body surface, we denote

¡

τx(x, z) ¢

and ¡τz(x, z) ¢

the components of the skin friction vector ¡τx ¢

along (x) and (z) respectively. The skin friction lines are defined by the following (time independent) differential equation:

dx τx(x, z) =

dz

τz(x, z) (2.19)

These equations define an infinity of solution curves called characteristic lines or trajecto- ries that are associated with the skin friction lines introduced above. In general, one and only one trajectory passes through a point on the surface. The only points that do not satisfy this rule are the singular points of the system where the skin friction vanishes, that is, where we simultaneously have

τx(x, z) =τz(x, z) = 0 (2.20)

Around such singularities, the shape of the skin friction lines are evaluated through a first order Taylor series expansion, written in tensorial form as

τ =Ax (2.21)

where¡τ = (τx, τz) ¢

and ¡x= (x, z)¢, (A) is the Jacobian of the above transformation. Depending on the values of (A) it is possible to classify the different types of singular points. One method of classification is according to the eigenvalues of the Jacobian, where real eigenvalues produces (nodes) if

³ |A|> 0 ´ ,saddles if ³ |A|< 0 ´ or a combination of both if ³ |A|=0 ´

. The most common singular points are presented in Figure 2.7. Singular points may be classified as two types: nodes and saddle points (see Fig- ure 2.7:(a) and 2.7:(b)). When the skin friction lines converge to, or diverge from a point, the point is called node - nodal point of separation or attachment - respectively

2.3 Shock/Boundary Layer Interaction S.J.VITHANA

Figure 2.7: Most typical patterns near critical points [13] (a) Attachment node (b) Separation saddle (c) focus (d) Nodal point of attachment (e) Nodal point of separation

(see Figure 2.7:(e) and 2.7:(d)). Nodal points can have one line to which all skin friction lines are tangent to, or none. In the latter case the node is calledfocus - of separation or attachment (see Figure 2.7:(c)). Nodal points of separation and attachment can be viewed as sinks and sources of skin friction, respectively. Nodal points of attachment are typically stagnation points on a forward facing surface (like a blunt nose).

There are cases where the skin friction lines deviate from a point as from a stagnation point. There are only two lines (normal to each other) through the point, which is called

saddle. Skin friction lines diverging from nodal points cannot cross, due to the presence of a saddle point between them. One of the lines through the saddle is a separation line. Nodal points of separation and attachment have other interesting features: they become edges of vortex cores. In some cases there is also a distinction between primary and sec- ondary lines of separation. Devices most commonly used for the study of three-dimensional separation include prolate spheroids, blunt and pointed cones at incidence, where non- axisymmetric vortex formation appear (all axisymmetric bodies at incidence are very prone to flow separation with consequent instability).

2.3 Shock/Boundary Layer Interaction S.J.VITHANA The existence of these critical points in the surface indicates the presence and types of separation. A flow is separated if the skin friction lines configuration contain at least one saddle point [58]. Accordingly, a separation line would form a separation surface which usually rolls up in a vortical structure, as demonstrated in Figure 2.8. A combination of saddle and nodes in the surface would give rise to the different types of separation, like the bubble of the horseshoe vortex type (1 saddle and 1 node).

boundary layer separation surface separated region surface streamlines trailing vortex boundary layer at "reattachment" external streamline N Y separation line Z X S

Figure 2.8: Horseshoe or bubble separation. S marks the saddle point at separation and N denotes the node point at reattachment [47]

A topological rule has been developed for streamlines on a vertical plane cutting a surface that extends to infinity both upstream and downstream [58].

³X N + 1 2 X N0 ´ −³X S+ 1 2 X S0 ´ = 0 (2.22)

The singular points in this rule are defined as half-nodes (N0), half-saddles (S0), saddles (S), and nodes (N). This rule has been used to describe the upstream plane of symmetry for an obstacle mounted on a wall [58].