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Chapter 2 Fluid Flow Topology
Topology and Terminology
There has been much work done in the field of flow topology in the past twenty years. All of which confirms that the starting point is to consider a steady viscous flow over a smooth three-dimensional body where the skin-friction lines or streamlines on the surface of the body form a continuous vector field. One translates this vector field into a mathematical model in terms of the surface velocity, shear stress and vorticity vector components as documented in Tobak and Peake [ref 1].
With the completion of the mathematical model, one must investigate where the magnitudes of the derived vector fields are identically zero. Such points in the vector field are called singular points. Singular points may be classified as two types: nodes and saddle points. The classification of nodes can be further divided into nodal points and foci, of either separation or attachment.
A nodal point is a point common to an infinite number of streamlines. If the streamlines converge to the nodal point, as seen in Figure 1a, it is said to be a nodal point of separation. Conversely, if the streamlines diverge from the nodal point it is said to be a nodal point of attachment, seen in Figure 2b.
Foci differ from nodal points in that an infinite number of streamlines spiral around the node. If the streamlines spiral away from the node, as seen in Figure 2c, the node it is defined as a foci of attachment. Streamlines, which spiral into the node, seen in Figure 2d, are defined as foci of separation.
A saddle point may be defined as a singular point in which only two particular lines intersect at the singular point, each of which is in the direction towards or away from the singular point. All other streamlines miss the singular point and follow the directions of the adjacent lines that pass through the singular point as seen in Figure 2e.
Different combinations of nodal/saddle points and how they work together have received much attention by Tobak and Peake [ref 1] and Chapman [ref 4]. For the purpose of this paper, we shall only be concerned with the specific singular point interaction in which a line of separation emerges. A line of separation is present when the streamlines emerging from the nodal points of attachment are prevented from crossing by the presence of a particular streamline emerging from the saddle point as defined by Lighthill [ref 10] and seen in Figure 3. Most researchers agree that the convergence of streamlines on either side of a particular line is a necessary condition for separation however; it should not be used solely to define it as this may occur in other situations as well.
According to Tobak and Peake [ref 1], lines of separation may be further subdivided into global and local lines of separation. A global line of separation is a streamline, which emerges from a saddle point and leads to global flow separation. On the other hand, if a streamline not originating from a saddle point has other lines converging on it, that streamline is defined as a local line of separation and leads to local flow separation. This terminology is not necessarily common amongst all researchers but will suffice for this paper.
Implications of Flow Topology
Singular Points acting in isolation or in combination fulfill certain characteristic functions that largely determine the distribution of streamlines on the surface (Tobak and Peake) [ref 1]. A nodal point of attachment typically represents a stagnation point on a forward-facing surface, such as a leading edge of a wing, where as, a nodal point of separation acts as a sink where the streamlines that have moved over the body have vanished. Saddle points typically act to separate the streamlines from adjacent nodes.
When studying the topology of fluid flow, especially of separated flows, it is often useful to consider the change of topology as different parameters are changed. One might want to examine how the topology changes with angle of attack, Mach number, Reynolds number or possibly geometric changes. The bifurcations are those, which change the structure of the singular points in the vector fields. One applies bifurcation theory to study the changes in singular points with respect to parameters and investigate if new singular points appear, singular points change from attachment to separation or vise versa, or if singular points change from a nodal point to a saddle point.
Because the patterns of skin-friction lines and external streamlines reflect the properties of continuous vector fields, we are able to characterize the patterns on the surface and on particular projections of the flow. Hunt et al [ref 11] have shown that the notions of singular points and the rules that they obey can be extended to apply to the flow above the surface on planes of symmetry and on crossflow planes. Most recently Delery [ref 2], discussed the collaboration of Legendre, a theoretician, and Werle’, an experimentalist, in their pursuit to construct a theoretical tool allowing the elucidation of the structure of largely separated three-dimensional flows. In this paper, Delery discusses the implications of flow topology off the surface as well as reviews the work done in the flow topology field to date.
Off-the-Surface Streamtraces
If one were to look down onto a wing and plot the axial and spanwise velocity components, u and w, at various heights above the wing these would be considered off-the-surface streamtraces. This was done for each of the 6, 7, 8 and 10 degree angle of attack cases at heights ranging from approximately on the surface to about one foot off the surface. The streamtraces were laid on top of a reversed axial flow contour for the corresponding height. The blue region indicates positive axial flow where the red region is used to display areas a negative or zero value of axial flow. Zero or negative values of axial flow are a good indication of separated flow because separatation is normally accompanied by flow reversal.
Off-the-Surface Grid and Data Generation
Amtec Tecplot was used to generate the off-the-surface grids and data. The imbedded Tecplot Slice function was used to extract 2-D planes out of the 3-D grid. The Slice function interpolates the data points in the grid to create a 2-D plane at a specified position on, in this case, the y-axis. Because the slice is created at a specified position, the height of the off-the-surface plane increases from the wing root to the wing tip. For the purpose of this paper the height of the surface at the wing root will designate the label. The planes generated are showed in Figure 4. Data for this grid, the u and w components of the velocity vector was calculated by dividing out the density from the momentum vector in the CFD solution files. Tecplot also has a streamtrace function that allows the user to plot 2-D vectors. This was taken advantage of to plot the off-the-surface streamtraces using the u and w components of velocity as the vector variables.
Abstract
Acknowledgments
Chapter 1 Introduction
1.1 BACKGROUND
1.2 APPROACH
Chapter 2 Fluid Flow Topology
2.1 TOPOLOGY AND TERMINOLOGY
2.2 IMPLICATIONS OF FLOW TOPOLOGY
Chapter 3 Investigation of Off-the-Surface Streamtraces
3.1 OFF-THE-SURFACE STREAMTRACES
3.2 OFF-THE-SURFACE GRID AND DATA GENERATION
3.3 INVESTIGATION OF OFF-THE-SURFACE STREAMTRACES
3.4 CONCLUSIONS FROM OFF-THE-SURFACE STREAMTRACES
Chapter 4 Investigation of Crossflow Velocity Traces
4.1 CROSSFLOW DEFINED
4.2 CROSSFLOW GRIDS AND DATA GENERATION
4.3 DEFINITION OF STREAMWISE STATIONS
4.4 CROSSFLOW TRACES IN DETAIL
4.5 CONCLUSIONS FROM CROSSFLOW VELOCITY TRACES
Chapter 5 Conclusions
References
Appendix
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The Relationship between Crossflow Velocity and Off-the-Surface Streamtrace Topology for a Moderately Swept Wing at Transonic Mach Numbers
