# What is the physical explanation for the formation and position of stagnation points on an object moving in a fluid?

I have some questions regarding the stagnation points, both on an airfoil and on a generic object.

• What is the physical explanation for the formation of two stagnation points on an object moving in a fluid (e.g. a cylinder)?
• Why in potential flow with zero circulation the rear stagnation point is located above the trailing edge of an airfoil (and not below)?

• Does the Kutta condition hold even on flatback airfoils like this one? They have a truncated trailing edge, so I think the "infinite velocity" problem would still arise (in theory, of course).
• Where is the rear stagnation point located? The image below gives their approximate position, but it's not clear where the rear stagnation point is.

(source)

The forward stagnation point is easy to explain: Imagine the flow path of molecules which approach an airfoil. Some will continue above and some below the airfoil contour. Between them there must be one flow path which will neither go above nor below, but hit the airfoil straight on. The point where it hits the airfoil is the stagnation point.

The rear stagnation point is similarly explained: When the flows from above and below meet, there will be a line dividing both. Continue this line forward until it hits the airfoil: This is where the rear stagnation point is located. A rear stagnation point is only possible in inviscid flow, which exists only in theory. In both points the speed of air relative to the airfoil is zero, and air molecules will never completely arrive at the stagnation point, but slow down on their way into it.

An airfoil of zero circulation and zero camber will have its rear stagnation point right at the trailing edge. Only when you add asymmetry by giving the airfoil positive camber, the flow has to develop an asymmetric pattern which shifts the rear stagnation point away from the trailing edge to satisfy the zero circulation condition. Give the airfoil negative camber, and the rear stagnation point will duly relocate below the trading edge.

The Kutta condition postulates that the rear stagnation point is fixed at the trailing edge. Make the trailing edge blunt and you need to define where precisely your stagnation point will be. Note that your choice of location will influence the circulation at a given angle of attack! The Kutta condition imposes an arbitrarily chosen stagnation point which just helps to explain real flows better. This does not mean that real flows will observe the Kutta condition closely.

If you look for the "real" stagnation point, you should really search for the point where local pressure reaches a maximum, fully knowing that a full deceleration to stagnation will not happen in reality. This can only be outside of the boundary layer, since the energy loss of the boundary layer flow will result in a pressure loss. Now the picture from the second paragraph will help: Find the line separating upper and lower flows and follow it upstream until it hits the airfoil. This procedure will land you straight at the lowest speed section of the boundary layer, and near the airfoil the pressure there is too low. But now move back again and watch how the slow center part of the boundary lower gets accelerated by shear with the neighboring flow. At some point pressure will peak: This is the closest you can come to for a rear stagnation point.

Below I plotted a quick XFOIL calculation of a NACA4418 with a 5% trailing edge gap to get something similar to your airfoil. Note how the $c_p$-value of the viscous flow (solid line) peaks at some point behind the blunt trailing edge. For the outer flow the airfoil looks as if it is slightly longer, the void behind the blunt trailing edge being filled by separated, slow flow. Also note that the upper boundary layer is thicker, shifting the "rear stagnation point" a little up.

XFOIL Plot of a NACA4412 with 5% blunt trailing edge at 4° AoA (own work).

• Thank you, excellent answer! One more thing: with "when you add asymmetry", do you also include a symmetrical airfoil with non-zero angle of attack? – lorenzownd May 16 '16 at 19:25
• @lorenzownd: Yes, absolutely. A symmetrical airfoil at a nonzero angle of attack will need the rear stagnation point shifted away from the trailing edge to stay at zero circulation. – Peter Kämpf May 16 '16 at 20:20