MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS Introduction to Streamlines, Stream Functions, and Velocity Potential January 28, 2011 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2 HARRIER INSTANTANEOUS STREAMLINES Streamline –Set of points that form a line that is everywhere tangent to local velocity vector –No flow across streamlines –For a steady flow, moving fluid element traces out a fixed path in space Stream tube –A set of streamlines that intersect a closed loop in space Water streamlines on F-16 model Harrier instantaneous streamlines

3 DIFFERENCES BETWEEN  and  1.Flow field variables are found by: –Differentiating  in the same direction as velocities –Differentiating  in direction normal to velocities 2.Potential function  applies for irrotational flow only 3.Stream function  applies for rotational or irrotational flows 4.Potential function  applies for 2D flows [  (x,y) or  (r,  )] and 3D flows [  (x,y,z) or  (r, ,  )] 5.Stream function  applies for 2D  (x,y) or  (r,  ) flows only 6.Stream lines (  =constant) and equipotential lines (  =constant) are mutually perpendicular –Slope of a line with  =constant is the negative reciprocal of the slope of a line with  =constant

4 STREAMLINE AND STREAM FUNCTION EXAMPLE

5  =0  =1  =2  =0  =1  =2

6 STREAMLINE PATTERN: MATLAB FUNCTION Physical interpretation of flow field 1.Flow caused by three intersecting streams 2.Flow against a 120º corner 3.Flow around a 60º corner Patterns (2) and (3) would not be realistic for viscous flow, because the ‘walls’ are not no-slip lines of zero velocity

7 STREAMLINE PATTERN: MATLAB FUNCTION

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9 u AND v VELOCITY COMPONENTS

10 VELOCITY MAGNITUDE

11 STATIC PRESSURE FIELD

12 TOTAL PRESSURE FIELD: P + ½  V 2 = P + ½  (u 2 +v 2 ) ½

13 ASIDE: MATLAB CAPABILITY FOR STREAMLINE PLOTTING Altitude 1 Altitude 2 Altitude 3

14 ASIDE: MATLAB CAPABILITY FOR STREAMLINE PLOTTING