Copyright by Dr.Zheyan Jin Chapter 3 Fundamentals of Inviscid,Incompressible Flow 3.5 Source and Sink Flows The total mass flow across the surface of the cylinder: ds 2刀 =SpV,(rde)=pV,rlSdo=2xpV,rl Line source mass per second 2(volume flow per second) p Source strength: A=Y=2mV or volume flow per second per unit length)
Copyright by Dr. Zheyan Jin Chapter 3 Fundamentals of Chapter 3 Fundamentals of Inviscid Inviscid, Incompressible Flow , Incompressible Flow 3.5 Source and Sink Flows ( mass per second ) The total mass flow across the surface of the cylinder: Line source r 2 rV l v V rd l V rl d V rl r r r m ( ) 2 2 0 2 0 V rl r 2 m v Vr θ dS r x l y z ( volume flow per second per unit length) Source strength: ( volume flow per second ) r Vr 2 or
Copyright by Dr.Zheyan Jin Chapter 3 Fundamentals of Inviscid,Incompressible Flow 3.5 Source and Sink Flows Velocity potential: -y- 8r 2π 1o0 ra0 =V。=0 Integrating the above equation with respect to r,we have =△1nr+f0) 2π Integrating the above equation with respect 中=const+f(r) to 0,we have The velocity potential is: 0= 2π
Copyright by Dr. Zheyan Jin Chapter 3 Fundamentals of Chapter 3 Fundamentals of Inviscid Inviscid, Incompressible Flow , Incompressible Flow 3.5 Source and Sink Flows Integrating the above equation with respect to θ, we have 0 1 r 2 V r r Vr ln ( ) 2 r f The velocity potential is: Velocity potential: Integrating the above equation with respect to r, we have const f ( r ) ln r 2
Copyright by Dr.Zheyan Jin Chapter 3 Fundamentals of Inviscid,Incompressible Flow 3.5 Source and Sink Flows Stream function: 1=y= r a0 20 ay=V。=0 Or Integrating the above equation with respect y const+f(0) to r,we have Integrating the above equation with respect to we have 0+f) 2π The velocity potential is: 业= 0 2π
Copyright by Dr. Zheyan Jin Chapter 3 Fundamentals of Chapter 3 Fundamentals of Inviscid Inviscid, Incompressible Flow , Incompressible Flow 3.5 Source and Sink Flows Integrating the above equation with respect to θ, we have 0 r 2 1 V r r Vr const f ( ) The velocity potential is: Stream function: Integrating the above equation with respect to r, we have ( ) 2 f r 2
Copyright by Dr.Zheyan Jin Chapter 3 Fundamentals of Inviscid,Incompressible Flow 3.5 Combination of a Uniform Flow with a Source and Sink Consider a polar coordinate system with a source of strength A located at the origin. Superimpose on this flow a uniform stream with velocity Vmoving from left to right. Stream function: Velocity Field: 1∂w w=Vrsing+Ao V,= =V Cos0+A r a0 2π %s、 aw Or -=-'.sin8
Copyright by Dr. Zheyan Jin Chapter 3 Fundamentals of Chapter 3 Fundamentals of Inviscid Inviscid, Incompressible Flow , Incompressible Flow 3.5 Combination of a Uniform Flow with a Source and Sink sin 2 cos r 1 V r V r Vr V 2 sin V r Stream function: Velocity Field: Consider a polar coordinate system with a source of strength Λlocated at the origin. Superimpose on this flow a uniform stream with velocity V ͚moving from left to right
Copyright by Dr.Zheyan Jin Chapter 3 Fundamentals of Inviscid,Incompressible Flow 3.5 Combination of a Uniform Flow with a Source and Sink The velocity of this flow is simply the direct sum of the two velocity fields-a result that is consistent with the linear nature of Laplace's equation. Stagnation points: 1∂w V,= =V.CoS0+A =0 r00 (,6 (One stagnation point) Vo=-7 r =-V.sin0=0 The streamline passing through w=V Slnπ+ 一π the stagnation point: 2πV 2π 2
Copyright by Dr. Zheyan Jin Chapter 3 Fundamentals of Chapter 3 Fundamentals of Inviscid Inviscid, Incompressible Flow , Incompressible Flow 3.5 Combination of a Uniform Flow with a Source and Sink 2 2 sin 2 rV V , 2 , V r (One stagnation point) Stagnation points: The velocity of this flow is simply the direct sum of the two velocity fieldsa result that is consistent with the linear nature of Laplace’s equation. The streamline passing through the stagnation point: sin 0 0 2 cos r 1 V r V r Vr V