058 0160 Professor Fred Stern Fall 2009 Chapter 8 1 Chapter 8 Inviscid Incompressible Flow a Useful Fantasy 8 1 Introduction For high Re external flow about streamlined bodies viscous effects are confined to boundary layer and wake region For regions where the B L is thin i e favorable pressure gradient regions Viscous Inviscid interaction is weak and traditional B L theory can be used For regions where B L is thick and or the flow is separated i e adverse pressure gradient regions more advanced boundary layer theory must be used including viscous Inviscid interactions For internal flows at high Re viscous effects are always important except near the entrance Recall that vorticity is generated in regions with large shear Therefore outside the B L and wake and if there is no upstream vorticity then 0 is a good approximation Note that for compressible flow this is not the case in regions of large entropy gradient Also we are neglecting noninertial effects and other mechanisms of vorticity generation Potential flow theory 1 Determine from solution to Laplace equation 2 0 B C at S B V n 0 0 n at S V Note F Surface Function 1 058 0160 Professor Fred Stern Chapter 8 2 Fall 2009 DF 1 F F 0 V F 0 V n Dt t F t for steady flow V n 0 2 Determine V from V and p x from Bernoulli equation Therefore primarily for external flow application we now consider inviscid flow theory 0 and incompressible flow const Euler equation V 0 DV p g Dt V V V p z t V2 V V V 2 Where V vorticity 2 fluid angular velocity V 1 p V 2 z V t 2 If 0 ie V 0 then V 1 p z B t t 2 Bernoulli s Equation for unsteady incompressible flow not f x Continuity equation shows that GDE for is the Laplace equation which is 2nd order linear PDE ie superposition principle is valid Linear combination of solution is also a solution V 2 0 1 2 2 0 2 0 2 1 2 0 2 1 2 2 0 2 1 2 0 Techniques for solving Laplace equation 1 superposition of elementary solution simple geometries 2 surface singularity method integral equation 3 FD or FE 4 electrical or mechanical analogs 5 Conformal mapping for 2D flow 6 Analytical for simple geometries separation of variable etc 2 058 0160 Professor Fred Stern Fall 2009 Chapter 8 3 8 2 Elementary plane flow solutions Recall that for 2D we can define a stream function such that u y v x z vx u y x y 2 0 x y i e 2 0 Also recall that and are orthogonal u y x v x y d x dx y dy udx vdy d x dx y dy vdx udy i e dy u 1 dx const v dy dx const 8 2 Elementary plane flow solutions Uniform stream u U y x const v 0 x y U x U y Note 2 2 0 is satisfied V U i i e Say a uniform stream is at an angle to the x axis 3 058 0160 Professor Fred Stern u U cos Chapter 8 4 Fall 2009 y x v U sin x y After integration we obtain the following expressions for the stream function and velocity potential U y cos x sin U x cos y sin 2D Source or Sink Imagine that fluid comes out radially at origin with uniform rate in all directions singularity in origin where velocity is infinite Consider a circle of radius r enclosing this source Let vr be the radial component of velocity associated with this source or sink Then form conservation of mass for a cylinder of radius r and unit width perpendicular to the paper L3 Q V d A S A Q 2 r b vr Or vr Q 2 br m v 0 r Q is the convenient constant with unit velocity length Where m 2 b m 0 for source and m 0 for sink Note that V is singular at 0 0 since vr vr In a polar coordinate system for 2 D flows we will use 1 V r r 4 058 0160 Professor Fred Stern Chapter 8 5 Fall 2009 And V 0 1 1 rvr v 0 r r r i e 1 r r 1 v Tangential velocity r r Such that V 0 by definition v r Radial velocity Therefore m 1 r r r 1 v 0 r r vr m ln r m ln x 2 y 2 i e m m tan 1 y x Doublets The doublet is defined as m 1 2 1 2 sink source sink source m tan tan 1 2 tan tan 1 2 1 tan tan m 1 2 5 058 0160 Professor Fred Stern Chapter 8 6 Fall 2009 r sin r sin tan 2 r cos a r cos a r sin r sin 2ar sin tan r cos a r cos a 2 r sin m 1 r sin r a2 r cos a r cos a tan 1 For small value of a 2ar sin 2amy y 2 2 2 2 r a r x y2 cos 2am Doublet Strength a 0 lim m tan 1 r By rearranging y 2 2 x2 y 2 2 2 2 x y It means that streamlines are circles with radius R and center at 0 R i e circles 2 are tangent to the origin with center on y axis The flow direction is from the source to the sink 6 058 0160 Professor Fred Stern Chapter 8 7 Fall 2009 2D vortex Suppose that value of the and for the source are reversal vr 0 1 K r r r Purely recirculating steady motion i e v f r integration results in K K ln r K constant 2D vortex is irrotational everywhere except at the origin where V and V are infinity v Circulation Circulation is defined by V d s For irrotational flow c C closed contour Or by using Stokes theorem if no singularity of the flow in A V d s V d A ndA 0 c A A Therefore for potential flow 0 in general However this is not true for the point vortex due to the singular point at vortex core where V and V are infinity If singularity exists Free vortex K r 2 2 K rd 2 K and K v e rd e 0 0 r 2 V ds 7 058 0160 Professor Fred Stern Chapter 8 8 Fall 2009 Note for point vortex flow still irrotational everywhere except at origin itself where V i e for a path not including 0 0 0 Also we can use Stokes theorem to show the existence of V ds ABC V d s C Since AB C V d s 0 ABCB A Therefore in general for irrotational motion V d x V d x d d x d dx V ds ds ds dx e s ds …
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