UI ME 5160 - Chapter 8 Inviscid Incompressible Flow

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058:0160 Chapter 8 Professor Fred Stern Fall 2009 1 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 ω=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 φ from solution to Laplace equation 02=∇φ B.C: at BS : .0 0Vnnφ∂=→ =∂ at ∞S : Vφ=∇ Note: F: Surface Function058:0160 Chapter 8 Professor Fred Stern Fall 2009 2 2 10.0.DF F FVF VnDt t F t∂∂=→ +∇=→ =−∂∇∂ for steady flow .0Vn= 2) Determine V from Vφ=∇ and p(x) from Bernoulli equation Therefore, primarily for external flow application we now consider inviscid flow theory (0=μ) and incompressible flow (const=ρ) Euler equation: .0.()VDVpgDtVVV p ztρρρργ∇==−∇ +∂+∇=−∇+∂ 2.2:2VVV VWhere V vorticity fluid angular velocityωω∇=∇ −×=∇× = = × 21()200 :VpVzVtIf ie V then Vρργρωωφ∂⇒+∇+ +=×∂=∇×= =∇ 1()2pzBttφρρφφγ∂++ ∇⋅∇+ =∂ Bernoulli’s Equation for unsteady incompressible flow, not f(x) Continuity equation shows that GDE for φ is the Laplace equation which is 2nd order linear PDE ie superposition principle is valid. (Linear combination of solution is also a solution) 212222 22112 1 222000()0 00Vφφφφφφφφφ φφφ∇⋅ =∇⋅∇ =∇ ==+⎧∇=∇=⇒∇ + =⇒∇+∇ =⇒⎨∇=⎩ 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)058:0160 Chapter 8 Professor Fred Stern Fall 2009 3 3 8.2 Elementary plane-flow solutions: Recall that for 2D we can define a stream function such that: xyvuψψ−== 0)()(2=−∇=∂∂−−∂∂=−=ψψψωyxyxzyxuv i.e. 02=∇ψ Also recall that φ and ψ are orthogonal. yxxyvuφψφψ=−=== udyvdxdydxdvdyudxdydxdyxyx+−=+=+=+=ψψψφφφ i.e. constconstdxdyvudxdy==−=−=ψφ1 8.2 Elementary plane flow solutions Uniform stream yxxyvconstUuφψφψ=−======∞0 i.e. yUxU∞∞==ψφ Note: 022=∇=∇ψφ is satisfied. ˆVUiφ∞=∇ = Say a uniform stream is at an angle α to the x-axis:058:0160 Chapter 8 Professor Fred Stern Fall 2009 4 4 cosuUyxψφα∞∂∂===∂∂ sinvUxyψφα∞∂∂==−=∂∂ After integration, we obtain the following expressions for the stream function and velocity potential: ()cos sinUy xψαα∞=− ()cos sinUx yφαα∞=+ 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, 3ALQVdAS⎡⎤=⋅⎢⎥⎣⎦∫ ()()2,2rrQrbvOrQvbrππ=⋅⋅= 0, ==⇒θvrmvr Where: 2Qmbπ= is the convenient constant with unit velocity × length (m>0 for source and m<0 for sink). Note that V is singular at (0,0) since rv →∞ In a polar coordinate system, for 2-D flows we will use: 1Vrrφφφθ∂∂=∇ = +∂∂058:0160 Chapter 8 Professor Fred Stern Fall 2009 5 5 And: .011() ()0rVrv vrr rθθ∇=∂∂+=∂∂ i.e.: rvrrvr∂∂−=∂∂==∂∂=∂∂==ψθφθψφθr1 velocityTangential1 velocityRadial Such that 0V∇⋅ = by definition. Therefore, rvrrvr∂∂−=∂∂==∂∂=∂∂==ψθφθψφθr101rm i.e. xymmyxmrm122tanlnln−==+==θψφ Doublets: The doublet is defined as: =−⇒−−=source2sink1source2sink1)(θθθθmψ mψ− 2tan1tan12tan1tantan21tanθθθθ)m()θ(θ+−=−=−ψ058:0160 Chapter 8 Professor Fred Stern Fall 2009 6 6 1222sin sintan tancos cossin sin2sincos costan( )sin sin1cos cos ; rrra rarrarrararrmrararaθθθθθθθθψθθθθθθθ==−+−−+−= =−+−+ For small value of a 122 2 2202sin 2lim tan ( )aar amy ymra r xyθλψ−→⎡⎤= − =− =−⎢⎥−+⎣⎦ cos2( ) am Doublet Strengthrλθλφ== By rearranging: 22222)2()2(ψλψλλ=++⇒+−= yxyxyψ It means that streamlines are circles with radius ψλ2=R and center at (0, -R) i.e. circles are tangent to the origin with center on y axis. The flow direction is from the source to the sink.058:0160 Chapter 8 Professor Fred Stern Fall 2009 7 7 2D vortex: Suppose that value of the ψ and φ for the source are reversal. 01rvKvrrrθφψθ=∂∂==−=∂∂ Purely recirculating steady motion, i.e. ()vfrθ=. integration results in: ln K=constantKθψ Krφ==− 2D vortex is irrotational everywhere except at the origin where V and V ×∇are infinity. Circulation Circulation is defined by: cC closed contourΓVd s ==⋅∫ For irrotational flow Or by using Stokes theorem: ( if no singularity of the flow in A) . 0cA A Γ Vd s VdA ndAω=⋅=∇×⋅ = =∫∫ ∫  Therefore, for potential flow 0=Γ 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 rK=θυ {{2200ˆˆ() 22 and VdsKv e rd e rd K Krππθθ θθθππΓΓ= ⋅ = = =∫∫058:0160 Chapter 8 Professor Fred Stern Fall 2009 8 8 Note: for point vortex, flow still irrotational everywhere except at origin itself where VÆ∞, i.e., for a path not including (0,0) 0Γ= Also, we can use Stokes theorem to show the existence of


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UI ME 5160 - Chapter 8 Inviscid Incompressible Flow

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