U zu x 5 6 45 x dx uu 5 6 2 2 45 x and ux known from

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UzUx05645.0xdxUU05620245.00)0(0xand U(x) known from potential flow solution Complete solution:  dxdU2
058:0160 Chapter 7 Professor Fred Stern Fall 2010 23  SUw H*Accuracy: mild px5% and strong adverse px(wnear 0) 15% i.Pohlhausen Velocity Profile:  432dcbafUuwith ya, b, c, d determined from boundary conditions 1) 0yu= 0, xyyUUu2) yUu, 0yu, 0yyuNo slip is automatically satisfied.   3431622GF  GFUu, 1212UpdxdUx22pressure gradient parameter related to 9072945315372Profiles are fairly realistic, except near separation. In guessed profile methods u/U directly used to solve momentum integral equation numerically, but accuracy not as good as empirical correlation methods; therefore, use Thwaites method to get etc., and then use to getand plot u/U. (experiment: separation= -5) separation
058:0160 Chapter 7 Professor Fred Stern Fall 2010 24 ii.Howarth linearly decelerating flow (example of exact solution of steady state 2D boundary layer) Howarth proposed a linearly decelerating external velocity distribution LxUxU1)(0as a theoretical model for laminar boundary layer study. Use Thwaites’s method to compute: a)Xsepb)1.0LxCfNote Ux = -U0/L Solution 11075.01145.06005506602LxULdxLxULxUxcan be evaluated for given L, ReL(Note: Lxx,00) 11075.062LxdxdU
058:0160 Chapter 7 Professor Fred Stern Fall 2010 25 123.009.0LXsepsep3% higher than exact solution =0.1199 1.0LxCfi.e. just before separation 0.066110.099Re22(0.099)ReffSCC Compute Rein terms if ReL2/12121210220602Re77.0Re257.0099.02Re257.0ReReRe257.0Re0661.00661.00661.011.01075.0LLfLLLLCLLULLULULTo complete solution must specify ReL
058:0160 Chapter 7 Professor Fred Stern Fall 2010 Consider the complex potential  ierazazF22222 2cos2Re2razF 2sin2Im2razFOrthogonal rectangular hyperbolas : asymptotes y = ± x : asymptotes x=0, y=0 2sin2cosˆ1ˆarvarvereVrrr02(flow direction as shown) jvvivvjivjivVrrrˆcossinˆsincosˆcosˆsinˆsinˆcosPotential flow slips along surface: (consider 901)determine asuch that 0Uvrat r=L, 9000)902cos(UaLUaLvr, i.e. Ua02)let  rvxUat x=L-r: )1()()()(:)()902cos(00LxUxLLUxLaxUOrxUxLavr
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058:0160 Chapter 7 Professor Fred Stern Fall 2010 27
058:0160 Chapter 7 Professor Fred Stern Fall 2010 28
058:0160 Chapter 7 Professor Fred Stern Fall 2010 29 7.6. Turbulent Boundary Layer1.Introduction: Transition to Turbulence

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