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Chapter 7 pp.14-23In potential situation, the flow goes through different region when px goes from xxx to advance with separation points dpending on magnitude of pxMomentum Integral EquationTo obtain general momentum integral relation which is valid for both laminar and turbulent flow:  dyvuy0continuityequation momentum of form BL For flat plate equation0dxdU 0**0211221dyUuHdyUuUudxdUUHdxdCUfwMomentum: ypxvuuuyx1const.212 Up00221xxUUpxxUUp  yxyxyxUvUuuvuuvuUu ,  continuityyxvu    vuvUyUuUuuUxUvUuuUUvuuuxyxyxyxy221       0 0001vuvUdyuUUdyuUuxdyxy,0 vuvU *200221xxxxwUUUUdyuUUdyUuUuUx dxdUUdxdCf122* dxdUUHdxdCf 22, *H xxfwUUHCU 2212Laminar flow One parameter SolutionsProblem method:Assume  432dcbafUu, and let ya, b, c, d determined from boundary conditions1) 0yxyyUUu2) yUu , 0yu, 0yyuNo slip is automatically satisfied.    3431622GF   GFUu, 1212 221UpUxx,     ww,,**1262,907274531537,120102*UwThus, by assuming the form of the profile we have expressed all the momentum integral xxx in terms of i.e.(MISSING THE NOTES)To determineand compute the solution we make use of momentum integral relation.1st multiply momentum integral equation by U HdxdUdxdUUw 22,  dxdUdxdUdxdUUSw22,21,Define a new parameter  222dxdU, and seek solution in terms of ;i.e.   HHSUw ,Define 2z,dxdUz    FHSdxdzU  22 645.0expression dcomplicate F(Th waits based on xxx xxx of experimental data)dxdUzdxdzU 645.0645.0 45.06 dxdUzdxdzUi.e.  45.0165zUdxdUCdxUzUx05645.0, 2zxdxUU05620245.0, 0)0(0xCompute solutions: dxdU2 SUwcomputed expression= 62.009.0 (This was based on curve fit to data.)Note: S=0 for 09.0i.e. 0wwhich is the condition for xxxxxx: 09.0To complete solution; that is, to evaluate velocity profile at at H we must determinefrom  9072745315372Assuming: evaluation : 10%Velocity profile: just OK but can be improved by using better approximation to the xxx xxx polynomial.Example 7.5: xxx xxx xxx flowa) Compute Xsepb) 1.0LxCfa) Sollution11075.01195.06005506602LxULdxLxULxUxcan be evaluated for given L, ReL (Lxx ,00)11075.062LxdxdU123.009.0 LXsepsep: 3% higher than exact solution (=0.1199)b) Solution 1.0LxCfi.e. just before separation      77.0257.0099.02,Re257.0099.02Re257.0ReReRe257.0Re0661.00661.00661.011.01075.0Re)099.0(2Re21099.00661.02121210220602LfLLLLffCLLULLULULCCSTo compute solution, i.e. get *,Uu, need to determine .Consider the complex potential ierazazF22222  2cos2Re2razF   2sin2Im2razF 2sin2cosˆ1ˆarvarvereVrrr,02,ˆcosˆsinˆ,ˆsinˆcosˆjiejier      jijijiVrrrˆcossinˆsincosˆcosˆsinˆsinˆcosPotential flow xxx along surface: (consider 90)1) determine a such that 0Uat r=L, 90 aLU 0, i.e. LUa02) let  xUat x=L-r   LxUxLLUxU 100Chapter 7: pp.26-30Integral Methods:Although xxxx, it is usefull to read section 6-8 of the text and study xxxx 6-8.The momentum-integral equation represents one equation:2)2(fccCdxdUUHdxdIn three unknowns:fCH ,, therefore xxxx xxxx the addition of at least two xxxx. The usual approach is to use a xxxx-shear item correction.),(HCCffAnd then an additional relation associated with the xxxx velocity xxxx xxxx xxxxx xxxx eg )( + xxxx’s xxxx log-law (eg (6-76)+(6-77)). Such a method is described in section 6-8.3. two alternative approaches are discussed in sections 6-8.4 and 6-8.5. here we shall xxxx xxxx xxxx method which has been xxxx as part 2 of HW#3.Turbulent flows*)2(21HdxdUUHdxdCfJust as for the laminar flow case we would like to xxxx)..,()()()(gexxxxUdxdUdxdUdxdUHHdxdUCCxttFor laminar flow all xxxx correlated xxxx terms  xxxx xxxx dxdUr2However for turbulent flow (,, HCf) cannot be correlated in terms  a single parameter. Additional parameters and relationships are required that model the influenceof the turbulent fluctuations. There are many xxxx all of which require a curtain amount of empirical data. As an example we will review the method of xxxx.Xxxx methodClosure of momentum integral equations for turbulent flow is accomplished through the use of the entrainment-integral xxxx.jixJxyxxxxxyˆˆ))(()(Continuity:


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