Chapter 7 pp 14 23 In potential situation the flow goes through different region when px goes from xxx to advance with separation points dpending on magnitude of px Momentum Integral Equation To obtain general momentum integral relation which is valid for both laminar and turbulent flow BL form of momentum equation u v continuity dy y 0 dU 0 dx w 1 d dU C f 2 H 2 U 2 dx U dx For flat plate equation u u 1 dy U U 0 u 1 dy U 0 H Momentum uu x vu y p 1 x y 1 1 U 2 const 0 p x 2UU x 0 p x UU x 2 2 u U u x v y uu x uv y Uu x Uv y u x v y continuity p 1 y 2uu x vu y UU x u y Uu x Uv y uU u 2 U u U x vU vu x y 0 1 y dy w u U 2 x U 0 Cf u U u dy U x U u dy vU vu x 0 0 vU vu 0 u U u dy U 2 x 2UU x U x 1 dy U x U 0 d 1 dU 2 2 dx U dx C f d dU 2 H H 2 dx U dx 0 w 1 C f x 2 H U x 2 2 U U Laminar flow One parameter Solutions Problem method Assume y u f a b 2 c 3 d 4 and let U a b c d determined from boundary conditions 1 y 0 u yy U Ux 2 y u U u y 0 u yy 0 No slip is automatically satisfied F 2 2 3 4 u F G 12 12 3 U G 1 6 2 1 U x p x U w w 2 37 2 w 2 12 6 10 120 315 745 9072 U 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 U d 1 d 2 w U d dU U 2 H S w U dx 2 dx U dx dx 2 Define a new parameter 2 dU dx 2 dU 2 and seek solution in terms of i e dx 2 w S H H U dU 2 Define z z dx dz U 2 S 2 H F dx F complicated expression 0 45 6 Th waits based on xxx xxx of experimental data dz dU 0 45 6 0 45 6 z dx dx dz dU U 6z 0 45 dx dx U x i e 1 d 2 6 5 6 zU 0 45 U dx C zU 0 45 z U 5 dx 0 x 0 45 U 5 dx 0 x 0 0 6 U 0 2 02 Compute solutions 2 dU dx w S computed expression 0 09 0 62 This was based on curve fit to data U Note S 0 for 0 09 i e w 0 which is the condition for xxx xxx 0 09 To complete solution that is to evaluate velocity profile at at H we must determine 37 2 315 745 9072 from 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 flow a Compute Xsep x 0 1 L b C f a Sollution 5 6 x L x U 1 dx 0 075 1 1 6 L U 0 L x 0 U 06 1 L can be evaluated for given L ReL 0 x 0 x L 2 0 95 x 5 0 6 2 dU x 0 075 1 1 dx L sep 0 09 X sep L 0 123 3 higher than exact solution 0 1199 b Solution x C f 0 1 i e just before separation L 0 0661 1 S 0 099 C f Re 2 2 0 099 Cf Re 2 0 075 L 1 0 1 6 1 0 0661 L U0 U0 2 L 0 0661 0 0661 2 U0 Re L L 0 257 L Re L 12 1 Re Re L 0 257 Re L 2 L 1 2 0 099 2 0 099 Cf Re L 2 0 77 0 257 0 257 To compute solution i e get u need to determine U Consider the complex potential a 2 a 2 2 i z r e 2 2 a Re F z r 2 cos 2 2 a Im F z r 2 sin 2 2 1 V r e r e r e r cos i sin j e sin i cos j 0 v r ar cos 2 2 v ar sin 2 F z V r cos i sin j sin i cos j r cos sin i r sin cos j Potential flow xxx along surface consider 90 1 determine a such that U 0 at r L 90 U U 0 aL i e a 0 L 2 let U x at x L r U x U x 0 L x U 0 1 L L Chapter 7 pp 26 30 Integral 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 d dU c C f 2 H dx U c dx 2 In three unknowns H C f therefore xxxx xxxx the addition of at least two xxxx The usual approach is to use a xxxx shear item correction C f C f H 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 1 d dU Cf 2 H 2 dx U dx H Just as for the laminar flow case we would like to xxxx dU C C t t dx dU H H dx dU dx dU U x xxxx e g dx 2 dU For laminar flow all xxxx correlated xxxx terms xxxx xxxx r dx However for turbulent flow C f H cannot be correlated in terms a single parameter Additional parameters and relationships are required that model the influence of 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 method Closure of momentum integral equations for turbulent flow is accomplished through the use of the entrainment integral xxxx y x xxxx y x J i j x 0 V dA Continuity CS dU dx udy vdy dx 0 0 0 dU d d enterainment …
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