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QUESTION 1 Question 1 (Total 25 marks) (a) Using dimensional analysis, express the characteristic length, velocity and time scales for near-wall flow in terms of the density, viscosity and wall shear-stress. Show that the length scale can also be expressed as where v is the kinematic V ur viscosity and uy is the friction velocity. (10 marks) (b) Derive the law of the wall and explain its applicability in a typical turbulent boundary layer. (15 marks)

Answer 1

(a) Using dimensional analysis, express the characteristic length, velocity and time scales for near-wall flow in terms of the density, viscosity and wall shear-stress. Show that the length scale can also be expressed as where v is the kinematic ut viscosity and u, is the friction velocity.

Answer: aj Now we are wing dimentional analepis We know that с [L] MOLTO [8] - n111? yº (v] - Millst [u]=MILY (T1 - M°LT [2] = MIL-T-2 8) (L) = [3]"[43b [7] Molto = ma (29 Mb Eby b M 24-26 1 -3Q- b-C=1,-6-20-0 By Selving them a=-112,6-1, C=-1/2 :. [L] - [8][u]' [z] equatim A+b+c=0 L = 4 See -26 ü[v] [sa] [4]b[y]° L'7' = M92-29 Mb by b mit atbtc :0 1 -39-6-(=1,-6-20= | by solving them a=-62, 6-0, 6-12 [v]= [3]2 [m]° [1] I S V- equation 2 [1] - [S] [] (c) T'zM9:34 Mb Eb7 M-26 By solving them atb1( 20 -39-6-(0-6-20 = 1

a= 0, b=11 (=-1 tu (879 [4] [] It- 시 H Equahon Now to prove L. UT V Me / M Р M Sv Lv=u S To prove this multiply equation ③ & 드 X LV - Sgr Р is Mtv = Velocity Lv= ples : L = u S.V L=re ut

typical turbulent (b) Derive the law of the wall and explain its applicability in a boundary layer.

• The Law of the Wall is inferred by accepting that the disturbance close to that limit is a capacity just of the stream conditions relating at that divider and is autonomous of the stream conditions further away.
• The law of the divider expresses that the normal speed of a violent stream at one point is relative to the logarithm of the good ways starting there to the divider or the limit of the liquid locale.
• The logarithmic law of the divider is a self comparable answer for the mean speed corresponding to the divider and is substantial for streams at high Reynolds numbers in a cover area with around steady shear pressure and far enough from the divider for thick impacts to be insignificant .
• The mean-speed profile (MVP) of a divider limited fierce stream is the capacity u(y) that gives the mean speed of the stream u, at some random separation to the divider y . MVPs are utilized to register transitions and different amounts of building interest, quite the tempestuous grating on that account, the MVPs of regular divider limited violent streams, similar to pipe stream and limit layer stream, have for some time been the subject of exploration.
• The exemplary laws are basically hypothesized as an appropriate series of expectations with respect to the asymptotic conduct of u in the restrictions of disappearing thickness and unending fierce area.
• Things being what they are, a determination of the exemplary laws can be finished, An adequate condition that explains the overall extent of gooey and limited area impacts in the energetics of divider choppiness.
• A limit layer might be laminar or violent. A laminar limit layer is one where the stream happens in layer . A violent limit layer frames just at bigger Reynolds numbers. The size of blending can't be dealt with by sub-atomic thickness alone.
• urbulent limit layers and the related little scope forms assume an indispensable job. The environmental limit layer and the OML contiguous the air-ocean interface intervene the trading of mass, force, vitality, and warmth between the air and the sea.
• The idea of blending inside a fierce limit layer relies upon the specific idea of the tempestuous swirls, or sound stream structures, that are available inside the stream. These sound stream structures are created inside the thick sublayer and disturbance age locale, and can be examined through their speed marks and their fleeting and spatial length scales.
• The violent speed field in the thick sublayer of the limit layer with attractions to a first estimate is homogeneous toward any path corresponding to the divider and is controlled by just three consistent amounts , the divider shear pressure, the pull speed, and the liquid thickness. This implies there exists a limited logarithmic connection between the tempestuous shear pressure and the longitudinal mean-speed angle, utilizing which as a conclusion condition for the conditions of movement, we build up a precise asymptotic conduct of the speed profile at the external edge of the thick sublayer. The got relationship gives a speculation of the logarithmic law to the instance of divider attractions.
• ASBL is identicalwith that in other close divider fierce flows with appropriated attractions on smoothed out surfaces.Among them, there are the limit layer in zero, good and unfriendly weight angle , the flow in a channel withporous dividers,

90 Beller zone Trorbulent. zone Dimensionlen velocity Cominer 30 Sublayer/ 20 10 1 - 2 5 10 2 5 LOV 2 104 5 103 5 2 Dimensionless Distance from the wall

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