Question

I will give rating for the complete answer ?

A two-dimensional thin shear layer is one where there is strong shearing in the cross- stream direction (i.e. y-direction) resulting in дх" ду where x is the stream-wise direction. For this flow, the Reynolds-stress u'v' is the only turbulent stress of prime importance. (U and V are the velocities in the x- and y-direction respectively.) (a) Boussinesq's eddy-viscosity concept may be expressed as -uu; = v( + ) - konj Eq. 2.1 where Vy is the eddy-viscosity, k is the turbulent kinetic energy and 8y is the Kronecker delta. Use Prandtl's mixing length hypothesis to show that u'v' can be approximated as follows. u'v' = 1 Eq. 22 where L, is the mixing length. (15 marks) (b) The equation of motion for a region of a turbulent boundary layer can be reduced to =0 Eq. 2.3 dy where v is the fluid kinematic viscosity. If the approximation for the mixing length is

Answer 1

(b) part of this question is incomplete so it is not possible to see the question and hence to answer.

solution: (a) Boussinesa eddy- viscosity concept ed as: may be express U; 4; Vr ( aui & aU; dijk Where Vr Eddy Viscosity 1 um um, Turbulent kinetic Energy k= 2 dij Kronecker Delta For thin shear layer, the relevant component of ear ② may be restated as : V au - uvi ay Prandte assumed that Vr ~ u lm, where u is a turbulent velocity scale and lm is referred to as the mixing length. furthermore, Pranate postulated that: au lm ( Where lm is {the mixing length Эy and hence 2 V VT lim อ) ay lim om uvi OV Әу