Assume that the local skin friction coefficient in turbulent flow is ...

Reading AssignmentAnderson, 3 rd edition: Chapter 15, pages 713 – 730, 738 – 740Chapter 16, pages 745 – 751, 781 – 786Anderson, 2 nd edition: Chapter 15, pages 637 – 655Chapter 16, pages 669 – 676, 705 - 709Problem 1The compressible, viscous momentum equations (i.e. the Navier-Stokes equations) given invector notation are:Du p τ τxx yx τzxρ =− ∂ Dt ∂ x + ∂ + ∂ + ∂ ∂x ∂y ∂zDv p τxyτyyτzyρ =− ∂ Dt ∂ y + ∂ + ∂ + ∂ ∂x ∂y ∂zDw p τ τxz yz τzzρ =− ∂ Dt ∂ z + ∂ + ∂ + ∂ ∂x ∂y ∂zwhere the viscous stresses are given by Stokes law (Equations 15.5 – 15.10 in 3 rd Edition ofAnderson). Assuming the flow is incompressible and that the dynamic viscosity is constant,show that the incompressible form of the Navier-Stokes equations can be written in vectornotation as:Problem 2rDVrρ =−∇ p+ µ ∇ 2 VDtStarting from the incompressible Navier-Stokes equations derived in Problem 1, derive thefollowing ‘Bernoulli-like’ equation:Fr∂Vρ ρ ρ ω µ ω∂ +∇ H+r 2K = r × r − ∇× r1p2V VtThe following vector calculus identities might be helpful:diIr r r 2 r rV ⋅∇ V =∇F V VrH1 I2rK− × ω2r∇ V =∇ ∇⋅V−∇× ωdi

Problem 3Starting from the incompressible Navier-Stokes equations derived in Problem 1, derive thefollowing equation for the evolution of the vorticity:rDωr r2r= bω⋅∇ gV+ ν∇ωDtProblem 4p aυ wp bLConsider the 2-D flow between two porous walls of length L at y =± h. Assume that the lengthof the walls is much larger than the half-height h so that the flow may be assumed to be constantin the x-direction. The upper wall produces a suction such that the velocity normal to the wall isυ wand is positive. Solve the incompressible Navier-Stokes equations to find the velocity andpressure everywhere in duct for the case in which the pressure inlet and exit of the pipe arep aand p b, respectively. Specifically, you should find the following solution for the x-velocity:uu0FHGRe Re2 y e − e yh= − 1+Re h sinh Rewhere u 0is a constant that you must determine as part of the solution and the Reynolds numberin this problem is defined asRe = υ h wνIKJShow that for the case of low Reynolds number (i.e. small blowing), the solution returns to theparabolic velocity profile for Poiseuille flow. Do this by carefully taking the limit of the velocityas Re → 0 .Show that for very large Reynolds number, the x-velocity (except near the walls) isapproximately,uu0FReH G I K J2 y ≈ 1+h