Eternal Flos An internal flo is surrounded by solid boundaries that can restrict the development of its boundary layer, for eample, a pipe flo. An eternal flo, on the other hand, are flos over bodies immersed in an unbounded fluid so that the flo boundary layer can gro freely in one direction. Eamples include the flos over airfoils, ship hulls, turbine blades, etc. One of the most important concepts in understanding the eternal flos is the boundary layer development. For simplicity, e are going to analyze a boundary layer flo over a flat plate ith no curvature and no eternal pressure variation. U U U U Dye streak laminar transition turbulent
Boundary Layer Definition Boundary layer thickness (): defined as the distance aay from the surface here the local velocity reaches to 99% of the free-stream velocity, that is u(y).99u. Somehat an easy to understand but arbitrary definition. Displacement thickness (*): Since the viscous force slos don the boundary layer flo, as a result, certain amount of the mass has been displaced (ejected) by the presence of the boundary layer (to satisfy the mass conservation requirement). Image that if e displace the uniform flo aay from the solid surface by an amount *, such that the flo rate ith the uniform velocity ill be the same as the flo rate being displaced by the presence of the boundary layer. Amount of fluid being displaced outard u * U ( U udy ), or * (1 ) dy U equals * U -u
Momentum Balance Eample: Determine the drag force acting on a flat plate hen a uniform flo past over it. Relate the drag to the surface shear stress. () (1) h s : all shear stresses Net Force change of linear momentum F V( ρv da) V( ρv da) + V( ρv da) all surfaces surface(1) surface() ( ). (Assume unit idth) ρ U U da+ ρ uda ρu h ρ udy (1) () From mass conservation: U h s udy, ρu h ρ U udy F ρ uu ( U ) dy. Force F is the surface acting on the fluid s
s Skin Friction The force acting on the plate is called the friction drag (D) (due to the presence of the skin friction). D -F ρ uu ( udy ) The drag is related to the deficit of momentum flu across the boundary layer. It can also be directly determined by the integration of the all shear stress over the entire plate surface: D da plate plate d Define momentum thickness ( θ ): thickness of a layer of fluid ith a uniform velocity U and its momentum flu is equal to the deficit of boundary layer momentum flu. u u U uu ( udy ), (1 ) dy U U ρ θ ρ θ
Wall Shear Stress and Momentum Thickness Therfore, the drag force can be related to the momentum thickness as D ρu θ, for a unit idth boundary layer and this relation is valid for laminar or turbulent flos. It is also knon that D plate d, dd dθ ρu d d Shear stress can be directly related to the gradient of dθ the momentum thickness along the streamise direction. d Recall that, for laminar flo, the all shear stress is defined as: µ ( u ) y y
Eample Assume a laminar boundary layer has a velocity profile as u(y)u (y/) for y and uu for y>, as shon. Determine the shear stress and the boundary layer groth as a function of the distance measured from the leading edge of the flat plate. uu y u(y)u (y/) dθ ρu d For a laminar flo U ( u y ) from the profile. µ y µ Substitute into the definition of the momentum thickness: θ u u y y Uy (1 ) dy (1 ) dy, since u U U θ. 6
Eample (cont.) dθ U 1 d ρ U, µ ρu d 6 d 6µ 1µ µ ρu ρu ρu Separation of variables: d, integrate 1( ), ν 1 U 3.46 3.46, here Re U Re ν ν 3.46, U 3 U ρµ U.89ρU 1 µ.89, Re Note: In general, the velocity distribution is not a straight line. A laminar flatplate boundary layer assumes a Blasius profile (chapter 9.3). The boundary layer thickness and the all shear stress behave as: 5. U ν 5. Re, (9.13)..33ρU Re, (9.14).
Laminar Boundary Layer Development 1 ( ).5 Boundary layer groth: Initial groth is fast Groth rate d/d 1/, decreasing donstream. 1.5 1 ( ) 5.5 1 Wall shear stress: 1/ As the boundary layer gros, the all shear stress decreases as the velocity gradient at the all becomes less steep.