![]() ![]() Let u be the velocity at this distance from the plate. Now let us consider an elemental strip of area bdy normal to the direction of flow and at a distance y from the plate. At a distance x from the leading edge let δ be the thickness of the boundary layer. Let U be the approach velocity of the fluid. Suppose δ** is the depth of flow with uniform velocity U so as to have a kinetic energy equal to the loss of kinetic energy due to the boundary layer.Ĭonsider a fluid moving past a plate of width b as shown in Fig. Mass flowing per second through the elemental strip (ρudy) Energy Thickness δ**:Īgain consider the flow through an elemental strip (1.dy) distant y from the boundary. ![]() The depth θ* is called momentum thickness. For a depth of flow θ* with a velocity V momentum per second per unit width The momentum thickness θ* may be visualized as the depth of flow with uniform velocity U, so as to have a momentum per second equal to the loss of momentum per second due to boundary layer. Momentum of this quantity in the absence of the boundary layer = (ρudy) U Momentum of this quantity = (ρudy) u = ρu 2dy Let us consider the above quantity of the fluid. Momentum Thickness θ * :Īgain consider the flow through an elemental strip of area (1.dy) distance y from the boundary. The depth δ* is called displacement thickness. Suppose the plate is displaced normal to itself by δ* and the velocity is uniform at the value U, then the mass of fluid passing through the strip of thickness δ* will be ρUδ *. ∴ Total reduction in mass flowing per second due to the plate – ∴ Reduction in mass flowing per second through the elemental strip If the plate had not been present the mass flowing per second through the above elemental strip would have been ρUdy. Mass flowing per second through the elemental strip = ρudy Consider an elemental strip (1 x dy) distance y from the plate. At this section the velocity varies from zero at the plate to U at a distance δ from the plate.Ĭonsider unit width of the plate. 16.4.Ĭonsider a fluid moving with a velocity U approaching a flat plate at rest as shown in Fig.16.5.Īt a section distance x from the leading edge, let δ be the thickness of the boundary layer. Hence for purpose of analysis the distance from the supporting surface at which the velocity of the fluid is 0.99 U may be taken as the thickness of the boundary layer. In fact the velocity varies reaching the value U asymptotically. In the actual cases, it is very difficult to determine this distance. The type of flow in the boundary layer is dependent on the value of the Reynold’s number ρUx/μ. In the case of a long plate the flow may pass through all the three stages, namely laminar, transition and turbulent after which a break away may take place as shown in Fig. If the plate is sufficiently smooth it is seen that within the thickness of the turbulent boundary layer there exits an extremely thin layer adjacent to the plate within which a laminar flow occurs and this thin layer is called laminar sub-layer.įor a plate of short length, the flow within the boundary layer may be laminar throughout its length. This part of the boundary layer is considerably thicker. This region is called the transition zone.īeyond the point B the flow in the boundary layer is turbulent. This condition prevails for a certain distance AB (which is nearly equal to OA). Beyond the point A the laminar boundary layer becomes unstable and the flow properties are between those of laminar and turbulent flows. For a certain distance from the leading edge O to A the laminar boundary layer prevails. 16.2 shows a fluid flowing past a horizontal plate and the boundary layer of the flowing fluid. It is proportional to the square root of the distance x from the leading edge and is also dependent on the Reynold’s number ρUx/μ.įig. The thickness δ of the boundary layer increases with the distance from the leading edge. This phenomenon is called separation or break away of the boundary layer. Under certain conditions, the boundary layer may leave the surface and coil up into a vortex or whirlpool. The type of flow within the boundary layer may be stream line flow or turbulent flow depending on the particular problem or the distance from the leading edge of the solid boundary. It is in the boundary layer the entire viscous or frictional resistance between the moving fluid and the solid boundary surface, occurs. This layer of fluid within which the velocity variation takes place is known as the boundary layer. Within this distance the velocity of the fluid will vary from zero at the surface of the solid boundary to the velocity U. At this section the velocity is disturbed for a distance δ from the boundary. As the fluid moves past a solid boundary, the velocity of the fluid is disturbed for a certain distance from the surface of the boundary. Consider a fluid moving with a velocity U.
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