![1. A flat plate 1.5m x 1.5m moves as 50km/hr in stationary air of density
1.15kg/m3. If the coefficients of drag and lift are 0.15 and 0.75 respectively.
Determine:
(i) The lift force
(ii) The drag force
(iii) The resultant force and
(iv) The power required to keep the plate in motion
[187.20 N, 37.44 N, 0.519 kW]
2. Find the difference in drag force exerted on a flat plate of size 2m x 2m
when the plate is moving at a speed of 4m/s normal to its plane in (i) water (ii)
air of density 1.24kg/m3. Coefficient of drag is 0.15 [36754.4 N].](https://image.slidesharecdn.com/unit5-250731063610-ebe7b34d/85/Boundary-layer-Boundary-layer-concept-Characteristics-of-boundary-layer-along-a-thin-flat-plate-Von-Karman-momentum-integral-equation-29-320.jpg)
Boundary layer Boundary layer concept Characteristics of boundary layer along a thin flat plate, Von Karman momentum integral equation,
- 1. BOUNDARY LAYER THEORY CONTENTS:- Boundary layer concept Characteristics of boundary layer along a thin flat plate, Von Karman momentum integral equation, Laminar and Turbulent Boundary layers Separation of Boundary Layer, Control of Boundary Layer, flow around submerged objects- Drag and Lift- Expression Magnus effect.
- 2. BOUNDARY LAYER When a real fluid will flow over a solid body or a solid wall, the particles of fluid will adhere to the boundary and there will be condition of no-slip. If we assume that boundary is stationary or velocity of boundary is zero, then the velocity of fluid particles adhere or very close to the boundary will also have zero velocity. If we move away from the boundary, the velocity of fluid particles will also be increasing. Velocity Gradient = Velocity of fluid particles will be changing from zero at the surface of stationary boundar y to the free stream velocity (U) of the fluid in a direction normal to the boundary.
- 3. Therefore, there will be presence of velocity gradient (du/dy) due to variation of velocity of fluid particles. The variation in the velocity of the fluid particles, from zero at the surface of stationary boundary to the free stream velocity (U) of the fluid, will take place in a narrow region in the vicinity of solid boundary and this narrow region of the fluid will be termed as boundary layer. Science and theory dealing with the problems of boundary layer flows will be termed as boundary layer theory. According to the boundary layer theory, fluid flow around the solid boundary might be divided in two regions as mentioned and displayed here in following figure.
- 4. First region A very thin layer of fluid, called the boundary layer, in the immediate region of the solid boundary, where the variation in the velocity of the fluid particles, from zero at the surface of stationary boundary to the free stream velocity (U) of the fluid, will take place. There will be presence of velocity gradient (du/dy) due to variation of velocity of fluid particles in this region and therefore fluid will provide one shear stress over the wall in the direction of motion. Shear stress applied by the fluid over the wall will be determined with the help of following equation. τ = µ (du/dy) Second region Second region will be the region outside of the boundary layer. Velocity of the fluid particles will be constant outside the boundary layer and will be similar with the free stream velocity (U) of the fluid. In this region, there will be no velocity gradient as velocity of the fluid particles will be constant outside the boundary layer and therefore there will be no shear stress exerted by the fluid over the wall beyond the boundary layer.
- 5. 1. Laminar boundary layer Length of the plate from the leading edge up to which laminar boundary layer exists will be termed as laminar zone. AB indicates the laminar zone in the figure. The distance B from leading edge is obtained from Reynolds number equal to 5X105 for a plate. The Reynold’s No. is given by - Re = BASIC DEFINITIONS Where x = Distance from leading edge up to which laminar boundary layer exists U = Free stream velocity of the fluid v = Kinematic viscosity of the fluid Re = ………… If U and are known then x can be calculated.
- 6. 2. Turbulent boundary layer It will lead to a transition from laminar to turbulent boundary layer. This small length over which the boundary layer flow changes from laminar to turbulent will be termed as transition zone, BC, in figure, indicates the transition zone. Further downstream the transition zone, boundary layer will be turbulent and the layer of boundary will be termed as turbulent boundary layer. FG, in the figure, indicates the turbulent boundary layer and CD represent the turbulent zone. If the length of plate is greater than the value of x which is determined from above equation, thickness of boundary layer will keep increasing in the downstream direction. Laminar boundary layer will become unstable and movement of fluid particles within it will be disturbed and irregular.
- 8. Boundary layer thickness is basically defined as the distance from the surface of the solid body, measured in the y-direction, up to a point where the velocity of flow is 0.99 times of the free stream velocity of the fluid. Boundary layer thickness will be displayed by the symbol δ. We can also define the boundary layer thickness as the distance from the surface of the body up to a point where the local velocity reaches to 99% of the free stream velocity of fluid. BOUNDARY LAYER THICKNESS
- 9. Displacement thickness Displacement thickness is basically defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in flow rate on account of boundary layer formation. • Displacement thickness will be displayed by the symbol δ*. • We can also define the displacement thickness as the distance, measured perpendicular to the boundary of the solid body, by which the free stream will be displaced due to the formation of boundary layer
- 10. Momentum thickness Momentum thickness is basically defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in momentum of the flowing fluid on account of boundary layer formation. Momentum thickness will be displayed by the symbol θ. Energy thickness Energy thickness is basically defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in kinetic energy of the flowing fluid on account of boundary layer formation. Energy thickness will be displayed by the symbol δ**.
- 11. Characteristics of boundary layer along a thin flat plate
- 12. Boundary layer Separation Effect of Pressure Gradient on Boundary Layer Separation The flow separation depends upon factors such as (i) The curvature of the surface (ii) The Reynolds number of flow (iii) The roughness of the surface
- 23. The following are some of the methods generally adopted to retard or arrest the flow separation: 1. Streamlining the body shape 2. Tripping the boundary layer from laminar to turbulent by provision of surface roughness 3. Sucking the retarded flow 4. Injecting high velocity fluid in the boundary layer 5. Providing slots near the leading edge 6. Guidance of flow in a confined passage 7. Providing a rotating cylinder near the leading edge 8. Energizing the flow by introducing optimum amount of swirl in the incoming flow Methods of preventing the Separation of Boundary Layer
- 24. Applications of Boundary Layer
- 25. Laminar and Turbulent boundary layer
- 26. When a fluid is flowing over a stationary body, a force is exerted by the fluid on the body. Similarly, when a body is moving in a stationary fluid, a force is exerted by the fluid on the body. Also, when both the body and fluid are moving at different velocities, a force is exerted by the fluid on the body. Some of the examples of the fluids flowing over stationary bodies or bodies moving in a stationary fluid are: (a) Flow of air over buildings, (b) Flow of water over bridges (c) Submarines, ships, airplanes and automobiles moving through water and air Forces on a submerged bodies
- 27. Force Exerted by a Flowing fluid on Stationary Bodies Consider a body held stationary in a real fluid which is flowing at a uniform velocity U as shown in the figure below The fluid will exert a force on the stationary body. The total force (FR) exerted by the fluid on the body is perpendicular to the surface of the body. Thus the total force is inclined to the direction of motion. The total force can be resolved into two components, or in the direction of motion and the other perpendicular to the direction of motion.
- 28. DRAG When a body is immersed in a fluid and is in relative motion with respect to it, the drag is defined as that component of the resultant or total force (FR) acting on the body which is in the direction of the relative motion. This is denoted by FD LIFT The component of the total or resultant force (FR) acting in the direction normal or perpendicular to the relative motion is called lift i.e. the force component perpendicular to drag. This component is denoted by FL. Lift force occurs only when the axis of the body is inclined to the direction of fluid flow. If the axis of the body is parallel to the direction of fluid flow, lift force is zero. In that case only drag force acts. If the fluid is assumed ideal and the body is symmetrical such as a sphere or a cylinder both drag and lift will be zero **Refer the notes for expression of drag and lift
- 29. 1. A flat plate 1.5m x 1.5m moves as 50km/hr in stationary air of density 1.15kg/m3. If the coefficients of drag and lift are 0.15 and 0.75 respectively. Determine: (i) The lift force (ii) The drag force (iii) The resultant force and (iv) The power required to keep the plate in motion [187.20 N, 37.44 N, 0.519 kW] 2. Find the difference in drag force exerted on a flat plate of size 2m x 2m when the plate is moving at a speed of 4m/s normal to its plane in (i) water (ii) air of density 1.24kg/m3. Coefficient of drag is 0.15 [36754.4 N].
- 30. Magnus effect. The force exerted on a fast spinning cylinder or sphere travelling through air or another fluid in a direction perpendicular to the axis of spin is known as the Magnus force. Magnus effect, generation of a sidewise force on a spinning cylindrical or spherical solid immersed in a fluid (liquid or gas) when there is relative motion between the spinning body and the fluid. Named after the German physicist and chemist H.G. Magnus, who first (1853) experimentally investigated the effect, it is responsible for the “curve” of a served tennis ball or a driven golf ball and affects the trajectory of a spinning artillery shell. The Magnus effect explains how a football player may bend the ball into a goal around a five-person wall. Magnus effect is an application of Newton’s third law of motion. As a result of the object pushing the air in one direction, the air pushes the object in the opposite direction.