ME 1404 Fluid Mechanics Drag and Lift Prof. M. K SINHA Mechanical Engineering Department N I T Jamshedpur 831014 Drag and Lift Forces • Any object immersed in a viscous fluid flow experiences a net force R from the shear stresses and pressure differences caused by the fluid motion. • The net force R acting on airfoil is resolved into a component D in flow direction U and the component L in a direction normal to U as shown in Fig.1. Fig. 1. Drag and Lift on airfoil Drag and Lift • Drag D is defined as the resistive force acting on an immersed body in the direction of flow. • Lift L is defined as a lifting force acting on an immersed body perpendicular to the direction of flow. • Drag is responsible for boundary layer, flow separation and wake. • For a symmetrical body there will be no any lift force. Development of Drag and Lift • Force acting on a body • Pressure of the fluid acting on a given element area dA is p and friction force per unit area is 훕 as shown in Fig. Types of Drag • The pressure force p dA acts normal to dA, while the friction force 훕 dA acts tangentially. • The drag Dp which is integration over the whole body surface of the component in the direction of flow velocity U of this force p dA is called form drag or pressure drag. • The drag Df is similar integration of 훕 dA and is called friction drag or skin drag or shear drag. Pressure and friction Drags • Mathematically we can write: • • The drag D on a body is the sum of the pressure drag and friction drag, whose proportions vary with the shape of the body. • By integrating the component of pdA and 훕dA normal to U, the lift L is obtained. Contributions of pressure and friction drags for various shapes • Table Drag and Lift Coefficients • For many engineering purposes, drag and lift forces are calculated from experimentally derived coefficients: • Drag and lift Coefficients are expressed as: • Where A is the projected area of the body on the plane vertical to the direction of flow. • Values of CD for bodies of various shapes are given in next table. Airfoils • An airfoil may be defined as a streamlined body designed to produce higher lift than low drag as shown in Fig. Airfoils • The angle 힪 between the flow direction U and the chord line is the angle of attack. • Projection area of airfoil A = bl where b is width and l is chord length of airfoil. • Aspect ratio = • D and L are drag and lift. • The line connecting the mid-points of upper and lower faces of the airfoil section is camber line or profile line. Airfoils • Airfoils are shaped to produce lift efficiently by accelerating flow over the upper surface to produce a low pressure region as shown in Fig. • Because the flow must again decelerate inevitably there must be a region of adverse pressure gradient near the rear of the upper surface. Pressur • Pressure profile of airfoil Airfoil Stall • Lift and drag coefficients of airfoil depend upon Reynolds number and angle of attack. • As angle of attack is increased the minimum pressure point moves forward on the upper surface and the minimum pressure becomes lower. • This increases adverse pressure gradient. • At some angle of attack, the adverse pressure gradient is strong enough to cause the boundary layer to separate completely from the upper surface. • This phenomenon is called stall of airfoil. Airfoil Stall • Flow around a stalled airfoil • The separated flow alters the pressure distribution, reducing lift sharply. • Increasing the angle of attack also causes the drag coefficient to increase. • At some angle of attack below stall the ratio of lift to drag, the lift -drag ratio, reaches a maximum value. Induced Drag • For wings (airfoils of finite span), lift and drag also are functions of aspect ratio. • Lift is reduced and drag increased compared with infinite span, because end effects cause the lift vector to rotate rearward. • For a given geometric angle of attack, this reduces effective angle of attack, reducing lift. • The additional component of lift acting in the flow direction increases drag. • The increase in drag due to lift is called induced drag. Finite-span airfoil • The drag of a finite-span airfoil is expressed as • Where AR is effective aspect ratio including the effect of planform shape and the subscript ᦎ refers to the infinite section drag coefficient at CL. • The lift coefficient must increase to support aircraft weight as speed is reduced. • Therefore, induced drag can increase rapidly at low flight speeds. Kutta-Joukowski lift theorem • Consider a cylinder placed in a uniform flow U rotates at angular velocity ω without flow separation as shown in Fig. Kutta-Joukowski lift theorem • Since the fluid on the cylinder surface moves at a circular velocity (u = r0ω) sticking to the cylinder due to viscosity of fluid. • The flow velocity at a given point on the cylinder surface( angle θ) is the tangential velocity vθ caused by uniform flow U and u or 2Usinθ + r0ω. • Apply Bernoulli’ equation for two points: • where pᦎis pressure of uniform flow and p is the pressure on the cylinder at a given point. • Also we can write • The lift acting on the cylinder per unit width is Kutta-Joukowski lift theorem • The circulation around the cylinder when it is placed in a uniform flow U with circular velocity u: • Then lift can be written in term of circulation • This equation is called Kutta – Joukowski lift equation. • This lift is used for design of baseball, tennis ball, golf ball, etc. Thank You References • Nakayma, Y. and Boucher, R. F., “Introduction to Fluid Mechanics”, Butterworth-Heinemann Publication, 2000. • Kreith and et.al."Fluid Mechanics" Mechanical Engineering Hand Book, CRC Press, 1999..