FLUID MECHANICS Chapter 9 Flow over Immersed Bodies
CHAP 9. FLOW OVER IMMERSED BODIES CONTENTS 9.1 General External Flow Characteristics 9.3 Drag 9.4 Lift
9.1 General External Flow Characteristics 9.1.1 Lift and Drag Concepts Body immersed in a moving fluid interaction between the body and fluid occurs. Forces at the fluid-body interface - Wall shear stress, τ w - pressure, p Drag ( ) ; Resultant force in the direction of the upstream velocity Lift ( ) ; Resultant force normal to the upstream velocity Components of fluid force on the small area element df ( pda)cos ( da)sin x df ( pda)sin ( da)cos y (9.1) (9.) C D w w Net components of the fluid force on the element x w y w D df pcosda sinda L df psinda cosda Drag coefficient & Lift coefficient, C 1 L 1 U A U A Find approximate value from numerical or experimental method (a) pressure force (b) Viscous force (c) Resultant force (lift and drag) Forces from the surrounding fluid on a two-dimensional object Pressure and shear forces on a small element of the surface of a body.
9.1 General External Flow Characteristics 9.1. Characteristics of Flow Past an Object Most important parameters for typical external flows Reynolds number ( Re = ρul/μ ) Mach number ( Ma = U/c ) Flow past flat plates of length l 1. Low Reynolds number flow ( Re = 0.1 ) - Viscous effects are relatively strong - Plate affects the uniform upstream flow far ahead. Moderate Reynolds number flow ( Re = 10 ) - Region in which viscous effects are important becomes smaller in all directions except downstream 3. Large Reynolds number flow ( Re = 10 7 ) - Flow is dominated by inertial effects - Viscous effects are negligible everywhere except in a region very close to the plate and in the relatively thin wake region behind the plate - Boundary Layer ; thin region of thickness ( x) l next to the plate in which the fluid velocity changes from the upstream value of to zero velocity on the plate Character of the steady, viscous flow past a flat plate parallel to the upstream velocity
9.1 General External Flow Characteristics Flow past a circular cylinder 1. Low Reynolds number flow ( Re = 0.1 ) - Viscous effects are important several diameters in any direction from the cylinder. - Streamlines are essentially symmetric about the center of the cylinder. Moderate Reynolds number flow ( Re = 10 ) - Viscous effects is smaller in front of the cylinder. - Separation location & Separation bubble ; With the increase in Reynolds number, fluid inertia becomes more important and cannot follow the curved path around to the rear of body. 3. Large Reynolds number flow ( Re = 10 7 ) - Area of Viscous effects is forced farther downstream and it involves only a thin ( D) boundary layer. - Unsteady (perhaps turbulent) wake region extends far downstream of the cylinder. - Fluid outside of the boundary layer and wake region flows as if it were inviscid. - Velocity gradients in the boundary layer and wake regions are larger than in the remainder of the flow field. (related to the shear stress) (a) Low Reynolds number flow (b) Moderate Reynolds number flow (c) Large Reynolds number flow Character of the steady, viscous flow past a circular cylinder
9..1 Boundary Layer Structure and Thickness on a Flat Plate Outside of boundary layer - Uniform flow - Irrotational flow - Consider as inviscid flow - Rectangular fluid particle Laminar boundary layer - Rotational flow - Viscous effect - Top of the particle has larger speed velocity gradients - Particle begins to distort Turbulent boundary layer - Rotational flow - Viscous effect - Flow becomes turbulent at some distance downstream - Particles become greatly distorted Distortion of a fluid particle as it flows within the boundary layer.
Boundary Layer Thickness y where u 0.99U Boundary Layer Displacement Thickness ; To make fictitious uniform inviscid flow has the same mass flowrate properties as the viscous flow * bu ( U u) b dy (9.3) 0 u * 1 dy 0 U u( U u) da b u( U u) dy 0 bu θ b u( U u) dy (9.4) u u θ 1 dy 0 U U 0 Boundary Layer Momentum Thickness ; Using when determining the drag on object ; Deficit in momentum flux because of the velocity deficit (a) Standard boundary layer thickness (b) Boundary layer displacement thickness Boundary layer thickness The boundary layer concept is based on that the layer is thin. x, * x, θ x
9.. Prandtl/Blasius Boundary Layer Solution Navier-Stokes Equation for viscous, incompressible flow For steady, -dimensional laminar flows with negligible gravitational effects, (9.5) (9.6) (9.7) u u 1 p u u u v x y x x y v v 1 p v v u v x y y x y Continuity equation for incompressible flow u v 0 x y Analytical solution has not been obtained! Much work is being done to obtain numerical solutions to these governing equation.
Prandtl/Blasius Boundary Layer Solution Boundary layer is thin Component of velocity normal to the plate is much smaller than that parallel to the plate. Rate of change of any parameter across the boundary layer is much greater than that along the flow direction. v u and x y Boundary layer equations u v (9.8) 0 x y u u (9.9) u v u x y y For boundary layer flow over a flat plate the pressure is constant throughout the fluid. Boundary conditions for the governing boundary layer equations (9.10) (9.11) u v 0 on y 0 u U as y
Dimensionless form the boundary layer velocity profiles u y g U Unknown function to be determined By applying an order of magnitude analysis for the governing equation, u v order of magnitude analysis U v 0 ~ x y x u u u U U U x x u v U v x y x U y U (9.1) (9.13) (9.14a) (9.14b) u Uf ( ) order of magnitude analysis 1/ ~ ~ ~ ~ Dimensionless similarity variable ( U / x) y 1/ Stream function ( xu ) f ( ), where f f ( ) U 1/ ( ) v f f 4x f ff 0 f f 0 at 0 and f 1 as 1/ Velocity components for two-dimensional flow in terms of the stream function u / y, v / x Substitute into the governing equations (boundary layer equations) and the boundary conditions
For f ( ) u/ U 0.9916, 5.0. Thus, Boundary layer thickness (9.15) 5 x x U 5 Re x Displacement and momentum thickness * 1.71 (9.16) x Re (9.17) (9.18) θ 0.664 x Re w x x Wall shear stress using the Blasius solution w ( u/ y) y 0 0.33U 3/ x η = y(u / νx) 1/ f(η) = u / U f(η) 0.0 0.0000 3.6 0.933 0.4 0.138 4.0 0.9555 0.8 0.647 4.4 0.9759 1. 0.3938 4.8 0.9878 1.6 0.5168 5.0 0.9916.0 0.698 5. 0.9943.4 0.790 5.6 0.9975.8 0.8115 6.0 0.9990 3. 0.8761 1.0000 Laminar Flow along a Flat Plate (the Blasius Solution) η
9..3 Momentum Integral Boundary Layer for a Flat Plate ; Approximate method to obtain the solution of the governing differential equation Drag caused by shear forces on a body ; Important for boundary layer theory Uniform flow past a flat plate and the fixed control volume X component of the momentum equation for the steady flow x (1) () F uv nˆ da uv nˆ da For a plate of width b, (9.19) F D da b dx x plate w plate w D U ( U ) da u da (9.0) Uh udy 0 (1) () D U bh b u dy From the conservation of mass, (9.1) U bh b Uu dy 0 0 Control volume used in derivation of the momentum integral equation
Drag in terms of the deficit of momentum flux across the outlet of the control volume (9.) D b u( U u)dy 0 If the flow were inviscid, the drag would be zero ( u U) Boundary layer flow on a flat plate is governed by a balance between shear drag and a decrease in the momentum of the fluid In terms of the momentum thickness, (9.3) D bu θ < Momentum deficit > Shear stress distribution dd dθ (9.4) bu dx dx (9.5) (9.6) dd dx b w dθ w U dx If we knew the detailed velocity profile in the boundary layer (i.e., the Blasius solution), we could obtain the drag.
9..6 Effect of Pressure Gradient In general, when a fluid flows past a flat plate, the pressure field is not uniform. If Reynolds number is large, the boundary layers will be relatively thin. Component of the pressure gradient along the body (streamwise dir.) is not zero. Variation in the free-stream velocity Ufs ; Fluid velocity at the edge of the boundary layer ; Cause of the pressure gradient in the direction For a flat plate parallel to the upstream flow with the negligible thickness of the plate, U U But for bodies of nonzero thickness, U Upstream velocity and pressure; U, p0 fs U If the fluid is inviscid ( 0), Re UD/ Streamlines would be symmetrical. Maximum pressure: Minimum pressure: p0 U / p0 3 U / fs (a) Streamline for the flow if there were no viscous effects (b) pressure distribution on the cylinder s (c) free-stream velocity on the surface cylinder s surface Inviscid flow past a circular cylinder
d Alembert s paradox We would find that no matter how small we make the viscosity (provided it is not precisely zero) we would measure a finite drag, essentially independent of the value of. Drag on an object in an inviscid fluid is zero, but the drag on an object in a fluid with vanishingly small(but nonzero) viscosity is not zero. For large Reynolds number flow of a viscous fluid past a circular cylinder, ( 0) Viscous effects confined to thin boundary layer near the surface Boundary layer theory ; Boundary layer is thin enough so that it does not greatly disturb the flow outside the boundary layer. Pressure variation is negligible across the thin boundary layer. Pressure within the boundary layer is that given by the inviscid flow field. U 0 at 0 Accelerate Decelerate U U at 90 U 0 at 180 fs fs fs This is accomplished by a balance between pressure and inertia effects Favorable pressure gradient ; Decrease in pressure in the direction of flow along the front half of the cylinder Adverse pressure gradient ; Increase in pressure in the direction of flow along the rear half of the cylinder
Fluid particle within the boundary layer ; Fluid particle travel from the front to the back of the cylinder coasts down the pressure hill from point A to C. Because of the viscous effects, the particle in the boundary layer experiences a loss of energy as it flows along. the particle does not have enough energy to coast all of the way up the pressure hill (from C to F) and to reach point F at the rear of the cylinder. Because of friction, the boundary layer fluid cannot travel from the front to the rear of the cylinder. Boundary layer separation (a) Boundary layer separation location Boundary layer characteristics on a circular cylinder (b) typical boundary layer velocity profiles at various locations on the cylinder (c) Surface pressure distribution for inviscid flow and boundary layer flow
9..7 Momentum Integral Boundary Layer Equation with Nonzero Pressure Gradient Momentum integral equation for boundary layer flows with zero pressure gradient Balance between the shear force on the plate ( w ) and rate of change of momentum of the fluid within the boundary layer U U fs If the free-stream velocity is not constant Bernoulli equation with negligible gravitational effects pu fs / const. along the streamline outside the boundary layer dp du fs (9.34) U fs dx dx Integral momentum equation for boundary layer flows d du fs (9.35) w ( U fs) * U fs dx dx Ufs Ufs ( x ), the pressure will not be constant.
9.3 Drag Drag Coefficient (9.36) CD C D 1 U A (shape, Re, Ma, Fr, / l) where, f 1 Re : Reynolds number Ma : Mach number Fr : Froude number / l : Relative roughness of the surface Friction Drag ; due to the shear stress w D U blc Df Pressure Drag ; due to the pressure p Dp pcosda Pressure drag coefficient Dp pcosda C p cosda (9.37) CDp 1 1 U A U A A
9.4 Lift Lift Coefficient (9.39) CL C L 1 U A (shape, Re, Ma, Fr, / l) where, Re : Reynolds number Ma : Mach number Fr : Froude number / l : Relative roughness of the surface