57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 1 Chapter 9 Flow over Immersed Bodies Fluid flows are broadly categorized: 1. Internal flows such as ducts/pipes, turbomachinery, open channel/river, which are bounded by walls or fluid interfaces: Chapter 8.. External flows such as flow around vehicles and structures, which are characterized by unbounded or partially bounded domains and flow field decomposition into viscous and inviscid regions: Chapter 9. a. Boundary layer flow: high Reynolds number flow around streamlines bodies without flow separation. b. Bluff body flow: flow around bluff bodies with flow separation.
57:000 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 009 Chapter 9 3. Free Shear flows such as jets, wakes, and mixing layers, which are also characterized by absence of walls and development and spreading in an unbounded or partially bounded ambient domain: advanced topic, which also uses boundary layer theory. Basic Considerations Drag is decomposed into form and contributions: skin-friction C D 1 1 ρv A S ( p p ) n C Dp îda + τ τ w S t îda C f
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 3 1 CL ( p p ) n ĵda 1 ρv A S t << 1 Cf > > C Dp streamlined body c t 1 CDp > > C f bluff body c Streamlining: One way to reduce the drag reduce the flow separation reduce the pressure drag increase the surface area increase the friction drag Trade-off relationship between pressure drag and friction drag Trade-off relationship between pressure drag and friction drag Benefit of streamlining: reducing vibration and noise
57:000 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 009 Qualitative Description of the Boundary Layer Chapter 4 9 Flow-field regions for high Re flow about slender bodies:
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 5 τ w shear stress τ w rate of strain (velocity gradient) u μ y y 0 large near the surface where fluid undergoes large changes to satisfy the no-slip condition Boundary layer theory and equations are a simplified form of the complete NS equations and provides τ w as well as a means of estimating C form. Formally, boundary-layer theory represents the asymptotic form of the Navier-Stokes equations for high Re flow about slender bodies. The NS equations are nd order nonlinear PDE and their solutions represent a formidable challenge. Thus, simplified forms have proven to be very useful. Near the turn of the last century (1904), Prandtl put forth boundary-layer theory, which resolved D Alembert s paradox: for inviscid flow drag is zero. The theory is restricted to unseparated flow. The boundary-layer equations are singular at separation, and thus, provide no information at or beyond separation. However, the requirements of the theory are met in many practical situations and the theory has many times over proven to be invaluable to modern engineering.
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 6 The assumptions of the theory are as follows: Variable order of magnitude u U O(1) v <<L O(ε) ε /L x 1/L O(1) y 1/ O(ε -1 ) ν ε The theory assumes that viscous effects are confined to a thin layer close to the surface within which there is a dominant flow direction (x) such that u U and v << u. However, gradients across are very large in order to satisfy the no slip condition; thus, >>. y x Next, we apply the above order of magnitude estimates to the NS equations.
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 7 u u p u u u + v + ν + x y x x y 1 1 ε ε -1 ε 1 ε - u + v + ν + x y y x y 1 ε ε 1 ε ε ε -1 v v p v v elliptic u v + x y 1 1 0 Retaining terms of O(1) only results in the celebrated boundary-layer equations u + v + ν x y x y p 0 y u v + 0 x y u u p u parabolic
57:000 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 009 Chapter 8 9 Some important aspects of the boundary-layer equations: 1) the y-momentum equation reduces to p 0 y i.e., p p e constant across the boundary layer from the Bernoulli equation: edge value, i.e., 1 inviscid flow value! p e + ρ Ue constant p i.e., e Ue ρue x x Thus, the boundary-layerr equations are solved subject to a specified inviscid pressure distribution ) continuity equation is unaffected 3) Although NS equations are fully elliptic, the boundary-layer equations are parabolic and can be solved using marching techniques 4) Boundary conditions u v 0 y 0 u U U e y + appropriate initial conditions @ x x i
57:000 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 009 Chapter 9 There are quite a few analytic solutions to the boundary- layer equations. Also numerical techniques are available for arbitrary geometries, including both two- and three- dimensional flows. Here, as an example, we consider the simple, but extremely important case of the boundary layer development over a flat plate. 9 Quantitative Relations Layer for the Laminar Boundary Laminar boundary-layer student of Prandtl over a flat (1908) plate: Blasius solution p Note: 0 x for a flat plate u v + 0 x y u u u u + v ν x y y u v 0 @ y 0 u U @ y We now introduce a dimensionless and a stream function, i.e., transverse coordinate η y U ν x y ψ νxu f ( η)
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 10 ψ ψ η y η y U f ( η) U u f u / v ψ x 1 νu x ( ηf f ) Substitution into the boundary-layer equations yields f f + f 0 Blasius Equation f f 0 @ η 0 f 1 @ η 1 The Blasius equation is a 3 rd order ODE which can be solved by standard methods (Runge-Kutta). Also, series solutions are possible. Interestingly, although simple in appearance no analytic solution has yet been found. Finally, it should be recognized that the Blasius solution is a similarity solution, i.e., the non-dimensional velocity profile f vs. η is independent of x. That is, by suitably scaling all the velocity profiles have neatly collapsed onto a single curve. Now, lets consider the characteristics of the Blasius solution: u U vs. y v U U V vs. y
57:000 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 009 5x Re x Re x Uν x Chapter 11 value of y where u/u.99 9
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 1 τ w μu f (0) νx / U i.e., c f τ ρu w 0.664 Re x θ x see below 1 C L f cf dx cf (L) L 0 1.38 Re L U L ν Wall shear stress: τ w ρμ x 3 0.33U or τw μ( U x) 0.33 Re x Other: * u x 1 dy 1.708 displacement thickness 0 U Rex measure of displacement of inviscid flow due to boundary layer u u x θ 1 dy 0.664 momentum thickness 0 U U Rex measure of loss of momentum due to boundary layer H shape parameter θ *.5916
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 13
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 14 Quantitative Relations for the Turbulent Boundary Layer -D Boundary-layer Form of RANS equations u v + 0 x y u u p + v x y x ρ ( u v ) u e Momentum Integral Analysis u + ν y y requires modeling Historically similarity and AFD methods used for idealized flows and momentum integral methods for practical applications, including pressure gradients. Modern approach: CFD. To obtain general momentum integral relation which is valid for both laminar and turbulent flow For flat plate or for general case ( momentum equation + (u v)continuity)dy y 0 τ w ρu 1 c f dθ dx + ( + H) du θ U du dx dp dx ρu du dx flat plate equation 0 dx
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 15 θ 0 u U 1 u U dy momentum thickness * H shape parameter θ * 1 0 u U dy displacement thickness Can also be derived by CV analysis as shown next for flat plate boundary layer. Momentum Equation Applied to the Boundary Layer Consider flow of a viscous fluid at high Re past a flat plate, i.e., flat palte fixed in a uniform stream of velocityui ˆ. Boundary-layer thickness arbitrarily defined by y 99% (where, 99% is the value of y at u 0.99U). Streamlines outside 99% will * deflect an amount (the displacement thickness). Thus the streamlines move outward from y H at x 0 to * y Y H + at x x1.
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 16 Conservation of mass: ρ V nda0 0 0 CS H ρudy + H+ ρudy Assume incompressible flow (constant density): UH Substituting ( U + u U ) dy UY + ( u U ) udy Y Y Y dy 0 0 0 Y * H + defines displacement thickness: * Y 0 u 1 dy U * is an important measure of effect of BL on external flow. Consider alternate derivation based on equivalent flow rate: * Lam/3 * Turb/8 Udy * 0 udy * and of inviscid flowactual flowrate, i.e., inviscid flow rate Flowrate between about displacement body viscous flow rate about actual body 0 Udy Inviscid flow about * body 0 * Udy 0 * udy 0 1 u U dy w/o BL - displacement effectactual discharge For 3D flow, in addition it must also be explicitly required that is a stream surface of the inviscid flow continued from outside of the BL. *
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 17 Conservation of x-momentum: H x + 0 0 CS Y ( ) ( ) F D ρuv nda ρu Udy ρu udy Y Drag D ρ U H ρu dy 0 Fluid force on plate - Plate force on CV (fluid) Again assuming constant density and using continuity: Y u H dy 0 U Y Y x / 0 0 0 D ρu u Udy ρ u dy τ dx D ρu Y u θ 1 0 U u U dy where, θ is the momentum thickness (a function of x only), an important measure of the drag. D θ 1 x CD c fdx ρu x x x 0 τ w d dθ cf cf ( xcd) 1 ρu dx dx dθ c f dθ τ w ρu dx dx w Per unit span Special case D momentum integral equation for p x 0
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 18 Simple velocity profile approximations: u U (y / y / ) u(0) 0 u() U u y ()0 no slip matching with outer flow Use velocity profile to get C f () and θ() and then integrate momentum integral equation to get (Re x ) * /3 θ /15 H */θ 5/ τ μu / w μu / dθ d c f ( /15) 1/ρU dx dx 15μdx d ρu 30μdx ρu / x 5.5/Re Re Ux / ν; x 1/ x / x 1.83/Re * 1/ x θ / x 0.73/Re 1/ x C C L 1/ D 1.46 / ReL f( ) 10% error, cf. Blasius
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 19 Approximate solution Turbulent Boundary-Layer Re t 5 10 5 3 10 6 for a flat plate boundary layer Re crit 100,000 c f dθ dx as was done for the approximate laminar flat plate boundary-layer analysis, solve by expressing c f c f () and θ θ() and integrate, i.e. assume log-law valid across entire turbulent boundary-layer u u * + * 1 yu ln κ ν B neglect laminar sub layer and velocity defect region at y, u U * U 1 u ln + B * u κ ν 1/ cf Re 1/ 1/ cf or.44ln Re + 5 cf c f () c f 1/ 6.0Re power-law fit
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 0 Next, evaluate dθ dx d dx 0 u U 1 u U dy can use log-law or more simply a power law fit 1/ 7 u y U 7 θ 7 θ( ) τ or w 1 dθ cf ρu ρu dx 1/ 6 d Re 9.7 dx 1/7 0.16Re x x 6/ 7 x almost linear 0.07 c f Re 1/7 x 0.031 7 C c L f 1/7 ReL 6 f ( ) 7 7 ρu d dx If the boundary layer is tripped by some roughness or leading edge disturbance, the boundary layer is turbulent from the beginning: x shear stress: 0.058 0.074 7 c C f 1/5 ( Re Re x 10 ) L 0.09ρU τ w 1/5 Re 1/5 0.37 Re x f 1/5 Rex x Note: can not be used to obtain c f () since τ w i.e., much faster growth rate than laminar boundary layer
57:000 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 009 Chapter 1 9 Alternate forms depending on experimental information and power-law fit used, etc. (i.e., dependent on Re range.) Some additional relations for largerr Re are.455 1700 C Total f ( log10 Re ).58 shear-stress L Re L coefficient L c f (.98log ReL. 73) as follows: Re > 10 7 Local shear-stress coefficient c f ( log Rex. 65). 3 ) Finally, a composi te formula that takes into account both the initial laminar boundary-layer (with translation at Re CR 500,000) and subsequent turbulent boundary layer.074 1700 is C f 1/51 10 5 < Re < 10 7 Re Re L L
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 Bluff Body Drag Drag of -D Bodies First consider a flat plate both parallel and normal to the flow C 1 1 ρv S A ( p p ) n î 0 Dp C f 1 1 ρv τ S A w t îda 1.33 1/ Re L.074 1/ 5 Re L laminar flow turbulent flow flow pattern where C p based on experimental data vortex wake typical of bluff body flow
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 3 1 C Dp ( p p ) n îda 1 S ρv A 1 C p A da S using numerical integration of experimental data C f 0 For bluff body flow experimental data used for C D.
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 4 In general, Drag f(v, L, ρ, μ, c, t, ε, T, etc.) from dimensional analysis c/l C D Drag t ε f Re, Ar,,, T, etc. 1 ρ L L V A scale factor
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 5 Potential Flow Solution: p + 1 ρv p + 1 a ψ U r sin θ r ρu u r 1 ψ r θ u r + 1 U θ p p u Cp ψ 1 uθ ρu r C r a 1 4 sin surface pressure p ( ) θ Flow Separation Flow separation: The fluid stream detaches itself from the surface of the body at sufficiently high velocities. Only appeared in viscous flow!! Flow separation forms the region called separated region
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 6 Inside the separation region: low-pressure, existence of recirculating/backflows viscous and rotational effects are the most significant! Important physics related to flow separation: Stall for airplane (Recall the movie you saw at CFD-PreLab!) Vortex shedding (Recall your work at CFD-Lab, AOA16! What did you see in your velocity-vector plot at the trailing edge of the air foil?)
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57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 35 Terminal Velocity Terminal velocity is the maximum velocity attained by a falling body when the drag reaches a magnitude such that the sum of all external forces on the body is zero. Consider a sphere using Newton Second law: Z F F + F F ma d b g when terminal velocity is attained F a 0 : or Fd + Fb Fg 1 ρv0 CD Ap ( γ Sphere γ fluid ) V Sphere For the sphere Ap π 4 d and V Sphere π d 6 3 The terminal velocity is: ( γ sphere γ fluid )( 43) V 0 d CDρ fluid 1 Magnus effect: Lift generation by spinning Breaking the symmetry causes the lift!
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 36 Effect of the rate of rotation on the lift and drag coefficients of a smooth sphere: Lift acting on the airfoil Lift force: the component of the net force (viscous+pressure) that is perpendicular to the flow direction
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 37 Variation of the lift-to-drag ratio with angle of attack: The minimum flight velocity: Total weight W of the aircraft be equal to the lift W F L 1 C L,max ρ V A V min min W ρc L,max A
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 38 Effect of Compressibility on Drag: CD CD(Re, Ma) Ma U a speed of sound rate at which infinitesimal disturbances are propagated from their source into undisturbed medium Ma < 1 Ma 1 Ma > 1 Ma >> 1 subsonic transonic (1 sonic flow) supersonic hypersonic < 0.3 flow is incompressible, i.e., ρ constant C D increases for Ma 1 due to shock waves and wave drag Ma critical (sphere).6 Ma critical (slender bodies) 1 For U > a: upstream flow is not warned of approaching disturbance which results in the formation of shock waves across which flow properties and streamlines change discontinuously
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