57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 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. Re 1: low Re flow (creeping or Stokes flow) Re > 1,000: Laminar BL Re > 5 10 5 (Re crit ): Turbulent BL b. Bluff body flow: flow around bluff bodies with flow separation.
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 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 skin-friction contributions: C D 1 1 ρv S A ( p p ) n îda + τ S w C Dp C f τ îda
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 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
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 4 Qualitative Description of the Boundary Layer Flow-field regions for high Re flow about slender bodies:
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 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 014 6 The assumptions of the theory are as follows: Variable order of magnitude u U O(1) v δ<<l O(ε) ε δ/l x 1/L O(1) y 1/δ O(ε -1 ) ν δ 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 δ 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 014 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 u p u u + v + ν x y x y p 0 y u v + 0 x y parabolic
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 8 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! pe + ρu e constant pe Ue i.e., ρue x x Thus, the boundary-layer 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 e y δ + appropriate initial conditions @ x i
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 9 There are quite a few analytic solutions to the boundarylayer equations. Also numerical techniques are available for arbitrary geometries, including both two- and threedimensional flows. Here, as an example, we consider the simple, but extremely important case of the boundary layer development over a flat plate. Quantitative Relations for the Laminar Boundary Layer Laminar boundary-layer over a flat plate: Blasius solution (1908) student of Prandtl 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 δ We now introduce a dimensionless transverse coordinate and a stream function, i.e., U y η y νx δ ψ νxu f ( η)
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 10 ψ ψ η ψ η ψ U f ( η) f u / U 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 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 vs. y
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 11 5x δ Re x U x Re x ν value of y where u/u.99
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 1 i.e., τ c w f τ ρu µ U f (0) νx / U w 0.664 Re x θ x see below : Local friction coeff. C Note: bb plate width LL plate length f b bl L c f dx c f ( L) : Friction drag coeff. 0 1.38 Re L U L ν Wall shear stress: τ w ρµ x 3 0.33U or τ w µ ( U x) 0.33 Re x Other: δ δ * u x 1 δy 1.708 displacement thickness 0 U Rex measure of displacement of inviscid flow due to boundary layer θ δ u u x 1 δy 0.664 momentum thickness 0 U U Rex measure of loss of momentum due to boundary layer H shape parameter θ δ *.5916
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57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 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 δ 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 014 15 θ δ 0 u U 1 u U δy momentum thickness * δ H shape parameter θ d * d 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 plate fixed in a uniform stream of velocityui ˆ. Boundary-layer thickness arbitrarily defined by y δ 99% (where, δ 99% is the value of y at u 0.99U). Streamlines outside δ 99% will * deflect an amountδ (the displacement thickness). Thus the streamlines move outward from y H at x 0 to * y Y δ H + δ at x x1.
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 16 Conservation of mass: ρ V nda0 0 0 CS H ρudy + H+ d 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 + δ d defines displacement thickness: * Y 0 u 1 dy U * δ is an important measure of effect of BL on external flow. Consider alternate derivation based on equivalent flow rate: δ δ* Lamδ/3 δ* Turbδ/8 δ δ Uδy * δ 0 uδy * δ and δ of inviscid flowactual flowrate, i.e., inviscid flow rate Flowrate between about displacement body viscous flow rate about actual body * δ δ δ δ * 0 Uδy Inviscid flow about δ* body 0 Uδy 0 uδy δ 0 1 u U δy 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. * δ
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 17 Conservation of x-momentum: H x + 0 0 CS Y ( ) ( ) F D ρuv nda ρu Udy ρu udy Y Drag D r U H ru 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 D ρu u / Udy ρ u dy τ dx D ρu 0 0 0 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 c f c f ( xcd) 1 ρu dx dx dθ c f dθ τ w ρu dx dx x w Per unit span Special case D momentum integral equation for dp/dx 0
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 18 Simple velocity profile approximations: u U (y / δ y δ / ) u(0) 0 u(δ) U u y (δ)0 no slip matching with outer flow Use velocity profile to get C f (δ) and θ(δ) and then integrate momentum integral equation to get δ(re x ) δ* δ/3 θ δ/15 H δ*/θ 5/ τ µ U / d w µ U / d dθ d c f ( d /15) 1/ρU dx dx 15µ dx d d ρu d 30µ dx ρu d / x 5.5 / Re Re Ux / ν ; x 1/ x d / 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 014 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 (δ) and θ θ(δ) and integrate, i.e. assume log-law valid across entire turbulent boundary-layer u u * + * 1 yu ln κ n B neglect laminar sub layer and velocity defect region at y δ, u U * + * U 1 δu ln B u κ n 1/ cf Re δ 1/ 1/ cf or.44ln Re + 5 cf δ c f (δ) c f 1/ 6 δ.0re power-law fit
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 0 Next, evaluate δθ δx δ δx δ 0 u U 1 u U δy can use log-law or more simply a power law fit 1/ 7 u y U δ 7 θ 7 δ θ( δ) τ or w 1 dθ cf ρu ρu dx 1/ 6 dd Red 9.7 dx δ 1/7 0.16 Re x x 6/ 7 δ x almost linear 0.07 c f 1/7 Rex 0.031 7 Cf c 1/7 f L Re 6 L Note: cannot be used to obtain c f (δ) since τ w ( ) 7 7 ρu dd dx i.e., much faster growth rate than laminar boundary layer These formulas are for a fully turbulent flow over a smooth flat plate from the leading edge; in general, give better results for sufficiently large Reynolds number Re L > 10 7.
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 1 1 uu UU yy δδ 7 (See Table 4-1 on page 13 of this lecture note) uu UU yy δδ yy δδ Comparison of dimensionless laminar and turbulent flat-plate velocity profiles (Ref: White, F. M., Fluid Mechanics, 7 th Ed., McGraw-Hill) Alternate forms by using the same velocity profile u/u (y/δ) 1/7 assumption but using an experimentally determined shear stress formula τ w 0.05ρU (ν/uδ) 1/4 are: δ x shear stress: 1/5 0.37 Re x f 1/5 Rex 0.09ρU τ w Re 0.058 0.074 c C f 1/5 Re 1/5 x L These formulas are valid only in the range of the experimental data, which covers Re L 5 10 5 10 7 for smooth flat plates. Other empirical formulas are by using the logarithmic velocity-profile instead of the 1/7-power law:
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 δδ LL cc ff(0.98 log RRRR LL 0.73) cc ff ( log RRRR xx 0.65).3 CC ff 0.455 (log 10 RRee LL ).58 These formulas are also called as the Prandtl-Schlichting skinfriction formula and valid in the whole range of Re L 10 9. For these experimental/empirical formulas, the boundary layer is usually tripped by some roughness or leading edge disturbance, to make the boundary layer turbulent from the leading edge. No definitive values for turbulent conditions since depend on empirical data and turbulence modeling. Finally, composite formulas that take into account both the initial laminar boundary layer and subsequent turbulent boundary layer, i.e. in the transition region (5 10 5 < Re L < 8 10 7 ) where the laminar drag at the leading edge is an appreciable fraction of the total drag: CC ff 0.031 1 7 RRee LL CC ff 0.074 1 5 RRee LL 1440 RRee LL 1700 RRee LL
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 3 CC ff 0.455 1700 (log 10 RRee LL ).58 RRee LL with transitions at Re t 5 10 5 for all cases. Local friction coefficient cc ff (top) and friction drag coefficient CC ff (bottom) for a flat plate parallel to the upstream flow.
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 4 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 τ 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 014 5 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 014 6 In general, Drag f(v, L, ρ, µ, c, t, ε, T, etc.) from dimensional analysis c/l C D Dρag t ε f Rε, Aρ,,, T, εtc. 1 ρ L L V A scale factor
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 7 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 ru 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 014 8 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 014 38 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 rvca 0 D p ( γsphere γ fluid ) V Sphere For the sphere Ap p 4 d and V Sphere p d 6 3 The terminal velocity is: ( γsphere γ fluid )( 43) V 0 d CDr 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 014 39 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 014 40 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 014 41 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 < 0.3 flow is incompressible, transonic (1 sonic flow) i.e., ρ constant supersonic hypersonic 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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