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Momentum Transfer Coefficients in Flow over a Solid Wall Surface

This lecture discusses the momentum exchange between a moving fluid and a solid surface, as well as the determination of momentum transfer coefficients. It also explores the angular dependence of skin friction and pressure coefficients and provides experimental values for overall drag coefficients in different flow scenarios.

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Momentum Transfer Coefficients in Flow over a Solid Wall Surface

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  1. Advanced Transport Phenomena Module 4 - Lecture 14 Momentum Transport: Flow over a Solid Wall Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

  2. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Applications: • Design of automobiles • Design of aircraft, etc. • Property of interest: • Momentum exchange between surface & surrounding fluid

  3. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Associated net force • “drag” in streamwise direction • ‘lift” in direction perpendicular to motion • Obtained by solving relevant conservation equations, subject to relevant boundary conditions, or • By experiments on full-scale or small-scale models

  4. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Momentum exchange between the moving fluid and a representative segment of a solid surface (confining wall or immersed body)

  5. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • X  approach stream direction • x  distance along surface • n  distance normal to surface • p (x,0)  local pressure • vx (x,n)  velocity field • tnx (x,0) = tw(x)  local wall shear stress • Associated momentum exchange: • Force on fluid • Equal & opposite force on solid

  6. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Solid surface motionless => vn (x,0) = 0 • vx (x,0) ≠ 0 => nonzero “slip” velocity • However, experimentally: local tangential velocity of fluid = that of solid, i.e., 0, under continuum conditions • Wall shear stress depends on local fluid-deformation rate: • Can be determined if local normal gradient of tangential fluid velocity can be measured

  7. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Dimensionless local momentum transfer coefficients: • Pressure coefficient: and • Skin-friction coefficient • Measured or predicted

  8. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Alternative definition of skin-friction coefficient: • In terms of properties at the edge of momentum transfer boundary layer • For an incompressible fluid ( ), in the absence of gravitational body-force effects, Bernoulli’s equation yields: • Reflects negligibility of viscous dissipation far from surface

  9. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS This equation implies that: and hence:

  10. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Experimentally determined angular dependence of the skin-friction- and pressure- coefficients around a circular cylinder in a cross-flow at Re= 1.7x105

  11. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Total (net) drag force D’ per unit length of cylinder: • Reference Force: where projected area of cylinder per unit length and • Calculated from Cp, Cf data

  12. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • By projecting pressure & shear forces in direction of approach flow: and (polar angle expressed in radians)

  13. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • cos q term  from (locally normal) pressure force (“form” drag) • sin q term  from (locally tangential) aerodynamic shear force (“friction” drag) • Thus, drag coefficient may be split into:

  14. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Experimental values for the overall drag coefficient (dimensionless total drag) for a cylinder (in cross-flow), over the Reynolds’ number ranger

  15. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Experimental values for the overall drag coefficient (dimensionless total drag) for a sphere over the Reynolds’ number range

  16. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Asymptotic theories: Re >> 1, Re << 1 • Re >> 1 case is of greatest engineering interest • e.g., flow past flat plate at zero incidence

  17. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Momentum Diffusion Boundary Layer Theory: Laminar Flow Past Flat Plate at Zero Incidence • 1904: L Prandtl • large but finite Reynolds number • vn and vx vanish at solid surface • Thin transition layer near surface across which vx abruptly drops to zero

  18. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Momentum Diffusion Boundary Layer Theory: Laminar Flow Past Flat Plate at Zero Incidence • Inside this “boundary layer”, velocity gradients large enough to make momentum diffusion important (though m is small) • Exterior: inviscid region

  19. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Division of flow field at Re1/2 >>1 into an inviscid “outer” region and a thin tangential momentum diffusion boundary layer (BL)(after L. Prandtl).

  20. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Momentum Diffusion Boundary Layer Theory: Laminar Flow Past Flat Plate at Zero Incidence • 1904: L Prandtl • For Re >> 1, within the BL: • vn << vx • Momentum diffusion important, but only in normal direction (tnx>> txx) • Pressure at any streamwise location x is nearly constant– i.e., p ≈ pe(x)

  21. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Momentum Diffusion Boundary Layer Theory: Laminar Flow Past Flat Plate at Zero Incidence • BL equations therefore simplified, solutions to match inner behavior of external inviscid flow • e.g., 2D steady flow of incompressible constant-property Newtonian fluid past a semi-infinite flat plate at zero incidence (Blasius, 1908)

  22. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Newtonian incompressible fluid flow past a flat plate; configuration, nome- nclature, and coordinate system

  23. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Laminar BL on a flat plate: • For thin flat plate, pressure constant everywhere => no need for y-momentum equation • 2 scalar PDE’s governing vx ≡ u(x,y), vy ≡ v(x,y)

  24. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Subject to boundary conditions: • Solved by Blasius using “combination of variables”

  25. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Laminar BL on a flat plate: • Blasius’ solution: and

  26. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Blasius derived & numerically solved nonlinear ODE governing and constructed tangential fluid-velocity profiles

  27. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Laminar BL on a flat plate:

  28. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Laminar BL on a flat plate: • = local BL thickness = y-location at which u/U = 0.99  occurs at • Therefore: (grows as square root of distance x from LE of plate)

  29. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Laminar BL on a flat plate: • Wall shear stress: • Local dimensionless skin-friction coefficient cf given by: • Total friction drag coefficient: (for plate of finite length L, set x = L)

  30. FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS • Laminar BL on a flat plate: • Effect of “blowing” or “suction” through porous solid wall: • cf values are modified • Blowing can reduce skin-friction drag where (cf)0 no-blowing momentum-transfer coefficient, and F(blowing) function of dimensionless variable ( = local mass injection rate)

  31. CONSERVATION EQUATION GOVERNING VELOCITY AND PRESURE FIELDS • Navier-Stokes (linear momentum conservation) law: • Nonlinear vector PDE • Equivalent to 3 independent, scalar 2nd order PDEs • Includes “Stokes’ Postulate”: bulk viscosity can be neglected • Total mass conservation (“continuity”):

  32. CONSERVATION EQUATION GOVERNING VELOCITY AND PRESURE FIELDS • Conservation equations provide 4 PDEs for 5 fields: v (3 scalar fields), p, • Hence, necessary to specify an EOS for closure • Unless is constant (incompressible flow; ) • “Caloric” EOS: h as a function of T, p • In addition to usual as a function of local state variables • Turbulent flows: • Conservation equations are time-averaged • replaced by

  33. TYPICAL BOUNDARY CONDITOINS By applying a “pillbox” control volume to straddle a moving interface, we can write: Gn normal component of mass flux t  tangential plane Mass balance: Momentum Balance: Tangential linear momentum:

  34. TYPICAL BOUNDARY CONDITOINS • These conservation equations allow: • Discontinuity in normal component of velocity, • Discontinuity in pressure across interface, • Discontinuity in tangential velocity (“slip”) across interface • Thus, the “classical” boundary conditions: are only sometimes true.

  35. TYPICAL INITIAL CONDITOINS • State of independent field variables at t = 0 • Start-up of a chemical reactor, separator, etc. • The present, if we want to predict future (e.g., weather, climate) • Governing conservation equations are first-order in time • Invariant wrt shift in origin (zero point) chosen for time • Principal of “local” action in time (determinacy) • Future cannot influence present! • Only applies in time-domain, not space

  36. SOLUTION METHODS • Coupled PDEs + bc’s + ic’s need not always be solved to extract valuable information • e.g., similitude analysis • Only relatively simple fluid-dynamic problems need to be solved to interpret instrument readings • e.g., flowmeters • Mathematical solutions have become possible with advent of powerful digital computers • Computational fluid mechanics, CFD • Discretizing by finite-difference, finite-element methods

  37. SOLUTION METHODS • Modularization: • In sub-regions, explicit results may be possible in terms of well-known special functions • e.g., Bessel functions, Legendre polynomials • Numerical: • Reduce problem to solution of one lor more nonlinear ODEs • Then solve numerically

  38. SOLUTION METHODS “Road map” of common methods of solution to problems in transport (convection /diffusion ) theory

  39. SOLUTION METHODS • Results should be independent of method chosen • But effort should be minimized! • Idealizations of complex problems serve a purpose • Capture concepts • Bring out qualitative features • Sanity check on more complex predictions

  40. Thank You

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