Momentum Transfer Coefficients in Flow over a Solid Wall Surface

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

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