1. Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 4 - Lecture 15 Momentum Transport: Steady Laminar Flow

2. PDEs governing steady velocity & pressure fields: (Navier-Stokes) and (Mass Conservation) “No-slip” condition at stationary solid boundaries: at fixed solid boundaries STEADY LAMINAR FLOW OF INCOMPRESSIBLE NEWTONIAN FLUID

3.  Special cases:  Fully-developed steady axial flow in a straight duct of constant, circular cross-section (Poiseuille)  2D steady flow at high Re-number past a thin flat plate aligned with stream (Prandtl, Blasius) STEADY LAMINAR FLOW OF INCOMPRESSIBLE NEWTONIAN FLUID

4. FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION Cylindrical polar-coordinate system for the analysis of viscous flow in a straight circular duct of constant cross section

5.  Wall coordinate: r = constant = a w (duct radius)  Fully developed => sufficiently far downstream of duct inlet that fluid velocity field is no longer a function of axial coordinate z  From symmetry, absence of swirl: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

6.  Conservation of mass ( = constant): v z independent of z implies:  PDEs required to find v z ( r), p(r,z)  Provided by radial & axial components of linear- momentum conservation (N-S) equations: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

7.  Pressure is a function of z alone, and if  p = p(z) and v z = v z ( r), then:  i.e., a function of z alone (LHS) equals a function of r alone (RHS)  Possible only if LHS & RHS equal the same constant, say C 1 FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

8. Hence: New pressure variable P defined such that: and, hence P varies linearly with z a s: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

9.  Integrating the 2 nd order ODE for v z twice:  Since v z is finite when r = 0, C 3 = 0  Since v z = 0 when  Hence, shape of velocity profile is parabolic: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

10. Since is a negative constant- i.e., non- hydrostatic pressure drops linearly along duct: and FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

11.  Total Flow Rate:  Sum of all contributions through annular rings each of area Substituting for v z (r) & integrating yields: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

12. (Hagen – Poiseuille Law– relates axial pressure drop to mass flow rate)  Basis for “capillary-tube flowmeter” for fluids of known Newtonian viscosity  Conversely, to experimentally determine fluid viscosity FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

13.  Total Flow Rate:  Average velocity, U, is defined by: Then: i.e., maximum (centerline) velocity is twice the average value, hence: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

14. Wall friction coefficient (non-dimensional):  w  wall shear stress C f  dimensionless coef (also called f  Fanning friction factor) Direct method of calculation: and FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

15. Wall friction coefficient (non-dimensional): Hence: equivalent to: Holds for all Newtonian fluids Flows stable only up to Re ≈ 2100 FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

16. Wall friction coefficient (non-dimensional): Experimental and theoretical friction coefficients for incompressible Newtonian fluid flow in straight smooth-walled circular duct of constant cross section FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

17.  Wall friction coefficient (non-dimensional):  Same result can be obtained from overall linear- momentum balance on macroscopic control volume A  z:  Axial force balance (for fully-developed flow where axial velocity is constant with z):  Solving for  w and introducing definition of P : FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

18. Wall friction coefficient (non-dimensional): Configuration and notation: steady flow of an incompressible Newtonian fluid In a straight circular duct of constant cross section FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

19.  Wall friction coefficient (non-dimensional):  Above Re = 2100, experimentally-measured friction coefficients much higher than laminar-flow predictions  Order of magnitude for Re > 20000  Due to transition to turbulence within duct  Causes Newtonian fluid to behave as if non- Newtonian  Augments transport of axial momentum to duct wall FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

20.  In fully-developed turbulent regime (Blasius):  C f varies as Re -1/4 for duct with smooth walls  C f sensitive to roughness of inner wall, nearly independent of Re FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

21.  Wall friction coefficient (non-dimensional):  Effective eddy momentum diffusivity  Can be estimated from time-averaged velocity profile & C f measurements  Hence, heat & mass transfer coefficients may be estimated (by analogy)  For fully-turbulent flow, perimeter-average skin friction & pressure drop can be estimated even for non- circular ducts by defining an “effective diameter”: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

22. where P  wetted perimeter  Not a valid approximation for laminar duct flow FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

23.  Circular jet discharging into a quiescent fluid  Sufficiently far from jet orifice, a fully-turbulent round jet has all properties of a laminar round jet, but, intrinsic kinematic viscosity of fluid   jet axial-momentum flow rate  Constant across any plane perpendicular to jet axis STEADY TURBULENT FLOWS: JETS

24.  Laminar round jet of incompressible Newtonian fluid: Far- Field  Schlichting BL approximation  PDE’s governing mass & axial momentum conservation in r, , z coordinates admit exact solutions by method of “combination of variables”, i.e., dependent variables are uniquely determined by the single independent variable: STEADY TURBULENT FLOWS: DISCHARGING JETS

25. Streamline pattern and axial velocity profiles in the far-field of a laminar (Newtonian) or fully turbulent unconfined rounded jet (adapted from Schlichting (1968)) STEADY TURBULENT FLOWS: DISCHARGING JETS

26. Total mass-flow rate past any station z far from jet mouth yielding i.e., mass flow in the jet increases with downstream distance  By entraining ambient fluid while being decelerated (by radial diffusion of initial axial momentum) STEADY TURBULENT FLOWS: DISCHARGING JETS

27.  Near-field behavior:  z/d j ≤ 10  Detailed nozzle shape important  “potential core”: within, jet profiles unaltered by peripheral & downstream momentum diffusion processes  Swirling jets:  Tangential swirl affects momentum diffusion & entrainment rates  Predicting flow structure huge challenge for any turbulence model TURBULENT JET MIXING

28.  Additional parameters:  Initially non-uniform density, viscous dissipation, chemical heat release, presence of a dispersed phase, etc.  Add complexity; far-field behavior can be simplified TURBULENT JET MIXING