Understanding Steady Laminar Flow in Circular Ducts

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

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

3. STEADY LAMINAR FLOW OF INCOMPRESSIBLE NEWTONIAN FLUID • 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)

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. FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION • Wall coordinate: r = constant = aw (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:

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

7. FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION • Pressure is a function of z alone, and if • p = p(z) and vz= vz( 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 C1

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

9. FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION • Integrating the 2nd order ODE for vz twice: • Since vz is finite when r = 0, C3 = 0 • Since vz = 0 when • Hence, shape of velocity profile is parabolic:

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

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

12. FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION (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

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

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

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

16. FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION 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

17. FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION • 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 fortwand introducing definition of P:

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

19. FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION • 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

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

21. FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION • Wall friction coefficient (non-dimensional): • Effective eddy momentum diffusivity • Can be estimated from time-averaged velocity profile & Cf 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”:

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

23. STEADY TURBULENT FLOWS: JETS • 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

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

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

26. STEADY TURBULENT FLOWS: DISCHARGING JETS 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)

27. TURBULENT JET MIXING • Near-field behavior: • z/dj ≤ 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

28. TURBULENT JET MIXING • 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