ME231

Fluid Mechanics

Credits:


3L0T1P0A (10 Credits)



Course Content:
Introduction; Reynolds Transport Theorem; Integral form of continuity, momentum and energy equations; Eulerian and Lagrangian viewpoints; Constitutive relations; Navier Stokes equations; Exact solutions; Potential flow; Boundary layer theory; Separation and drag; Turbulent flow: Reynolds averaged equations; Turbulent flows in pipes and channels; compressible flows.
Lecturewise Breakup:
I. Introduction: (2 Lectures)

Fluid statics; Definition of continuum, statement of physical laws (mass, momentum, energy and second law of thermodynamics), Reynolds transport theorem. Distinction between a systems approach and a control volume approach.
II. Simplified Global Analysis: (6 Lectures)
III. Dimensional Analysis: (1 Lecture)
IV. Detailed Analysis: (4 Lectures)

Discussion on local versus global approach to solving flow problems; Derivation of conservation of mass equation using RTT, coordinatefree form, curvilinear coordinates, incompressibility Newton's second law of motion via RTT, body and surface forces, Eulerian and Lagrangian form of acceleration; Meaning of the material derivative; flow kinematics: streamlines. Calculation of material derivatives through examples Expression of surface forces in terms of the stress tensor; Properties of the stress tensor; Discussion on the most general form of the constitutive relation for a linear homogeneous isotropic material Construction of the strainrate tensor, stressstrain rate relationship, thermodynamic and mechanical pressures, Stokes hypothesis.
V. NavierStokes Equations: (3 Lectures)

Special form of NS equations for constant property fluids, two dimensional Cartesian coordinates, steady flow Mathematical properties of NS equations, Discussion on nonuniqueness of the solution of NS equations; Calculation of volume flow rate, forces and moments from the local solution Boundary conditions for velocity and pressure, stream function and vorticity; vorticity transport equation; Surface tension and continuity condition for the traction vector at material interfaces.
VI. Exact Solutions: (6 Lectures)

Creeping flow and fully developed flow approximations. Flow between parallel plates, friction factor relationship Flow between concentric rotating cylinders, TaylorCouette flow, application to viscometry Stability considerations; Flow in a tube of square crosssection General discussion on separationofvariables for solving PDEs; Application to square tubes and partiallyfilled tubes; Friction factor relations; Hydraulic diameter approach Unsteady flow in a circular tube: effect of transients; (Introduction to Bessel functions.) Discussion on the realizability of the predicted flow field Stokes 1st and 2nd problems; extension to a general transient problem via Duhamel's theorem Stokes flow past a sphere; Analytical solution, streamline patterns, nature of pressure gradient, expression for drag Data for drag and lift coefficients for spheres and cylinders as a function of Reynolds number. Determination of the trajectory of particles moving in a fluid medium Theory of Lubrication: Reynolds equation, journalbearing problem, load bearing capacity.
VII. Potential Theory: (7 Lectures)

Inviscid, incompressible irrotational flow, utility and applications, Kelvin's theorem, governing equations; Bernoulli's equation (steady and unsteady) Method of potentials: stream function and velocity potential, flow kinematics in terms of streamlines and isopotential contours; CauchyRiemann conditions, complex potential, complex velocity, solution by complex potentials using the method of superposition; boundary conditions Elementary complex potentials for uniform flow, sources, sinks vortices; flow in a sector. Superposition of source and uniform flow, doublets, superposition of doublet and uniform flow Flow past a circular cylinder. Flow past a cylinder with circulation, calculation of forces, Blasius integral laws, KuttaJoukowski theorem Development of complex potentials by conformal transformation, flow past an ellipseshaped object; flow past a vertically oriented flat plate. Flow at sharp corners Thin aerofoil theory: complex potential, flow patterns, Kutta condition, development of lift, angle of stall. Experimental results for aerofoils and comparison with theory in terms of pressure distribution and lift coefficients for circular cylinders and aerofoils.
VIII. BoundaryLayer Theory: (6 Lectures)

Prandtl's wind tunnel experiments, BL approximation, notion of an impressed pressure field; Separation explained in terms of boundarylayers. CD versus Re curve for a circular cylinder; Vortex shedding, Wake structure; Qualitative description of KelvinHelmholtz instability Boundarylayer growth in favorable and adverse pressure gradients; Dissipation capacity of a boundarylayer; Application to the design of wind tunnels; BL control by suction and blowing Derivation of BL equations and boundary conditions; Momentumintegral (MI) approach; displacement and momentum thicknesses MI approach for a flat plate, nonzero pressure gradient flows, Importance of a fourth order polynomial for inflexion point profiles.
IX. Turbulent BoundaryLayers : (3 Lectures)

Notion of instability and transition, fully developed boundarylayers, effect on viscous drag, heat transfer and point of separation; BL control; Reynolds decomposition, timeaveraged NS equations, closure, Reynolds stresses, crosscorrelation and its physical significance, Boussinesq approximation, BL equations for a turbulent flat plate BL Prandtl's mixing length theory, twolayer model, derivation of the loglaw; 1/7th power law approximation. Calculation of wall shear stress from BL measurements Numerical examples, Explanation of Moody's chart in terms of the loglaw, effect of wall roughness, effect of pressure gradient.
X. Compressible Flow : (4 Lectures)

High speed gas flow, special features, speed of sound, onedimensional form of the governing equations; Isentropic gas relations; velocity measurement using a pitot tube at all Mach numbers Flow through nozzles, areavelocity relations, numerical examples; Nonideal flow in nozzles, formation of shocks; Application to supersonic velocity measurement by a pitot tube; Shockboundary layer interaction.
Laboratory Sessions:
I. Flow visualization around streamlined and bluff objects (including multimedia resources).
II. Measurement of viscosity of liquids and gases.
III. Force acting on a circular cylinder placed in crossflow.
IV. Probes and transducers – pitot static tube, 5hole probe, manometers, hotwire anemometer, Wind tunnels (This material is to be covered in a one hour lecture.).
V. Measurement of velocity and velocity fluctuations in a turbulent mixing layer.
VI. Boundarylayer flow over a flat plate (laminar and turbulent).
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