ASEN - 5051 FUNDAMENTALS OF FLUID MECHANICS

Prof. J.M. Forbes

This course is a rigorous introduction to the fundamentals of fluid mechanics. It provides a solid foundation for students intending to study fluids at the advanced level, but is sufficiently broad that it serves as a valuable survey for many other students. No prior knowledge of fluids is assumed, but prior exposure to ordinary and partial differential equations is required.

1. CARTESIAN TENSORS

Second order tensors; matrix equivalents; Kronecker delta, alternating tensor, epsilon-delta relation; gradient, divergence, curl; relationships containing symmetric and antisymmetric tensors; eigenvalues and eigenvectors; principal axes; Gauss and Stokes theorems.

2. KINEMATICS OF FLUID FLOWS

Eulerian vs. Lagrangian descriptions; material derivative; strain and deformation; strain rate tensor; vorticity; circulation; solid body rotation; irrotational vortex; Rankine vortex/tornadoes; stream function and velocity potential.

3. CONSERVATION LAWS

Mass/continuity and momentum; Leibnitz's theorem and Reynolds Transport Theorem; constitutive relations and assumptions leading to Newtonian fluid; Navier Stokes equation; Euler's equation; Equations in a rotating frame; conservation of mechanical energy; Thermal energy equation; Bernoulli equation.

4. VORTICITY DYNAMICS

Vortexlines, vortex tubes, streamtubes, solenoidal property; Kelvin's circulation theorem; vorticity; irrotational flow; vortex lines and vortex sheets; viscosity and rotational and irrotational vortices; the vorticity equation; tilting and stretching; vorticity conservation concepts in boundary layers and separation.

5. THEORY AND APPLICATION OF IRROTATIONAL FLOWS

Complex potential; complex potentials for various special-case flows; flow past a half-body; determination of stream function, stagnation points, pressure distribution; flow past a circular cylinder; flow around a rotating cylinder; Kutta-Zhukhovski lift theorem; Blasius theorem; uniqueness and simply-connected domains; application of conformal mapping to low around cylinders and plates; Zhukovski transformation and airfoil; Kutta condition.

6. TOPICS IN GEOPHYSICAL FLUID DYNAMICS

Boussinesq approximation; Brunt-Vaisala frequency; hydrostatic approximation; geostrophy; Taylor-Proudman Theorem; shallow-water equations; conservation of potential vorticity; quasi-geostrophy; Rossby waves; Ekman layer; Ekman transport and Ekman pumping; normal modes; internal waves; Kelvin waves; barotropic and baroclinic instability.

7. DYNAMIC SIMILARITY AND NONDIMENSIONAL PARAMETERS

Buckingham's Pi Theorem; applications to pipe flow, drag on a sphere in uniform flow; Reynolds, Froude, and other nondimensional flow parameters.

8. VISCOUS (LAMINAR) FLOWS

Plane Couette and Poiseuille flows; circular Poiseuille flow; impulsively-started plate (Stokes' first problem); similarity solutions; high and low Reynolds number flows.

9. BOUNDARY LAYERS AND TURBULENCE

Boundary layer equations; various definitions of boundary layer thickness; boundary layer on a flat plate (Blasius solution); turbulent boundary layers; separation; introduction to concepts of turbulence; Reynolds stresses, turbulent diffusion.

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