SESM3004 Fluid Mechanics

Lecture 7: Unsteady Laminar FlowImpulsive Shearing Motion of a Plane WallLaminar Boundary LayersImpulsive shearing motion of a plane wall

At , the wall is impulsively set into motion.

Non-steady plane-parallel flow:

Governing equation:

i.c.: b.c.:

(diffusion equation) We seek solution in the form:

(1)

Finally (substituting the above partial derivatives into equation (1)),

b.c.:

First integration:

Second integration:

Applying the first boundary condition:

Applying the second boundary condition:

Final expression:

is typical thickness of the fluid layer involved into motion, it is called the thickness of the laminar boundary layererror function (a tabulated function)Velocity profile for water ( )

for t=1s (green curve),100s (red), and 10000s (brown).

Videos: http://www.youtube.com/watch?v=jx8j2OVZKjc http://www.youtube.com/watch?v=TADz6fqfXo0 http://www.youtube.com/watch?v=6WiplPYOv6M Friction forceFriction force exerted by the fluid and applied to the wall,

Laminar boundary layers

if

Conclusion:Re>>1 is typical for most engineering flows: where the flow may be separated into two regions: (i) thin viscous boundary layer and (ii) inviscid bulk flow.

Comment: for very high Re( , for a flow above a plane wall), the boundary layer becomes turbulent.

Lecture 8: Irrotational (or potential) flowIrrotational motion. Kelvin circulation theoremLaplace equationBernoullis equationIrrotational motion

-- definition of vorticityThe flow is irrotational if for every point.

The irrotational flow is always inviscid; i.e. Re>>1, but not turbulent and far from the boundaries where the viscous laminar layers are formed. Consequence of the Kelvins circulation theorem: if for a fluid particle, at initial moment, vorticity is zero, the vorticity remains zero for an entire trajectory of a particle (for every point on a streamline).

Typical example of an irrotational flows is depicted above. This is the flow around a stationary body immersed in a fluid (the flow must be uniform at the initial moment) orthe motion of a body in a stationary fluid.

The flow is irrotational only at distances of several viscous boundary layers from the immersed body.Kelvin circulation theorem

Considered flows are irrotational, i.e. . Any vector field with can be represented as gradient of a scalar, . is the velocity potential.Laplace equation

Governing equations for an inviscid incompressible fluid subject to gravity field:Let us rewrite the governing equations in terms of velocity potential.Continuity equation turns into

-- Laplace equation

Boundary conditionsSolutions of the Laplace equation determine the velocity potential, hence, the velocity field.The boundary conditions are needed to find a solution.Rigid wall: the fluid is inviscid, the no-slip boundary condition cannot be satisfied. The no-slip condition is replaced by a no-penetration boundary condition. vn=0 is imposed on the wall (in terms of the velocity potential, ).

Free surface: pressure is equal to the atmospheric pressure.

At infinity: the velocity field must be bounded.

Bernoullis equation

The Navier-Stokes equation turns intoor

-- Bernoullis equationAs the velocity field is already determined by the Laplace equation, this equation can be used to find the pressure.

William Thomson, 1st Baron Kelvin (or Lord Kelvin), (26 June 1824 in Belfast 17 December 1907 in Largs) was a mathematical physicist and engineer.Lord Kelvin: "Heavier-than-air flying machines are impossible.(1895)

Daniel Bernoulli (Groningen, 8 February 1700 Basel, 8 March 1782) was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family.

Pierre-Simon, marquis de Laplace (23 March 1749 5 March 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics.