CEE 262A

HYDRODYNAMICS

Lecture 7

Conservation Laws Part III

The Boussinesq approximation

The x3 momentum equation reads (after neglecting rotation):

i.e., part of the pressure is associated with offsetting the weight of the fluid above.

We can subtract out a significant part of this as follows:

0 3Let ' ,x x t

Reference density

Background density variation – exists in the absence of motion

Perturbation density – association with motion

The preferred ordering (which is often valid in oceans, estuaries and lakes) is 0 3 ' ,x x t

Likewise we write the pressure as 0 3 ' ,p p x p x t

such that in the absence of motion

00

3

0p

gx

If p0 is defined by this eqn., then we can subtract out the background hydrostatic pressure gradient and the weight force associated with the density field that exists in the absence of motion.

Start with

Then define p0 such that

Then since p=p0+p',

Substitution into NSE gives

00

3

0p

gx

We can use the ordering of the density field to make an important simplification/approximation:

0 0'Du Du

Dt Dt

i.e., the mass of each fluid particle that determines what acceleration results from a given force is approximately constant. On the other hand we retain the effect of density variations in the buoyancy (gravity) term (’ g). This requires that

' 'Du

gDt

Dug

Dt

Particle accelerations << g

This approximation is known as the Boussinesq approximation

If '<<, we require that

Navier-Stokes equation with the Boussinesq approximation

We also need to make the same approximation in the mass-conservation equation, i.e.

which implies that, as a consequence of the Boussinesq approximation,

Note that we assumed this a priori when writing the viscous term as given above...

01

Dt

D

The difference between incompressibility and the Boussinesq approximation

If a flow is incompressible, this implies that the density following a fluid particle is identically zero, which gives the equations for conservation of momentum and mass as

Under the Boussinesq approximation, the density following a fluid particle is not constant, but its time rate of change is much smaller than that due to changes resulting from velocity gradients. This enables one to write the momentum and mass conservation equations as

The Boussinesq approximation does two things: it linearizes the acceleration term in the Navier-Stokes equations and enables use of the continuity equation while retaining the effects of density in the momentum equation.

Incompressible

Boussinesq

How do we cope with free surfaces?

1 2, ,x x tx3=0

x3

-x3

From before, we had p=p0+p', where p0 was the pressure fieldin the absence of motion, while p' was that associated with motion.We can define an alternate splitting of the pressure as p=ph+0, where:

ph = Hydrostatic pressure arising from weight of fluid(can include motion this time)

0= Dynamic, or nonhydrostatic, pressure arising from fluid motion

Defining the hydrostatic pressure as satisfying the balance

we can integrate both sides to obtain ph:

Surface pressurePressure due to depth and free surface: BAROTROPICPRESSURE

Pressure due to density variations:BAROCLINICPRESSURE

Now take gradients of ph:

Where we have assumed that

(Why is this not obvious?)

Adopting very commonly-used shorthand notation for the horizontalgradient, such that

we have

Surface pressure gradienti.e. Atmospheric pressure.

Barotropic pressure gradient due to free-surface gradient.

Baroclinic pressure gradient due to density gradient.

Substitution into the Navier-Stokes equation with the Boussinesqapproximation gives

Or, component-wise:

Why does water level go down when atmospheric pressure goes up?

An example from SF Bay/Delta (observations):

10 cm (water)

15 cm

Answer: The ocean likes to tend towards steady state which has

Thus, the response to an imposed pressure on the surface would give

1 1

1 1

1 1

1 1

01 1

0 1 11 1

0

0

s

x b x b

s

x a x a

x b x b

sx a x a

s

pgx x

pg dx dxx x

g p

p

g

The “inverse barometer” - water level goes down when pressure goes up