Equations of Motion and Turbulent Processes Part I

CEE510 Estuarine CirculationBy David A. Jay

Important Concepts –This section of notes introduces several key ideas:•

The distinction between the Eulerian

and Langrangian

points

of view •

The concept of the control volume

•

Use of the control volume to derive conservation expressions, conservation of mass in the present case.

These ideas are used in all areas of fluid mechanics

The Euler-Lagrange Transform –

•

Newton’s second law, F = MA, is formulated for particles, which must be tracked over time (a Lagrangian

viewpoint).•

Conceptually, this is possible for water particles, but it is very inconvenient!

•

Therefore, we develop a dynamic picture based on flow properties of a field (an Eulerian

viewpoint)

•

We must know how each property F(x, y, z, t) at a particular (x, y, z) in a fluid continuum relates to the value of F following a particle that is at (x, y, z) initially –

that is the Euler-Lagrange transform

The Euler-Lagrange Transform (Continued) –

•

We use a Taylor Series expansion to follow motion over a short time, dt; for an F:F(t+dt, x+dx, y+dy, z+dz) = F(x, y, z, t) + ∑F/∑x

dx +

∑F/∑y dy + ∑F/∑z

dz + ∑F/∑t

dt +0(dx2, dy2, dz2, dxdy, dxdz, dxdt, dydt, dzdt)

•

define dx/dt= u, dy/dt

= v and dw/dt

= w•

This gives:

•

F(x+dx, y+dy, z+dz, t+ t) F(x,y,z,t) + ∑F/∑xU dt + ∑F/∑yv dt + ∑F/∑zw dt

+ ∑F/∑t

dt

•

Note that dx, dy, dz

and dt

are infinitesimal, so second derivatives are very small –

neglect

•

F is a smoothly varying function (“well-behaved”)

The Euler-Lagrange Transform (Continued) –•

We can write the total derivative:

DtDF

DtDF

tF

dtdz

zF

dtdy

yF

dtdx

xF

FtF

zFw

yFv

xFu

tF

U

U is the velocity vector {u,v,w}

•

DF/Dt is the “total derivative”•

It expresses the total time change in F following a parcel pf

water that is at (x,y,z) initially•

The non-linear terms are called “the convective accelerations”

•

If F = velocity vector U: UUUU

tDt

D

The Euler-Lagrange Transform (Continued) –

•

The same result can be achieved using the chain rule:

DF[x(t), y(t), z(t),t]/Dt

•

The total derivative is very useful for properties like salinity; following a parcel:

DS/Dt

= 0 says that S does not change following a water parcel, as

long as there is no mixing. Salinity at a point (x,y,z) may change, as it is occupied

by different parcels of water

tF

dtdz

zF

dtdy

yF

dtdx

xF

tF

zFw

yFv

xFu

Conservation of Mass and the Control Volume –•

Use a control volume

to define mass conservation

•

The control volume V is an infinitesimal box fixed in space, through which the fluid in question moves. It has sides dx, dy

and dz, in the x, y and z directions•

Because density varies, the mass in V may change

•

How do we quantify this using V and a Taylor series?

•

-∑(u)/∑x (dxdydz) is the change in the x direction (approximately) •

-∑(v)/∑y (dxdydz) is the change in the y direction•

-∑(w)/∑z (dxdydz) is the change in the z direction

Conservation of Mass (continued) –•

The rate of change of mass in V =dx

dy

dz

is:

•

This can be re-arranged to:

•

Then :

•

For an incompressible flow: D/Dt

= 0 (density following a parcel of water is constant, even if (x,y,z,t) varies

•

So we have the continuity equation: “U =0

dxdydztt

V

dxdydz

zw

yv

xu

0)()()()(

U

tw

zv

yu

xt

010 UUU

DtD

t

The Continuity Equation –•

We can write the continuity equation:

•

How good is D/Dt

= 0? ––

following a parcel of water will change if mixing occurs –

how much? –

varies only ~3% in the world ocean; SPM variations are usually smaller (with exceptions); SPM is actually a separate phase

–

The other approximations in the equation of motion (e.g., representing turbulent motion are less exact)

–

Incompressibility is used in most models, even of the atmosphere, despite compressibility of air

0

zw

yv

xuU

Conservation of Momentum –•

Cauchy’s Law is Newton’s F = M A in a continuum version•

Cauchy’s Law says that the acceleration of a fluid element can be specified in terms of body forces and surface forces:

mass x acceleration = body forces + surface forces

•

Write this as:

We write the equation of motion in vector form as follows:where T ′

is the stress tensor (Ti,j

′

) and f = (0, 0, g) is gravity

The stress tensor is a second order tensor with 9 components –What does this mean?

T fDtDU

The Stress Tensor –

•

It is the divergence

of the stress that is the total of the surface forces acting on a fluid element

•

Also, we want to separate out the normal forces (pressure) and the tangential or surface forces that deform the fluid element;

•

Write the stress Ti,j

′

as the sum of deviatoric

Ti,j

and pressure parts:

Ti,j

′

= -pij + Ti,j

ij = Kroneker delta•

The minus sign in front of p is there because pressure is defined inward, not outward

•

T31 or Tzx is the surface force acting in x direction on the z-face of a fluid element, and similarly in the other directions:

The Stress Tensor –•

Formally, we write the deviatoric

stress T in terms of rate of strain tensor and a viscosity:

Tij = Aijkl ∑Uk

/∑xl

is the rate of strain

and the tensor Aijkl is the viscosity. •

By intuition and symmetry arguments, we know that viscosity is isotropic in water, though not necessarily in solids or liquid mud

•

From Newton’s experiment, we define the viscosity:

•

Assume that m

is spatially uniform, and put this back into the momentum equation

The Navier-Stokes Equations –

•

Here is the momentum equation in vector form:

•

The component equations:

kgpDtD -UU 2 2

2

2

2

2

22

zyx

uxp

zuw

yuv

xuu

tu 21

vxp

zvw

yvv

xvu

tv 21

wgxp

zww

ywv

xwu

tw 21

•

Kinematic

and dynamic viscosity

are properties of the fluid; essentially constant! Note that:

( U) =

U = 2 U

In terms of values:

10-3

kg/(ms)

= /

in m2/s

10-6m2/s

The Hydrostatic Assumption –

•

Most of the flows we will talk about are shear flows•

We assume that vertical velocities are small, and simplify the z-

equation to:

•

This is called the hydrostatic assumption; it means:–

The pressure is simply due to the weight of water overhead–

That these two terms are much bigger than the others –

think about how much your ears hurt underwater!

–

Example: if w changes by 1 cm/s2

= 0.01 m/s2

(a rapid change for a shear flow), this acceleration is 1000 times less than g = 10m/s2.

•

This assumption is not always valid, e.g. in flow over an airfoil –

flight is not hydrostatic

•

But pressure is ALWAYS isotropic; i.e., same in all directions!

gzp

10

On the Nature of Turbulence –

Big whorls have little whorlsThat feed on their velocity

And little whorls have lesser whorlsAnd so on to viscosityL.F. Richardson (1922)

Turbulence and the Reynolds Equations –•

The Reynolds equations simplify the Navier-Stokes equation by separating the chaotic, turbulent part from the mean flow. Formally, with primes being turbulent variations and overbars

averages:

U = U +

U

u = + u

v = + v

w = + wTo do this right, we need to understand turbulence!

1. What is turbulence? What are its properties? It is or has:–

A property of the flow, not the fluid, and changes with the state of the flow

–

Highly irregular in space and time–

Three dimensional, even when the flow that causes it is 2-D–

Chaotic (sensitive to initial conditions) and not predictable in

detail–

Turbulent eddies that may are almost as large as the depth of flow•

For a flow with depth d = 10 m, u

~ 0.01 to 0.1 m/s. This gives periods t = d/u

of up to 100 to 1000 s.

u v w

More Properties of Turbulence –

•

Turbulence is rotational (has vorticity), and eddies wraps up on themselves

•

Vorticity

usually enters the flow at the bed, where the flow is slowed, and this makes eddies.

•

Turbulence is dissipative –

it removes energy from the flow, and this energy cannot be recovered, because it is lost to heat and increased entropy

•

Turbulence occurs at high Reynolds (Re) numbers. Re is a ratio of the convective acceleration terms to the viscous term in the equation of motion:

Re = ul/n

where l

is a length scale, u is mean flow,

and n

is molecular diffusivity

Where Does Turbulence Occur? –•

Turbulence is generated where there is shear. Vertical shear often generates turbulence, e.g.:–

Tidal shear at the bed in a river or estuary.–

On the continental shelf, the Coriolis

force (caused by the rotation of the earth) is also important. A turbulent bottom boundary layer with

rotation is called a benthic Ekman

layer–

In the open ocean, shears is greatest at the surface because of the wind. There is a surface turbulent Ekman

layer–

In stratified estuaries and buoyant plumes, shear occurs between

the layers with different densities The density difference inhibits mixing, But some mixing occurs:

•

If the shear is strong the turbulence generated here may penetrate all the way to the surface and bed,

•

It may be confined to a small fraction of the total depth, if stratification is strong relative to the shear.

•

Sometimes, horizontal shear is important, e.g. in the jet-like ebb flow of water out of the mouth of a river or estuary with a narrow mouth

The Reynolds Averaged Equations –•

The Reynolds equations: –

Separate turbulent and mean quantities–

Allow us to work with time-averaged variables, without worrying about short-

term fluctuations

–

Are formed by separating average and fluctuating variables–

Then averaging the Navier-Stokes over time•

Averaging vector velocity over a time period T of a few minutes:

Tt

t

dtT

0

0

1 UU UUu

•

We then define 6 averaging rules:0u

2121 uuuu

tu

tu

uccu uu

3121321 uuuuuuu

The Reynolds Equations (More) –

•

Add continuity to the momentum equation, e.g. for the x-

equation:

•

Yielding (using 0

instead of

from the Boussinesq

approximation):

0

1 2

0

zw

yv

xuu

uxp

zuw

yuv

xuu

tu

uzyxx

puwz

uvy

uxt

u

2

2

2

2

2

2

0

2 1

•

Substitute mean and fluctuating variables and time average. The only complicated term are the quadratic terms, e.g., u 2

in the x-equation:

uuu 22222 '''2'' uuuuuuuuuuu

The Reynolds Equations (More) –

•

In the y-

and z-equations, the analogous terms are:

•

This gives the Reynolds equations, with a new term at the right, the Reynolds stresses:

'''''' vuvuvuuvvuvuuv

'''''' wuwuwuuwwuwuuw

''1 2 uuvpt

UUUU

UUUUUUUU but

''1Thus 2 uuvpDtD

UU

0

UUzw

yv

xu 0''''

zw

yv

xuu

•

The same averaging of the linear continuity equation gives separate mean and turbulent equations:

The Reynolds Stresses –•

What is the meaning of the Reynolds stresses??, e.g. the stress in the x direction on the z face, which is represented by:

•

From Delo

and Smits, http://elecpress.monash.edu.au/ijfd/1997_vol1/paper3/figures/Figure1.html

0'' wu

•

The Reynolds stresses do NOT vanish, even though u, v, and w

all average separately to zero

•

This means that these pairs of variables are correlated; i.e., vary together •

Because they vary together, they transfer momentum•

Near-bed turbulence occurs as bursts with:

Typical bursts:

'' wu

What are the Reynolds Stresses? –•

Turbulent stresses result from the correlations of fluctuating (turbulent) parts of u and w (u’

and w’)

•

Represented here as u’

and v’•

Viscous stresses results from actual molecular effects

From Jeff Parsons, UW

The Reynolds Stresses (more) –•

Suppose near the bed w > 0 and u < 0 at a point:–

This means that a “slow”

parcel of water from near the bed is being pushed up higher in the flow,

–

It will exchange momentum (be mixed into) the ambient water which is moving faster. This ambient water is therefore slowed.

–

On the average, another parcel of water must be pushed down, to take the place of the first (otherwise, mass is lost near the bed).

–

This parcel will have more momentum than other parcels near the bed –

This process transfers momentum toward the bed, where it is dissipated•

The Reynolds stress are typically much larger than viscous stress, so we usually write the Reynolds equations w/o the viscous stresses

•

They appear because we time-averaged a non-linear equation –

they are the price for working with time-averaged variables•

The Reynolds stresses are a function of the flow, not the fluid!•

From now on, we will write the mean variables without overbars; they will be understood to be averaged to remove turbulence

A Turbulence Closure –•

So far, we’ve complicated the equations by adding the Reynolds stress, and we don’t know how to represent them

•

This is the turbulence closure problem

–

writing the Reynolds stresses explicitly means that we have more unknowns that equations

•

A turbulence closure specifies what the Reynolds stresses are, so we have the same number of equations and unknowns

•

All turbulence closures are approximate, most are complicated!

•

We will use a simple closure that uses an “eddy diffusivity”

to specify the magnitude of turbulent momentum transfer

Defining An Eddy Diffusivity –

•

Kv

is for vertical mixing, KH

is for horizontal mixing•

Because vertical mixing is larger than horizontal, we neglect the horizontal (KH

) part of•

We will also ignore horizontal mixing

(The above is actually cheating a bit, but it explains the idea)

'''' uwzuK

xwK

zuKwu vHv

xv

yuKvu H ''

•

By analogy to molecular diffusion, we write the Reynolds stresses as the product of an eddy diffusivity K and a velocity gradient, e.g.:

'' wu

A Practical form of the Reynolds Equations –•

Neglecting horizontal mixing, the momentum and continuity equations are:

zuK

zxp

zuw

yuv

xuu

tu

m1

zvK

zyp

zvw

yvv

xvu

tv

m1

gzp

10 0

zw

yv

xu

The scalar transport equation for sediment concentration C is:

zCK

zzCww

zCv

xCu

tC

DtDC

S

Where: wS

is the settling velocity of sediment particles For salinity or temperature, which are disso/ved, wS

=0