Marine Boundary Layers

Shear Stress

Velocity Profiles in the Boundary Layer

Laminar Flow/Turbulent Flow

“Law of the Wall”

Rough and smooth boundary conditions

motionof

thresholdU criticalbz

Suspended Load

Bed Load

Shear Stress

LT

M

LT

ML

Area

Force222

1

In cgs units: Force is in dynes = g * cm / s2

Shear stress is in dynes/cm2

(N/m2 in MKS)

X

Z

Y

xx

xz

xy

Each plane has three components – i.e., for the x plane:

For three dimensions: nine components

What are the key components in the marine boundary layer?

XX, YY, ZZ component – is the pressure force, doesn’t act to move particles

XZ, YZ component – the flow is not shearing in the z-direction (in the mean)

XY, YX component – assume uniform flow (flow not rotating in the mean)

End up with two components:

, shear on the z-plane in x and y directions

As we get close to the seabed and rotate into flow:

τb

zy zx

Simplest boundary layer case:

Laminar Flow – smooth boundary

no turbulence generated

layers of fluid slipping past each other

In this case:

Z

X

F

h

“No-slip” condition

constantA

Fzx

Deformation of fluid layers is at same rate for shearing force

linear velocity profile

Integrating:

Boundary conditions:

Description of velocity profile:

Kdz

du

What force (or shear stress) was needed to pull plate A and create this velocity profile?

z

uzx

Molecular viscosity of the fluid (resistance of the fluid to deformation)

PSalTf ,,

Provides transfer of momentum between adjacent fluid layers

Another way to think about shear stress:

Transfer of momentum perpendicular to the surface on which stress is applied.

momentumuz

u

z

u

zx

zx

kinematic viscosity

Velocity gradient Fluid momentum gradient

Diffusion of momentum

Turbulent Flows

A random (statistically irregular) component added to the mean flow

Define u = instantaneous velocity

u’ = random turbulent velocity

ū = mean velocity u = ū + u’

Dyer, 1986

NOTE! Beware of averaging time scale.

Turbulent fluctuations follow a Gaussian distribution:

Turbulence intensity can be described by the RMS fluctuation

2'usqrt

Turbulent eddies transfer momentum, much the same way as molecular diffusion, but at appreciably greater rates.

Frequency of occurrence

u’

Average of u’==0

Van Dyke, “An Album of Fluid Motions”, 1982

Transfer of momentum can be described by:

“eddy” viscosity - Az – transfer of momentum in z-direction

(note: in Wright, 1995 chapter)

Az >>

dz

udAzzx )(

dz

udAzzx

Eddy fluctuations and momentum transfer:

u’, v’, w’ - responsible for the transfer of momentum

Middleton & Southard, 1984

Z

ū

• Parcel has lower momentum at z2 by ρΔu

• flux of momentum:w’•(ρΔu)

• As z2 and z1 approach each other,

u2 - u1 = Δu u’

• flux of momentum:w’•(ρu’) or u’w’

This rate of change of momentum represents the resistance to motion, or the shear stress, and averaged over time:

''wuzx Reynolds Stress

Since turbulent fluctuations difficult to characterize, simplifying assumptions can be made:

u’ u turbulent fluctuations are proportional to the mean flow

u’, v’, w’ are of similar magnitude

2uzx

2uCdzx Quadratic Stress Law

Summarize: Three ways to describe shear stress in the turbulent bottom boundary layer.

• Eddy Viscosity

•Reynolds Stress

•Quadratic Stress Law2uCdzx

''wuzx

dz

udAzzx