Governing equations for velocity profile and sediment particles behaviour

The velocity profile:

Intro:

The particle motion is above all depending on the global motion of water. Both waves and steady current can be present. These 2 primary kind of dynamics implicate different kind of study.Following details may refer to a particular situation, in order to make things easier.

As an example of that difficulty, whether you add a steady current with a wave -current or do the opposite you won't get the same result!! That reveals the deeply non-linear nature of the governing phenomenons.

Typical velocity amplitude:

Here we'll study the case of a laminar wave-current flow.

For such a flow, the velocity above the boundary layer can be written as harmonic.

The shear stress can be written as follws: ( for turbulent flow, the Eddy viscosity replaces this viscosity, and it has then to be modelised)

The momentum equation gives (*):

Let's consider D= Ulim - U(z), the deficit of velocity at a given depth, solving (*) the solution is:

As the solution is finite, and also identifying the harmonic sum with awcos(wt), the result is:

Eventually, the profile is:

The shear stress:

This study of theory is based on a purely oscillatory flow. Of course it's an ideal vision, but it'a although sufficient to be a good basis for further study.

Firstly, an important parameter is the bed shear stress, t (o,t) . Indeed, it represents for a great part the forces on the particles, and consequently helps to understand their own motion.Generally, for a laminar harmonic motion, the bed shear strass is also harmonic, and in advance of about 45° with u(,t).

The motion equation is again used :

and so,

(Experiments have shown that we can consider that this result is still good with turbulent flow.)

The wave friction factor:

Définition:

This semi-experimental factor is defined by jonsson in amplitude, with the following equation:

Kajiura established another solution, depending on time:

This definition allows to introduce the advance of t if C1 is complex.

® explicit form for fw :

Jonsson has shown using dimensionnal analysis that :

Then, several explicit expression can be written according to the physical model being used.

For a smooth bed, fw=fw (Re)

Over a rippled bed, fw=fw (r/a)

Forces applied and motion of sediment particles:

Forces :

These forces are numerous and have diffrent origins:

-Granular forces between particles.

-their weight.

-Fluid forces( Drag, Pressure )

Concretely, to caracterise the balance of forces applied, a few specific ratios are available.

The motion:

Once understood the difficult problems caused by definition ( when does the bed move? What can quantitatively be called the bed ?) , there are simple tools to caractérized the bed behavior.

-The mobility number:

It reveals the ratio between the drag force and the weight.

The Shields parameter:

In 1936, Shields introduces an analog parameter where the shear stress appears instead of the drag force ( shear stress constant).

In the case of a wave current, the shear stress amplitude is taken, and with the parameters defined above the definition gives:

The critical Shields parameter:

It is defined as the Shield parameter above wich the motion starts.

This value depends on the nature of the fluid and of the bed. For example, a smooth sandy bed and water, it reaches 0.05.

Next section: a Macro- view of sediment transport.