Governing equations for velocity profile and sediment particles behaviour
Velocity profile ___ Shear stress ___ Forces applied
The velocity profile:
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 :
(Experiments have shown that we can consider that this result is still good with turbulent flow.)
The wave friction factor:
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:
These forces are numerous and have diffrent origins:
-Granular forces between particles.
-Fluid forces( Drag, Pressure )
Concretely, to caracterise the balance of forces applied, a few specific ratios are available.
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.