User:Egm6322.s09.bit.gk/project

 See my comments below. Egm6322.s09 14:29, 1 May 2009 (UTC)

Modeling Turbidity Current through Multiphase Flow Approach =Introduction=

Background
Turbidity currents are rapidly moving currents of sediment-laden fluid moving down a slope either through air,water or another fluid which could not only be self sustaining but also self reinforcing using the  bed sediment as fuel.This can be divided into two main sections:
 * Ignitive conditions on a larger slope which is reinforced by turbulence (continental slope)
 * Supply-driven,deposition dominated condition (hyperpycnal flow)

Motivation
Turbidity Currents can be observed and witnessed in varied natural environments,
 * Marine turbidities occur from tectonic movement, storm waves,Tsunamis,Earthquakes
 * Inflow of turbid water
 * Wave action
 * Subaqueous slumps
 * Dredging operation
 * Lakes
 * Reservoir

Physical Model
A 2-phase flow approach is considered to tackle the problem of simulating Turbidity currents in deep sea environment which otherwise is quite arduous and hazardous to physically implement practical methods of calculating and establishing the presence of deep sea avalanches or self reinforcing turbidity currents.

The project hopes to successfully simulate the conditions near edge of the continental shelf and the beginning the submarine canyon where the initial forcing conditions would cease to exist as the flow would be fully developed flow and the effects of gravity would be dominant as the only driving force to create auto suspension of sediment which creates the necessary turbulence to keep the sediment entrained until the flow runs out of sediment or reaches the ocean bottom where an eventual hydraulic jump is created and dissipates the turbulent energy.

Objective
The Objective of the present study is analysis and assessment of the conditions under which self sustaining currents occur and to study the sediment transport of a non cohesive bed under these conditions of turbidity current using a two phase model by including both the Fluid-Sediment and Sediment-Sediment interactions by incorporating the $$\kappa-\epsilon$$ model

Previous work
Among the earliest of works done studying the effects of gravity flow and turbulence effecting sediment transport was undertaken by R.A Bagnold ,later on Gary Parker used the conditions necessary to obtain sediment entrainment due to gravity effects and corroborated them with his analysis and gave the concept of "ignition".H.M.Pantin gave an small correction factor to the Bagnold condition of Auto suspension which he called the efficiency factor

Literature Review
Previous contributions to the study of turbidity currents have been taking place for quite some time and in particular Bagnold's paper is reviewed below

Motivation:

The suspension and sedimentation of fine sediment grains by turbulent water streams flowing under the influence of gravity was until then treated kinematically, but it was clearly evident that centre of gravity of the suspended sediment was above the bed because of suspension and dynamic effects of the fluid as the weight of the solid cannot be ignored. R.A Bagnold was the one of the earliest to look into this problem with his approach directed towards the energetics.

Objective:

The objective of this paper was to suggest a criterion by looking at the dynamics of the fluid flow which could be studied and reproduced under laboratory condition.

Physical model:

The physical condition that was under consideration was the gently sloping ocean bed which enabled the turbidity suspension and sedimentation of sand and silt with turbulence being a cause for re-suspension.

Model Formulation:

The mass m of the suspended sediment steadily falls downwards but the centre of gravity of the suspended sediment stays above the bed, indicating turbulence must be acting in the opposing direction keeping the sediment suspended .This force should be acting against the immersed weight of a body with fall velocity $$\nu$$ .The power that is required to do so given by

$$\frac{\rho -\sigma }{\sigma }gm\nu$$ Where $$\sigma$$ is the sediment density and $$\rho$$ is the fluid density. The power expended by the fluid is replenished by the tangential pull acting on the excess sediment by the down slope gravity in the direction of the flow. This is given by

$$\frac{\rho -\sigma }{\sigma }gm\bar{U}sin \beta$$ Where $$\bar{U}$$ is the speed of the solids travelling downstream and $$\beta$$ is the angle of the inclination with the horizontal. Therefore the net power expended by the fluid in keeping the sediment suspended is given by

$$\frac{\rho -\sigma }{\sigma }gm\bar{U}\left (\nu/\bar{U}-sin \beta \right )$$ When we decrease the size of the sediment this becomes zero independent of the magnitude of the mass, therefore,

$$\nu=\bar{U}sin \beta$$ Solution:

As the bed slope increases the power provided by the tangential gravity is sufficient not only to keep the sediment suspended but also contributes to the power needed fluid flow against the drag at the boundary. The total power due to the available sediment is given by

$$\left (\sigma-\rho \right )g\bar Ch\bar U sin\beta$$ Where, $$\bar C$$ is the mean sediment concentration and h thickness of the current. To maintain suspension the power expended is

$$\left (\sigma-\rho \right )\bar Ch\nu$$ Therefore, the excess power needed for fluid flow against drag is given as

$$\left (\sigma-\rho \right )g\bar Ch\left (\bar U sin \beta-\nu  \right )$$ The Power needed to maintain a fluid flow of mean velocity $$\bar u$$ is $$\tau_0\bar u$$ where, $$\tau_0$$ is the boundary stress. From Francis 1957 the relationship is given as,

$$\tau_0\bar u=\frac{\rho\bar u^3}{33}Log^{-2}_{10}\frac{13.2h}{k}$$ From this Bagnold concluded that the criterion for self sustained turbidity current is given by,

$$\left (\sigma-\rho \right )g\bar Ch\bar U \left (sin \beta-\nu/\bar U   \right ) \geq \frac{\rho\bar u^3}{33}Log^{-2}_{10}\frac{13.2h}{k}$$ Database:

No database was used for this paper.

Result:

Since the chances of observing the data in nature is rare, the criterion derived by Bagnold would go a long way in simulating the self sustained environment experimentally, which could be checked under laboratory conditions.

=Approach=
 * Bagnold's criteria for auto suspension:

$$\frac{Us}{\nu_s}> 1$$

Where $$U$$ is the mean down slope velocity ,s is the slope and $$\nu_s$$ is the fall velocity

This follows from the above literature review ,which essence comes says that during the down slope if the current entrains more sediment,C and the impelling force $$gRCsin \theta$$ increases,the current accelerates and the bed stress increases leading to the increase in turbulence in a self reinforcing cycle that leads to highly erosive turbidity currents.This comes to end only if the bed is devoid any further sediment or the if damping occurs at high sediemnt concentration leading to a decrease in the turbulence.


 * Pantin's equation for auto suspension

$$\frac{e_xUs}{\nu_s}> 1$$ where he incorporates an efficiency factor.

=Concept and Theory= A pseudo 1-dimensional multiphase approach simulating an initial horizontal bed followed by a slope indicating a flow under gravity at the continental slope with Ghost grids on either sides on x-axis.

=Modeling with PDE=

Model Formulation
A two phase approach indicates fluid and mud as two distinct phases which includes Fluid-Fluid interactions from the fluid phase equations and the sediment-sediment interaction from the sediment phase equations.The $$\kappa-\epsilon$$ model is used as the closure for the turbulence model and the bridge between the fluid and the sediment phase.WE consider 6 equations which include the continuity equation in the fluid and the sediment phase given as

$$\frac{\partial }{\partial t}\rho^f\left ( 1-\bar c \right )+\frac{\partial }{\partial z}\rho^f\left ( 1-\bar c \right )\bar w^f=0$$

where the overbar indicates the Favre averaged values of concentration c and velocity w in the vertical direction.

Similarly for the sediment phase ,

$$\frac{\partial }{\partial t}\rho^s \bar c +\frac{\partial }{\partial z}\rho^f\bar c \bar w^f=0$$

since the concentration should be unity.

The fluid momentum equation for the horizontal velocity is given by, $$\frac{\partial }{\partial t}\rho^f\left ( 1-\bar c \right )\bar u^f=-\frac{\partial }{\partial z}\rho^f\left ( 1-\bar c \right )\bar u^f\bar w^f-\left ( 1-\bar c \right )\frac{\partial \bar P^f}{\partial x}+\frac{\partial \tau^f_{xz}}{\partial x}-\rho^f\left ( 1-\bar c \right )Sg-\beta \bar c \left ( \bar u^f-\bar u^s \right )$$

and for the vertical velocity

$$\frac{\partial }{\partial t}\rho^f\left ( 1-\bar c \right )\bar w^f=-\frac{\partial }{\partial z}\rho^f\left ( 1-\bar c \right )\bar w^f\bar w^f-\left ( 1-\bar c \right )\frac{\partial \bar P^f}{\partial x}+\frac{\partial \tau^f_{zz}}{\partial x}-\rho^f\left ( 1-\bar c \right )Sg-\beta \bar c \left ( \bar w^f-\bar w^s \right )+\beta\nu_{ft}\frac{\partial \bar c}{\partial x}$$

The first term on the left hand side is the unsteady term .The first term on the right hand side indicates the advective term followed by the pressure gradient term and the total stress ,which not only includes the small scale viscous stresses but also the large scale Reynolds stresses.The next two terms are the gravity and drag forces.The final term is the important one identify as it the turbulent suspension term which differentiates the present model from most other models.This vertical diffusion term calculates the concentration profile.

Similarly we have the sediment phase momentum equations as follows,

$$\frac{\partial }{\partial t}\rho^f\bar c \bar u^s=-\frac{\partial }{\partial z}\rho^s\bar c \bar u^s\bar w^s-\bar c \frac{\partial \bar P^f}{\partial x}+\frac{\partial \tau^s_{xz}}{\partial x}-\rho^s\bar c Sg+\beta \bar c \left ( \bar u^f-\bar u^s \right )$$

$$\frac{\partial }{\partial t}\rho^s \bar c \bar w^s=-\frac{\partial }{\partial z}\rho^s \bar c \bar w^s\bar w^s- \bar c \frac{\partial \bar P^f}{\partial x}+\frac{\partial \tau^s_{zz}}{\partial x}-\rho^s \bar c g+\beta \bar c \left ( \bar w^f-\bar w^s \right )-\beta\nu_{ft}\frac{\partial \bar c}{\partial x}$$

 It would be useful for the general readers to explicitly point out the unknowns to be solved for, and to identify the actual PDEs that will be discretized by finite difference; see below. Egm6322.s09 14:52, 1 May 2009 (UTC)

Initial and Boundary value


Initial Conditions:

The initial concentration profile is as shown in the figure,with the maximum concentration being 63.5% .The non uniform concentration profile is is better setup of the numerical code also a the varying concentration profile helps in being a non rigid bed for better entrainment and a better concentration profile.If the bed was to be rigid ,then the initial stress to suspend the bed would be very high leading to numerical instability.

The initial velocities are taken as follows:

$$\bar u^f=\bar u^s=\bar w^s=0$$

Boundary Conditions:

Lateral Boundary condition:Fluid Pressure

$$\frac{1}{\rho^f}\frac{\partial \bar P^f}{\partial x}=gS=-\frac{u^2_*}{r_b}$$

Bottom Boundary Condition:

$$\bar u^s=K_s=\bar w^s=0$$

Top Boundary Condition:

$$\bar P^f=0$$ and $$C_{min}=5\times10^{-4}$$

Left and Right Boundary Conditions

The boundary conditions for the ghost grids are periodic.

Classification
These are non linear second order PDE'S

 Any further details on the classification of these PDEs other than just the nonlinearity and the order? Egm6322.s09 14:52, 1 May 2009 (UTC)

Method of solution
The closure's for the turbulence  model is the $$\kappa-\epsilon$$ model .The method of finite difference scheme are used to solve the above PDE'S

 Did you actually solve these nonlinear PDEs by finite difference? You may use the Newton-Raphson method, and would need a linearization of these PDEs; perhaps then one could identify the classification of the linearized PDEs (hyperbolic? parabolic?). Egm6322.s09 14:52, 1 May 2009 (UTC)

Results
The parameters that were tweaked to get the results were the slope ,the frictional velocity or the forcing condition and the sediment diameter .From this the shields parameter,the Bagnold's condition and the mean downstream velocity.The following table illustrates the following





The first figure illustrates for the case of sediment of diameter 0.21 mm and frictional velocity of 0.06m/s and a slope of 14 degrees.The plots include the time series of velocities ,the velocity flux ,the concentration profile and the flux profile at different heights ,while the second one is the same expect for a more steeper profile.We can see from the different plots how the velocity profile jumps after time 150 sec which is where the forcing was stopped and gravity was allowed to take effect.

 A more detailed explanation of these figures would be useful for the general readers. Egm6322.s09 14:52, 1 May 2009 (UTC)

=Future Work= A alternate criteria to justify the presence of turbidity currents taking the particle-particle collisions into account =References=