We shall study the open channel with periodic in streamwise x and spanwise z direction and in vertical direction "y" at the lower boundary no-slip and at the upper boundary stress-free boundary conditions. The statistics of the organism distribution are examined by varying the cell swimming speed, aspect ratio and gyrotaxis at different resolutions.
The Orientation of micro-organisms is determined by the balance of viscous and gravitational torques. The trajectory of swimming cell is directed by the orientation, this torque balance results in directed locomotion, called gyrotaxis. Gyrotaxis affects pattern formation in suspension of swimming algae. In this thesis we studied passive swimming particles of spherical and elongated shape in open channel flow.
We considered a typical algal cell, Chlamydomonas, has shape approximately spherical with a pair of flagella at one end, due to this cell swims in a direction roughly parallel to its axis. These cells are bottom heavy and they tend to swim vertically upwards in the fluid otherwise at rest; if they started to swim at an angle to the vertical, the gravitational couple would immediately rotate them to the vertical. However if the fluid medium flows with the horizontal component of vorticity, it will exert the viscous torque on the cell and rotate it away from the vertical. If the vorticity is not too large, there will be a balance between the viscous and gravitational torques, and the cell swim at a fixed angle to the vertical. This mechanism is called gyrotaxis of spherical cells.
Equation 1‑1
Where "" is the horizontal component of vorticity in the flow and "B" is the constant that is determined by the geometry of the cell and viscosity of the suspending
Figure 1.1
In the figure 1.1, "p" is the unit vector in the swimming direction, "h" is the displacement of the center of gravity "G" from the center of the cell "C", so that relative to the Cartesian coordinates, "Lv" the viscous torque and "Lg" is the gravitational torque.
Swimming at small Reynolds number is very different. One of the keys to being able to swim is to have a stroke that is not symmetric in time otherwise the reversible viscous fluid flow prevents ground from being gained.
There are two stages: an effective or power stroke in which the micro organism move forward and recovery stroke in which the flagella are returned to their original positions at the expense of backward motion. The flagella do not perform their stroke symmetrically and hence the cells revolve about an axis with the swimming direction. The cell takes advantage of this rotation in that its eye spot is able to survey the surrounding light field and hence control its phototaxis.
In general, the viscous torque on non-spherical cells contains contributions from the rate of strain tensor, as well as all three components of the vorticity vector.
1.2 Micro-organisms
The term organism represents an individual that is capable to grow, metabolize nutrients, and usually able to reproduce. It can be unicellular or multicellular. These organisms are divided into five different groups named prokaryotes, protists, fungi, plants, and animals. These groups are also called kingdoms.
1.3 What is Taxis?
We studied the pattern formation of micro-organisms in this thesis. The spontaneous pattern formation in suspensions of microorganisms is called Bioconvection. Micro-organisms responsed to different stimuli with their movement in particular direction. Taxis means arrangement. It is a Greek word. According to Henderson's dictionary of biological terms its definition is
"A movement of free motile, usually simple organisms, especially Protista, or part of an organism. Towards (positive), or away from (negative), a source of stimulation, such as light, temperature, chemicals; an orientation behavior related to a directional stimulus."
Taxes include change in surroundings and mechanisms of organism's movement in response to change in surroundings. Organisms move in random manner in the absence of taxes. Most organisms use a combination of random movement and taxes. Natural selection ensures that optimal tactics are always employed.
There are some typical examples of taxis: chemotaxis, in this organisms move towards or opposite to the chemical concentration gradient. Chemotactic bacteria experience the change in nutrient concentration with time. If there is any change in sensed concentration level, they respond by appropriate change in their tumbling probability.
Phototaxis, in this organisms are sensitive to light intensity, its direction or polarization. Phototactic organisms need light for photosynthesis so they swim towards it.
Geotaxis, in this organisms move towards or opposite to the gravity. This is also known as gravitaxis. The organisms that are bottom heavy tend to move upwards due to anisotropic mass distribution of organelles within their bodies. This upward movement is known as negative gravitaxis or negative geotaxis.
The orientation of organisms is due to gradients in local fluid velocity. Their swimming is vorticity sensitive. The rotational viscous drag and the distance between the center of volume and the center of mass is responsible for the angle between the axis of the cell and the vertical direction. Swimming by this mechanism is termed as gyrotaxis.
Rheotaxis, in this organisms try to keep position in a stream rather than being swept downstream by the flow because of their shape. Some organisms exhibit negative rheotaxis where they will avoid flow.
Micro-organisms need any mechanism to come close for sexual mating. So chemotaxis can effectively drive sexual aggregation. Geotaxis and gyrotaxis result in pattern formation. In the absence of wind shear or thermal convection, gyrotaxis might work to extract more nutrients from the bed of a pond than a mere geotactic instability, involving organisms that do not exhibit gyrotaxis, by increasing the width of up flowing fluid and creating higher wall shear stress.
1.4 Wall bounded Flows
By definition Turbulent flow is not stationary but stationary in the mean i.e. fluctuating around the mean value. The fluid motion is irregular and shows a random variation in both space and time. The flow field should show large vorticity intensity and vortices should span over a large range of scale.
Turbulent flows are categorized into internal flows and external flows. Fully developed flows through pipes and ducts are common examples of internal flows. In these flows mean velocity profile and friction laws are of important concern which illustrate the shear stress exerted on the wall by the fluid. External flows include flow around aircraft and ships etc.
Boundary layer flows are complex as compared to flows in free shear layers because, walls present in bounded flows imposes constraint for example viscosity of the fluid causes no slip condition. This no slip condition or viscous constraint causes a viscosity characteristic length of the order of ʋ/w where is the kinematic viscosity and w represents characteristic of the level of turbulent velocity fluctuations. At high Reynolds numbers, /w is smaller than boundary layer thickness δ, so we can say that /w will not influence the entire flow.
Friction velocity
Equation 1-2
Viscous length scale
Equation 1-3
The Reynolds number defines on the basis of viscous scales
Equation 1-4
The percentage of viscous stress to the total stress decreases from 100% at the wall where y+ = 0 to the 50% at y+=12 and less than 10% by y+=50.
Equation 1-5
Where represents the Reynolds stress and viscous stress.
We divide the regions near wall on the basis of y+.
In 1925 Prandtl postulated that in inner layer u+ is only the function of y+ for y/δ << 1.
Equation 1-6
In viscous sub layer, the deviation from the linear relation u+=y+ are negligible for y+<5, but significant for y+ > 12.
The log law or the logarithmic law of wall due to von Karman
Equation 1-7
Where "k" is von Karman constant and "B" is the constant of integration. In the literature, there is little variation in the values of these constants, but generally these are within 5%.
K = 0.41 and B=5.2
Buffer layer, is the region between viscous sub layer ( y+ < 5) and log-law region (y+> 30). It is transition region between viscosity-dominated and turbulence-dominated parts of the flow.
In the outer layer where velocity profile is not expected to depend on the viscosity for high Reynolds numbers.
Equation 1-8
Where
1.5 Isotropic Turbulence
Kolmogorov's idea is that the parameters those are responsible for the size of dissipating eddies are relevant to the smallest eddies. These parameters are the rate of energy dissipation Ñ” and the viscosity Ê‹ that does the smearing out of the velocity gradients. In turbulent flow at high Reynolds number, the statistics of the small scale motions have universal form those can be determined by Ñ” and Ê‹.
With these parameters we can form length, time and velocity scales.
Equation 1-9
Where "η" is the kolmogorov length scale
Equation 1-10
Equation 1-11
We considered these length, time and velocity as the smallest length, time and velocity in our problem. For channel flow, these scales vary as the distance from the wall varies.
To find the swimming speed of micro-organisms we used the Kolmogorov velocity scale.
Refrence:
The Orientation of Spheroidal Microorganisms Swimming in a Flow Field T. J. Pedley and J. O. Kessler Proc. R. Soc. Lond. B 1987 231, 47-70
Turbulent Flows by Stephen B. Pope
A First Course in Turbulence by H. Tennekes and J.L. Lumley
http://science.yourdictionary.com/organism
http://en.wikipedia.org/wiki/Rheotaxis
Chapter 3
3.1 Particle laden Flow
Particle laden flow is one in which particles are dispersed. This makes it a two phase flow, the fluid forms the continuum phase while the particles form the dispersed phase. Multiphase flows have more complicated dynamics than single phase flow. Single phase flow can be characterized solely by the Reynolds number, but to characterize the two phase flow we need volume fraction of particles "" and Stokes number "St". These non-dimensional numbers are defined as
Equation 3-1
Equation 3-2
Where
N, Vp, V, Tp, Ts, represent the number of particles, the volume of a single particle, the total volume occupied by both phases and the characteristic time scale of the turbulent flow, the friction time scale considered for wall bounded respectively.
Figure 3-1
For small , the particles have negligible effect on the turbulence, and the interaction between particles and turbulence is termed as one way coupling. In such cases particle dispersion will depend only on the state of turbulence. If the value of increases, the momentum transfer from the particles is large enough to change the turbulence structure. This is known as two way coupling. For very high value of in addition to the two way coupling, particle-particle collision takes place, which is called four way coupling. In our simulations, we have low stokes, low inertia and passive particles so we have the case of one way coupling
3.2 Equation of Motion
Equation of motion of spherical particles in turbulent flows has been developed and presented in literature. These particles are small as compared to the smallest characteristic scale of the flow. Corrsin and Lumley proposed the following equation of motion for small sphere with radius "a" and mass "mp" moving with speed V(t) with its centre located at Y(t)
Equation 3-3
Where ui(x, t) is the mass of the fluid displaced by the sphere, and "µ" and ""are dynamic and kinematic viscosity respectively. In the above equation is the material and time derivative respectively.
Interpretations of the terms used in the above equation are followings:
The first term represents the contribution of the pressure gradient of the flow on the force imposed by the flow on the particle, also known as pressure drag.
The second term is the added mass or virtual mass. This is an inertia imposed by the fluid as the accelerating particle has to move a volume of the surrounding body while it moves through the fluid.
The third term is the Stokes drag in the linear form. To achieve more accuracy we used in a nonlinear Stokes drag
Where,
3.3 Fluid Dynamics
An open channel of size 4Ï€h Ã- 2h Ã- 2Ï€h is considered with dimensions streamwise "x", wall normal direction "y", and spanwise "z" direction respectively ;With "h" being half channel height. Particles are injected in a turbulent flow with different initial velocities on a pseudo random pattern as the initial position. We solve the non-dimensional Navier- Stokes equations for incompressible viscous fluid.
Equation 3‑4
Equation 3-5
The flow is defined by the non-dimensional parameter, Re. The flow is periodic in streamwise and spanwise directions and is driven by a time dependent pressure gradient. Waves with a drift can be reproduce by something like
Equation 3-6
At the lower wall one can apply non-homogenous wall-normal velocity to reproduce transpiration. The upper boundary of the domain can be treated as free surface with symmetry boundary condition.
3.4 Governing equations for the swimmers
The flow is seeded with many particles, typically between 2 Ã- 105 and 106. Swimmers are point particles which advected with the local flow velocity and the swimming speed us
Equation 3-7
In this expression, u is the local fluid velocity, which may depend on local fluid velocity and shear and p is the local particle orientation. Note that one could modify the transport by including inertial effects, so that particles have a characteristic time scale to adjust to the local fluid velocity (this is zero for the expression above). This should not be the case for very small organisms.
To close the system one needs to define rules for us and an evolution equation for the orientation p. Assuming ellipsoid shape, the angular velocity of the organisms is defined by the inertia-free balance of gravitational and viscous torques. The deterministic part of the cells' rate of change of direction is given by
Equation 3-9
The first term in the above equation describes the reorientation due to cells' being bottom heavy, k is unit vector in the vertical direction and B ï¡ µ/(Ïpgh) is the gyrotactic reorientation parameter with h the distance between the centre of spheroid and its centre of mass. The second term represents reorientation due to viscous torque on the cell caused by vorticity "ï·" and the third term is reorientation due to rate of strain of the linear shear flow, ï¡0 = (a2 - b2)/(a2 +b2) the eccentricity of the spheroids, and E symmetric part of the deformation tensor. Rotational diffusion is added as stochastic process of given mean and standard deviation.
We did the some cases with particles having zero velocity when injected in the fluid so the equation reduces to the following equation
Equation 3-10
If the cells are symmetric not the bottom heavy, the gyrotactic reorientation parameter B will be equal to zero because "h" which is the distance between the centre of spheroid and its centre of mass equal to zero in this case means the centre of spheroid and its centre of mass lie at same the same point. For spherical cells a=b so ï¡0 will be equal to zero so that the rate of strain does not affect the orientation due to these the equation ---- reduces to
Equation 3-11
This shows that the reorientation due to viscous torque on the cell caused by vorticity "ï·" But for the elongated shape having eccentricity equal to 1 and without gyrotaxis we have the following equation
Equation 3-12
In this case, the first term represents reorientation due to viscous torque on the cell caused by vorticity "ï·" and the second term is reorientation due to rate of strain of the linear shear flow.
Refrences
1. S. Elghobashi, "On predicting particle-laden turbulent flows", Applied Scientific Research, 52, 309-329 (1994)
2. M.R. Maxey and J.Riley, _Equation of motion for a small rigid sphere in a nonuniform _ow_, Phys. Fluids 26, 883 (1983).
Chapter 4
Numerical Simulation
We used a program called "SIMSON" for all simulations in our work. This solver is implemented in FORTRAN 77/90. An efficient spectral integration technique is used to solve the Navier Stokes equations for incompressible channel flows. We can run this program either as a solver for direct numerical simulations (DNS) in which all length and time scales are resolved or in large-eddy simulations (LES) mode where a number of different subgrid-scale models are available. The evaluation of multiple passive scalars can also be computed. The code can be run distributed or with shared memory parallelization through the Message Passing Interface (MPI) or OpenMP, respectively.
Fluid Phase
Equations for fluid are solved in spectral space, i.e. Fourier series for streamwise and spanwise directions and Chebyshev series for wall normal directions. Their results are transferred back to the physical space using backward Fourier and Chebyshev transformations. The streamwise and spanwise directions are spatically discretized using Fourier expansions, while Chebyshev polynomial on Gauss-Lobatto points is used for wall normal directions. For temporal integrations, two semi-implicit schemes are used. The third order Runge-Kutta (RK3) scheme is used for the integration of advection and forcing terms, while diffusive terms are integrated by an implicit second-order Crank-Nicolson scheme. The basic numerical method is similar to the Fourier-Chebyshev method used by Kim et al. (1987) classically used for canonical turbulent flows.
Particle Phase
Particle advection is also solved in SIMSON. Fluid velocity and its derivative must be known at the particle position for the calculation of the force at the particle. For that, the particle position is projected onto the horizontal planes, both above and beneath. On each horizontal plane four grid points surrounding the particle projections are found and fluid velocities are then interpolated from these grid points on to the particle projections using a second order accurate linear interpolation. Then fluid velocities are interpolated onto the particle position using another linear interpolation, this time in wall normal direction.
For Particle integration we used the RK3 integration scheme, the same scheme that is used to integrate the fluid. For this particle equation of motion is solved at each of four RK sub steps and the interpolation of fluid velocities is taken at each sub step. Vorticity and velocity fields are used to get velocity gradients in Fourier space on each and every horizontal plane. Two explicit schemes, First order accurate Euler forward, and second order accurate Adams Bashforth integration techniques are also implemented and tested in SIMSON. These explicit schemes have a major advantage over RK3 technique, and that is the particle integration has to be performed at each time step, i.e. every fourth iteration, while RK3 technique requires the particle integration to be run at each RK sub step, i.e. at each iteration. This evidently could increase the efficiency of the software by decreasing the amount of required computations at a given time step. This is in particular important for cases with many particles, as the amount of CPU time spent in the particle interpolation and communication part is getting large.
Chapter 2
Literature Review
In recent years, the interest to investigate spontaneous pattern formation in suspensions of motile microorganisms is increased. These organisms are evolved million of years ago. Weather they are in our stomach or affecting the global weather by photosynthesis in the sea. They form certain patterns which is definitely an important part of their life. It is crucial that we understand how and why these organisms, at the base of the whole food chain, behave. After all, they consist of the majority of the Earth's biomass and variation in their numbers could have a catastrophic consequences e.g. positive or negative feedback effects in global warming.
There is also the possibility of harnessing the power of micro-organisms. Some algae and bacteria produce alcohol as an unwanted byproduct but to us this is a valuable commodity not least for its use as a fuel. Plastics, fertilizers, waste treatment plants and solid fuels are other possible applications for algae and their products. This thesis is aim to explain the patterns observed in suspensions of swimming micro-organisms.
The first detailed observations of Bioconvection were carried out at the beginning of 19th century
By Wager (1911) but the subject was not taken up until the work of Platt (1961), who apparently coined the term 'bioconvection'.
