Control of Chaotic Flows and Fluid Forces

Control of Chaotic Flows and Fluid Forces (Part 1)

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Control of Chaotic Flows and Fluid Forces

Introduction

Simulation of incompressible flows in bio medical flows, image processing, multiphase flows, blood vessel flows, porous media flows, chemical reactions, bridges, rise towers and chimneys. These are just a few examples of the many engineering applications of fluid flowing through bluff bodies and reducing fluid forces. Due to the existence of non-linear components, studying fluid flow issues using analytical results of fluid dynamics equations is difficult. The LBM provides an advantage over standard computational methodologies because of its ease of implementation for complicated geometries.

Guillaume and LaRue experience different drag forces for two structurs experimentally. The flow characteristics are considerably change while changing the spacing between the two transversely arranged structures Sumner et al.

Kolar et al., studied through an experiment the wake pattern behaviour near downstream of the two structures arranged side-by-side. And observed mostly the anti-phase mode at g^* = 2, Re = 23 and 100.

Control of Chaotic Flows and Fluid Forces Around Two Offset Cylinders in Presence of Control Plate

Zdravkovich systematically studied the influence of gap spacing between the two structures arranged in various arrangements in an open-jet wind tunnel’ Sayers tested the lift and drag coefficients on a single cylinder in a group of four equally sized cylinders at spacing ratios ranging from 1.1 to 5.0 and Reynolds number (Re = 3.0\times 10^{4}). Norberg examine the flow and pressure over rectangular cylinders in various directions experimentally. The authar examined the effect of aspect ratio on static pressure distribution and calculated it. The authar discovered relevant wake frequencies for struhal number (St) and several wake frequencies for Re ranges from 2 to 3 using hot wire calculations. For some calculations, the authar used the fourth-order Runge-Kutta method.

Please also read Reflection of Waves in Micropolar Cubic Medium

In an experimental study with high Re, Prasad et al., investigated the flow past a circular cylinder. A small flat plate was positioned upstream. A flat plate was placed parallel to the cylinder. A 38\% reduction in drag was observed. It was also discovered that a small spacing between the plate and the cylinder was more effective. With a lower spacing ratio, the maximum drag reduction was achieved.
Rathakrishnan conducted an experiment to see what effect a splitter plate has on a bluff body. The discovery suggests that there may be a method for reducing drag and modifying vortex shedding characteristics for bluff bodies with detached splitter plates other than near wake manipulation. For centre line positioning, an upstream splitter plate was more effective at reducing drag. With a zero pitch angle, the splitter plate performs best.

A bluff structure is the most common design in many practical applications. In the flow around a bluff body, reattachment, separation and vortex shedding are common. The research on bluff body flow is important in both engineering and science because of the rapid development of computer technology and the effectiveness of mathematical methods. Depending on the aspect ratio, the fluid-dynamic properties of a cylinder can vary dramatically Ozono et al.

Please also read Reflection of Waves in Micropolar Cubic Medium (Part 2)

Construction, buildings, heat exchangers, bridges, stacks, chimneys and other fields have examples at relatively high Reynolds numbers (Re). Electronic device cooling, computer equipment cooling, fibre cooling and micro-electro-mechanical systems are all examples of low Re applications (MEMS). A rectangular and circular cylinder’s adjacent wakes are thought to be topologically identical. The areas of separation for a circular cylinder, on the other hand, are determined by the approaching velocity, whereas the partition sites for a rectangular cylinder are fixed at the front surface corners. These structures are known to interact with fluids like water and to be subjected to flow-induced forces that can cause collapse due to local and global instability Xie et al.

Islam et al., investigated the flow through a 2-D square cylinder channel with a detached flat plate numerically. For different Re and spacing ratios, they represented the wake structure mechanism, Vortex shedding frequency and fluid forces acting on the cylinder. They discovered that the Re varied depending on the flow velocity and cylinder diameter, ranging from 75 to 200. They used the LBM to examine the vortex generation, lift coefficient spectra, force reduction, and lift and drag coefficient time-trace analysis. Vortex shedding behind the main cylinder, according to their observations, is caused by wake development and is influenced by Re and spacing ratios. They also notice that as Re is changed, the amplitude of C_L and C_D increases, as they are directly proportional.

Control of Chaotic Flows and Fluid Forces Around Two Offset Cylinders in Presence of Control Plate

Inoue et al., investigated sound generation in a 2-D uniform flow past a tandem square cylinder at Re = 150. They experimented with different gap spacing between the cylinders using sound generation techniques. They found that sound production was impossible to hear at small gap spacing and increased significantly when the sound magnitude appeared at large gap spacing.

Lattice Boltzmann Method

Fluid dynamics studies the effect of all pressures on a moving fluid, including external pressures. Over the last few decades, fluid dynamics research has grown in importance, both mathematically and in terms of application. Fluid dynamics explains atmospheric circulation, ocean currents and even blood circulation. Also include jet and rocket engines, windmills, oil pipelines, pollution control equipment.

Fluid flow problems require solving the conservation equations for mass, momentum, and energy. Because the presence of a nonlinear component makes solving fluid dynamics equations analytically difficult, numerical approaches to solving real-world problems or complicated fluid flow problems have gained popularity in recent years. The computer has recently had a significant impact on fluid flow problem solving. In many cases, the NS Eqs can now be numerically solved across the flow field, and progress is being made even in turbulent flows. The velocity, pressure, temperature and density of a flowing fluid are all interrelated in these equations.

MD is simple to use and can handle both complex geometries and the addition of additional ingredients with ease. MD simulations of large-scale systems are computationally demanding, requiring significant computational resources and data storage.

Collision step

Molecules interact during collisions, and various models predict these interactions. Each linkage has a per-collision distribution function value (f_i^{in}), when the collision phase begins. It will be converted to an outgoing distribution function (f_i^{out}) after the collision.

Streaming or propagation step

All outgoing distribution functions from collision are transferred to neighbouring lattices in the direction of their velocities during streaming, and they become distribution functions at their new lattices.
This step do not change distribution function values.

LBM is a powerful approach in the field of hydrodynamics Flekkoy and Hermann. LBM is a versatile tool for modeling fluid flows, from micro-scale to large-scale Reynolds numbers. Its emergence from the LGA model spurred significant research interest. LGA and LBM simulate the flow of gases and fluids by irritating the basic behaviour of gases, which causes them to move forward and scatter as they collide. In the physical theory of gas physics, the Lattice Boltzmann approach achieved a stronger theoretical basis.

LBM offers a simpler implementation than traditional numerical schemes, enabling effortless handling of complex boundaries.

The LBM differs from other traditional numerical schemes because of its kinetic nature. We can easily calculate the pressure of the LBM by using the equation of state. LBM employs a minimal velocity set in phase space and utilizes single-particle distribution functions instead of Boolean variables. LBM is an explicit method that can model flow with low Mach numbers. Boltzmann velocity-discrete equation with relaxation type collision operator in its normal form. LBM is based on a paradigm shift, in contrast to other traditional computational methods.

LGA

Hardy, Pomeau, and de Pazzis pioneered LGA or LGCA in the 1970s, revolutionizing the field. LGA is a fluid flow simulator that evolved from cellular automata (CA).

HPP model

In 1973, Pomeau, Hardy, and de Pazzis introduced the HPP model. There are four different types of velocities, in square-shape model. Particles are unable to move in a diagonal direction.

FHP model

Frisch, Hasslacher, and Pomeau introduced the Hexagonal Grid Model in 1986, and it became known as the FHP model.

Boltzmann Transport Equation

(1)   \begin{equation*} f(r^{*} + \vec\zeta dt,\vec\zeta + F dt, t + dt) - f(r^{*},\vec\zeta, t) = 0.\end{equation*}

If there exists any collision in between particles in the interval dr^{*}d\vec\zeta, there will be a net difference in molecule number.

(2)   \begin{equation*} f(r^{*}+\vec\zeta dt,\vec \zeta+Fdt, t+dt)dr^{*}d\vec\zeta-f(r^{*}, \vec\zeta, t)dr^{*}d\vec\zeta = \Omega(f)dr^{*}d\vec\zeta dt.\end{equation*}

Dividing both side by dr^{*}d\vec\zeta dt and appiying limite dt\to\ 0.

(3)   \begin{equation*} \frac{df}{dt} = \Omega(f).\end{equation*}

Distribution function change with respect to time due to collison inside particles.

(4)   \begin{equation*} \frac{df}{dt}=\frac{\partial f}{\partial r^{*}}\frac{dr^{*}}{dt}+\frac{\partial f}{\partial \vec\zeta}\frac{d\vec\zeta}{dt}+\frac{\partial f}{\partial t}\frac{dt}{dt}.\end{equation*}


or

(5)   \begin{equation*} \frac{df}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial r^{*}}\frac{dr^{*}}{dt}+\frac{\partial f}{\partial \vec\zeta}\frac{d\vec\zeta}{dt}.\end{equation*}


Where \frac{\partial r^{*}}{\partial t}=\vec\zeta, represent velocity of single particle.

\frac{\partial \vec\zeta}{\partial t}=\vec{a}, represent acceleration of single partical.

(6)   \begin{equation*} \frac{df}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial r^{*}}\vec\zeta+\frac{\partial f}{\partial \vec\zeta}\vec{a},\end{equation*}

Substitute Newton’s second law \vec{a}=\frac{\vec{F}}{m}​ for \vec{a}.

(7)   \begin{equation*} \frac{df}{dt}=\frac{\partial f}{\partial t}+\vec\zeta \frac{\partial f}{\partial r^{*}}+\frac{\vec{F}}{m} \frac{\partial f}{\partial \vec\zeta}.\end{equation*}


Comparing Eq. 3 and Eq. 7:

(8)   \begin{equation*} \frac{\partial f}{\partial t}+\vec\zeta \frac{\partial f}{\partial r^{*}}+\frac{\vec{F}}{m} \frac{\partial f}{\partial \vec\zeta}= \Omega.\end{equation*}

Equation 8 is the fundamental Boltzmann transport equation, which describes the distribution function f(r^{*},\vec\zeta,t). In the absence of external forces, this equation simplifies to:

(9)   \begin{equation*} \frac{\partial f}{\partial t}+\vec\zeta.\nabla f=\Omega.\end{equation*}

Discrete Boltzmann BGK Equation

(10)   \begin{equation*} \frac{\partial f}{\partial t}+\vec\zeta.\nabla f=-\frac{1}{\tau}(f-f^{eq}). \end{equation*}

Control of Chaotic Flows and Fluid Forces

You can also read next part Control of Chaotic Flows and Fluid Forces Part 2

Control of Chaotic Flows and Fluid Forces

Control of Chaotic Flows and Fluid Forces

Control of Chaotic Flows and Fluid Forces

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