Control of Chaotic Flows and Fluid Forces (Previous Part 3)

Using these results in above equation we get the mass conservation

(1)

Now by combining Eq. (??) and (??), we get

(2)

As

So Eq. (2) becomes

(3)

For D2Q9, we have the gas dynamical relation

Now by replacing by in above equation. We get

(4)

Let

(5)

(6)

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We use the relation for the tensor

So from Eq. (5) we have

##### Consider

(7)

Let

So Eq. (7) becomes

(8)

Combining Eq. (6) and (8), we get

Here \textit{E} represent the error term, we neglect this to get quadratic Navier Stokes equation.

(9)

##### Consider the Navier stokes equation

Comparing above equation with Eq. (9), we get

Replacing with in Eq. (9) we get

Above equation can also be represented as

We concluded that in macroscopic limit LBE Recovers Navier-Stokes equations

**Boundary conditions in LBM**

PDE solutions are unable to be determined uniquely without using proper initial and BCs. LBM does not allow for the application of BCs directly. It’s necessary to convert the BCs into streaming-in distribution functions. In LBM BCs Kruger ., there are two types of groups: lattice links and lattice nodes, referred to as link-wise and wet node, respectively.

**Convective boundary conditions**

The convective BC’s are the linear combination of the function values, and its derivative at the computational domain boundary is a mixed form of BCs from some problems this boundary is also known as the impedance boundary.

The following is a mathematical representation of convective BCs.

(10)

Where is the uniform and , are in-flow velocity, such type of BC is used in the computational domain outlet, and the velocity is chosen in such a way that one can maintain general conservation.

**Lift coefficient**

The lift coefficient is the proportion of lift produced by a lifting body to the fluid density around it.

(11)

where, is the lift force of body.

**Drag coefficient**

The drag coefficient is an important tool for determining an object’s aerodynamic efficiency, regardless of its size or shape. The flow is streamline when the drag coefficient is low.

(12)

where, is the drag force of body and is the density of fluid.

Control of Chaotic Flows and Fluid Part 1