Control of Chaotic Flows and Fluid Forces (Previous Part 1)

LBM is a simple concept. After the collision step, the next step is streaming. When these two steps are completed, one step is completed at a time, and these steps are repeated to achieve the desired accuracy.

**Equilibrium Distribution Function**

The process of solving transport equations is similar. The primary factor that varies from problem to problem, we can use LBM to solve a variety of real-world problems. The Maxwellian EDF is written as

(1)

By ignoring the 3rd order and higher terms and applying Taylor expansion to the above equation, we get

(2)

Using substitutions and .

is the weighting factors and is the speed of sound.

(3)

The following discretized equilibrium distribution function.

(4)

**Discrete velocity sets**

Discrete velocity set is represented as , where is the dimensions, is the discrete velocities. Commonly used velocity sets are , and .

**1-Dimensional**

There are velocity vectors. The distribution function of central partical has velocity.

**2-Dimensional**

There are velocity vectors. Again the distribution function of central partical has velocity.

velocity discrete set Perumal and Dass.

Let

(5)

**3-Dimensional**

There are velocity vectors. Again the distribution function of central partical has velocity.

velocity discrete set Mei .,

(6)

**The Chapman-Enskog Expansion**

Chapman and Enskog, both mathematical physicists, revealed the same methods for obtaining macroscopic equations using the Boltzmann equation in 1917. Chapman eventually discovered a new technique called Chapman-Enskog expansion by combining the two ways in his book.

The equilibrium distribution function for , the discrete velocities and weight functions.

(7)

(8)

(9)

(10)

### Control of Chaotic Flows and Fluid Forces Part 1

For Part 4 Control of Chaotic Flows and Fluid Part 4

Read Control of Chaotic Flows Part 5