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Definition of Initial Conditions

In this tutorial, we cover the definition of initial conditions. They can be defined either in lattice units or in physical units. Lattice units can be confusing, especially to people just starting in this field. The relationships between lattice and physical units can be found in this paper:

In this tutorial, a two-dimensional plain channel is set up. Not only are the boundaries specified to obtain a defined pressure drop over the channel length, but also are the initial conditions set in a consistent manner.

To run this test case use the generic channel test case and set in
`seeder.lua`

:

```
case2d = true
usePeriodic = false
qValues = false
useObstacle = false
fixHeight = true
useRefine = false
```

For the viscous, laminar two-dimensional plain channel flow, an analytical solution of the incompressible Navier-Stokes equation can be derived. From the analytical solution the pressure drop, the velocity profile and the shear stress distribution can be computed.

Before starting, we need to define the flow regime and physical reference values.

```
-- Density of the fluid [kg/m^3]
rho0_phy = 1.0
-- Reynolds number of the flow
Re = 60
-- Inflow velocity [m/s]
vel_phy = 1.0
-- Kinematic viscosity of the fluid calculated from Re [m^2/s]
nu_phy = vel_phy * height_phy / Re
```

For the Lattice_Boltzmann simulation, basic simulation parameters such as a lattice velocity, the timestep and the lattice viscosity need to be specified.

```
if (scaling == 'acoustic') then
-- Lattice Mach number
Ma_lat = 0.05
-- Lattice velocity
vel_lat = Ma_lat * cs_lat
-- Physical timestep computed from physical and lattice velocity
dt = vel_lat / vel_phy*dx
-- Lattice viscosity
nu_lat = nu_phy*dt /dx /dx
-- Relaxation parameter
omega = 1.0/(3.0*nu_lat + 0.5)
else
-- Diffusive scaling
-- Relaxation parameter
omega = 1.7
-- Lattice viscosity
nu_lat = ( 1.0/omega - 0.5 ) / 3.0
-- Physical timestep computed from physical and lattice velocity
dt = nu_lat/nu_phy*dx*dx
-- Lattice velocity
vel_lat = vel_phy*dt/dx
-- Lattice Mach number
Ma_lat = vel_lat * cs_lat
end
--------------------------------------------------------------------------------
--! [Reference LB values]
-- Square of lattice speed of sound
cs2 = 1.0/3.0
-- Lattice density
rho0_lat = 1.0
-- Zero lattice density
rho0_lat0 = 0.0
--! [Reference LB values]
```

Depending on the model used, the reference pressure differs. For the
incompressible model, the reference pressure is 0, while for the compressible
model the reference pressure is `rho0*cs2`

```
--! [Reference pressure dependent on fluid kind]
if physicsModel == 'fluid_incompressible' then
p0 = 0.0
else
p0 = rho0_lat*cs2*dx*dx/dt/dt
end
--! [Reference pressure dependent on fluid kind]
```

From the solution of the Navier-Stokes equation, the following relations for the velocity distribution across the channel height can be obtained

```
--! [Velocity function]
function velX(x,y,z)
velX_phy = vel_phy * ( 1.0 - ( 2.0*y/height_phy )^2 )
return velX_phy
end
--! [Velocity function]
```

Similarly for the pressure drop along the channel length

```
--! [Pressure function]
function pressureRef(x,y,z)
press_drop = vel_phy*8.*nu_phy*rho0_phy/height_phy^2*length
return p0 + press_drop*0.5 - press_drop/length*x
end
--! [Pressure function]
```

and the shear stress across the channel height

```
--! [Shear stress function]
function Sxy(x,y,z)
tauxy= -nu_phy*rho0_phy*8./height_phy^2*vel_phy*y
S_xy = tauxy/nu_phy/rho0_phy
return S_xy
end
--! [Shear stress function]
```

Now the physics table establishes the connection between the lattice reference
values and the physical values and gives *Musubi* means of transferring between
these two unit systems. See mus_physics_module for more information.

```
physics = {
dt = dt,
rho0 = rho0_phy
}
```

For the Lattice_Boltzmann algorithm, a reference density and the kinematic viscosity (for compressible also the bulk viscosity) need to be defined. See mus_fluid_module for more information.

```
fluid = {
kinematic_viscosity = nu_phy,
bulk_viscosity = bulk_visc
}
```

Now the initial conditions for each element in the simulation domain is defined by setting each physical quantity and connecting it to a lua function, which we defined above.

```
--! [Initial conditions]
initial_condition = { pressure = ic_pressure,
velocityX = ic_velX,
velocityY = 0.0,
velocityZ = 0.0,
Sxx = ic_Sxx,
Syy = ic_Syy,
Sxy = ic_Sxy
}
--! [Initial conditions]
```

The whole code of musubi.lua is shown in the chapter_02.

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