Name: Putika Ashfar Khoiri (28J16118)

Email: putika@civil.osaka-u.ac.jp

Computational of Fluid Dynamics Assignment 2

The Chimera Grid Concept and Application

The chimera grid scheme and similar scheme are using overset grids to resolve complex geometry of

flow features, are generally classified into complex structure grid category. Overset grids allow

structure grids to be used in good quality, such as orthogonally, smoothness and with ease to control

grid spacing in viscous layer. Furthermore, overset grids can be employed as a solution adaptation

procedure. However, the non-conservative interpolations to update variables in the overlapped

region, without strict satisfaction of the governing equation, can give rise to spurious solutions,

especially through regions of sharp gradients.

The chimera method is an outgrowth of the attempt to generalize a powerful solution approach (the

structured and body-conforming grid method) to complex situation. It has two principal elements; (1)

Decomposition of a chosen computation domain into sub-domains, and (2) communication of

solutions data among these sub-domains through an interpolation procedure.

Fig 1. Construct of Chimera Grid (Zheng, 2003)

Summary of the grid-related portion of the chimera method (see fig 1); (1) The overlapping of two

grids, respectively designated as major and minor grids with the latter being told as conforming

component, (2) The hole creation boundary specified as a level surface in the minor grid and the

fringe-point boundary in the major grid, (3) The outer boundary in the minor grid. Boundary condition

in each grid must be made available, since the separate grids are to be solved independently.

Boundary conditions on the interpolated hole boundary and interpolated outer boundary are supplied

from the grid in which the boundaries are contained.

The gap region is created by arbitrary overset grids is inevitably of irregular shape. Unstructured grid

can lend its strength to handle this irregularity shaped space (see fig 2)

Fig 2. Arbitrary grid overlapping by non-structured grid in Chimera grid (Zheng, 2003)

The chimera grid method makes use of body fitted grids in which are an essential asset known to give

viscous solutions accurately and economically. In the gridding process the complex geometrical model

is divided up into sub-domains, which in general are associated with components of a configuration.

Each sub-domain is gridded independently and required to have overlapped region between the sub-

domains. The computational volumetric domain has been decomposed into many sub-domains filled

with structured grids. The sub-domains overlapped with each other. Hole boundaries and outer

boundaries are the two ways through which information is communicated from grid to another.

For example, the chimera grid method is used for integrated space shuttle geometry (complex

geometry) (see fig. 3)

Fig 3. NASA shuttle space appearance model construct in chimera grid (overset grid)

A surface element consists of several edges. An edge is referred to as an isolated edge, if it is belonging

to only one element of the surface. For general surface there are many edges which belong to two

neighbouring elements and clearly these edges are not isolated edges. Furthermore, for a closed

surface, there are no isolated edges. The corresponding isolated edges are bound the openings of the

surface. Therefore, in order to close the surface, it is natural to establish the isolated edge first, then

to identify facets which fill the openings.

Additional reference (is attached to email)

(1) Y. Zheng, M.-S Liou. A novel approach of three-dimensional hybrid grid methodology: Part 1.

Grid Generation. Computer Methods in Applied Mechanics and Engineering.192

(2003).4147-4171

ITTC (International Towing Tank Conference) – Practical Guides for Ship CFD

Application (2014)

The guidelines summarize the CFD process into three steps.

1. Pre-processing -> The definition of the geometry, computational domain and grids

2. Computation -> Analyse the data

3. Post-processing -> Visualisation, analysis, verification and the validation of the results.

The process must have to begin with calculate

1. The Reynold Number (Re)

In model scale, the Re is 106 and 109 in the full scale.

𝑅𝑒 = 𝜌𝑉𝐿

𝜇

Where, V = ship speeds, L = length between perpendicular, ρ = density, and µ =

viscosity.

2. Froude Number (Fr)

Its typically 0.1 for free surface flow simulations, and rarely above 1

𝐹𝑟 =𝑉

𝑔𝐿

3. Mach Number (Ma)

It recommends to have the value of Ma ≤ 0.3. a = speed of sounds.

𝑀𝑎 =𝑉

𝑎

1. Pre-processing

1.1 Geometry

Indicative tolerance for the geometry is ± 10-5 L, with L = ship length for the hull for Re

around 106. This tolerance may need to be reduced when smaller scale appendages,

such as rudder or propeller are to be included in the flow computations or when it

requires smaller grid cells than the indicate CAD tolerance.

1.2 Computational domain and boundary conditions

- The computational domain should include inflow and outflow surfaces. A Dirichlet

condition (i.e. a known velocity field) should be imposed on the inflow boundary.

- Outflow face use different boundary condition. The option is use Neumann

condition (zero gradient) for velocity and pressure grid.

1.3 Grid

1.3.1 Structured grid

It shows regular connectivity in 2D quadrilateral and hexahedra in 3D. Mapping

the relation between the grid in the physical space (x, y, z) and the

computational space (i, j, k)

1.3.2 Unstructured grid

It shows irregular connectivity and the cells have several possible shapes

(tetrahedral, hexahedra, and other polyhedral). It doesn’t show a defined

topology and can be generated much more easily.

1.3.3 Hybrid grid and other techniques

It can use both structured and unstructured grids in different parts of the

domain. Other methods for this interpolation scheme may be defined by

interface where the interpolation method is defined across grid boundary

surfaces. For other methods, the interpolation scheme is defined by overlaps,

where the interpolation is defined across local grid volume.

1.3.4 Cell types

Quadrilateral (2D, 4 sided) and hexahedral (3D, 6 sided) cells are support by

almost all CFD codes. The usage is depending on:

o What cell types and the solver support

o The objective of the computations

o The hardware resources (are computational resources are available to run

case involving large grids?).

For complex geometries which high quality structured grid is difficult to

generate, it preferable using a hybrid unstructured grid or overlapping grids.

1.3.5 Grid resolutions

It should be decided based on the chosen turbulence model and on the

temporal or spatial scale of the flow features of interest. Always use orthogonal

grids to solve free surface flow simulations if it is possible.

1.3.6 Near-wall region

We have to define:

Friction velocity (u*)

𝑢∗ = √𝜏𝑤

𝜌

Non-dimensional wall velocity (u+)

𝑢+ =𝑢

𝑢∗

Non-dimensional wall distance (y+)

𝑦+ =𝜌𝑢∗

𝜇𝑦

Where 𝜏𝑤= skin friction on the wall, u = local stream velocity component, y= wall

normal coordinate, Cf= friction coefficient

𝑦 =𝑦+𝐿

𝑅𝑒𝐿√

𝐶𝑓

2

Recommendation value for grid design at the wall

First point Expansion

ratio Points within

boundary layer

Near wall y+ ≤ 1 1.2 20

Wall function 30 < y+ < 100 1.2 15

The volume of Cf for different conditions as follows:

Laminar flow

Transitional flow turbulence flow in

smooth surface turbulence flow in rough

surface

𝐶𝑓 =1.328

√𝑅𝑒𝐿

𝐶𝑓 =0.455

[ln(𝑅𝑒𝐿)]2.58−

1700

𝑅𝑒𝐿 𝐶𝑓 =

0.455

[ln(𝑅𝑒𝐿)]2.58

𝐶𝑓

=1

[1.89 − 1.62ln (𝜀𝐿)]

2.5

1.3.7 Grid Quality

Typically, the 3x3 determinant for structured grids should be greater than 0.3.

However, it may be necessary to have a few small cells where 3x3 determinant

is no better than 0.15.

2. Computation

2.1 Turbulence model

- RANS -> This approach assume that the turbulence is modelled with RANS where the

instaneous velocity is split into the sum of its statically average and turbulence

fluctuation which is modelled by the turbulence model.

- DNS -> Direct numerical simulation (DNS) solve the Navier-Stokes equations to the

resolution of the smallest turbulent scales but needs grids and time resolution which

are not yet achieve for high Re.

- LES -> Large Eddy Simulation (LES) is currently used for high demanding flow where

the transient nature of the turbulence needs to be resolved to smaller scales.

- DES -> Detached Eddy Simulation (DES) is a hybrid method that reduces the required

computational effort by solving the (unsteady) RANS equation in the near-wall

region and applying LES in the result of the domain.

2.2 Near-wall modelling

It requires grid nodes highly refined towards walls (y+ =1) in range 30 < y+ < 100. Two

approach are available:

1. Use of near wall turbulence models.

2. Use of wall function to bridge the solutions in the fully turbulent.

2.3 Surface roughness

1. If the physical roughness height (ks) is less than the thickness of the laminar sub-

layer, then the flow in the boundary layer is not affected.

2. If 𝑘𝑠 =2𝑣

𝑢∗, the effect of surface roughness is negligible.

2.4 Numerical schemes

1. First-order upwind (FOU) schemes

It introduces large numerical diffusion and very stable

2. Second-order upwind (SOU) schemes

It uses flux limiter to suppress unphysical oscillations in the solutions. It also

recommends for all convection-diffusion transport equations.

2.5 Time step

Courant-Friedrichs-Lewy (CFL) is the condition to resolve the flow features interest,

where Cmax ≥ 1 |𝑢|∆𝑡

∆𝑥< 𝐶𝑚𝑎𝑥,

for ∆𝑡 ≤ 0.01𝐿

𝑈 (turbulence model) and ∆𝑡 ≤ 0.001

𝐿

𝑈 ( Re stress turbulence model)

2.6 Convergence

Convergence must be evaluated at each time-step. The residual (L) for a discretised

equation is defined as (L1-Norm). (L2-Norm) is used to define the residual

3. Post processing

3.1 Visualisation

A number of post processing plots should be used as a minimum sub-set of information

to ensure that the correct settings have been used for each computations

3.2 Verification and validation

Refers to the ITTC procedure 7.6 – 03 – 01 – 01 for “methodology and procedures for

estimating the uncertainty in a simulation result”.