Moment Coefficient#

The Moment Coefficient, $C_M$, is a dimensionless parameter that provides a generalized representation of the resultant moment experienced by an object within a fluid flow. It offers a means to evaluate the cumulative effect of pressure coefficients, $c_p$ across different regions of an object’s surface and how these pressures translate into aerodynamic moment forces.

$C_M$ is a fundamental tool for torsional effects for the design and analysis of aerodynamic components.

Definition#

Similarly to the force coefficient, this coefficient is defined as a resulting moment coefficient of a body.

It is defined as a sum of the resulting moment for each triangle of each surface of the body:

\[\vec{C_{M}} = \frac{\sum \vec{M_{res}}}{q V_{nom}} = \frac{\sum \vec{r_o} \times \vec{f_{i}}}{q V_{nom}}\]

\[\vec{f_i} = c_{pi} q \vec{A_i}\]

\[\vec{C_{M}} = \frac{\sum (\vec{r_o} \times \vec{A_i}) c_{pi}}{V_{nom}}\]

The position vector $r_o$ is defined for each triangle, from a common arbitrary points $o$. One can also define it for each axis direction:

\[C_{M_x} = \frac{\sum M_{res_x}}{q V_{nom}} = \frac{\sum (r_{oy} A_{iz} - r_{oz} A_{iy}) c_{pi}}{V_{nom}}\]

\[C_{M_y} = \frac{\sum M_{res_y}}{q V_{nom}} = \frac{\sum (r_{oz} A_{ix} - r_{ox} A_{iz}) c_{pi}}{V_{nom}}\]

\[C_{M_z} = \frac{\sum M_{res_z}}{q V_{nom}} = \frac{\sum (r_{ox} A_{iy} - r_{oy} A_{ix}) c_{pi}}{V_{nom}}\]

We define the nominal volume ($V_{nom}$) as a user input. This is done to let the user define how they want to calculate its value. For example, considering a rectangular tall building:

The nominal volume could be calculated with:

\[V_{nom} = b h l\]

Use Case#

A common application of the moment coefficient requires sectioning the body in different sub-bodies. To do so, the same logic applied to the force coefficient is used to determine the respective sub-body of each of the body’s triangles. If its center lies inside the sub-body volume, then it belongs to it.

The result is a sectionated body in different sub-bodies for each interval. When sectioning the body, the respective nominal volume should be the same as the sub-body nominal volume.

Note

Check out the definitions section for more information about surface, body and sub-body definitions.

Like the other coefficients, we can apply statistical analysis to the moment coefficient.

By definition, the moment coefficient is a property of a body.

It is used for primary and secondary structures design, such as canopies. It can also be used for evaluating the resultant wind torsional effect over a building or the building paviments. It can be seen as the resulting torsion effect of the wind induced stress over a body.

Artifacts#

In order to use the moment coefficient module, the user has to provide a set of artifacts:

A lnas file: It contains the information about the mesh.
HDF time series: It contains the pressure coefficient signals indexed by each of the mesh triangles.
Parameters file: It contains the coordinate for the arbitrary point when evaluating the force moment, as well as other configs parameters.

Which outputs the following data:

Dimensionless time series: moment coefficient time series for each body.
Statistical results: maximum, minimum, RMS and average values for the moment coefficient time series, for each body.
VTK File: contains the statistical values inside the original mesh (VTK).

An illustration of the moment coefficient module pipeline can be seen below:

Usage#

An example of the parameters file required for calculating the moment coefficient can be seen below:

bodies:
  marquise:
    surfaces: [list_of_surfaces_of_marquise]
  lanternim:
    surfaces: [list_of_surfaces_of_lanternim]
  building:
    surfaces: [building]
moment_coefficient:
  measurement_1:
    bodies:
      - name: marquise
        lever_origin: [0, 10, 10]
    directions: ["x", "y", "z"]
    # Nominal volume to use for calculations
    nominal_volume: 10
    statistics:
      - stats: "mean"
      - stats: "rms"
      - stats: "skewness"
      - stats: "kurtosis"
      - stats: "min"
        params:
          method_type: "Absolute"
      - stats: "max"
        params:
          method_type: "Absolute"
    # Apply transformations before indexing the regions
    transformation:
      translation: [0, 0, 0]
      rotation: [0, 0, 0]
      fixed_point: [0, 0, 0]
  measurement_2:
    bodies:
      - name: building
        lever_origin: [0, 10, 10]
        sub_bodies: # Optional, default is the whole body
          z_intervals: [0, 10, 20]
    directions: ["x", "y", "z"]
    # Nominal volume to use for calculations
    nominal_volume: 10
    statistics:
      - stats: "mean"
      - stats: "rms"
      - stats: "skewness"
      - stats: "kurtosis"
      - stats: "min"
        params:
          method_type: "Absolute"
      - stats: "max"
        params:
          method_type: "Absolute"
    # Apply transformations before indexing the regions
    transformation:
      translation: [0, 0, 0]
      rotation: [0, 0, 0]
      fixed_point: [0, 0, 0]

To invoke and run the calculation, the following command can be used:

uv run python -m cfdmod.use_cases.pressure.Cm \
   --output {OUTPUT_PATH} \
   --cp     {CP_SERIES_PATH} \
   --mesh   {LNAS_PATH} \
   --config {CONFIG_PATH}

Or it can be generated together with the pressure data conversion:

uv run python -m cfdmod.use_cases.pressure \
   --output {OUTPUT_PATH} \
   --cp     {CP_SERIES_PATH} \
   --mesh   {LNAS_PATH} \
   --config {CONFIG_PATH} \
   --Cm

Another way to run the moment coefficient calculation, is through the notebook

Data format#

Note

The rule for determining the region_idx is based on the region index and the body name. Input mesh can have multiple bodies, and each of them can be applied a specific zoning/region rule. Because of that, region_idx has to be composed by the zoning region index joined by “-” and the body name. This also guarantee that even if different bodies lie on the same region, the interpreted region for each of them will be different

Note

For more information about the normalized time scale ($t^*$), check the Normalization section

$C_{mx}(t)$#
time_idx/region_idx	Normalized time ($t^*$)	0-Body1	1-Body1	0-Body2
0	10000	1.25	1.15	-1.1
1	11000	1.5	0.9	-1.15

$C_{my}(t)$#
time_idx/region_idx	Normalized time ($t^*$)	0-Body1	1-Body1	0-Body2
0	10000	1.25	1.15	-1.1
1	11000	1.5	0.9	-1.15

$C_{mz}(t)$#
time_idx/region_idx	Normalized time ($t^*$)	0-Body1	1-Body1	0-Body2
0	10000	1.25	1.15	-1.1
1	11000	1.5	0.9	-1.15

$C_{mx} (stats)$#
region_idx	max	min	mean	std	skewness	kurtosis
0-Body1	1.25	0.9	1.1	0.2	0.1	0.15
1-Body1	1.15	0.95	1.13	0.19	0.11	0.13

$C_{my} (stats)$#
region_idx	max	min	mean	std	skewness	kurtosis
0-Body1	1.25	0.9	1.1	0.2	0.1	0.15
1-Body1	1.15	0.95	1.13	0.19	0.11	0.13

$C_{mz} (stats)$#
region_idx	max	min	mean	std	skewness	kurtosis
0-Body1	1.25	0.9	1.1	0.2	0.1	0.15
1-Body1	1.15	0.95	1.13	0.19	0.11	0.13

$Regions(indexing)$#
region_idx	point_idx
0-Body1	0
1-Body1	1

$Regions(definition)$#
region_idx	x_min	x_max	y_min	y_max	z_min	z_max	Lx	Ly	Lz
0-Body1	0	100	0	50	0	20	0.5	0.8	0.1
1-Body1	100	200	0	50	0	20	0.5	0.8	0.1

Moment Coefficient

Contents

Moment Coefficient#

Definition#

Use Case#

Artifacts#

Usage#

Data format#