Non-Dimensional Inputs

Non-Dimensional Inputs#

Note

Thanks to Units, manual dimensionalization of inputs is rarely needed. In most cases, you can simply provide the dimensional value and the unit system will be inferred automatically.

Most input variables in the Flow360 Python API accept dimensional values and non-dimensionalization is automatically performed during the preprocessing. However, there are still input variables defined using C-style string expression that only accept non-dimensional values, such as AngleExpression, HeatFlux, etc. A string expression cannot carry units and none are inferred for it, so the number you write is already the non-dimensional one. In this section, we demonstrate how to compute the non-dimensional value for these input variables.

Caution

Several of the fields in the table below accept either a dimensional value or a string expression, and only the string form is non-dimensional. Freestream(velocity=(29, 0, 0) * u.m / u.s) is 29 m/s, while Freestream(velocity=("29", "0", "0")) is 29 times \(U_\text{scale}\) — for an AerospaceCondition that is Mach 29. Only the string form can vary in space or time.

Theoretically, the reference values for non-dimensionalization can be arbitrary as long as the resulting equations are identical to the original ones, but in practice, the reference values are usually selected based on some typical parameters of problems and flow characteristics to avoid confusion. The following table shows the usage of reference values to obtain the non-dimensional input variables:

Usage of reference values for non-dimensional inputs in Flow360#

Property

Ref. value for nondim.

Usage in Flow360 Python API

Density

\(\rho_\infty\)

NavierStokesInitialCondition.rho

Pressure

\(\rho_\infty U_\text{scale}^2\) where \(U_\text{scale}\) is the reference velocity scaling

NavierStokesInitialCondition.p, TotalPressure.value

Velocity

\(U_\text{scale}\) (reference velocity scaling)

Wall.velocity, Freestream.velocity, NavierStokesInitialCondition.u, NavierStokesInitialCondition.v, NavierStokesInitialCondition.w

Angular speed

\(U_\text{scale}/L_\text{gridUnit}\) where \(U_\text{scale}\) is the reference velocity scaling

AngleExpression.value

Temperature

\(T_\infty\)

Temperature.value, Inflow.total_temperature, HeatEquationInitialCondition.temperature

Volumetric heat source

\(\frac{\rho_{\infty} U_\text{scale}^3}{L_{gridUnit}}\) where \(U_\text{scale}\) is the reference velocity scaling

Solid.volumetric_heat_source, PorousMedium.volumetric_heat_source

Heat flux

\(\rho_{\infty} U_\text{scale}^3\) where \(U_\text{scale}\) is the reference velocity scaling

HeatFlux.value

Length

\(L_{gridUnit}\) (the mesh unit)

the coordinates x, y, z inside the spatially varying string expressions (not AngleExpression, which admits only t)

Time

\(L_{gridUnit}/U_\text{scale}\) where \(U_\text{scale}\) is the reference velocity scaling

the variable t inside any string expression (AngleExpression also accepts t_seconds for seconds)

Note

  1. All reference values can be accessed via the Python API as shown in the Reference Quantities Table.

  2. Many input variables are non-dimensionalized with the reference velocity scaling \(U_\text{scale}\) (see Introduction). In the examples below, we assume \(U_\text{scale}\) has been computed as shown there.

Example: Convert RPM to non-dimensional rotating speed omega#

The RPM determines the angular speed, from it we can calculate the non-dimensional omega used in defining the AngleExpression.

(1)#\[\text{omega} = \frac{\omega}{U_\text{scale}/L_\text{gridUnit}}.\]

where \(U_\text{scale}\) is the reference velocity scaling.

Assume the RPM = 800, the non-dimensional omega_radians value then becomes:

1omega = 800 * fl.u.rpm / (U_scale / project.length_unit)
2omega_radians = omega.to(fl.u.rad).value

Example: Compute non-dimensional volumetric_heat_source#

For conjugate heat transfer simulations, the non-dimensional heat sources (volumetric_heat_source) of a solid zone are found from the dimensional \(Q_s\) as:

(2)#\[\text{volumetricHeatSource}= \frac{Q_s}{Q_{ref}} = \frac{Q_s L_{gridUnit}}{\rho_{\infty} U_\text{scale}^3}\]

where \(U_\text{scale}\) is the reference velocity scaling.

Assume the \(Q_s=10\;\text{W}/\text{m}^3\), the non-dimensional volumetric_heat_source value can be obtained as:

1density = operating_condition.thermal_state.density
2Q_s = 10 * fl.u.W / fl.u.m**3
3volumetric_heat_source = Q_s * project.length_unit / (density * U_scale ** 3).value

Example: Compute non-dimensional HeatFlux#

The non-dimensional heat flux for a wall boundary condition can be calculated by dividing the dimensional heat flux \(q\) by the reference value:

Note

Sign convention. A positive heat flux removes energy from the fluid (heat flows from the fluid into the wall, cooling the fluid), while a negative heat flux adds energy to the fluid (heat flows from the wall into the fluid, heating it). See the wall boundary condition page.

(3)#\[\text{heatFlux} = \frac{q}{q_{ref}} = \frac{q}{\rho_{\infty} U_\text{scale}^3}\]

where \(U_\text{scale}\) is the reference velocity scaling.

Assume the \(q=10\;\text{W}/\text{m}^2\), the non-dimensional heat_flux value can be obtained as:

1density = operating_condition.thermal_state.density
2q = 10 * fl.u.W / fl.u.m**2
3heat_flux = q / (density * U_scale ** 3).value

Example: Compute a non-dimensional spatially varying boundary velocity#

Wall.velocity and Freestream.velocity accept either a dimensional value or a per-component C-style string expression, and only the string form can vary in space or time. The string form is non-dimensional: each component is divided by \(U_\text{scale}\), and the coordinates x, y, z inside it are divided by \(L_{gridUnit}\).

Consider an atmospheric boundary layer profile, a log law with a friction velocity \(u_* = 1.2\;\text{m}/\text{s}\), a roughness length \(z_0 = 0.03\;\text{m}\) and von Karman constant \(\kappa = 0.41\), capped at \(29\;\text{m}/\text{s}\), on a mesh built in metres:

(4)#\[u(z) = \min\left(\frac{u_*}{\kappa}\ln\frac{z + z_0}{z_0},\; 29\;\text{m}/\text{s}\right)\]

Both the coefficient and the cap have to be divided by \(U_\text{scale}\), and \(z_0\) by \(L_{gridUnit}\):

 1u_star, z_0, kappa = 1.2 * fl.u.m / fl.u.s, 0.03 * fl.u.m, 0.41
 2coefficient = (u_star / kappa / U_scale).value
 3cap = (29 * fl.u.m / fl.u.s / U_scale).value
 4z_0_nondim = (z_0 / project.length_unit).value
 5
 6velocity = (
 7    f"min({coefficient} * log((max(0.0, z) + {z_0_nondim}) / {z_0_nondim}), {cap})",
 8    "0",
 9    "0",
10)

Caution

Writing the dimensional numbers straight into the string is a silent error, not a validation failure. With the cap left as 1.0 the profile tops out at one \(U_\text{scale}\) — for an AerospaceCondition that is Mach 1, roughly 340 m/s rather than the intended 29 m/s. The case still converges and the contours still look plausible.

Define the angle of attack alpha and sideslip angle beta#

According to Flow360’s definitions of the angle of attack \(\alpha\) and the sideslip angle \(\beta\), with respect to the grid coordinates, the following values of velocity components are imposed at a Freestream farfield boundary:

(5)#\[\begin{split}U_{\infty} &= \text{Mach} \cdot cos(\beta) \cdot cos(\alpha) \\ V_{\infty} &= - \text{Mach} \cdot sin(\beta) \\ W_{\infty} &= \text{Mach} \cdot cos(\beta) \cdot sin(\alpha)\end{split}\]

where, the velocity components are nondimensionalized by the reference velocity scaling \(U_\text{scale}\). The effects of these two angles are used to compute the forces in stability axes rather than body axes, \(CL\) and \(CD\), as follows:

(6)#\[\begin{split}C_{L} &= C_{Fz}\cdot cos(\alpha) - C_{Fx}\cdot sin(\alpha)\\ C_{D} &= C_{Fx} \cdot \cos(\alpha) \cos(\beta) - C_{Fy} \cdot \sin(\beta) + C_{Fz} \cdot \sin(\alpha) \cos(\beta)\end{split}\]

Angle of attack \(\alpha\) and sideslip angle \(\beta\) can be expressed by readjusting the above equations in the following way:

(7)#\[\begin{split}\alpha = tan^{-1} \left( \frac{W_\infty}{U_\infty} \right)\\ \beta = sin^{-1} \left( -\frac{V_\infty}{\text{Mach}} \right)\end{split}\]