8.1.2. Nondimensional Inputs

In Flow360, most variables are nondimensional. The nondimensionalization reduces the number of free parameters and helps to provide better understanding of the underlying physics. Some common dimensional variables used to define the inputs are shown in the table below:

\(L_{gridUnit}\) (SI unit = \(\text{m}\))

physical length represented by unit length in the given mesh file, e.g.

if your grid is in feet, \(L_{gridUnit}=1 \text{ feet}=0.3048 \text{ meter}\);

if your grid is in millimeters, \(L_{gridUnit}=1 \text{ millimeter}=0.001 \text{ meter}\).

\(C_\infty\) (SI unit = \(\text{m}/\text{s}\))

freestream speed of sound

\(\rho_\infty\) (SI unit = \(\text{kg}/\text{m}^3\))

density of freestream

\(\mu_\infty\) (SI unit = \(\text{N} \cdot \text{s}/\text{m}^2\))

dynamic viscosity of freestream

\(p_\infty\) (SI unit = \(\text{N}/\text{m}^2\))

static pressure of freestream

\(T_\infty\) (SI unit = \(K\))

temperature of freestream

\(U_\text{ref} \triangleq \text{MachRef}\times C_\infty\) (SI unit = \(\text{m}/\text{s}\))

reference velocity

\(k_\text{ref} \triangleq \frac{\rho_{\infty} C_{\infty}^3 L_{gridUnit}}{T_{\infty}}\) (SI unit = \(\text{kg}\cdot\text{m}/(\text{s}^3\cdot\text{K})\))

reference thermal conductivity

\(\dot{q}_\text{ref} \triangleq \frac{\rho_{\infty} C_{\infty}^3}{L_{gridUnit}}\) (SI unit = \(\text{kg}/(\text{s}^3\cdot\text{m})\))

reference volumetric heat source

A nondimensional variable is obtained by dividing its dimensional counterpart by an appropriately selected constant like Eq.(8.1.3).

(8.1.3)\[\text{nondimensional variable} = \frac{\text{dimensional variable}}{\text{reference value}}\]

Note

Any value presented here in symbolic format (for example \(A_\text{ref}\)) refers to a dimensional value, whereas any value written in text format (for example “geometry->refArea”) refers to a nondimensional value

Theoretically, the reference values for nondimensionalization 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 list shows some commonly used nondimensional variables in Flow360.json file:

Table 8.1.2 Reference values for nondimensional inputs in Flow360

Property	Ref. value for nondim.	Examples in Flow360.json
Length	\(L_\text{gridUnit}\)	geometry->momentLength, BETDisks->radius
Area	\(L_\text{gridUnit}^2\)	geometry->refArea
Dynamic viscosity	\(\rho_\infty C_\infty L_\text{gridUnit}\)	freestream->muRef
Angular speed	\(C_\infty/L_\text{gridUnit}\)	slidingInterfaces->omegaRadians, BETDisks->omega
Time	\(L_\text{gridUnit}/C_\infty\)	timeStepping->timeStepSize
Mass flow rate	\(\rho C_{\infty} L_{gridUnit}^2\)	MassOutflow->massFlowRate, MassInflow->massFlowRate

8.1.2.1. Compute Reynolds

The Reynolds number in Flow360 is based on the units used in the grid. The Reynolds value can be calculated using the following equation, with the \(L_\text{ref}\) taking the value of the experimental reference length in grid units and \(L_{gridUnit}\) value being the unit length of the grid. For reference, the experimental Reynolds number is \(\rho_\infty U_\text{ref} L_\text{ref} / \mu_\infty\).

(8.1.4)\[\text{freestream->Reynolds} = \frac{\rho_\infty U_\text{ref} \text{L}_\text{gridUnit}}{\mu_\infty} = Re_\text{EXP} \cdot \frac{{L}_\text{gridUnit}}{\text{L}_\text{ref}}\]

As an example for an experimental Reynolds number of \(20*10^6\), mesh unit in inches with an experimental reference length \(L_\text{ref}\) of 10 inches, the Flow360 input Reynolds number is calculated as follows:

(8.1.5)\[\text{freestream->Reynolds} = 20 * 10^6 \cdot \frac{1 \text{ inch}}{10 \text{ inches}} = 20 * 10^5\]

8.1.2.2. Compute muRef

The reference dynamic viscosity muRef is a nondimensional number than can be used instead of Reynolds number to define the freestream and is dependent on the grid units. The muRef parameter can be calculated using the following equation:

(8.1.6)\[\text{freestream->muRef} = \frac{\mu_\infty}{\rho_\infty C_\infty L_\text{gridUnit}}\]

For example, for a mesh in meters at sea-level ISA conditions the reference dynamic viscosity can be calculated as follows:

(8.1.7)\[\text{freestream->muRef} = \frac{1.789 * 10^{-5} \text{kg}/\text{ms}}{1.225 \text{kg}/\text{m}^3 * 340.27 \text{m}/\text{s} * 1 \text{m}} = 4.29191 * 10^{-8}\]

8.1.2.3. 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:

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

where, the velocity components are nondimensionalized by the freestream speed of sound \(C_{\infty}\). The effects of these two angles are used to compute the forces in stability axes rather than body axes, \(CL\) and \(CD\), as follows:

(8.1.9)\[\begin{split}CL = CFz\cdot cos(\alpha) - CFx\cdot sin(\alpha)\\ CD = CFx\cdot cos(\alpha) cos(\beta) - CFz\cdot sin(\alpha) cos(\beta)\end{split}\]

8.1.2.4. Compute timeStepSize

The definition of timeStepSize can be found at timeStepping. Assume the physical time step size is 2 seconds, speed of sound of freestream is 340 m/s and grid unit is feet, therefore we have:

(8.1.10)\[\text{timeStepSize} = \frac{2 \ \text{s} \times 340 \ \text{m/s}}{0.3048 \ \text{m}} = 2230.971128608\]

8.1.2.5. Convert RPM to nondimensional rotating speed omega

The RPM determines the angular speed, and the nondimensional omega can be calculated by using the same equation as the dimensional angular speed from slidingInterfaces (list). Assume the RPM = 800, speed of sound of freestream is 340 m/s and grid unit is 1 millimeter, so omegaRadians can be written as:

(8.1.11)\[\text{omegaRadians} = \frac{800 \times 2\pi}{60 \ \text{s}} \times \frac{0.001 \ \text{m}}{340 \ \text{m/s}} = 0.00024639942\]

Alternatively, omegaDegrees is given by:

(8.1.12)\[\text{omegaDegrees} = \frac{800 \times 360}{60 \ \text{s}} \times \frac{0.001 \ \text{m}}{340 \ \text{m/s}} = 0.01411764706\]

8.1.2.6. Compute thermalConductivity, heatCapacity and volumetricHeatSource

For conjugate heat transfer simulations, the nondimensional thermal conductivity and heat source of a solid zone are found from the dimensional ones (\(k_s\), \(\dot{q}_s\)) as:

(8.1.13)\[\begin{split}\text{thermalConductivity} &= \frac{k_s}{k_{ref}} = \frac{k_s T_{\infty}}{\rho_{\infty} C_{\infty}^3 L_{gridUnit}} \\ \text{volumetricHeatSource} &= \frac{\dot{q}_s}{\dot{q}_{ref}} = \frac{\dot{q}_s L_{gridUnit}}{\rho_{\infty} C_{\infty}^3}\end{split}\]

The heat capacity per unit volume is defined as density (\(\rho_s\)) times the heat capacity per unit mass (\(c_s\)) of the material. It is nondimensionalized as:

(8.1.14)\[\text{heatCapacity} = \frac{\rho_s c_s}{\rho_{ref} c_{ref}} = \frac{\rho_s c_s T_{\infty}}{\rho_{\infty}C_{\infty}^2}\]