Nondimensional Inputs

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:

$L_{g r i d U n i t}$ (SI unit = $m$ ): physical length represented by unit length in the given mesh file, e.g.

if your grid is in feet, $L_{g r i d U n i t} = 1 feet = 0.3048 meter$ ;

if your grid is in millimeters, $L_{g r i d U n i t} = 1 millimeter = 0.001 meter$ .
$C_{\infty}$ (SI unit = $m / s$ ): freestream speed of sound
$ρ_{\infty}$ (SI unit = $kg / m^{3}$ ): density of freestream
$μ_{\infty}$ (SI unit = $N \cdot s / m^{2}$ ): dynamic viscosity of freestream
$p_{\infty}$ (SI unit = $N / m^{2}$ ): static pressure of freestream
$T_{\infty}$ (SI unit = $K$ ): temperature of freestream
$U_{ref} \equiv MachRef \times C_{\infty}$ (SI unit = $m / s$ ): reference velocity

A nondimensional variable is obtained by dividing its dimensional counterpart by an appropriately selected constant like Eq.(41).

(41)#

nondimensional variable = \frac{dimensional variable}{reference value}

Note

Any value presented here in symbolic format (for example $A_{ref}$ ) 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 50 Reference values for nondimensional inputs in Flow360#
Property	Ref. value for nondim.	Examples in Flow360.json
Length	$L_{gridUnit}$	geometry->momentLength, BETDisks->radius
Area	$L_{gridUnit}^{2}$	geometry->refArea
Dynamic viscosity	$ρ_{\infty} C_{\infty} L_{gridUnit}$	freestream->muRef
Angular speed	$C_{\infty} / L_{gridUnit}$	volumeZones->referenceFrame->omegaRadians, BETDisks->omega
Time	$L_{gridUnit} / C_{\infty}$	timeStepping->timeStepSize
Mass flow rate	$ρ C_{\infty} L_{g r i d U n i t}^{2}$	MassOutflow->massFlowRate, MassInflow->massFlowRate
Thermal conductivity	$\frac{ρ_{\infty} C_{\infty}^{3} L_{g r i d U n i t}}{T_{\infty}}$	volumeZones->thermalConductivity
Volumetric heat source	$\frac{ρ_{\infty} C_{\infty}^{3}}{L_{g r i d U n i t}}$	volumeZones->volumetricHeatSource
Heat flux	$ρ_{\infty} C_{\infty}^{3}$	HeatFluxWall->heatFlux

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 $L_{ref}$ taking the value of the experimental reference length in grid units and $L_{g r i d U n i t}$ value being the unit length of the grid. For reference, the experimental Reynolds number is $ρ_{\infty} U_{ref} L_{ref} / μ_{\infty}$ .

(42)#

freestream->Reynolds = \frac{ρ_{\infty} U_{ref} L_{gridUnit}}{μ_{\infty}} = R e_{EXP} \cdot \frac{L_{gridUnit}}{L_{ref}}

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

(43)#

freestream->Reynolds = 20 * 10^{6} \cdot \frac{1 inch}{10 inches} = 20 * 10^{5}

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:

(44)#

freestream->muRef = \frac{μ_{\infty}}{ρ_{\infty} C_{\infty} L_{gridUnit}}

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

(45)#

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

Define the angle of attack `alpha` and sideslip angle `beta`#

According to Flow360’s definitions of the angle of attack $α$ and the sideslip angle $β$ , with respect to the grid coordinates, the following values of velocity components are imposed at a Freestream farfield boundary:

(46)#

\begin{aligned} U_{\infty} & = Mach \cdot c o s (β) \cdot c o s (α) \\ V_{\infty} & = - Mach \cdot s i n (β) \\ W_{\infty} & = Mach \cdot c o s (β) \cdot s i n (α) \end{aligned}

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, $C L$ and $C D$ , as follows:

(47)#

\begin{array}{r} C L = C F z \cdot c o s (α) - C F x \cdot s i n (α) \\ C D = C F x \cdot c o s (α) c o s (β) - C F z \cdot s i n (α) c o s (β) \end{array}

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:

(48)#

timeStepSize = \frac{2 s \times 340 m/s}{0.3048 m} = 2230.971128608

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:

(49)#

omegaRadians = \frac{800 \times 2 π}{60 s} \times \frac{0.001 m}{340 m/s} = 0.00024639942

Alternatively, omegaDegrees is given by:

(50)#

omegaDegrees = \frac{800 \times 360}{60 s} \times \frac{0.001 m}{340 m/s} = 0.01411764706

Compute `heatFlux`#

The nondimensional heat flux for a wall boundary condition can be calculated by dividing the dimensional heat flux $q$ by the reference value:

(53)#

heatFlux = \frac{q}{q_{r e f}} = \frac{q}{ρ_{\infty} C_{\infty}^{3}}

Nondimensional Inputs

Contents

Nondimensional Inputs#

Compute Reynolds#

Compute muRef#

Define the angle of attack alpha and sideslip angle beta#

Compute timeStepSize#

Convert RPM to nondimensional rotating speed omega#

Compute conjugate heat transfer related properties#

Compute heatFlux#