Non-Dimensional Outputs

Non-Dimensional Outputs#

Note

With User Variables, manual dimensionalization is rarely needed — simply request the desired unit system and the conversion is automatic. The formulas below describe the underlying logic.

All solver outputs are non-dimensional. To recover physical (SI) values, multiply by the appropriate reference quantity from the Reference Quantities Table.

Reference values for non-dimensional outputs#

Property

Ref. value for nondim.

Examples of non-dimensional outputs in Flow360

Length

\(L_\text{gridUnit}\)

wallDistance

Density

\(\rho_\infty\)

primitiveVars

Velocity

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

primitiveVars

Pressure

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

primitiveVars, nodeForcesPerUnitArea

Temperature

\(T_\infty\)

T

Heat Flux

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

heatFlux

BET, Actuator Disk and Porous Media Force

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

force in BET output

BET, Actuator Disk and Porous Media Moment

\(\rho_\infty U_\text{scale}^2 L_\text{gridUnit}^3\) where \(U_\text{scale}\) is the reference velocity scaling

moment in BET output

Note

Many output variables use the reference velocity scaling \(U_\text{scale}\) (not \(U_\text{ref}\)). In the examples below we assume \(U_\text{scale}\) has been computed as shown in the Introduction.

Visualization Field Conversions#

Flow360 exports ParaView (.pvtu) and Tecplot (.szplt) files. All exported fields are non-dimensional. The subsections below show how to recover dimensional values.

../../../_images/download_results_WebUI.png

Accessing case results.#

Velocity#

Reference value: \(U_\text{scale}\).

If velocityX = 0.6 and \(U_\text{scale} = 340\;\text{m/s}\), then \(v_x = 0.6 \times 340 = 204\;\text{m/s}\).

Pressure#

Reference value: \(\rho_\infty\,U_\text{scale}^2\).

If p = 0.65, \(\rho_\infty = 1.225\;\text{kg/m}^3\), and \(U_\text{scale} = 340\;\text{m/s}\):

(1)#\[p_\text{dim} = 0.65 \times 1.225 \times 340^2 = 92\,046.5\;\text{Pa}\]

Node Force Per Unit Area#

nodeForcesPerUnitArea is the total (pressure + friction) force at a node divided by the surface area attributed to that node. Integrating over the whole surface yields the total force. The reference value is the same as for pressure: \(\rho_\infty\,U_\text{scale}^2\).

Temperature#

Reference value: \(T_\infty\). Multiply the non-dimensional temperature by the freestream temperature to obtain Kelvin.

Surface Coefficient Definitions#

The following coefficients appear in both visualization files and CSV files. They all use the reference velocity \(U_\text{ref}\) (accessed via case.params.reference_velocity), which is different from \(U_\text{scale}\).

Tip

Flow360’s unit system carries SI units through arithmetic, so you can convert any coefficient to physical units in one step. For example, once q_ref and A_ref are retrieved from the API:

tau_wall = (Cf * q_ref).to('Pa')           # wall shear stress
p_dim    = (Cp * q_ref + p_inf).to('Pa')   # static pressure

We define the dynamic pressure for coefficients as:

(2)#\[q_\text{ref} = \tfrac{1}{2}\,\rho_\infty\,U_\text{ref}^2\]

Skin Friction Coefficient#

The skin friction coefficient vector \(\mathbf{C_f}\) and its magnitude \(C_f\):

(3)#\[\mathbf{C_f} = \frac{\mathbf{\tau_\text{wall}}}{q_\text{ref}}\]

To recover the dimensional wall shear stress:

(4)#\[\tau_\text{wall}\;[\text{Pa}] = C_f \cdot q_\text{ref}\]

Pressure Coefficient#

(5)#\[C_p = \frac{p - p_\infty}{q_\text{ref}}\]

To recover the dimensional static pressure:

(6)#\[p\;[\text{Pa}] = C_p \cdot q_\text{ref} + p_\infty\]

Total Pressure Coefficient#

(7)#\[C_{p_t} = \frac{p_t - p_{t,\infty}}{q_\text{ref}}\]

To recover the dimensional total pressure:

(8)#\[p_t\;[\text{Pa}] = C_{p_t} \cdot q_\text{ref} + p_{t,\infty}\]

The total pressure coefficient can also be computed from the primitive variables as described in equations 26 and 29 of this paper:

(9)#\[C_{p_t} = \frac{\gamma\,p\,\bigl(1 + \tfrac{\gamma-1}{2}\,\text{Ma}^2\bigr)^{\gamma/(\gamma-1)} - \bigl(1 + \tfrac{\gamma-1}{2}\,\text{Ma}_\infty^2\bigr)^{\gamma/(\gamma-1)}}{\tfrac{\gamma}{2}\,\text{Ma}_\infty^2}\]

where \(\gamma = 1.4\) for standard air.

Note

Ensure Mach and primitiveVars are included in your volume output configuration.

CSV File Outputs#

Actuator Disk#

The file actuatorDisk_output_v2.csv reports power, force, and moment for each disk. The power column contains a coefficient:

(10)#\[C_p = \frac{\text{Power}\;[\text{W}]}{\rho_\infty\,U_\text{scale}^3\,L_\text{gridUnit}^2}\]

To obtain dimensional power:

actuator_disk_output = case.results.actuator_disks.averages
density = case.params.operating_condition.thermal_state.density
L_grid  = project.length_unit

C_p   = actuator_disk_output['Disk0_Power']
power = C_p * density * U_scale**3 * L_grid**2      # W

Attention

Actuator Disk forces and moments use \(U_\text{scale}\), not \(U_\text{ref}\). See Converting to Physical Units for the general conversion formulas.

BET Loading#

The file bet_forces_v2.csv contains:

  1. Integrated forces and moments per disk (Disk0_Force_x, Disk0_Moment_x, …). These are non-dimensional raw values:

    (11)#\[\text{Force}^* = \frac{F\;[\text{N}]}{\rho_\infty\,U_\text{scale}^2\,L_\text{gridUnit}^2}\]
    (12)#\[\text{Moment}^* = \frac{M\;[\text{N·m}]}{\rho_\infty\,U_\text{scale}^2\,L_\text{gridUnit}^3}\]

    Note

    These values represent forces/moments on the solid. Forces on the fluid are the negative of the above.

    Note

    Equations Eq.(11) and Eq.(12) also apply to Porous Media outputs in porous_media_output_v2.csv.

    Attention

    BET forces and moments use \(U_\text{scale}\), not \(U_\text{ref}\). The x, y, z components are in the global inertial frame defined by the mesh.

  1. Sectional thrust \(C_t\) and torque \(C_q\) per blade at each radial station (Disk0_Blade0_R0_ThrustCoeff, …).

    (13)#\[C_t(r) = \frac{\text{Thrust/span}\;[\text{N/m}]}{\tfrac{1}{2}\,\rho_\infty\,(\Omega r)^2\,\text{chord}_\text{ref}} \cdot \frac{r}{R}\]
    (14)#\[C_q(r) = \frac{\text{Torque/span}\;[\text{N}]}{\tfrac{1}{2}\,\rho_\infty\,(\Omega r)^2\,\text{chord}_\text{ref}\,R} \cdot \frac{r}{R}\]

    Here \(r\) is the dimensional distance from the rotation axis, \(\text{chord}_\text{ref}\) the dimensional reference chord, and \(R\) the rotor radius.

    Note

    \(C_t\) and \(C_q\) are sectional loadings: \(C_t(r) = \text{d}C_T/\text{d}(r/R)\) and \(C_q(r) = \text{d}C_Q/\text{d}(r/R)\).

    Important

    All right-hand-side quantities in Eq.(11)–Eq.(14) are dimensional. The non-dimensional radius in the CSV must first be converted: \(r\) = Disk0_Blade0_R0_Radius \(\times\, L_\text{gridUnit}\).

    Warning

    For steady-state BET Disk simulations, \(C_t\) and \(C_q\) are written only for Blade0; all other blades report zeros because the loading is identical. For unsteady BET Line simulations each blade has distinct values.

Python example — dimensional force, moment, and sectional loading:

bet_forces = case.results.bet_forces.averages
density    = case.params.operating_condition.thermal_state.density
L_grid     = project.length_unit

# --- Integrated force & moment (Disk 0) ---
force_x  = bet_forces["Disk0_Force_x"]  * density * U_scale**2 * L_grid**2     # N
moment_x = bet_forces["Disk0_Moment_x"] * density * U_scale**2 * L_grid**3     # N·m

# --- Sectional loading (Disk 0, Blade 0, radial station 1) ---
bet_model  = case.params.models[4]   # index of the BETDisk model
chord_ref  = bet_model.chord_ref
omega      = bet_model.omega
R          = bet_model.entities.stored_entities[0].outer_radius

r1    = bet_forces["Disk0_Blade0_R1_Radius"]
Ct_r1 = bet_forces["Disk0_Blade0_R1_ThrustCoeff"]
Cq_r1 = bet_forces["Disk0_Blade0_R1_TorqueCoeff"]

thrust_r1 = Ct_r1 * 0.5 * density * omega**2 * r1 * L_grid * R * chord_ref     # N/m
torque_r1 = Cq_r1 * 0.5 * density * omega**2 * r1 * L_grid * R**2 * chord_ref  # N

Aeroacoustic Output#

The file total_acoustics_v3.csv reports the non-dimensional acoustic pressure signal at each observer location. If AeroacousticOutput.write_per_surface_output is True, per-surface files surface_<name>_acoustics_v3.csv are also written.

Columns: time, physical_step, observer_0_pressure, observer_0_thickness, observer_0_loading, … for N+1 observers.

Note

The observation time may fall outside the simulation time range because the acoustic signal takes a finite propagation time from the surface to each observer. Early and late time entries are zero-padded.

Heat Transfer#

The file surface_heat_transfer_v2.csv contains surface-integrated heat flux in non-dimensional form.

To recover the dimensional heat transfer rate, multiply by \(\rho_\infty\,U_\text{scale}^3\,L_\text{gridUnit}^2\):

surface_ht = case.results.surface_heat_transfer.averages
density    = case.params.operating_condition.thermal_state.density
L_grid     = project.length_unit

q_nd = surface_ht["fluid/Interface_solid_HeatTransferRate"]
q    = q_nd * density * U_scale**3 * L_grid**2   # W

Visualization Tips#

The sections below are not about conversion formulas but about using specific output fields effectively.

Skin Friction — Detecting Separation#

The CfVec vector is useful for locating boundary-layer separation. Fully attached flow follows the surface along the streamwise direction; separated flow produces local recirculation with reversed skin friction.

For flow predominantly in the x-direction, regions of negative CfVecX indicate separation. Visualizing with a three-level scale (e.g. −1e-6, 0, 1e-6) highlights separated vs. attached regions:

../../../_images/surfaceRecirculation.png

x-component of skin friction showing attached (yellow) and separated (purple) flow at high angle of attack.#

Surface streamlines can also reveal recirculation. Since wall velocity is zero on a NoSlipWall, use CfVec{X,Y,Z} as the integration variable instead of velocity:

../../../_images/surfaceRecirculationVec.png

Surface streamlines showing recirculation regions.#

Total Pressure — Visualizing Separation and Boundary Layers#

\(C_{p_t}\) reveals separation regions in volume slices and is effective for visualizing the boundary layer:

../../../_images/CPT.png

Total pressure coefficient on a slice through a partially stalled wing.#

../../../_images/CPT_ZOOMED.png

Zoomed view showing the developing boundary layer (blue) near the leading edge.#

Q-Criterion — Visualizing Vortices#

The qcriterion field identifies vortices via isosurfaces. Recommended isosurface values:

  • Aircraft: \(\text{Ma}^2 / \text{span}^2\)

  • Rotors: \(\text{Ma}_\text{tip}^2 / D^2\)

Larger values show only strong vortices; smaller values reveal weaker structures.

../../../_images/qcriterion.png

qcriterion isosurface showing tip vortex and vortices from the separation region.#

../../../_images/qcriterion_mesh.png

Volume mesh coarsening away from the wing causes rapid vortex dissipation.#

After refining the far-field mesh:

../../../_images/qcriterion_finer.png

Smoother isosurface with improved vortex resolution on a refined mesh.#

../../../_images/qcriterion_mesh_finer.png

Refined volume mesh reduces numerical dissipation.#

For more Q-criterion visualizations:

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