.. _nondim_input: 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: :math:`L_{gridUnit}` (SI unit = :math:`\text{m}`) | physical length represented by unit length in the given mesh file, e.g. | if your grid is in feet, :math:`L_{gridUnit}=1 \text{ feet}=0.3048 \text{ meter}`; | if your grid is in millimeters, :math:`L_{gridUnit}=1 \text{ millimeter}=0.001 \text{ meter}`. :math:`C_\infty` (SI unit = :math:`\text{m}/\text{s}`) freestream speed of sound :math:`\rho_\infty` (SI unit = :math:`\text{kg}/\text{m}^3`) density of freestream :math:`\mu_\infty` (SI unit = :math:`\text{N} \cdot \text{s}/\text{m}^2`) dynamic viscosity of freestream :math:`p_\infty` (SI unit = :math:`\text{N}/\text{m}^2`) static pressure of freestream :math:`T_\infty` (SI unit = :math:`K`) temperature of freestream :math:`U_\text{ref} \equiv \text{MachRef}\times C_\infty` (SI unit = :math:`\text{m}/\text{s}`) reference velocity A nondimensional variable is obtained by dividing its dimensional counterpart by an appropriately selected constant like :eq:`def_nondim`. .. math:: :label: def_nondim \text{nondimensional variable} = \frac{\text{dimensional variable}}{\text{reference value}} .. note:: Any value presented here in symbolic format (for example :math:`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: .. _tab_nondim_input: .. csv-table:: Reference values for nondimensional inputs in Flow360 :file: Tables/nondim_input.csv :widths: 10,10,60 :header-rows: 1 :delim: @ .. _nondim_Reynolds: Compute :code:`Reynolds` ------------------------ The :code:`Reynolds` number in Flow360 is based on the units used in the grid. The :code:`Reynolds` value can be calculated using the following equation, with the :math:`L_\text{ref}` taking the value of the experimental reference length in grid units and :math:`L_{gridUnit}` value being the unit length of the grid. For reference, the experimental Reynolds number is :math:`\rho_\infty U_\text{ref} L_\text{ref} / \mu_\infty`. .. math:: :label: ReynoldsLgridUnit2 \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 :math:`20*10^6`, mesh unit in inches with an experimental reference length :math:`L_\text{ref}` of 10 inches, the Flow360 input :code:`Reynolds` number is calculated as follows: .. math:: :label: ReynoldsExample \text{freestream->Reynolds} = 20 * 10^6 \cdot \frac{1 \text{ inch}}{10 \text{ inches}} = 20 * 10^5 .. _nondim_muRef: Compute :code:`muRef` --------------------- The reference dynamic viscosity :code:`muRef` is a nondimensional number than can be used instead of :code:`Reynolds` number to define the freestream and is dependent on the grid units. The :code:`muRef` parameter can be calculated using the following equation: .. math:: :label: muRefDef2 \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: .. math:: :label: muRefDefEx \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} .. _alphaBetaAngles: Define the angle of attack :code:`alpha` and sideslip angle :code:`beta` ------------------------------------------------------------------------ According to Flow360's definitions of the angle of attack :math:`\alpha` and the sideslip angle :math:`\beta`, with respect to the grid coordinates, the following values of velocity components are imposed at a :code:`Freestream` farfield boundary: .. math:: :label: EQ_FreestreamBC 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) where, the velocity components are nondimensionalized by the freestream speed of sound :math:`C_{\infty}`. The effects of these two angles are used to compute the forces in stability axes rather than body axes, :math:`CL` and :math:`CD`, as follows: .. math:: :label: EQ_CL_CD CL = CFz\cdot cos(\alpha) - CFx\cdot sin(\alpha)\\ CD = CFx\cdot cos(\alpha) cos(\beta) - CFz\cdot sin(\alpha) cos(\beta) .. _nondim_timeStepSize: Compute :code:`timeStepSize` ---------------------------- The definition of :code:`timeStepSize` can be found at :ref:`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: .. math:: \text{timeStepSize} = \frac{2 \ \text{s} \times 340 \ \text{m/s}}{0.3048 \ \text{m}} = 2230.971128608 .. _non_dim_omega: Convert RPM to nondimensional rotating speed :code:`omega` ------------------------------------------------------------------ The RPM determines the angular speed, and the nondimensional :code:`omega` can be calculated by using the same equation as the dimensional angular speed from :ref:`slidingInterfacesParameters`. Assume the RPM = 800, speed of sound of freestream is 340 m/s and grid unit is 1 millimeter, so :code:`omegaRadians` can be written as: .. math:: \text{omegaRadians} = \frac{800 \times 2\pi}{60 \ \text{s}} \times \frac{0.001 \ \text{m}}{340 \ \text{m/s}} = 0.00024639942 Alternatively, :code:`omegaDegrees` is given by: .. math:: \text{omegaDegrees} = \frac{800 \times 360}{60 \ \text{s}} \times \frac{0.001 \ \text{m}}{340 \ \text{m/s}} = 0.01411764706 .. _non_dim_thermal: Compute conjugate heat transfer related properties ---------------------------------------------------------------------------------------------------- For conjugate heat transfer simulations, the nondimensional thermal conductivity and heat source of a solid zone are found from the dimensional ones (:math:`k_s`, :math:`Q_s`) as: .. math:: \text{thermalConductivity} &= \frac{k_s}{k_{ref}} = \frac{k_s T_{\infty}}{\rho_{\infty} C_{\infty}^3 L_{gridUnit}} \\ \text{volumetricHeatSource} &= \frac{Q_s}{Q_{ref}} = \frac{Q_s L_{gridUnit}}{\rho_{\infty} C_{\infty}^3} The heat capacity per unit volume is defined as density (:math:`\rho_s`) times the specific heat capacity (:math:`c_s`) of the material. It is nondimensionalized as: .. math:: \text{heatCapacity} = \frac{\rho_s c_s}{\rho_{ref} c_{ref}} = \frac{\rho_s c_s T_{\infty}}{\rho_{\infty}C_{\infty}^2} Compute :code:`heatFlux` ---------------------------------------------------------------------------------------------------- The nondimensional heat flux for a wall boundary condition can be calculated by dividing the dimensional heat flux :math:`q` by the reference value: .. math:: \text{heatFlux} = \frac{q}{q_{ref}} = \frac{q}{\rho_{\infty} C_{\infty}^3}