.. _XV15BETDisk_caseStudy:

.. |deg|    unicode:: U+000B0 .. DEGREE SIGN
   :ltrim:

XV-15 Rotor Blade Analysis using the Blade Element Disk Method
==============================================================

Highlights
--------------
In this case study we are going to show you how to:

* Simulate an XV-15 tiltrotor airplane in both hovering helicopter mode and forward flight airplane mode using the steady BET Disk solver in Flow360
* Convert dimensional input quantities to non-dimensional quantities in the Flow360 configuration json file
* Convert non-dimensional quantities in the output to dimensional quantities

Introduction
-------------

The development of the XV-15 tiltrotor was launched by the NASA Ames Research Center in collaboration with Bell Helicopters in the late 1960s and early 1970s. It has become the foundation of many VTOL aircrafts today. Unlike typical helicopter rotors, tiltrotor blades demand a compromised design to operate efficiently in both helicopter and propeller modes. Hence, the blades of the XV-15 tiltrotor have high twist, high solidity and small rotor radius. In this case study, we will apply our Blade Element Theory (BET) Model based solver in Flow360 to evaluate the performance of the XV-15 rotor in both helicopter hovering mode and airplane mode.

.. _fig_XV15_landing:

.. figure:: Figures/XV-15_N703NA_USCG.jpg
   :scale: 70%
   :align: center

   The Bell Textron XV-15 Tiltrotor aircraft prepares to land (source: `United States Coast Guard <https://commons.wikimedia.org/wiki/File:XV-15_N703NA_USCG.jpg>`_)

The definition of the cross sections of the rotor blade is shown in :numref:`tab_xv15_airfoils`. The main geometric characteristics of the XV-15 rotor blades are listed in :numref:`tab_xv15_parameters`.

.. grid:: 2

    .. grid-item::
        :columns: 6
        :margin: auto

        .. _tab_xv15_airfoils:
        .. csv-table:: Cross sections of XV-15 rotor blade
           :file: Tables/xv15_airfoils.csv
           :widths: 20, 20
           :header-rows: 1
           :delim: @

    .. grid-item::
        :columns: 6
        :margin: auto

        .. _tab_xv15_parameters:
        .. csv-table:: Geometric characteristics of XV-15 rotor
           :file: Tables/xv15_parameters.csv
           :widths: 20, 20
           :header-rows: 1
           :delim: @

The radial chord and twist distribution is shown in :numref:`fig_xv15_twist_chord_definition`.

.. _fig_xv15_twist_chord_definition:

.. figure:: Figures/xv15_twist_chord_definition.png
   :scale: 80%
   :align: center

   XV-15 rotor blade's twist and chord radial distribution

More information about the geometry and a high-fidelity detached eddy simulation done using Flow360 can be found at

`Feilin Jia, John Moore and Qiqi Wang, 2022, Assessment of Detached Eddy Simulation and Sliding Mesh Interface in Predicting Tiltrotor Performance in Helicopter and Airplane Modes <https://arxiv.org/pdf/2201.11560.pdf>`_

In this case study, the steady Navier-Stokes solver coupled with the Blade Element Theory Disk method is used to simulate the XV-15 rotor blades under two flight conditions: helicopter hovering mode and airplane mode. In order run cases in a few minutes, compared to the high fidelity unsteady simulations with full rotor geometry in the above paper, the :ref:`mesh for this BET Disk example <BETexamplefiles>` is much coarser. The mesh used in the current case study contains only 592K nodes. An overview of a slice of the mesh is shown below

.. container::

   |mesh1| |mesh2|

   |mesh3| |mesh4|

   .. |mesh1| image:: Figures/mesh_592K_zoom1.png
      :width: 49%

   .. |mesh2| image:: Figures/mesh_592K_zoom2.png
      :width: 49%

   .. |mesh3| image:: Figures/mesh_592K_zoom3.png
      :width: 49%

   .. |mesh4| image:: Figures/mesh_592K_zoom4.png
      :width: 49%

   .. rst-class:: center1

   Overview of volume mesh used in BET Disk simulations of XV-15 rotor, successively zooming in from top left->top right->bottom left->bottom right.

The freestream quantities are shown below. These quantities are needed to set up the non-dimensional variables in the flow360 case configuration and translate the non-dimensional output variables into dimensional values. More information on non-dimensionalization in Flow360 can be found at :ref:`nondimensionalization_Flow360`. In the above mesh, the grid unit :math:`L_{gridUnit}=1\,\text{inch}=0.0254\,\text{meter}`. In this case study, the freestream conditions are set to standard sea level values as shown in :numref:`tab_air_isa`

.. _tab_air_isa:
.. csv-table:: Sea-level air properties
   :file: Tables/air_sea_level.csv
   :widths: 20, 20
   :align: center
   :header-rows: 1
   :delim: @

.. caution::

   In some simulations, the freestream is not set to be at standard sea level on purpose. For example, in the "case 1a" from `3rd AIAA CFD High Lift Prediction Workshop <https://hiliftpw.larc.nasa.gov/Workshop3/testcases.html>`_, the viscosity of the freestream is adjusted to analyze the full-scale geometry at wind tunnel conditions, so please set the freestream properties based on your requirements.

.. _BET_XV15_hovering_caseStudy:

Helicopter Hovering Mode
---------------------------

In helicopter hovering mode, the freestream velocity is zero. Five blade collective angles are considered in the current study: :math:`0^o, 3^o, 5^o, 10^o, 13^o` at :math:`r/R=0.75`, corresponding to low, medium and high disk loadings. The flow conditions are:

* Tip Mach Number, defined as :math:`U_\text{tip}/C_{\infty}`, is 0.69, so :math:`U_\text{tip}=0.69C_\infty`.
* Reynolds Number (based on reference chord (14 inch) and blade tip speed) = :math:`4.95\times 10^6`.
* Reference Temperature = 288.15 K.

Here are some points to set up the case configuration file:

1. :code:`freestream::Mach` is set to 0, because freestream velocity is zero.
2. :code:`freestream::MachRef` has to be a non-zero number because the above "Mach" is 0. Theoretically, this value could be arbitrary, but we set it equal to the tip Mach number (0.69) for convenience.
3. Either :code:`freestream::Reynolds` or :code:`freestream::muRef` should be given to set a proper freestream viscosity. Both options are explained below:

   * **Option 1**: set :code:`freestream::Reynolds`. The :code:`freestream::Reynolds` is based on grid unit as the reference length, thus it is mesh dependent. It's definition is :math:`\rho_\infty U_\text{ref} L_\text{gridUnit}/{\mu_\infty}`. In the case description, we know the Reynolds number based on tip speed and reference chord is :math:`4.95\times 10^6`, so we must convert the value from the reference chord used to the :math:`L_{gridUnit}` in the mesh.

   .. math::
      :label: hoverReynoldsChord

      \frac{\rho_\infty U_\text{tip} \text{chord}_\text{ref}}{\mu_\infty} = \frac{\rho_\infty \left(0.69 C_\infty\right) \text{chord}_\text{ref}}{\mu_\infty} = 4.95\times 10^6

thus the :code:`freestream::Reynolds` is calculated by:

   .. math::
      :label: hoverReynoldsGridUnit

      \text{freestream::Reynolds}&=\frac{\rho_\infty U_\text{ref} L_\text{gridUnit}}{\mu_\infty} = \frac{\rho_\infty\cdot \text{MachRef} \cdot C_\infty L_\text{gridUnit}}{\mu_\infty} \\
      &=\frac{\rho_\infty \left(0.69 C_\infty\right) \text{chord}_\text{ref}}{\mu_\infty}\times\frac{\text{MachRef}}{0.69}\times\frac{L_\text{gridUnit}}{\text{chord}_\text{ref}} \\
      &= 4.95\times 10^6 \times \frac{0.69}{0.69}\times \frac{1\, \text{inch}}{14\, \text{inch}} = 3.3536\times10^5

The :code:`freestream::Reynolds` is :math:`3.3536\times10^5`

   * **Option 2**: set :code:`freestream::muRef`. The "muRef" here is a non-dimensional quantity, it is mesh dependent because it depends on :math:`L_{gridUnit}`. Its definition is

   .. math::
      :label: muRefDef

      \text{freestream::muRef} = \frac{\mu_\infty}{\rho_\infty C_\infty L_\text{gridUnit}}

If you already know the Reynolds number then you could use the following equation to get the :code:`freestream::muRef`:

   .. math::
      :label: muRefEq

      \text{freestream::muRef} &= \frac{\rho_\infty U_\text{tip}\text{Chord}_\text{ref}/Re}{\rho_\infty C_\infty L_\text{gridUnit}} =\frac{\rho_\infty \left(0.69 C_\infty\right) \text{14 inch}}{\rho_\infty C_\infty \text{1 inch} \cdot 4.95\times10^6}  \\
      &=\frac{0.69\cdot \text{14 inch}}{\text{1 inch} \cdot 4.95\times 10^6} = 1.95152\times10^{-6}

Therefore, the :code:`freestream::muRef` is :math:`1.95152\times10^{-6}`

4. :code:`BETDisks::omega` is the non-dimensional angular speed of the rotor blades. Its definition is

   .. math::
      :label: BETDisk_omega_def

      \text{BETDisks::omega} = \Omega\cdot\frac{L_\text{gridUnit}}{C_\infty}

Because we know :math:`\Omega=\frac{U_\text{tip}}{\text{Radius}}`,

   .. math::
      :label: BETDisk_omega_calc

      \text{BETDisks::omega} &= \frac{U_\text{tip}}{\text{Radius}}\cdot\frac{L_\text{gridUnit}}{C_\infty} = \frac{0.69C_\infty\cdot \text{1 inch}}{\text{150 inch} \cdot C_\infty}\\
      &= 0.69/150 = 0.0046

Therefore, :code:`BETDisks::omega` is 0.0046

After setting up the case configuration file, the case is ready to submit. The 592K-node mesh file, and its corresponding mesh and case configuration files can be downloaded via the following links:

.. _BETexamplefiles:

* Mesh file: `XV15_BETDisk_R150_592K.lb8.ugrid <https://simcloud-public-1.s3.amazonaws.com/XV15_BETDisk/XV15_BETDisk_R150_592K.lb8.ugrid>`_
* Mesh configuration file: `Flow360Mesh.json <https://simcloud-public-1.s3.amazonaws.com/XV15_BETDisk/Flow360Mesh.json>`_
* Case configuration file for :math:`\theta_{75}=0^o`: `Flow360_hover_pitch0.json <https://simcloud-public-1.s3.amazonaws.com/XV15_BETDisk/Flow360_hover_pitch0.json>`_
* Case configuration file for :math:`\theta_{75}=3^o`: `Flow360_hover_pitch3.json <https://simcloud-public-1.s3.amazonaws.com/XV15_BETDisk/Flow360_hover_pitch3.json>`_
* Case configuration file for :math:`\theta_{75}=5^o`: `Flow360_hover_pitch5.json <https://simcloud-public-1.s3.amazonaws.com/XV15_BETDisk/Flow360_hover_pitch5.json>`_
* Case configuration file for :math:`\theta_{75}=10^o`: `Flow360_hover_pitch10.json <https://simcloud-public-1.s3.amazonaws.com/XV15_BETDisk/Flow360_hover_pitch10.json>`_
* Case configuration file for :math:`\theta_{75}=13^o`: `Flow360_hover_pitch13.json <https://simcloud-public-1.s3.amazonaws.com/XV15_BETDisk/Flow360_hover_pitch13.json>`_

.. note::

   The only difference between the configuration files of the above 5 cases is the :code:`BETDisks/twists` distribution.

For detailed instructions on how to upload a mesh, run a case and download the results, please refer to the :ref:`Quick Start <quickstart>` section of this documentation - these details will not be covered in this case study.

Some tips on setting the input quantities related to BET can be found in the :ref:`bet_input` section. Please note that there is no hub in the high fidelity detached eddy simulation of the XV-15 rotor blades in `our paper <https://arxiv.org/pdf/2201.11560.pdf>`_ mentioned above, so to match the model used in the high fidelity simulations, the chord length in :math:`r<0.09R` should be set to 0 in the BET simulation configuration files. I.e. the chord length is set to 0 right before the first cross section (:math:`r=0.09R`). This setting leads to a radial distribution similar to "chords_distribution_1" shown in :ref:`bet_input`.

The forces and moments related to the BET Disk are saved in the "bet_forces_v2.csv" file. A detailed description can be found at :ref:`betDiskLoadingNote`. Here we will convert those non-dimensional values into dimensional values:

1. thrust and torque.

Because the axial direction of the rotor is in the Z axis, the thrust is saved as "Disk0_Force_z" and the torque is saved as "Disk0_Moment_z" in the .csv file. The dimensional thrust and torque can be calculated by :eq:`defBETForce` and :eq:`defBETMoment`:

   .. math::
      :label: thrust_convert

      \text{Thrust} &= \text{Disk0_Force_z}\cdot \rho_\infty C^2_\infty L^2_\text{gridUnit} \\
      &= \text{Disk0_Force_z}\cdot 1.225 kg/m^3 \times 340.3^2 m^2/s^2 \times 0.0254^2 m^2 \\
      &= \text{Disk0_Force_z}\cdot 91.5224 N

   .. math::
      :label: torque_convert

      \text{Torque} &= \text{Disk0_Moment_z}\cdot \rho_\infty C^2_\infty L^3_\text{gridUnit} \\
      &= \text{Disk0_Moment_z}\cdot 1.225 kg/m^3 \times 340.3^2 m^2/s^2 \times 0.0254^3 m^3 \\
      &= \text{Disk0_Moment_z}\cdot 2.324669 N\cdot m

The convergence history of the dimensional thrust and torque using steady BET Disk solver is shown in following figures:

.. container::

   |hover_history_thrust| |hover_history_torque|

   .. |hover_history_thrust| image:: Figures/xv15_BETDisk_history_thrust_hover.svg
      :width: 49%

   .. |hover_history_torque| image:: Figures/xv15_BETDisk_history_torque_hover.svg
      :width: 49%

   .. rst-class:: center1

   Loading convergence of BET Disk simulation in hovering helicopter mode at various pitch angles.

2. sectional thrust and torque.

In the "bet_forces_v2.csv" file, the sectional thrust coefficients are provided. The process to convert the non-dimensional Ct into its physical dimension, i.e. sectional thrust per unit span in SI is very similar to the one shown above. We need the dimensional quantities mentioned in :eq:`defBETCt` and :eq:`defBETCq` to compute the dimensional sectional thrust:

* Radius of the rotor disk :math:`R = 150\times L_{gridUnit} = 3.81\,\text{meter}`
* rotating angular speed :math:`\Omega = V_{tip}/R = Mach_{tip}*C_{\infty}/R = 61.6237\, \text{rad/second}`
* reference chord :math:`\text{chord}_{\text ref}=14\times L_{gridUnit} = 0.3556\, \text{meter}`

If we assume that we want to calculate the sectional thrust and torque at the first disk's first blade's second radial location, i.e. Disk0_Blade0_R1:

.. math::
   :label: sectionalThrust_convert

   &C_t = \text{Disk0_Blade0_R1_ThrustCoeff}

   &C_q = \text{Disk0_Blade0_R1_TorqueCoeff}

   &r = \text{Disk0_Blade0_R1_Radius}*L_\text{gridUnit}

   &\text{Thrust per unit blade span}_\text{Disk0_Blade_R1} = C_t\times\frac{1}{2}\rho_\infty\times\Omega^2 r^2 \text{chord}_\text{ref}\times R/r

   &\text{Torque per unit blade span}_\text{Disk0_Blade_R1} = C_q\times\frac{1}{2}\rho_\infty\times\Omega^2 r^2 \text{chrod}_\text{ref} R\times R/r

As a comparison example between the high fidelity full-rotor unsteady simulation and the BET Disk steady simulation, the physical sectional thrust on a blade per unit span, for the :math:`\theta_{75}=10^o` case, is shown below

.. container::

   |hover_sectionalThrust| |hover_history_thrust_compare|

   .. |hover_sectionalThrust| image:: Figures/xv15_BETDisk_hover_pitch10_sectionalThrust.svg
      :width: 49%

   .. |hover_history_thrust_compare| image:: Figures/xv15_BETDisk_hover_pitch10_thrust_history.svg
      :width: 49%

   .. rst-class:: center1

   Sectional thrust distribution and history of total thrust in hovering mode, :math:`\theta_{75}=10^o`.

The biggest differences between these high fidelity simulations and these BET Disk simulations are near the tip regions where blade-vortex interactions are strong. The flow around the tip can be highly three dimensional, making a BET Disk approximation locally inaccurate. This effect is proportional to the blade disk loading, thus it is larger at hovering or near-hovering conditions when the thrust is higher then in forward flight. Even then, the total thrust of the three blades in hover, compared to the Flow360 high fidelity unsteady simulation, is ~8% different. This level of accuracy makes the BET Disk a useful tool in the preliminary design stages.

To provide another overview of the BET disk accuracy; If we look at the propeller efficiency in hovering mode, the thrust coefficient, the torque coefficient and the figure of merit defined in :eq:`CT_CQ_definition` and compare them with several experimental data and numerical prediction of high-fidelity DES simulations, we get the graphs below:

.. math::
   :label: CT_CQ_definition

   C_T &= \frac{\text{Thrust}}{\rho_\infty \left(\Omega R\right)^2A}

   C_Q &= \frac{\text{Torque}}{\rho_\infty \left(\Omega R\right)^2AR}

   \text{Figure of Merit} &= \frac{C_T^{3/2}}{\sqrt{2}C_Q}

where :math:`R` is rotor disk radius and :math:`A` is rotor disk area, i.e. :math:`\pi R^2`

.. container::

   |hover_CT_CQ| |hover_CT_FOM|

   .. |hover_CT_CQ| image:: Figures/xv15_BETDisk_CT_CQ_hover.svg
      :width: 49%

   .. |hover_CT_FOM| image:: Figures/xv15_BETDisk_CT_FOM_hover.svg
      :width: 49%

   .. rst-class:: center1

   Comparison on thrust and torque coefficient and figure of merit in hovering mode at various pitch angles.

Airplane Mode
---------------

In airplane mode, four blade collective angles are considered: :math:`26^o, 27^o, 28^o, 28.8^o` at :math:`r/R=0.75`. The flow conditions are:

* Tip Mach Number = 0.54.
* Reynolds Number (based on reference chord and tip speed, with no account for the inflow velocity) = :math:`4.5\times 10^6`.
* Reference chord = 14 inch.
* Reference Temperature= 288.15 K.
* Advance ratio (defined as inflow speed over tip speed) = 0.337

The mesh file and mesh configuration file are the :ref:`same as the files<BETexamplefiles>` used in the helicopter hovering example above. The case configuration files, however, are different:

* Case configuration file for :math:`\theta_{75}=26^o`: `Flow360_airplane_pitch26.json <https://simcloud-public-1.s3.amazonaws.com/XV15_BETDisk/Flow360_airplane_pitch26.json>`_
* Case configuration file for :math:`\theta_{75}=27^o`: `Flow360_airplane_pitch27.json <https://simcloud-public-1.s3.amazonaws.com/XV15_BETDisk/Flow360_airplane_pitch27.json>`_
* Case configuration file for :math:`\theta_{75}=28^o`: `Flow360_airplane_pitch28.json <https://simcloud-public-1.s3.amazonaws.com/XV15_BETDisk/Flow360_airplane_pitch28.json>`_
* Case configuration file for :math:`\theta_{75}=28.8^o`: `Flow360_airplane_pitch28.8.json <https://simcloud-public-1.s3.amazonaws.com/XV15_BETDisk/Flow360_airplane_pitch28.8.json>`_

The convergence history of the thrust coefficients and torque coefficients using the steady BET Disk solver are shown below

.. container::

   |airplane_T_history| |airplane_Q_history|

   .. |airplane_T_history| image:: Figures/xv15_BETDisk_history_thrust_airplane.svg
      :width: 49%

   .. |airplane_Q_history| image:: Figures/xv15_BETDisk_history_torque_airplane.svg
      :width: 49%

   .. rst-class:: center1

   Convergence history of thrust coefficient and torque coefficient in airplane mode at various pitch angles.

   As an example, similarly to the helicopter hovering case shown above, the physical sectional thrust for the :math:`\theta_{75}=26^o` case is shown below. We have comparisons to the high fidelity DES simulation both on a blade per unit span basis and for the total force.

   .. note::

    Please note the non-zero origin of the Y axis on the right plot of total force.


.. container::

   |airplane_sectionalThrust| |airplane_thrust_history|

   .. |airplane_sectionalThrust| image:: Figures/xv15_BETDisk_airplane_pitch26_sectionalThrust.svg
      :width: 49%

   .. |airplane_thrust_history| image:: Figures/xv15_BETDisk_airplane_pitch26_thrust_history.svg
      :width: 49%

   .. rst-class:: center1

   Sectional thrust distribution and thrust history in airplane mode at :math:`\theta_{75}=26^o`.


To provide an overview of power efficiency in airplane mode at various pitching angles, the figure below shows the comparison of the thrust coefficient, torque coefficient and propeller propulsive efficiency, defined in :eq:`propulsive_efficiency_eta`, between the BET Disk and high fidelity simulations.

.. math::
   :label: propulsive_efficiency_eta

   \eta = \frac{C_T \cdot V_\infty}{C_Q \cdot V_\text{tip}}

.. container::

   |airplane_CTCQ| |airplane_CTeta|

   .. |airplane_CTCQ| image:: Figures/xv15_BETDisk_CT_CQ_airplane.svg
      :width: 49%

   .. |airplane_CTeta| image:: Figures/xv15_BETDisk_CT_eta_airplane.svg
      :width: 49%

   .. rst-class:: center1

   Comparison of thrust coefficient, torque coefficient and propulsive efficiency in airplane mode at various pitch angles.