Yuval Tassa
7b7cf5e86c
PiperOrigin-RevId: 571341445 Change-Id: I3a02bb01099cacb62e0c599e6198ed90b48baad0
442 lines
23 KiB
ReStructuredText
442 lines
23 KiB
ReStructuredText
Fluid forces
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============
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Proper simulation of fluid dynamics is beyond the scope of MuJoCo, and would be too slow for the applications we aim to
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facilitate. Nevertheless we provide two phenomenological models which are sufficient for simulating behaviors
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such as flying and swimming. These models are *stateless*, in the sense that no additional states are assigned to the
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surrounding fluid, yet are able to capture the salient features of rigid bodies moving through a fluid medium.
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Both models are enabled by setting the :ref:`density<option-density>` and :ref:`viscosity<option-viscosity>` attributes
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to positive values. These parameters correspond to the density :math:`\rho` and viscosity :math:`\beta` of the medium.
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1. The :ref:`Inertia-based model<flInertia>`, uses only viscosity and density, inferring geometry from body
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equivalent-inertia boxes.
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2. The :ref:`Ellipsoid-based model <flEllipsoid>` is more elaborate, using an ellipsoid approximation of geoms.
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In addition to the global viscosity and density of the medium, this model exposes 5 tunable parameters per
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interacting geom.
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.. tip::
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As detailed in the :ref:`Numerical Integration<geIntegration>` section, implicit integration significantly improves
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simulation stability in the presence of velocity-dependent forces. Both of the fluid-force models described below
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exhibit this property, so the ``implicit`` or ``implicitfast`` :ref:`intergrators<option-integrator>` are
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recommended when using fluid forces. The required analytic derivatives for both models are fully implemented.
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.. _flInertia:
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Inertia model
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-------------
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In this model, the shape of each body, for fluid dynamics purposes, is assumed to be the *equivalent inertia box*,
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which can also be visualized. Each forward-facing (relative to the linear velocity) face of the box experiences force
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along its normal direction. All faces also experience torque due to the angular velocity; this torque is obtained by
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integrating the force resulting from the rotation over the surface area. In this sub-section, let :math:`v` and
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:math:`\omega` denote the linear and angular body velocity in the body local frame (aligned with the equivalent
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inertia box), and :math:`s` the 3D vector of box sizes. When the contributions from all faces are added, the resulting
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force and torque applied to the body by a fluid of density :math:`\rho`, in local body coordinates, have the
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:math:`i`-th component
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.. math::
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\begin{aligned}
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\text{density force} : \quad &- {1 \over 2} \rho s_j s_k |v_i| v_i \\
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\text{density torque} : \quad &- {1 \over 64} \rho s_i \left(s_j^4 + s_k^4 \right) |\omega_i| \omega_i \\
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\end{aligned}
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This model implicitly assumes high Reynolds numbers, with lift-to-drag ratio equal to the tangent of the angle of
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attack. One can also specify a non-zero :ref:`wind<option-wind>`, which is a 3D vector subtracted from the body linear
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velocity in the fluid dynamics computation.
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Each body also experiences a force and a torque proportional to the viscosity :math:`\beta` and opposite to its linear and
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angular velocity. Note that viscosity can be used independent of density, to make the simulation more damped. We use the
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formulas for a sphere at low Reynolds numbers, with diameter :math:`d` equal to the average of the equivalent inertia
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box sizes. The resulting 3D force and torque in local body coordinates are
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.. math::
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\begin{aligned}
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\text{viscosity force} : \quad &- 3 \beta \pi d v \\
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\text{viscosity torque} : \quad &- \beta \pi d^3 \omega \\
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\end{aligned}
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.. _flEllipsoid:
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Ellipsoid model
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---------------
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.. cssclass:: caption-small
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.. figure:: ../images/computation/fruitfly.png
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:figwidth: 50%
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:align: right
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The flight-capable Drosophila Melanogaster model in this figure will be described in a
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forthcoming publication.
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In this section we describe and derive a stateless model of the forces exerted onto a moving rigid body by the
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surrounding fluid, based on an ellipsoidal approximation of geom shape. This model provides finer-grained control of the
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different types of fluid forces than the inertia-based model of the previous section. The motivating use-case for this
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model is insect flight, see figure on the right.
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Summary
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~~~~~~~
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The model is activated per-geom by setting the :ref:`fluidshape<body-geom-fluidshape>` attribute to ``ellipsoid``, which
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also disables the inertia-based model for the parent body. The
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5 numbers in the :ref:`fluidcoef<body-geom-fluidcoef>` attribute correspond to the following semantics
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.. list-table::
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:width: 60%
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:align: left
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:widths: 1 5 2 1
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:header-rows: 1
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* - Index
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- Description
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- Symbol
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- Default
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* - 0
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- Blunt drag coefficient
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- :math:`C_{D, \text{blunt}}`
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- 0.5
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* - 1
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- Slender drag coeficient
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- :math:`C_{D, \text{slender}}`
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- 0.25
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* - 2
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- Angular drag coeficient
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- :math:`C_{D, \text{angular}}`
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- 1.5
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* - 3
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- Kutta lift coeficient
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- :math:`C_K`
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- 1.0
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* - 4
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- Magnus lift coeficient
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- :math:`C_M`
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- 1.0
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Elements of the model are a generalization of :cite:t:`andersen2005b` to 3 dimensions.
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The force :math:`\mathbf{f}_{\text{fluid}\rightarrow \text{solid}}` and torque
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:math:`\mathbf{g}_{\text{fluid} \rightarrow \text{solid}}` exerted by the fluid onto the solid are
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the sum of of the terms
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.. math::
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\begin{align*}
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\mathbf{f}_{\text{fluid} \rightarrow \text{solid}} &= \mathbf{f}_A + \mathbf{f}_D + \mathbf{f}_M + \mathbf{f}_K \\
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\mathbf{g}_{\text{fluid} \rightarrow \text{solid}} &= \mathbf{g}_A + \mathbf{g}_D
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\end{align*}
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Where subscripts :math:`A`, :math:`D`, :math:`M` and :math:`K`, denote Added mass, viscous Drag, Magnus lift and
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Kutta lift, respectively. The :math:`D`, :math:`M` and :math:`K` terms are scaled by the respective
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:math:`C_D`, :math:`C_M` and :math:`C_K` coefficients above, while the added mass term cannot be scaled.
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Notation
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~~~~~~~~
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We describe the motion of the object in an inviscid, incompressible quiescent fluid of density :math:`\rho`. The
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arbitrarily-shaped object is described in the model as the equivalent ellipsoid of semi-axes
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:math:`\mathbf{d} = \{d_x, d_y, d_z\}`.
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The problem is described in a reference frame aligned with the sides of the ellipsoid and moving with it. The
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body has velocity :math:`\mathbf{v} = \{v_x, v_y, v_z\}` and angular velocity
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:math:`\boldsymbol{\omega} = \{\omega_x, \omega_y, \omega_z\}`. We will also use
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.. math::
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\begin{align*}
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d_\text{max} &= \max(d_x, d_y, d_z) \\
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d_\text{min} &= \min(d_x, d_y, d_z) \\
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d_\text{mid} &= d_x + d_y + d_z - d_\text{max} - d_\text{min}
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\end{align*}
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The Reynolds number is the ratio between inertial and viscous forces within a flow and is defined as :math:`Re=u~l/\beta`, where
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:math:`\beta` is the kinematic viscosity of the fluid, :math:`u` is the characteristic speed of the flow (or, by change of frame, the
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speed of the body), and :math:`l` is a characteristic size of the flow or the body.
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We will use :math:`\Gamma` to denote circulation, which is the line integral of the velocity field around a closed curve
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:math:`\Gamma = \oint \mathbf{v} \cdot \textrm{d} \mathbf{l}` and, due to Stokes' Theorem,
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:math:`\Gamma = \int_S \nabla \times \mathbf{v} \cdot \textrm{d}\mathbf{s}`.
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In fluid dynamics notation the symbol :math:`\boldsymbol{\omega}` is often used for the
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vorticity, defined as :math:`\nabla \times \mathbf{v}`, rather than the angular velocity. For a rigid-body motion, the
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vorticity is twice the angular velocity.
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Finally, we use the subscripts :math:`i, j, k` to denote triplets of equations that apply symmetrically to
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:math:`x, y, z`. For example :math:`a_i = b_j + b_k` is shorthand for the 3 equations
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.. math::
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\begin{align*}
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a_x &= b_y + b_z \\
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a_y &= b_x + b_z \\
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a_z &= b_x + b_y
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\end{align*}
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.. _flProjection:
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Ellipsoid projection
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~~~~~~~~~~~~~~~~~~~~
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We provide the following result without proof. For the derivation, contact the development team.
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.. admonition:: Lemma
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:class: note
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Given an ellipsoid at the origin with semi-axes :math:`(d_x, d_y, d_z)` aligned
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with the coordinate axes :math:`(x, y, z)`, and a unit vector :math:`\mathbf{u} = (u_x, u_y, u_z)`,
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the area projected by the ellipsoid onto the plane normal to :math:`\mathbf{u}` is
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.. math::
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A^{\mathrm{proj}}_{\mathbf{u}} = \pi \sqrt{\frac{d_y^4 d_z^4 u_x^2 + d_x^4 d_z^4 u_y^2 + d_x^4 d_y^4 u_z^2}{d_y^2 d_z^2 u_x^2 + d_x^2 d_z^2 u_y^2 + d_x^2 d_y^2 u_z^2}}
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Added mass
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~~~~~~~~~~
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For a body moving in a fluid, added mass or virtual mass measures the inertia of the fluid that is moved due to the
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body's motion. It can be derived from potential flow theory (i.e. it is present also for inviscid flows).
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Following Chapter 5 of :cite:t:`lamb1932`, the forces :math:`\mathbf{f}_{V}` and torques :math:`\mathbf{g}_{V}` exerted
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onto a moving body due to generation of motion in the fluid from rest can be written as:
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.. math::
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\begin{align*}
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\mathbf{f}_{A} &= - \frac{\textrm{d}}{\textrm{d} t} \nabla_{\mathbf{v}} \mathcal{T} + \nabla_{\mathbf{v}} \mathcal{T} \times \boldsymbol{\omega} \\
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\mathbf{g}_{A} &= - \frac{\textrm{d}}{\textrm{d} t} \nabla_{\boldsymbol{\omega}} \mathcal{T} + \nabla_{\mathbf{v}} \mathcal{T} \times \mathbf{v} + \boldsymbol{\omega} \times \nabla_{\boldsymbol{\omega}} \mathcal{T}
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\end{align*}
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where :math:`\mathcal{T}` is the kinetic energy of the fluid alone. These forces are often described as added or
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virtual mass because they are due to the inertia of the fluid that is to moved or deflected by the accelerating body. In
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fact, for a body with constant linear velocity these forces reduce to zero. We consider the body as having three planes
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of symmetry because under this assumption the kinetic energy greatly simplifies and can be written as:
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.. math::
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2 \mathcal{T} = m_{A, x} v_x^2 + m_{A, y} v_y^2 + m_{A, z} v_z^2 +
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I_{A, x} \omega_x^2 + I_ {A, y} \omega_y^2 + I_{A, y} \omega_z^2
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For convenience we introduce the added-mass vector :math:`\mathbf{m}_A = \{m_{A, x}, m_{A, y}, m_{A, z}\}` and added-moment of
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inertia vector :math:`\mathbf{I}_A = \{I_{A, x}, I_{A, y}, I_{A, z}\}`. Each of these quantities should estimate the inertia
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of the moved fluid due the motion of the body in the corresponding direction and can be derived from potential flow
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theory for some simple geometries.
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For a body with three planes of symmetry, we can write in compact form the forces and torques due to added inertia:
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.. math::
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\begin{align*}
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\mathbf{f}_{A} &= - \mathbf{m}_A \circ \dot{\mathbf{v}} + \left(\mathbf{m}_A \circ \mathbf{v} \right) \times \boldsymbol{\omega} \\
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\mathbf{g}_{A} &= - \mathbf{I}_A \circ \dot{\boldsymbol{\omega}} + \left(\mathbf{m}_A \circ \mathbf{v} \right) \times \mathbf{v} + \left(\mathbf{I}_A \circ \boldsymbol{\omega} \right) \times \boldsymbol{\omega}
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\end{align*}
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Here :math:`\circ` denotes an element-wise product, :math:`\dot{\mathbf{v}}` is the linear acceleration and
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:math:`\dot{\boldsymbol{\omega}}` is the angular acceleration. :math:`\mathbf{m}_A \circ \mathbf{v}` and
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:math:`\mathbf{I}_A \circ \boldsymbol{\omega}` are the virtual linear and angular momentum respectively.
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For an ellipsoid of semi-axis :math:`\mathbf{d} = \{d_x, d_y, d_z\}` and volume :math:`V = 4 \pi d_x d_y d_z / 3`, the
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virtual inertia coefficients were derived by :cite:t:`tuckerman1925`. Let:
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.. math::
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\kappa_i = \int_0^\infty \frac{d_i d_j d_k}{\sqrt{(d_i^2 + \lambda)^3 (d_j^2 + \lambda) (d_k^2 + \lambda)}} \textrm{d} \lambda
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It should be noted that these coefficients are non-dimensional (i.e. if all semi-axes are multiplied by the same scalar
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the coefficients remain the same). The virtual masses of the ellipsoid are:
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.. math::
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m_{A, i} = \rho V \frac{\kappa_i}{2 - \kappa_i}
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And the virtual moments of inertia are:
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.. math::
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I_{A, i} = \frac{\rho V}{5} \frac{(d_j^2 - d_k^2)^2 (\kappa_k-\kappa_j)}{2(d_j^2 - d_k^2) + (d_j^2 + d_k^2) (\kappa_j-\kappa_k)}
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Viscous drag
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~~~~~~~~~~~~
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The drag force acts to oppose the motion of the body relative to the surrounding flow. We found that viscous forces
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serve also to reduce the stiffness of the equations of motion extended with the fluid dynamic terms. For this reason, we
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opted to err on the conservative side and chose approximations of the viscous terms that may overestimate dissipation.
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Despite being ultimately caused by viscous dissipation, for high Reynolds numbers the drag is independent of the
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viscosity and scales with the second power of the velocity. It can be written as:
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.. math::
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\begin{align*}
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\mathbf{f}_\text{D} = - C_D~\rho~ A_D ~ \|\mathbf{v}\|~ \mathbf{v}\\
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\mathbf{g}_\text{D} = - C_D \rho~ I_D ~ \|\boldsymbol{\omega}\| ~ \boldsymbol{\omega}
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\end{align*}
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Where :math:`C_D` is a drag coefficient, and :math:`A_D` is a reference surface area (e.g. a measure of the projected
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area on the plane normal to the flow), and :math:`I_D` a reference moment of inertia.
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.. youtube:: nljr0X79vI0
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:align: right
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:width: 50%
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Even for simple shapes, the terms :math:`C_D`, :math:`A_D` and :math:`I_D` need to be tuned to the problem-specific
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physics and dynamical scales :cite:p:`duan2015`. For example, the drag coefficient :math:`C_D` generally decreases with
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increasing Reynolds numbers, and a single reference area :math:`A_D` may not be sufficient to account for the skin
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drag for highly irregular or slender bodies. For example, experimental fits are derived from problems ranging from
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falling playing cards :cite:p:`wang2004,andersen2005a,andersen2005b` to particle transport :cite:p:`loth2008,
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bagheri2016`. See screen capture of the
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`cards.xml <https://github.com/deepmind/mujoco/blob/main/model/card/cards.xml>`__ model on the right.
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We derive a formula for :math:`\mathbf{f}_\text{D}` based on two surfaces :math:`A^\text{proj}_\mathbf{v}` and
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:math:`A_\text{max}`. The first, :math:`A^\text{proj}_\mathbf{v}`, is the cylindrical projection of the body onto a
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plane normal to the velocity :math:`\mathbf{v}`. The second is the maximum projected surface
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:math:`A_\text{max} = 4 \pi d_{max} d_{min}`.
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.. math::
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\mathbf{f}_\text{D} = - \rho~ \big[ C_{D, \text{blunt}} ~ A^\text{proj}_\mathbf{v} ~ +
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C_{D, \text{slender}}\left(A_\text{max} - A^\text{proj}_\mathbf{v} \right) \big] ~ \|\mathbf{v}\|~ \mathbf{v}
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The formula and derivation for :math:`A^\text{proj}_\mathbf{v}` is given in the :ref:`lemma<flProjection>` above.
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We propose an analogous model for the angular drag. For each Cartesian axis we consider the moment of inertia of the
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maximum swept ellipsoid obtained by the rotation of the body around the axis. The resulting diagonal entries of the
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moment of inertia are:
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.. math::
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\mathbf{I}_{D,ii} = \frac{8\pi}{15} ~d_i ~\max(d_j, ~d_k)^4 .
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Given this reference moment of inertia, the angular drag torque is computed as:
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.. math::
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\mathbf{g}_\text{D} = - \rho ~ \boldsymbol{\omega} ~ \Big( \big[ C_{D, \text{angular}} ~ \mathbf{I}_D ~ +
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C_{D, \text{slender}} \left(\mathbf{I}_\text{max} - \mathbf{I}_D \right) \big] \cdot \boldsymbol{\omega} \Big)
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Here :math:`\mathbf{I}_\text{max}` is a vector with each entry equal to the maximal component of :math:`\mathbf{I}_D`.
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The viscosity :math:`\beta`
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For Reynolds numbers around or below :math:`O(10)`, the drag is best approximated as linear in the flow velocity
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(e.g. Stokes' law). For example, for a sphere the drag force :cite:p:`stokes1850` and torque :cite:p:`lamb1932` are:
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.. math::
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\begin{align*}
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\mathbf{f}_\text{S} &= - 6 \pi r_D \rho ~ \beta \mathbf{v}\\
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\mathbf{g}_\text{S} &= - 8 \pi r_D^3 \rho ~ \beta \boldsymbol{\omega}
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\end{align*}
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Here, :math:`r_D` is the radius of the sphere and :math:`\beta` is the kinematic viscosity of the medium (e.g.
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:math:`1.48~\times 10^{-5}~m^2/s` for ambient-temperature air and :math:`0.89 \times 10^{-4}~m^2/s` for water). Here,
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for simplicity, we estimate the radius of the equivalent sphere as :math:`r_D = (d_x + d_y + d_z)/3`. To make a
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quantitative example, Stokes' law become accurate for room-temperature air if
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:math:`u\cdot l \lesssim 2 \times 10^{-4}~m^2/s`, where :math:`u` is the speed and :math:`l` a characteristic length of
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the body.
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Viscous lift
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~~~~~~~~~~~~
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The Kutta-Joukowski theorem calculates the lift :math:`L` of a two-dimensional body translating in a uniform flow with
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speed `u` as :math:`L = \rho u \Gamma`. Here, :math:`\Gamma` is the circulation around the body. In the next
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subsections we define two sources of circulation and the resulting lift forces.
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Magnus force
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^^^^^^^^^^^^
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.. cssclass:: caption-small
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.. figure:: ../images/computation/magnus.png
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:figwidth: 45%
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:align: right
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Smoke flow visualization of the flow past a rotating cylinder (WikiMedia Commons, CC BY-SA 4.0). Due to viscosity,
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the rotating cylinder deflects the incoming flow upward and receives a downwards force (red arrow).
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The Magnus effect describes the motion of a rotating object moving through a fluid. Through viscous effects, a spinning
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object induces rotation in the surrounding fluid. This rotation deflects the trajectory of the fluid past the object
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(i.e. it causes linear acceleration), and the object receives an equal an opposite reaction. For a cylinder, the Magnus
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force per unit length of the cylinder can be computed as :math:`F_\text{M} / L = \rho v \Gamma`, where :math:`\Gamma`
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is the circulation of the flow caused by the rotation and :math:`v` the velocity of the object. We estimate this force
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for an arbitrary body as:
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.. math::
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\mathbf{f}_{\text{M}} = C_M ~\rho~ V~ \boldsymbol{\omega}\times\mathbf{v} ,
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where :math:`V` is the volume of the body and :math:`C_M` is a coefficient for the force, typically set to 1.
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It's worth making an example. To reduce the number of variables, suppose a body rotating in only one direction, e.g.
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:math:`\boldsymbol{\omega} = \{0, 0, \omega_z\}`, translating along the other two, e.g. :math:`\mathbf{v} = \{v_x, v_y, 0\}`. The
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sum of the force due to added mass and the force due to the Magnus effect along, for example, :math:`x` is:
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.. math::
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\frac{f}{\pi \rho d_z} = v_y \omega_z \left(2 d_x \min\{d_x, d_z\} - (d_x + d_z)^2\right)
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Note that the two terms have opposite signs.
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Kutta condition
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^^^^^^^^^^^^^^^
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A stagnation point is a location in the flow field where the velocity is zero. For a body moving in a flow (in 2D, in
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the frame moving with the body) there are two stagnation points: in the front, where the stream-lines separate to either
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sides of the body, and in the rear, where they reconnect. A moving body with a sharp trailing (rear) edge will generate
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in the surrounding flow a circulation of sufficient strength to hold the rear stagnation point at the trailing edge.
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This is the Kutta condition, a fluid dynamic phenomenon that can be observed for solid bodies with sharp corners, such
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as slender bodies or the trailing edges of airfoils.
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.. cssclass:: caption-small
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.. figure:: ../images/computation/kutta_cond_plate.svg
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:figwidth: 95%
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:align: left
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Sketch of the Kutta condition. Blue lines are streamlines and the two magenta points are the stagnation points. The
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dividing streamline, which connects the two stagnation points, is marked in green. The dividing streamline and the
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body inscribe an area where the flow is said to be "separated" and recirculates within. This circulation produces an
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upward force acting on the plate.
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For a two-dimensional flow sketched in the figure above, the circulation due to the Kutta condition can be estimated as:
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:math:`\Gamma_\text{K} = C_K ~ d_x ~ \| \mathbf{v}\| ~ \sin(2\alpha)`,
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where :math:`C_K` is a lift coefficient, and :math:`\alpha` is the angle between the velocity vector and its projection
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onto the surface. The lift force per unit length can be computed with the Kutta–Joukowski theorem as
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:math:`\mathbf{f}_K / L = \rho \Gamma_\text{K} \times \mathbf{v}`.
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In order to extend the lift force equation to three-dimensional motions, we consider the normal
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:math:`\mathbf{n}_{s, \mathbf{v}} = \{\frac{d_y d_z}{d_x}v_x, \frac{d_z d_x}{d_y}v_y, \frac{d_x d_x}{d_z}v_z\}`
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to the cross-section of the body which generates the body's projection :math:`A^\text{proj}_\mathbf{v}` onto a plane
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normal to the velocity given in the :ref:`lemma<flProjection>` above and the corresponding unit vector
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:math:`\hat{\mathbf{n}}_{s, \mathbf{v}}`.
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We use this direction to decompose :math:`\mathbf{v} = \mathbf{v}_\parallel ~+~ \mathbf{v}_\perp` with
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:math:`\mathbf{v}_\perp = \left(\mathbf{v} \cdot \hat{\mathbf{n}}_{s, \mathbf{v}}\right) \hat{\mathbf{n}}_{s, \mathbf{v}}`.
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We write the lift force as:
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.. math::
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\begin{align*}
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\mathbf{f}_\text{K} &= \frac{C_K~\rho~ A^\text{proj}_\mathbf{v}}{\|\mathbf{v}\|}
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\left( \mathbf{v} \times \mathbf{v}_\parallel\right)\times \mathbf{v} \\
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&= C_K~\rho~ A^\text{proj}_\mathbf{v} \left(\hat{\mathbf{v}} \cdot \hat{\mathbf{n}}_{s, \mathbf{v}}\right)
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\left( \hat{\mathbf{n}}_{s, \mathbf{v}} \times \mathbf{v} \right)\times \mathbf{v}
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\end{align*}
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Here, :math:`\hat{\mathbf{v}}` is the unit-normal along :math:`\mathbf{v}`. Note that the direction of :math:`\hat{\mathbf{n}}_{s,
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\mathbf{v}}` differs from :math:`\hat{\mathbf{v}}` only on the planes where the semi-axes of the body are unequal. So for
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example, for spherical bodies :math:`\hat{\mathbf{n}}_{s, \mathbf{v}} \equiv \hat{\mathbf{v}}` and by construction
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:math:`\mathbf{f}_\text{K} = 0`.
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Let's unpack the relation with an example. Suppose a body with :math:`d_x = d_y` and :math:`d_z \ll d_x`. Note that the vector
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:math:`\hat{\mathbf{n}}_{s, \mathbf{v}} \times \hat{\mathbf{v}}` gives the direction of the circulation induced by the
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deflection of the flow by the solid body. Along :math:`z`, the circulation will be proportional to :math:`\frac{d_y d_z}{d_x}v_x v_y
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- \frac{d_z d_x}{d_y}v_x v_y = 0` (due to :math:`d_x = d_y`). Therefore, on the plane where the solid is blunt, the motion
|
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produces no circulation.
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||
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Now, for simplicity, let :math:`v_x = 0`. In this case also the circulation along :math:`y`, proportional
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to :math:`\frac{d_y d_z}{d_x}v_x v_z - \frac{d_y d_x}{d_y}v_x v_z`, is zero. The only non-zero component of the circulation
|
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will be along :math:`x` and be proportional to :math:`\left(\frac{d_x d_z}{d_y} - \frac{d_x d_y}{d_z}\right) v_y v_z \approx
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\frac{d_x^2}{d_z} v_y v_z`.
|
||
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||
We would have :math:`\mathbf{v}_\parallel = \{v_x, 0, v_z\}` and
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:math:`\Gamma \propto \{d_z v_y v_z, ~ 0,~ - d_x v_x v_y \} / \|\mathbf{v}\|`.
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The motion produces no circulation on the plane where the solid is blunt, and on the other two planes
|
||
the circulation is
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||
:math:`\Gamma \propto r_\Gamma ~ \|\mathbf{v}\|~ \sin(2 \alpha) ~ = ~2 r_\Gamma ~\|\mathbf{v}\| ~\sin(\alpha)~\cos(\alpha)`
|
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with :math:`\alpha` the angle between the velocity and its projection on the body on the plane (e.g. on the plane
|
||
orthogonal to :math:`x` we have :math:`\sin(\alpha) = v_y/\|\mathbf{v}\|` and
|
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:math:`\cos(\alpha) = v_z/\|\mathbf{v}\|`), and :math:`r_\Gamma`, the lift surface on the plane (e.g. :math:`d_z` for
|
||
the plane orthogonal to :math:`x`). Furthermore, the direction of the circulation is given by the cross product (because
|
||
the solid boundary "rotates" the incoming flow velocity towards its projection on the body).
|
||
|
||
Acknowledgements
|
||
~~~~~~~~~~~~~~~~
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||
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The design and implementation of the model in this section are the work of Guido Novati.
|
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|
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References
|
||
~~~~~~~~~~
|
||
|
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.. bibliography::
|