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We offer two models for open boundaries, with the choice depending on the specific problem and flow characteristics near the boundary:
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1.[**Method of characteristics**](@ref method_of_characteristics): The method of characteristics is typically used in problems where tracking of wave propagation
where ``V_a``, ``V_b`` denote the volume of particles ``a`` and ``b`` respectively.
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Note that although in the continuous case ``\nabla p_{\text{background}} = 0``, the discretization is not 0th-order consistent for **non**-uniform particle distribution,
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which means that there is a non-vanishing contribution only when particles are disordered.
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That also means that ``p_{\text{background}}`` occurs as prefactor to correct the trajectory of a particle resulting in uniform pressure distributions.
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Suggested is a background pressure which is in the order of the reference pressure but can be chosen arbitrarily large when the time-step criterion is adjusted.
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The inviscid momentum equation with an additional convection term for a particle moving with ``\tilde{v}`` is
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```math
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\frac{\tilde{\mathrm{d}} \left( \rho v \right)}{\mathrm{d}t} = -\nabla p + \nabla \cdot \bm{A},
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```
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where the tensor ``\bm{A} = \rho v\left(\tilde{v}-v\right)^T`` is a consequence of the modified
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advection velocity and can be interpreted as the convection of momentum with the relative velocity ``\tilde{v}-v``.
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The discretized form of the momentum equation for a particle ``a`` reads as
with the density ``\rho_a``, ``\rho_b`` and the pressure ``p_a``, ``p_b`` of particles ``a`` and ``b`` respectively. ``\bm{A}_a`` and ``\bm{A}_b`` are the convection tensors for particle ``a`` and ``b`` respectively and are given, e.g. for particle ``a``, as ``\bm{A}_a = \rho v_a\left(\tilde{v}_a-v_a\right)^T``.
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