reconstructLagrangianAcceleration

There are currently 4 main versions.

• Version 1: \( \frac{d\mathbf{v}}{dt} = ((\mathbf{v}) \cdot \nabla) \cdot (\mathbf{v})\) -> quasistationary approach
• Version 2: \( \frac{d\mathbf{v}}{dt} = \frac{( \mathbf{v}(t) - \mathbf{v}(t-\Delta t) )}{\Delta t} + \left( \mathbf{v}_{Euler}^T \nabla \right) \mathbf{v}\) -> dynamic approach, in case of EULER or EULERIMPL, \( \mathbf{v}_{Euler}\) is given by %ind_v_Euler(1)%
• Version 3: \( \frac{d\mathbf{v}}{dt} = \nabla^T \cdot \left( ( \mathbf{v}) ( \mathbf{v} )^T \right) - ( \mathbf{v} ) \left( \nabla^T \mathbf{v} \right)\) -> local quasistationary approach with chain rule in order to isolate the div(v)-part. Formally the same as version 1.
• Version 4: another quasistationary approach, formally the same as versions 1 and 3. \( \frac{du}{dt} = \frac{1}{2} \left( (u)^2 \right)_x + \left( (u)(v) \right)_y + \left( (u)(w) \right)_z - (u) (v)_y - (u) (w)_z\) \( \frac{dv}{dt} = \left( (v)(u) \right)_x + \frac{1}{2} \left( (v)^2 \right)_y + \left( (v)(w) \right)_z - (v) (u)_x - (v) (w)_z\) \( \frac{dw}{dt} = \left( (w)(u) \right)_x + \left( (w)(v) \right)_y + \frac{1}{2} \left( (w)^2 \right)_z - (w) (u)_x - (w) (v)_y\)
• Version 5: void
• Version 6: \( \frac{d\mathbf{v}}{dt} = \frac{ \mathbf{v}(t) - \left( \mathbf{v}(t-\Delta t) - ( (Delta t) \mathbf{v}_{Euler}^T \nabla ) \mathbf{v}(t-\Delta t) \right) }{\Delta t}\)

Note: \( (\mathbf{v})\) and its components \( (u)\), \( (v)\), \( (w)\) might stand for 2 things.

• Usually, it is the true velocity of the fluid, that is \( (\mathbf{v}) = \mathbf{v}\).
• However, if the user defines a reference velocity (see fifth digit of FLIQUID_ConsistentPressure_Version), then we have \( (\mathbf{v}) = \mathbf{v} - \mathbf{v}_{ref}\).

Note: the way of reconstruction of the Lagrangian derivatives has impact ONLY if

• regularization of the pressure system is requested by RegularizeDPA. Here, it impacts the way the target pressure gradient is computed.
• version 4 or 8 is used for ComputationOfPHI (fourth digit of FLIQUID_ConsistentPressure_Version ).
• %BND_none% is used as a boundary condition, as this condition is based on the AlternativeDPA-algorithm, and so this digit impacts the computation of the target pressure gradient.