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Hydrodynamic/acoustic splitting

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In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the perturbed compressible ones as, Failed to parse (syntax error): \rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) \\ \vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t) \\ p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)

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