POLYFLOW - On the meaning of the pressure (also negative pressure) for an incompressible fluid
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Simulations defined under carefully selected boundary conditions may still exhibit pressure values below the atmospheric one. What does this mean? The pressure field can exhibit very large values near corners. Why? There can be several explanations to this observation. Pressure definition and Lagrange multiplier For an incompressible fluid, the pressure variable (p) does not necessarily correspond to the pressure that could be experimentally measured with an appropriate device. The experimentally measured quantity corresponds actually to a given component of the force density applied on a given surface (defined by its normal vector). For further details, see the public FAQ located at <a target=_blank href="http://www.fluentusers.com/polyflow/FAQ/poly40.htm">http://www.fluentusers.com/polyflow/FAQ/poly40.htm</a>http://www.fluentusers.com/polyflow/FAQ/poly40.htm There is probably a historical reason for the confusion: for an incompressible Newtonian fluid in a simple Poiseuille flow, the calculated pressure corresponds to the one that could be measured. Notwithstanding, for a general complex flow, the pressure variable should at first be understood as the unknown (Lagrange multiplier) associated with the incompressibility constraint (divergence free velocity). Pressure in extensional flow In a flow involving extension, it is still possible that the properly calculated pressure field be below the atmospheric value. To illustrate this, let us consider a simple extensional flow along the z-direction for a Newtonian fluid filament. We assume that a zero force is applied on the border. The total stress tensor S is diagonal, and is given by: Szz = -p + 2 eta . e-dot , Sxx = Syy = -p - eta . e-dot , where eta is the viscosity, while e-dot is the extension rate (positive). Since we assume that a zero force is applied on the border of the filament, we have: p = -eta . e-dot Hence, the pressure can be as negative as one wants.Integration issues The total stress tensor S in an extensional flow is obtained as the difference between two large quantities. In the previous example, it is obtained as the difference betwen the pressure p and the xx-component of the extra-stress tensor. For a Newtonian flow calculation, the pressure p and the velocity v are primary unknowns; the extra-stress tensor is calculated as a post-processor by derivating the velocity field on individual elements. This process may be affected by the discretisation. Also, numerical integration rules differ for the calculation of the primary unknowns and for the post-calculated quantities.In the vicinity of singularities, such as discontinuity of boundary conditions or re-entrant corners, the stress tensor may show unbounded values (although they are still integrable for a Newtonian fluid). This has been shown in a former paper by H.K. Moffat, "Viscous and resistive eddies near a sharp corner", J. Fluid Mech. 18 (1964) 1-18. |
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