Field Histories: Pressure Time Derivative

A field history is a type of field monitor that stores a finite number of past values (or samples) of a field function, captured at moments defined by its update policy, for a given set of input regions or boundaries (or both).

This tutorial demonstrates the field histories feature of Simcenter STAR-CCM+ using the simulation file that was created in the tutorial, Rigid Body Motion: Rotating Fan.

Field histories provide the ability to compute derivatives of solution quantities, for example using turbulent kinetic energy time derivatives to identify power fluctuations. A pressure time derivative, d p / d t , can help investigate acoustics, among other uses.

In this tutorial, a pressure time derivative is calculated for a rotating radial fan [1002].

From basic transport equations, the substantial or material derivative for pressure is:

D p D t = p t + v p

Hence to obtain the pressure derivative in a simulation with mesh motion, you must evaluate:

p t = D p D t v p

In order to compute the velocity-gradient term v p , you must first obtain the mesh velocity in each cell. The mesh velocity in each cell can be obtained using a field history that samples Position, and for which the resulting field functions are the current time-step, HistoryofPositionSample0, and the two previous time-steps, HistoryofPositionSample1 and HistoryofPositionSample2. These are combined in a field function MeshVelocity as:

(3*$${HistoryofPositionSample0} - 4*$${HistoryofPositionSample1} + $${HistoryofPositionSample2})/(2*${TimeStep})

The velocity-gradient term can be computed by a field function, Pconvective_current:

dot($${MeshVelocity},grad(${Pressure}))

Where Pressure is the built-in solution field function.

For computing the final pressure time derivative d(Pressure)/dt, the expression is:

(3*${HistoryofPressureSample0} - 4*${HistoryofPressureSample1} + ${HistoryofPressureSample2})/(2*${TimeStep}) - ${Pconvective_current}

For MeshVelocity and d(Pressure)/dt, a second order backward-differencing formulation is used. This formulation comes from [1002].

[1002]
James, M.L., Smith, G.M., and Wolford, J.C. 1977. "Applied Numerical Methods for Digital Computation with FORTRAN and CSMP", 2nd ed. Harper & Row.