Numerical simulation of pipe flows with uncertain parameters
Authors: Sonja Schmelter, André Fiebach (PTB)
Pipe flows are influenced by a variety of different parameters. Among these are uncertain initial and boundary conditions (for the temperature, the pressure, or the velocity profile, for example), manufacturing tolerances and inaccuracies (like wall roughness, angles, or radii) as well as material properties (density, viscosity, etc.), which may influence the accuracy of volume flow measurements. In practice, it has to be ensured that the meter is working within the prescribed tolerances also for disturbed inflow conditions. Thus, the disturbances due to installation effects have to be decayed below certain acceptable tolerances.
Within the EMRP NEW-04 project "Novel mathematical and statistical approaches to uncertainty evaluation'' the non-intrusive generalized polynomial chaos method [LeMaitre_Knio, Xiu_Karn] is used in conjunction with classical flow simulations. The idea of the polynomial chaos method is that every random variable with finite second order moment can be developed in a series of orthogonal polynomials. There exists a connection between the distribution of the random variable and the orthogonal ansatz polynomials. For a uniformly distributed random variable, the family of Legendre polynomials is used, whereas for a normal distribution the corresponding functions are the Hermite polynomials, for example.
The expectation value as well as the variance of the output of interest can directly be derived from the modes of the series expansion. The modes are determined by orthogonal projection in conjunction with numerical cubature, which means that the deterministic problem is evaluated at fixed and a-priori known nodes. Hence, an existing solver can be used.
In the following the method is illustrated using a pipe flow with uncertain inflow conditions, see [Schmelter].
Possibly uncertain parameters and inlet conditions for a pipe flow configuration
The below figure shows the spread of the velocity profiles and their expectation value as well as variance for the propagation of an idealized disturbed inlet profile at the pipe entrance and outlet, respectively.
Since the maximum of the inlet profiles tends to the left, also the two moments tend to the left at the inlet. The disturbance of the inlet profile decays with increasing pipe length. Further downstream the flow profile gets fully developed so that only the volumetric velocity is varying. Thus, the expectation value approaches the norm profile of Gersten and Herwig [Gersten_Herwig]. The variance reflects the different volumetric velocities.
The polynomial chaos method is applicable to a wide range of problems, where the influence of uncertainties within process conditions or variations of material parameters needs to be quantified. In many cases the method needs less evaluations of the underlying deterministic problem than a classical Monte Carlo approach. Therefore it is suitable for computationally expensive problems.
- D. Xiu and G. E. Karniadakis. “The Wiener-Askey polynomial chaos for stochastic differential equations”. In: SIAM 24 (2002), pp. 1118–1139.
- K. Gersten and G. Herwig. Strömungsmechanik: Grundlagen der Impuls- , Wärme- und Stoffübertragung aus asymptotischer Sicht. Braunschweig: Vieweg, 1992.
- O. P. Le Maître and O. M. Knio. Spectral Methods for Uncertainty Quantification. Dordrecht: Springer, 2010.
- S. Schmelter et al. “Numerical prediction of pipe flows with uncertain inflow conditions”. In: Computational Fluid Dynamics 29 (6-8) (2015), pp. 411-422.