106 research outputs found
Constraint-consistent Runge-Kutta methods for one-dimensional incompressible multiphase flow
New time integration methods are proposed for simulating incompressible
multiphase flow in pipelines described by the one-dimensional two-fluid model.
The methodology is based on 'half-explicit' Runge-Kutta methods, being explicit
for the mass and momentum equations and implicit for the volume constraint.
These half-explicit methods are constraint-consistent, i.e., they satisfy the
hidden constraints of the two-fluid model, namely the volumetric flow
(incompressibility) constraint and the Poisson equation for the pressure. A
novel analysis shows that these hidden constraints are present in the
continuous, semi-discrete, and fully discrete equations.
Next to constraint-consistency, the new methods are conservative: the
original mass and momentum equations are solved, and the proper shock
conditions are satisfied; efficient: the implicit constraint is rewritten into
a pressure Poisson equation, and the time step for the explicit part is
restricted by a CFL condition based on the convective wave speeds; and
accurate: achieving high order temporal accuracy for all solution components
(masses, velocities, and pressure). High-order accuracy is obtained by
constructing a new third order Runge-Kutta method that satisfies the additional
order conditions arising from the presence of the constraint in combination
with time-dependent boundary conditions.
Two test cases (Kelvin-Helmholtz instabilities in a pipeline and liquid
sloshing in a cylindrical tank) show that for time-independent boundary
conditions the half-explicit formulation with a classic fourth-order
Runge-Kutta method accurately integrates the two-fluid model equations in time
while preserving all constraints. A third test case (ramp-up of gas production
in a multiphase pipeline) shows that our new third order method is preferred
for cases featuring time-dependent boundary conditions
Structure-Preserving Hyper-Reduction and Temporal Localization for Reduced Order Models of Incompressible Flows
A novel hyper-reduction method is proposed that conserves kinetic energy and
momentum for reduced order models of the incompressible Navier-Stokes
equations. The main advantage of conservation of kinetic energy is that it
endows the hyper-reduced order model (hROM) with a nonlinear stability
property. The new method poses the discrete empirical interpolation method
(DEIM) as a minimization problem and subsequently imposes constraints to
conserve kinetic energy. Two methods are proposed to improve the robustness of
the new method against error accumulation: oversampling and Mahalanobis
regularization. Mahalanobis regularization has the benefit of not requiring
additional measurement points. Furthermore, a novel method is proposed to
perform structure-preserving temporal localization with the principle interval
decomposition: new interface conditions are derived such that energy and
momentum are conserved for a full time-integration instead of only during
separate intervals. The performance of the new structure-preserving
hyper-reduction methods and the structure-preserving temporal localization
method is analysed using two convection-dominated test cases; a shear-layer
roll-up and two-dimensional homogeneous isotropic turbulence. It is found that
both Mahalanobis regularization and oversampling allow hyper-reduction of these
test cases. Moreover, the Mahalanobis regularization provides comparable
robustness while being more efficient than oversampling
Constraint-consistent Runge-Kutta methods for one-dimensional incompressible multiphase flow
New time integration methods are proposed for simulating incompressible multiphase flow in
pipelines described by the one-dimensional two-fluid model. The methodology is based on ‘halfexplicit’
Runge-Kutta methods, being explicit for the mass and momentum equations and implicit
for the volume constraint. These half-explicit methods are constraint-consistent, i.e., they satisfy
the hidden constraints of the two-fluid model, namely the volumetric flow (incompressibility)
constraint and the Poisson equation for the pressure. A novel analysis shows that these hidden
constraints are prese
Analysis of shock relations for steady potential flow models
Potential flow models remain to be practically relevant, for both physical and numerical reasons. Detailed knowledge of their difference with rotational and viscous flow models is still important. In the present paper, this knowledge is reviewed and extended. Normal and oblique shock relations for the steady full potential equation and steady transonic small disturbance equation are derived. Among others, the deficiencies in conservation of mass and momentum across shock waves are analyzed in detail for these potential flow models. By comparison with the shock relations for the Euler equations guidelines are offered for the applicability of potential flow models in numerical simulations. Furthermore, the analytical expressions derived here may serve for verification of numerical methods
An adaptive minimum spanning tree multi-element method for uncertainty quantification of smooth and discontinuous responses
A novel approach for non-intrusive uncertainty propagation is proposed. Our
approach overcomes the limitation of many traditional methods, such as
generalised polynomial chaos methods, which may lack sufficient accuracy when
the quantity of interest depends discontinuously on the input parameters. As a
remedy we propose an adaptive sampling algorithm based on minimum spanning
trees combined with a domain decomposition method based on support vector
machines. The minimum spanning tree determines new sample locations based on
both the probability density of the input parameters and the gradient in the
quantity of interest. The support vector machine efficiently decomposes the
random space in multiple elements, avoiding the appearance of Gibbs phenomena
near discontinuities. On each element, local approximations are constructed by
means of least orthogonal interpolation, in order to produce stable
interpolation on the unstructured sample set. The resulting minimum spanning
tree multi-element method does not require initial knowledge of the behaviour
of the quantity of interest and automatically detects whether discontinuities
are present. We present several numerical examples that demonstrate accuracy,
efficiency and generality of the method.Comment: 20 pages, 18 figure
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