scholarly journals Regularization of Operator DAEs

2015 ◽  
Author(s):  
Robert Altmann
Keyword(s):  
General Framework ◽  
Stokes Equations ◽  
Differential Algebraic Equations ◽  
Navier Stokes ◽  
Algebraic Equations ◽  
Spatial Discretization ◽  
Navier Stokes Equations ◽  
First Order ◽  
Index 2 ◽  
Differentiation Index

A general framework for the regularization of constrained PDEs, also called operator differential-algebraic equations (DAEs), is presented. For this, we consider semi-explicit systems of first order which includes the Navier-Stokes equations. The proposed reformulation is consistent in the sense that the solution of the PDE remains untouched. However, one can observe improved numerical properties in terms of the sensitivity to perturbations and the fact that a spatial discretization leads to a DAE of lower index, i.e., of differentiation index $1$ instead of differentiation index 2.

scholarly journals A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations

2016 ◽  
Vol 20 (4) ◽  
pp. 1016-1044 ◽  
Author(s):  
Xiaodong Liu ◽  
Yidong Xia ◽  
Hong Luo ◽  
Lijun Xuan
Keyword(s):  
Comparative Study ◽  
Discontinuous Galerkin ◽  
Stokes Equations ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Navier Stokes ◽  
Third Order ◽  
Navier Stokes Equations ◽  
Index 2 ◽  
Runge Kutta

AbstractA comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flows to DNS of turbulent flows, are presented to assess the performance of these schemes. Numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.

On the energy dissipative spatial discretization of the barotropic quasi-gasdynamic and compressible Navier–Stokes equations in polar coordinates

2018 ◽  
Vol 33 (3) ◽  
pp. 199-210 ◽  
Author(s):  
Alexander Zlotnik
Keyword(s):  
Stokes System ◽  
Stokes Equations ◽  
Physical Property ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Uniform Mesh ◽  
Viscous Stress ◽  
Spatial Discretization ◽  
Navier Stokes Equations ◽  
Viscous Stress Tensor

Abstract The barotropic quasi-gasdynamic system of equations in polar coordinates is treated. It can be considered a kinetically motivated parabolic regularization of the compressible Navier–Stokes system involving additional 2nd order terms with a regularizing parameter τ > 0. A potential body force is taken into account. The energy equality is proved ensuring that the total energy is non-increasing in time. This is the crucial physical property. The main result is the construction of symmetric spatial discretization on a non-uniform mesh in a ring such that the property is preserved. The unknown density and velocity are defined on the same mesh whereas the mass flux and the viscous stress tensor are defined on the staggered meshes. Additional difficulties in comparison with the Cartesian coordinates are overcome, and a number of novel elements are implemented to this end, in particular, a self-adjoint and positive definite discretization for the Navier–Stokes viscous stress, special discretizations of the pressure gradient and regularizing terms using enthalpy, non-standard mesh averages for various products of functions, etc. The discretization is also well-balanced. The main results are valid for τ = 0 as well, i.e., for the barotropic compressible Navier–Stokes system.

scholarly journals A multilayer shallow model for dry granular flows with the -rheology: application to granular collapse on erodible beds

2016 ◽  
Vol 798 ◽  
pp. 643-681 ◽  
Author(s):  
E. D. Fernández-Nieto ◽  
J. Garres-Díaz ◽  
A. Mangeney ◽  
G. Narbona-Reina
Keyword(s):  
Transient Flow ◽  
Stokes Equations ◽  
Low Cost ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Runout Distance ◽  
Multilayer Model ◽  
First Order ◽  
Constant Friction ◽  
Erodible Beds

In this work we present a multilayer shallow model to approximate the Navier–Stokes equations with the ${\it\mu}(I)$-rheology through an asymptotic analysis. The main advantages of this approximation are (i) the low cost associated with the numerical treatment of the free surface of the modelled flows, (ii) the exact conservation of mass and (iii) the ability to compute two-dimensional profiles of the velocities in the directions along and normal to the slope. The derivation of the model follows Fernández-Nieto et al. (J. Comput. Phys., vol. 60, 2014, pp. 408–437) and introduces a dimensional analysis based on the shallow flow hypothesis. The proposed first-order multilayer model fully satisfies a dissipative energy equation. A comparison with steady uniform Bagnold flow – with and without the sidewall friction effect – and laboratory experiments with a non-constant normal profile of the downslope velocity demonstrates the accuracy of the numerical model. Finally, by comparing the numerical results with experimental data on granular collapses, we show that the proposed multilayer model with the ${\it\mu}(I)$-rheology qualitatively reproduces the effect of the erodible bed on granular flow dynamics and deposits, such as the increase of runout distance with increasing thickness of the erodible bed. We show that the use of a constant friction coefficient in the multilayer model leads to the opposite behaviour. This multilayer model captures the strong change in shape of the velocity profile (from S-shaped to Bagnold-like) observed during the different phases of the highly transient flow, including the presence of static and flowing zones within the granular column.

scholarly journals Detection of multiple solutions using a mid-cell back substitution technique applied to computational fluid dynamics

2000 ◽  
Vol 214 (11) ◽  
pp. 1401-1407
Author(s):  
S R Kendall ◽  
H V Rao
Keyword(s):  
Multiple Solutions ◽  
Computational Models ◽  
Stokes Equations ◽  
Compressible Fluids ◽  
Navier Stokes ◽  
Algebraic Equations ◽  
Navier Stokes Equations ◽  
Flow Modelling ◽  
Spurious Solutions ◽  
Linear Algebraic Equations

Computational models for fluid flow based on the Navier-Stokes equations for compressible fluids led to numerical procedures requiring the solution of simultaneous non-linear algebraic equations. These give rise to the possibility of multiple solutions, and hence there is a need to monitor convergence towards a physically meaningful flow field. The number of possible solutions that may arise is examined, and a mid-cell back substitution technique (MCBST) is developed to detect and avoid convergence towards apparently spurious solutions. The MCBST was used successfully for flow modelling in micron-sized flow passages, and was found to be particularly useful in the early stages of computation, optimizing the speed of convergence.

Calculation of Rotordynamic Coefficients and Leakage for Annular Gas Seals by Means of Finite Difference Techniques

1989 ◽  
Vol 111 (3) ◽  
pp. 545-552 ◽  
Author(s):  
R. Nordmann ◽  
F. J. Dietzen ◽  
H. P. Weiser
Keyword(s):  
Finite Difference ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Zeroth Order ◽  
Navier Stokes Equations ◽  
First Order ◽  
Fluid Forces ◽  
Rotordynamic Coefficients ◽  
Gas Seals ◽  
Finite Difference Techniques

The compressible flow in a seal can be described by the Navier-Stokes equations in connection with a turbulence model (k–ε model) and an energy equation. By introducing a perturbation analysis in these differential equations we obtain zeroth order equations for the centered position and first order equations for small motions of the shaft about the centered position. These equations are solved by a finite difference technique. The zeroth order equations describe the leakage flow. Integrating the pressure solution of the first order equations yields the fluid forces and the rotordynamic coefficients, respectively.

A Discrete Flux Scheme for Aerodynamic and Hydrodynamic Flows

2011 ◽  
Vol 9 (5) ◽  
pp. 1257-1283 ◽  
Author(s):  
S. C. Fu ◽  
R. M. C. So ◽  
W. W. F. Leung
Keyword(s):  
Thermal Energy ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Moment Equation ◽  
Zero Order ◽  
Distribution Functions ◽  
Navier Stokes ◽  
Order Moment ◽  
Navier Stokes Equations ◽  
First Order

AbstractThe objective of this paper is to seek an alternative to the numerical simulation of the Navier-Stokes equations by a method similar to solving the BGK-type modeled lattice Boltzmann equation. The proposed method is valid for both gas and liquid flows. A discrete flux scheme (DFS) is used to derive the governing equations for two distribution functions; one for mass and another for thermal energy. These equations are derived by considering an infinitesimally small control volume with a velocity lattice representation for the distribution functions. The zero-order moment equation of the mass distribution function is used to recover the continuity equation, while the first-order moment equation recovers the linear momentum equation. The recovered equations are correct to the first order of the Knudsen number(Kn);thus, satisfying the continuum assumption. Similarly, the zero-order moment equation of the thermal energy distribution function is used to recover the thermal energy equation. For aerodynamic flows, it is shown that the finite difference solution of the DFS is equivalent to solving the lattice Boltzmann equation (LBE) with a BGK-type model and a specified equation of state. Thus formulated, the DFS can be used to simulate a variety of aerodynamic and hydrodynamic flows. Examples of classical aeroacoustics, compressible flow with shocks, incompressible isothermal and non-isothermal Couette flows, stratified flow in a cavity, and double diffusive flow inside a rectangle are used to demonstrate the validity and extent of the DFS. Very good to excellent agreement with known analytical and/or numerical solutions is obtained; thus lending evidence to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid flow simulations.

Rotordynamic Coefficients of Circumferentially-Grooved Liquid Seals Using the Averaged Navier-Stokes Equations

1997 ◽  
Vol 119 (3) ◽  
pp. 556-567 ◽  
Author(s):  
Mihai Arghir ◽  
Jean Freˆne
Keyword(s):  
Coordinate Transformation ◽  
Stokes Equations ◽  
Turbulent Viscosity ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
High Reynolds ◽  
Good Agreement

The paper presents a method to calculate the rotordynamic coefficients of circumferentially-grooved liquid seals operating in centered position and turbulent flow regimes. The method is based on the integration of the averaged Navier-Stokes equations and uses a coordinate transformation proposed by Dietzen and Nordmann (1987). The effect of the coordinate transformation on the components of the stress tensor is included in the first order transport equations. To ensure grid independent solutions, numerical boundary conditions for the first-order velocities were formulated using the logarithmic law. The perturbation of the turbulent viscosity was also considered. A pressure recovery effect at the exit section was included in the first order mathematical model. The method is validated by calculations for straight and circumferentially-grooved seals. Comparisons with experimental and theoretical results show a good agreement for straight seals and for seals with few grooves, and a reasonable agreement for severe industrial cases (high Reynolds numbers and large number of grooves).

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