scholarly journals Closed-form shock solutions

2014 ◽  
Vol 745 ◽  
Author(s):  
B. M. Johnson
Keyword(s):  
Mach Number ◽  
Closed Form ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Numerical Algorithms ◽  
Ideal Gas ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Closed Form Solutions ◽  
Nonlinear Character

AbstractIt is shown here that a subset of the implicit analytical shock solutions discovered by Becker and by Johnson can be inverted, yielding several exact closed-form solutions of the one-dimensional compressible Navier–Stokes equations for an ideal gas. For a constant dynamic viscosity and thermal conductivity, and at particular values of the shock Mach number, the velocity can be expressed in terms of a polynomial root. For a constant kinematic viscosity, independent of Mach number, the velocity can be expressed in terms of a hyperbolic tangent function. The remaining fluid variables are related to the velocity through simple algebraic expressions. The solutions derived here make excellent verification tests for numerical algorithms, since no source terms in the evolution equations are approximated, and the closed-form expressions are straightforward to implement. The solutions are also of some academic interest as they may provide insight into the nonlinear character of the Navier–Stokes equations and may stimulate further analytical developments.

scholarly journals Implicit MAC scheme for compressible Navier–Stokes equations: low Mach asymptotic error estimates

2020 ◽  
Author(s):  
David Maltese ◽  
Antonín Novotný
Keyword(s):  
Mach Number ◽  
Initial Data ◽  
Error Estimate ◽  
Strong Solution ◽  
Stokes Equations ◽  
Main Tool ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Mach Number ◽  
Mac Scheme

Abstract We investigate the error between any discrete solution of the implicit marker-and-cell (MAC) numerical scheme for compressible Navier–Stokes equations in the low Mach number regime and an exact strong solution of the incompressible Navier–Stokes equations. The main tool is the relative energy method suggested on the continuous level in Feireisl et al. (2012, Relative entropies, suitable weak solutions, and weak–strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech., 14, 717–730). Our approach highlights the fact that numerical and mathematical analyses are not two separate fields of mathematics. The result is achieved essentially by exploiting in detail the synergy of analytical and numerical methods. We get an unconditional error estimate in terms of explicitly determined positive powers of the space–time discretization parameters and Mach number in the case of well-prepared initial data and in terms of the boundedness of the error if the initial data are ill prepared. The multiplicative constant in the error estimate depends on a suitable norm of the strong solution but it is independent of the numerical solution itself (and of course, on the discretization parameters and the Mach number). This is the first proof that the MAC scheme is unconditionally and uniformly asymptotically stable in the low Mach number regime.

scholarly journals Effects of surfactant on propagation and rupture of a liquid plug in a tube

2019 ◽  
Vol 872 ◽  
pp. 407-437 ◽  
Author(s):  
M. Muradoglu ◽  
F. Romanò ◽  
H. Fujioka ◽  
J. B. Grotberg
Keyword(s):  
Shear Stress ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Thin Liquid Film ◽  
Front Tracking ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Liquid Plug ◽  
Front Tracking Method ◽  
Airway Walls

Surfactant-laden liquid plug propagation and rupture occurring in lower lung airways are studied computationally using a front-tracking method. The plug is driven by an applied constant pressure in a rigid axisymmetric tube whose inner surface is coated by a thin liquid film. The evolution equations of the interfacial and bulk surfactant concentrations coupled with the incompressible Navier–Stokes equations are solved in the front-tracking framework. The numerical method is first validated for a surfactant-free case and the results are found to be in good agreement with the earlier simulations of Fujioka et al. (Phys. Fluids, vol. 20, 2008, 062104) and Hassan et al. (Intl J. Numer. Meth. Fluids, vol. 67, 2011, pp. 1373–1392). Then extensive simulations are performed to investigate the effects of surfactant on the mechanical stresses that could be injurious to epithelial cells, such as pressure and shear stress. It is found that the liquid plug ruptures violently to induce large pressure and shear stress on airway walls and even a tiny amount of surfactant significantly reduces the pressure and shear stress and thus improves cell survivability. However, addition of surfactant also delays the plug rupture and thus airway reopening.

scholarly journals Numerical modelling of bank failures during reservoir draw-down

2018 ◽  
Vol 40 ◽  
pp. 03001 ◽  
Author(s):  
Nils Reidar B. Olsen ◽  
Stefan Haun
Keyword(s):  
Stokes Equations ◽  
Limit Equilibrium ◽  
Numerical Algorithms ◽  
High Viscosity ◽  
Navier Stokes ◽  
Bank Failures ◽  
Navier Stokes Equations ◽  
Order Of Magnitude ◽  
Sediment Layers ◽  
Hydropower Reservoir

Numerical algorithms are presented for modeling bank failures during reservoir flushing. The algorithms are based on geotechnical theory and the limit equilibrium approach to find the location and the depth of the slides. The actual movements of the slides are based on the solution of the Navier-Stokes equations for laminar flow with high viscosity. The models are implemented in the SSIIM computer program, which also can be used for modelling erosion of sediments from reservoirs. The bank failure algorithms are tested on the Bodendorf hydropower reservoir in Austria. Comparisons with measurements show that the resulting slides were in the same order of magnitude as the observed ones. However, some scatter on the locations were observed. The algorithms were stable for thick sediment layers, but instabilities were observed for thin sediment layers.

Validation of a Numerical Method for Unsteady Flow Calculations

1993 ◽  
Vol 115 (1) ◽  
pp. 110-117 ◽  
Author(s):  
M. Giles ◽  
R. Haimes
Keyword(s):  
Heat Transfer ◽  
Flat Plate ◽  
Numerical Method ◽  
Stokes Equations ◽  
Companion Paper ◽  
Ideal Gas ◽  
Euler Method ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Loss Prediction

This paper describes and validates a numerical method for the calculation of unsteady inviscid and viscous flows. A companion paper compares experimental measurements of unsteady heat transfer on a transonic rotor with the corresponding computational results. The mathematical model is the Reynolds-averaged unsteady Navier–Stokes equations for a compressible ideal gas. Quasi-three-dimensionality is included through the use of a variable streamtube thickness. The numerical algorithm is unusual in two respects: (a) For reasons of efficiency and flexibility, it uses a hybrid Navier–Stokes/Euler method, and (b) to allow for the computation of stator/rotor combinations with arbitrary pitch ratio, a novel space–time coordinate transformation is used. Several test cases are presented to validate the performance of the computer program, UNSFLO. These include: (a) unsteady, inviscid flat plate cascade flows (b) steady and unsteady, viscous flat plate cascade flows, (c) steady turbine heat transfer and loss prediction. In the first two sets of cases comparisons are made with theory, and in the third the comparison is with experimental data.

scholarly journals On the Low Mach Number Limit for Quantum Navier--Stokes Equations

2020 ◽  
Vol 52 (6) ◽  
pp. 6105-6139
Author(s):  
Paolo Antonelli ◽  
Lars Eric Hientzsch ◽  
Pierangelo Marcati
Keyword(s):  
Mach Number ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Mach Number ◽  
Low Mach Number Limit

scholarly journals A numerical evaluation of the asymptotic theory of receptivity for subsonic compressible boundary layers

2015 ◽  
Vol 771 ◽  
pp. 520-546 ◽  
Author(s):  
Nicola De Tullio ◽  
Anatoly I. Ruban
Keyword(s):  
Mach Number ◽  
Boundary Layers ◽  
Asymptotic Theory ◽  
Stokes Equations ◽  
Roughness Element ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Compressible Boundary Layers ◽  
Tollmien Schlichting ◽  
Low Amplitude

The capabilities of the triple-deck theory of receptivity for subsonic compressible boundary layers have been thoroughly investigated through comparisons with numerical simulations of the compressible Navier–Stokes equations. The analysis focused on the two Tollmien–Schlichting wave linear receptivity problems arising due to the interaction between a low-amplitude acoustic wave and a small isolated roughness element, and the low-amplitude time-periodic vibrations of a ribbon placed on the wall of a flat plate. A parametric study was carried out to look at the effects of roughness element and vibrating ribbon longitudinal dimensions, Reynolds number, Mach number and Tollmien–Schlichting wave frequency. The flat plate is considered isothermal, with a temperature equal to the laminar adiabatic-wall temperature. Numerical simulations of the full and the linearised compressible Navier–Stokes equations have been carried out using high-order finite differences to obtain, respectively, the steady basic flows and the unsteady disturbance fields for the different flow configurations analysed. The results show that the asymptotic theory and the Navier–Stokes simulations are in good agreement. The initial Tollmien–Schlichting wave amplitudes and, in particular, the trends indicated by the theory across the whole parameter space are in excellent agreement with the numerical results. An important finding of the present study is that the behaviour of the theoretical solutions obtained for $\mathit{Re}\rightarrow \infty$ holds at finite Reynolds numbers and the only conditions needed for the theoretical predictions to be accurate are that the receptivity process be linear and the free-stream Mach number be subsonic.

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