The structure of a weak shock wave undergoing reflexion from a wall

1968 ◽  
Vol 31 (3) ◽  
pp. 501-528 ◽  
Author(s):  
M. B. Lesser ◽  
R. Seebass
Keyword(s):  
Shock Wave ◽  
Boundary Condition ◽  
Burgers Equation ◽  
Stokes Equations ◽  
Weak Shock ◽  
Weak Shock Wave ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
First Order ◽  
Isothermal Wall

The Navier–Stokes equations are used to study the unsteady structure of a weak shock wave reflecting from a plane wall. Both an adiabatic and an isothermal wall are considered. Incident and reflected shock structures are found by expanding the dependent variables in asymptotic series in the shock strength; the first-order terms are shown to satisfy an equation analogous to Burgers equation. The structure of the wave during reflexion is obtained from an expansion in which the first-order terms satisfy the acoustic equations. The isothermal wall boundary condition requires the introduction of a thermal layer adjacent to the wall. In this case viscosity and convection play a role secondary to the wall temperature boundary condition in determining the structure of the reflected wave. The presentation is simplified by introducing a generalized Burgers equation that gives the same first-order results as the Navier–Stokes equations. Correct second-order results are obtained from this equation simply by applying a correction to the result for the temperature.

A Conservative Isothermal Wall Boundary Condition for the Compressible Navier–Stokes Equations

2005 ◽  
Vol 30 (2) ◽  
pp. 177-192 ◽  
Author(s):  
G. B. Jacobs ◽  
D. A. Kopriva ◽  
F. Mashayek
Keyword(s):  
Boundary Condition ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Wall Boundary Condition ◽  
Wall Boundary ◽  
Isothermal Wall

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

scholarly journals Involving the Navier–Stokes equations in the derivation of boundary conditions for the lattice Boltzmann method

2011 ◽  
Vol 369 (1944) ◽  
pp. 2184-2192 ◽  
Author(s):  
Joris C. G. Verschaeve
Keyword(s):  
Boundary Condition ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Grid Spacing ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Boltzmann Method

By means of the continuity equation of the incompressible Navier–Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.

Self-Excited Oscillation of Transonic Flow Around an Airfoil in Two-Dimensional Channel

1989 ◽  
Author(s):  
Kazuomi Yamamoto ◽  
Yoshimichi Tanida
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Transonic Flow ◽  
Stokes Equations ◽  
Flow Region ◽  
Boundary Layer Separation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Layer Separation ◽  
Excited Oscillation

A self-excited oscillation of transonic flow in a simplified cascade model was investigated experimentally, theoretically and numerically. The measurements of the shock wave and wake motions, and unsteady static pressure field predict a closed loop mechanism, in which the pressure disturbance, that is generated by the oscillation of boundary layer separation, propagates upstream in the main flow and forces the shock wave to oscillate, and then the shock oscillation disturbs the boundary layer separation again. A one-dimensional analysis confirms that the self-excited oscillation occurs in the proposed mechanism. Finally, a numerical simulation of the Navier-Stokes equations reveals the unsteady flow structure of the reversed flow region around the trailing edge, which induces the large flow separation to bring about the anti-phase oscillation.

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.

Transonic nozzle flow of dense gases

1993 ◽  
Vol 247 ◽  
pp. 661-688 ◽  
Author(s):  
A. Kluwick
Keyword(s):  
Stokes Equations ◽  
Equations Of State ◽  
Weak Shock ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Perturbation Equation ◽  
Navier Stokes Equations ◽  
Sonic Point ◽  
Specific Heats ◽  
Dense Gases

The paper deals with the flow properties of dense gases in the throat area of slender nozzles. Starting from the Navier–Stokes equations supplemented with realistic equations of state for gases which have relatively large specific heats a novel form of the viscous transonic small-perturbation equation is derived. Evaluation of the inviscid limit of this equation shows that three sonic points rather than a single sonic point may occur during isentropic expansion of such media, in contrast to the case of perfect gases. As a consequence, a shock-free transition from subsonic to supersonic speeds cannot, in general, be achieved by means of a conventional converging–diverging nozzle. Nozzles leading to shock-free flow fields must have an unusual shape consisting of two throats and an intervening antithroat. Additional new results include the computation of the internal thermoviscous structure of weak shock waves and a phenomenon referred to as impending shock splitting. Finally, the relevance of these results to the description of external transonic flows is discussed briefly.

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