Shocks generated in a confined gas due to rapid heat addition at the boundary. II. Strong shock waves

1984 ◽  
Vol 393 (1805) ◽  
pp. 331-351 ◽  
Keyword(s):  
Shock Wave ◽  
Ad Hoc ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Strong Shock ◽  
Mean Free Path ◽  
Present Theory ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Heat Addition

A study is made of the birth and evolution of a strong shock wave in an inert gas due to rapid energy deposition at a boundary. The gas is confined between infinite parallel planes separated by a distance large compared with the molecular mean free path. Heat flux at the wall rises from zero to a finite constant value during an interval that is a modest multiple of the intermolecular collision time. The thermomechanical response of the gas near the boundary is described by the complete Navier-Stokes equations in a layer with a thickness that is a few molecular mean free paths. Numerical solutions show how a spatial pressure variation is generated adjacent to the boundary, which then propagates away as an almost steady shock wave. If heat addition is continued a thicker high-temperature expanding layer develops in which the pressure remains uniform. This expanding layer acts like a piston, or a contact surface, the speed of which is calculated to leading order. In this way the present theory provides a rational basis for the ad hoc piston analogies used by earlier authors. In particular it shows the importance of power as a crucial factor in the determination of shock strength.

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.

A numerical study of the interaction between unsteay free-stream disturbances and localized variations in surface geometry

1989 ◽  
Vol 209 ◽  
pp. 285-308 ◽  
Author(s):  
R. J. Bodonyi ◽  
W. J. C. Welch ◽  
P. W. Duck ◽  
M. Tadjfar
Keyword(s):  
Numerical Solutions ◽  
Free Stream ◽  
Stokes Equations ◽  
Numerical Study ◽  
Navier Stokes ◽  
Surface Geometry ◽  
Navier Stokes Equations ◽  
S Waves ◽  
Tollmien Schlichting

A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

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.

Pulsatile Waterhammer Subject to Laminar Friction

1972 ◽  
Vol 94 (2) ◽  
pp. 467-472 ◽  
Author(s):  
D. A. P. Jayasinghe ◽  
H. J. Leutheusser
Keyword(s):  
Stokes Equations ◽  
Fluid Motion ◽  
Present Theory ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Arbitrary Velocity ◽  
Velocity Input ◽  
Special Case ◽  
Signalling Problem

This paper deals with elastic waves which may be generated in a fluid by the sudden movement of a flow boundary. In particular, an analysis of the classical piston, or signalling problem is presented for the special case of arbitrary velocity input into a stationary fluid contained in a circular, semi-infinite waveguide. The decay of the pulse, as well as the resulting flow development in the inlet region of the pipe are analyzed by means of an asymptotic expansion of the suitably nondimensionalized Navier-Stokes equations for a compressible, nonheat-conducting Newtonian fluid. The results differ significantly from those of the more conventional one-dimensional approach based on the so-called telegrapher’s equation of mathematical physics. The present theory realistically predicts the growth of a boundary layer both in time and position and, hence, it appears to represent the transient fluid motion in a manner which is physically more appealing.

scholarly journals Mathematical Model for the Electrokinetic Instability of Electrolyte Flow

2019 ◽  
Vol 224 ◽  
pp. 02003
Author(s):  
Andrey Shobukhov
Keyword(s):  
Electric Potential ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Fluid Velocity ◽  
Steady State Solution ◽  
Stability Loss ◽  
Navier Stokes ◽  
Ion Distribution ◽  
Navier Stokes Equations ◽  
Dilute Aqueous Solution

We study a one-dimensional model of the dilute aqueous solution of KCl in the electric field. Our model is based on a set of Nernst-Planck-Poisson equations and includes the incompressible fluid velocity as a parameter. We demonstrate instability of the linear electric potential variation for the uniform ion distribution and compare analytical results with numerical solutions. The developed model successfully describes the stability loss of the steady state solution and demonstrates the emerging of spatially non-uniform distribution of the electric potential. However, this model should be generalized by accounting for the convective movement via the addition of the Navier-Stokes equations in order to substantially extend its application field.

scholarly journals Aerodynamic Dissipation

1958 ◽  
Vol 8 ◽  
pp. 966-974
Author(s):  
H. E. Petschek
Keyword(s):  
Stokes Equations ◽  
Viscosity Coefficient ◽  
Mean Free Path ◽  
Navier Stokes ◽  
Small Scale ◽  
Minimum Length ◽  
Ionized Gas ◽  
Navier Stokes Equations ◽  
Flow Equations ◽  
Rate Of Dissipation

Analyses of aerodynamic dissipation in ordinary un-ionized gases are all based upon the Navier-Stokes equations. These equations relate the rate of dissipation to the local gradients in velocity and temperature through the viscosity and heat conduction coefficients. Although it is true that in many flow situations the magnitude of the total dissipation in the gas does not depend on the magnitude of the viscosity coefficient, this coefficient does determine the minimum scale of variations observed in the gas and the form of the Navier-Stokes equations determines the type of phenomena which are observed on a small scale. In order to discuss dissipation in an ionized gas in the presence of a magnetic field, it is therefore necessary to re-examine the derivation of the basic flow equations. This paper attempts to do this for a case of a completely ionized gas and demonstrates that the basic microscopic dissipation mechanism is appreciably different. For example, it is shown that the minimum length in which the properties of the flow field can change noticeably is appreciably less than one mean free path.

Reliability of Convective Diffusion Process in Stenosis Blood Vessels

2008 ◽  
Vol 3 (1) ◽  
Author(s):  
R.K. Saket ◽  
Anil Kumar
Keyword(s):  
Mass Transport ◽  
Diffusion Process ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Wall Stress ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Flow Equations ◽  
Recirculation Flow ◽  
Convection Diffusion Equation

This paper presents a convective dominated reliable diffusion process in an axi-symmetric tube with a local constriction simulating a stenos artery considering the porosity effects. The investigations demonstrate the effects of wall shear stress and recirculation flow on the concentration distribution in the vessels lumen and on wall mass transfer keeping the porosity in view. The flow is governed by the incompressible Navier-Stokes equations for Newtonian fluid in porous medium. The convection diffusion equation has been used for the mass transport. The effect of porosity is examined on the velocity field and wall stress. The numerical solutions of the flow equations and the coupled mass transport equations have been obtained using a finite difference method. This paper explains the reliable effects of flow porosity on the mass transport.

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