Shock formation in the presence of entropy gradients

2001 ◽  
Vol 431 ◽  
pp. 161-188 ◽  
Author(s):  
HAO LIN ◽  
ANDREW J. SZERI
Keyword(s):  
Flow Field ◽  
Sound Speed ◽  
Compression Wave ◽  
Method Of Characteristics ◽  
Principal Result ◽  
Polar Coordinates ◽  
Shock Formation ◽  
Analytical Criterion ◽  
Energy Focusing ◽  
Normal Compression

The steepening of a normal compression wave into a shock in a homentropic flow field is understood well through the method of characteristics. In a non-homentropic flow field, however, shock formation from a compression wave is more complex. The effects of entropy (or sound speed) gradients on shock formation from a compression wave are determined using a wave front expansion in Cartesian and in spherical polar coordinates. The latter problem has application to the intense energy focusing of sonoluminescence, particularly when applied to a spherically collapsing gas. The principal result is an analytical criterion for the time and place of shock formation, for a wave propagating into a field of smoothly varying entropy.

Chemical amplification at the wave head of a finite amplitude gasdynamic disturbance

1977 ◽  
Vol 81 (2) ◽  
pp. 257-264 ◽  
Author(s):  
J. F. Clarke
Keyword(s):  
Shock Wave ◽  
Sound Speed ◽  
Method Of Characteristics ◽  
Ignition Time ◽  
Finite Amplitude ◽  
General State ◽  
Expansion Waves ◽  
Shock Wave Formation ◽  
Reactive Mixture ◽  
Background State

Consider a background state which consists of a spatially uniform chemically reactive mixture in a general state of disequilibrium. The analytical method of characteristics is used to show that a plane finite amplitude disturbance propagates through this system at the frozen sound speed and, if the degree of disequilibrium is sufficient, is amplified by the chemical reaction. Some comments are made about the time to shock-wave formation and its relation to the homogeneous explosion ignition time, and also about expansion waves, which are found to have a tendency towards fixed-strength ‘quenching waves’, their strength being proportional to the extent of the ambient disequilibrium.

Shock Waves in Mathematical Models of the Aorta

1970 ◽  
Vol 37 (1) ◽  
pp. 34-37 ◽  
Author(s):  
George Rudinger
Keyword(s):  
Mathematical Model ◽  
Blood Flow ◽  
Shock Waves ◽  
Method Of Characteristics ◽  
Wave Form ◽  
Shock Formation ◽  
Rapid Flow ◽  
Latex Tube ◽  
Flow Changes ◽  
The Mathematical Model

If the nonlinear equations for nonsteady blood flow are solved by the method of characteristics, shock discontinuities may develop as a result of omitting from the mathematical model some aspect of the system that becomes significant at rapid flow changes. As an illustration, the flow from the heart into the aorta at the beginning of systole is analyzed. An equation is derived which yields shock formation distances between a few centimeters and several meters depending on the elastic properties of the aorta. Since knowledge of the actual wave form would be useful for computer programming, a few exploratory experiments were performed with an unrestrained latex tube. They indicated wave transitions extending over several tube diameters, but maximum steepening of the wave has not yet been achieved.

The Okubo-Weiss criterion in hydrodynamic flows:Geometric aspects and further extension

2022 ◽  
Author(s):  
Bhimsen Shivamoggi ◽  
G Heijst ◽  
Leon Kamp
Keyword(s):  
Flow Field ◽  
Polar Coordinates ◽  
Coriolis Parameter ◽  
Laminar Jet ◽  
Type Criterion ◽  
Burgers Vortex ◽  
2D Flow ◽  
Flow Configurations ◽  
Elliptic Regions ◽  
Oseen Vortex

Abstract The Okubo [5]-Weiss [6] criterion has been extensively used as a diagnostic tool to divide a two-dimensional (2D) hydrodynamical flow field into hyperbolic and elliptic regions and to serve as a useful qualitative guide to the complex quantitative criteria. The Okubo-Weiss criterion is frequently validated on empirical grounds by the results ensuing its application. So, we will explore topological implications into the Okubo-Weiss criterion and show the Okubo-Weiss parameter is, to within a positive multiplicative factor, the negative of the Gaussian curvature of the underlying vorticity manifold. The Okubo-Weiss criterion is reformulated in polar coordinates, and is validated via several examples including the Lamb- Oseen vortex, and the Burgers vortex. These developments are then extended to 2D quasi- geostrophic (QG) flows. The Okubo-Weiss parameter is shown to remain robust under the -plane approximation to the Coriolis parameter. The Okubo-Weiss criterion is shown to be able to separate the 2D flow-field into coherent elliptic structures and hyperbolic flow configurations very well via numerical simulations of quasi-stationary vortices in QG flows. An Okubo-Weiss type criterion is formulated for 3D axisymmetric flows, and is validated via application to the round Landau-Squire Laminar jet flow.

Fluid Compressibility Effects in Steam Hammer Analyses

2015 ◽  
Author(s):  
Beniamino Rovagnati ◽  
John H. Gray
Keyword(s):  
Incompressible Flow ◽  
Sound Speed ◽  
Pressure Wave ◽  
Method Of Characteristics ◽  
Wave Length ◽  
Valve Closure ◽  
The Method Of Characteristics ◽  
Fluid Transient ◽  
Transient Loads ◽  
Compressibility Effects

In the power industry, steam hammer piping analyses (and fluid transient loads) are often based on incompressible flow principles. Consequently, it is common to use either computer programs based on the Method of Characteristics for incompressible flow or simple hand calculations such as that described by E. C. Goodling (ASME PVP, Vol. 149–157, 1989). Goodling’s paper provides a simplified method to visualize and estimate transient loads resulting from stop valve closure in a main steam system. To account for compressibility effects, the calculated loads are sometimes adjusted by using a correction factor directly applied to the fluid loads with the fluid sound speed assumed constant in time and space. For example, Goodling’s paper suggests that a load increase of 5% may be used. In this paper, a sample steam hammer problem is solved using the Method of Characteristics for compressible flows, where the fluid is assumed to behave as a perfect gas. The effects of steam compressibility are then discussed. Specifically the shortening of the characteristic valve closure pressure wave length, resulting from the increasing magnitude of the sound speed as the pressure wave moves upstream from the closing valve, and resulting higher loads in straight pipe segments shorter than the pressure wave length, are then discussed. It is shown that in these shorter pipe segments fluid transient loads may almost double those calculated using the MOC for incompressible flows or the Goodling methodology (without correction factors) if the distance upstream from the closing valve is of sufficient length.

Computational Analysis of the Transonic Flow Field of Two-Dimensional Minimum Length Nozzles

1991 ◽  
Vol 113 (3) ◽  
pp. 479-488 ◽  
Author(s):  
B. M. Argrow ◽  
G. Emanuel
Keyword(s):  
Supersonic Flow ◽  
Flow Field ◽  
Method Of Characteristics ◽  
Reynolds Numbers ◽  
Boundary Layer Separation ◽  
Minimum Length ◽  
Two Dimensional ◽  
Sonic Line ◽  
Inlet Geometry ◽  
Straight Sonic Line

The method of characteristics is used to generate supersonic wall contours for two-dimensional, straight sonic line (SSL) and curved sonic line (CSL) minimum length nozzles for exit Mach numbers of two, four and six. These contours are combined with subsonic inlets to determine the influence of the inlet geometry on the sonic-line shape, its location, and on the supersonic flow field. A modified version of the VNAP2 code is used to compute the inviscid and laminar flow fields for Reynolds numbers of 1,170, 11,700, and 23,400. Supersonic flow field phenomena, including boundary-layer separation and oblique shock waves, are observed to be a result of the inlet geometry. The sonic-line assumptions made for the SSL prove to be superior to those of the CSL.

Supersonic Nozzle Design

1946 ◽  
Vol 13 (4) ◽  
pp. A265-A270 ◽  
Author(s):  
A. E. Puckett
Keyword(s):  
Flow Field ◽  
Method Of Characteristics ◽  
Supersonic Nozzle ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Nozzle Design ◽  
Small Areas ◽  
Engineering Problems ◽  
Two Dimensional Flow ◽  
Flow Changes

Abstract A two-dimensional flow field in which the velocity is everywhere supersonic can always be represented approximately by a number of small adjacent quadrilateral flow fields in each of which the velocity and pressure are constant. These quadrilaterals must be separated by lines representing waves in the flow; changes in velocity and pressure through any wave can be computed. By increasing the number of small areas into which the complete flow field is divided, the accuracy of this approximate solutions may be increased without limit. This constitutes the “method of characteristics” solution, which has been known for many years. This method may be applied to the graphical computation of flow in a supersonic nozzle, with the particular aim of producing uniform supersonic flow at the end of the nozzle. It is pointed out that such a computation is essentially simple and rapid, and its essential features are presented in a form which, it is hoped, may be easily applied to engineering problems.

On the formation of bubbles in gas-particulate fluidized beds

1979 ◽  
Vol 94 (2) ◽  
pp. 353-367 ◽  
Author(s):  
Jerome B. Fanucci ◽  
Nathan Ness ◽  
Ruey-Hor Yen
Keyword(s):  
Fluidized Beds ◽  
Method Of Characteristics ◽  
Bubble Formation ◽  
Shock Formation ◽  
Fluid Viscosity ◽  
Phase Flow ◽  
Fluid Density ◽  
Pressure Function ◽  
Two Phase ◽  
Particulate Size

The method of characteristics is applied to the nonlinear equations describing two-phase flow in a fluidized bed. The method shows how a small disturbance changes with time and distance and can, eventually, produce a flow discontinuity similar to a shock wave in gases. The parameters entering the analysis are the amplitude of the initial disturbance, the wavelength of the original disturbance, the particulate pressure function, the particulate size, the uniform fluidization voidage, the uniform fluidization velocity, the fluid viscosity, the particulate density, and the fluid density. A parametric study shows that the following factors delay shock formation: a decrease in particulate size, an increase in bed density, an increase in fluid viscosity, and a decrease in particulate density. Experimental data on bubble formation in gas-particulate fluidized beds show that these same factors delay bubble formation. It is concluded, therefore, that the shock front and the bubble front are one and the same thing.

Patterns of Secondary Flow Field in a Circular Tube Caused by Corona Wind Using the Method of Characteristics

2009 ◽  
Author(s):  
Reza Baghaei Lakeh ◽  
Majid Molki
Keyword(s):  
Flow Field ◽  
Electric Field ◽  
Charge Density ◽  
Corona Discharge ◽  
Secondary Flow ◽  
Computational Methods ◽  
Method Of Characteristics ◽  
Body Force ◽  
Circular Tube ◽  
Secondary Flow Field

Corona discharge is widely known as an effective method for improving the characteristics of the flow field and enhancing heat transfer. Distribution of charge density and electric field form a Coulomb body force which acts on the charged particles within the fluid and generates a secondary flow field. The thermal enhancing effects of corona wind are normally dominant in low Reynolds numbers or free convection problems. Although the governing differential equations of corona discharge are relatively simple, solving these equations by conventional computational methods does not yield a smooth solution for charge density and electric field. In particular, the results obtained from finite-volume method suffer from dispersion errors and fluctuations which lead to distorted values of electric body force, and consequently a distorted secondary flow. In this study, the corona discharge in a circular tube with the electrode positioned at the tube centerline is considered. An exact solution for charge density, electric field, and potential distribution along the radius of the tube has been derived analytically using a Lagrangian formulation for the charge density and the Method of Characteristics. It was found that the results of this method do not show any fluctuations or dispersion effects on charge density and electric field. The solution of the electric field provided a body force which was used in the Navier-Stokes equations to obtain the secondary flow in the cross section of the tube. In this paper, the electric and fluid flow fields are presented. The results are compared with those obtained by other computational methods and the differences are discussed.

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