Supersonic flow of a vibrationally relaxing gas past a circular cone

1978 ◽  
Vol 85 (3) ◽  
pp. 519-542 ◽  
Author(s):  
James Kao ◽  
J. P. Hodgson
Keyword(s):  
Supersonic Flow ◽  
Numerical Methods ◽  
Numerical Results ◽  
Circular Cone ◽  
Atmospheric Air ◽  
Relaxing Gas ◽  
Characteristic Network

The steady supersonic flow of a vibrationally relaxing gas past a cone is studied using numerical methods. Near the tip of the cone the flow is obtained by means of a coordinate expansion and built on to this is a characteristic network used to obtain the remainder of the flow. Of particular interest is the development of the frozen shock at the tip into a relaxation-dominated wave at distances large compared with the width of the wave. The numerical results are presented in a concise similarity form which will permit accurate extrapolation to very weak waves in atmospheric air.

scholarly journals Numerical methods for solving the multi-term time-fractional wave-diffusion equation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Fawang Liu ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Pinghui Zhuang ◽  
Qingxia Liu
Keyword(s):  
Numerical Methods ◽  
Theoretical Analysis ◽  
Diffusion Equation ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
Numerical Results ◽  
Diffusion Equations ◽  
Time Space ◽  
Methods And Techniques ◽  
Wave Diffusion

AbstractIn this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Observations Regarding Numerical Results Obtained by the Floquet-Method

2013 ◽  
Author(s):  
Till J. Kniffka ◽  
Horst Ecker
Keyword(s):  
Numerical Methods ◽  
Monodromy Matrix ◽  
Mathieu Equation ◽  
Numerical Results ◽  
The Other ◽  
Benchmark Problem ◽  
Stability Studies ◽  
Floquet Method ◽  
System Equations ◽  
Time Periodic

Stability studies of parametrically excited systems are frequently carried out by numerical methods. Especially for LTP-systems, several such methods are known and in practical use. This study investigates and compares two methods that are both based on Floquet’s theorem. As an introductary benchmark problem a 1-dof system is employed, which is basically a mechanical representation of the damped Mathieu-equation. The second problem to be studied in this contribution is a time-periodic 2-dof vibrational system. The system equations are transformed into a modal representation to facilitate the application and interpretation of the results obtained by different methods. Both numerical methods are similar in the sense that a monodromy matrix for the LTP-system is calculated numerically. However, one method uses the period of the parametric excitation as the interval for establishing that matrix. The other method is based on the period of the solution, which is not known exactly. Numerical results are computed by both methods and compared in order to work out how they can be applied efficiently.

Surface pressures on a wing moving with supersonic speed

1974 ◽  
Vol 341 (1625) ◽  
pp. 177-193 ◽  
Keyword(s):  
Supersonic Flow ◽  
Delta Wing ◽  
Leading Edge ◽  
Ideal Gas ◽  
Numerical Results ◽  
Coordinate Surface ◽  
Supersonic Speed ◽  
Extrapolation Procedure ◽  
Specific Heats ◽  
Limiting Values

Estimates for pressures on the surface of a given delta wing at zero incidence in a steady uniform stream of air are obtained by numerically integrating two semi-characteristic forms of equations which govern the inviscid supersonic flow of an ideal gas with constant specific heats. In one form of the equations coordinate surfaces are fixed in space so that the surface of the wing, which has round sonic leading edges, is a coordinate surface. In the other, two families of coordinates are chosen to be stream-surfaces. For each form of the equations, a finite difference method has been used to compute the supersonic flow around the wing. Convergence of the numerical results, as the mesh is refined, is slow near the leading edge of the wing and an extrapolation procedure is used to predict limiting values for the pressures on the surface of the wing at two stations where theoretical and experimental results have been given earlier by another worker. At one station differences between the results given here and the results given earlier are significant. The two methods used here produce consistent values for the pressures on the surface of the wing and, on the basis of this numerical evidence together with other cited numerical results, it is concluded that the pressures given here are close to the true theoretical values.

scholarly journals A PRACTICAL PERFORMANCE INDEX FOR COMPARING OF OPTIMIZATION SOFTWARE

2014 ◽  
pp. 7-12
Author(s):  
Andrea Attanasio ◽  
Patrizia Beraldi ◽  
Francesca Guerriero
Keyword(s):  
Numerical Methods ◽  
Performance Index ◽  
Relative Efficiency ◽  
Numerical Results ◽  
Optimization Software ◽  
Practical Performance

In this paper we propose a new practical performance index for ranking of numerical methods. This index may be very helpful especially when several methods are tested on a large number of instances, since it provides a concise and precise idea of the relative efficiency of a method with the respect to the others. In order to evaluate the efficiency of the proposed rule, we have applied it to the numerical results presented on previously published papers.

On the breakdown of characteristics solutions in flows with vibrational relaxation

1967 ◽  
Vol 27 (1) ◽  
pp. 49-57 ◽  
Author(s):  
B. S. H. Rarity
Keyword(s):  
Relaxation Time ◽  
Mach Number ◽  
Supersonic Flow ◽  
Steady Flow ◽  
Speed Of Sound ◽  
Vibrational Relaxation ◽  
Precise Measure ◽  
Derivatives Of ◽  
Relaxing Gas ◽  
Flow Case

The breakdown of the characteristics solution in the neighbourhood of the leading frozen characteristic is investigated for the flow induced by a piston advancing with finite acceleration into a relaxing gas and for the steady supersonic flow of a relaxing gas into a smooth compressive corner. It is found that the point of breakdown moves outwards along the leading characteristic as the relaxation time decreases and that there is no breakdown of the solution on the leading characteristic if the gas has a sufficiently small, but non-zero, relaxation time. A precise measure of this relaxation time is derived. The paper deals only with points of breakdown determined by initial derivatives of the piston path or wall shape. In the steady-flow case, the Mach number based on the frozen speed of sound must be greater than unity.

scholarly journals Solving Dual Fuzzy Nonlinear Equations via Shamanskii Method

2018 ◽  
Vol 7 (3.28) ◽  
pp. 89 ◽  
Author(s):  
Ibrahim Mohammed Sulaiman ◽  
Mustafa Mamat ◽  
Nurnadiah Zamri ◽  
Puspa Liza Ghazali
Keyword(s):  
Numerical Methods ◽  
Numerical Method ◽  
Nonlinear Equations ◽  
Computational Cost ◽  
Numerical Results ◽  
Parametric Form ◽  
Solution Point ◽  
New Ideas

New ideas on numerical methods for solving fuzzy nonlinear equations have spread quickly across the globe. However, most of the methods available are based on Newton’s approach whose performance is impaired by either discontinuity or singularity of the Jacobian at the solution point. Also, the study of dual fuzzy nonlinear equations is yet to be explored by many researchers. Thus, in this paper, a numerical method to investigate the solution of dual fuzzy nonlinear equations is proposed. This method reduces the computational cost of Jacobian evaluation at every iteration. The fuzzy coefficients are presented in its parametric form. Numerical results obtained have shown that the proposed method is efficient. 

scholarly journals VIII.—Note as to the Dynamical Equivalent of Temperature in Liquid Water, and the Specific Heat of Atmospheric Air and Steam, being a Supplement to a Paper On the Mechanical Action of Heat

1853 ◽  
Vol 20 (2) ◽  
pp. 191-193
Author(s):  
William John Macquorn Rankine
Keyword(s):  
Cast Iron ◽  
Specific Heat ◽  
Liquid Water ◽  
Royal Society ◽  
Constant Pressure ◽  
Mechanical Action ◽  
Numerical Results ◽  
Detailed Account ◽  
Atmospheric Air ◽  
Apparent Specific Heat

(33*.) In my paper on the Mechanical Action of Heat, published in the 1st Part of the 20th Volume of the Transactions of the Royal Society of Edinburgh, some of the numerical results depend upon the dynamical equivalent of a degree of temperature in liquid water. The value of that quantity which I then used, was calculated from the experiments of De la Roche and Bérard on the apparent specific heat of atmospheric air under constant pressure, as compared with liquid water.The experiments of Mr Joule on the production of heat by friction, give, for the specific heat of liquid water, an equivalent about one-ninth part greater than that which is determined from those of De la Roche and Bérard. I was formerly disposed to ascribe this discrepancy in a great measure to the smallness of the differences of temperature measured by Mr Joule, and to unknown causes of loss of power in his apparatus, such as the production of sound and of electricity; but, subsequently to the publication of my paper, I have seen the detailed account of Mr Joule's last experiments in the Philosophical Transactions for 1850, which has convinced me, that the uncertainty arising from the smallness of the elevations of temperature, is removed by the multitude of experiments (being forty on water, fifty on mercury, and twenty on cast iron); that the agreement amongst the results from substances so different, shews that the error by unknown losses of power is insensible, or nearly so; and that the necessary conclusion is, that the dynamical value assigned by Mr Joule to the specific heat of liquid water, viz.:—772 feet per degree of Fahrenheit, does not err by more than two or at the utmost, three feet; and therefore, that the discrepancy originates chiefly in the experiments of De la Roche and Bérard.

NUCLEAR COLLISIONS MODELS WITH BOLTZMANN OPERATORS

2000 ◽  
Vol 10 (04) ◽  
pp. 479-505 ◽  
Author(s):  
S. DELLACHERIE ◽  
R. SENTIS
Keyword(s):  
Numerical Methods ◽  
Asymptotic Analysis ◽  
Numerical Results ◽  
Nuclear Collisions

We describe a model related to nuclear collisions using Boltzmann operators. An asymptotic analysis is performed concerning the gain operator for the outgoing particles. Some numerical methods related to this model are also described and numerical results are given.

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