Application of artificial viscosity in establishing supercritical solutions to the transonic integral equation

1982 ◽  
Vol 380 (1778) ◽  
pp. 77-97
Keyword(s):  
Integral Equation ◽  
Transonic Flow ◽  
Iterative Scheme ◽  
Artificial Viscosity ◽  
Linear Solution ◽  
Number Range ◽  
Flow Equations ◽  
Viscosity Term ◽  
Initial Iterate ◽  
Shock Fitting

The nonlinear singular integral equation of transonic flow is examined in the free-stream Mach number range where only solutions with shocks are known to exist. It is shown that, by the addition of an artificial viscosity term to the integral equation, even the direct iterative scheme, with the linear solution as the initial iterate, leads to convergence. Detailed tables indicating how the solution varies with changes in the parameters of the artificial viscosity term are also given. In the best cases (when the artificial viscosity is smallest), the solutions compare well with known results, their characteristic feature being the representation of the shock by steep gradients rather than by abrupt discontinuities. However, ‘sharp-shock solutions’ have also been obtained by the implementation of a quadratic iterative scheme with the ‘artificial viscosity solution’ as the initial iterate; the converged solution with a sharp shock is obtained with only a few more iterates. Finally, a review is given of various shock-capturing and shock-fitting schemes for the transonic flow equations in general, and for the transonic integral equation in particular, frequent comparisons being made with the approach of this paper.

Averaging and integral manifolds

1970 ◽  
Vol 2 (2) ◽  
pp. 197-222 ◽  
Author(s):  
W. A. Coppel ◽  
K. J. Palmer
Keyword(s):  
Autonomous System ◽  
Method Of Characteristics ◽  
Integral Manifold ◽  
Artificial Viscosity ◽  
Original Method ◽  
Method Of Averaging ◽  
System Of Differential Equations ◽  
Viscosity Term ◽  
Integral Manifolds ◽  
The Individual

An integral manifold for a system of differential equations is a manifold such that any solution of the equations which has a point on it is entirely contained on it. The method of averaging establishes the existence of such a manifold for a system which is a perturbation of an autonomous system with a periodic orbit. The existence of the manifold is established here under more general hypotheses, namely for perturbations which are ‘integrally small’. The method differs from the original method of Bogolyubov and Mitropolskii and operates directly with the individual solutions. This is made possibly by the use of an appropriate norm, and is equivalent to solving the partial differential equation which occurs in work by Moser and Sacker by the method of characteristics rather than by the introduction of an artificial viscosity term. Moreover, detailed smoothness properties of the manifold are obtained. For periodic perturbations the integral manifold is a torus and these smoothness properties are just sufficient to permit the application of Denjoy's theorem.

scholarly journals Newton Type Iteration for Tikhonov Regularization of Nonlinear Ill-Posed Problems in Hilbert Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Monnanda Erappa Shobha ◽  
Santhosh George
Keyword(s):  
Integral Equation ◽  
Iterative Method ◽  
Iterative Scheme ◽  
Initial Guess ◽  
Optimal Order ◽  
Source Condition ◽  
Adaptive Scheme ◽  
Actual Solution ◽  
Ill Posed ◽  
General Source

Recently, Vasin and George (2013) considered an iterative scheme for approximately solving an ill-posed operator equationF(x)=y. In order to improve the error estimate available by Vasin and George (2013), in the present paper we extend the iterative method considered by Vasin and George (2013), in the setting of Hilbert scales. The error estimates obtained under a general source condition onx0-x^(x0is the initial guess andx^is the actual solution), using the adaptive scheme proposed by Pereverzev and Schock (2005), are of optimal order. The algorithm is applied to numerical solution of an integral equation in Numerical Example section.

scholarly journals Spiral Shocks in Accretion Disks with SPH

1997 ◽  
Vol 163 ◽  
pp. 770-770
Author(s):  
James Rhys Murray
Keyword(s):  
Accretion Disks ◽  
Equations Of Motion ◽  
Fixed Ratio ◽  
Artificial Viscosity ◽  
Spiral Density Waves ◽  
Viscosity Term ◽  
Particle Hydrodynamics ◽  
Smoothed Particle ◽  
Low Mass ◽  
Continued Use

AbstractSmoothed Particle Hydrodynamics (SPH) is now seen as a numerical scheme well suited to the study of accretion disks. SPH simulations have been conducted of cataclysmic variable disks (Lubow 1991, Murray 1996, Armitage and Livio 1996), galactic disks (Artymowicz and Lubow 1989), and protostellar disks (Artymowicz and Lubow 1994). It is therefore important to test the technique against theory and other numerical results to obtain an estimate of the accuracy and reliability of SPH in this context. Previously SPH has been tested against standard stationary and time-dependent results of viscous thin disk theory (Murray 1996). Strictly these tests relate to disks where ‘viscous’ terms dominate pressure terms in the equations of motion.In this paper we describe tests of the code more appropriate for hot disks where pressure forces are relatively more important than viscosity. Specifically we consider the form of the spiral density waves that can be excited in a disk by a perturbing gravitational potential. Very low mass perturbing bodies excite linear spiral waves which redistribute angular momentum in the disk. For increasingly massive perturbers, the disk response becomes nonlinear and eventually shocks form. In the standard formulation of SPH, an artificial viscosity term is added to the SPH equations to improve shock capture. This is equivalent to introducing a fixed ratio of shear to bulk viscosity into the equations of motion. In Eulerian schemes, artificial viscosity has been discarded in favour of other more accurate, less dissipative schemes for resolving shocks. The continued use of artificial viscosity in SPH has become a source of ‘friction’ between numericists. The simulations described here demonstrate the scheme’s ability to resolve spiral shocks, and show that SPH is a valuable tool for probing the structure of tidally perturbed accretion disks.

Transonic flow of moist air around a thin airfoil with non-equilibrium and homogeneous condensation

2000 ◽  
Vol 403 ◽  
pp. 173-199 ◽  
Author(s):  
ZVI RUSAK ◽  
JANG-CHANG LEE
Keyword(s):  
Transonic Flow ◽  
Droplet Growth ◽  
Inviscid Fluid ◽  
Small Disturbance ◽  
Moist Air ◽  
Integration Rule ◽  
Flow Equations ◽  
Homogeneous Condensation ◽  
Thin Airfoil ◽  
Non Equilibrium

A new small-disturbance model for a steady transonic flow of moist air with non-equilibrium and homogeneous condensation around a thin airfoil is presented. The model explores the nonlinear interactions among the near-sonic speed of the flow, the small thickness ratio and angle of attack of the airfoil, and the small amount of water vapour in the air. The condensation rate is calculated according to classical nucleation and droplet growth models. The asymptotic analysis gives the similarity parameters that govern the flow problem. Also, the flow field can be described by a non-homogeneous (extended) transonic small-disturbance (TSD) equation coupled with a set of four ordinary differential equations for the calculation of the condensate (or sublimate) mass fraction. An iterative numerical scheme which combines Murman & Cole's (1971) method for the solution of the TSD equation with Simpson's integration rule for the estimation of the condensate mass production is developed. The results show good agreement with available numerical simulations using the inviscid fluid flow equations. The model is used to study the effects of humidity and of energy supply from condensation on the aerodynamic performance of airfoils.

Numerical Calibration of 3D Printed Five-Hole Probes for the Transonic Flow Regime

2019 ◽  
Author(s):  
Maximilian Passmann ◽  
Stefan aus der Wiesche ◽  
Franz Joos
Keyword(s):  
Flow Regime ◽  
Transonic Flow ◽  
Calibration Method ◽  
Calibration Procedure ◽  
Tip Clearance ◽  
Outer Diameter ◽  
Number Range ◽  
Half Angle ◽  
3D Printed ◽  
Transonic Turbine

Abstract This paper presents a method for a cost- and time-effective calibration procedure for five-hole probes for the transonic flow regime based on additive manufacturing and a numerical calibration routine. The computational setup and calibration routine are described in detail. The calibration procedure is tested on a custom-built L-shaped conical probe of 30° half-angle with a flat tip and an outer diameter of 2.4mm. The probe tip is manufactured in stainless steel using DMLS. Numerical calibration is carried out over a Mach number range of 0.2 to 1.4 and pitch and yaw angles of ±45°. The numerical calibration charts are validated with wind tunnel tests and the expected accuracy of the numerical calibration method is quantified. Exemplary results of area traverses up- and downstream of a linear transonic turbine cascade with tip clearance are presented and discussed briefly.

Effect of Artificial Viscosity on the Expansion of Dis-Continuities in a Rotating Interplanetary Medium with Material Pressure

2015 ◽  
Vol 6 (1) ◽  
Author(s):  
Ajai Singh Yadav ◽  
Seema Singh ◽  
K. K. Srivastava
Keyword(s):  
Interplanetary Medium ◽  
Viscosity Coefficient ◽  
Artificial Viscosity ◽  
Material Strength ◽  
Characteristic Method ◽  
Newtonian Viscosity ◽  
Viscosity Term ◽  
Solution Of Equations ◽  
Velocity Effect ◽  
Weak Shocks

The solution of equations by seeking quasi-similar solution, in which the viscosity coefficient is taken to be at most a function of time but independent of space co-ordinates. an attempt is made to account for the material strength by including Newtonian- Viscosity term. In the present paper the characteristic method (Chester, Witham) is applied to obtain expressions of the density, the pressure, the particle velocity just behind the shock propagating in a rotating atmosphere. The effect of cariolis force is taken into account. Since the velocity effect has a tendency to smoothen out such discontinuities, the artificial viscosity coefficient suggested by Rithchmyer and Von Newmann is introduced. The problem is discussed for two different cases (i) for weak shocks and (ii) for strong shocks respectively.

Potential of Michell’s Integral for Ship Wave Resistance

2014 ◽  
Author(s):  
Takashi Tsubogo
Keyword(s):  
Integral Equation ◽  
Froude Number ◽  
Computing Time ◽  
Wave Resistance ◽  
Boundary Integral ◽  
Hull Form ◽  
Number Range ◽  
Gradient Effect ◽  
Depth Direction ◽  
Wigley Hull

The Michell’s integral (Michell 1898) for the wave making resistance of a thin ship has not been used widely in practice, since its accuracy is questioned especially for a Froude number range about 0.2 to 0.35 for conventional ships. We examine calculations by Michell’s integral for some ship forms, e.g. a parabolic strut, Wigley hull and so on. As a result, one reason of the disagreement with experiments is revealed. It must be the gradient of hull form in the depth direction. Then a thin ship theory including the hull gradient effect in the depth direction is presented, which improves slightly in low Froude numbers but needs more computing time than Michell’s integral so as to solve a boundary integral equation.

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