scholarly journals Propagation of Quasi-plane Nonlinear Waves in Tubes

10.14311/368 ◽  
2002 ◽  
Vol 42 (4) ◽  
Author(s):  
P. Koníček ◽  
M. Bednařík ◽  
M. Červenka
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Nonlinear Waves ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Circular Duct ◽  
Generalized Burgers Equation ◽  
Kzk Equation ◽  
Model Equations ◽  
Diffraction Effects

This paper deals with possibilities of using the generalized Burgers equation and the KZK equation to describe nonlinear waves in circular ducts. A new method for calculating of diffraction effects taking into account boundary layer effects is described. The results of numerical solutions of the model equations are compared. Finally, the limits of validity of the used model equations are discussed with respect to boundary conditions and the radius of the circular duct. The limits of applicability of the KZK equation and the GBE equation for describing nonlinear waves in tubes are discussed.

An Approximate Solution of the Laminar Boundary Layer on a Flat Plate with Uniform Suction

1955 ◽  
Vol 59 (538) ◽  
pp. 697-698
Author(s):  
S. J. Peerless ◽  
D. B. Spalding
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Present Method ◽  
Integral Method ◽  
Boundary Layer Thickness ◽  
Numerical Solutions ◽  
Approximate Methods ◽  
Boundary Layer Problems ◽  
Momentum Integral ◽  
Similar Solutions

Boundary layer problems may be divided into two classes: (a) those for which similar solutions can be found, i.e. where the boundary conditions are such that similar profiles differing only in scale factor exist at different sections; and (b) those where the boundary conditions do not effect similarity, so that the development of the boundary layer must be calculated in stages. The latter class are known as “continuation problems,” and very few numerical solutions have been obtained because of the labour involved.Approximate methods of solving continuation problems are known, using the Karman momentum integral method (e.g. Ref. 1) or variants. Some of these methods make use of velocity profiles calculated for “similar” boundary layers. This note presents a new approximate method which uses “similar” profiles but avoids using the momentum integral. Instead of characterising the boundary layer thickness by the “momentum thickness,” which needs to be calculated yet is of less direct interest, the wall shear stress is used; this stress usually has to be calculated in any case and the present method is therefore comparatively simple.

Analytical and numerical solutions of generalized Burgers' equation via Buckingham's Pi-theorem

2005 ◽  
Vol 83 (10) ◽  
pp. 1035-1049
Author(s):  
I A Hassanien ◽  
A A Salama ◽  
H A Hosham
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Similarity Solutions ◽  
Initial Value ◽  
Generalized Burgers Equation ◽  
One Step ◽  
Pi Theorem ◽  
Scaling Invariant

A generalized dimensional analysis performed by using Buckingham's Pi-theorem for the generalized Burgers' equation is presented. The application of the Buckingham Pi-theorem is used to reduce the governing partial differential equation with the boundary and initial conditions to an ordinary differential equation with appropriate corresponding conditions. By using a scaling invariant we simplify the similarity solutions, which are discussed for a specific choice of boundary conditions, and yield analytical solutions, which are in closed form. Also, using extended one-step methods of order five we solve the final ordinary differential equations. This criterion for solvability involves converting the boundary value problem to an initial value problem. PACS Nos.: 02.60.Lj, 47.27.Jv

scholarly journals On the existence of solution of the Greenspan-Carrier equation

1973 ◽  
Vol 15 (3) ◽  
pp. 373-384
Author(s):  
H. P. Heinig ◽  
K. Kuen Tam
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Existence Of Solution ◽  
Conducting Fluid ◽  
Rigid Plate ◽  
Different Boundary Conditions ◽  
Electrically Conducting ◽  
Boundary Layer Equations ◽  
Electrically Conducting Fluid

We are concerned with the flow of a viscous incompressible electrically conducting fluid of constant properties past a semi-infinite rigid plate. The governing boundary layer equations were derived by Greenspan and Carrier [2] in 1959. Numerical solutions of these equations subject to different boundary conditions have been considered by Stewartson and Wilson [5], Wilson [8], and recently by Bramley [1].

scholarly journals Asymptotic Structure of Rotating MHD Winds and Its Relation to Wind Boundary Conditions

1997 ◽  
Vol 182 ◽  
pp. 275-290
Author(s):  
J. Heyvaerts ◽  
C.A. Norman
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
First Integrals ◽  
Simplified Model ◽  
Necessary Condition ◽  
General Boundary Conditions ◽  
Asymptotic Structure ◽  
Arbitrary Boundary ◽  
Wind Source

Approximate asymptotic solutions for rotating MHD winds are obtained analytically in terms of the first integrals of the motion. It is shown that the paraxial region of such winds is a line-shaped boundary layer which has, even at large distances, the structure of a pressure-supported current pinch. A necessary condition for cylindrically focused asymptotics to be possible is derived. A simplified model by which the asymptotic structure of such winds can be obtained in terms of general boundary conditions at the wind source is introduced. Results of semi numerical solutions of the model are reported. The model is analytically solved in the limit of very fast rotators, giving in this particular case an explicit and complete description of the wind outputs and asymptotic structure in terms of arbitrary boundary conditions at the wind source.

Similarity solutions for two-dimensional steady laminar gravity currents

1988 ◽  
Vol 192 ◽  
pp. 75-96 ◽  
Author(s):  
P. W. M. Brighton
Keyword(s):  
Boundary Layer ◽  
Shear Flow ◽  
Boundary Conditions ◽  
Similarity Solution ◽  
Numerical Solutions ◽  
Similarity Solutions ◽  
Surface Velocity ◽  
Two Dimensional ◽  
Dense Fluid ◽  
Gaussian Plume

Similarity solutions have been found for steady two-dimensional laminar flow in which dense fluid is emitted upwards from a horizontal plane into a laminar shear flow or into a uniform flow. The solutions also apply to a light fluid released at an upper horizontal surface. The Navier-Stokes equations and the diffusion-advection equation are simplified by making the Boussinesq approximation and the boundary-layer approximation, which here also implies that pressure is hydrostatic.For an oncoming linear shear flow representing flow near a solid surface, a similarity solution is obtained with depth proportional to $x^{\frac{1}{3}}$ where x is the horizontal coordinate. Horizontal velocity and concentration of dense fluid both increase as $x^{\frac{1}{3}}$, so that the solution represents fluid propagating upstream along the surface, and diffusing vertically to be swept downstream again. Numerical solutions for vertical profiles of velocity and concentration are presented for a Schmidt or Prandtl number σ between 0.71 and infinity. Two alternative sets of boundary conditions are possible. In one set, the pressure above the boundary layer is unchanged but the velocity profile is displaced upwards. In the second, this displacement is forced to be zero with the result that a pressure gradient is generated in the outer flow. These two boundary conditions are known to apply to disturbances in a laminar boundary-layer on horizontal lengthscales respectively greater or smaller than the triple-deck scale.With a uniform velocity upstream and a stress-free boundary, representing flow at a free surface, similarity solutions exist only for a plume growing downstream from the source of a buoyancy flux B, with depth increasing as x½ and concentration decreasing as x−½. When gravity has negligible effects, so that B = 0, the solution is a Gaussian plume. With finite B, there is an adverse gradient of hydrostatic pressure and the plume is decelerated so that it is deeper than in neutral flow. Numerical solutions for σ = 0.71 reveal that there is a maximum buoyancy flux Bcrit above which no similarity solution exists. This occurs with a non-zero value of the surface velocity. For B < Bcrit it is found that there are in fact two possible solutions. One has surface velocity greater than at the critical flux and tends to the passive Gaussian plume as B → 0. In the other, surface velocity decreases from Bcrit, reaching zero at a non-zero value of B. Similar behaviour is found in an asymptotic solution for very large σ.

scholarly journals On a stochastic Burgers equation with Dirichlet boundary conditions

2003 ◽  
Vol 2003 (43) ◽  
pp. 2735-2746 ◽  
Author(s):  
Ekaterina T. Kolkovska
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Martingale Problem ◽  
Weak Limit ◽  
Dirichlet Boundary Conditions ◽  
One Dimensional ◽  
Stochastic Burgers Equation ◽  
The One

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

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