Nonlinear Fluid Dynamics Problems Research Articles

Surprising solutions for some challenging problems arising from boundary layer theory by a new technique: the homotopy contraction mapping technique (HCMT)

In this work, the homotopy structure $$ (1 - p)L\,[f] = - p\,N\,[f] $$ is considered to solve some newly developed nonlinear problems in fluid dynamics. In summary, deformation in homotopy series solutions occurs from the initial trial function to the actual solution. The main challenging part in many similarity equations associated with the boundary layer theory is to identify a certain quantity of the exact solution such as $$ f^{{\prime \prime }} (0) $$ . This quantity is initially unknown and is automatically guessed through the zeroth function. Therefore, it makes sense that in the previous homotopy series solutions one would need at least another parameter i.e. the controlling parameter ( $$ \hbar $$ ) to handle the convergence through the so-called $$ \hbar $$ -curves analysis and eventually get an approximate value for $$ f^{{\prime \prime }} (0) $$ . Here it is accounted the basic homotopy structure with No controlling parameter ( $$ \hbar $$ ) and it is shown for the 1st time that the zeroth order solution in homotopy series may be potential to contain some certain quantities of the exact solution, here is to be $$ f^{{\prime \prime }} (0) $$ ; i.e. $$ f_{0}^{{\prime \prime }} (0) $$ could be $$ f^{{\prime \prime }} (0) $$ . This hypothesis is checked through a theorem, being totally linked to the Fixed Point Property (FPP), a topological invariant; i.e. being preserved by any homeomorphism. The theorem enables us Not only to check the validity of the hypothesis, but also to extract the value of the quantity as the unique description of the homotopy series solutions truncated at any order of approximation. The technique is initially introduced in a simple and straightforward manner and then it is applied to some fluid mechanical problems to be further compared with traditional homotopy analysis method, homotopy perturbation method and Numerical solutions. The outcomes indicated an excellent improvement by the present approach as only a few series terms were accounted to obtain highly accurate solutions.

Two upwinding schemes for nonlinear problems in fluid dynamics

The correct modeling for processes involving convection, without introducing excessive artificial damping while retaining high accuracy, stability, boundedness and simplicity of implementation continues being nowadays a challenging task for CFD practitioners. The objective of this study is to present and evaluate the performance of two new upwinding schemes, namely SDPUS-C1 and EPUS, for nonlinear convection term discretization. Both SDPUS-C1 and EPUS schemes satisfy the TVD principle of Harten and are based on the NVD formulation of Leonard. Firstly, a description of the schemes is presented and then the numerical results are provided for one- and two-dimensional hyperbolic conservation laws. Finally, as an application, the SDPUS-C1 and EPUS schemes are employed for the simulation of two-dimensional incompressible fluid flows involving moving free surfaces. The numerical experiments show that the proposed upwinding schemes perform very well.

The Korteweg-de Vries equation in a cylindrical pipe

A mathematical model of nonlinear wave propagation in a pipeline is constructed. The Korteweg-de Vries equation is derived by determining asymptotic solutions and changing variables. A particular solution to the model equations is found that has the fluid velocity function in the form of a solitary wave. Thus, the class of nonlinear fluid dynamics problems described by the KdV equation is expanded.

An implicit multidomain spectral collocation method for stiff highly non-linear fluid dynamics problems

An implicit, multidomain, Chebyshev spectral collocation technique for solving highly non-linear stiff problems is presented. First, the accuracy and efficiency of the explicit—implicit Adams—Bashforth Crank—Nicolson (ABCN) and the implicit Crank—Nicolson (CN) methods in obtaining solutions to the Burgers equation are compared. Then, the implicit CN scheme is used to solve the highly non-linear Reynolds equation of lubrication for a gas bearing separating two textured surfaces. Surface texture is approximated by fifty sinusoidal waves each of which is assigned a separate subdomain. An additional subdomain is used to resolve the trailing edge boundary layer. The combined method is shown to be well suited for the simulation of highly non-linear phenomena that exhibit high frequency spatial variations in the desired solution and that possess very sharp interior or boundary layers.

Unconditionally Stable Explicit Algorithms for Nonlinear Fluid Dynamics Problems

This paper describes novel explicit algorithms that are unconditionally stable. The algorithms are applied to some 1D convection and diffusion problems, including nonlinear problems. Algorithms such as these are of particular interest for massively parallel computers, where one is trying to minimize communications while at the same time maintain the stability properties normally associated with implicit schemes. It is shown how these stable algorithms can be applied in higher spatial dimensions and how they can be extended to problems defined on unstructured meshes.

Fast iterative algorithms based on optimized explicit time stepping

The Generalized Nonlinear Minimal Residual (GNLMR) method is shown to consistently accelerate and stabilize iterative algorithms for solving nonlinear problems by using the optimized explicit multistepping. The examples presented in this paper illustrate the beneficial effects of the optimized multistep algorithm on the computational efficiency and the convergence rate as applied to several nonlinear problems in fluid dynamics. The significant reduction in computing time when using the multiple optimized acceleration factors is only negligibly weighed down by the computation costs due to the requirements for additional computer storage.

On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods (II). Application to transonic flow simulations

On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods (II). Application to transonic flow simulations

Domain decomposition methods for nonlinear problems in fluid dynamics

The numerical solution of finite element approximations of complicated two- and three-dimensional nonlinear problems can be a most formidable task. In order to overcome this difficulty related to dimensionality, domain-splitting methods can be very effective, particularly in view of obtaining a fast and economical conjugate gradient solver, which can be used to precondition the solution of nonlinear problems by optimization methods via nonlinear least squares or weighted residual formulations. A new technique of this type will be introduced and analysed and its efficiency will be discussed from numerical experiments concerning the numerical simulation of transonic flows for compressible inviscid fluids and incompressible viscous flows modelled by the Navier-Stokes equations.

On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods (I) least square formulations and conjugate gradient solution of the continuous problems

Abstract In this paper we consider the solution of nonlinear problems in Fluid Dynamics by least square formulations and conjugate gradient algorithms. We discuss first the solution by these methods of a fairly academic example and then their applications to the transonic potential flows of compressible, inviscid fluids and to the Navier-Stokes equations of incompressible viscous fluids. The solution of the corresponding discrete problems (via finite element approximation) will be reported in forthcoming papers containing numerical examples.

A new approach to some nonlinear fluid dynamics problems

The fluid equations for two-dimensional electron-ion turbulence in the collision dominated regime are reformulated in terms of potentials. External magnetic and gravitational fields are included as well as electron-ion collisions and collisions with neutrals. The utility of the technique in connection with some ionospheric turbulence problems is briefly discussed.