Multidimensional Navier-Stokes Equations Research Articles

A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform

Abstract This investigation explores the analytical solutions to the time-fractional multi-dimensional Navier–Stokes (NS) problem using advanced approaches, namely the Aboodh residual power series method and the Aboodh transform iteration method, within the context of the Caputo operator. The NS equation governs the motion of fluid flow and is essential in fluid dynamics, engineering, and atmospheric sciences. Given the equation’s extensive and diverse applicability across several disciplines, we are motivated to conduct a thorough analysis to understand the complex dynamics associated with the nonlinear events it describes. For this purpose, we effectively handle the challenges posed by fractional derivatives by utilizing the Aboodh approach. This will enable us to obtain accurate analytical approximations for the time fractional multi-dimensional NS equation. By conducting thorough analysis and computational simulations, we provide evidence of the efficiency and dependability of the suggested methodologies in accurately representing the dynamic behavior of fractional fluid flow systems. This work enhances our comprehension of the utilization of fractional calculus in fluid dynamics and provides valuable analytical instruments for examining intricate flow phenomena. Its interdisciplinary nature ensures that the findings are applicable to various scientific and engineering fields, making the research highly versatile and impactful.

Numerical analysis of multi-dimensional Navier–Stokes equation based on Yang–Abdel–Cattani fractional operator

The Navier–Stokes equation is a key governing equation for the motion of viscous fluid flow. The main target of our work is to obtain the solution to multi-dimensional Navier–Stokes equations in the Yang–Abdel–Cattani fractional sense. The results of the model are in terms of the three-parameter Prabhakar function. Three examples are discussed that depend on the recommended method for a novel Yang–Abdel–Cattani arbitrary order operator. The obtained results are plotted with the help of MATLAB R2016a mathematical software.

An analytical approach of multi-dimensional Navier-Stokes equation in the framework of natural transform

<abstract><p>This article introduces a new iterative transform method and homotopy perturbation transform method along with a natural transform to analyze the multi-dimensional Navier-Stokes equations. To solve the fractional-derivative, the Caputo-Fabrizio definition of the fractional derivative was employed. Four examples were considered to examine the efficacy and accuracy of the proposed methods. The efficiency and accuracy were also demonstrated by the solution comparison via graphs. The proposed methods' convergence and uniqueness are also discussed. The methods mentioned above are straightforward and support a high rate of convergence.</p></abstract>

New finite difference unequal-sized Hermite WENO scheme for Navier-Stokes equations

In this paper, a new finite difference unequal-sized Hermite weighted essentially non-oscillatory (US-HWENO) scheme is proposed for solving multi-dimensional Navier-Stokes equations on structured meshes, which could achieve sixth-order accuracy in one dimension and fifth-order accuracy in two and three dimensions, respectively. The high-order spatial reconstruction procedure uses a convex combination of one quintic polynomial and two linear polynomials defined on three unequal-sized central or biased spatial stencils. Compared with the classical finite difference, finite volume, or mixed finite difference and finite volume HWENO schemes [32] , [33] , the new scheme uses same largest stencil and only three stencils in spatial reconstruction procedures, and the first-order derivative values in auxiliary equations can be obtained directly rather than using complex HWENO-type reconstruction methodologies [32] , [33] . Moreover, the associated viscosity terms could be directly computed from the function values and first-order derivative values by using the characteristic of the WENO-type spatial reconstructions. Meanwhile, the linear weights can be any positive numbers on condition that their summation equals one, which avoids the problem of negative linear weights in classical HWENO schemes [24] , [25] . So the new finite difference US-HWENO scheme is more suitable for solving Navier-Stokes equations, and can be easily implemented to multi-dimensions on structured meshes. Some benchmark viscous examples are illustrated to verify the good performance of this new finite difference US-HWENO scheme.

Explicit-Iterative Scheme for the Time Integration of a System of Navier–Stokes Equations

We propose a new explicit-iterative scheme for time integrating the multidimensional Navier–Stokes equations of a compressible environment through splitting into convective and diffusive stages, which are performed sequentially at each time step. The convective stage is implemented using the Godunov scheme; and the diffusive stage, using the Chebyshev explicit-iterative LI-M scheme, which has no restriction on the time step. The resulting scheme ensures that the basic conservation laws hold on arbitrary irregular grids. The explicit nature of the calculations ensures that the scheme can be efficiently used in various parallel technologies.

Time Advancement of the Navier-Stokes Equations: <i>p</i>-Adaptive Exponential Methods

An adaptive exponential time advancement framework is developed for solving the multidimensional Navier-Stokes equations with a variable-order discontinuous Galerkin (DG) discretization on hybrid unstructured curved grids. The adaptive framework is realized with cell-wise, variable-order DG refinements and a dynamic assembly of elemental Jacobian matrices. The accuracy and performance gain are investigated for several benchmark cases up to a realistic, three-dimensional rotor flow. Numerical results are shown to be more efficient than the use of uniform-order exponential DG for simulating viscous flows.

One-dimensional two-fluid model for wavy flow beyond the Kelvin–Helmholtz instability: Limit cycles and chaos

A 1D TFM numerical simulation of near horizontal stratified two-phase flow is performed where the TFM, including surface tension and viscous stresses, is simplified to a two-equation model using the fixed-flux approximation. As the angle of inclination of the channel increases so does the driving body force, so the flow becomes KH unstable, and waves grow and develop nonlinearities. It is shown that these waves grow until they reach a limit cycle due to viscous dissipation at wave fronts. Upon further inclination of the channel, chaos is observed. The appearance of chaos in a 1D TFM implies a nonlinear process that transfers energy intermittently from long wavelengths where energy is produced to short wavelengths where energy is dissipated by viscosity, so that an averaged energy equilibrium in frequency space is attained. This is comparable to the well-known turbulent stability mechanism of the multi-dimensional Navier–Stokes equations, i.e., chaos implies Lyapunov stability, but in this case it is strictly a two-phase phenomenon.

Macroscopic model and truncation error of discrete Boltzmann method

A derivation procedure to secure the macroscopically equivalent equation and its truncation error for discrete Boltzmann method is proffered in this paper. Essential presumptions of two time scales and a small parameter in the Chapman–Enskog expansion are disposed of in the present formulation. Equilibrium particle distribution function instead of its original non-equilibrium form is chosen as key variable in the derivation route. Taylor series expansion encompassing fundamental algebraic manipulations is adequate to realize the macroscopically differential counterpart. A self-contained and comprehensive practice for the linear one-dimensional convection–diffusion equation is illustrated in details. Numerical validations on the incurred truncation error in one- and two-dimensional cases with various distribution functions are conducted to verify present formulation. As shown in the computational results, excellent agreement between numerical result and theoretical prediction are found in the test problems. Straightforward extensions to more complicated systems including convection–diffusion–reaction, multi-relaxation times in collision operator as well as multi-dimensional Navier–Stokes equations are also exposed in the Appendix to point out its expediency in solving complicated flow problems.

Local existence of solution to free boundary value problem for compressible Navier-Stokes equations

This paper is concerned with the free boundary value problem for multi-dimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local (in time) existence of a weak solution is established; in particular, the density is positive and the solution is regular away from the free boundary.

A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients

Local solutions of the multidimensional Navier-Stokes equations for isentropic compressible flow are constructed with spherically symmetric initial data between a solid core and a free boundary connected to a surrounding vacuum state. The viscosity coefficients $\lambda, \mu$ are proportional to $\rho^\theta$, $0<\theta<\gamma$, where $\rho$ is the density and $\gamma > 1$ is the physical constant of polytropic fluid. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.

GLOBAL SOLUTIONS TO THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE HEAT-CONDUCTING FLOW WITH SYMMETRY AND FREE BOUNDARY

ABSTRACT Global solutions of the multidimensional Navier-Stokes equations for compressible heat-conducting flow are constructed, with spherically symmetric initial data of large oscillation between a static solid core and a free boundary connected to a surrounding vacuum state. The free boundary connects the compressible heat-conducting fluids to the vacuum state with free normal stress and zero normal heat flux. The fluids are initially assumed to fill with a finite volume and zero density at the free boundary, and with bounded positive density and temperature between the solid core and the initial position of the free boundary. One of the main features of this problem is the singularity of solutions near the free boundary. Our approach is to combine an effective difference scheme to construct approximate solutions with the energy methods and the pointwise estimate techniques to deal with the singularity of solutions near the free boundary and to obtain the bounded estimates of the solutions and the free boundary as time evolves. The convergence of the difference scheme is established. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.

Reduction of modes for the control of viscous flows

This paper applies the Karhunen–Loève Galerkin procedure to the solution of optimal control problems of viscous fluid flow through a grooved channel. Through the Karhunen–Loève Galerkin procedure, one can reduce the Navier–Stokes equation to a small number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena. This technique is well suited for the problem of control or optimization, where one has to solve the governing equation repeatedly but one can also estimate the approximate solution space based on the range of control variables. In our previous work [H.M. Park, Y.D. Jang, Int. J. Engrg. Sci. 38 (2000) 785–805], we showed that the Karhunen–Loève Galerkin procedure could solve an optimal control problem of one-dimensional Burgers equation very efficiently. The present paper extends the applicability of the Karhunen–Loève Galerkin procedure to the solution of optimal control problems of multi-dimensional Navier–Stokes equations.

Boundary control of the Navier–Stokes equation by empirical reduction of modes

The Karhunen–Loève Galerkin procedure is a type of Galerkin methods that employs the empirical eigenfunctions of the Karhunen–Loève decomposition as basis functions. This technique can reduce nonlinear partial differential equations to sets of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena. Previously [1], it has been shown that one dimensional Burgers equation is reduced to a low dimensional model by this method, which is employed to solve boundary optimal control problems very efficiently. The present paper demonstrates that the Karhunen–Loève Galerkin procedure can be extended to solve problems of the boundary optimal control of multidimensional Navier–Stokes equations. Since the reduction of modes in the multidimensional case is much larger than that in the one dimensional case, the present technique is found to be more powerful when applied to the control problems of the Navier–Stokes equation than those of the Burgers equation.

Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity. I

Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity. I

Stabilization mechanism of lifted jet diffusion flames

Flame lift and stabilization are studied using numerical simulations of diffusion flames resulting from amethane jet injected into an air background. The numerical model solves the time-dependent, axisymmetric, multidimensional Navier-Stokes equations coupled to submodels for chemical reaction and heat release, soot formation and radiation transport. Simulations are conducted for an undiluted methane jet and for two nitrogen-diluted jets (CH 4 :N 2 /3:1 and CH 4 :N 2 /1:1). The jet exit velocities range from 20 to 50 m/s through a 1-cm-diameter nozzle, coflowing into a 30-cm/s air stream. The flame liftoff height increases linearly with jet exit velocity and the stabilization height increases as the nitrogen dilution of the jet increases. The computations show that the flame is stabilized on a vortical structure in the inner shear layer, which is on the stoichiometric surface at a height where the local axial velocity is approximately equal to the turbulent burning velocity. There is no appreciable chemical heat release in the region below the stabilization point, although a stoichiometric surface exists in that region. The flame base moves upward with the vortical structure to which it is attached, and then quickly jumps down to attach to a new, lower vortex, resulting in an oscillating (1–2 cm) flame liftoff height. The results corroborate parts of both the premixedness and extinction stabilization theories, and suggest that the liftoff mechanism is a result of complex fluid-chemical interactions, parts of which are incorporated in the simplified theories.

An unfactored implicit scheme with multigrid acceleration for the solution of the Navier-Stokes equations

An unfactored implicit difference scheme for the steady state solution of the multidimensional Navier-Stokes equations of a compressible fluid is presented. The hyperbolic part is approximated by a high resolution scheme based on flux-vector splitting and upwind-biased differences to avoid the necessity of artificial dissipation terms and to construct a diagonal dominant solution matrix. Consequently, an iterative inversion of the solution matrix can be performed without any time step restriction. The rate of convergence is improved by using the indirect multigrid concept in form of the FAS scheme. The method is formulated for a body-fitted, curvilinear coordinate system. The computational results for laminar subsonic, transonic and supersonic steady-state flows which are compared with analytical and other numerical results as well as with experimental data illustrate the efficiency and the accuracy of the algorithm.