Governing Equations in Fluid Mechanics

In this paper, a special integrable negative-order Calogero–Bogoyavlenskii–Schiff equation (nCBS) in fluid mechanics is studied by means of the symmetry reduction method and consistent tanh expansion method. The Painlevé integrability is investigated to confirm the compatibility conditions. This integrable nCBS equation has been transformed into different reduction equations and the corresponding invariant solutions with arbitrary functions are obtained. The corresponding structures of the invariant solutions for the nCBS equation are also shown graphically. At last, new types of soliton–cnoidal interaction solutions for the nCBS equation are presented through the consistent tanh expansion method on the basis of the truncated Painlevé expansion.

The three-dimensional incompressible Navier-Stokes equations in conservative form are discretized with the finite volume method, and are solved in a time-accurate manner, using the pseudo-compressibility technique. A third-upwind differencing scheme, based on the Roe’s flux-splitting, is used to compute the convective term in the MUSCL scheme way which satisfies the TVD condition so as to avoid excessive numerical diffusivity. The viscous term is evaluated by a Gauss integration in the finite volume method way, resulting in the center differencing. An implicit factorization approximate method, based on the Euler implicit scheme, is used for the discretization of time derivative. In the time-accurate formulation, the time derivatives of velocity components in the momentum equations are differenced using a first-order, two-point, backward-difference formula. To satisfy the equation of continuity at each physical time step, the artificial compressibility relation is also introduced and several subiterations in pseudotime are required. The cell-centered layout is adopted, in which flow variable node is placed at the center of grid cell and the grid cell is used as control volume.

sented in these previous studies. Because the linearized Euler equations in conservative form are used in the present work, new source terms have to be derived for the conservative set of equations. A formal derivation of source terms for the linearized Euler equations in conservative form is presented, and the system of equations is compared to those of Goldstein and Bailly et al. Simplified versions of the derived source terms are also developed. To test the derived source terms, a direct simulation of a forced two-dimensional mixing layer is carried out. The direct simulation provides the mean flow and source field for the inhomogeneous linearized Euler equations. The solutions to the linearized Euler equations with source terms are compared to the solution of the direct simulation and show good agreement. All simulations are performed using Tam and Webb’s fourth-order dispersion-relation-preserving scheme and a four-step fourth-order Runge‐Kutta time-marching technique. Artificial selective damping introduced through the numerical scheme is used to avoid spurious waves. Absorbing boundary conditions based on characteristic variables are used at the free boundaries and a buffer layer is added at the outflow.

This paper offers new insights into gravitational interactions within a general non-inertial reference frame. By utilizing symbolic tensor calculus, the study establishes a unified framework that connects time derivatives in non-inertial frames to those in inertial frames. The research introduces new first integrals of motion for a system of many particles in arbitrary non-inertial and barycentric rotating reference frames. These first integrals provide a kinematic and geometric visualization of motion in non-inertial frames. Additionally, a generalized potential energy function is presented for broader applicability. For the gravitational two-body problem, the paper delivers a closed-form, coordinate-free solution for the motion of each body relative to the original frame. Consequently, sufficient conditions for stability against collisions are established within the context of the two-body problem in a non-inertial reference frame. Furthermore, the paper examines the relative orbital motion of spacecraft, presenting a closed-form and coordinate-free solution in the local vertical local horizontal (LVLH) non-inertial frame, which is centered on the center of mass of the main spacecraft.

Cartesian cut cell approach for simulating incompressible flows with rigid bodies of arbitrary shape

Mixed lump-stripe, bright rogue wave-stripe, dark rogue wave-stripe and dark rogue wave solutions of a generalized Kadomtsev–Petviashvili equation in fluid mechanics

Modeling and simulation of a downdraft biomass gasifier 1. Model development and validation

Of current interest, the paper commented [i.e., Eur. Phys. J. Plus (2020) 135:272] has investigated the $$(3+1)$$ -dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation in ocean dynamics, fluid mechanics and plasma physics. However, the story that the paper commented has caused could be made more complete. Hereby, for the same equation, via the Bell polynomials, we work out the same bilinear form as that in the paper commented, but the ways of derivation are different. With symbolic computation, we also construct two auto-Backlund transformations for the same equation with some solitons. Our results are with respect to the amplitude or elevation of the relevant wave in ocean dynamics, fluid mechanics and plasma physics.

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.

Under investigation in this paper is the (2 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation, which can be utilized to describe certain nonlinear phenomena in fluid mechanics. We obtain the higher-order lump, breather and hybrid solutions, and analyze the effects of the constant coefficients $$h_1$$ , $$h_2$$ , $$h_4$$ and $$h_5$$ in that equation on those solutions, since the higher-order lump solutions are generalized via the long-wave limit method, and since the higher-order breather solutions and hybrid solutions composed of the solitons, breathers and lumps are derived. With the help of the analytic and graphic analysis, we get the following: (1) amplitudes of the humps and valleys of the first-order lumps are related to $$h_1$$ , $$h_2$$ , $$h_4$$ and $$h_5$$ , proportional to $$h_4$$ while inversely proportional to $$h_2$$ . Velocities of the first-order lumps are proportional to $$h_4$$ . The second-order lumps describe the interaction between the two first-order lumps, which is elastic since those lumps keep their shapes, velocities and amplitudes unchanged after the interaction. Effects of $$h_2$$ and $$h_4$$ on the second-order lumps are graphically illustrated. (2) Amplitudes of the first-order breathers are proportional to $$h_2$$ . Interaction between the breather waves is graphically presented. Effects of $$h_2$$ and $$h_1$$ on the amplitudes and shapes of the second-order breathers are graphically discussed. (3) Elastic interactions are graphically illustrated, between the first-order breathers and one solitons, the first-order lumps and one solitons, as well as the first-order breathers and first-order lumps. Also graphically illustrated, amplitudes of all those three kinds of hybrid solutions are inversely proportional to $$h_2$$ , and velocity of the one soliton is positively correlated to $$h_4$$ .

This paper investigates the quasi-periodic breathers of the (2+1)-dimensional Hirota equation, which can describe the wave phenomena in nonlinear optics and fluid mechanics. By combining the theta function and Hirota bilinear method, the quasi-periodic breather solutions are constructed. The solvability problem of the quasi-periodic breathers is transformed into a least squares problem, which can be solved via a new algorithm called the global Levenberg-Marquardt (LM) method. Different types of quasi-periodic breathers are obtained, including quasi-periodic Kuznetsov-Ma breather, quasi-periodic Akhmediev breather as well as regular quasi-periodic breather. By introducing an analysis method related to the characteristic line, the dynamical characteristics of the quasi-periodic breathers are derived. In addition, the data-driven quasi-periodic breathers are studied by using the physics-informed neural network-adaptive residual distribution optimization (PINN-ARDO) method.

Traffic flow analysis was considered as one of the main steps in transportation planning and engineering. Various methods have been developed for this purpose. Most of these methods are based on available equations in traffic engineering. Mathematical methods used for analyzing traffic flow has always been very complicated and resource consuming. In 1995, using finite elements method for analyzing traffic flow was first introduced and developed over the few years. Hence it is a new look at traffic flow analysis which still needs lots of complex equations to be solved. In 2007, for the first time, it was suggested to simulate traffic flow as a fluid; using finite elements equations in fluid mechanics (i.e. model the vehicles in the road as a fluid in a canal). Results were considerably matching the real world case studies data. In this paper which is a part of study on the finite elements methods in transportation modeling and planning, ten different routes (e.g. bridge, intersection, roundabout) in Shiraz city (one of large scale cities of Iran) have been modeled in one of fluid mechanics modeling platform. Outputs compared with real world data by some adjustment and calibration which have been made to make it possible to match data and calculate the errors. Results show acceptable errors regarding modeling traffic flow by fluid mechanics simulation platforms. Additionally, a sensitivity analysis has been performed for acquisition of each element’s role (e.g. number of lanes, maximum allowed speed and AADT) in model accuracy in Neural Network Algorithm platform.

In this work, we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics, scientific fields, and ocean engineering. This equation will be reduced to the Korteweg–de Vries equation via using the perturbation analysis. We derive the corresponding vectors, symmetry reduction and explicit solutions for this equation. We readily obtain Bäcklund transformation associated with truncated Painlevé expansion. We also examine the related conservation laws of this equation via using the multiplier method. Moreover, we investigate the reciprocal Bäcklund transformations of the derived conservation laws for the first time.

This paper describes a numerical method for designing closed aerofoils that correspond to given surface speed distributions including flows with shock waves. The prescribed surface speed distribution in general contains three parameters whose values are found as part of the solution. The inverse aerofoil design problem is not well posed unless the prescribed speed satisfies three constraints that can be used to fix the value of the three adjustable parameters. The computational method is based on the numerical solution of the full potential equation in conservation form by an approximate factorization–multigrid scheme. The nature of the constraints and several strategies for introducing the necessary freedom in the prescribed speed are discussed along with typical solutions illustrating the method.

This paper presents a numerical simulation method developed for separated flow in cascades using the Euler equations and demonstrates the feasibility of this method. MacCormack’s two-steps explicit finite difference scheme is used to discretize the equations in conservation form, and the artificial viscosity is added to the discretized inviscid equations by means of the self-adapted filter technique. The initial separation boundary is given according to simple experimental results. The numerical simulation results including subsonic and transonic turbine cascades flow with or without separation show that the fundamental idea of this numerical method is reasonable and simple. The present study indicates that for solving certain engineering problems it is a simple and effective tool for adding some viscosity corrections to inviscid flow model, especially the current when the Navier-Stokes equations have not been solved very effectively for various complicated flows in turbomachinery.