Analytical solution of plane Poiseuille flow within Burnett hydrodynamics

In the current paper, low-speed isothermal microscale gas flows have been investigated utilizing the augmented Burnett equations. There has been limited success to analytically solve the Burnett equations till date. We propose an analytical solution to Burnett equations, which is shown to satisfy the full set of augmented Burnett equations up to Kn of 2.2 with an error of 1 %. Detailed validation shows that the solution represents the integral flow parameters accurately up to Kn ~ 2.2 and local field properties up to Kn ~ 0.5. The range over which the proposed Burnett analytical solution is applicable is substantially better than existing analytical solutions, without incorporating any wall scaling functions into constitutive relations and variation of slip coefficients in the boundary conditions. Normalized mass flow rate, friction factor, and axial velocity profile results show very good agreement with the experimental and simulation data. The analytical solution is also able to predict the change in the curvature of streamwise pressure profile.     

Numerical Simulation of Low Mach Number Reactive Flows

A new formulation for the numerical solution of low Mach number compressible flow problems is presented and analyzed. In this formulation the thermal part (energy and species equations) is solved implicitly and decoupled from the momentum equation, whereas the hydrodynamic part (momentum-continuity) is advanced in time using a high order splitting approach which results in overall high order accuracy in time and minimal errors in mass conservation. These errors are analyzed using both analytical tools and benchmark numerical examples. Results from two-dimensional simulations with one-step global reaction in opposed jet flame and porous particle configurations are also presented.     

The use of Richardson extrapolation for the numerical solution of low Mach number flow in confined regions

We use artificial compressibility together with Richardson extrapolation in the Mach numberM as a method for solving the time dependent Navier-Stokes equation for very low Mach number flow and for incompressible flow. The question of what boundary conditions one should use for low Mach number flow, especially at inflow and outflow boundaries, is investigated theoretically, and boundary layer suppressing boundary conditions are derived. For the case of linearization around a constant flow we show that the low Mach number solution will converge with the rateO(M²) to the true incompressible solution, provided that we choose the boundary conditions correctly. The results of numerical calculations for the time dependent, nonlinear equations and for flow situations with time dependent inflow velocity profiles are presented. The convergence rateM ² to incompressible solution is numerically confirmed. It is also shown that using Richardson extrapolation toM ²= 0 in order to derive a solution with very small divergence can with good result be carried through withM ² as large as 0.1 and 0.05. As the time step in numerical methods must be chosen approximately such thatt · (i/(M x)–v/x ²) is in the stability region of the time stepping method, and asM ²=0.05 is sufficiently small to yield good results, the restriction on the time step due to the Mach number is not serious. Therefore the equations can be integrated very fast by explicit time stepping methods. This method for solving very low Mach number flow and incompressible flow is well suited to parallel processing.     

Accurate initial conditions for the direct numerical simulation of temporal compressible binary shear layers with high density ratio

Accurate initial conditions based on a 2D temporal self-similarity hypothesis are developed for the direct numerical simulation of compressible binary free shear layers with high density ratio. Sample results illustrate effects of density ratio, convective Mach number and free streams temperature on the similarity solution. Direct numerical simulation of a stiff test case (density ratio ∼ 32) shows how the amplitude of early acoustic waves is strongly reduced, with regards to those generated by classical analytical distributions of velocity, temperature and mass fractions. Hence, stable simulations using higher order centered schemes can be achieved for a reasonable number of grid points, even at moderate Reynolds number.     

Magnetohydrodynamic three-dimensional flow of nanofluids with slip and thermal radiation over a nonlinear stretching sheet: a numerical study

A numerical simulation for mixed convective three-dimensional slip flow of water-based nanofluids with temperature jump boundary condition is presented. The flow is caused by nonlinear stretching surface. Conservation of energy equation involves the radiation heat flux term. Applied transverse magnetic effect of variable kind is also incorporated. Suitable nonlinear similarity transformations are used to reduce the governing equations into a set of self-similar equations. The subsequent equations are solved numerically by using shooting method. The solutions for the velocity and temperature distributions are computed for several values of flow pertinent parameters. Further, the numerical values for skin-friction coefficients and Nusselt number in respect of different nanoparticles are tabulated. A comparison between our numerical and already existing results has also been made. It is found that the velocity and thermal slip boundary condition showed a significant effect on momentum and thermal boundary layer thickness at the wall. The presence of nanoparticles stabilizes the thermal boundary layer growth.

     

Nonlocal vibration and instability analysis of carbon nanotubes conveying fluid considering the influences of nanoflow and non-uniform velocity profile

In this article, the influences of non-uniform velocity profile attributable to slip boundary condition and viscosity of fluid on the dynamic instability of carbon nanotubes (CNTs) conveying fluid are investigated. The nonlocal elasticity theory and the Euler–Bernoulli beam theory are employed to derive partial differential equation of nanotubes conveying fluid. Furthermore, a dimensionless momentum correction factor (MCF) is obtained as a function of Knudsen number (Kn) so as to insert the effects of non-uniform velocity profile into the equation of motion. In continuation, complex eigen-frequencies of the system are attained with respect to different boundary conditions, the momentum correction factor, slip boundary condition and nonlocal parameter. The results delineate that considering the effects of non-uniform velocity profile could diminish predicted critical velocity of flow. Therefore, the divergence instability occurs in the lower values of flow velocity. In addition, the MCF decreases through enhancement of Kn; hence, the effects of non-uniform velocity profile are more noticeable for liquid fluid than gas fluid.     

Analytical and numerical solutions of the unsteady 2D flow of MHD fractional Maxwell fluid induced by variable pressure gradient

The nonlocal property of the fractional derivative can supply more precise mathematical models for depicting flow dynamics of complex fluid which cannot be modelled appropriately by normal integer order differential equations. This paper studies the analytical and numerical methods of unsteady 2D flow of Magnetohydrodynamic (MHD) fractional Maxwell fluid in a rectangular pipe driven by variable pressure gradient. The governing equation is formulated with Caputo time dependent fractional derivatives whose orders are distributed in interval (0, 2). A challenge is to firstly obtain the exact solution by combining modified separation of variables method with Mikusiński-type operational calculus. Meanwhile, the numerical solution is also obtained by the implicit finite difference method whose validity has been confirmed by the comparison with the exact solution constructed. Different to the most classical works, both the stability and convergence analysis of two-dimensional multi-term time fractional momentum equation are derived. Based on numerical analysis, the results show that the velocity increases with the rise of the fractional parameter and relaxation time. While an increase in the values of Hartmann number leads to a slower velocity in the rectangular pipe.     

An algorithm for velocity calculation in close proximity to body surfaces in panel methods

To model incompressible flow over a body of arbitrary geometry when using vortex methods, it is necessary to construct an irrotational field to impose the impermeability condition at the surface of the object. In order to achieve this impermeability, this paper uses a boundary integral equation based on the single-layer representation for the velocity potential. Specifically, we formulate this exterior Neumann problem in terms of a source/sink boundary integral equation. The solution to this integral equation is then coupled with an interpolation procedure which smoothes the transition between near-wall and interior regimes. We describe the numerical scheme embedding this strategy and discuss its accuracy and efficiency. For validation purposes, we consider the potential and vortical flow over a circular cylinder, for which an analytical solution and the commonly used method of images are available.     

On the discontinuous Galerkin method for the simulation of compressible flow with wide range of Mach numbers

The paper is concerned with the numerical simulation of compressible flow with wide range of Mach numbers. We present a new technique which combines the discontinuous Galerkin space discretization, a semi-implicit time discretization and a special treatment of boundary conditions in inviscid convective terms. It is applicable to the solution of steady and unsteady compressible flow with high Mach numbers as well as low Mach number flow at incompressible limit without any modification of the Euler or Navier–Stokes equations.     

A Local Pressure Boundary Condition Spectral Collocation Scheme for the Three-Dimensional Navier–Stokes Equations

A spectral collocation scheme for the three-dimensional incompressible \(({\varvec{u}},p)\) formulation of the Navier–Stokes equations, in domains \(\varOmega \) with a non-periodic boundary condition, is described. The key feature is the high order approximation, by means of a local Hermite interpolant, of a Neumann boundary condition for use in the numerical solution of the pressure Poisson system. The time updates of the velocity \({\varvec{u}}\) and pressure \(p\) are decoupled as a result of treating the pressure gradient in the momentum equation explicitly in time. The pressure update is computed from a pressure Poisson equation. Extension of the overall methodology to the Boussinesq system is also described. The uncoupling of the pressure and velocity time updates results in a highly efficient scheme that is simple to implement and well suited for simulating moderate to high Reynolds and Rayleigh number flows. Accuracy checks are presented, along with simulations of the lid-driven cavity flow and a differentially heated cavity flow, to demonstrate the scheme produces accurate three-dimensional results at a reasonable computational cost.     

Simulation of transport processes in the Couette flow problem under incomplete accommodation of the tangential momentum of gas molecules by channel walls

The Couette flow problem has been used as an example for the proposed analytical method for calculating macro parameters of gas in channels, the thickness of which is comparable with the mean free path of gas molecules by means of simple numerical procedures. As a basic equation, the linearized BGK (Bhatnagar-Gross-Krook) model of the Boltzmann kinetic equation is used, and the boundary condition on the channel walls is taken to be the model of a specular-diffuse reflection. For different values of the channel thickness and the coefficient of accommodation of the tangential momentum of gas molecules by the channel walls, the profiles of the gas mass velocity in the channel have been constructed and the values of the nonzero component of the viscous stress tensor and the gas mass flow per unit channel width have been calculated. A comparison with similar results published in the press has been made.     

A Fourth Order Scheme for Incompressible Boussinesq Equations

A fourth order finite difference method is presented for the 2D unsteady viscous incompressible Boussinesq equations in vorticity-stream function formulation. The method is especially suitable for moderate to large Reynolds number flows. The momentum equation is discretized by a compact fourth order scheme with the no-slip boundary condition enforced using a local vorticity boundary condition. Fourth order long-stencil discretizations are used for the temperature transport equation with one-sided extrapolation applied near the boundary. The time stepping scheme for both equations is classical fourth order Runge–Kutta. The method is highly efficient. The main computation consists of the solution of two Poisson-like equations at each Runge–Kutta time stage for which standard FFT based fast Poisson solvers are used. An example of Lorenz flow is presented, in which the full fourth order accuracy is checked. The numerical simulation of a strong shear flow induced by a temperature jump, is resolved by two perfectly matching resolutions. Additionally, we present benchmark quality simulations of a differentially-heated cavity problem. This flow was the focus of a special session at the first MIT conference on Computational Fluid and Solid Mechanics in June 2001.     

Analysis of unsteady compressible boundary layer flow via finite elements

This paper is concerned with the discrete formulation and numerical solution of unsteady compressible boundary layer flows using the Galerkin-finite element method. Linear interpolation functions for the velocity, density, temperature and pressure are used in the momentum equation and equations of continuity, energy and state. The coupled nonlinear finite element equations are approximated by a third order Taylor series expansion as temporal operator to integrate in time with Newton-Raphson type iterations performed until convergence within each time step. As an example, a boundary layer problem of a perfect gas behind a normal shock wave is solved. A comparison of the results with those by other method indicates a favorable agreement.     

Analytical Solution of the Minimum Time Slew Maneuver Problem for an Axially Symmetric Spacecraft in the Class of Conical Motions

The conventional problem of the time-optimal slew of a spacecraft considered as a solid body with a single symmetry axis subject to arbitrary boundary conditions for the attitude and angular velocity is considered in the quaternion statement. By making certain changes of variables, the original dynamic Euler equations are simplified, and the problem turns into the optimal slew problem for a solid body with a spherical distribution of mass containing one additional scalar differential equation. For this problem, a new analytical solution in the class of conical motions is found; in this solution, the initial and terminal attitudes of the space vehicle belong to the same cone realized under a bounded control. A modification of the optimal slew problem in the class of generalized conical motions is made that makes it possible to obtain its analytical solution under arbitrary boundary conditions for the attitude and angular velocity of the spacecraft. A numerical example of a spacecraft’s conical motion and examples demonstrating the proximity of the solutions of the conventional and modified optimal slew problems of an axially symmetric spacecraft are discussed.     

The decomposition method for Cauchy advection-diffusion problems

In this paper, the solution of Cauchy problems for the advection-diffusion equation is obtained using the decomposition method. In the case when the flow velocity is constant, an analytical solution can be derived, whilst for variable flow velocity, symbolic numerical computations need to be performed.     

Boundary domain integral method for high Reynolds viscous fluid flows in complex planar geometries

The article presents new developments in boundary domain integral method (BDIM) for computation of viscous fluid flows, governed by the Navier–Stokes equations. The BDIM algorithm uses velocity–vorticity formulation and is based on Poisson velocity equation for flow kinematics. This results in accurate determination of boundary vorticity values, a crucial step in constructing an accurate numerical algorithm for computation of flows in complex geometries, i.e. geometries with sharp corners. The domain velocity computations are done by the segmentation technique using large segments. After solving the kinematics equation the vorticity transport equation is solved using macro-element approach. This enables the use of macro-element based diffusion–convection fundamental solution, a key factor in assuring accuracy of computations for high Reynolds value laminar flows. The versatility and accuracy of the proposed numerical algorithm is shown for several test problems, including the standard driven cavity together with the driven cavity flow in an L shaped cavity and flow in a Z shaped channel. The values of Reynolds number reach 10,000 for driven cavity and 7500 for L shaped driven cavity, whereas the Z shaped channel flow is computed up to Re = 400. The comparison of computational results shows that the developed algorithm is capable of accurate resolution of flow fields in complex geometries.     

Analytical solution of the optimal slew problem for an axisymmetric spacecraft in the class of conical motions

The traditional problem is discussed of an optimal spacecraft slew in terms of minimum energy costs. The spacecraft is considered as a rigid body with one symmetry axis under arbitrary boundary conditions for the angular position and angular velocity of the spacecraft in the quaternion formulation. Using substitutions of variables, the original problem is simplified (in terms of dynamic Euler equations) to the optimal slew problem for a rigid body with a spherical mass distribution. The simplified problem contains one additional scalar differential equation. A new analytical solution is presented for this problem in the class of conical motions, leading to constraints on the initial and final values of the angular velocity vector. In addition, the optimal slew problem is modified in the class of conical motions to derive an analytical solution under arbitrary boundary conditions for the angular position and angular velocity of the spacecraft. A numerical example is given for the conical motion of the spacecraft, as well as examples showing the closeness of the solutions of the traditional and modified optimal slew problems for an axisymmetric spacecraft.     

Gas-kinetic numerical study of complex flow problems covering various flow regimes

The Boltzmann simplified velocity distribution function equation, as adapted to various flow regimes, is described on the basis of the Boltzmann–Shakhov model from the kinetic theory of gases in this study. The discrete velocity ordinate method of gas-kinetic theory is studied and applied to simulate complex multi-scale flows. On the basis of using the uncoupling technique on molecular movements and collisions in the DSMC method, the gas-kinetic finite difference scheme is constructed by extending and applying the unsteady time-splitting method from computational fluid dynamics, which directly solves the discrete velocity distribution functions. The Gauss-type discrete velocity numerical quadrature technique for flows with different Mach numbers is developed to evaluate the macroscopic flow parameters in the physical space. As a result, the gas-kinetic numerical algorithm is established for studying the three-dimensional complex flows with high Mach numbers from rarefied transition to continuum regimes. On the basis of the parallel characteristics of the respective independent discrete velocity points in the discretized velocity space, a parallel strategy suitable for the gas-kinetic numerical method is investigated and, then, the HPF (High Performance Fortran) parallel programming software is developed for simulating gas dynamical problems covering the full spectrum of flow regimes. To illustrate the feasibility of the present gas-kinetic numerical method and simulate gas transport phenomena covering various flow regimes, the gas flows around three-dimensional spheres and spacecraft-like shapes with different Knudsen numbers and Mach numbers are investigated to validate the accuracy of the numerical methods through HPF parallel computing. The computational results determine the flow fields in high resolution and agree well with the theoretical and experimental data. This computing, in practice, has confirmed that the present gas-kinetic algorithm probably provides a promising approach for resolving hypersonic aerothermodynamic problems with the complete spectrum of flow regimes from the gas-kinetic point of view for solving the mesoscopic Boltzmann model equation.     

Effects of slip on sheet-driven flow and heat transfer of a non-Newtonian fluid past a stretching sheet

The flow and heat transfer of an electrically conducting non-Newtonian fluid due to a stretching surface subject to partial slip is considered. The constitutive equation of the non-Newtonian fluid is modeled by that for a third grade fluid. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHFcase) in presence of a uniform heat source or sink. Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective second order numerical scheme has been adopted to solve the obtained differential equations. The important finding in this communication is the combined effects of the partial slip, magnetic field, heat source (sink) parameter and the third grade fluid parameters on the velocity, skin friction coefficient and the temperature field. It is interesting to find that slip decreases the momentum boundary layer thickness and increases the thermal boundary layer thickness, whereas the third grade fluid parameter has an opposite effect on the thermal and velocity boundary layers.