A hybrid DSMC/Navier–Stokes frame to solve mixed rarefied/nonrarefied hypersonic flows over nano‐plate and micro‐cylinder

We extend a hybrid DSMC/Navier–Stokes (NS) approach to unify the DSMC and the NS simulators in one framework capable of solving the mixed non‐equilibrium and near‐equilibrium flow regions efficiently. Furthermore, we use a one‐way state‐based coupling (Dirichlet–Dirichlet boundary‐condition coupling) to transfer the required information from the continuum region to the rarefied one. The current hybrid DSMC–NS frame is applied to the hypersonic flows over nanoflat plate and microcylinder cases. The achieved solutions are compared with the pure DSMC and NS solutions. The results show that the current hybrid approach predicts the surface heat transfer rate and shear stress magnitudes very accurately. Some important conclusions can be drawn from this study. For example, although the shock wave region would be a non‐equilibrium region, it is not necessary to use a pure DSMC simulator to solve it entirely. This is important when the researchers wish to predict the surface properties such as velocity slip, temperature jump, wall heat flux rate, and friction drag magnitudes accurately. Our investigation showed that our hybrid solution time would be at least 40% (for the flat plate) and 35% (for the cylinder) of the time that must be spent by a pure DSMC solver to attain the same accuracy.Copyright © 2013 John Wiley & Sons, Ltd.     

Investigation on a coupled Navier–Stokes–Direct Simulation Monte Carlo method for the simulation of plume flowfield of a conical nozzle

To assess the plume effects of space thrusters, the accurate plume flowfield is indispensable. The plume flow of thrusters involves both continuum and rarefied flow regimes. Coupled Navier–Stokes–Direct Simulation Monte Carlo (NS–DSMC) method is a major approach to the simulation of continuum‐rarefied flows. An axisymmetric coupled NS–DSMC solver, possessing adaptive‐interface and two‐way coupling features, is investigated in this paper for the simulation of the nozzle and plume flows of thrusters. The state‐based coupling scheme is adopted, and the gradient local Knudsen number is used to indicate the breakdown of continuum solver. The nitrogen flows in a conical nozzle and its plume are chosen as the reference case to test the coupled solver. The threshold value of the continuum breakdown parameter is studied based on both theoretical kinetic velocity sampling and coupled numerical tests. Succeeding comparisons between coupled and full DSMC results demonstrate their conformities, meanwhile, the former saves 58.8% computational time. The pitot pressure evaluated from the coupled simulation result is compared with the experimental data proposed in literatures, revealing that the coupled method makes precise predictions on the experimental pitot pressure. Copyright © 2014 John Wiley & Sons, Ltd.     

Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A novel efficient hybrid DSMC–dynamic collision limiter algorithm for multiscale transitional flows

A new 2D parallel multispecies polyatomic particle–based hybrid flow solver is developed by coupling the Direct Simulation Monte Carlo (DSMC) method with a novel Dynamic Collision Limiter (DCL) approach to solve multiscale transitional flows. The hybrid DSMC‐DCL solver can solve nonequilibrium multiscale flows with length scales ranging from continuum to rarefied. The DCL method, developed in this work, dynamically assigns different number of collisions in cells, which is based on the local value of K‐S parameter such that the number of collisions per time step is limited in near‐equilibrium flow regions. Present hybrid solver uses the Kolmogorov‐Smirnov statistical test as the continuum breakdown parameter, based on which, the solution domain is decomposed into near‐equilibrium and nonequilibrium flow regions. Direct Simulation Monte Carlo is used where nonequilibrium flow regions are encountered, while the DCL method is used where flow regions are found to be in near‐equilibrium state. In this work, we have studied hypersonic flow of nitrogen over a blunt body with an aerospike and supersonic flow of argon through a micronozzle. The results obtained by the hybrid DSMC‐DCL solver are compared and shown to agree well with the experimental data and with those obtained from DSMC, with significant savings in the computational cost.     

Coupled Navier–Stokes/molecular dynamics simulations in nonperiodic domains based on particle forcing

This paper focuses on coupling methods for hybrid Navier–Stokes/molecular dynamics (MD) simulations. The computational domain is split in a continuum flow region, where a finite‐volume discretisation of the Navier–Stokes equations is used, and one or more particle domains, where molecular level modelling of the flow is employed. The domains are defined with a partial overlap, in which the flow states are coupled through an exchange of the velocity components. For the steady flows considered, an under‐relaxed Newton iteration method is used to drive the coupled system to convergence. The main focus of the present work is on methods to impose nonperiodic boundary conditions on the particle domain(s). A particle forcing is applied in the direction normal to the particle domain boundary to impose the boundary normal velocity component. A novel aspect of the present work is the extension of this method to more general nonplanar particle domain boundaries. The main contribution of the paper is the development of a particle forcing method in the direction tangential to the domain boundary, which is based on the equivalent continuum‐flow boundary shear stresses along with an iterative forcing strength adjustment based on the extrapolated particle boundary velocity. Furthermore, an adaptation scheme is presented, which uses the finite‐volume flux residuals of the particle bin averaged velocity field as a truncation criterion for the iterative force‐update scheme. It is demonstrated that by comparing the residual reduction for the momentum equation in the nonhomogeneous directions during the molecular dynamics simulations with that for a homogeneous direction, the forcing iteration at which the statistical noise in the velocity field dominates the uncertainty in the forcing strength can be determined. At this point the iteration can be truncated. It is shown that with adaptive schemes of this type, the total number of MD evaluations required in a coupled Navier–Stokes/MD simulation can be reduced relative to a hybrid scheme with a fixed number of forcing‐strength updates. Copyright © 2011 John Wiley & Sons, Ltd.     

3D free‐surface flow computation using a RANSE/Fourier–Kochin coupling

A coupling method for numerical calculations of steady free‐surface flows around a body is presented. The fluid domain in the neighbourhood of the hull is divided into two overlapping zones. Viscous effects are taken in account near the hull using Reynolds‐averaged Navier–Stokes equations (RANSE), whereas potential flow provides the flow away from the hull. In the internal domain, RANSE are solved by a fully coupled velocity, pressure and free‐surface elevation method. In the external domain, potential‐flow theory with linearized free‐surface condition is used to provide boundary conditions to the RANSE solver. The Fourier–Kochin method based on the Fourier–Kochin formulation, which defines the velocity field in a potential‐flow region in terms of the velocity distribution at a boundary surface, is used for that purpose. Moreover, the free‐surface Green function satisfying this linearized free‐surface condition is used. Calculations have been successfully performed for steady ship‐waves past a serie 60 and then have demonstrated abilities of the present coupling algorithm. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical studies on non‐linear free surface flow using generalized Schwarz–Christoffel transformation

This paper deals with a technique to transform a free surface flow problem in the physical domain with an unknown boundary to a standard domain that has a fixed boundary. All the difficulties in the physical domain are reduced to finding an unknown mapping function that can be solved iteratively in a standard domain. A derivation is first presented to express an analytic function in terms of the boundary value of its imaginary part. Using a relationship between boundaries of the standard and the physical domains, a formula for the generalized Schwarz–Christoffel transformation is then developed. Based on the generalized Schwarz–Christoffel integral and the Hilbert transform, a pair of non‐linear boundary integro‐differential equations in an infinite strip is formulated for solving fully non‐linear free surface flow problems. The boundary integral equations are then discretized with quadratic elements in an untruncated standard domain and solved by the Levenberg–Marquardt algorithm. Several examples of supercritical flow past obstructions are provided to demonstrate the flexibility and the accuracy of the proposed numerical scheme. Copyright © 2000 John Wiley & Sons, Ltd.     

DSMC／EPSM混合算法研究

将EPSM算法与DSMC方法结合，构造了可模拟含近连续流区及过渡流区的DSMC/EPSM混合算法。运用混合算法模拟了马赫数等于5时超音速竖板绕流及马赫数等于4时超音速平板绕流，并将结果与DSMC算法的结果进行比较，证明了DSMC／EPSM混合算法的有效性，同时将EPSM算法与DSMC算法的效率进行了比较。     

Empirical slip and viscosity model performance for microscale gas flow

For the simple geometries of Couette and Poiseuille flows, the velocity profile maintains a similar shape from continuum to free molecular flow. Therefore, modifications to the fluid viscosity and slip boundary conditions can improve the continuum based Navier–Stokes solution in the non‐continuum non‐equilibrium regime. In this investigation, the optimal modifications are found by a linear least‐squares fit of the Navier–Stokes solution to the non‐equilibrium solution obtained using the direct simulation Monte Carlo (DSMC) method. Models are then constructed for the Knudsen number dependence of the viscosity correction and the slip model from a database of DSMC solutions for Couette and Poiseuille flows of argon and nitrogen gas, with Knudsen numbers ranging from 0.01 to 10. Finally, the accuracy of the models is measured for non‐equilibrium cases both in and outside the DSMC database. Flows outside the database include: combined Couette and Poiseuille flow, partial wall accommodation, helium gas, and non‐zero convective acceleration. The models reproduce the velocity profiles in the DSMC database within an L₂ error norm of 3% for Couette flows and 7% for Poiseuille flows. However, the errors in the model predictions outside the database are up to five times larger. Copyright © 2005 John Wiley & Sons, Ltd.     

An atomistic–continuum hybrid scheme for numerical simulation of droplet spreading on a solid surface

We present an atomistic–continuum hybrid method to investigate spreading dynamics of drops on solid surfaces. The Navier–Stokes equations are solved by the finite-volume method in a continuum domain comprised of the main body of the drop, and atomistic molecular dynamics simulations are used in a particle domain in the vicinity of the contact line. The spatial coupling between the continuum and particle domains is achieved through constrained dynamics of flux continuities in an overlap domain.     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

Methods for implementing the stream boundary condition in DSMC computations

In the direct simulation Monte‐Carlo (DSMC) method for simulating rarefied gas flows, the velocities of simulator particles that cross a simulation boundary and enter the simulation space are typically generated using the acceptance–rejection procedure that samples the velocities from a truncated theoretical velocity distribution that excludes low and high velocities. This paper analyses an alternative technique, where the velocities of entering particles are obtained by extending the simulation procedures to a region adjacent to the simulation space, and considering the movement of particles generated within that region during the simulation time step. The alternative method may be considered as a form of acceptance–rejection procedure, and permits the generation of all possible velocities, although the population of high velocities is depleted with respect to the theoretical distribution. Nevertheless, this is an improvement over the standard acceptance–rejection method. Previous implementations of the alternative method gave a number flux lower than the theoretical number required. Two methods for obtaining the correct number flux are presented. For upstream boundaries in high‐speed flows, the alternative method is more computationally efficient than the acceptance–rejection method. However, for downstream boundaries, the alternative method is extremely inefficient. The alternative method, with the correct theoretical number flux, should therefore be used in DSMC computations in favour of the acceptance–rejection method for upstream boundaries in high‐speed flows. Copyright © 2003 John Wiley & Sons, Ltd.     

A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm

The velocity–vorticity formulation is selected to develop a time‐accurate CFD finite element algorithm for the incompressible Navier–Stokes equations in three dimensions.The finite element implementation uses equal order trilinear finite elements on a non‐staggered hexahedral mesh. A second order vorticity kinematic boundary condition is derived for the no slip wall boundary condition which also enforces the incompressibility constraint. A biconjugate gradient stabilized (BiCGSTAB) sparse iterative solver is utilized to solve the fully coupled system of equations as a Newton algorithm. The solver yields an efficient parallel solution algorithm on distributed‐memory machines, such as the IBM SP2. Three dimensional laminar flow solutions for a square channel, a lid‐driven cavity, and a thermal cavity are established and compared with available benchmark solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

DSMC (direct simulation Monte Carlo) analysis of microchannel flow with gas–liquid boundary

High heat capacity and constant operation temperature make a 2-phase heat remover tool promising for solving high heat dissipation problems in MEMS devices. However, microscale analysis of the flow with the conventional Navier–Stokes equation is inadequate, because the non-continuum effect is important when the characteristic dimension is comparable to the local mean free path. DSMC is a direct, particle-based numerical simulation method that uses no continuum assumption. In this paper, the gas–liquid boundary effects in microchannel flow are studied using this method. Modified DSMC code is used to simulate low-speed flow—under which viscous heating produces no significant temperature change—and MD results are incorporated into the DSMC boundary condition. Steady Couette flow simulation results show that the gas–liquid boundary affects the density distribution and the temperature dependence of the slip velocity. Unsteady simulation results show that mass transfer by diffusion is faster than momentum transfer by collision.     

A numerical scheme for Euler–Lagrange simulation of bubbly flows in complex systems

An Eulerian–Lagrangian approach is developed for the simulation of turbulent bubbly flows in complex systems. The liquid phase is treated as a continuum and the Navier–Stokes equations are solved in an unstructured grid, finite volume framework for turbulent flows. The dynamics of the disperse phase is modeled in a Lagrangian frame and includes models for the motion of each individual bubble, bubble size variations due to the local pressure changes, and interactions among the bubbles and with boundaries. The bubble growth/collapse is modeled by the Rayleigh–Plesset (RP) equation. Three modeling approaches are considered: (a) one‐way coupling, where the influence of the bubble on the fluid flow is neglected, (b) two‐way coupling, where the momentum‐exchange between the fluid and the bubbles is modeled, and (c) volumetric coupling, where the volumetric displacement of the fluid by the bubble motion and the momentum‐exchange are modeled. A novel adaptive time‐stepping scheme based on stability‐analysis of the non‐linear bubble dynamics equations is developed. The numerical approach is verified for various single bubble test cases to show second‐order accuracy. Interactions of multiple bubbles with vortical flows are simulated to study the effectiveness of the volumetric coupling approach in predicting the flow features observed experimentally. Finally, the numerical approach is used to perform a large‐eddy simulation in two configurations: (i) flow over a cavity to predict small‐scale cavitation and inception and (ii) a rising dense bubble plume in a stationary water column. The results show good predictive capability of the numerical algorithm in capturing complex flow features. Copyright © 2010 John Wiley & Sons, Ltd.     

Simulations for gas flows in microgeometries using the direct simulation Monte Carlo method

Micro gas flows are often encountered in MEMS devices and classical CFD could not accurately predict the flow and thermal behavior due to the high Knudsen number. Therefore, the gas flow in microgeometries was investigated using the direct simulation Monte Carlo (DSMC) method. New treatments for boundary conditions are verified by simulations of micro-Poiseuille flow, compared with the previous boundary treatments, and slip analytical solutions of the continuum theory. The orifice flow and the corner flow in microchannels are simulated using the modified DSMC codes. The predictions were compared with existing experimental phenomena as well as predictions using continuum theory. The results showed that the channel geometry significantly affects the microgas flow. In the orifice flow, the flow separation occurred at very small Reynolds numbers. In the corner flow, no flow separation occurred even with a high driving pressure. The DSMC results agreed well with existing experimental information.     

An improved Rhie–Chow interpolation scheme for the smoothed‐interface immersed boundary method

Rhie–Chow interpolation is a commonly used method in CFD calculations on a co‐located mesh in order to suppress non‐physical pressure oscillations arising from chequerboard effects. A fully parallelized smoothed‐interface immersed boundary method on a co‐located grid is described in this paper. We discuss the necessity of modifications to the original Rhie–Chow interpolation in order to deal with a locally refined mesh. Numerical simulation with the modified scheme of Choi shows that numerical dissipation due to Rhie–Chow interpolation introduces significant errors at the immersed boundary. To address this issue, we develop an improved Rhie–Chow interpolation scheme that is shown to increase the accuracy in resolving the flow near the immersed boundary. We compare our improved scheme with the modified scheme of Choi by parallel simulations of benchmark flows: (i) flow past a stationary cylinder; (ii) flow past an oscillating cylinder; and (iii) flow past a stationary elliptical cylinder, where Reynolds numbers are tested in the range 10–200. Our improved scheme is significantly more accurate and compares favourably with a staggered grid algorithm. We also develop a scheme to compute the boundary force for the direct‐forcing immersed boundary method efficiently. Copyright © 2016 John Wiley & Sons, Ltd.     

Time‐domain BEM solution of convection–diffusion‐type MHD equations

The two‐dimensional convection–diffusion‐type equations are solved by using the boundary element method (BEM) based on the time‐dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady‐state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time‐domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time‐dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley & Sons, Ltd.     

A combined analytical–numerical method for treating corner singularities in viscous flow predictions

A combined analytical–numerical method based on a matching asymptotic algorithm is proposed for treating angular (sharp corner or wedge) singularities in the numerical solution of the Navier–Stokes equations. We adopt an asymptotic solution for the local flow around the angular points based on the Stokes flow approximation and a numerical solution for the global flow outside the singular regions using a finite‐volume method. The coefficients involved in the analytical solution are iteratively updated by matching both solutions in a small region where the Stokes flow approximation holds. Moreover, an error analysis is derived for this method, which serves as a guideline for the practical implementation. The present method is applied to treat the leading‐edge singularity of a semi‐infinite plate. The effect of various influencing factors related to the implementation are evaluated with the help of numerical experiments. The investigation showed that the accuracy of the numerical solution for the flow around the leading edge can be significantly improved with the present method. The results of the numerical experiments support the error analysis and show the desired properties of the new algorithm, i.e. accuracy, robustness and efficiency. Based on the numerical results for the leading‐edge singularity, the validity of various classical approximate models for the flow, such as the Stokes approximation, the inviscid flow model and the boundary layer theory of varying orders are examined. Although the methodology proposed was evaluated for the leading‐edge problem, it is generally applicable to all kinds of angular singularities and all kinds of finite‐discretization methods. Copyright © 2004 John Wiley & Sons, Ltd.     

A pseudo‐viscoelastic approach for transient polymeric flow exiting a channel

The influence of elasticity of a fluid exiting a channel is examined on transient coating downstream. A hybrid spectral/boundary element approach is proposed to solve the problem. The flow inside the channel is assumed to be fully developed. A viscoelastic instability of one‐dimensional plane Couette flow is first determined for a large class of Oldroyd fluids with added viscosity, which typically represent polymer solutions composed of a Newtonian solvent and a polymeric solute. The Johnson–Segalman equation is used as the constitutive model. The velocity profile inside the channel is taken as the exit profile for the emerging free‐surface flow. The flow is assumed to be Newtonian as it emerges from the channel. An estimate of the magnitude of the rate‐of‐strain tensor components in the free‐surface region reveals that they are generally smaller than the shear rate inside the channel. The evolution of the flow front is simulated using the boundary element method. For the channel flow, the problem is reduced to a nonlinear dynamical system using the Galerkin projection method. Stability analysis indicates that the channel velocity may be linear or non‐linear depending on the range of the Weissenberg number. The evolution of the coating flow at the exit is examined for steady as well as transient (monotonic and oscillatory) channel flow. It is found that adverse flow can exist as a result of fluid elasticity, which can hinder the process of blade coating. Copyright © 2003 John Wiley & Sons, Ltd.