Slip Wall Boundary Condition Research Articles

Bona–Smith-Type systems in Bounded Domains with Slip-Wall Boundary Conditions: Theoretical Justification and a Conservative Numerical Scheme

Bona–Smith-Type systems in Bounded Domains with Slip-Wall Boundary Conditions: Theoretical Justification and a Conservative Numerical Scheme

Equations for small amplitude shallow water waves over small bathymetric variations

A generalized version of the a b c d -Boussinesq class of systems is derived to accommodate variable bottom topography in two-dimensional space. This extension allows for the conservation of suitable energy functionals in some cases and enables the description of water waves in closed basins with well-justified slip–wall boundary conditions. The derived systems possess a form that ensures their solutions adhere to important principles of physics and mathematics. By demonstrating their consistency with the Euler equations and estimating their approximation error, we establish the validity of these new systems. Their derivation is based on the assumption of small bathymetric variations. With practical applications in mind, we assess the effectiveness of some of these new systems through comparisons with standard benchmarks. The results indicate that the assumptions made during the derivation are not overly restrictive. The applications of the new systems encompass a wide range of scenarios, including the study of tsunamis, tidal waves and waves in ports and lakes.

Numerical study of the wall boundary condition in large-eddy simulation of the flow past a sharp-edged bluff body

Based on the fact that the drag coefficient is relatively insensitive to the Reynolds number in many experiments on the flow past a sharp-edged bluff body (SEBB) at high Reynolds numbers, it is speculated that the effect of the wall viscosity on the aerodynamics of the SEBB can be neglected to some extent. To validate this hypothesis for the large-eddy simulation (LES), the slip-wall boundary condition was applied to the SEBB, and this LES was termed slip-wall LES (SWLES). The flow past a square cylinder was chosen to validate the SWLES. The aerodynamic parameters, such as the drag coefficient and Strouhal number, agreed well with the experimental data and those of the traditional wall-resolved LES (WRLES). The surface pressure distributions and energy spectra were close to those of the WRLES. The Kelvin–Helmholtz instability structures, flow separation, and von Kármán vortex shedding were captured accurately by the SWLES. The computational cost was compared with that of the WRLES. An interpretation of the SWLES was discussed based on flow visualizations. After the validation, the SWLES was applied to the flow past a photovoltaic system. The wind loads of the system were compared with the wind tunnel experimental data at different incident angles. The surface pressure distributions, flow structures, and wake were analyzed. The present study demonstrated that the SWLES has potential applications for the flow around an SEBB with a much lower cost than that of the WRLES.

Effect of nonlinear thermal radiation and homogeneous-heterogeneous reactions on peristaltic propulsion of Johnson-Segalman fluid

The basic aim of this research is to investigate the main features of the peristaltic flow of Johnson-Segalman fluid in a curved flow channel in the presence of a homogeneous-heterogeneous reaction. The fluid is considered electrically conducting with a radial magnetic field effect. The constitutive relation for energy is formulated with the addition of viscous dissipation and nonlinear thermal radiation. The slip and flexible wall boundary conditions are taken into consideration. The lubrication technique is used for the mathematical the modelling of the problem. The solution of the resulting nonlinear system of ODE’s is computed numerically through NDSolve command of Mathematica, and graphical results are prepared for flow variables. It has been clearly noticed that flow is accelerated in the planer medium as compared to the curved channel, and the velocity of the fluid increases with a larger Weissenberg number. Moreover, the thermal profile has an inverse relation to the parameters, i.e., radiation, temperature ratio, and curvature. Furthermore, it is also observed that the magnitude of heat transfer coefficient rises significantly when a comparatively less curved channel is utilized, however, a reduction in concentration profile is witnessed in this case. Alternative impact of homogeneous and heterogeneous reactions on concentration profile are observed through graphical results. Application of the study include the usage of peristaltic pump in chemical industry and in pharmacology industry for the purpose of drug deliveries.

Simulations of Wind Formation in Idealised Mountain–Valley Systems Using OpenFOAM

An OpenFOAM computational fluid dynamics model setup is proposed for simulating thermally driven winds in mountain–valley systems. As a first step, the choice of Reynolds Averaged Navier–Stokes k−ε turbulence model is validated on a 3D geometry by comparing its results vs. large-eddy simulations reported in the literature. Then, a numerical model of an idealised 2D mountain–valley system with mountain slope angle of 20° is developed to simulate thermally driven winds. A couple of top surface boundary conditions (BC) and various combinations of temperature initial conditions (IC) are tested. A transient solver for buoyant, turbulent flow of incompressible fluids is used. Contrary to classical approaches where buoyancy is set as a variable of the problem, here temperature linearly dependent with altitude is imposed as BC on the slope and successfully leads to thermally driven wind generation. The minimum fluid domain height needed to properly simulate the thermally driven winds and the effects of the different setups on the results are discussed. Slip wall BC on the top surface of the fluid domain and uniform temperature IC are found to be the most adequate choices. Finally, valleys with different widths are simulated to see how the mountain–valley geometry affects the flow behaviour, both for anabatic (daytime, up-slope) and katabatic (nighttime, down-slope) winds. The simulations correctly reproduce the acceleration and deceleration of the flow along the slope. Increasing the valley width does not significantly affect the magnitude of the thermally driven wind but does produce a displacement of the generated convective cell.

Performance of a Propeller Coated with Hydrophobic Material

Computational and experimental methods were used to study a propeller coated with hydrophobic material and a propeller with a conventional surface for comparison. In CFD simulations, the blade surface mesh was arranged in a way to set non-slip or free slip wall boundary conditions with different proportions to define the level of surface slip. The conventional and the hydrophobic material propellers defined by different surface slip rates were simulated under different advance speed coefficients and different rotational speeds. Propeller performance results, blade pressure, and the Liutex vorticity distribution were studied. An experimental platform was established to study the velocity field around the propeller using a Particle Image Velocimetry (PIV) device. The CFD calculation results were compared with the PIV results. It was found that the calculation results using a 75% surface slip rate were closer to the experimental results. The calculation results show that the propeller coated with hydrophobic material has improved thrust and efficiency compared with the propeller with conventional material. The hydrophobic material can significantly reduce the low-speed region downstream of the propeller hub. The hub and the tip vortices shown by the Liutex are also significantly reduced. Those changes help to improve the propulsion efficiency.

An entropy–stable discontinuous Galerkin approximation of the Spalart–Allmaras turbulence model for the compressible Reynolds Averaged Navier–Stokes equations

We present an entropy–stable formulation for the compressible Reynolds Averaged Navier–Stokes (RANS) Discontinuous Galerkin (DG) equations and the Spalart–Allmaras one–equation closure. The model is designed to satisfy an entropy law, which includes free– and no–slip wall boundary conditions. We then construct a high–order DG approximation of the model that satisfies the summation–by–parts simultaneous–approximation–term (SBP-SAT) property. With the help of a discrete stability analysis, we construct two approximations: a kinetic energy preserving scheme based on Pirozzoli's two–point flux and a thermodynamic entropy conserving one based on Chandrashekar's split–form. The schemes are applicable to, and the stability proofs hold for, three–dimensional unstructured meshes with curvilinear hexahedral elements. We test the convergence of the schemes on a manufactured solution for increasing polynomial orders and mesh refinement levels, to then assess their numerical stability by propagating a flow from a random initial condition, and finally solve the flow around a two–dimensional flat plate and a NACA 0012 airfoil, comparing numerical results with those available in the literature. The proposed schemes are entropy–stable, and provide accurate solutions for the selected test cases.

A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis

The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche’s method. Convergence of the numerical method is analysed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.

A Parametric Study on the Fluid Dynamics and Performance Characteristic of Micronozzle Flows

Abstract This study investigates the fluid dynamics and performance characteristics in micronozzle flows with changes in various geometric parameters using Navier–Stokes simulation based on slip wall boundary conditions. The various geometric parameters considered for the study are (1) area ratio with fixed throat dimension and (2) the semidivergence angle variation with no change in area ratio. The simulation results show that the flow choking for micronozzle happens not at the geometric throat; rather pushed downstream to the divergent channel of the nozzle. This is due to the thick boundary layer growth, which reduces the effective flow area and shifts the minimum allowable flow area downstream to the throat. The distance to which the choking point shifts downstream to the throat reduces with Maxwell's slip wall conditions compared to the conventional no-slip wall condition. The downstream movement of the choking point from the throat reduces with an increase in area ratio and with increase in divergence angle with fixed area ratio. This is due to the fact that the increase in area ratio and divergence angle increases the nozzle height at any particular section in the divergent portion of the nozzle. As a result of this, the boundary layer profile also moves upward and the restriction of potential core by the thick boundary layer reduces, which in turn leads to an increase in the effective minimum flow area downstream to the throat.

The thermal –fluid investigation of effects of different wall boundary conditions on platinum catalytic micro-channel combined with a thermoelectric system

In this work, the effects of different regimes of boundary conditions such as convective heat transfer coefficient, insulated wall jointed by slip flow, thermal jump and slip-jump cases are investigated. In all cases of solution, an output voltage/power is achieved from a thermoelectric that is mounted on upper wall of catalytic micro-channel. The solution of this problem is done with coupled fluid (transport species model) and solid (wall of micro-channel and bottom legs of thermoelectric) sections. Furthermore, the electrical values are calculated with an algebraic code prepared into FORTRAN software. The variations of normalized temperature are compared with experimental data that show acceptable agreements with maximum error of 38% for non-slip and 18% for slip boundary conditions. Therefore, the slip wall boundary condition is more precise than non-slip condition. The values of generated power of thermoelectric have the maximum difference of 18% between slip and non-slip condition at inlet velocity of 0.3 m/s. The complete conversion of methane is occurred in insulated wall condition while the lowest value is achieved at thermal-jump condition with 85.46% methane conversion. Increasing inlet velocity the generated electrical power of thermoelectric decreases that is as results of decreasing the average temperature of hot legs of thermoelectric. Furthermore, non-continuity of temperature and continuity of velocity profiles increases output voltage and generated power of thermoelectric.

Wall effect on the flow structures of three-dimensional rotating detonation wave

The wall effect on the flow structures of the three-dimensional rotating detonation wave in an annular chamber is investigated by utilizing premixed hydrogen/air at a stoichiometric ratio as the reactant and a 7-species-8-reaction finite-rate chemical reaction model for evaluating the chemical source term. Three cases are constructed with different flow solvers, i.e., Euler, Navier-Stokes and filtered Navier-Stokes with Improved Delayed Detached-Eddy Simulation to identify the wall effects. Both slip and non-slip wall boundary conditions are considered to quantify the viscous effect near the wall. It is found that the flow structures of the rotating detonation wave change notably when the effect of the viscous wall is taken into consideration, resulting in the extension of the deflagration zone to the upstream along the wall, the severe deformation of the detonation front at the unbounded side, and the formation of the reactant deficit. The mechanism of such alteration in the flow structures is analyzed, and the formation procedure of the reactant deficit zone is proposed. The comparison of specific impulse shows that our simulation is about 10% lower than the ideal two-dimensional rotating detonation engine model, and the practical experiment model is about 5.8% lower than our simulation. The wall effect, the three-dimensional effect and the turbulent mixing would lead to the performance losses.

Modelling of a turbulent lean premixed combustor using a Reynolds-averaged Navier–Stokes approach

This work aims to characterize lean premixed turbulent swirling flames representative of gas turbines and jet engines via a numerical study of flame topologies in a laboratory-scale burner. The state of the art of the numerical studies concerning these types of flames is first reviewed, with respect to Reynolds-averaged Navier–Stokes and large eddy simulations. Then, a turbulent, isothermal flow study is performed within the radial swirler. The impact of mesh refinement levels and boundary conditions on the swirl number and overall flow structure is investigated. The mesh refinement level and slip wall boundary condition alter the computed swirl number significantly. The computed swirl number converges to a value of 0.7, which is larger than the geometrical one, 0.4. Furthermore, using Reynolds-averaged Navier–Stokes transport equations, closed by the realizable $$k-\epsilon$$ model, coupled with a two-equation premixed combustion model for methane/air mixtures, two combustion regimes are analysed. These regimes correspond to the outer recirculation zone flame and an unstable regime. The flow structure is characterized in terms of velocity fields, turbulence and combustion properties. A reaction progress variable comparison is also performed, using existing experimental results, yielding qualitatively similar structures for both studied regimes. Some discrepancies between numerical and experimental results concerning the stable regime may be observed: The computed progress variable at the outer recirculation zone, 0.5, is smaller than the experimental value, 0.8, and the average flame brush thickness, 1 mm, is found to be smaller than the measured, 3 mm.

Entropy–stable discontinuous Galerkin approximation with summation–by–parts property for the incompressible Navier–Stokes/Cahn–Hilliard system

We develop an entropy–stable two–phase incompressible Navier–Stokes/Cahn–Hilliard discontinuous Galerkin (DG) flow solver method. The model poses the Cahn–Hilliard equation as the phase field method, a skew–symmetric form of the momentum equation, and an artificial compressibility method to compute the pressure. We design the model so that it satisfies an entropy law, including free– and no–slip wall boundary conditions with non–zero wall contact angle. We then construct a high–order DG approximation of the model that satisfies the summation–by–parts simultaneous–approximation–term property. With the help of a discrete stability analysis, the scheme has two modes: an entropy–conserving approximation with central advective fluxes and the Bassi–Rebay 1 (BR1) method for diffusion, and an entropy–stable approximation with an exact Riemann solver for advection and interface stabilization added to the BR1 method. The scheme is applicable to, and the stability proofs hold for, three–dimensional unstructured meshes with curvilinear hexahedral elements. We test the convergence of the schemes on a manufactured solution, and their robustness by solving a flow initialized from random numbers. In the latter, we find that a similar scheme that does not satisfy an entropy inequality had 30% probability to fail, while the entropy–stable scheme never does. We also solve the static and rising bubble test problems, and to challenge the solver capabilities we compute a three–dimensional pipe flow in the annular regime.

An improved CFD simulation for investigation of the sand particles flow behavior in the sand shooting process

The flow behavior of sand particles has been examined by computational fluid dynamics (CFD) and experimental validations in sand shooting process. To avoid over-estimating the drag force between gas and solid phases, the energy minimization multiscale model(EMMS)has been incorporated. An applicable value of grid size is determined by grid dependency study. Three specularity coefficients are simulated to get an insight into the influence on the flow hydrodynamics of sand particles. The specularity coefficient affects tangential velocity and volume distribution of sand particles. No slip wall boundary condition is in better agreement with experimental data. With validation of sand shooting experiment, agreement is achieved between experiment images and simulation results. A comparison between EMMS drag model and Gidaspow model has been discussed in terms of simulating particles flow behavior with different particle diameters. The EMMS drag model has an accuracy prediction especially for smaller particles and dense flow.

Instability of the Kolmogorov flow in a wall-bounded domain

In the magnetohydrodynamics (MHD) experiment performed by Bondarenko and his co-workers in 1979, the Kolmogorov flow loses stability and transits into a secondary steady state flow at the Reynolds number R = O(103), as Hartmann boundary layer friction dominates the energy dissipation of the fluid motion. This problem is modelled as a two-dimensional MHD flow bounded between horizontal walls under slip wall boundary condition. However, the rigorous analysis showing the existence of the secondary steady state flow was missing in earlier investigations and has now been established. The theoretical solution has a very good agreement with the flow measured in laboratory experiment at R = O(103). Further transition of the secondary flow is observed numerically. Especially, well developed turbulence arises at R = O(104) via an inverse energy cascade scenario.

Numerical investigation of a viscous regularization of the Euler equations by entropy viscosity

The Navier–Stokes viscous fluxes are a well-known viscous regularization of the Euler equations. However, since these fluxes do not add any viscosity to the mass equation, the positivity of density is violated. This paper investigates a new class of viscous regularization of the Euler equations, which was recently proposed by Guermond & Popov (2014). In contrast to the Navier–Stokes fluxes, the new regularization adds a viscous term to the mass equation. Since non-physical viscous terms are used, it is important to show that the exact solution’s properties, such as the location of shocks, contact and rarefaction waves are not violated. The present study concerns a careful numerical investigation of the new viscous regularization in a number of well-known 1D and 2D benchmark problems. Also, a direct numerical comparison with respect to the physical Navier–Stokes regularization is shown. The numerical tests show that the entropy viscosity method can achieve high order accuracy for any polynomial degrees. Detailed algorithms for the implementation of a slip wall boundary condition are presented in a weak and a strong form.

Error transport equation boundary conditions for the Euler and Navier–Stokes equations

Discretization error is usually the largest and most difficult numerical error source to estimate for computational fluid dynamics, and boundary conditions often contribute a significant source of error. Boundary conditions are described with a governing equation to prescribe particular behavior at the boundary of a computational domain. Boundary condition implementations are considered sufficient when discretized with the same order of accuracy as the primary governing equations; however, careless implementations of boundary conditions can result in significantly larger numerical error. Investigations into different numerical implementations of Dirichlet and Neumann boundary conditions for Burgers' equation show a significant impact on the accuracy of Richardson extrapolation and error transport equation discretization error estimates. The development of boundary conditions for Burgers' equation shows significant improvements in discretization error estimates in general and a significant improvement in truncation error estimation. The latter of which is key to accurate residual-based discretization error estimation. This research investigates scheme consistent and scheme inconsistent implementations of inflow and outflow boundary conditions up to fourth order accurate and a formulation for a slip wall boundary condition for truncation error estimation are developed for the Navier–Stokes and Euler equations. The scheme consistent implementation resulted in much smoother truncation error near the boundaries and more accurate discretization error estimates.

MEMS-Scale Turbomachinery Based Vacuum Roughing Pump

This study forms part of a program to develop a micro-electro-mechanical systems (MEMS) scale turbomachinery based vacuum pump and investigates the roughing portion of such a system. Such a machine would have many radial stages with the exhaust stages operating near atmospheric conditions while the inlet stages operate at near vacuum conditions. In low vacuum such as those to the inlet of a roughing pump, the flow can still be treated as a continuum; however, the no-slip boundary condition is not accurate. The Knudsen number becomes a dominant nondimensional parameter in these machines due to their small size and low pressures. As the Knudsen number increases, slip-flow becomes present at the walls. The study begins with a basic overview on implementing the slip wall boundary condition in a commercial code by specifying the wall shear stress based on the mean-free-path of the gas molecules. This is validated against an available micro-Poiseuille classical solution at Knudsen numbers between 0.001 and 0.1 with reasonable agreement found. The method of specifying the wall shear stress is then applied to a generic MEMS scale roughing pump stage that consists of two stators and a rotor operating at a nominal absolute pressure of 500 Pa. The zero flow case was simulated in all cases as the pump down time for these machines is small due to the small volume being evacuated. Initial transient two-dimensional (2D) simulations are used to evaluate three boundary conditions, classical no-slip, specified-shear, and slip-flow. It is found that the stage pressure rise increased as the flow began to slip at the walls. In addition, it was found that at lower pressures the pure slip boundary condition resulted in very similar predictions to the specified-shear simulations. As the specified-shear simulations are computationally expensive it is reasonable to use slip-flow boundary conditions. This approach was used to perform three-dimensional (3D) simulations of the stage. Again the stage pressure increased when slip-flow was present compared with the classical no-slip boundaries. A characteristic of MEMS scale turbomachinery are the large relative tip gaps requiring 3D simulations. A tip gap sensitivity study was performed and it was found that when no-slip boundaries were present the pressure ratio increased significantly with decreasing tip gap. When slip-flow boundaries were present, this relationship was far weaker.

Simulation of mixed electroosmotic/pressure-driven flows by utilizing dissipative particle dynamics

In this paper, we present an extension of dissipative particle dynamics method in order to study the mixed electroosmotic/pressure-driven micro- or nano-flows. This method is based on the Poisson–Boltzmann equation and has a great potential to resolve the electric double layer (EDL). Hence, apart from studying the bulk flow, it also provides a strong capability in order to resolve the complex phenomena occur inside the EDL. We utilize the proposed method to study the pure electroosmotic and also the mixed electroosmotic/pressure-driven flow through the straight micro-/nano-channels. The obtained results are in good agreement with the available analytical solutions. Furthermore, we study the electroosmotic flow and motion of DNA molecules through a T-shaped micro-channel. We show that neglecting the EDL and utilizing the slip wall boundary condition model can result in crucially misleading hydrodynamic characteristics if the EDL is comparable to the width of the channel. Finally, we utilize the presented method in order to study the complex flow patterns, which are created due to the heterogeneous distribution of the electric potential of the walls. These complex flow patterns usually are utilized in order to enhance the efficiency of mixing process in micro-/nano-length scales. In addition, we show that they can also be utilized effectively in order to separate the different macro-molecules such as polymers, DNA molecules and so on, according to their length of chain.

Numerical evaluation of turbulence models for dense to dilute gas–solid flows in vertical conveyor

A two-fluid model (TFM) of multiphase flows based on the kinetic theory and small frictional limit boundary condition of granular flow was used to study the behavior of dense to dilute gas–solid flows in vertical pneumatic conveyor. An axisymmetric 2-dimensional, vertical pipe with 5.6m length and 0.01m internal diameter was chosen as the computation domain, same to that used for experimentation in the literature. The chosen particles are spherical, of diameter 1.91mm and density 2500kg/m3. Turbulence interaction between the gas and particle phases was investigated by Simonin's and Ahmadi's models and their numerical results were validated for dilute to dense conveying of particles. Flow regimes transition and pressure drop were predicted. Voidage and velocity profiles of each phase were calculated in radial direction at different lengths of the conveying pipe. It was found that the voidage has a minimum, and gas and solid velocities have maximum values along the center line of the conveying pipe and pressure drop has a minimum value in transition from dense slugging to dilute stable flow regime. Slug length and pressure fluctuation reduction were predicted with increasing gas velocity, too. It is shown that solid phase turbulence plays a significant role in numerical prediction of hydrodynamics of conveyor and the capability of particles turbulence models depends on tuning parameters of slip-wall boundary condition.