No-slip Wall Research Articles

Comparison of Laminar and Turbulent K–Omega Shear Stress Transport Models Under Realistic Boundary Conditions Using Clinical Data for Arterial Stenosis

Abstract Early diagnosis of cardiovascular diseases (CVDs), including arterial stenosis, enables targeted treatments that reduce CVD mortality. It is vital to improve the accuracy of early diagnostic tools. Current computational studies of stenosis use mathematical models, such as laminar and k–omega shear stress transport (SST) models, available in ansys (Fluent and CFX), openfoam, and comsol software packages. Users can adjust boundary conditions, such as inlet velocity and outlet pressure using user-defined functions (UDFs) with different expressions and constant values. However, currently there is no rule over what to impose at these boundaries, and previous studies have used various assumptions, such as rigid artery wall, one-way fluid–structure interaction (FSI) or two-way FSI, and the blood's Newtonian or non-Newtonian material properties. This variety in construction has associated deviations of the models from the clinical data and lessens the value of the models as potential diagnostic or predictive tools for medical practitioners. In this study, we examine arterial stenosis models, with severities of 20%, 40%, and 50%, compared with the healthy artery analyzed in terms of strain energy to the artery wall. Additionally, we investigate elastic walls using one-way FSI, comparing with laminar and k–omega SST. These boundary conditions are based on clinical data. The results regarding the strain energy (mJ) behavior along the artery wall show that the k–omega SST model outperforms the laminar model for short arterial segments and under the Newtonian assumption with a no-slip boundary wall and turbulent flow.

Asymmetric vertical transport in weakly forced shallow flows

In this paper, we report on an investigation of the vertical transport of tracer particles released within a shallow, continuously-forced flow by means of numerical simulations. The investigation is motivated by the shallow flows encountered in many environmental situations and inspired by the laboratory experiments conducted in electromagnetically forced shallow fluid layers. The flow is confined to a thin fluid layer by stress-free top and no-slip bottom walls. The dynamics and the transport properties of the shallow flow are investigated under various flow conditions characterized by a Reynolds number related to the forcing, ReF, and the aspect ratio of vertical and horizontal length scales δ. The forcing generates an array of vortices that becomes unsteady when ReFδ2≳10. These vortices are accompanied by upwellings in their cores which are surrounded by narrower, stronger downwellings. Hence, upwellings occur where the horizontal flow is vorticity-dominated, while downwellings where it is strain-dominated. The magnitude of the asymmetry in strength and size of the vertical flows and their correlation with horizontal structures depends on the flow conditions and significantly influences the vertical spreading of particles within the fluid volume. Under conditions leading to a large asymmetry, particles within updrafts are transported slowly upwards, while particles within downdrafts rapidly move downwards. In addition, particles are trapped for longer within the updrafts than downdrafts because of their correlation with vorticity-dominated regions. However, when the flow becomes fully three-dimensional and highly unsteady for large ReFδ2 values, this transport asymmetry subsides because the updrafts and downdrafts exhibit similar strength and size in such flow conditions. Consequently, similar amounts of particles are transported upwards and downwards at similar rates.

Inertial migration of rigid particles in shear-thinning fluids under asymmetric wall slip conditions

The determination of flow-induced equilibrium positions in pressure-driven flows in microchannels is of great practical importance in particle manipulation. In the computational analysis presented in this paper, the inertial ordering of neutrally buoyant rigid spheres in shear-thinning fluid flow through a hydrophobic microchannel is investigated. The combined effect of the viscosity index n of a power-law fluid and fluid slippage at the wall on the lateral focusing of microspheres is examined in detail. Using the finite element method, the Eulerian flow field between partially slipping parallel walls is simulated, and the Lagrangian movement of particles is continuously tracked. The Navier slip model is used to ensure a finite fluid velocity at the wall, and it is tuned by modifying the slip-length. It is observed that inertial particles concentrate at a standard equilibrium position of 0.6 times the channel half-width H, irrespective of fluid slip due to the symmetry of the flow field. However, this equilibrium position shifts closer to the walls as the viscosity index increases; for instance, when n = 0.5, particles stabilize at 0.75H. As a consequence of asymmetry in hydrodynamic behavior due to different fluid slippages at the upper and lower walls, the particle migration path is altered. In a channel with a no-slip upper wall and a partially slipping lower wall (β/H = 0.4), particles settle closer to the lower wall at 0.8H. Most importantly, the lateral movement of a particle released at a given vertical position can be altered by tailoring the wall hydrophobicity and viscosity index, thus enabling multiple equilibrium locations to be achieved.

The hydraulically smooth limit of flow over surfaces with spanwise heterogeneity

Wall slip is conventionally obtained through riblets or grooves. As riblets are macroscopic objects, it is usually not practical to discuss slip patterns at the hydraulically smooth limit. Per Nikuradse, the smooth limit is when the characteristic length of the surface pattern is comparable to the viscous length scale. Recent studies show the possibility of slip at the microscopic scale, making slip patterns at the hydraulically smooth limit relevant. In this study, we leverage a high-fidelity pseudo-spectral code and study flow over surfaces featuring alternating strips of slip and no-slip wall conditions. The wavelength of the surface pattern, denoted as l, varies from 1.5 times the half channel height to 2 viscous units (plus units), eventually approaching the anticipated hydraulically smooth limit. The presence of surface slip gives rise to a slip velocity at the wall, denoted as Us, which contributes to drag reduction. The surface spanwise heterogeneity leads to secondary flows and intensifies turbulent mixing, consequently leading to drag increase. This drag increase effect can be parameterized using the “roughness function” ΔU+. The sum of Us+ and −ΔU+ determines whether the surface increases or reduces drag. Here, the superscript ＋ denotes normalization by the wall units. In most cases, the slip velocity at the wall Us+ predominates over −ΔU+, resulting in drag reduction. However, when l is a few viscous units, the roughness function ΔU+ does not vanish and overwhelms the slip velocity, giving rise to a net drag increase. Considering that the wall is a mixture of slip and no-slip conditions, this drag increase at the anticipated hydraulically smooth limit is unexpected. To gain an insight into the mechanism responsible for this drag increase, we derive a Navier–Stokes-based decomposition of the roughness function. Here, we generalize the definition of the roughness function such that it is a function of the wall-normal coordinate, thereby overcoming the difficulty of measuring the roughness function when the log region is narrow and hard to define. Analysis shows that when l is a few plus units, secondary flows contribute to a slightly positive ΔU+, while turbulent and viscous contributions, by and large, cancel out, ultimately leading to an overall drag increase at the anticipated hydraulically smooth limit. The evidence in the paper suggests that the hydraulically smooth limit does not exist for certain surfaces.

Investigation of flow over an inclination pile under a Reynolds number of 3900 by numerical simulation

Flow over an inclination pile was studied numerically by solving the Navier–Stokes equations with large eddy simulation closure under a Reynolds number of 3900, and the inclination angles were in the range of −45° ≤ θ ≤ 45°. First, flow over a finite-length vertical pile with symmetric boundary conditions at both the upper and lower boundaries is used for verification, and the results are in good agreement with the previous results. Then, the lower boundary is set as a no-slip wall to simulate a rigid seabed, and 19 different inclination angles are numerically simulated to investigate the effects of the rigid seabed on the hydrodynamic characteristics and the wake structure of the cylindrical wake with different inclination angles. It is found that there is a significant difference between the forces on the piles, which were inclined forward and backward due to the end boundary conditions. When the pile is inclined forward, the force on the pile and the frequency of vortex shedding are lower than that of a vertical pile, which is more conducive to the stability of the structure. When the pile is inclined backward, the lower sidewall induces a strong spiral upward flow of the fluid at the lower end around the pile, which leads to the shedding of the wake vortex at a very close position to the pile. The return zone tightening phenomenon occurs at inclination angles greater than 25° backward, and this phenomenon leads to a sharp increase in the forces on the pile. Backward inclination angles greater than 30° are extremely unfavorable for structural forces.

Moving air-water interface on no-slip solid walls for high-speed planing hulls

ABSTRACT In this study, the classical ‘moving-contact-line’ problem associated with the no-slip boundary condition (BC) is examined, with a particular focus on large-scale, high Reynolds number turbulent ship flows. Numerical ventilation is one of the main issues reported for the computational prediction of the high-speed small planing craft using the Volume-of-Fluid (VOF) method. A numerical strategy is presented to resolve this issue with a wave blanking distance defined and used when solving the VOF equations, which is chosen based on the y + values and the velocity profiles in the boundary layer. A series of numerical tests are conducted using a slamming plate and a stepped high-speed planing hull. The numerical experiments show that if the blanking distance is y + < 30 (inside the buffer and viscous sublayers), the air-water interface on the wall will be unstable and numerical ventilation will occur. For the blanking distance y + > 30 (outside the buffer layer), the air-water contact line is smooth and air entrainment can be avoided. It is suggested that the blank distance needs to satisfy 30.0 < y + < 200.0 in consideration of accuracy and stability, and a value of y + ∼ 100.0 can be used in practice.

Hydrodynamic synchronization of elastic cilia: How surface effects determine the characteristics of metachronal waves.

Cilia are hairlike microactuators whose cyclic motion is specialized to propel extracellular fluids at low Reynolds numbers. Clusters of these organelles can form synchronized beating patterns, called metachronal waves, which presumably arise from hydrodynamic interactions. We model hydrodynamically interacting cilia by microspheres elastically bound to circular orbits, whose inclinations with respect to a no-slip wall model the ciliary power and recovery stroke, resulting in an anisotropy of the viscous flow. We derive a coupled phase-oscillator description by reducing the microsphere dynamics to the slow timescale of synchronization and determine analytical metachronal wave solutions and their stability in a periodic chain setting. In this framework, a simple intuition for the hydrodynamic coupling between phase oscillators is established by relating the geometry of flow near the surface of a cell or tissue to the directionality of the hydrodynamic coupling functions. This intuition naturally explains the properties of the linear stability of metachronal waves. The flow near the surface stabilizes metachronal waves with long wavelengths propagating in the direction of the power stroke and, moreover, metachronal waves with short wavelengths propagating perpendicularly to the power stroke. Performing simulations of phase-oscillator chains with periodic boundary conditions, we indeed find that both wave types emerge with a variety of linearly stable wave numbers. In open chains of phase oscillators, the dynamics of metachronal waves is fundamentally different. Here the elasticity of the model cilia controls the wave direction and selects a particular wave number: At large elasticity, waves traveling in the direction of the power stroke are stable, whereas at smaller elasticity waves in the opposite direction are stable. For intermediate elasticity both wave directions coexist. In this regime, waves propagating towards both ends of the chain form, but only one wave direction prevails, depending on the elasticity and initial conditions.

Three-dimensional shock–flame interactions: effect of wall friction

Multi-dimensional numerical simulations were performed to study the interactions between shock wave and premixed flame. The three-dimensional (3D) fully-compressible, reactive Navier–Stokes equations were solved using a high-order numerical method on a dynamically adapting mesh. The effect of wall friction on the shock–flame interaction was examined by varying wall boundary condition on sidewall. The simulations agree with previous experiment in terms of flame perturbation and flame evolution. The results show two effects of wall friction on flame–shock interaction: (1) flame stretching and (2) damping of local flame perturbation very close to the no-slip wall. The stretch effect leads to non-uniform development of the perturbated flame and consequently a significantly higher growth rate in both global flame perturbation and averaged pressure in the no-slip case compared to the free-slip case. By contrast, the damping effect locally moderates the flame perturbation in close proximity to the no-slip wall because less vorticity is deposited on this part of flame during shock–flame interaction. Quantitative analysis of vortex dynamics suggests that vorticity generation that is produced by baroclinic torque and enhanced by the dilation term during shock–flame interaction causes a growth of flame perturbation via Richtmyer–Meshkov instability. Nevertheless, the vorticity generation is greatly weakened in close proximity to the no-slip wall by the viscous torque and viscous dissipation terms due to friction.

Fluid dynamics of a flapping wing interacting with the boundary layer at a flat wall

In this paper, we consider the fluid dynamics of a flapping wing interacting with a boundary layer developed at a no-slip flat wall. Direct numerical simulations are carried out via implementing the non-iterative immersed boundary-lattice Boltzmann method, over a Reynolds number range of 10≤Re≤1000, for a fixed Strouhal number of St = 0.3 and for a given symmetric plunging and pitching flapping motion. The interactions between the wing and the boundary layer are modulated by varying the mean distance of the wing to the wall H0. The results indicate that the presence of the boundary layer at the wall amplifies the fluctuations in both lift and drag due to the boundary layer separation, in contrast to the pure ground effect. This separation also leads to the decrease in both average lift and average drag over one flapping cycle when H0 is low. When it comes to the flow patterns in the wake, it generally gets more complex for a low H0 and/or a high Re. Secondary vortices can be observed for Re≥500 in the present configuration, which either evolve by themselves or interact with the vortices in the wake while being convected downstream and dissipated via viscosity. In the end, a dynamic mode decomposition analysis is performed to explore further the flow structures in the wake. One observes the sheltering effect of the boundary layer that the vortices in the wake are prevented from penetrating the boundary layer, while this effect will not hold if the vortex intensity is sufficiently high, such as the low order mode of the case for Re≥1000 in this study.

Collective motion in a sheet of microswimmers

Self-propelled particles such as bacteria or algae swimming through a fluid are non-equilibrium systems where particle motility breaks microscopic detailed balance, often resulting in large-scale collective motion. Previous theoretical work has identified long-ranged hydrodynamic interactions as the driver of collective motion in unbounded suspensions of rear-actuated (“pusher”) microswimmers. In contrast, most experimental studies of collective motion in microswimmer suspensions have been carried out in restricted geometries where both the swimmers’ motion and their long-range flow fields become altered due to the proximity of a boundary. Here, we study numerically a minimal model of microswimmers in such a restricted geometry, where the particles move in the midplane between two no-slip walls. For pushers, we demonstrate collective motion with short-ranged order, in contrast with the long-ranged flows observed in unbounded systems. For front-actuated (“puller”) microswimmers, we discover a long-wavelength density instability resulting in the formation of dense microswimmer clusters. Both types of collective motion are fundamentally different from their previously studied counterparts in unbounded domains. Our results show that this difference is dictated by the geometrical restriction of the swimmers’ motion, while hydrodynamic screening due to the presence of a wall is subdominant in determining the suspension’s collective state.

The Impact of Inclined Magnetic Field on Streamlines in a Constricted Lid-driven Cavity

The influence of oriented magnetic field on the incompressible and electrically conducting flow is investigated in a square cavity with a moving top wall and a no-slip constricted bottom wall. Radial basis function (RBF) approximation is employed to velocity-stream function-vorticity formulation of MHD equations. Numerical results are shown in terms of streamlines for different values of Hartmann number M, orientation angle of magnetic field ϴ and the height of the constricted bottom wall hc with a fixed Reynolds number. It is obtained that the number of vortices increases as either hc or M increases. However, the increase in ϴ leads to decrease the number of vortices. Formation of vortices depends on not only the strength and the orientation of the magnetic field but also the constriction of the bottom wall.

Development of an Interpolated RANS-LES solver for wall-bounded turbulent fluid flows

Scale-resolving hybrid RANS-LES models are being increasingly used to numerically simulate wall-bounded turbulent flows since they are computationally efficient compared to LES (Large Eddy Simulation) while also being more accurate compared to RANS (Reynolds-Averaged Navier–Stokes) models. Existing hybrid RANS-LES models typically either suffer from modeled stress depletion or they can introduce a sharp jump in the mean Reynolds stress across the RANS-LES zone. We present a novel interpolated RANS-LES (IRL) solver implemented in OpenFOAM, in which the RANS equations (for eddy-viscosity) and LES equations (for filtered velocity) are evolved simultaneously on the same computational grid. Based on the distance from the no-slip wall, the flow domain is demarcated into a near-wall, free-stream, and a “hybrid” region, sandwiched between the near-wall and free-stream regions. Spalart Allmaras (S-A) model has been used to evolve the RANS eddy viscosity, while the Wall Adapting Local Eddy Viscosity (WALE) model is used for evolving the filtered LES velocity. Using a novel scheme based on solving a partial differential equation, the turbulent eddy-viscosity is interpolated between the RANS eddy viscosity in the near wall region and the effective eddy viscosity from the resolved LES fluctuations in the free-stream region. The mean subgrid stress in the LES momentum equation is then corrected in the near-wall and hybrid regions using the interpolated turbulent eddy-viscosity. IRL’s grid resolution requirements are the same as DES’s (Detached Eddy Simulation). Grid convergence and sensitivity studies have been carried out for two canonical flow geometries. For turbulent channel flow simulations, the IRL solver does not display the typical mean stress depletion observed in DES while also predicting resolved Reynolds stresses accurately in the free-stream region. For simulations of flow over backward facing step, the IRL solver can predict the friction factor and coefficient of pressure better than that DES, especially after the reattachment point.

Temporal characteristics of hypersonic flows over a double wedge with Reynolds number

Laminar hypersonic flows at Mach 7.10 with unit Reynolds numbers of 5.2×104, 1.04×105, and 4.14×105 m−1 over a 30°/55° double-wedge configuration were studied to investigate the spatial–temporal characteristics of the flow in a time-accurate manner. Close comparisons were made between previous kinetic and current continuum approaches to test the validity of the continuum assumption, especially considering the presence of large gradients associated with shock–shock and shock–boundary layer interactions, as well as spanwise instabilities. Previous results from direct simulation Monte Carlo, which inherently predicts rarefied effects such as velocity slip and temperature jumps, were found to be in very close agreement with the current work, even for the lowest Reynolds number. The impact of velocity slip and temperature jumps on flow and surface parameters was investigated, and comparisons were made with a no-slip and constant temperature wall model. The temporal and spatial variation of two- and three-dimensional flows were thoroughly investigated using two-dimensional (2D), three-dimensional (3D) periodic sidewall boundary conditions, and a full 3D configuration consistent with existing experimental data. Close comparisons among the 2D and 3D cases were made. The characteristics of 2D periodic oscillations were reported for the moderate Reynolds number case for the first time. The presence of spanwise instabilities, even at a relatively low free stream pressure of about 100 Pa, establishes that the flow field depends on spanwise effects and is fully 3D. High-fidelity numerical schlieren videos captured strong spanwise oscillations for 3D configurations.

Single-particle dynamics in a low-Reynolds-number fluid under spherical confinement

Non-colloidal dynamics of a single particle suspended in a low-Reynolds-number fluid under spherical confinement was studied numerically. We calculated hydrodynamic mobilities of a sphere, a prolate spheroid and an oblate spheroid parallel and transverse to the particle-cavity line of centres. The mobilities show maximum in the cavity centre and decay as the particle moves towards the no-slip wall. For prolate and oblate spheroids, their mobilities are also affected by the angle between the particle's axis of revolution and the particle-cavity line of centres due to particle anisotropy. It was observed that the effect of particle anisotropy becomes stronger as the confinement level increases. When the external force on the particle is not parallel or transverse to the particle-cavity line of centres, a drift velocity perpendicular to the force occurs because of the confinement-induced anisotropy of the mobility in the cavity. The normalized drift velocity depends on the particle location, size, shape and orientation of the non-spherical particle. We also studied the motion of a non-neutrally buoyant particle under external forces in a rotating flow inside the cavity. Cooperation between the external force, rotation-induced centrifugal or centripetal force and the force from particle–wall interactions leads to multiple modes of particle motion. A fundamental understanding of single-particle dynamics in this work forms the basis for studying more complex particle dynamics in intracellular transport, and can guide particle manipulation in microfluidic applications ranging from droplet-based microreactors to microfluidic encapsulation.

Thermo-osmotic flow in slit channels with boundary slip: giant flow amplification between polarized graphene surfaces

The thermo-osmotic flow (TOF) of an electrolyte solution in a slit channel with a Navier slip condition at the channel walls is studied. An analytical expression for the TOF velocity profile, based on the long-wavelength and Debye–Hückel approximations, is derived and compared to numerical solutions based on the finite-element method. The TOF between graphene surfaces whose charge is created via polarization through an applied electric field is considered as a special case. Using the relationship between the surface charge and the slip length obtained from molecular dynamics simulations, a giant flow amplification is uncovered. Specifically, for such flow in a channel with a width of 10 nm, compared to the flow between no-slip walls, a flow velocity enhancement by a factor of up to 250 is predicted.

Higher order lubrication model between slip walls

A higher order lubrication model between slip walls is proposed for predicting the flow fields that cannot be described by the standard lubrication models based on the thin-gap approximation. The analysis shows that when considering the non-negligible pressure gradient in the surface-normal direction, the local pressure can be separated into (i) the base contribution by the modified Reynolds lubrication equation and (ii) the higher order component varying in both longitudinal and wall-normal directions, which takes the form proportional to the longitudinal derivative of the local velocity of the Couette–Poiseuille flow. For both (i) and (ii), the effect of the slip boundaries appears as the apparent displacements of the no-slip solid walls, and for (i) additional terms (to the no-slip case) also appear. The validity of the higher order slip-wall lubrication model is established by comparing the analytical prediction of the pressure with the fully resolved numerical results in a relatively wide region between a no-slip corrugated wall and a flat plate with varying slip length: the contribution of the higher order term is identified as the decreased lubrication pressure due to velocity slip. The model also successfully predicts the trend of pressure change between the varying slip case and a more realistic system with constant slip length for a channel, where the thin-gap approximation does not hold.

Moment analysis for predicting effective transport properties in hierarchical retentive porous media

We report on a new homogenization approach to solve, with drastically improved speed and accuracy, the general advection-diffusion equation in hierarchical porous media with localized diffusion and adsorption/desorption processes, thus opening the way to a much deeper understanding of the band broadening process in chromatographic systems. The proposed robust and efficient moment-based approach allows us to compute the exact local and integral concentration moments and hence provides exact solutions for the effective velocity and dispersion coefficients of migrating solute particles. Innovative to the proposed method is also that it not only produces the exact effective transport parameters of the long-time asymptotic solution, but also their entire transient. The analysis of the transient behaviour can be used, for example, to properly identify the time and length scales needed to achieve the macro-transport conditions. If the hierarchical porous media can be represented as the periodic repetition of a unit lattice cell, the method only requires the solution of the time-dependent advection-diffusion equations for the zeroth order and first-order exact local moments, exclusively on the unit cell. This implies an enormous reduction of the computational efforts and a significant improvement of the accuracy of the results when compared to the direct numerical simulation (DNS) approaches which require flow domains that are long enough to achieve steady-state conditions, and hence often cover tens to hundreds of unit cells. The reliability of the proposed method is verified by comparing its predictions with DNS results, in one, two and three dimensions, in both transient and asymptotic conditions. The influence of top and bottom no-slip walls on the separation performance of chromatographic columns with micromachined porous and nonporous pillars is discussed in detail.

Dynamics of flames in tubes with no-slip walls and the mechanism of tulip flame formation1

ABSTRACT A hydrogen/air flame propagation and the development of tulip-shaped flame in 2D tubes of different aspect ratios with both closed ends and in a half-open rectangular channel were studied using high resolution direct numerical simulations of the fully compressible Navier – Stokes equations coupled with a detailed chemistry. Flame propagation in a 3D rectangular channel was studied using large eddy simulations and compared with the results of direct numerical simulations of flame propagation in a 2D rectangular channel with the same aspect ratio. It is shown that the interaction of the rarefaction wave generated by the flame at the deceleration stage with the “positive” flow of unburned gas generated by the flame at the previous accelerating stage leads to a significant decrease of the velocity of the unburned gas flow in the near field zone ahead of the flame front. As a result, the thickness of the boundary layer in the near-field zone ahead of the flame increases significantly, and the profile of the axial velocity of the unburned gas in the near-field zone ahead of the flame front takes the form of a tulip or an inverted tulip, which leads to corresponding changes in the velocities of different parts of the flame front, the flame front inversion, and the formation of a tulip-shaped flame.

Comparative Evaluation of the Immersed-Solid Method for Simulating the Flow Field around Hydrofoil

The wall boundary is important in computational-fluid-dynamics simulations. If extremely small leakage, changing leakage or a moving body exists in the simulation case, the difficulty in meshing and solving near-wall flow increases. The immersed-solid method, which inserts a rigid, solid body into the entire fluid domain, was a choice to solve the wall-boundary-solution problems mentioned above, without considering mesh deformation. The purpose of this paper is to verify the effectiveness of the immersed-solid method in the simulation of extremely small leakage, changing leakage or a moving body, and to provide a theoretical basis for the use of the submerged-solid method in engineering. In this study, the NACA0015 hydrofoil was used to check the hydrodynamic characteristics in using the immersed-solid method. The comparative study was conducted at the incidence angle of 8 degrees and a Reynolds number of 5.0 × 105, by using the immersed-solid and traditional no-slip-wall boundary. The results show that the flow striking and separation with pressure rise and drop can be correctly captured using an immersed-solid setup with boundary tracking. However, the accuracy of pressure and velocity field using the immersed-solid method was insufficient. The turbulence-kinetic energy was much higher around the immersed-solid foil body. Generally, the immersed-solid method can qualitatively predict the correct hydrodynamic characteristics. Its convergence ability is better, and it can save approximately 20% of CPU time, even if the grid density is 4.39 times of the traditional no-slip wall. Therefore, the immersed-solid method can be a good choice for engineering-flow cases with complex wall problems.

A high order moving boundary treatment for convection-diffusion equations

In this paper, a new type of inverse Lax-Wendroff boundary treatment is designed for high order finite difference schemes for solving general convection-diffusion equations on time-varying domain. This new method can achieve high order accuracy on Dirichlet boundary conditions with moving boundary. To ensure stability of the boundary treatment, we give a convex combination of the boundary treatments for the diffusion-dominated and the convection-dominated cases. A group convex combination of weights is carefully designed to avoid zero denominator, resulting in a unified algorithm for pure convection, convection-dominated, convection-diffusion, diffusion-dominated and pure diffusion cases. In order to match the time levels when constructing values of ghost points in the two intermediate stages of the third order Runge-Kutta method, we propose a new approximation to the mixed derivatives at the boundaries to ensure high order accuracy and to improve computational efficiency. In particular, we extend the boundary treatment to the compressible Navier-Stokes equations, which satisfies the isothermal no-slip wall boundary condition at any Reynolds number. We provide numerical tests for one- and two-dimensional problems involving both scalar equations and systems, demonstrating that our boundary treatment is high order accurate for problems with smooth solutions and also performs well for problems involving interactions between viscous shocks and moving rigid bodies.